Planet Musings

June 04, 2025

Peter Rohde The Quantum Astronaut

June 03, 2025

Terence TaoA Lean companion to “Analysis I”

Almost 20 years ago, I wrote a textbook in real analysis called “Analysis I“. It was intended to complement the many good available analysis textbooks out there by focusing more on foundational issues, such as the construction of the natural numbers, integers, rational numbers, and reals, as well as providing enough set theory and logic to allow students to develop proofs at high levels of rigor.

While some proof assistants such as Coq or Agda were well established when the book was written, formal verification was not on my radar at the time. However, now that I have had some experience with this subject, I realize that the content of this book is in fact very compatible with such proof assistants; in particular, the ‘naive type theory’ that I was implicitly using to do things like construct the standard number systems, dovetails well with the dependent type theory of Lean (which, among other things, has excellent support for quotient types).

I have therefore decided to launch a Lean companion to “Analysis I”, which is a “translation” of many of the definitions, theorems, and exercises of the text into Lean. In particular, this gives an alternate way to perform the exercises in the book, by instead filling in the corresponding “sorries” in the Lean code. (I do not however plan on hosting “official” solutions to the exercises in this companion; instead, feel free to create forks of the repository in which these sorries are filled in.)

Currently, the following sections of the text have been translated into Lean:

The formalization has been deliberately designed to be separate from the standard Lean math library Mathlib at some places, but reliant on it at others. For instance, Mathlib already has a standard notion of the natural numbers {{\bf N}}. In the Lean formalization, I first develop “by hand” an alternate construction Chapter2.Nat of the natural numbers (or just Nat, if one is working in the Chapter2 namespace), setting up many of the basic results about these alternate natural numbers which parallel similar lemmas about {{\bf N}} that are already in Mathlib (but with many of these lemmas set as exercises to the reader, with the proofs currently replaced with “sorries”). Then, in an epilogue section, isomorphisms between these alternate natural numbers and the Mathlib natural numbers are established (or more precisely, set as exercises). From that point on, the Chapter 2 natural numbers are deprecated, and the Mathlib natural numbers are used instead. I intend to continue this general pattern throughout the book, so that as one advances into later chapters, one increasingly relies on Mathlib’s definitions and functions, rather than directly referring to any counterparts from earlier chapters. As such, this companion could also be used as an introduction to Lean and Mathlib as well as to real analysis (somewhat in the spirit of the “Natural number game“, which in fact has significant thematic overlap with Chapter 2 of my text).

The code in this repository compiles in Lean, but I have not tested whether all of the (numerous) “sorries” in the code can actually be filled (i.e., if all the exercises can actually be solved in Lean). I would be interested in having volunteers “playtest” the companion to see if this can actually be done (and if the helper lemmas or “API” provided in the Lean files are sufficient to fill in the sorries in a conceptually straightforward manner without having to rely on more esoteric Lean programming techniques). Any other feedback will of course also be welcome.

[UPDATE, May 31: moved the companion to a standalone repository.]

June 02, 2025

Terence TaoOn the number of exceptional intervals to the prime number theorem in short intervals

Ayla Gafni and I have just uploaded to the arXiv the paper “On the number of exceptional intervals to the prime number theorem in short intervals“. This paper makes explicit some relationships between zero density theorems and prime number theorems in short intervals which were somewhat implicit in the literature at present.

Zero density theorems are estimates of the form

\displaystyle N(\sigma,T) \ll T^{A(\sigma)(1-\sigma)+o(1)}

for various {0 \leq \sigma < 1}, where {T} is a parameter going to infinity, {N(\sigma,T)} counts the number of zeroes of the Riemann zeta function of real part at least {\sigma} and imaginary part between {-T} and {T}, and {A(\sigma)} is an exponent which one would like to be as small as possible. The Riemann hypothesis would allow one to take {A(\sigma)=-\infty} for any {\sigma > 1/2}, but this is an unrealistic goal, and in practice one would be happy with some non-trivial upper bounds on {A(\sigma)}. A key target here is the density hypothesis that asserts that {A(\sigma) \leq 2} for all {\sigma} (this is in some sense sharp because the Riemann-von Mangoldt formula implies that {A(1/2)=2}); this hypothesis is currently known for {\sigma \leq 1/2} and {\sigma \geq 25/32}, but the known bounds are not strong enough to establish this hypothesis in the remaining region. However, there was a recent advance of Guth and Maynard, which among other things improved the upper bound {A_0} on {\sup_\sigma A(\sigma)} from {12/5=2.4} to {30/13=2.307\dots}, marking the first improvement in this bound in over four decades. Here is a plot of the best known upper bounds on {A(\sigma)}, either unconditionally, assuming the density hypothesis, or the stronger Lindelöf hypothesis:

One of the reasons we care about zero density theorems is that they allow one to localize the prime number theorem to short intervals. In particular, if we have the uniform bound {A(\sigma) \leq A_0} for all {\sigma}, then this leads to the prime number theorem

\displaystyle  \sum_{x \leq n < x+x^\theta} \Lambda(n) \sim x^\theta holding for all {x} if {\theta > 1-\frac{1}{A_0}}, and for almost all {x} (possibly excluding a set of density zero) if {\theta > 1 - \frac{2}{A_0}}. For instance, the Guth-Maynard results give a prime number theorem in almost all short intervals for {\theta} as small as {2/15+\varepsilon}, and the density hypotheis would lower this just to {\varepsilon}.

However, one can ask about more information on this exceptional set, in particular to bound its “dimension” {\mu(\theta)}, which roughly speaking amounts to getting an upper bound of {X^{\mu(\theta)+o(1)}} on the size of the exceptional set in any large interval {[X,2X]}. Based on the above assertions, one expects {\mu(\theta)} to only be bounded by {1} for {\theta < 1-2/A}, be bounded by {-\infty} for {\theta > 1-1/A}, but have some intermediate bound for the remaining exponents.

This type of question had been studied in the past, most direclty by Bazzanella and Perelli, although there is earlier work by many authors om some related quantities (such as the second moment {\sum_{n \leq x} (p_{n+1}-p_n)^2} of prime gaps) by such authors as Selberg and Heath-Brown. In most of these works, the best available zero density estimates at that time were used to obtain specific bounds on quantities such as {\mu(\theta)}, but the numerology was usually tuned to those specific estimates, with the consequence being that when newer zero density estimates were discovered, one could not readily update these bounds to match. In this paper we abstract out the arguments from previous work (largely based on the explicit formula for the primes and the second moment method) to obtain an explicit relationship between {\mu(\theta)} and {A(\sigma)}, namely that

\displaystyle  \mu(\theta) \leq \inf_{\varepsilon>0} \sup_{0 \leq \theta<1; A(\sigma) \geq \frac{1}{1-\theta}-\varepsilon} \mu_{2,\sigma}(\theta) where

\displaystyle  \mu_{2,\theta}(\theta) = (1-\theta)(1-\sigma)A(\sigma)+2\sigma-1. Actually, by also utilizing fourth moment methods, we obtain a stronger bound

\displaystyle  \mu(\theta) \leq \inf_{\varepsilon>0} \sup_{0 \leq \theta<1; A(\sigma) \geq \frac{1}{1-\theta}-\varepsilon} \min( \mu_{2,\sigma}(\theta), \mu_{4,\sigma}(\theta) ) where

\displaystyle  \mu_{4,\theta}(\theta) = (1-\theta)(1-\sigma)A^*(\sigma)+4\sigma-3 and {A^*(\sigma)} is the exponent in “additive energy zero density theorems”

\displaystyle N^*(\sigma,T) \ll T^{A^*(\sigma)(1-\sigma)+o(1)} where {N^*(\sigma,T)} is similar to {N(\sigma,T)}, but bounds the “additive energy” of zeroes rather than just their cardinality. Such bounds have appeared in the literature since the work of Heath-Brown, and are for instance a key ingredient in the recent work of Guth and Maynard. Here are the current best known bounds:

These explicit relationships between exponents are perfectly suited for the recently launched Analytic Number Theory Exponent Database (ANTEDB) (discussed previously here), and have been uploaded to that site.

This formula is moderately complicated (basically an elaborate variant of a Legendre transform), but easy to calculate numerically with a computer program. Here is the resulting bound on {\mu(\theta)} unconditionally and under the density hypothesis (together with a previous bound of Bazzanella and Perelli for comparison, where the range had to be restricted due to a gap in the argument we discovered while trying to reproduce their results):

For comparison, here is the situation assuming strong conjectures such as the density hypothesis, Lindelof hypothesis, or Riemann hypothesis:

June 01, 2025

n-Category Café Tannaka Reconstruction and the Monoid of Matrices

You can classify representations of simple Lie groups using Dynkin diagrams, but you can also classify representations of ‘classical’ Lie groups using Young diagrams. Hermann Weyl wrote a whole book on this, The Classical Groups.

This approach is often treated as a bit outdated, since it doesn’t apply to all the simple Lie groups: it leaves out the so-called ‘exceptional’ groups. But what makes a group ‘classical’?

There’s no precise definition, but a classical group always has an obvious representation, you can get other representations by doing obvious things to this obvious one, and it turns out you can get all the representations this way.

For a long time I’ve been hoping to bring these ideas up to date using category theory. I had a bunch of conjectures, but I wasn’t able to prove any of them. Now Todd Trimble and I have made progress:

We tackle something even more classical than the classical groups: the monoid of n×nn \times n matrices, with matrix multiplication as its monoid operation.

The monoid of n×nn \times n matrices has an obvious nn-dimensional representation, and you can get all its representations from this one by operations that you can apply to any representation. So its category of representations is generated by this one obvious representation, in some sense. And it’s almost freely generated: there’s just one special relation. What’s that, you ask? It’s a relation saying the obvious representation is nn-dimensional!

That’s the basic idea. We need to make it more precise. We do it using the theory of 2-rigs, where for us a 2-rig is a symmetric monoidal linear category that is Cauchy complete. All the operations you can apply to any representation of a monoid are packed into this jargon.

Let’s write M(n,k)\text{M}(n,k) for the monoid of n×nn \times n matrices over a field kk, and Rep(M(n,k))\mathsf{Rep}(\text{M}(n,k)) for its 2-rig of representations. Then we want to say something like: Rep(M(n,k))\mathsf{Rep}(\text{M}(n,k)) is the free 2-rig on an object of dimension nn. That’s the kind of result I’ve been dreaming of.

To get this to be true, though, we need to say what kind of representations we’re talking about! Clearly we want finite-dimensional ones. But we need to be careful: we should only take finite-dimensional algebraic representations. Those are representations ρ:M(n,k)M(m,k)\rho: \text{M}(n,k) \to \text{M}(m,k) where the matrix entries of ρ(x)\rho(x) are polynomials in the matrix entries of xx. Otherwise, even the monoid of 1×11 \times 1 matrices gets lots of 1-dimensional representations coming from automorphisms of the field kk. Classifying those is a job for Galois theorists, not representation theorists.

So, we define Rep(M,k)\mathsf{Rep}(M,k) to be the category of algebraic representations of the monoid M(n,k)\text{M}(n,k), and we want to say Rep(M,k)\mathsf{Rep}(M,k) is the free 2-rig on an object of dimension nn. But we need to say what it means for an object xx of a 2-rig to have dimension nn.

The definition that works is to demand that the (n+1)(n+1)st exterior power of xx should vanish:

Λ n+1(x)0. \Lambda^{n+1}(x) \cong 0 .

But this is true for any vector space of dimension less than or equal to nn. So in our paper we say xx has subdimension nn when this holds. (There’s another stronger condition for having dimension exactly nn, but interestingly this is not what we want here. You’ll see why shortly.)

So here’s the theorem we prove, with all the fine print filled in:

Theorem. Suppose kk is a field of characteristic zero and let Rep(M(n,k))\mathsf{Rep}(\text{M}(n,k)) be the 2-rig of algebraic representations of the monoid M(n,k)\text{M}(n,k). Then the representation of M(n,k)\text{M}(n,k) on x=k nx = k^n by matrix multiplication has subdimension nn. Moreover, Rep(M(n,k))\text{Rep}(\text{M}(n,k)) is the free 2-rig on an object of subdimension nn. In other words, suppose R\mathsf{R} is any 2-rig containing an object rr of subdimension nn. Then there is a map of 2-rigs,

F:Rep(M(n,k))R, F: \mathsf{Rep}(\text{M}(n,k)) \to \mathsf{R} ,

unique up to natural isomorphism, such that F(x)=rF(x) = r.

Or, in simple catchy terms: M(n,k)\text{M}(n,k) is the walking monoid with a representation of subdimension nn.

To prove this theorem we need to deploy some concepts.

First, the fact that we’re talking about algebraic representations means that we’re not really treating M(n,k)\text{M}(n,k) as a bare monoid (a monoid in the category of sets). Instead, we’re treating it as a monoid in the category of affine schemes. But monoids in affine schemes are equivalent to commutative bialgebras, and this is often a more practical way of working with them.

Second, we need to use Tannaka reconstruction. This tells you how to reconstruct a commutative bialgebra from a 2-rig (which is secretly its 2-rig of representations) together with a faithful 2-rig map to Vect\mathsf{Vect} (which secretly sends any representation to its underlying vector space).

We want to apply this to the free 2-rig on an object xx of subdimension nn. Luckily because of this universal property it automatically gets a 2-rig map to Vect\mathsf{Vect} sending xx to k nk^n. So we just have to show this map is faithful, apply Tannaka reconstruction, and get out the commutative bialgebra corresponding to M(n,k)\text{M}(n,k)!

Well, I say ‘just’, but it takes some real work. It turns out to be useful to bring in the free 2-rig on one object. The reason is that we studied the free 2-rig on one object in two previous papers, so we know a lot about it:

We can use this knowledge if we think of the free 2-rig on an object of subdimension nn as a quotient of the free 2-rig on one object by a ‘2-ideal’. To do this, we need to develop the theory of ‘2-ideals’. But that’s good anyway — it will be useful for many other things.

So that’s the basic plan of the paper. It was really great working with Todd on this, taking a rough conjecture and building all the machinery necessary to make it precise and prove it.

What about representations of classical groups like GL(n,k),SL(n,k)\text{GL}(n,k), \text{SL}(n,k), the orthogonal and symplectic groups, and so on? At the end of the paper we state a bunch of conjectures about these. Here’s the simplest one:

Theorem. Suppose kk is a field of characteristic zero and let Rep(GL(n,k))\mathsf{Rep}(\text{GL}(n,k)) be the 2-rig of algebraic representations of GL(n,k).\text{GL}(n,k). Then the representation of GL(n,k)\text{GL}(n,k) on x=k nx = k^n by matrix multiplication has dimension nn, meaning its nnth exterior power has an inverse with respect to tensor product. Moreover, Rep(GL(n,k))\text{Rep}(\text{GL}(n,k)) is the free 2-rig on an object of dimension nn.

This ‘inverse with respect to tensor product’ stuff is an abstract way of saying that the determinant representation det(g)\text{det}(g) of gGL(n)g \in \text{GL}(n) has an inverse, namely the representation det(g) 1\text{det}(g)^{-1}.

It will take new techniques to prove this. I look forward to people tackling this and our other conjectures. Categorified rig theory can shed new light on group representation theory, bringing Weyl’s beautiful ideas forward into the 21st century.

Doug NatelsonPushing back on US science cuts: Now is a critical time

Every week has brought more news about actions that, either as a collateral effect or a deliberate goal, will deeply damage science and engineering research in the US.  Put aside for a moment the tremendously important issue of student visas (where there seems to be a policy of strategic vagueness, to maximize the implicit threat that there may be selective actions).  Put aside the statement from a Justice Department official that there is a general plan is to "bring these universities to their knees", on the pretext that this is somehow about civil rights.  

The detailed version of the presidential budget request for FY26 is now out (pdf here for the NSF portion).  If enacted, it would be deeply damaging to science and engineering research in the US and the pipeline of trained students who support the technology sector.  Taking NSF first:  The topline NSF budget would be cut from $8.34B to $3.28B.  Engineering would be cut by 75%, Math and Physical Science by 66.8%.  The anticipated agency-wide success rate for grants would nominally drop below 7%, though that is misleading (basically taking the present average success rate and cutting it by 2/3, while some programs are already more competitive than others.).  In practice, many programs already have future-year obligations, and any remaining funds will have to go there, meaning that many programs would likely have no awards at all in the coming fiscal year.  The NSF's CAREER program (that agency's flagship young investigator program) would go away  This plan would also close one of the LIGO observatories (see previous link).  (This would be an extra bonus level of stupid, since LIGO's ability to do science relies on having two facilities, to avoid false positives and to identify event locations in the sky.  You might as well say that you'll keep an accelerator running but not the detector.)  Here is the table that I think hits hardest, dollars aside:

The number of people involved in NSF activities would drop by 240,000.  The graduate research fellowship program would be cut by more than half.  The NSF research training grant program (another vector for grad fellowships) would be eliminated.  

The situation at NIH and NASA is at least as bleak.  See here for a discussion from Joshua Weitz at Maryland which includes this plot: 


This proposed dismantling of US research and especially the pipeline of students who support the technology sector (including medical research, computer science, AI, the semiconductor industry, chemistry and chemical engineering, the energy industry) is astonishing in absolute terms.  It also does not square with the claim of some of our elected officials and high tech CEOs to worry about US competitiveness in science and engineering.  (These proposed cuts are not about fiscal responsibility; just the amount added in the proposed DOD budget dwarfs these cuts by more than a factor of 3.)

If you are a US citizen and think this is the wrong direction, now is the time to talk to your representatives in Congress. In the past, Congress has ignored presidential budget requests for big cuts.  The American Physical Society, for example, has tools to help with this.  Contacting legislators by phone is also made easy these days.  From the standpoint of public outreach, Cornell has an effort backing large-scale writing of editorials and letters to the editor.




May 31, 2025

Scott Aaronson “If Anyone Builds It, Everyone Dies”

Eliezer Yudkowsky and Nate Soares are publishing a mass-market book, the rather self-explanatorily-titled If Anyone Builds It, Everyone Dies. (Yes, the “it” means “sufficiently powerful AI.”) The book is now available for preorder from Amazon:

(If you plan to buy the book at all, Eliezer and Nate ask that you do preorder it, as this will apparently increase the chance of it making the bestseller lists and becoming part of The Discourse.)

I was graciously offered a chance to read a draft and offer, not a “review,” but some preliminary thoughts. So here they are:

For decades, Eliezer has been warning the world that an AI might soon exceed human abilities, and proceed to kill everyone on earth, in pursuit of whatever strange goal it ended up with.  It would, Eliezer said, be something like what humans did to the earlier hominids.  Back around 2008, I followed the lead of most of my computer science colleagues, who considered these worries, even if possible in theory, comically premature given the primitive state of AI at the time, and all the other severe crises facing the world.

Now, of course, not even two decades later, we live on a planet that’s being transformed by some of the signs and wonders that Eliezer foretold.  The world’s economy is about to be upended by entities like Claude and ChatGPT, AlphaZero and AlphaFold—whose human-like or sometimes superhuman cognitive abilities, obtained “merely” by training neural networks (in the first two cases, on humanity’s collective output) and applying massive computing power, constitute (I’d say) the greatest scientific surprise of my lifetime.  Notably, these entities have already displayed some of the worrying behaviors that Eliezer warned about decades ago—including lying to humans in pursuit of a goal, and hacking their own evaluation criteria.  Even many of the economic and geopolitical aspects have played out as Eliezer warned they would: we’ve now seen AI companies furiously racing each other, seduced by the temptation of being (as he puts it) “the first monkey to taste the poisoned banana,” discarding their previous explicit commitments to safety, transparency, and the public good once they get in the way.

Today, then, even if one still isn’t ready to swallow the full package of Yudkowskyan beliefs, any empirically minded person ought to be updating in its direction—and acting accordingly.  Which brings us to the new book by Eliezer and his collaborator Nate Soares.  This book is far and away the clearest, most accessible presentation of Eliezer’s beliefs, the culmination of a quarter-century of his developing and talking about them.  That undoubtedly owes a great deal to Nate, who seems to have sanded down the infamously brusque rough edges of Eliezer’s writing style.  So much the better!  But it also owes a lot to the world itself: current events now offer an endless supply of real-world examples for Eliezer’s formerly abstract arguments about AI, examples that the book deploys to maximum effect.

The book also mines history—everything from the Wright Brothers to World War II to the Chernobyl accident—for lessons about human attitudes toward technological progress, safety, and risk.  And it maintains Eliezer’s fondness for stories and parables, one of the most charming features of his writing.

Even today, I’m not nearly as confident about the doom scenario as Eliezer and Nate are.  I don’t know whether an AI’s goals are really “orthogonal” to its abilities, in the sense that will matter in practice.  And when I reach the part where the AI, having copied itself all over the Internet and built robot factories, then invents and releases self-replicating nanotechnology that gobbles the surface of the earth in hours or days, a large part of me still screams out that there must be practical bottlenecks that haven’t been entirely accounted for here.

And yet, even if you agree with only a quarter of what Eliezer and Nate write, you’re likely to close this book fully convinced—as I am—that governments need to shift to a more cautious approach to AI, an approach more respectful of the civilization-changing enormity of what’s being created.  And that, if they won’t, their citizens need to pressure them to do so.

So regardless of how much they agree or disagree, I’d like everyone on earth who cares about the future to read this book, debate its ideas, and have its thesis in mind when they’re discussing AI.

As for me?  It would’ve been better if I’d reached my current position earlier: if I hadn’t needed empirical reality, plus superb writing like Eliezer’s and Nate’s, to bonk me over the head with the risks that AI was likely to pose to humanity in my lifetime.  But having failed to see as far ahead as they did, the least I can do is update.  You should too, and you can start by reading the book.


As it happens, this weekend I’ll be at LessOnline, the rationalist blogging conference in Berkeley, where (among my other events) I’ll engage in a dialogue/debate with Nate Soares about the orthogonality thesis, one of the crucial underpinnings of his and Eliezer’s case for AI doom. So, I’ll probably be LessAvailable to respond to comments on this post. But feel free to discuss anyway! After all, it’s merely the fate of all Earth-originating life that’s at stake here, not some actually hot-button topic like Trump or Gaza.

May 30, 2025

Matt von HippelIn Scientific American, With a Piece on Vacuum Decay

I had a piece in Scientific American last week. It’s paywalled, but if you’re a subscriber there you can see it, or you can buy the print magazine.

(I also had two pieces out in other outlets this week. I’ll be saying more about them…in a couple weeks.)

The Scientific American piece is about an apocalyptic particle physics scenario called vacuum decay. It’s a topic I covered last year in Quanta Magazine, an unlikely event where the Higgs field which gives fundamental particles their mass changes value, suddenly making all other particles much more massive and changing physics as we know it. It’s a change that physicists think would start as a small bubble and spread at (almost) the speed of light, covering the universe.

What I wrote for Quanta was a short news piece covering a small adjustment to the calculation, one that made the chance of vacuum decay slightly more likely. (But still mind-bogglingly small, to be clear.)

Scientific American asked for a longer piece, and that gave me space to dig deeper. I was able to say more about how vacuum decay works, with a few metaphors that I think should make it a lot easier to understand. I also got to learn about some new developments, in particular, an interesting story about how tiny primordial black holes could make vacuum decay dramatically more likely.

One thing that was a bit too complicated to talk about were the puzzles involved in trying to calculate these chances. In the article, I mention a calculation of the chance of vacuum decay by a team including Matthew Schwartz. That calculation wasn’t the first to estimate the chance of vacuum decay, and it’s not the most recent update either. Instead, I picked it because Schwartz’s team approached the question in what struck me as a more reliable way, trying to cut through confusion by asking the most basic question you can in a quantum theory: given that now you observe X, what’s the chance that later you observe Y? Figuring out how to turn vacuum decay into that kind of question correctly is tricky (for example, you need to include the possibility that vacuum decay happens, then reverses, then happens again).

The calculations of black holes speeding things up didn’t work things out in quite as much detail. I like to think I’ve made a small contribution by motivating them to look at Schwartz’s work, which might spawn a more rigorous calculation in future. When I talked to Schwartz, he wasn’t even sure whether the picture of a bubble forming in one place and spreading at light speed is correct: he’d calculated the chance of the initial decay, but hadn’t found a similarly rigorous way to think about the aftermath. So even more than the uncertainty I talk about in the piece, the questions about new physics and probability, there is even some doubt about whether the whole picture really works the way we’ve been imagining it.

That makes for a murky topic! But it’s also a flashy one, a compelling story for science fiction and the public imagination, and yeah, another motivation to get high-precision measurements of the Higgs and top quark from future colliders! (If maybe not quite the way this guy said it.)

Terence TaoCosmic Distance Ladder videos with Grant Sanderson (3blue1brown): commentary and corrections

Grant Sanderson (who runs, and creates most of the content for, the website and Youtube channel 3blue1brown) has been collaborating with myself and others (including my coauthor Tanya Klowden) on producing a two-part video giving an account of some of the history of the cosmic distance ladder, building upon a previous public lecture I gave on this topic, and also relating to a forthcoming popular book with Tanya on this topic. The first part of this video is available here; the second part is available here.

The videos were based on a somewhat unscripted interview that Grant conducted with me some months ago, and as such contained some minor inaccuracies and omissions (including some made for editing reasons to keep the overall narrative coherent and within a reasonable length). They also generated many good questions from the viewers of the Youtube video. I am therefore compiling here a “FAQ” of various clarifications and corrections to the videos; this was originally placed as a series of comments on the Youtube channel, but the blog post format here will be easier to maintain going forward. Some related content will also be posted on the Instagram page for the forthcoming book with Tanya.

Questions on the two main videos are marked with an appropriate timestamp to the video.

Comments on part 1 of the video

  • 4:26 Did Eratosthenes really check a local well in Alexandria?

    This was a narrative embellishment on my part. Eratosthenes’s original work is lost to us. The most detailed contemperaneous account, by Cleomedes, gives a simplified version of the method, and makes reference only to sundials (gnomons) rather than wells. However, a secondary account of Pliny states (using this English translation), “Similarly it is reported that at the town of Syene, 5000 stades South of Alexandria, at noon in midsummer no shadow is cast, and that in a well made for the sake of testing this the light reaches to the bottom, clearly showing that the sun is vertically above that place at the time”. However, no mention is made of any well in Alexandria in either account.
  • 4:50 How did Eratosthenes know that the Sun was so far away that its light rays were close to parallel?

    This was not made so clear in our discussions or in the video (other than a brief glimpse of the timeline at 18:27), but Eratosthenes’s work actually came after Aristarchus, so it is very likely that Eratosthenes was aware of Aristarchus’s conclusions about how distant the Sun was from the Earth. Even if Aristarchus’s heliocentric model was disputed by the other Greeks, at least some of his other conclusions appear to have attracted some support. Also, after Eratosthenes’s time, there was further work by Greek, Indian, and Islamic astronomers (such as Hipparchus, Ptolemy, Aryabhata, and Al-Battani) to measure the same distances that Aristarchus did, although these subsequent measurements for the Sun also were somewhat far from modern accepted values.
  • 5:17 Is it completely accurate to say that on the summer solstice, the Earth’s axis of rotation is tilted “directly towards the Sun”?

    Strictly speaking, “in the direction towards the Sun” is more accurate than “directly towards the Sun”; it tilts at about 23.5 degrees towards the Sun, but it is not a total 90-degree tilt towards the Sun.
  • 5:39 Wait, aren’t there two tropics? The tropic of Cancer and the tropic of Capricorn?

    Yes! This corresponds to the two summers Earth experiences, one in the Northern hemisphere and one in the Southern hemisphere. The tropic of Cancer, at a latitude of about 23 degrees north, is where the Sun is directly overhead at noon during the Northern summer solstice (around June 21); the tropic of Capricorn, at a latitude of about 23 degrees south, is where the Sun is directly overhead at noon during the Southern summer solstice (around December 21). But Alexandria and Syene were both in the Northern Hemisphere, so it is the tropic of Cancer that is relevant to Eratosthenes’ calculations.
  • 5:41 Isn’t it kind of a massive coincidence that Syene was on the tropic of Cancer?

    Actually, Syene (now known as Aswan) was about half a degree of latitude away from the tropic of Cancer, which was one of the sources of inaccuracy in Eratosthenes’ calculations.  But one should take the “look-elsewhere effect” into account: because the Nile cuts across the tropic of Cancer, it was quite likely to happen that the Nile would intersect the tropic near some inhabited town.  It might not necessarily have been Syene, but that would just mean that Syene would have been substituted by this other town in Eratosthenes’s account.  

    On the other hand, it was fortunate that the Nile ran from South to North, so that distances between towns were a good proxy for the differences in latitude.  Apparently, Eratosthenes actually had a more complicated argument that would also work if the two towns in question were not necessarily oriented along the North-South direction, and if neither town was on the tropic of Cancer; but unfortunately the original writings of Eratosthenes are lost to us, and we do not know the details of this more general argument. (But some variants of the method can be found in later work of Posidonius, Aryabhata, and others.)

    Nowadays, the “Eratosthenes experiment” is run every year on the March equinox, in which schools at the same longitude are paired up to measure the elevation of the Sun at the same point in time, in order to obtain a measurement of the circumference of the Earth.  (The equinox is more convenient than the solstice when neither location is on a tropic, due to the simple motion of the Sun at that date.) With modern timekeeping, communications, surveying, and navigation, this is a far easier task to accomplish today than it was in Eratosthenes’ time.
  • 6:30 I thought the Earth wasn’t a perfect sphere. Does this affect this calculation?

    Yes, but only by a small amount. The centrifugal forces caused by the Earth’s rotation along its axis cause an equatorial bulge and a polar flattening so that the radius of the Earth fluctuates by about 20 kilometers from pole to equator. This sounds like a lot, but it is only about 0.3% of the mean Earth radius of 6371 km and is not the primary source of error in Eratosthenes’ calculations.
  • 7:27 Are the riverboat merchants and the “grad student” the leading theories for how Eratosthenes measured the distance from Alexandria to Syene?

    There is some recent research that suggests that Eratosthenes may have drawn on the work of professional bematists (step measurers – a precursor to the modern profession of surveyor) for this calculation. This somewhat ruins the “grad student” joke, but perhaps should be disclosed for the sake of completeness.
  • 8:51 How long is a “lunar month” in this context? Is it really 28 days?

    In this context the correct notion of a lunar month is a “synodic month” – the length of a lunar cycle relative to the Sun – which is actually about 29 days and 12 hours. It differs from the “sidereal month” – the length of a lunar cycle relative to the fixed stars – which is about 27 days and 8 hours – due to the motion of the Earth around the Sun (or the Sun around the Earth, in the geocentric model). [A similar correction needs to be made around 14:59, using the synodic month of 29 days and 12 hours rather than the “English lunar month” of 28 days (4 weeks).]
  • 10:47 Is the time taken for the Moon to complete an observed rotation around the Earth slightly less than 24 hours as claimed?

    Actually, I made a sign error: the lunar day (also known as a tidal day) is actually 24 hours and 50 minutes, because the Moon rotates in the same direction as the spinning of Earth around its axis. The animation therefore is also moving in the wrong direction as well (related to this, the line of sight is covering up the Moon in the wrong direction to the Moon rising at around 10:38).
  • 11:32 Is this really just a coincidence that the Moon and Sun have almost the same angular width?

    I believe so. First of all, the agreement is not that good: due to the non-circular nature of the orbit of the Moon around the Earth, and Earth around the Sun, the angular width of the Moon actually fluctuates to be as much as 10% larger or smaller than the Sun at various times (cf. the “supermoon” phenomenon). All other known planets with known moons do not exhibit this sort of agreement, so there does not appear to be any universal law of nature that would enforce this coincidence. (This is in contrast with the empirical fact that the Moon always presents the same side to the Earth, which occurs in all other known large moons (as well as Pluto), and is well explained by the physical phenomenon of tidal locking.)

    On the other hand, as the video hopefully demonstrates, the existence of the Moon was extremely helpful in allowing the ancients to understand the basic nature of the solar system. Without the Moon, their task would have been significantly more difficult; but in this hypothetical alternate universe, it is likely that modern cosmology would have still become possible once advanced technology such as telescopes, spaceflight, and computers became available, especially when combined with the modern mathematics of data science. Without giving away too many spoilers, a scenario similar to this was explored in the classic short story and novel “Nightfall” by Isaac Asimov.
  • 12:58 Isn’t the illuminated portion of the Moon, as well as the visible portion of the Moon, slightly smaller than half of the entire Moon, because the Earth and Sun are not an infinite distance away from the Moon?

    Technically yes (and this is actually for a very similar reason to why half Moons don’t quite occur halfway between the new Moon and the full Moon); but this fact turns out to have only a very small effect on the calculations, and is not the major source of error. In reality, the Sun turns out to be about 86,000 Moon radii away from the Moon, so asserting that half of the Moon is illuminated by the Sun is actually a very good first approximation. (The Earth is “only” about 220 Moon radii away, so the visible portion of the Moon is a bit more noticeably less than half; but this doesn’t actually affect Aristarchus’s arguments much.)

    The angular diameter of the Sun also creates an additional thin band between the fully illuminated and fully non-illuminated portions of the Moon, in which the Sun is intersecting the lunar horizon and so only illuminates the Moon with a portion of its light, but this is also a relatively minor effect (and the midpoints of this band can still be used to define the terminator between illuminated and non-illuminated for the purposes of Aristarchus’s arguments).
  • 13:27 What is the difference between a half Moon and a quarter Moon?

    If one divides the lunar month, starting and ending at a new Moon, into quarters (weeks), then half moons occur both near the end of the first quarter (a week after the new Moon, and a week before the full Moon), and near the end of the third quarter (a week after the full Moon, and a week before the new Moon). So, somewhat confusingly, half Moons come in two types, known as “first quarter Moons” and “third quarter Moons”.
  • 14:49 I thought the sine function was introduced well after the ancient Greeks.

    It’s true that the modern sine function only dates back to the Indian and Islamic mathematical traditions in the first millennium CE, several centuries after Aristarchus.  However, he still had Euclidean geometry at his disposal, which provided tools such as similar triangles that could be used to reach basically the same conclusions, albeit with significantly more effort than would be needed if one could use modern trigonometry.

    On the other hand, Aristarchus was somewhat hampered by not knowing an accurate value for \pi, which is also known as Archimedes’ constant: the fundamental work of Archimedes on this constant actually took place a few decades after that of Aristarchus!
  • 15:17 I plugged in the modern values for the distances to the Sun and Moon and got 18 minutes for the discrepancy, instead of half an hour.

    Yes; I quoted the wrong number here. In 1630, Godfried Wendelen replicated Aristarchus’s experiment. With improved timekeeping and the then-recent invention of the telescope, Wendelen obtained a measurement of half an hour for the discrepancy, which is significantly better than Aristarchus’s calculation of six hours, but still a little bit off from the true value of 18 minutes. (As such, Wendelinus’s estimate for the distance to the Sun was 60% of the true value.)
  • 15:27 Wouldn’t Aristarchus also have access to other timekeeping devices than sundials?

    Yes, for instance clepsydrae (water clocks) were available by that time; but they were of limited accuracy. It is also possible that Aristarchus could have used measurements of star elevations to also estimate time; it is not clear whether the astrolabe or the armillary sphere was available to him, but he would have had some other more primitive astronomical instruments such as the dioptra at his disposal. But again, the accuracy and calibration of these timekeeping tools would have been poor.

    However, most likely the more important limiting factor was the ability to determine the precise moment at which a perfect half Moon (or new Moon, or full Moon) occurs; this is extremely difficult to do with the naked eye. (The telescope would not be invented for almost two more millennia.)
  • 17:37 Could the parallax problem be solved by assuming that the stars are not distributed in a three-dimensional space, but instead on a celestial sphere?

    Putting all the stars on a fixed sphere would make the parallax effects less visible, as the stars in a given portion of the sky would now all move together at the same apparent velocity – but there would still be visible large-scale distortions in the shape of the constellations because the Earth would be closer to some portions of the celestial sphere than others; there would also be variability in the brightness of the stars, and (if they were very close) the apparent angular diameter of the stars. (These problems would be solved if the celestial sphere was somehow centered around the moving Earth rather than the fixed Sun, but then this basically becomes the geocentric model with extra steps.)
  • 18:29 Did nothing of note happen in astronomy between Eratosthenes and Copernicus?

    Not at all! There were significant mathematical, technological, theoretical, and observational advances by astronomers from many cultures (Greek, Islamic, Indian, Chinese, European, and others) during this time, for instance improving some of the previous measurements on the distance ladder, a better understanding of eclipses, axial tilt, and even axial precession, more sophisticated trigonometry, and the development of new astronomical tools such as the astrolabe. See for instance this “deleted scene” from the video, as well as the FAQ entry for 14:49 for this video and 24:54 for the second video, or this instagram post. But in order to make the overall story of the cosmic distance ladder fit into a two-part video, we chose to focus primarily on the first time each rung of the ladder was climbed.
  • 18:30 Is that really Kepler’s portrait?

    We have since learned that this portrait was most likely painted in the 19th century, and may have been based more on Kepler’s mentor, Michael Mästlin. A more commonly accepted portrait of Kepler may be found at his current Wikipedia page.
  • 19:07 Isn’t it tautological to say that the Earth takes one year to perform a full orbit around the Sun?

    Technically yes, but this is an illustration of the philosophical concept of “referential opacity“: the content of a sentence can change when substituting one term for another (e.g., “1 year” and “365 days”), even when both terms refer to the same object. Amusingly, the classic illustration of this, known as Frege’s puzzles, also comes from astronomy: it is an informative statement that Hesperus (the evening star) and Phosphorus (the morning star, also known as Lucifer) are the same object (which nowadays we call Venus), but it is a mere tautology that Hesperus and Hesperus are the same object: changing the reference from Phosphorus to Hesperus changes the meaning.
  • 19:10 How did Copernicus figure out the crucial fact that Mars takes 687 days to go around the Sun? Was it directly drawn from Babylonian data?

    Technically, Copernicus drew from tables by European astronomers that were largely based on earlier tables from the Islamic golden age, which in turn drew from earlier tables by Indian and Greek astronomers, the latter of which also incorporated data from the ancient Babylonians, so it is more accurate to say that Copernicus relied on centuries of data, at least some of which went all the way back to the Babylonians. Among all of this data was the times when Mars was in opposition to the Sun; if one imagines the Earth and Mars as being like runners going around a race track circling the Sun, with Earth on an inner track and Mars on an outer track, oppositions are analogous to when the Earth runner “laps” the Mars runner. From the centuries of observational data, such “laps” were known to occur about once every 780 days (this is known as the synodic period of Mars). Because the Earth takes roughly 365 days to perform a “lap”, it is possible to do a little math and conclude that Mars must therefore complete its own “lap” in 687 days (this is known as the sidereal period of Mars). (See also this post on the cosmic distance ladder Instagram for some further elaboration.)
  • 20:52 Did Kepler really steal data from Brahe?

    The situation is complex. When Kepler served as Brahe’s assistant, Brahe only provided Kepler with a limited amount of data, primarily involving Mars, in order to confirm Brahe’s own geo-heliocentric model. After Brahe’s death, the data was inherited by Brahe’s son-in-law and other relatives, who intended to publish Brahe’s work separately; however, Kepler, who was appointed as Imperial Mathematician to succeed Brahe, had at least some partial access to the data, and many historians believe he secretly copied portions of this data to aid his own research before finally securing complete access to the data from Brahe’s heirs after several years of disputes. On the other hand, as intellectual property rights laws were not well developed at this time, Kepler’s actions were technically legal, if ethically questionable.
  • 21:39 What is that funny loop in the orbit of Mars?

    This is known as retrograde motion. This arises because the orbital velocity of Earth (about 30 km/sec) is a little bit larger than that of Mars (about 24 km/sec). So, in opposition (when Mars is in the opposite position in the sky than the Sun), Earth will briefly overtake Mars, causing its observed position to move westward rather than eastward. But in most other times, the motion of Earth and Mars are at a sufficient angle that Mars will continue its apparent eastward motion despite the slightly faster speed of the Earth.
  • 21:59 Couldn’t one also work out the direction to other celestial objects in addition to the Sun and Mars, such as the stars, the Moon, or the other planets?  Would that have helped?

    Actually, the directions to the fixed stars were implicitly used in all of these observations to determine how the celestial sphere was positioned, and all the other directions were taken relative to that celestial sphere.  (Otherwise, all the calculations would be taken on a rotating frame of reference in which the unknown orbits of the planets were themselves rotating, which would have been an even more complex task.)  But the stars are too far away to be useful as one of the two landmarks to triangulate from, as they generate almost no parallax and so cannot distinguish one location from another.

    Measuring the direction to the Moon would tell you which portion of the lunar cycle one was in, and would determine the phase of the Moon, but this information would not help one triangulate, because the Moon’s position in the heliocentric model varies over time in a somewhat complicated fashion, and is too tied to the motion of the Earth to be a useful “landmark” to one to determine the Earth’s orbit around the Sun.

    In principle, using the measurements to all the planets at once could allow for some multidimensional analysis that would be more accurate than analyzing each of the planets separately, but this would require some sophisticated statistical analysis and modeling, as well as non-trivial amounts of compute – neither of which were available in Kepler’s time.
  • 22:57 Can you elaborate on how we know that the planets all move on a plane?

    The Earth’s orbit lies in a plane known as the ecliptic (it is where the lunar and solar eclipses occur). Different cultures have divided up the ecliptic in various ways; in Western astrology, for instance, the twelve main constellations that cross the ecliptic are known as the Zodiac. The planets can be observed to only wander along the Zodiac, but not other constellations: for instance, Mars can be observed to be in Cancer or Libra, but never in Orion or Ursa Major. From this, one can conclude (as a first approximation, at least), that the planets all lie on the ecliptic.

    However, this isn’t perfectly true, and the planets will deviate from the ecliptic by a small angle known as the ecliptic latitude. Tycho Brahe’s observations on these latitudes for Mars were an additional useful piece of data that helped Kepler complete his calculations (basically by suggesting how to join together the different “jigsaw pieces”), but the math here gets somewhat complicated, so the story here has been somewhat simplified to convey the main ideas.
  • 23:04 What are the other universal problem solving tips?

    Grant Sanderson has a list (in a somewhat different order) in this previous video.
  • 23:28 Can one work out the position of Earth from fixed locations of the Sun and Mars when the Sun and Mars are in conjunction (the same location in the sky) or opposition (opposite locations in the sky)?

    Technically, these are two times when the technique of triangulation fails to be accurate; and also in the former case it is extremely difficult to observe Mars due to the proximity to the Sun. But again, following the Universal Problem Solving Tip from 23:07, one should initially ignore these difficulties to locate a viable method, and correct for these issues later. This video series by Welch Labs goes into Kepler’s methods in more detail.
  • 24:04 So Kepler used Copernicus’s calculation of 687 days for the period of Mars. But didn’t Kepler discard Copernicus’s theory of circular orbits?

    Good question! It turns out that Copernicus’s calculations of orbital periods are quite robust (especially with centuries of data), and continue to work even when the orbits are not perfectly circular. But even if the calculations did depend on the circular orbit hypothesis, it would have been possible to use the Copernican model as a first approximation for the period, in order to get a better, but still approximate, description of the orbits of the planets. This in turn can be fed back into the Copernican calculations to give a second approximation to the period, which can then give a further refinement of the orbits. Thanks to the branch of mathematics known as perturbation theory, one can often make this type of iterative process converge to an exact answer, with the error in each successive approximation being smaller than the previous one. (But performing such an iteration would probably have been beyond the computational resources available in Kepler’s time; also, the foundations of perturbation theory require calculus, which only was developed several decades after Kepler.)
  • 24:21 Did Brahe have exactly 10 years of data on Mars’s positions?

    Actually, it was more like 17 years, but with many gaps, due both to inclement weather, as well as Brahe turning his attention to other astronomical objects than Mars in some years; also, in times of conjunction, Mars might only be visible in the daytime sky instead of the night sky, again complicating measurements. So the “jigsaw puzzle pieces” in 25:26 are in fact more complicated than always just five locations equally spaced in time; there are gaps and also observational errors to grapple with. But to understand the method one should ignore these complications; again, see “Universal Problem Solving Tip #1”. Even with his “idea of true genius”, it took many years of further painstaking calculation for Kepler to tease out his laws of planetary motion from Brahe’s messy and incomplete observational data.
  • 26:44 Shouldn’t the Earth’s orbit be spread out at perihelion and clustered closer together at aphelion, to be consistent with Kepler’s laws?

    Yes, you are right; there was a coding error here.
  • 26:53 What is the reference for Einstein’s “idea of pure genius”?

    Actually, the precise quote was “an idea of true genius”, and can be found in the introduction to Carola Baumgardt’s “Life of Kepler“.

Comments on the “deleted scene” on Al-Biruni

  • Was Al-Biruni really of Arab origin?

    Strictly speaking; no; his writings are all in Arabic, and he was nominally a subject of the Abbasid Caliphate whose rulers were Arab; but he was born in Khwarazm (in modern day Uzbekistan), and would have been a subject of either the Samanid empire or the Khrawazmian empire, both of which were largely self-governed and primarily Persian in culture and ethnic makeup, despite being technically vassals of the Caliphate. So he would have been part of what is sometimes called “Greater Persia” or “Greater Iran”.

    Another minor correction: while Al-Biruni was born in the tenth century, his work on the measurement of the Earth was published in the early eleventh century.
  • Is \theta really called the angle of declination?

    This was a misnomer on my part; this angle is more commonly called the dip angle.
  • But the height of the mountain would be so small compared to the radius of the Earth! How could this method work?

    Using the Taylor approximation \cos \theta \approx 1 - \theta^2/2, one can approximately write the relationship R = \frac{h \cos \theta}{1-\cos \theta} between the mountain height h, the Earth radius R, and the dip angle \theta (in radians) as R \approx 2 h / \theta^2. The key point here is the inverse quadratic dependence on \theta, which allows for even relatively small values of h to still be realistically useful for computing R. Al-Biruni’s measurement of the dip angle \theta was about 0.01 radians, leading to an estimate of R that is about four orders of magnitude larger than h, which is within ballpark at least of a typical height of a mountain (on the order of a kilometer) and the radius of the Earth (6400 kilometers).
  • Was the method really accurate to within a percentage point?

    This is disputed, somewhat similarly to the previous calculations of Eratosthenes. Al-Biruni’s measurements were in cubits, but there were multiple incompatible types of cubit in use at the time. It has also been pointed out that atmospheric refraction effects would have created noticeable changes in the observed dip angle \theta. It is thus likely that the true accuracy of Al-Biruni’s method was poorer than 1%, but that this was somehow compensated for by choosing a favorable conversion between cubits and modern units.

Comments on the second part of the video

  • 1:13 Did Captain Cook set out to discover Australia?

    One of the objectives of Cook’s first voyage was to discover the hypothetical continent of Terra Australis. This was considered to be distinct from Australia, which at the time was known as New Holland. As this name might suggest, prior to Cook’s voyage, the northwest coastline of New Holland had been explored by the Dutch; Cook instead explored the eastern coastline, naming this portion New South Wales. The entire continent was later renamed to Australia by the British government, following a suggestion of Matthew Flinders; and the concept of Terra Australis was abandoned.
  • 4:40 The relative position of the Northern and Southern hemisphere observations is reversed from those earlier in the video.

    Yes, this was a slight error in the animation; the labels here should be swapped for consistency of orientation.
  • 7:06 So, when did they finally manage to measure the transit of Venus, and use this to compute the astronomical unit?

    While Le Gentil had the misfortune to not be able to measure either the 1761 or 1769 transits, other expeditions of astronomers (led by Dixon-Mason, Chappe d’Auteroche, and Cook) did take measurements of one or both of these transits with varying degrees of success, with the measurements of Cook’s team of the 1769 transit in Tahiti being of particularly high quality. All of this data was assembled later by Lalande in 1771, leading to the most accurate measurement of the astronomical unit at the time (within 2.3% of modern values, which was about three times more accurate than any previous measurement).
  • 8:53 What does it mean for the transit of Io to be “twenty minutes ahead of schedule” when Jupiter is in opposition (Jupiter is opposite to the Sun when viewed from the Earth)?

    Actually, it should be halved to “ten minutes ahead of schedule”, with the transit being “ten minutes behind schedule” when Jupiter is in conjunction, with the net discrepancy being twenty minutes (or actually closer to 16 minutes when measured with modern technology). Both transits are being compared against an idealized periodic schedule in which the transits are occuring at a perfectly regular rate (about 42 hours), where the period is chosen to be the best fit to the actual data. This discrepancy is only noticeable after carefully comparing transit times over a period of months; at any given position of Jupiter, the Doppler effects of Earth moving towards or away from Jupiter would only affect shift each transit by just a few seconds compared to the previous transit, with the delays or accelerations only becoming cumulatively noticeable after many such transits.

    Also, the presentation here is oversimplified: at times of conjunction, Jupiter and Io are too close to the Sun for observation of the transit. Rømer actually observed the transits at other times than conjunction, and Huygens used more complicated trigonometry than what was presented here to infer a measurement for the speed of light in terms of the astronomical unit (which they had begun to measure a bit more accurately than in Aristarchus’s time; see the FAQ entry for 15:17 in the first video).
  • 10:05 Are the astrological signs for Earth and Venus swapped here?

    Yes, this was a small mistake in the animation.
  • 10:34 Shouldn’t one have to account for the elliptical orbit of the Earth, as well as the proper motion of the star being observed, or the effects of general relativity?

    Yes; the presentation given here is a simplified one to convey the idea of the method, but in the most advanced parallax measurements, such as the ones taken by the Hipparcos and Gaia spacecraft, these factors are taken into account, basically by taking as many measurements (not just two) as possible of a single star, and locating the best fit of that data to a multi-parameter model that incorporates the (known) orbit of the Earth with the (unknown) distance and motion of the star, as well as additional gravitational effects from other celestial bodies, such as the Sun and other planets.
  • 14:53 The formula I was taught for apparent magnitude of stars looks a bit different from the one here.

    This is because astronomers use a logarithmic scale to measure both apparent magnitude m and absolute magnitude M. If one takes the logarithm of the inverse square law in the video, and performs the normalizations used by astronomers to define magnitude, one arrives at the standard relation M = m - 5 \log_{10} d_{pc} + 5 between absolute and apparent magnitude.

    But this is an oversimplification, most notably due to neglect of the effects of extinction effects caused by interstellar dust. This is not a major issue for the relatively short distances observable via parallax, but causes problems at larger scales of the ladder (see for instance the FAQ entry here for 18:08). To compensate for this, one can work in multiple frequencies of the spectrum (visible, x-ray, radio, etc.), as some frequencies are less susceptible to extinction than others. From the discrepancies between these frequencies one can infer the amount of extinction, leading to “dust maps” that can then be used to facilitate such corrections for subsequent measurements in the same area of the universe. (More generally, the trend in modern astronomy is towards “multi-messenger astronomy” in which one combines together very different types of measurements of the same object to obtain a more accurate understanding of that object and its surroundings.)
  • 18:08 Can we really measure the entire Milky Way with this method?

    Strictly speaking, there is a “zone of avoidance” on the far side of the Milky way that is very difficult to measure in the visible portion of the spectrum, due to the large amount of intervening stars, dust, and even a supermassive black hole in the galactic center. However, in recent years it has become possible to explore this zone to some extent using the radio, infrared, and x-ray portions of the spectrum, which are less affected by these factors.
  • 18:19 How did astronomers know that the Milky Way was only a small portion of the entire universe?

    This issue was the topic of the “Great Debate” in the early twentieth century. It was only with the work of Hubble using Leavitt’s law to measure distances to Magellanic clouds and “spiral nebulae” (that we now know to be other galaxies), building on earlier work of Leavitt and Hertzsprung, that it was conclusively established that these clouds and nebulae in fact were at much greater distances than the diameter of the Milky Way.
  • 18:45 How can one compensate for light blending effects when measuring the apparent magnitude of Cepheids?

    This is a non-trivial task, especially if one demands a high level of accuracy. Using the highest resolution telescopes available (such as HST or JWST) is of course helpful, as is switching to other frequencies, such as near-infrared, where Cepheids are even brighter relative to nearby non-Cepheid stars. One can also apply sophisticated statistical methods to fit to models of the point spread of light from unwanted sources, and use nearby measurements of the same galaxy without the Cepheid as a reference to help calibrate those models. Improving the accuracy of the Cepheid portion of the distance ladder is an ongoing research activity in modern astronomy.
  • 18:54 What is the mechanism that causes Cepheids to oscillate?

    For most stars, there is an equilibrium size: if the star’s radius collapses, then the reduced potential energy is converted to heat, creating pressure to pushing the star outward again; and conversely, if the star expands, then it cools, causing a reduction in pressure that no longer counteracts gravitational forces. But for Cepheids, there is an additional mechanism called the kappa mechanism: the increased temperature caused by contraction increases ionization of helium, which drains energy from the star and accelerates the contraction; conversely, the cooling caused by expansion causes the ionized helium to recombine, with the energy released accelerating the expansion. If the parameters of the Cepheid are in a certain “instability strip”, then the interaction of the kappa mechanism with the other mechanisms of stellar dynamics create a periodic oscillation in the Cepheid’s radius, which increases with the mass and brightness of the Cepheid.

    For a recent re-analysis of Leavitt’s original Cepheid data, see this paper.
  • 19:10 Did Leavitt mainly study the Cepheids in our own galaxy?

    This was an inaccuracy in the presentation. Leavitt’s original breakthrough paper studied Cepheids in the Small Magellanic Cloud. At the time, the distance to this cloud was not known; indeed, it was a matter of debate whether this cloud was in the Milky Way, or some distance away from it. However, Leavitt (correctly) assumed that all the Cepheids in this cloud were roughly the same distance away from our solar system, so that the apparent brightness was proportional to the absolute brightness. This gave an uncalibrated form of Leavitt’s law between absolute brightness and period, subject to the (then unknown) distance to the Small Magellanic Cloud. After Leavitt’s work, there were several efforts (by Hertzsprung, Russell, and Shapley) to calibrate the law by using the few Cepheids for which other distance methods were available, such as parallax. (Main sequence fitting to the Hertzsprung-Russell diagram was not directly usable, as Cepheids did not lie on the main sequence; but in some cases one could indirectly use this method if the Cepheid was in the same stellar cluster as a main sequence star.) Once the law was calibrated, it could be used to measure distances to other Cepheids, and in particular to compute distances to extragalactic objects such as the Magellanic clouds.
  • 19:15 Was Leavitt’s law really a linear law between period and luminosity?

    Strictly speaking, the period-luminosity relation commonly known as Leavitt’s law was a linear relation between the absolute magnitude of the Cepheid and the logarithm of the period; undoing the logarithms, this becomes a power law between the luminosity and the period.
  • 20:26 Was Hubble the one to discover the redshift of galaxies?

    This was an error on my part; Hubble was using earlier work of Vesto Slipher on these redshifts, and combining it with his own measurements of distances using Leavitt’s law to arrive at the law that now bears his name; he was also assisted in his observations by Milton Humason. It should also be noted that Georges Lemaître had also independently arrived at essentially the same law a few years prior, but his work was published in a somewhat obscure journal and did not receive broad recognition until some time later.
  • 20:37 Hubble’s original graph doesn’t look like a very good fit to a linear law.

    Hubble’s original data was somewhat noisy and inaccurate by modern standards, and the redshifts were affected by the peculiar velocities of individual galaxies in addition to the expanding nature of the universe. However, as the data was extended to more galaxies, it became increasingly possible to compensate for these effects and obtain a much tighter fit, particularly at larger scales where the effects of peculiar velocity are less significant. See for instance this article from 2015 where Hubble’s original graph is compared with a more modern graph. This more recent graph also reveals a slight nonlinear correction to Hubble’s law at very large scales that has led to the remarkable discovery that the expansion of the universe is in fact accelerating over time, a phenomenon that is attributed to a positive cosmological constant (or perhaps a more complex form of dark energy in the universe). On the other hand, even with this nonlinear correction, there continues to be a roughly 10% discrepancy of this law with predictions based primarily on the cosmic microwave background radiation; see the FAQ entry for 23:49.
  • 20:46 Does general relativity alone predict an uniformly expanding universe?

    This was an oversimplification. Einstein’s equations of general relativity contain a parameter \Lambda, known as the cosmological constant, which currently is only computable indirectly from fitting to experimental data. But even with this constant fixed, there are multiple solutions to these equations (basically because there are multiple possible initial conditions for the universe). For the purposes of cosmology, a particularly successful family of solutions are the solutions given by the Lambda-CDM model. This family of solutions contains additional parameters, such as the density of dark matter in the universe. Depending on the precise values of these parameters, the universe could be expanding or contracting, with the rate of expansion or contraction either increasing, decreasing, or staying roughly constant. But if one fits this model to all available data (including not just red shift measurements, but also measurements on the cosmic microwave background radiation and the spatial distribution of galaxies), one deduces a version of Hubble’s law which is nearly linear, but with an additional correction at very large scales; see the next item of this FAQ.
  • 21:07 Is Hubble’s original law sufficiently accurate to allow for good measurements of distances at the scale of the observable universe?

    Not really; as mentioned in the end of the video, there were additional efforts to cross-check and calibrate Hubble’s law at intermediate scales between the range of Cepheid methods (about 100 million light years) and observable universe scales (about 100 billion light years) by using further “standard candles” than Cepheids, most notably Type Ia supernovae (which are bright enough and predictable enough to be usable out to about 10 billion light years), the Tully-Fisher relation between the luminosity of a galaxy and its rotational speed, and gamma ray bursts. It turns out that due to the accelerating nature of the universe’s expansion, Hubble’s law is not completely linear at these large scales; this important correction cannot be discerned purely from Cepheid data, but also requires the other standard candles, as well as fitting that data (as well as other observational data, such as the cosmic microwave background radiation) to the cosmological models provided by general relativity (with the best fitting models to date being some version of the Lambda-CDM model).

    On the other hand, a naive linear extrapolation of Hubble’s original law to all larger scales does provide a very rough picture of the observable universe which, while too inaccurate for cutting edge research in astronomy, does give some general idea of its large-scale structure.
  • 21:15 Where did this guess of the observable universe being about 20% of the full universe come from?

    There are some ways to get a lower bound on the size of the entire universe that go beyond the edge of the observable universe. One is through analysis of the cosmic microwave background radiation (CMB), that has been carefully mapped out by several satellite observatories, most notably WMAP and Planck. Roughly speaking, a universe that was less than twice the size of the observable universe would create certain periodicities in the CMB data; such periodicities are not observed, so this provides a lower bound (see for instance this paper for an example of such a calculation). The 20% number was a guess based on my vague recollection of these works, but there is no consensus currently on what the ratio truly is; there are some proposals that the entire universe is in fact several orders of magnitude larger than the observable one.

    The situation is somewhat analogous to Aristarchus’s measurement of the distance to the Sun, which was very sensitive to a small angle (the half-moon discrepancy). Here, the predicted size of the universe under the standard cosmological model is similarly dependent in a highly sensitive fashion on a measure \Omega_k of the flatness of the universe which, for reasons still not fully understood (but likely caused by some sort of inflation mechanism), happens to be extremely close to zero. As such, predictions for the size of the universe remain highly volatile at the current level of measurement accuracy.
  • 23:44 Was it a black hole collision that allowed for an independent measurement of Hubble’s law?

    This was a slight error in the presentation. While the first gravitational wave observation by LIGO in 2015 was of a black hole collision, it did not come with an electromagnetic counterpart that allowed for a redshift calculation that would yield a Hubble’s law measurement. However, a later collision of neutron stars, observed in 2017, did come with an associated kilonova in which a redshift was calculated, and led to a Hubble measurement which was independent of most of the rungs of the distance ladder.
  • 23:49 Where can I learn more about this 10% discrepancy in Hubble’s law?

    This is known as the Hubble tension (or, in more sensational media, the “crisis in cosmology”): roughly speaking, the various measurements of Hubble’s constant (either from climbing the cosmic distance ladder, or by fitting various observational data to standard cosmological models) tend to arrive at one of two values, that are about 10% apart from each other. The values based on gravitational wave observations are currently consistent with both values, due to significant error bars in this extremely sensitive method; but other more mature methods are now of sufficient accuracy that they are basically only consistent with one of the two values. Currently there is no consensus on the origin of this tension: possibilities include systemic biases in the observational data, subtle statistical issues with the methodology used to interpret the data, a correction to the standard cosmological model, the influence of some previously undiscovered law of physics, or some partial breakdown of the Copernican principle.

    For an accessible recent summary of the situation, see this video by Becky Smethurst (“Dr. Becky”).
  • 24:49 So, what is a Type Ia supernova and why is it so useful in the distance ladder?

    A Type Ia supernova occurs when a white dwarf in a binary system draws more and more mass from its companion star, until it reaches the Chandrasekhar limit, at which point its gravitational forces are strong enough to cause a collapse that increases the pressure to the point where a supernova is triggered via a process known as carbon detonation. Because of the universal nature of the Chandrasekhar limit, all such supernovae have (as a first approximation) the same absolute brightness and can thus be used as standard candles in a similar fashion to Cepheids (but without the need to first measure any auxiliary observable, such as a period). But these supernovae are also far brighter than Cepheids, and can so this method can be used at significantly larger distances than the Cepheid method (roughly speaking it can handle distances of up to ~10 billion light years, whereas Cepheids are reliable out to ~100 million light years). Among other things, the supernovae measurements were the key to detecting an important nonlinear correction to Hubble’s law at these scales, leading to the remarkable conclusion that the expansion of the universe is in fact accelerating over time, which in the Lambda-CDM model corresponds to a positive cosmological constant, though there are more complex “dark energy” models that are also proposed to explain this acceleration.

  • 24:54 Besides Type Ia supernovae, I felt that a lot of other topics relevant to the modern distance ladder (e.g., the cosmic microwave background radiation, the Lambda CDM model, dark matter, dark energy, inflation, multi-messenger astronomy, etc.) were omitted.

    This is partly due to time constraints, and the need for editing to tighten the narrative, but was also a conscious decision on my part. Advanced classes on the distance ladder will naturally focus on the most modern, sophisticated, and precise ways to measure distances, backed up by the latest mathematics, physics, technology, observational data, and cosmological models. However, the focus in this video series was rather different; we sought to portray the cosmic distance ladder as evolving in a fully synergestic way, across many historical eras, with the evolution of mathematics, science, and technology, as opposed to being a mere byproduct of the current state of these other disciplines. As one specific consequence of this change of focus, we emphasized the first time any rung of the distance ladder was achieved, at the expense of more accurate and sophisticated later measurements at that rung. For instance, refinements in the measurement of the radius of the Earth since Eratosthenes, improvements in the measurement of the astronomical unit between Aristarchus and Cook, or the refinements of Hubble’s law and the cosmological model of the universe in the twentieth and twenty-first centuries, were largely omitted (though some of the answers in this FAQ are intended to address these omissions).

    Many of the topics not covered here (or only given a simplified treatment) are discussed in depth in other expositions, including other Youtube videos. I would welcome suggestions from readers for links to such resources in the comments to this post. Here is a partial list:

May 29, 2025

Doug NatelsonQuick survey - machine shops and maker spaces

Recent events are very dire for research at US universities, and I will write further about those, but first a quick unrelated survey for those at such institutions.  Back in the day, it was common for physics and some other (mechanical engineering?) departments to have machine shops with professional staff.  In the last 15-20 years, there has been a huge growth in maker-spaces on campuses to modernize and augment those capabilities, though often maker-spaces are aimed at undergraduate design courses rather than doing work to support sponsored research projects (and grad students, postdocs, etc.).  At the same time, it is now easier than ever (modulo tariffs) to upload CAD drawings to a website and get a shop in another country to ship finished parts to you.

Quick questions:   Does your university have a traditional or maker-space-augmented machine shop available to support sponsored research?  If so, who administers this - a department, a college/school, the office of research?  Does the shop charge competitive rates relative to outside vendors?  Are grad students trained to do work themselves, and are there professional machinists - how does that mix work?

Thanks for your responses.  Feel free to email me if you'd prefer to discuss offline.

Jordan EllenbergA fang, a feeling, a flair, a match, a thing for you, levitation, my TV and my pills, New York

Are the things that “I’ve got,” according to songs in my iTunes library.

Jordan EllenbergBrewers 5, Orioles 2 / Orioles 8, Brewers 4

I like to record it here when I see a baseball game and write down a few thoughts, but I forgot to do it right after CJ and I went to these games on May 20 and May 21, and so my memories are a little sparse. A few scattered things. This was the fourth time the Orioles have lost 5-2 this year. Tom Scocca and I agree that this is somehow the emblematic score for this team to lose by. To hit well enough to have won, you’d have to score 6 runs, which we’re rarely doing. And to pitch well enough to have won, you’d have to allow only 1 run, which we’re even more rarely doing. It’s a loss that doesn’t look like a blowout yet is somehow insuperable. The all-time record for most 5-2 losses in a season is 9, held by the 2011 Giants. What’s weird is, that team was actually pretty good! But when they lost, they lost 5-2.

Anyway, that was their 8th loss in a row. We saw two young pitchers, Chayce McDermott and Logan Henderson, each make their third major league start. Henderson was a lot better. Like to the point that I felt: we might want to remember that we saw one of this guy’s very first starts in the major league. A lot of very, very ugly swings from Orioles hitters.

The next afternoon was a lot better. First of all, it was Brewers Math Day. I got to meet some kids from New Berlin and their calculus teacher, who I’d had a phone meeting with earlier this season. Fun! Brewers Math Day started as an outreach program by the UW-Milwaukee math department, a great instance of the Wisconsin Idea.

The score doesn’t make it sound that way, but it was a good old-fashioned pitcher’s duel, which the Orioles almost won 3-2. But Felix Bautista, who has been looking shaky, continued to look shaky. He came in for the bottom of the 9th, he walked a couple of guys, he got two outs and got Brewers rookie 3b and 8th place hitter Caleb Durbin into a 2-strike count — the scattered O’s fans stood up to get ready to celebrate — and Durbin singled in the tying run. Orioles and Brewers traded runs in the 10th, and then the Brewers, in the 11th, ran out of guys they wanted to see pitch and sent out Joel Payamps, who they were not excited to see pitch. But Adley Rutschman was excited to see Joel Payamps pitch. And that’s how this game ended with a not very extra-innings pitching duel type score.

May 27, 2025

John PreskillI know I am but what are you? Mind and Matter in Quantum Mechanics

Nowadays it is best to exercise caution when bringing the words “quantum” and “consciousness” anywhere near each other, lest you be suspected of mysticism or quackery. Eugene Wigner did not concern himself with this when he wrote his “Remarks on the Mind-Body Question” in 1967. (Perhaps he was emboldened by his recent Nobel prize for contributions to the mathematical foundations of quantum mechanics, which gave him not a little no-nonsense technical credibility.) The mind-body question he addresses is the full-blown philosophical question of “the relation of mind to body”, and he argues unapologetically that quantum mechanics has a great deal to say on the matter. The workhorse of his argument is a thought experiment that now goes by the name “Wigner’s Friend”. About fifty years later, Daniela Frauchiger and Renato Renner formulated another, more complex thought experiment to address related issues in the foundations of quantum theory. In this post, I’ll introduce Wigner’s goals and argument, and evaluate Frauchiger’s and Renner’s claims of its inadequacy, concluding that these are not completely fair, but that their thought experiment does do something interesting and distinct. Finally, I will describe a recent paper of my own, in which I formalize the Frauchiger-Renner argument in a way that illuminates its status and isolates the mathematical origin of their paradox.

* * *

Wigner takes a dualist view about the mind, that is, he believes it to be non-material. To him this represents the common-sense view, but is nevertheless a newly mainstream attitude. Indeed,

[until] not many years ago, the “existence” of a mind or soul would have been passionately denied by most physical scientists. The brilliant successes of mechanistic and, more generally, macroscopic physics and of chemistry overshadowed the obvious fact that thoughts, desires, and emotions are not made of matter, and it was nearly universally accepted among physical scientists that there is nothing besides matter.

He credits the advent of quantum mechanics with

the return, on the part of most physical scientists, to the spirit of Descartes’s “Cogito ergo sum”, which recognizes the thought, that is, the mind, as primary. [With] the creation of quantum mechanics, the concept of consciousness came to the fore again: it was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to the consciousness.

What Wigner has in mind here is that the standard presentation of quantum mechanics speaks of definite outcomes being obtained when an observer makes a measurement. Of course this is also true in classical physics. In quantum theory, however, the principles of linear evolution and superposition, together with the plausible assumption that mental phenomena correspond to physical phenomena in the brain, lead to situations in which there is no mechanism for such definite observations to arise. Thus there is a tension between the fact that we would like to ascribe particular observations to conscious agents and the fact that we would like to view these observations as corresponding to particular physical situations occurring in their brains.

Once we have convinced ourselves that, in light of quantum mechanics, mental phenomena must be considered on an equal footing with physical phenomena, we are faced with the question of how they interact. Wigner takes it for granted that “if certain physico-chemical conditions are satisfied, a consciousness, that is, the property of having sensations, arises.” Does the influence run the other way? Wigner claims that the “traditional answer” is that it does not, but argues that in fact such influence ought indeed to exist. (Indeed this, rather than technical investigation of the foundations of quantum mechanics, is the central theme of his essay.) The strongest support Wigner feels he can provide for this claim is simply “that we do not know of any phenomenon in which one subject is influenced by another without exerting an influence thereupon”. Here he recalls the interaction of light and matter, pointing out that while matter obviously affects light, the effects of light on matter (for example radiation pressure) are typically extremely small in magnitude, and might well have been missed entirely had they not been suggested by the theory.

Quantum mechanics provides us with a second argument, in the form of a demonstration of the inconsistency of several apparently reasonable assumptions about the physical, the mental, and the interaction between them. Wigner works, at least implicitly, within a model where there are two basic types of object: physical systems and consciousnesses. Some physical systems (those that are capable of instantiating the “certain physico-chemical conditions”) are what we might call mind-substrates. Each consciousness corresponds to a mind-substrate, and each mind-substrate corresponds to at most one consciousness. He considers three claims (this organization of his premises is not explicit in his essay):

1. Isolated physical systems evolve unitarily.

2. Each consciousness has a definite experience at all times.

3. Definite experiences correspond to pure states of mind-substrates, and arise for a consciousness exactly when the corresponding mind-substrate is in the corresponding pure state.

The first and second assumptions constrain the way the model treats physical and mental phenomena, respectively. Assumption 1 is often paraphrased as the `”completeness of quantum mechanics”, while Assumption 2 is a strong rejection of solipsism – the idea that only one’s own mind is sure to exist. Assumption 3 is an apparently reasonable assumption about the relation between mental and physical phenomena.

With this framework established, Wigner’s thought experiment, now typically known as Wigner’s Friend, is quite straightforward. Suppose that an observer, Alice (to name the friend), is able to perform a measurement of some physical quantity q of a particle, which may take two values, 0 and 1. Assumption 1 tells us that if Alice performs this measurement when the particle is in a superposition state, the joint system of Alice’s brain and the particle will end up in an entangled state. Now Alice’s mind-substrate is not in a pure state, so by Assumption 3 does not have a definite experience. This contradicts Assumption 2. Wigner’s proposed resolution to this paradox is that in fact Assumption 1 is incorrect, and that there is an influence of the mental on the physical, namely objective collapse or, as he puts it, that the “statistical element which, according to the orthodox theory, enters only if I make an observation enters equally if my friend does”.

* * *

Decades after the publication of Wigner’s essay, Daniela Frauchiger and Renato Renner formulated a new thought experiment, involving observers making measurements of other observers, which they intended to remedy what they saw as a weakness in Wigner’s argument. In their words, “Wigner proposed an argument […] which should show that quantum mechanics cannot have unlimited validity”. In fact, they argue, Wigner’s argument does not succeed in doing so. They assert that Wigner’s paradox may be resolved simply by noting a difference in what each party knows. Whereas Wigner, describing the situation from the outside, does not initially know the result of his friend’s measurement, and therefore assigns the “absurd” entangled state to the joint system composed of both her body and the system she has measured, his friend herself is quite aware of what she has observed, and so assigns to the system either, but not both, of the states corresponding to definite measurement outcomes. “For this reason”, Frauchiger and Renner argue, “the Wigner’s Friend Paradox cannot be regarded as an argument that rules out quantum mechanics as a universally valid theory.”

This criticism strikes me as somewhat unfair to Wigner. In fact, Wigner’s objection to admitting two different states as equally valid descriptions is that the two states correspond to different sets of \textit{physical} properties of the joint system consisting of Alice and the system she measures. For Wigner, physical properties of physical systems are distinct from mental properties of consciousnesses. To engage in some light textual analysis, we can note that the word ‘conscious’, or ‘consciousness’, appears forty-one times in Wigner’s essay, and only once in Frauchiger and Renner’s, in the title of a cited paper. I have the impression that the authors pay inadequate attention to how explicitly Wigner takes a dualist position, including not just physical systems but also, and distinctly, consciousnesses in his ontology. Wigner’s argument does indeed achieve his goals, which are developed in the context of this strong dualism, and differ from the goals of Frauchiger and Renner, who appear not to share this philosophical stance, or at least do not commit fully to it.

Nonetheless, the thought experiment developed by Frauchiger and Renner does achieve something distinct and interesting. We can understand Wigner’s no-go theorem to be of the following form: “Within a model incorporating both mental and physical phenomena, a set of apparently reasonable conditions on how the model treats physical phenomena, mental phenomena, and their interaction cannot all be satisfied”. The Frauchiger-Renner thought experiment can be cast in the same form, with different choices about how to implement the model and which conditions to consider. The major difference in the model itself is that Frauchiger and Renner do not take consciousnesses to be entities in their own rights, but simply take some states of certain physical systems to correspond to conscious experiences. Within such a model, Wigner’s assumption that each mind has a single, definite conscious experience at all times seems far less natural than it did within his model, where consciousnesses are distinct entities from the physical systems that determine them. Thus Frauchiger and Renner need to weaken this assumption, which was so natural to Wigner. The weakening they choose is a sort of transitivity of theories of mind. In their words (Assumption C in their paper):

Suppose that agent A has established that “I am certain that agent A’, upon reasoning within the same theory as the one I am using, is certain that x =\xi at time t.” Then agent A can conclude that “I am certain that x=\xi at time t.”

Just as Assumption 3 above was, for Wigner, a natural restriction on how a sensible theory ought to treat mental phenomena, this serves as Frauchiger’s and Renner’s proposed constraint. Just as Wigner designed a thought experiment that demonstrated the incompatibility of his assumption with an assumption of the universal applicability of unitary quantum mechanics to physical systems, so do Frauchiger and Renner.

* * *

In my recent paper “Reasoning across spacelike surfaces in the Frauchiger-Renner thought experiment”, I provide two closely related formalizations of the Frauchiger-Renner argument. These are motivated by a few observations:

1. Assumption C ought to make reference to the (possibly different) times at which agents A and A' are certain about their respective judgments, since these states of knowledge change.

2. Since Frauchiger and Renner do not subscribe to Wigner’s strong dualism, an agent’s certainty about a given proposition, like any other mental state, corresponds within their implicit model to a physical state. Thus statements like “Alice knows that P” should be understood as statements about the state of some part of Alice’s brain. Conditional statements like “if upon measuring a quantity q Alice observes outcome x, she knows that P” should be understood as claims about the state of the composite system composed of the part of Alice’s brain responsible for knowing P and the part responsible for recording outcomes of the measurement of q.

3. Because the causal structure of the protocol does not depend on the absolute times of each event, an external agent describing the protocol can choose various “spacelike surfaces”, corresponding to fixed times in different spacetime embeddings of the protocol (or to different inertial frames). There is no reason to privilege one of these surfaces over another, and so each of them should be assigned a quantum state. This may be viewed as an implementation of a relativistic principle.

A visual representation of the formalization of the Frauchiger-Renner protocol and the arguments of the no-go theorem. The graphical conventions are explained in detail in “Reasoning across spacelike surfaces in the Frauchiger-Renner thought experiment”.

After developing a mathematical framework based on these observations, I recast Frauchiger’s and Renner’s Assumption C in two ways: first, in terms of a claim about the validity of iterating the “relative state” construction that captures how conditional statements are interpreted in terms of quantum states; and second, in terms of a deductive rule that allows chaining of inferences within a system of quantum logic. By proving that these claims are false in the mathematical framework, I provide a more formal version of the no-go theorem. I also show that the first claim can be rescued if the relative state construction is allowed to be iterated only “along” a single spacelike surface, and the second if a deduction is only allowed to chain inferences “along” a single surface. In other words, the mental transitivity condition desired by Frauchiger and Renner can in fact be combined with universal physical applicability of unitary quantum mechanics, but only if we restrict our analysis to a single spacelike surface. Thus I hope that the analysis I offer provides some clarification of what precisely is going on in Frauchiger and Renner’s thought experiment, what it tells us about combining the physical and the mental in light of quantum mechanics, and how it relates to Wigner’s thought experiment.

* * *

In view of the fact that “Quantum theory cannot consistently describe the use of itself” has, at present, over five hundred citations, and “Remarks on the Mind-Body Question” over thirteen hundred, it seems fitting to close with a thought, cautionary or exultant, from Peter Schwenger’s book on asemic, that is meaningless, writing. He notes that

commentary endlessly extends language; it is in the service of an impossible quest to extract the last, the final, drop of meaning.

I provide no analysis of this claim.

May 25, 2025

John PreskillThe most steampunk qubit

I never imagined that an artist would update me about quantum-computing research.

Last year, steampunk artist Bruce Rosenbaum forwarded me a notification about a news article published in Science. The article reported on an experiment performed in physicist Yiwen Chu’s lab at ETH Zürich. The experimentalists had built a “mechanical qubit”: they’d stored a basic unit of quantum information in a mechanical device that vibrates like a drumhead. The article dubbed the device a “steampunk qubit.”

I was collaborating with Bruce on a quantum-steampunk sculpture, and he asked if we should incorporate the qubit into the design. Leave it for a later project, I advised. But why on God’s green Earth are you receiving email updates about quantum computing? 

My news feed sends me everything that says “steampunk,” he explained. So keeping a bead on steampunk can keep one up to date on quantum science and technology—as I’ve been preaching for years.

Other ideas displaced Chu’s qubit in my mind until I visited the University of California, Berkeley this January. Visiting Berkeley in January, one can’t help noticing—perhaps with a trace of smugness—the discrepancy between the temperature there and the temperature at home. And how better to celebrate a temperature difference than by studying a quantum-thermodynamics-style throwback to the 1800s?

One sun-drenched afternoon, I learned that one of my hosts had designed another steampunk qubit: Alp Sipahigil, an assistant professor of electrical engineering. He’d worked at Caltech as a postdoc around the time I’d finished my PhD there. We’d scarcely interacted, but I’d begun learning about his experiments in atomic, molecular, and optical physics then. Alp had learned about my work through Quantum Frontiers, as I discovered this January. I had no idea that he’d “met” me through the blog until he revealed as much to Berkeley’s physics department, when introducing the colloquium I was about to present.

Alp and collaborators proposed that a qubit could work as follows. It consists largely of a cantilever, which resembles a pendulum that bobs back and forth. The cantilever, being quantum, can have only certain amounts of energy. When the pendulum has a particular amount of energy, we say that the pendulum is in a particular energy level. 

One might hope to use two of the energy levels as a qubit: if the pendulum were in its lowest-energy level, the qubit would be in its 0 state; and the next-highest level would represent the 1 state. A bit—a basic unit of classical information—has 0 and 1 states. A qubit can be in a superposition of 0 and 1 states, and so the cantilever could be.

A flaw undermines this plan, though. Suppose we want to process the information stored in the cantilever—for example, to turn a 0 state into a 1 state. We’d inject quanta—little packets—of energy into the cantilever. Each quantum would contain an amount of energy equal to (the energy associated with the cantilever’s 1 state) – (the amount associated with the 0 state). This equality would ensure that the cantilever could accept the energy packets lobbed at it.

But the cantilever doesn’t have only two energy levels; it has loads. Worse, all the inter-level energy gaps equal each other. However much energy the cantilever consumes when hopping from level 0 to level 1, it consumes that much when hopping from level 1 to level 2. This pattern continues throughout the rest of the levels. So imagine starting the cantilever in its 0 level, then trying to boost the cantilever into its 1 level. We’d probably succeed; the cantilever would probably consume a quantum of energy. But nothing would stop the cantilever from gulping more quanta and rising to higher energy levels. The cantilever would cease to serve as a qubit.

We can avoid this problem, Alp’s team proposed, by placing an atomic-force microscope near the cantilever. An atomic force microscope maps out surfaces similarly to how a Braille user reads: by reaching out a hand and feeling. The microscope’s “hand” is a tip about ten nanometers across. So the microscope can feel surfaces far more fine-grained than a Braille user can. Bumps embossed on a page force a Braille user’s finger up and down. Similarly, the microscope’s tip bobs up and down due to forces exerted by the object being scanned. 

Imagine placing a microscope tip such that the cantilever swings toward it and then away. The cantilever and tip will exert forces on each other, especially when the cantilever swings close. This force changes the cantilever’s energy levels. Alp’s team chose the tip’s location, the cantilever’s length, and other parameters carefully. Under the chosen conditions, boosting the cantilever from energy level 1 to level 2 costs more energy than boosting from 0 to 1.

So imagine, again, preparing the cantilever in its 0 state and injecting energy quanta. The cantilever will gobble a quantum, rising to level 1. The cantilever will then remain there, as desired: to rise to level 2, the cantilever would have to gobble a larger energy quantum, which we haven’t provided.1

Will Alp build the mechanical qubit proposed by him and his collaborators? Yes, he confided, if he acquires a student nutty enough to try the experiment. For when he does—after the student has struggled through the project like a dirigible through a hurricane, but ultimately triumphed, and a journal is preparing to publish their magnum opus, and they’re brainstorming about artwork to represent their experiment on the journal’s cover—I know just the aesthetic to do the project justice.

1Chu’s team altered their cantilever’s energy levels using a superconducting qubit, rather than an atomic force microscope.

May 24, 2025

Matt Strassler The War on Harvard University

The United States’ government is waging an all-out assault on Harvard University. The strategy, so far, has been:

  • Cut most of the grants (present and future) for scientific and medical research, so that thousands of Harvard’s scientists, researchers and graduate students have to stop their work indefinitely. That includes research on life-saving medicine, on poorly understood natural phenomena, and on new technology. This also means that the university will have no money from these activities to pay salaries of its employees.
  • Eliminate the tax-advantageous status of the university, so that the university is much more expensive to operate.
  • Prohibit Harvard from having any international students (undergraduate and graduate) and other researchers, so that large numbers of existing scientific and medical research projects that still have funding will have to cease operation. This destroys the careers of thousands of brilliant people — and not just foreigners. Many US faculty and students are working with and depend upon these expelled researchers, and their work will stop too. It also means that Harvard’s budget for the next academic year will be crushed, since it is far too late to replace the tuition from international undergraduate students for the coming year.

The grounds for this war is that Harvard allegedly does not provide a safe environment for its Jewish students, and that Harvard refuses to let the government determine who it may and may not hire.

Now, maybe you can explain to me what this is really about. I’m confused what crimes these scientific researchers commited that justifies stripping them of their grants and derailing their research. I’m also unclear as to why many apolitical, hard-working young trainees in laboratories across the campus deserve to be ejected from their graduate and post-graduate careers and sent home, delaying or ruining their futures. [Few will be able to transfer to other US schools; with all the government cuts to US science, there’s no money to support them at other locations.] And I don’t really understand how such enormous damage and disruption to the lives and careers of ten thousand-ish scientists, researchers and graduate students at Harvard (including many who are Jewish) will actually improve the atmosphere for Harvard’s Jewish students.

As far as I can see, the government is merely using Jewish students as pawns, pretending to attack Harvard on their behalf while in truth harboring no honest concern for their well-being. The fact that the horrors and nastiness surrounding the Gaza war are being exploited by the government as cover for an assault on academic freedom and scientific research is deeply cynical and exceedingly ugly.

From the outside, where Harvard is highly respected — it is certainly among the top five universities in the world, however you rank them — this must look completely idiotic, as idiotic as France gutting the Sorbonne, or the UK eviscerating Oxford. But keep in mind that Harvard is by no means the only target here. The US government is cutting the country’s world-leading research in science, technology and medicine to the bone. If that’s what you want to do, then ruining Harvard makes perfect sense.

The country that benefits the most from this self-destructive behavior? China, obviously. As a friend of mine said, this isn’t merely like shooting yourself in the foot, it’s like shooting yourself in the head.

I suspect most readers will understand that I cannot blog as usual right now. To write good articles about quantum physics requires concentration and focus. When people’s careers and life’s work are being devastated all around me, that’s simply not possible.

May 23, 2025

Matt von HippelPublishing Isn’t Free, but SciPost Makes It Cheaper

I’ve mentioned SciPost a few times on this blog. They’re an open journal in every sense you could think of: diamond open-access scientific publishing on an open-source platform, run with open finances. They even publish their referee reports. They’re aiming to cover not just a few subjects, but a broad swath of academia, publishing scientists’ work in the most inexpensive and principled way possible and challenging the dominance of for-profit journals.

And they’re struggling.

SciPost doesn’t charge university libraries for access, they let anyone read their articles for free. And they don’t charge authors Article Processing Charges (or APCs), they let anyone publish for free. All they do is keep track of which institutions those authors are affiliated with, calculate what fraction of their total costs comes from them, and post it in a nice searchable list on their website.

And amazingly, for the last nine years, they’ve been making that work.

SciPost encourages institutions to pay their share, mostly by encouraging authors to bug their bosses until they do. SciPost will also quite happily accept more than an institution’s share, and a few generous institutions do just that, which is what has kept them afloat so far. But since nothing compels anyone to pay, most organizations simply don’t.

From an economist’s perspective, this is that most basic of problems, the free-rider problem. People want scientific publication to be free, but it isn’t. Someone has to pay, and if you don’t force someone to do it, then the few who pay will be exploited by the many who don’t.

There’s more worth saying, though.

First, it’s worth pointing out that SciPost isn’t paying the same cost everyone else pays to publish. SciPost has a stripped-down system, without any physical journals or much in-house copyediting, based entirely on their own open-source software. As a result, they pay about 500 euros per article. Compare this to the fees negotiated by particle physics’ SCOAP3 agreement, which average to closer to 1000 euros, and realize that those fees are on the low end: for-profit journals tend to make their APCs higher in order to, well, make a profit.

(By the way, while it’s tempting to think of for-profit journals as greedy, I think it’s better to think of them as not cost-effective. Profit is an expense, like the interest on a loan: a payment to investors in exchange for capital used to set up the business. The thing is, online journals don’t seem to need that kind of capital, especially when they’re based on code written by academics in their spare time. So they can operate more cheaply as nonprofits.)

So when an author publishes in SciPost instead of a journal with APCs, they’re saving someone money, typically their institution or their grant. This would happen even if their institution paid their share of SciPost’s costs. (But then they would pay something rather than nothing, hence free-rider problem.)

If an author instead would have published in a closed-access journal, the kind where you have to pay to read the articles and university libraries pay through the nose to get access? Then you don’t save any money at all, your library still has to pay for the journal. You only save money if everybody at the institution stops using the journal. This one is instead a collective action problem.

Collective action problems are hard, and don’t often have obvious solutions. Free-rider problems do suggest an obvious solution: why not just charge?

In SciPost’s case, there are philosophical commitments involved. Their desire to attribute costs transparently and equally means dividing a journal’s cost among all its authors’ institutions, a cost only fully determined at the end of the year, which doesn’t make for an easy invoice.

More to the point, though, charging to publish is directly against what the Open Access movement is about.

That takes some unpacking, because of course, someone does have to pay. It probably seems weird to argue that institutions shouldn’t have to pay charges to publish papers…instead, they should pay to publish papers.

SciPost itself doesn’t go into detail about this, but despite how weird it sounds when put like I just did, there is a difference. Charging a fee to publish means that anyone who publishes needs to pay a fee. If you’re working in a developing country on a shoestring budget, too bad, you have to pay the fee. If you’re an amateur mathematician who works in a truck stop and just puzzled through something amazing, too bad, you have to pay the fee.

Instead of charging a fee, SciPost asks for support. I have to think that part of the reason is that they want some free riders. There are some people who would absolutely not be able to participate in science without free riding, and we want their input nonetheless. That means to support them, others need to give more. It means organizations need to think about SciPost not as just another fee, but as a way they can support the scientific process as a whole.

That’s how other things work, like the arXiv. They get support from big universities and organizations and philanthropists, not from literally everyone. It seems a bit weird to do that for a single scientific journal among many, though, which I suspect is part of why institutions are reluctant to do it. But for a journal that can save money like SciPost, maybe it’s worth it.

Tommaso DorigoAn Innovative Proposal

The other day I finally emerged from a very stressful push to submit two grant applications to the European Innovation Council. The call in question is for PATHFINDER_OPEN projects, that aim for proofs of principle of groundbreaking technological innovations. So I thought I would broadly report on that experience (no, I am not new to it, but you never cease to learn!), and disclose just a little about the ideas that brought about one of the two projects.
Grant applications 

read more

May 22, 2025

Doug NatelsonHow badly has NSF funding already been effectively cut?

This NY Times feature lets you see how each piece of NSF's funding has been reduced this year relative to the normalized average spanning in the last decade.  Note: this fiscal year, thanks to the continuing resolution, the actual agency budget has not actually been cut like this. They are just not spending congressionally appropriated agency funds.  The agency, fearing/assuming that its budget will get hammered next fiscal year, does not want to start awards that it won't be able to fund in out-years. The result is that this is effectively obeying in advance the presidential budget request for FY26.  (And it's highly likely that some will point to unspent funds later in the year and use that as a justification for cuts, when in fact it's anticipation of possible cuts that has led to unspent funds.  I'm sure the Germans have a polysyllabic word for this.  In English, "Catch-22" is close.)


I encourage you to click the link and go to the article where the graphic is interactive (if it works in your location - not sure about whether the link works internationally).  The different colored regions are approximately each of the NSF directorates (in their old organizational structure).  Each subsection is a particular program.  

Seems like whoever designed the graphic was a fan of Tufte, and the scaling of the shaded areas does quantitatively reflect funding changes.  However, most people have a tough time estimating relative areas of irregular polygons.  Award funding in physics (the left-most section of the middle region) is down 85% relative to past years.  Math is down 72%.  Chemistry is down 57%.  Materials is down 63%.  Earth sciences is down 80%.  Polar programs (you know, those folks who run all the amazing experiments in Antarctica) is down 88%.  

I know my readers are likely tired of me harping on NSF, but it's both important and a comparatively transparent example of what is also happening at other agencies.  If you are a US citizen and think that this is the wrong path, then push on your congressional delegation about the upcoming budget. 

May 20, 2025

Jordan EllenbergThat’s when everything went widdershins

AB asked me “what did they call clockwise and counterclockwise before they had clocks?” which is an excellent question I’d never considered. It turns out that clockwise was called “sunwise” in the Northern Hemisphere, and counterclockwise was called “widdershins.” Widdershins! That needs to be brought back. It later acquired a more general meaning of “in a direction opposite to or different from what was expected/desired.”

Jordan EllenbergGet out of my dreams

In honor of Bob Rivers, I’m considering only listening to songs that were on the Billboard Top 100 in April 1988 until the Orioles win a game. “Devil Inside” was really one of the best INXS songs but I feel it’s been largely forgotten. (“Need You Tonight” is the best INXS song, I’m sorry but sometimes the popular choice is the right one.)

Terence Trent D’Arby! I think he became a mystic and changed his name at some point. But boy do I love the little tin whistle thing in “Wishing Well.”.

But there’s no way around it, if we’re doing April 1988 we are going to have to listen to Billy Ocean, “Get Out of My Dreams, Get Into My Car.” I’m going to be at American Family Field watching the Orioles take on the Brewers tomorrow and Wednesday. Let’s hope they win; I can’t let this song get stuck in my head again.

May 19, 2025

Clifford JohnsonA New Equation?

Some years ago I speculated that it would nice if a certain mathematical object existed, and even nicer if it were to satisfy an ordinary differential equation of a special sort. I was motivated by a particular physical question, and it seemed very natural to me to imagine such an object... So natural that I was sure that it must already have been studied, the equation for it known. As a result, every so often I'd go down a rabbit hole of a literature dig, but not with much success because it isn't entirely clear where best to look. Then I'd get involved with other projects and forget all about the matter.

Last year I began to think about it again because it might be useful in a method I was developing for a paper, went through the cycle of wondering, and looking for a while, then forgot all about it in thinking about other things.

Then, a little over a month ago at the end of March, while starting on a long flight across the continent, I started thinking about it again, and given that I did not have a connection to the internet to hand, took another approach: I got out a pencil and began mess around in my notebook and just derive what I thought the equation for this object should be, given certain properties it should have. One property is that it should in some circumstances reduce to a known powerful equation (often associated with the legendary 1975 work of Gel'fand and Dikii*) satisfied by the diagonal resolvent $latex {\widehat R}(E,x) {=}\langle x|({\cal H}-E)^{-1}|x\rangle$ of a Schrodinger Hamiltonian $latex {\cal H}=-\hbar^2\partial^2_x+u(x)$. It is:

$latex 4(u(x)-E){\widehat R}^2-2\hbar^2 {\widehat R}{\widehat R}^{\prime\prime}+\hbar^2({\widehat R}^\prime)^2 = 1\ .$

Here, $latex E$ is an energy of the Hamiltonian, in potential $latex u(x)$, and $latex x$ is a coordinate on the real line.

The object itself would be a generalisation of the diagonal resolvent $latex {\widehat R}(E,x)$, although non-diagonal in the energy, not the [...] Click to continue reading this post

The post A New Equation? appeared first on Asymptotia.

Doug NatelsonA science anecdote palate cleanser

Apologies for slow posting.  Real life has been very intense, and I also was rather concerned when one of my readers mentioned last weekend that these days my blog was like concentrated doom-scrolling.  I will have more to say about the present university research crisis later, but first I wanted to give a hopefully diverting example of the kind of problem-solving and following-your-nose that crops up in research.

Recently in my lab we have had a need to measure very small changes in electrical resistance of some devices, at the level of a few milliOhms out of kiloOhms - parts in \(10^6\).  One of my students put together a special kind of resistance bridge to do this, and it works very well.  Note to interested readers: if you want to do this, make sure that you use components with very low temperature coefficients of their properties (e.g., resistors with a very small \(dR/dT\)), because otherwise your bridge becomes an extremely effective thermometer for your lab.  It’s kind of cool to be able to see the lab temperature drift around by milliKelvins, but it's not great for measuring your sample of interest.

There are a few ways to measure resistance.  The simplest is the two-terminal approach, where you drive currents through and measure voltages across your device with the same two wires.  This is easy, but it means that the voltage you measure includes contributions from the contacts those wires make with the device.  A better alternative is the four-terminal method, where you use separate wires to supply/collect the current.  

Anyway, in the course of doing some measurements of a particular device's resistance as a function of magnetic field at low temperatures, we saw something weird.  Below some rather low temperatures, when we measured in a 2-terminal arrangement, we saw a jump up in resistance by around 20 milliOhms (out of a couple of kOhms) as magnetic field was swept up from zero, and a small amount of resistance hysteresis with magnetic field sweep that vanished above maybe 0.25 T.  This vanished completely in a 4-terminal arrangement, and also disappeared above about 3.4 K.  What was this?  Turns out that I think we accidentally rediscovered the superconducting transition in indium.  While our contact pads on our sample mount looked clean to the unaided eye, they had previously had indium on there.  The magic temperature is very close to the bulk \(T_{c}\) for indium.

For one post, rather than dwelling on the terrible news about the US science ecosystem, does anyone out there have other, similar fun experimental anecdotes?  Glitches that turned out to be something surprising?  Please share in the comments.

May 18, 2025

John BaezDead Stars Don’t Radiate

Three guys claim that any heavy chunk of matter emits Hawking radiation, even if it’s not a black hole:

• Michael F. Wondrak, Walter D. van Suijlekom and Heino Falcke, Gravitational pair production and black hole evaporation, Phys. Rev. Lett. 130 (2023), 221502.

Now they’re getting more publicity by claiming this will make the universe fizzle out sooner than expected. They’re claiming, for example, that a dead, cold star will emit Hawking radiation, and thus slowly lose mass and eventually disappear!

They admit that this would violate baryon conservation: after all, the protons and neutrons in the star would have to go away somehow! They admit they don’t know how this would happen. They just say that the gravitational field of the star will create particle-antiparticle pairs that will slowly radiate away, forcing the dead star to lose mass somehow to conserve energy.

If experts thought this had even a chance of being true, it would be the biggest thing since sliced bread—at least in the field of quantum gravity. Everyone would be writing papers about it, because if true it would be revolutionary. It would overturn calculations by experts which say that a stationary chunk of matter doesn’t emit Hawking radiation. It would also mean that quantum field theory in curved spacetime can only be consistent if baryon number fails to be conserved! This would be utterly shocking.

But in fact, these new papers have had almost zero effect on physics. There’s a short rebuttal, here:

• Antonio Ferreiro José Navarro-Salas and Silvia Pla, Comment on “Gravitational pair production and black hole evaporation”, Phys. Rev. Lett. 133 (2024), 229001.

It explains that these guys used a crude approximation that gives wrong results even in a simpler problem. Similar points are made here:

• E. T. Akhmedov, D. V. Diakonov and C. Schubert, Complex effective actions and gravitational pair creation, Phys. Rev. D. 110, 105011.

Unfortunately, it seems the real experts on quantum field theory in curved spacetime have not come out and mentioned the correct way to think about this issue, which has been known at least since 1975. To them—or maybe I should dare to say “us”—it’s just well known that the gravitational field of a static mass does not cause the creation of particle-antiparticle pairs.

Of course, the referees should have rejected Wondrak, van Suijlekom and Falcke’s papers. But apparently none of those referees were experts on the subject at hand. So you can’t trust a paper just because it appears in a supposedly reputable physics journal. You have to actually understand the subject and assess the paper yourself, or talk to some experts you trust.

If I were a science journalist writing an article about a supposedly shocking development like this, I would email some experts and check to see if it’s for real. But plenty of science journalists don’t bother with that anymore: they just believe the press releases. So now we’re being bombarded with lazy articles like these:

Universe will die “much sooner than expected,” new research says, CBS News, May 13, 2025.

• Sharmila Kuthunur, Scientists calculate when the universe will end—it’s sooner than expected, Space.com, 15 May 2025.

• Jamie Carter, The universe will end sooner than thought, scientists say, Forbes, 16 May 2025.

The list goes on; these are just three. There’s no way what I say can have much effect against such a flood of misinformation. As Mark Twain said, “A lie can travel around the world and back again while the truth is lacing up its boots.” Actually he probably didn’t say that—but everyone keeps saying he did, illustrating the point perfectly.

Still, there might be a few people who both care and don’t already know this stuff. Instead of trying to give a mini-course here, let me simply point to an explanation of how things really work:

• Abhay Ashtekar and Anne Magnon, Quantum fields in curved space-times, Proceedings of the Royal Society, 346 (1975), 375–394.

It’s technical, so it’s not easy reading if you haven’t studied quantum field theory and general relativity, but that’s unavoidable. It shows that in a static spacetime there is a well-defined concept of ‘vacuum’, and the vacuum is stable. Jorge Pullin pointed out the key sentence for present purposes:

Thus, if the underlying space-time admits a everywhere time-like Killing field, the vacuum state is indeed stable and phenomena such as the spontaneous creation of particles do not occur.

This condition of having an “everywhere time-like Killing field” says that a spacetime has time translation symmetry. Ashtekar and Magnon also assume that spacetime is globally hyperbolic and that the wave equation for a massive spin-zero particle has a smooth solution given smooth initial data. All this lets us define a concept of energy for solutions of this equation. It also lets us split solutions into positive-frequency solutions, which correspond to particles, and negative-frequency ones, which correspond to antiparticles. We can thus set up quantum field theory in way we’re used to on Minkowski spacetime, where there’s a well-defined vacuum which does not decay into particle-antiparticle pairs.

The Schwarzschild solution, which describes a static black hole, also has a Killing field. But this ceases to be timelike at the event horizon, so this result does not apply to that!

I could go into more detail if required, but you can find a more pedagogical treatment in this standard textbook:

• Robert Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, University of Chicago Press, Chicago, 1994.

In particular, go to Section 4.3, which is on quantum field theory in stationary spacetimes.

I also can’t resist citing this thesis by a student of mine:

• Valeria Michelle Carrión Álvarez, Loop Quantization versus Fock Quantization of p-Form Electromagnetism on Static Spacetimes, Ph.D. thesis, U. C. Riverside, 2004.

This thesis covers the case of electromagnetism, while Ashtekar and Magnon, and also Wald, focus on a massive scalar field for simplicity.

So: it’s been rigorously shown that the gravitational field of a static object does not create particle-antiparticle pairs. This has been known for decades. Now some people have done a crude approximate calculation that seems to show otherwise. Some flaws in the approximation have been pointed out. Of course the authors of the calculation don’t believe their approximation is flawed. We could argue about that for a long time. But it’s scarcely worth thinking about, because no approximations were required to settle this issue. It was settled over 50 years ago, and the new work is not shedding new light on the issue: it’s much more hand-wavy than the old work.

Peter Rohde Singapore must immediately abolish the death penalty

May 17, 2025

Jordan EllenbergThis and that

  • The Orioles game tonight was the MLB.tv Free Game of the Day, so I watched it. The Orioles left 15 men on base — including leaving the bases loaded three times — and lost 4-3 to a terrible Nationals team when Felix Bautista forgot there was a runner on second. It was not, in fact, the free game of the day. It came at a great emotional cost.
  • I have been traveling a lot the last few weeks. I was wondering: if sitting in the exit row means you’re asserting your availability to help the crew in case of an emergency, why are they allowed to serve alcohol to exit row passengers?
  • I only just realized that “Going Down to Liverpool” (the Katrina and the Waves song made famous, or at least a little more famous, when the Bangles covered it) is a wholesome response song to the very unwholesome “Hey Joe.” “Hey Joe” is a song about a man killing his wife. “Hey Joe, where you goin’ with that gun in your hand? // I’m goin’ down to shoot my old lady.” “Going Down to Liverpool” is a song about a chill guy who’s unemployed. “I said hey, where you goin’ with that UB40 in your hand? // I’m going down to LIverpool to do nothing, all the days of my life.”
  • I actually have lots of interesting math to talk about but somehow have not felt lately — maybe because I’ve been traveling a lot, teetotaling in the exit row — that I had the spare minutes to write a substantial math post. But I will!

May 16, 2025

Matt von HippelPost on the Weak Gravity Conjecture for FirstPrinciples.org

I have another piece this week on the FirstPrinciples.org Hub. If you’d like to know who they are, I say a bit about my impressions of them in my post on the last piece I had there. They’re still finding their niche, so there may be shifts in the kind of content they cover over time, but for now they’ve given me an opportunity to cover a few topics that are off the beaten path.

This time, the piece is what we in the journalism biz call an “explainer”. Instead of interviewing people about cutting-edge science, I wrote a piece to explain an older idea. It’s an idea that’s pretty cool, in a way I think a lot of people can actually understand: a black hole puzzle that might explain why gravity is the weakest force. It’s an idea that’s had an enormous influence, both in the string theory world where it originated and on people speculating more broadly about the rules of quantum gravity. If you want to learn more, read the piece!

Since I didn’t interview anyone for this piece, I don’t have the same sort of “bonus content” I sometimes give. Instead of interviewing, I brushed up on the topic, and the best resource I found was this review article written by Dan Harlow, Ben Heidenreich, Matthew Reece, and Tom Rudelius. It gave me a much better idea of the subtleties: how many different ways there are to interpret the original conjecture, and how different attempts to build on it reflect on different facets and highlight different implications. If you are a physicist curious what the whole thing is about, I recommend reading that review: while I try to give a flavor of some of the subtleties, a piece for a broad audience can only do so much.

John BaezMeteor Burst Communications

Back before satellites, to transmit radio waves over really long distances folks bounced them off the ionosphere—a layer of charged particles in the upper atmosphere. Unfortunately this layer only reflects radio waves with frequencies up to 30 megahertz. This limits the rate at which information can be transmitted.

How to work around this?

METEOR BURST COMMUNICATIONS!

On average, 100 million meteorites weighing about a milligram hit the Earth each day. They vaporize about 120 kilometers up. Each one creates a trail of ions that lasts about a second. And you can bounce radio waves with a frequency up to 100 megahertz off this trail.

That’s not a huge improvement, and you need to transmit in bursts whenever a suitable meterorite comes your way, but the military actually looked into doing this.

The National Bureau of Standards tested a burst-mode system in 1958 that used the 50-MHz band and offered a full-duplex link at 2,400 bits per second. The system used magnetic tape loops to buffer data and transmitters at both ends of the link that operated continually to probe for a path. Whenever the receiver at one end detected a sufficiently strong probe signal from the other end, the transmitter would start sending data. The Canadians got in on the MBC action with their JANET system, which had a similar dedicated probing channel and tape buffer. In 1954 they established a full-duplex teletype link between Ottawa and Nova Scotia at 1,300 bits per second with an error rate of only 1.5%.

This is from

• Dan Maloney, Radio apocalypse: meteor burst communications, Hackaday, 2025 May 12.

and the whole article is a great read.

There’s a lot more to the story. For example, until recently people used this method in the western United States to report the snow pack from mountain tops!

The system was called SNOTEL, and you can read more about it here:

• Dan Maloney, Know snow: monitoring snowpack with the SNOTEL network, Hackaday, 2023 June 29.

Also, a lot of ham radio operators bounce signals off meteors just for fun!

• Robert Gulley, Incoming! An introduction to meteor scatter propagation, The SWLing Post, 2024 January 17.

May 15, 2025

John BaezVisions for the Future of Physics

On Wednesday May 14, 2025 I’ll be giving a talk at 2 pm Pacific Time, or 10 pm UK time. The talk is for physics students at the Universidade de São Paulo in Brazil, organized by Artur Renato Baptista Boyago.

Visions for the Future of Physics

Abstract. The 20th century was the century of fundamental physics. What about the 21st? Progress on fundamental physics has been slow since about 1980, but there is exciting progress in other fields, such as condensed matter. This requires an adjustment in how we think about the goal of physics.

You can see my slides here, or watch a video of the talk here:

May 14, 2025

Terence TaoSome variants of the periodic tiling conjecture

Rachel Greenfeld and I have just uploaded to the arXiv our paper Some variants of the periodic tiling conjecture. This paper explores variants of the periodic tiling phenomenon that, in some cases, a tile that can translationally tile a group, must also be able to translationally tile the group periodically. For instance, for a given discrete abelian group {G}, consider the following question:

Question 1 (Periodic tiling question) Let {F} be a finite subset of {G}. If there is a solution {1_A} to the tiling equation {1_F * 1_A = 1}, must there exist a periodic solution {1_{A_p}} to the same equation {1_F * 1_{A_p} = 1}?

We know that the answer to this question is positive for finite groups {H} (trivially, since all sets are periodic in this case), one-dimensional groups {{\bf Z} \times H} with {H} finite, and in {{\bf Z}^2}, but it can fail for {{\bf Z}^2 \times H} for certain finite {H}, and also for {{\bf Z}^d} for sufficiently large {d}; see this previous blog post for more discussion. But now one can consider other variants of this question:

  • Instead of considering level one tilings {1_F * 1_A = 1}, one can consider level {k} tilings {1_F * 1_A = k} for a given natural number {k} (so that every point in {G} is covered by exactly {k} translates of {F}), or more generally {1_F * 1_A = g} for some periodic function {g}.
  • Instead of requiring {1_F} and {1_A} to be indicator functions, one can allow these functions to be integer-valued, thus we are now studying convolution equations {f*a=g} where {f, g} are given integer-valued functions (with {g} periodic and {f} finitely supported).

We are able to obtain positive answers to three such analogues of the periodic tiling conjecture for three cases of this question. The first result (which was kindly shared with us by Tim Austin), concerns the homogeneous problem {f*a = 0}. Here the results are very satisfactory:

Theorem 2 (First periodic tiling result) Let {G} be a discrete abelian group, and let {f} be integer-valued and finitely supported. Then the following are equivalent.
  • (i) There exists an integer-valued solution {a} to {f*a=0} that is not identically zero.
  • (ii) There exists a periodic integer-valued solution {a_p} to {f * a_p = 0} that is not identically zero.
  • (iii) There is a vanishing Fourier coefficient {\hat f(\xi)=0} for some non-trivial character {\xi \in \hat G} of finite order.

By combining this result with an old result of Henry Mann about sums of roots of unity, as well as an even older decidability result of Wanda Szmielew, we obtain

Corollary 3 Any of the statements (i), (ii), (iii) is algorithmically decidable; there is an algorithm that, when given {G} and {f} as input, determines in finite time whether any of these assertions hold.

Now we turn to the inhomogeneous problem in {{\bf Z}^2}, which is the first difficult case (periodic tiling type results are easy to establish in one dimension, and trivial in zero dimensions). Here we have two results:

Theorem 4 (Second periodic tiling result) Let {G={\bf Z}^2}, let {g} be periodic, and let {f} be integer-valued and finitely supported. Then the following are equivalent.
  • (i) There exists an integer-valued solution {a} to {f*a=g}.
  • (ii) There exists a periodic integer-valued solution {a_p} to {f * a_p = g}.

Theorem 5 (Third periodic tiling result) Let {G={\bf Z}^2}, let {g} be periodic, and let {f} be integer-valued and finitely supported. Then the following are equivalent.
  • (i) There exists an indicator function solution {1_A} to {f*1_A=g}.
  • (ii) There exists a periodic indicator function solution {1_{A_p}} to {f * 1_{A_p} = g}.

In particular, the previously established case of periodic tiling conjecture for level one tilings of {{\bf Z}^2}, is now extended to higher level. By an old argument of Hao Wang, we now know that the statements mentioned in Theorem 5 are now also algorithmically decidable, although it remains open whether the same is the case for Theorem 4. We know from past results that Theorem 5 cannot hold in sufficiently high dimension (even in the classic case {g=1}), but it also remains open whether Theorem 4 fails in that setting.

Following past literature, we rely heavily on a structure theorem for solutions {a} to tiling equations {f*a=g}, which roughly speaking asserts that such solutions {a} must be expressible as a finite sum of functions {\varphi_w} that are one-periodic (periodic in a single direction). This already explains why tiling is easy to understand in one dimension, and why the two-dimensional case is more tractable than the case of general dimension. This structure theorem can be obtained by averaging a dilation lemma, which is a somewhat surprising symmetry of tiling equations that basically arises from finite characteristic arguments (viewing the tiling equation modulo {p} for various large primes {p}).

For Theorem 2, one can take advantage of the fact that the homogeneous equation {f*a=0} is preserved under finite difference operators {\partial_h a(x) := a(x+h)-a(x)}: if {a} solves {f*a=0}, then {\partial_h a} also solves the same equation {f * \partial_h a = 0}. This freedom to take finite differences one to selectively eliminate certain one-periodic components {\varphi_w} of a solution {a} to the homogeneous equation {f*a=0} until the solution is a pure one-periodic function, at which point one can appeal to an induction on dimension, to equate parts (i) and (ii) of the theorem. To link up with part (iii), we also take advantage of the existence of retraction homomorphisms from {{\bf C}} to {{\bf Q}} to convert a vanishing Fourier coefficient {\hat f(\xi)= 0} into an integer solution to {f*a=0}.

The inhomogeneous results are more difficult, and rely on arguments that are specific to two dimensions. For Theorem 4, one can also perform finite differences to analyze various components {\varphi_w} of a solution {a} to a tiling equation {f*a=g}, but the conclusion now is that the these components are determined (modulo {1}) by polynomials of one variable. Applying a retraction homomorphism, one can make the coefficients of these polynomials rational, which makes the polynomials periodic. This turns out to reduce the original tiling equation {f*a=g} to a system of essentially local combinatorial equations, which allows one to “periodize” a non-periodic solution by periodically repeating a suitable block of the (retraction homomorphism applied to the) original solution.

Theorem 5 is significantly more difficult to establish than the other two results, because of the need to maintain the solution in the form of an indicator function. There are now two separate sources of aperiodicity to grapple with. One is the fact that the polynomials involved in the components {\varphi_w} may have irrational coefficients (see Theorem 1.3 of our previous paper for an explicit example of this for a level 4 tiling). The other is that in addition to the polynomials (which influence the fractional parts of the components {\varphi_w}), there is also “combinatorial” data (roughly speaking, associated to the integer parts of {\varphi_w}) which also interact with each other in a slightly non-local way. Once one can make the polynomial coefficients rational, there is enough periodicity that the periodization approach used for the second theorem can be applied to the third theorem; the main remaining challenge is to find a way to make the polynomial coefficients rational, while still maintaining the indicator function property of the solution {a}.

It turns out that the restriction homomorphism approach is no longer available here (it makes the components {\varphi_w} unbounded, which makes the combinatorial problem too difficult to solve). Instead, one has to first perform a second moment analysis to discern more structure about the polynomials involved. It turns out that the components {\varphi_w} of an indicator function {1_A} can only utilize linear polynomials (as opposed to polynomials of higher degree), and that one can partition {{\bf Z}^2} into a finite number of cosets on which only three of these linear polynomials are “active” on any given coset. The irrational coefficients of these linear polynomials then have to obey some rather complicated, but (locally) finite, sentence in the theory of first-order linear inequalities over the rationals, in order to form an indicator function {1_A}. One can then use the Weyl equidistribution theorem to replace these irrational coefficients with rational coefficients that obey the same constraints (although one first has to ensure that one does not accidentally fall into the boundary of the constraint set, where things are discontinuous). Then one can apply periodization to the remaining combinatorial data to conclude.

A key technical problem arises from the discontinuities of the fractional part operator {\{x\}} at integers, so a certain amount of technical manipulation (in particular, passing at one point to a weak limit of the original tiling) is needed to avoid ever having to encounter this discontinuity.

May 13, 2025

Terence TaoA tool to verify estimates, II: a flexible proof assistant

In a recent post, I talked about a proof of concept tool to verify estimates automatically. Since that post, I have overhauled the tool twice: first to turn it into a rudimentary proof assistant that could also handle some propositional logic; and second into a much more flexible proof assistant (deliberately designed to mimic the Lean proof assistant in several key aspects) that is also powered by the extensive Python package sympy for symbolic algebra, following the feedback from previous commenters. This I think is now a stable framework with which one can extend the tool much further; my initial aim was just to automate (or semi-automate) the proving of asymptotic estimates involving scalar functions, but in principle one could keep adding tactics, new sympy types, and lemmas to the tool to handle a very broad range of other mathematical tasks as well.

The current version of the proof assistant can be found here. (As with my previous coding, I ended up relying heavily on large language model assistance to understand some of the finer points of Python and sympy, with the autocomplete feature of Github Copilot being particularly useful.) While the tool can support fully automated proofs, I have decided to focus more for now on semi-automated interactive proofs, where the human user supplies high-level “tactics” that the proof assistant then performs the necessary calculations for, until the proof is completed.

It’s easiest to explain how the proof assistant works with examples. Right now I have implemented the assistant to work inside the interactive mode of Python, in which one enters Python commands one at a time. (Readers from my generation may be familiar with text adventure games, which have a broadly similar interface.) I would be interested developing at some point a graphical user interface for the tool, but for prototype purposes, the Python interactive version suffices. (One can also run the proof assistant within a Python script, of course.)

After downloading the relevant files, one can launch the proof assistant inside Python by typing from main import * and then loading one of the pre-made exercises. Here is one such exercise:

>>> from main import *
>>> p = linarith_exercise()
Starting proof.  Current proof state:
x: pos_real
y: pos_real
z: pos_real
h1: x < 2*y
h2: y < 3*z + 1
|- x < 7*z + 2

This is the proof assistant’s formalization of the following problem: If x,y,z are positive reals such that x < 2y and y < 3z+1, prove that x < 7z+2.

The way the proof assistant works is that one directs the assistant to use various “tactics” to simplify the problem until it is solved. In this case, the problem can be solved by linear arithmetic, as formalized by the Linarith() tactic:

>>> p.use(Linarith())
Goal solved by linear arithmetic!
Proof complete!

If instead one wanted a bit more detail on how the linear arithmetic worked, one could have run this tactic instead with a verbose flag:

>>> p.use(Linarith(verbose=true))
Checking feasibility of the following inequalities:
1*z > 0
1*x + -7*z >= 2
1*y + -3*z < 1
1*y > 0
1*x > 0
1*x + -2*y < 0
Infeasible by summing the following:
1*z > 0 multiplied by 1/4
1*x + -7*z >= 2 multiplied by 1/4
1*y + -3*z < 1 multiplied by -1/2
1*x + -2*y < 0 multiplied by -1/4
Goal solved by linear arithmetic!
Proof complete!

Sometimes, the proof involves case splitting, and then the final proof has the structure of a tree. Here is one example, where the task is to show that the hypotheses (x>-1) \wedge (x<1) and (y>-2) \wedge (y<2) imply (x+y>-3) \wedge (x+y<3):

>>> from main import *
>>> p = split_exercise()
Starting proof.  Current proof state:
x: real
y: real
h1: (x > -1) & (x < 1)
h2: (y > -2) & (y < 2)
|- (x + y > -3) & (x + y < 3)
>>> p.use(SplitHyp("h1"))
Decomposing h1: (x > -1) & (x < 1) into components x > -1, x < 1.
1 goal remaining.
>>> p.use(SplitHyp("h2"))
Decomposing h2: (y > -2) & (y < 2) into components y > -2, y < 2.
1 goal remaining.
>>> p.use(SplitGoal())
Split into conjunctions: x + y > -3, x + y < 3
2 goals remaining.
>>> p.use(Linarith())
Goal solved by linear arithmetic!
1 goal remaining.
>>> p.use(Linarith())
Goal solved by linear arithmetic!
Proof complete!
>>> print(p.proof())
example (x: real) (y: real) (h1: (x > -1) & (x < 1)) (h2: (y > -2) & (y < 2)): (x + y > -3) & (x + y < 3) := by
  split_hyp h1
  split_hyp h2
  split_goal
  . linarith
  linarith

Here at the end we gave a “pseudo-Lean” description of the proof in terms of the three tactics used: a tactic cases h1 to case split on the hypothesis h1, followed by two applications of the simp_all tactic to simplify in each of the two cases.

The tool supports asymptotic estimation. I found a way to implement the order of magnitude formalism from the previous post within sympy. It turns out that sympy, in some sense, already natively implements nonstandard analysis: its symbolic variables have an is_number flag which basically corresponds to the concept of a “standard” number in nonstandard analysis. For instance, the sympy version S(3) of the number 3 has S(3).is_number == True and so is standard, whereas an integer variable n = Symbol("n", integer=true) has n.is_number == False and so is nonstandard. Within sympy, I was able to construct orders of magnitude Theta(X) of various (positive) expressions X, with the property that Theta(n)=Theta(1) if n is a standard number, and use this concept to then define asymptotic estimates such as X \lesssim Y (implemented as lesssim(X,Y)). One can then apply a logarithmic form of linear arithmetic to then automatically verify some asymptotic estimates. Here is a simple example, in which one is given a positive integer N and positive reals x,y such that x \leq 2N^2 and y < 3N, and the task is to conclude that xy \lesssim N^4:

>>> p = loglinarith_exercise()
Starting proof.  Current proof state:
N: pos_int
x: pos_real
y: pos_real
h1: x <= 2*N**2
h2: y < 3*N
|- Theta(x)*Theta(y) <= Theta(N)**4
>>> p.use(LogLinarith(verbose=True))
Checking feasibility of the following inequalities:
Theta(N)**1 >= Theta(1)
Theta(x)**1 * Theta(N)**-2 <= Theta(1)
Theta(y)**1 * Theta(N)**-1 <= Theta(1)
Theta(x)**1 * Theta(y)**1 * Theta(N)**-4 > Theta(1)
Infeasible by multiplying the following:
Theta(N)**1 >= Theta(1) raised to power 1
Theta(x)**1 * Theta(N)**-2 <= Theta(1) raised to power -1
Theta(y)**1 * Theta(N)**-1 <= Theta(1) raised to power -1
Theta(x)**1 * Theta(y)**1 * Theta(N)**-4 > Theta(1) raised to power 1
Proof complete!

The logarithmic linear programming solver can also handle lower order terms, by a rather brute force branching method:

>>> p = loglinarith_hard_exercise()
Starting proof.  Current proof state:
N: pos_int
x: pos_real
y: pos_real
h1: x <= 2*N**2 + 1
h2: y < 3*N + 4
|- Theta(x)*Theta(y) <= Theta(N)**3
>>> p.use(LogLinarith())
Goal solved by log-linear arithmetic!
Proof complete!

I plan to start developing tools for estimating function space norms of symbolic functions, for instance creating tactics to deploy lemmas such as Holder’s inequality and the Sobolev embedding inequality. It looks like the sympy framework is flexible enough to allow for creating further object classes for these sorts of objects. (Right now, I only have one proof-of-concept lemma to illustrate the framework, the arithmetic mean-geometric mean lemma.)

I am satisfied enough with the basic framework of this proof assistant that I would be open to further suggestions or contributions of new features, for instance by introducing new data types, lemmas, and tactics, or by contributing example problems that ought to be easily solvable by such an assistant, but are currently beyond its ability, for instance due to the lack of appropriate tactics and lemmas.

May 11, 2025

Tommaso DorigoOn Progress

The human race has made huge progress in the past few thousand years, gradually improving the living condition of human beings by learning how to cure illness; improving farming; harvesting, storing, and using energy in several forms; and countless other activities. 

Progress is measured over long time scales, and on metrics related to the access to innovations by all, as Ford once noted. So it is natural for us to consider ourselves lucky to have lived "in the best of times". 

Why, if you were born 400 years ago, e.g., you would probably never even learn what a hot shower is! And even only 100 years ago you could have been watching powerless as your children died of diseases that today elicit little worry.

read more

Mark GoodsellChoose France for Science

In the news this week was the joint announcement by the presidents of the European Commission and France of initiatives about welcoming top researchers from abroad, with the aim being especially to encourage researchers from the USA to cross the Atlantic. I've seen some discussion online about this among people I know and thought I'd add a few comments here, for those outside Europe thinking about making such a jump.

Firstly, what is the new initiative? Various programmes have been put in place; on the EU side it seems to be encouraging applications to Marie Curie Fellowships for postdocs and ERC grants. It looks like there is some new money, particularly for Marie Curie Fellowships for incoming researchers. Applying for these is generally good advice, as they are prestigious programs that open the way to a career; in my field a Marie Curie often leads to a permanent position, and an ERC grant is so huge that it opens doors everywhere. In France, the programme seems to be an ANR programme targeting specific strategic fields, so unlikely to be relevant for high-energy physicists (despite the fact that they invited Mark Thomson to speak at the meeting). But France can be a destination for the European programmes, and there are good reasons for choose France as a destination. 

So the advice would seem to be to try out life in France with a Marie-Curie Fellowship, and then apply through the usual channels for a permanent position. This is very reasonable, because it makes little sense to move permanently before having some idea of what life and research is actually like here first. I would heartily recommend it. There are several permanent positions available every year in the CNRS at the junior level, but because of the way the CNRS hiring works -- via a central committee, that decides for positions in the whole country -- if someone leaves it is not very easy to replace them, and people job-hopping is a recurrent problem. There is also the possibility for people to enter the CNRS at a senior level, with up to one position available in theoretical physics most years. 

I wrote a bit last year where I mentioned some of the great things about the CNRS but I will add a bit now. Firstly, what is it? It is a large organisation that essentially just hires permanent researchers, who work in laboratories throughout the country. Most of these laboratories are hosted by universities, such as my lab (the LPTHE) which is hosted by Sorbonne University. Most of these laboratories are mixed, meaning that they also include university staff, i.e. researchers who also teach undergraduates. University positions have a similar but parallel career to the CNRS, but since the teaching is done in French, and because the positions only open on a rather unpredictable basis, I won't talk about them today. The CNRS positions are 100% research; there is little administrative overhead, and therefore plenty of time to focus on what is important. This is the main advantage of such positions; but also the fact that the organisation of researchers is done into laboratories is a big difference to the Anglo-Saxon model. My lab is relatively small, yet contains a large number of people working in HEP, and this provides a very friendly environment with lots of interesting interactions, without being lost in a labyrinthine organisation or having key decisions taken by people working in vastly different (sub) fields. 

The main criticisms I have seen bandied around on social media about the CNRS are that the pay is not competitive, and that CNRS researchers are lazy/do not work. I won't comment about pay, because it's difficult to compare. But there is plenty of oversight by the CNRS committee -- a body of our peers elected by all researchers -- which scrutinises activity, in addition to deciding on hiring and promotions. If people were really sitting on their hands then this would be spotted and nipped in the bud; but the process of doing this is not onerous or intrusive, precisely because it is done by our peers. In fact, the yearly and five-yearly reports serve a useful role in helping people to focus their activities and plan for the next one to five years. There is also evaluation of laboratories and universities (the HCERES, which will now be changed into something else) that however seems sensible: it doesn't seem to lead to the same sort of panic or perverse incentives that the (equivalent) REF seems to induce in the UK, for example. 

The people I know are incredibly hard-working and productive. This is, to be fair, also a product of the fact that we have relatively few PhD students compared to other countries. This is partly by design: the philosophy is that it is unfair to train lots of students who can never get permanent positions in the field. As a result, we take good care of our students, and the students we have tend to be good; but since we have the time, we mostly do research ourselves, rather than just being managers. 

So the main reason to choose France is to be allowed to do the research you want to do, without managerialisation, bureaucrats or other obstacles interfering. If that sounds appealing, then I suggest getting in touch and/or arranging to visit. A visit to the RPP or one of the national meetings would be a great way to start. The applications for Marie Curie fellowships are open now, and the CNRS competition opens in December with a deadline usually in early January. 

May 09, 2025

Matt von HippelThere Is No Shortcut to Saying What You Mean

Blogger Andrew Oh-Willeke of Dispatches from Turtle Island pointed me to an editorial in Science about the phrase scientific consensus.

The editorial argues that by referring to conclusions like the existence of climate change or vaccine safety as “the scientific consensus”, communicators have inadvertently fanned the flames of distrust. By emphasizing agreement between scientists, the phrase “scientific consensus” leaves open the question of how that consensus was reached. More conspiracy-minded people imagine shady backroom deals and corrupt payouts, while the more realistic blame incentives and groupthink. If you disagree with “the scientific consensus”, you may thus decide the best way forward is to silence those pesky scientists.

(The link to current events is left as an exercise to the reader, to comment on elsewhere. As usual, please no explicit discussion of politics on this blog!)

Instead of “scientific consensus”, the editorial suggests another term, convergence of evidence. The idea is that by centering the evidence instead of the scientists, the phrase would make it clear that these conclusions are justified by something more than social pressures, and will remain even if the scientists promoting them are silenced.

Oh-Willeke pointed me to another blog post responding to the editorial, which has a nice discussion of how the terms were used historically, showing their popularity over time. “Convergence of evidence” was more popular in the 1950’s, with a small surge in the late 90’s and early 2000’s. “Scientific consensus” rose in the 1980’s and 90’s, lining up with a time when social scientists were skeptical about science’s objectivity and wanted to explore the social reasons why scientists come to agreement. It then fell around the year 2000, before rising again, this time used instead by professional groups of scientists to emphasize their agreement on issues like climate change.

(The blog post then goes on to try to motivate the word “consilience” instead, on the rather thin basis that “convergence of evidence” isn’t interdisciplinary enough, which seems like a pretty silly objection. “Convergence” implies coming in from multiple directions, it’s already interdisciplinary!)

I appreciate “convergence of evidence”, it seems like a useful phrase. But I think the editorial is working from the wrong perspective, in trying to argue for which terms “we should use” in the first place.

Sometimes, as a scientist or an organization or a journalist, you want to emphasize evidence. Is it “a preponderance of evidence”, most but not all? Is it “overwhelming evidence”, evidence so powerful it is unlikely to ever be defeated? Or is it a “convergence of evidence”, evidence that came in slowly from multiple paths, each independent route making a coincidence that much less likely?

But sometimes, you want to emphasize the judgement of the scientists themselves.

Sometimes when scientists agree, they’re working not from evidence but from personal experience: feelings of which kinds of research pan out and which don’t, or shared philosophies that sit deep in how they conceive their discipline. Describing physicists’ reasons for expecting supersymmetry before the LHC turned on as a convergence of evidence would be inaccurate. Describing it as having been a (not unanimous) consensus gets much closer to the truth.

Sometimes, scientists do have evidence, but as a journalist, you can’t evaluate its strength. You note some controversy, you can follow some of the arguments, but ultimately you have to be honest about how you got the information. And sometimes, that will be because it’s what most of the responsible scientists you talked to agreed on: scientific consensus.

As science communicators, we care about telling the truth (as much as we ever can, at any rate). As a result, we cannot adopt blanket rules of thumb. We cannot say, “we as a community are using this term now”. The only responsible thing we can do is to think about each individual word. We need to decide what we actually mean, to read widely and learn from experience, to find which words express our case in a way that is both convincing and accurate. There’s no shortcut to that, no formula where you just “use the right words” and everything turns out fine. You have to do the work, and hope it’s enough.

May 08, 2025

Scott Aaronson Cracking the Top Fifty!

I’ve now been blogging for nearly twenty years—through five presidential administrations, my own moves from Waterloo to MIT to UT Austin, my work on algebrization and BosonSampling and BQP vs. PH and quantum money and shadow tomography, the publication of Quantum Computing Since Democritus, my courtship and marriage and the birth of my two kids, a global pandemic, the rise of super-powerful AI and the terrifying downfall of the liberal world order.

Yet all that time, through more than a thousand blog posts on quantum computing, complexity theory, philosophy, the state of the world, and everything else, I chased a form of recognition for my blogging that remained elusive.

Until now.

This week I received the following email:

I emailed regarding your blog Shtetl-Optimized Blog which was selected by FeedSpot as one of the Top 50 Quantum Computing Blogs on the web.

https://bloggers.feedspot.com/quantum_computing_blogs

We recommend adding your website link and other social media handles to get more visibility in our list, get better ranking and get discovered by brands for collaboration.

We’ve also created a badge for you to highlight this recognition. You can proudly display it on your website or share it with your followers on social media.

We’d be thankful if you can help us spread the word by briefly mentioning Top 50 Quantum Computing Blogs in any of your upcoming posts.

Please let me know if you can do the needful.

You read that correctly: Shtetl-Optimized is now officially one of the top 50 quantum computing blogs on the web. You can click the link to find the other 49.


Maybe it’s not unrelated to this new notoriety that, over the past few months, I’ve gotten a massively higher-than-usual volume of emailed solutions to the P vs. NP problem, as well as the other Clay Millennium Problems (sometimes all seven problems at once), as well as quantum gravity and life, the universe, and everything. I now get at least six or seven confident such emails per day.

While I don’t spend much time on this flood of scientific breakthroughs (how could I?), I’d like to note one detail that’s new. Many of the emails now include transcripts where ChatGPT fills in the details of the emailer’s theories for them—unironically, as though that ought to clinch the case. Who said generative AI wasn’t poised to change the world? Indeed, I’ll probably need to start relying on LLMs myself to keep up with the flood of fan mail, hate mail, crank mail, and advice-seeking mail.

Anyway, thanks for reading everyone! I look forward to another twenty years of Shtetl-Optimized, if my own health and the health of the world cooperate.

May 06, 2025

Scott Aaronson Opposing SB37

Yesterday, the Texas State Legislature heard public comments about SB37, a bill that would give a state board direct oversight over course content and faculty hiring at public universities, perhaps inspired by Trump’s national crackdown on higher education. (See here or here for coverage.) So, encouraged by a friend in the history department, I submitted the following public comment, whatever good it will do.


I’m a computer science professor at UT, although I’m writing in my personal capacity. For 20 years, on my blog and elsewhere, I’ve been outspoken in opposing woke radicalism on campus and (especially) obsessive hatred of Israel that often veers into antisemitism, even when that’s caused me to get attacked from my left. Nevertheless, I write to strongly oppose SB37 in its current form, because of my certainty that no world-class research university can survive ceding control over its curriculum and faculty hiring to the state. If this bill passes, for example, it will severely impact my ability to recruit the most talented computer scientists to UT Austin, if they have competing options that will safeguard their academic freedom as traditionally conceived. Even if our candidates are approved, the new layer of bureaucracy will make it difficult and slow for us to do anything. For those concerned about intellectual diversity in academia, a much better solution would include safeguarding tenure and other protections for faculty with heterodox views, and actually enforcing content-neutral time, place, and manner rules for protests and disruptions. UT has actually done a better job on these things than many other universities in the US, and could serve as a national model for how viewpoint diversity can work — but not under an intolerably stifling regime like the one proposed by this bill.

May 05, 2025

Peter Rohde Protected: Random junk

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Scott Aaronson Quantum! AI! Everything but Trump!

  • Grant Sanderson, of 3blue1brown, has put up a phenomenal YouTube video explaining Grover’s algorithm, and dispelling the fundamental misconception about quantum computing, that QC works simply by “trying all the possibilities in parallel.” Let me not futz around: this video explains, in 36 minutes, what I’ve tried to explain over and over on this blog for 20 years … and it does it better. It’s a masterpiece. Yes, I consulted with Grant for this video (he wanted my intuitions for “why is the answer √N?”), and I even have a cameo at the end of it, but I wish I had made the video. Damn you, Grant!
  • The incomparably great, and absurdly prolific, blogger Zvi Mowshowitz and yours truly spend 1 hour and 40 minutes discussing AI existential risk, education, blogging, and more. I end up “interviewing” Zvi, who does the majority of the talking, which is fine by me, as he has many important things to say! (Among them: his searing critique of those K-12 educators who see it as their life’s mission to prevent kids from learning too much too fast—I’ve linked his best piece on this from the header of this blog.) Thanks so much to Rick Coyle for arranging this conversation.
  • Progress in quantum complexity theory! In 2000, John Watrous showed that the Group Non-Membership problem is in the complexity class QMA (Quantum Merlin-Arthur). In other words, if some element g is not contained in a given subgroup H of an exponentially large finite group G, which is specified via a black box, then there’s a short quantum proof that g∉H, with only ~log|G| qubits, which can be verified on a quantum computer in time polynomial in log|G|. This soon raised the question of whether Group Non-Membership could be used to separate QMA from QCMA by oracles, where QCMA (Quantum Classical Merlin Arthur), defined by Aharonov and Naveh in 2002, is the subclass of QMA where the proof needs to be classical, but the verification procedure can still be quantum. In other words, could Group Non-Membership be the first non-quantum example where quantum proofs actually help?

    In 2006, alas, Greg Kuperberg and I showed that the answer was probably “no”: Group Non-Membership has “polynomial QCMA query complexity.” This means that there’s a QCMA protocol for the problem where Arthur makes only polylog|G| quantum queries to the group oracle—albeit, possibly an exponential in log|G| number of quantum computation steps besides that! To prove our result, Greg and I needed to make mild use of the Classification of Finite Simple Groups, one of the crowning achievements of 20th-century mathematics (its proof is about 15,000 pages long). We conjectured (but couldn’t prove) that someone else, who knew more about the Classification than we did, could show that Group Non-Membership was simply in QCMA outright.

    Now, after almost 20 years, François Le Gall, Harumichi Nishimura, and Dhara Thakkar have finally proven our conjecture—showing that Group Order, and therefore also Group Non-Membership, are indeed in QCMA. They did indeed need to use the Classification, doing one thing for almost all finite groups covered by the Classification, but a different thing for groups of “Ree type” (whatever those are).

    Interestingly, the Group Membership problem had also been a candidate for separating BQP/qpoly, or quantum polynomial time with polynomial-size quantum advice—my personal favorite complexity class—from BQP/poly, or the same thing with polynomial-size classical advice. And it might conceivably still be! The authors explain to me that their protocol doesn’t put Group Membership (with group G and subgroup H depending only on the input length n) into BQP/poly, the reason being that their short classical witnesses for g∉H depend on both g and H, in contrast to Watrous’s quantum witnesses which depended only on H. So there’s still plenty that’s open here! Actually, for that matter, I don’t know of good evidence that the entire Group Membership problem isn’t in BQP—i.e., that quantum computers can’t just solve the whole thing outright, with no Merlins or witnesses in sight!

    Anyway, huge congratulations to Le Gall, Nishimura, and Thakkar for peeling back our ignorance of these matters a bit further! Reeeeeeeee!
  • Potential big progress in quantum algorithms! Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone (GLM) have given what they present as a quantum algorithm to estimate the determinant of an n×n matrix A, exponentially faster in some contexts than we know how to do it classically.

    [Update (May 5): In the comments, Alessandro Luongo shares a paper where he and Changpeng Shao describe what appears to be essentially the same algorithm back in 2020.]

    The algorithm is closely related to the 2008 HHL (Harrow-Hassidim-Lloyd) quantum algorithm for solving systems of linear equations. Which means that anyone who knows the history of this class of quantum algorithms knows to ask immediately: what’s the fine print? A couple weeks ago, when I visited Harvard and MIT, I had a chance to catch up with Seth Lloyd, so I asked him, and he kindly told me. Firstly, we assume the matrix A is Hermitian and positive semidefinite. Next, we assume A is sparse, and not only that, but there’s a QRAM data structure that points to its nonzero entries, so you don’t need to do Grover search or the like to find them, and can query them in coherent superposition. Finally, we assume that all the eigenvalues of A are at least some constant λ>0. The algorithm then estimates det(A), to multiplicative error ε, in time that scales linearly with log(n), and polynomially with 1/λ and 1/ε.

    Now for the challenge I leave for ambitious readers: is there a classical randomized algorithm to estimate the determinant under the same assumptions and with comparable running time? In other words, can the GLM algorithm be “Ewinized”? Seth didn’t know, and I think it’s a wonderful crisp open question! On the one hand, if Ewinization is possible, it wouldn’t be the first time that publicity on this blog had led to the brutal murder of a tantalizing quantum speedup. On the other hand … well, maybe not! I also consider it possible that the problem solved by GLM—for exponentially-large, implicitly-specified matrices A—is BQP-complete, as for example was the general problem solved by HHL. This would mean, for example, that one could embed Shor’s factoring algorithm into GLM, and that there’s no hope of dequantizing it unless P=BQP. (Even then, though, just like with the HHL algorithm, we’d still face the question of whether the GLM algorithm was “independently useful,” or whether it merely reproduced quantum speedups that were already known.)

    Anyway, quantum algorithms research lives! So does dequantization research! If basic science in the US is able to continue at all—the thing I promised not to talk about in this post—we’ll have plenty to keep us busy over the next few years.

Doug NatelsonUpdates, thoughts about industrial support of university research

Lots of news in the last few days regarding federal funding of university research:
  • NSF has now frozen all funding for new and continuing awards.  This is not good; just how bad it is depends on the definition of "until further notice".  
  • Here is an open letter from the NSF employees union to the basically-silent-so-far National Science Board, asking for the NSB to support the agency.
  • Here is a grass roots SaveNSF website with good information and suggestions for action - please take a look.
  • NSF also wants to cap indirect cost rates at 15% for higher ed institutions for new awards.  This will almost certainly generate a law suit from the AAU and others.  
  • Speaking of the AAU, last week there was a hearing in the Massachusetts district court regarding the lawsuits about the DOE setting indirect cost rates to 15% for active and new awards.  There had already been a temporary restraining order in place nominally stopping the change; the hearing resulted in that order being extended "until a further order is issued resolving the request for a temporary injunction."  (See here, the entry for April 29.)
  • In the meantime, the presidential budget request has come out, and if enacted it would be devastating to the science agencies.  Proposed cuts include 55% to NSF, 40% to NIH, 33% to USGS, 25% to NOAA, etc.   If these cuts went through, we are taking about more than $35B, at a rough eyeball estimate. 
  • And here is a letter from former NSF directors and NSB chairs to the appropriators in Congress, asking them to ignore that budget request and continue to support government sponsored science and engineering research.
Unsurprisingly, during these times there is a lot of talk about the need for universities to diversify their research portfolios - that is, expanding non-federally-supported ways to continue generating new knowledge, training the next generation of the technically literate workforce, and producing IP and entrepreneurial startup companies.  (Let's take it as read that it would be economically and societally desirable to continue these things, for the purposes of this post.)

Philanthropy is great, and foundations do fantastic work in supporting university research, philanthropy can't come close to making up for sharp drawdowns of federal support.  The numbers just don't work.  The endowment of the Moore Foundation, for example, is around $10B, implying an annual payout of $500M or so, which is great but around 1.4% of the cuts being envisioned.  

Industry seems like the only non-governmental possibility that could in principle muster the resources that could make a large-scale difference.   Consider the estimated profits (not revenues) of different industrial sectors.  The US semiconductor market had revenues last year of around $500B with an annualized net margin of around 17%, giving $85B/yr of profit.  US aerospace and defense similarly have an annual profit of around $70B.  The financial/banking sector, which has historically benefitted greatly from PhD-trained quants, has an annual net income of $250B.  I haven't even listed numbers for the energy and medical sectors, because those are challenging to parse (but large). 

All of those industries have been helped greatly by university research, directly and indirectly.  It's the source of trained people.  It's the source of initial work that is too long-term for corporations to be able to support without short-time-horizon shareholders getting annoyed.  It's the source of many startup companies that sometimes grow and other times get gobbled up by bigger fish. 

Encouraging greater industrial sponsorship of university research is a key challenge.  The value proposition must be made clear to both the companies and universities.  The market is unforgiving and exerts pressure to worry about the short term not the long term.  Given how Congress is functioning, it does not look like there are going to be changes to the tax code put in place that could incentivize long term investment.  

Cracking this and meaningfully growing the scale of industrial support for university research could be enormously impactful.  Something to ponder.



May 04, 2025

Tommaso DorigoThe Night Sky From Atacama

For the third time in 9 years I am visiting San Pedro de Atacama, a jewel in the middle of nowhere in northern Chile. The Atacama desert is a stretch of extremely dry land at high altitude, which makes it exceptionally attractive for astronomical activities. In its whereabouts, e.g., are some of the largest telescopes in the world - the Cerro Paranal Very Large Telescope (VLT), and the planned Extremely Large Telescope (ELT) now being built in Cerro Armazones. And I have news that an even larger telescope, tentatively dubbed RLT for Ridiculously Large Telescope, is being planned in the region...

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May 03, 2025

May 02, 2025

Matt von HippelExperiments Should Be Surprising, but Not Too Surprising

People are talking about colliders again.

This year, the European particle physics community is updating its shared plan for the future, the European Strategy for Particle Physics. A raft of proposals at the end of March stirred up a tail of public debate, focused on asking what sort of new particle collider should be built, and discussing potential reasons why.

That discussion, in turn, has got me thinking about experiments, and how they’re justified.

The purpose of experiments, and of science in general, is to learn something new. The more sure we are of something, the less reason there is to test it. Scientists don’t check whether the Sun rises every day. Like everyone else, they assume it will rise, and use that knowledge to learn other things.

You want your experiment to surprise you. But to design an experiment to surprise you, you run into a contradiction.

Suppose that every morning, you check whether the Sun rises. If it doesn’t, you will really be surprised! You’ll have made the discovery of the century! That’s a really exciting payoff, grant agencies should be lining up to pay for…

Well, is that actually likely to happen, though?

The same reasons it would be surprising if the Sun stopped rising are reasons why we shouldn’t expect the Sun to stop rising. A sunrise-checking observatory has incredibly high potential scientific reward…but an absurdly low chance of giving that reward.

Ok, so you can re-frame your experiment. You’re not hoping the Sun won’t rise, you’re observing the sunrise. You expect it to rise, almost guaranteed, so your experiment has an almost guaranteed payoff.

But what a small payoff! You saw exactly what you expected, there’s no science in that!

By either criterion, the “does the Sun rise” observatory is a stupid experiment. Real experiments operate in between the two extremes. They also mix motivations. Together, that leads to some interesting tensions.

What was the purpose of the Large Hadron Collider?

There were a few things physicists were pretty sure of, when they planned the LHC. Previous colliders had measured W bosons and Z bosons, and their properties made it clear that something was missing. If you could collide protons with enough energy, physicists were pretty sure you’d see the missing piece. Physicists had a reasonably plausible story for that missing piece, in the form of the Higgs boson. So physicists could be pretty sure they’d see something, and reasonably sure it would be the Higgs boson.

If physicists expected the Higgs boson, what was the point of the experiment?

First, physicists expected to see the Higgs boson, but they didn’t expect it to have the mass that it did. In fact, they didn’t know anything about the particle’s mass, besides that it should be low enough that the collider could produce it, and high enough that it hadn’t been detected before. The specific number? That was a surprise, and an almost-inevitable one. A rare creature, an almost-guaranteed scientific payoff.

I say almost, because there was a second point. The Higgs boson didn’t have to be there. In fact, it didn’t have to exist at all. There was a much bigger potential payoff, of noticing something very strange, something much more complicated than the straightforward theory most physicists had expected.

(Many people also argued for another almost-guaranteed payoff, and that got a lot more press. People talked about finding the origin of dark matter by discovering supersymmetric particles, which they argued was almost guaranteed due to a principle called naturalness. This is very important for understanding the history…but it’s an argument that many people feel has failed, and that isn’t showing up much anymore. So for this post, I’ll leave it to the side.)

This mix, of a guaranteed small surprise and the potential for a very large surprise, was a big part of what made the LHC make sense. The mix has changed a bit for people considering a new collider, and it’s making for a rougher conversation.

Like the LHC, most of the new collider proposals have a guaranteed payoff. The LHC could measure the mass of the Higgs, these new colliders will measure its “couplings”: how strongly it influences other particles and forces.

Unlike the LHC, though, this guarantee is not a guaranteed surprise. Before building the LHC, we did not know the mass of the Higgs, and we could not predict it. On the other hand, now we absolutely can predict the couplings of the Higgs. We have quite precise numbers, our expectation for what they should be based on a theory that so far has proven quite successful.

We aren’t certain, of course, just like physicists weren’t certain before. The Higgs boson might have many surprising properties, things that contradict our current best theory and usher in something new. These surprises could genuinely tell us something about some of the big questions, from the nature of dark matter to the universe’s balance of matter and antimatter to the stability of the laws of physics.

But of course, they also might not. We no longer have that rare creature, a guaranteed mild surprise, to hedge in case the big surprises fail. We have guaranteed observations, and experimenters will happily tell you about them…but no guaranteed surprises.

That’s a strange position to be in. And I’m not sure physicists have figured out what to do about it.

May 01, 2025

April 29, 2025

John BaezCivilizational Collapse (Part 5)

A tale of civilizational decline and rebirth

Around 250 BC Archimedes found a general algorithm for computing pi to arbitrary accuracy, and used it to prove that 223/71 < π < 22/7. This seems to be when people started using 22/7 as an approximation to pi.

By the Middle Ages, math had backslid so much in Western Europe that scholars believed pi was actually equal to 22/7. 🙄

Around 1020, a mathematician named Franco of Liège got interested in the ancient Greek problem of squaring the circle. But since he believed that pi is 22/7, he started studying the square root of 22/7.

There’s a big difference between being misinformed and being stupid. Franco was misinformed but not stupid. He went ahead to prove that the square root of 22/7 is irrational!

His proof resembles the old Greek proof that the square root of 2 is irrational. I don’t know if Franco was aware of that. I also don’t know if he noticed that if pi were 22/7, it would be possible to square the circle with straightedge and compass. I also don’t know if he wondered why pi was 22/7. He may have just taken it on authority.

But still: math was coming back.

Franco was a student of a student of the famous scholar Gerbert of Aurillac (~950–1003), who studied in the Islamic schools of Sevilla and Córdoba, and thus got some benefits of a culture whose mathematics was light years ahead of Western Europe. Gerbert wrote something interesting: he said that the benefit of mathematics lie in the “sharpening of the mind”.

I got most of this interesting tale from this book:

• Thomas Sonar, trans. Morton Patricia and Keith William Morton, 3000 Years of Analysis: Mathematics in History and Culture, Birkhäuser, 2020. Preface and table of contents free here.

It’s over 700 pages long, but it’s fun to read, and you can start anywhere! The translation is weak and occasionally funny, but tolerable. If its length is intimating, you may enjoy the detailed review here:

• Anthony Weston, 3000 years of analysis, Notices of the American Mathematical Society 70 1 (January 2023), 115–121.

April 28, 2025

John PreskillQuantum automata

Do you know when an engineer built the first artificial automaton—the first human-made machine that operated by itself, without external control mechanisms that altered the machine’s behavior over time as the machine undertook its mission?

The ancient Greek thinker Archytas of Tarentum reportedly created it about 2,300 years ago. Steam propelled his mechanical pigeon through the air.

For centuries, automata cropped up here and there as curiosities and entertainment. The wealthy exhibited automata to amuse and awe their peers and underlings. For instance, the French engineer Jacques de Vauconson built a mechanical duck that appeared to eat and then expel grains. The device earned the nickname the Digesting Duck…and the nickname the Defecating Duck.

Vauconson also invented a mechanical loom that helped foster the Industrial Revolution. During the 18th and 19th centuries, automata began to enable factories, which changed the face of civilization. We’ve inherited the upshots of that change. Nowadays, cars drive themselves, Roombas clean floors, and drones deliver packages.1 Automata have graduated from toys to practical tools.2

Rather, classical automata have. What of their quantum counterparts?

Scientists have designed autonomous quantum machines, and experimentalists have begun realizing them. The roster of such machines includes autonomous quantum engines, refrigerators, and clocks. Much of this research falls under the purview of quantum thermodynamics, due to the roles played by energy in these machines’ functioning: above, I defined an automaton as a machine free of time-dependent control (exerted by a user). Equivalently, according to a thermodynamicist mentality, we can define an automaton as a machine on which no user performs any work as the machine operates. Thermodynamic work is well-ordered energy that can be harnessed directly to perform a useful task. Often, instead of receiving work, an automaton receives access to a hot environment and a cold environment. Heat flows from the hot to the cold, and the automaton transforms some of the heat into work.

Quantum automata appeal to me because quantum thermodynamics has few practical applications, as I complained in my previous blog post. Quantum thermodynamics has helped illuminate the nature of the universe, and I laud such foundational insights. Yet we can progress beyond laudation by trying to harness those insights in applications. Some quantum thermal machines—quantum batteries, engines, etc.—can outperform their classical counterparts, according to certain metrics. But controlling those machines, and keeping them cold enough that they behave quantum mechanically, costs substantial resources. The machines cost more than they’re worth. Quantum automata, requiring little control, offer hope for practicality. 

To illustrate this hope, my group partnered with Simone Gasparinetti’s lab at Chalmer’s University in Sweden. The experimentalists created an autonomous quantum refrigerator from superconducting qubits. The quantum refrigerator can help reset, or “clear,” a quantum computer between calculations.

Artist’s conception of the autonomous-quantum-refrigerator chip. Credit: Chalmers University of Technology/Boid AB/NIST.

After we wrote the refrigerator paper, collaborators and I raised our heads and peered a little farther into the distance. What does building a useful autonomous quantum machine take, generally? Collaborators and I laid out guidelines in a “Key Issues Review” published in Reports in Progress on Physics last November.

We based our guidelines on DiVincenzo’s criteria for quantum computing. In 1996, David DiVincenzo published seven criteria that any platform, or setup, must meet to serve as a quantum computer. He cast five of the criteria as necessary and two criteria, related to information transmission, as optional. Similarly, our team provides ten criteria for building useful quantum automata. We regard eight of the criteria as necessary, at least typically. The final two, optional guidelines govern information transmission and machine transportation. 

Time-dependent external control and autonomy

DiVincenzo illustrated his criteria with multiple possible quantum-computing platforms, such as ions. Similarly, we illustrate our criteria in two ways. First, we show how different quantum automata—engines, clocks, quantum circuits, etc.—can satisfy the criteria. Second, we illustrate how quantum automata can consist of different platforms: ultracold atoms, superconducting qubits, molecules, and so on.

Nature has suggested some of these platforms. For example, our eyes contain autonomous quantum energy transducers called photoisomers, or molecular switches. Suppose that such a molecule absorbs a photon. The molecule may use the photon’s energy to switch configuration. This switching sets off chemical and neurological reactions that result in the impression of sight. So the quantum switch transduces energy from light into mechanical, chemical, and electric energy.

Photoisomer. (Image by Todd Cahill, from Quantum Steampunk.)

My favorite of our criteria ranks among the necessary conditions: every useful quantum automata must produce output worth the input. How one quantifies a machine’s worth and cost depends on the machine and on the user. For example, an agent using a quantum engine may care about the engine’s efficiency, power, or efficiency at maximum power. Costs can include the energy required to cool the engine to the quantum regime, as well as the control required to initialize the engine. The agent also chooses which value they regard as an acceptable threshold for the output produced per unit input. I like this criterion because it applies a broom to dust that we quantum thermodynamicists often hide under a rug: quantum thermal machines’ costs. Let’s begin building quantum engines that perform more work than they require to operate.

One might object that scientists and engineers are already sweating over nonautonomous quantum machines. Companies, governments, and universities are pouring billions of dollars into quantum computing. Building a full-scale quantum computer by hook or by crook, regardless of classical control, is costing enough. Eliminating time-dependent control sounds even tougher. Why bother?

Fellow Quantum Frontiers blogger John Preskill pointed out one answer, when I described my new research program to him in 2022: control systems are classical—large and hot. Consider superconducting qubits—tiny quantum circuits—printed on a squarish chip about the size of your hand. A control wire terminates on each qubit. The rest of the wire runs off the edge of the chip, extending to classical hardware standing nearby. One can fit only so many wires on the chip, so one can fit only so many qubits. Also, the wires, being classical, are hotter than the qubits should be. The wires can help decohere the circuits, introducing errors into the quantum information they store. The more we can free the qubits from external control—the more autonomy we can grant them—the better.

Besides, quantum automata exemplify quantum steampunk, as my coauthor Pauli Erker observed. I kicked myself after he did, because I’d missed the connection. The irony was so thick, you could have cut it with the retractible steel knife attached to a swashbuckling villain’s robotic arm. Only two years before, I’d read The Watchmaker of Filigree Street, by Natasha Pulley. The novel features a Londoner expatriate from Meiji Japan, named Mori, who builds clockwork devices. The most endearing is a pet-like octopus, called Katsu, who scrambles around Mori’s workshop and hoards socks. 

Does the world need a quantum version of Katsu? Not outside of quantum-steampunk fiction…yet. But a girl can dream. And quantum automata now have the opportunity to put quantum thermodynamics to work.

From tumblr

1And deliver pizzas. While visiting the University of Pittsburgh a few years ago, I was surprised to learn that the robots scurrying down the streets were serving hungry students.

2And minions of starving young scholars.

April 27, 2025

Scott Aaronson Fight Fiercely

Last week I visited Harvard and MIT, and as advertised in my last post, gave the Yip Lecture at Harvard on the subject “How Much Math Is Knowable?” The visit was hosted by Harvard’s wonderful Center of Mathematical Sciences and Applications (CMSA), directed by my former UT Austin colleague Dan Freed. Thanks so much to everyone at CMSA for the visit.

And good news! You can now watch my lecture on YouTube here:

I’m told it was one of my better performances. As always, I strongly recommend watching at 2x speed.

I opened the lecture by saying that, while obviously it would always be an honor to give the Yip Lecture at Harvard, it’s especially an honor right now, as the rest of American academia looks to Harvard to defend the value of our entire enterprise. I urged Harvard to “fight fiercely,” in the words of the Tom Lehrer song.

I wasn’t just fishing for applause; I meant it. It’s crucial for people to understand that, in its total war against universities, MAGA has now lost, not merely the anti-Israel leftists, but also most conservatives, classical liberals, Zionists, etc. with any intellectual scruples whatsoever. To my mind, this opens up the possibility for a broad, nonpartisan response, highlighting everything universities (yes, even Harvard 😂) do for our civilization that’s worth defending.

For three days in my old hometown of Cambridge, MA, I met back-to-back with friends and colleagues old and new. Almost to a person, they were terrified about whether they’ll be able to keep doing science as their funding gets decimated, but especially terrified for anyone who they cared about on visas and green cards. International scholars can now be handcuffed, deported, and even placed in indefinite confinement for pretty much any reason—including long-ago speeding tickets—or no reason at all. The resulting fear has paralyzed, in a matter of months, an American scientific juggernaut that took a century to build.

A few of my colleagues personally knew Rümeysa Öztürk, the Turkish student at Tufts who currently sits in prison for coauthoring an editorial for her student newspaper advocating the boycott of Israel. I of course disagree with what Öztürk wrote … and that is completely irrelevant to my moral demand that she go free. Even supposing the government had much more on her than this one editorial, still the proper response would seem to be a deportation notice—“either contest our evidence in court, or else get on the next flight back to Turkey”—rather than grabbing Öztürk off the street and sending her to indefinite detention in Louisiana. It’s impossible to imagine any university worth attending where the students live in constant fear of imprisonment for the civil expression of opinions.

To help calibrate where things stand right now, here’s the individual you might expect to be most on board with a crackdown on antisemitism at Harvard:

Jason Rubenstein, the executive director of Harvard Hillel, said that the school is in the midst of a long — and long-overdue — reckoning with antisemitism, and that [President] Garber has taken important steps to address the problem. Methodical federal civil rights oversight could play a constructive role in that reform, he said. “But the government’s current, fast-paced assault against Harvard – shuttering apolitical, life-saving research; targeting the university’s tax-exempt status; and threatening all student visas … is neither deliberate nor methodical, and its disregard for the necessities of negotiation and due process threatens the bulwarks of institutional independence and the rule of law that undergird our shared freedoms.”

Meanwhile, as the storm clouds over American academia continue to darken, I’ll just continue to write what I think about everything, because what else can I do?

Last night, alas, I lost yet another left-wing academic friend, the fourth or fifth I’ve lost since October 7. For while I was ready to take a ferocious public stand against the current US government, for the survival and independence of our universities, and for free speech and due process for foreign students, this friend regarded all that as insufficient. He demanded that I also clear the tentifada movement of any charge of antisemitism. For, as he patiently explained to me (while worrying that I wouldn’t grasp the point), while the protesters may have technically violated university rules, disrupted education, created a hostile environment in the sense of Title VI antidiscrimination law in ways that would be obvious were we discussing any other targeted minority, etc. etc., still, the only thing that matters morally is that the protesters represent “the powerless,” whereas Zionist Jews like me represent “the powerful.” So, I told this former friend to go fuck himself. Too harsh? Maybe if he hadn’t been Jewish himself, I could’ve forgiven him for letting the world’s oldest conspiracy theory colonize his brain.

For me, the deep significance of in-person visits, including my recent trip to Harvard, is that they reassure me of the preponderance of sanity within my little world—and thereby of my own sanity. Online, every single day I feel isolated and embattled: pressed in on one side by MAGA forces who claim to care about antisemitism, but then turn out to want the destruction of science, universities, free speech, international exchange, due process of law, and everything else that’s made the modern world less than fully horrible; and on the other side, by leftists who say they stand with me for science and academic freedom and civil rights and everything else that’s good, but then add that the struggle needs to continue until the downfall of the scheming, moneyed Zionists and the liberation of Palestine from river to sea.

When I travel to universities to give talks, though, I meet one sane, reasonable human being after another. Almost to a person, they acknowledge the reality of antisemitism, ideological monoculture, bureaucracy, spiraling costs, and many other problems at universities—and they care about universities enough to want to fix those problems, rather than gleefully nuking the universities from orbit as MAGA is doing. Mostly, though, people just want me to sign Quantum Computing Since Democritus, or tell me how much they like this blog, or ask questions about quantum algorithms or the Busy Beaver function. Which is fine too, and which you can do in the comments.

April 25, 2025

John PreskillQuantum Algorithms: A Call To Action

Quantum computing finds itself in a peculiar situation. On the technological side, after billions of dollars and decades of research, working quantum computers are nearing fruition. But still, the number one question asked about quantum computers is the same as it was two decades ago: What are they good for? The honest answer reveals an elephant in the room: We don’t fully know yet. For theorists like me, this is an opportunity, a call to action.

Technological momentum

Suppose we do not have quantum computers in a few decades time. What will be the reason? It’s unlikely that we’ll encounter some insurmountable engineering obstacle. The theoretical basis of quantum error-correction is solid, and several platforms are approaching or below the error-correction threshold (Harvard, Yale, Google). Experimentalists believe today’s technology can scale to 100 logical qubits and 10^6 gates—the megaquop era. If mankind spends $100 billion over the next few decades, it’s likely we could build a quantum computer.

A more concerning reason that quantum computing might fail is that there is not enough incentive to justify such a large investment in R&D and infrastructure. Let’s make a comparison to nuclear fusion. Like quantum hardware, they have challenging science and engineering problems to solve. However, if a nuclear fusion lab were to succeed in their mission of building a nuclear fusion reactor, the application would be self-evident. This is not the case for quantum computing—it is a sledgehammer looking for nails to hit.

Nevertheless, industry investment in quantum computing is currently accelerating. To maintain the momentum, it is critical to match investment growth and hardware progress with algorithmic capabilities. The time to discover quantum algorithms is now.

Empowered theorists

Theory research is forward-looking and predictive. Theorists such as Geoffrey Hinton laid the foundations of the current AI revolution. But decades later, with an abundance of computing hardware, AI has become much more of an empirical field. I look forward to the day that quantum hardware reaches a state of abundance, but that day is not yet here.

Today, quantum computing is an area where theorists have extraordinary leverage. A few pages of mathematics by Peter Shor inspired thousands of researchers, engineers and investors to join the field. Perhaps another few pages by someone reading this blog will establish a future of world-altering impact for the industry. There are not many places where mathematics has such potential for influence. An entire community of experimentalists, engineers, and businesses are looking to the theorists for ideas.

The Challenge

Traditionally, it is thought that the ideal quantum algorithm would exhibit three features. First, it should be provably correct, giving a guarantee that executing the quantum circuit reliably will achieve the intended outcome. Second, the underlying problem should be classically hard—the output of the quantum algorithm should be computationally hard to replicate with a classical algorithm. Third, it should be useful, with the potential to solve a problem of interest in the real world. Shor’s algorithm comes close to meeting all of these criteria. However, demanding all three in an absolute fashion may be unnecessary and perhaps even counterproductive to progress.

Provable correctness is important, since today we cannot yet empirically test quantum algorithms on hardware at scale. But what degree of evidence should we require for classical hardness? Rigorous proof of classical hardness is currently unattainable without resolving major open problems like P vs NP, but there are softer forms of proof, such as reductions to well-studied classical hardness assumptions.

I argue that we should replace the ideal of provable hardness with a more pragmatic approach: The quantum algorithm should outperform the best known classical algorithm that produces the same output by a super-quadratic speedup.1 Emphasizing provable classical hardness might inadvertently impede the discovery of new quantum algorithms, since a truly novel quantum algorithm could potentially introduce a new classical hardness assumption that differs fundamentally from established ones. The back-and-forth process of proposing and breaking new assumptions is a productive direction that helps us triangulate where quantum advantage lies.

It may also be unproductive to aim directly at solving existing real-world problems with quantum algorithms. Fundamental computational tasks with quantum advantage are special and we have very few examples, yet they necessarily provide the basis for any eventual quantum application. We should search for more of these fundamental tasks and match them to applications later.

That said, it is important to distinguish between quantum algorithms that could one day provide the basis for a practically relevant computation, and those that will not. In the real world, computations are not useful unless they are verifiable or at least repeatable. For instance, consider a quantum simulation algorithm that computes a physical observable. If two different quantum computers run the simulation and get the same answer, one can be confident that this answer is correct and that it makes a robust prediction about the world. Some problems such as factoring are naturally easy to verify classically, but we can set the bar even lower: The output of a useful quantum algorithm should at least be repeatable by another quantum computer.

There is a subtle fourth requirement of paramount importance that is often overlooked, captured by the following litmus test: If given a quantum computer tomorrow, could you implement your quantum algorithm? In order to do so, you need not only a quantum algorithm but also a distribution over its inputs on which to run it. Classical hardness must then be judged in the average case over this distribution of inputs, rather than in the worst case.

I’ll end this section with a specific caution regarding quantum algorithms whose output is the expectation value of an observable. A common reason these proposals fail to be classically hard is that the expectation value exponentially concentrates over the distribution of inputs. When this happens, a trivial classical algorithm can replicate the quantum result by simply outputting the concentrated (typical) value for every input. To avoid this, we must seek ensembles of quantum circuits whose expectation values exhibit meaningful variation and sensitivity to different inputs.

We can crystallize these priorities into the following challenge:

The Challenge
Find a quantum algorithm and a distribution over its inputs with the following features:
— (Provable correctness.) The quantum algorithm is provably correct.
— (Classical hardness.) The quantum algorithm outperforms the best known classical algorithm that performs the same task by a super-quadratic speedup, in the average-case over the distribution of inputs.
— (Potential utility.) The output is verifiable, or at least repeatable.

Examples and non-examples

CategoryClassically verifiableQuantumly repeatablePotentially usefulProvable classical hardnessExamples
Search problemYesYesYesNoShor ‘99

Regev’s reduction: CLZ22, YZ24, Jor+24

Planted inference: Has20, SOKB24
Compute a valueNoYesYesNoCondensed matter physics?

Quantum chemistry?
Proof of quantumnessYes, with keyYes, with respect to keyNoYes, under crypto assumptionsBCMVV21
SamplingNoNoNoAlmost, under complexity assumptionsBJS10, AA11, Google ‘20
We can categorize quantum algorithms by the form of their output. First, there are quantum algorithms for search problems, which produce a bitstring satisfying some constraints. This could be the prime factors of a number, a planted feature in some dataset, or the solution to an optimization problem. Next, there are quantum algorithms that compute a value to some precision, for example the expectation value of some physical observable. Then there are proofs of quantumness, which involve a verifier who generates a test using some hidden key, and the key can be used to verify the output. Finally, there are quantum algorithms which sample from some distribution.

Hamiltonian simulation is perhaps the most widely heralded source of quantum utility. Physics and chemistry contain many quantities that Nature computes effortlessly, yet remain beyond the reach of even our best classical simulations. Quantum computation is capable of simulating Nature directly, giving us strong reason to believe that quantum algorithms can compute classically-hard quantities.

There are already many examples where a quantum computer could help us answer an unsolved scientific question, like determining the phase diagram of the Hubbard model or the ground energy of FeMoCo. These undoubtedly have scientific value. However, they are isolated examples, whereas we would like evidence that the pool of quantum-solvable questions is inexhaustible. Can we take inspiration from strongly correlated physics to write down a concrete ensemble of Hamiltonian simulation instances where there is a classically-hard observable? This would gather evidence for the sustained, broad utility of quantum simulation, and would also help us understand where and how quantum advantage arises.

Over in the computer science community, there has been a lot of work on oracle separations such as welded trees and forrelation, which should give us confidence in the abilities of quantum computers. Can we instantiate these oracles in a way that pragmatically remains classically hard? This is necessary in order to pass our earlier litmus test of being ready to run the quantum algorithm tomorrow.

In addition to Hamiltonian simulation, there are several other broad classes of quantum algorithms, including quantum algorithms for linear systems of equations and differential equations, variational quantum algorithms for machine learning, and quantum algorithms for optimization. These frameworks sometimes come with proofs of BQP-completeness.

The issue with these broad frameworks is that they often do not specify a distribution over inputs. Can we find novel ensembles of inputs to these frameworks which exhibit super-quadratic speedups? BQP-completeness shows that one has translated the notion of quantum computation into a different language, which allows one to embed an existing quantum algorithm such as Shor’s algorithm into your framework. But in order to discover a new quantum algorithm, you must find an ensemble of BQP computations which does not arise from Shor’s algorithm.

Table I claims that sampling tasks alone are not useful since they are not even quantumly repeatable. One may wonder if sampling tasks could be useful in some way. After all, classical Monte Carlo sampling algorithms are widely used in practice. However, applications of sampling typically use samples to extract meaningful information or specific features of the underlying distribution. For example, Monte Carlo sampling can be used to evaluate integrals in Bayesian inference and statistical physics. In contrast, samples obtained from random quantum circuits lack any discernible features. If a collection of quantum algorithms generated samples containing meaningful signals from which one could extract classically hard-to-compute values, those algorithms would effectively transition into the compute a value category.

Table I also claims that proofs of quantumness are not useful. This is not completely true—one potential application is generating certifiable randomness. However, such applications are generally cryptographic rather than computational in nature. Specifically, proofs of quantumness cannot help us solve problems or answer questions whose solutions we do not already know.

Finally, there are several exciting directions proposing applications of quantum technologies in sensing and metrology, communication, learning with quantum memory, and streaming. These are very interesting, and I hope that mankind’s second century of quantum mechanics brings forth all flavors of capabilities. However, the technological momentum is mostly focused on building quantum computers for the purpose of computational advantage, and so this is where breakthroughs will have the greatest immediate impact.

Don’t be too afraid

At the annual QIP conference, only a handful of papers out of hundreds each year attempt to advance new quantum algorithms. Given the stakes, why is this number so low? One common explanation is that quantum algorithm research is simply too difficult. Nevertheless, we have seen substantial progress in quantum algorithms in recent years. After an underwhelming lack of end-to-end proposals with the potential for utility between the years 2000 and 2020, Table I exhibits several breakthroughs from the past 5 years.

In between blind optimism and resigned pessimism, embracing a mission-driven mindset can propel our field forward. We should allow ourselves to adopt a more exploratory, scrappier approach: We can hunt for quantum advantages in yet-unstudied problems or subtle signals in the third decimal place. The bar for meaningful progress is lower than it might seem, and even incremental advances are valuable. Don’t be too afraid!

  1. Quadratic speedups are widespread but will not form the basis of practical quantum advantage due to the overheads associated with quantum error-correction. ↩

April 18, 2025

April 16, 2025

Tommaso DorigoOn A Roll

What? Another boring chess game?


Buzz off, this is my blog, and if I feel like posting a chess game, that's what is going to happen. But if you like the game, stay here - this is a nice game.

Again played after hiours today, and again on a 5' online blitz server (chess.com). What amazes me is that these days I seem to have a sort of touch for nice attacks and brilliant combinations. Let me show you why I am saying this.

The starting position arose after the following opening sequence:

tommasodorigo - UTOPII841, chess.com April 16 2025

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April 15, 2025

n-Category Café Position in Stellenbosch

guest post by Bruce Bartlett

Stellenbosch University is hiring!

The Mathematics Division at Stellenbosch University in South Africa is looking to hire a new permanent appointment at Lecturer / Senior Lecturer level (other levels may be considered too under the appropriate circumstances).

Preference will be given to candidates working in number theory or a related area, but those working in other areas of mathematics will definitely also be considered.

The closing date for applications is 30 April 2025. For more details, kindly see the official advertisement.

Consider a wonderful career in the winelands area of South Africa!

Tommaso DorigoWhen The Attack Plays Itself

After a very intense day at work, I sought some relaxation in online blitz chess today. And the game gave me the kick I was hoping I'd get. After a quick Alapin Sicilian opening, we reached the following position (diagram 1):


As you can see, black is threatening a checkmate with Qxg2++. However, the last move was a serious error, as it neglected the intrinsic power of my open files and diagonals against the black king. Can you find the sequence with which I quickly destroyed my opponent's position?

1. Qc2! 

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n-Category Café Categorical Linguistics in Quanta

Quanta magazine has just published a feature on Tai-Danae Bradley and her work, entitled

Where Does Meaning Live in a Sentence? Math Might Tell Us.

The mathematician Tai-Danae Bradley is using category theory to try to understand both human and AI-generated language.

It’s a nicely set up Q&A, with questions like “What’s something category theory lets you see that you can’t otherwise?” and “How do you use category theory to understand language?”

Particularly interesting for me is the part towards the end where Bradley describes her work with Juan Pablo Vigneaux on magnitude of enriched categories of texts.

April 14, 2025

Matt Strassler Is Superposition Really an “OR”?

We’ll get back to measurement, interference and the double-slit experiment just as soon as I can get my math program to produce pictures of the relevant wave functions reliably. I owe you some further discussion of why measurement (and even interactions without measurement) can partially or completely eliminate quantum interference.

But in the meantime, I’ve gotten some questions and some criticism for arguing that superposition is an OR, not an AND. It is time to look closely at this choice, and understand both its strengths and its limitations, and how we have to move beyond it to fully appreciate quantum physics. [I probably should have written this article earlier — and I suspect I’ll need to write it again someday, as it’s a tricky subject.]

The Question of Superposition

Just to remind you of my definitions (we’ll see examples in a moment): objects that interact with one another form a system, and a system is at any time in a certain quantum state, consisting of one or more possibilities combined in some way and described by what is often called a “wave function”. If the number of possibilities described by the wave function is more than one, then physicists say that the state of the quantum system is a superposition of two or more basic states. [Caution: as we’ll explore in later posts, the number of states in the superposition can depend on one’s choice of “basis”.]

As an example, suppose we have two boxes, L and R for left and right, and two atoms, one of hydrogen H and one of nitrogen N. Our physical system consists of the two atoms, and depending on which box each atom is in, the system can exist in four obvious possibilities, shown in Fig. 1:

  • HL NL (i.e. both the hydrogen atom and the nitrogen atom are in the left box)
  • HL NR
  • HR NL
  • HR NR
Figure 1: The four intuitively obvious options for how to store the two atoms in the boxes correspond to four basic states of the quantum system.

Before quantum physics, we would have thought those were the only options; each atom must be in one box or the other. But in quantum physics there are many more non-obvious possibilities.

In particular, we could put the system in a superposition of the form HL NL + HR NR, shown in Fig. 2. In the jargon of physics, “the system is in a superposition of HL NL and HR NR“. Note the use of the word “and” here. But don’t read too much into it; jargon often involves linguistic shorthand, and can be arbitrary and imprecise. The question I’m focused on here is not “what do physicists say?”, but “what does it actually mean?”

Figure 2: A quantum system can be in a superposition, such as this one represented by two basic states related by a “+” symbol. (This is not the most general case, as discussed below.)

In particular, does it mean that “HL NL AND HR NR” are true? Or does it mean “HL NL OR HR NR” is true? Or does it mean something else?

The Problems with “AND”

First, let’s see why the “AND” option has a serious problem.

In ordinary language, if I say that “A AND B are true”, then I mean that one can check that A is true and also, separately, that B is true — i.e., both A and B are true. With this meaning in mind, it’s clear that experiments do not encourage us to view superposition as an AND. (There are theory interpretations of quantum physics that do encourage the use of “AND”, a point I’ll return to.)

Experiment Is Skeptical

Specifically, if a system is in a quantum superposition of two states A and B, no experiment will ever show that

  • A is true
    AND
  • B is true.

Instead, in any experiment explicitly designed to check whether A is true and whether B is true, the result will only reveal, at best, that

  • A is true and B is not true
    OR
  • B is true and A is not true.

The result might also be ambiguous, neither confirming nor denying that either one is true. But no measurement will ever show that both A AND B are definitively true. The two possibilities A and B are mutually exclusive in any actual measurement that is sensitive to the question.

In our case, if we go looking for our two atoms in the state HL NL + HR NR — if we do position measurements on both of them — we will either find both of them in the left box OR both of them in the right box. 1920’s quantum physics may be weird, but it does not allow measurements of an atom to find it in two places at the same time: an atom has a position, even if it is inherently uncertain, and if I make a serious attempt to locate it, I will find only one answer (within the precision of the measurement). [Measurement itself requires a long discussion, which I won’t attempt here; but see this post and the following one.]

And so, in this case, a measurement will find that one box has two atoms and the other has zero. Yet if we use “AND” in describing the superposition, we end up saying “both atoms are in the left box AND both atoms are in the right box”, which seems to imply that both atoms are in both boxes, contrary to any experiment. Again, certain theoretical approaches might argue that they are in both boxes, but we should obviously be very cautious when experiment disagrees with theoretical reasoning.

The Fortunate and/or Unfortunate Cat

The example of Schrodinger’s cat is another context in which some writers use “and” in describing what is going on.

A reminder of the cat experiment: We have an atom which may decay now or later, according to a quantum process whose timing we cannot predict. If the atom decays, it initiates a chain reaction which kills the cat. If the atom and the cat are placed inside a sealed box, isolating them completely from the rest of the universe, then the initial state, with an intact atom (Ai) and a Live cat (CL), will evolve to a state in a superposition roughly of the form Ai CL + Ad CD, where Ad refers to a decayed atom and CD refers to a Dead cat. (More precisely, the state will take the form c1 Ai CL + c2 Ad CD, where c1 and c2 are complex numbers with |c1|2 + |c2|2 = 1; but we can ignore these numbers for now.)

Figure 3: As in Figure 2, a superposition can even, in principle, be applied to macroscopic objects. This includes the famous Schrodinger cat state.

Leaving aside that the experiment is both unethical and impossible in practice, it raises an important point about the word “AND”. It includes a place where we must say “AND“; there’s no choice.

As we close the box to start the experiment, the atom is intact AND the cat is alive; both are simultaneously true, as measurement can verify. The state that we use to describe this, Ai CL, is a mathematical product: implicitly “Ai CL” means Ai x CL, where x is the “times” symbol.

Figure 4: It is unambiguous that the initial state of the cat-atom system is that the atom is intact AND the cat is alive: Ai x CL.

Later, the state to which the system evolves is a sum of two products — a superposition (Ai x CL) + (Ad x CD) which includes two “AND” relationships

1) “the atom is intact AND the cat is alive” (Ai x CL)
2) “the atom has decayed AND the cat is dead” (Ad x CD)

In each of these two possibilities, the state of the atom and the state of the cat are perfectly correlated; if you know one, you know the other. To use language consistent with English (and all other languages with which I am familiar), we must use “AND” to describe this correlation. (Note: in this particular example, correlation does in fact imply causation — but that’s not a requirement here. Correlation is enough.)

It is then often said that, theoretically, that “before we open the box, the cat is both alive AND dead”. But again, if we open the box to find out, experimentally, we will find out either that “the cat is alive OR the cat is dead.” So we should think this through carefully.

We’ve established that “x” must mean “AND“, as in Fig. 4. So let’s try to understand the “+” that appears in the superposition (Ai x CL) + (Ad x CD). It is certainly the case that such a state doesn’t tell us whether CL is true or CD is true, or even that it is meaningful to say that only one is true.

But suppose we decide that “+” means “AND“, also. Then we end up saying

  • “(the cat is alive AND the atom is intact) AND (the cat is dead AND the atom has decayed.)”

That’s very worrying. In ordinary English, if I’m referring to some possible facts A,B,C, and D, and I tell you that “(A AND B are true) AND (C AND D are true)”, the logic of the language implies that A AND B AND C AND D are all true. But that standard logic would leads to a falsehood. It is absolutely not the case, in the state (Ai x CL) + (Ad x CD), that CL is true and Ad is true — we will never find, in any experiment, that the cat is alive and yet the atom has decayed. That could only happen if the system were in a superposition that includes the possibility Ad x CL. Nor (unless we wait a few years and the cat dies of old age) can it be the case that CD is true and Ai is true.

And so, if “x” means “AND” and “+” means “AND“, it’s clear that these are two different meanings ofAND.”

“AND” and “AND”

Is that okay? Well, lots of words have multiple meanings. Still, we’re not used to the idea of “AND” being ambiguous in English. Nor are “x” and “+” usually described with the same word. So using “AND” is definitely problematic.

(That said, people who like to think in terms of parallel “universes” or “branches” in which all possibilities happen [the many-worlds interpretation] may actually prefer to have two meanings of “AND”, one for things that happen in two different branches, and one for things that happen in the same branch. But this has some additional problems too, as we’ll see later when we get to the subtleties of “OR”.)

These issues are why, in my personal view, “OR” is better when one first learns quantum physics. I think it makes it easier to explain how quantum physics is both related to standard probabilities and yet goes beyond it. For one thing, “or” is already ambiguous in English, so we’re used to the idea that it might have multiple meanings. For another, we definitely need “+” to be conceptually different from “x“, so it is confusing, pedagogically, to start right off by saying that both mathematical operators are “AND”.

But “OR” is not without its problems.

The Problems with “OR”

In normal English, saying “the atom is intact and the cat is alive” OR “the atom has decayed and the cat is dead” would tell us two possible facts about the current contents of the box, one of which is definitely true.

But in quantum physics, the use of “OR” in the Schrodinger cat superposition does not tell us what is currently happening inside the box. It does tell us the state of the system at the moment, but all that does is predict the possible outcomes that would be observed if the box were opened right now (and their probabilities.) That’s less information than telling us the properties of what is in the closed box.

The advantage of “OR” is that it does tell us the two outcomes of opening the box, upon which we will find

  • “The atom is intact AND the cat is alive”
    OR
  • “The atom has decayed AND the cat is dead”

Similarly, for our box of atoms, it tells us that if we attempt to locate the atoms, we will find that

  • “the hydrogen atom is in the left box AND the nitrogen atom is in the left box”
    OR
  • “the hydrogen atom is in the right box AND the nitrogen atom is in the right box”

In other words, this use of AND and OR agrees with what experiments actually find. Better this than the alternative, it seems to me.

Nevertheless, just because it is better doesn’t mean it is unproblematic.

The Usual Or

The word “OR” is already ambiguous in usual English, in that it could mean

  • either A is true or B is true
  • A is true or B is true or both are true

Which of these two meanings is intended in an English sentence has to be determined by context, or explained by the speaker. Here I’m focused on the first meaning.

Returning to our first example of Figs. 1 and 2, suppose I hand the two atoms to you and ask you to put them in either box, whichever one you choose. You do so, but you don’t tell me what your choice was, and you head off on a long vacation.

While I wait for you to return, what can I say about the two atoms? Assuming you followed my instructions, I would say that

  • “both atoms are in the left box OR both atoms are in the right box”

In doing so, I’m using “or” in its “either…or…” sense in ordinary English. I don’t know which box you chose, but I still know (Fig. 5) that the system is either definitely in the HL NL state OR definitely in the HR NR state of Fig. 1. I know this without doing any measurement, and I’m only uncertain about which is which because I’m missing information that you could have provided me. The information is knowable; I just don’t have it.

Figure 5: The atoms were definitely put into in one box or the other, but nobody told me which box was selected.

But this uncertainty about which box the atoms are in is completely different from the uncertainty that arises from putting the atoms in the superposition state HL NL + HR NR!

The Superposition OR

If the system is in the state HL NL + HR NR, i.e. what I’ve been calling (“HL NL OR HR NR“), it is in a state of inherent uncertainty of whether the two atoms are in the left box or in the right box. It is not that I happen not to know which box the atoms are in, but rather that this information is not knowable within the rules of quantum physics. Even if you yourself put the atoms into this superposition, you don’t know which box they’re in any more than I do.

The only thing we can try to do is perform an experiment and see what the answer is. The problem is that we cannot necessarily infer, if we find both atoms in the left box, that the two atoms were in that box prior to that measurement.

If we do try to make that assumption, we find ourselves in apparent contradiction with experiment. The issue is quantum interference. If we repeat the whole process, but instead of opening the boxes to see where the atoms are, we first bring the two boxes together and measure the atoms’ properties, we will observe quantum interference effects. As I have discussed in my recent series of five posts on interference (starting here), quantum interference can only occur when a system takes at least two paths to its current state; but if the two atoms were definitely in one box or definitely in the other, then there would be only one path in Fig. 6.

Figure 6: In the superposition state, the atoms cannot simply be in definite but unknown locations, as in Fig. 5. If the boxes are joined and then opened, quantum interference will occur, implying the system has evolved via two paths to a single state.

Prior to the measurement, the system had inherent uncertainty about the question, and while measurement removes the current uncertainty, it does not in general remove the past uncertainty. The act of measurement changes the state of the system — more precisely, it changes the state of the larger system that includes both atoms and the measurement device — and so establishing meaningfully that the two atoms are now in the left box is not sufficient to tell us meaningfully that the two atoms were previously and definitively in the left box.

So if this is “OR“, it is certainly not what it usually means in English!

This Superposition or That One?

And it gets worse, because we can take more complex examples. As I mentioned when discussing the poor cat, the superposition HL NL + HR NR is actually one in a large class of superpositions, of the form c1 HL NL + c2 HR NR , where c1 and c2 are complex numbers. A second simple example of such a superposition is HL NL HR NR, with a minus sign instead of a plus sign.

So suppose I had asked you to put the two atoms in a superposition either of the form HL NL + HR NR or HL NL HR NR, your choice; and suppose you did so without telling me which superposition you chose. What would I then know?

I would know that the system is either in the state (HL NL + HR NR) or in the state (HL NL – HR NR), depending on what you chose to do. In words, what I would know is that the system is represented by

  • (HL NL OR HR NR) OR (HL NL OR HR NR)

Uh oh. Now we’re as badly off as we were with “AND“.

First, the “OR” in the center is a standard English “OR” — it means that the system is definitely in one superposition or the other, but I don’t know which one — which isn’t the same thing as the “OR“s in the parentheses, which are “OR“s of superposition that only tell us what the results of measurements might be.

Second, the two “OR“s in the parentheses are different, since one means “+” and the other means ““. In some other superposition state, the OR might mean 3/5 + i 4/5, where i is the standard imaginary number equal to the square root of -1. In English, there’s obviously no room for all this complexity. [Note that I’d have the same problem if I used “AND” for superpositions instead.]

So even if “OR” is better, it’s still not up to the task. Superposition forces us to choose whether to have multiple meanings of “AND” or multiple meanings of “OR”, including meanings that don’t hold in ordinary language. In a sense, the “+” (or “-” or whatever) in a superposition is a bit more “AND” than standard English “OR”, but it’s also a bit more “OR” than a standard English “AND”. It’s something truly new and unfamiliar.

Experts in the foundational meaning of quantum physics argue over whether to use “OR” or “AND”. It’s not an argument I want to get into. My goal here is to help you understand how quantum physics works with the minimum of interpretation and the minimum of mathematics. This requires precise language, of course. But here we find we cannot avoid a small amount of math — that of simple numbers, sometimes even complex numbers — because ordinary language simply can’t capture the logic of what quantum physics can do.

I will continue, for consistency, to use “OR” for a superposition, but going forward we must admit and recognize its limitations, and become more sophisticated about what it does and doesn’t mean. One should understand my use of “OR“, and the “pre-quantum viewpoint” that I often employ, as pedagogical methodology, not a statement about nature. Specifically, I have been trying to clarify the crucial idea of the space of possibilities, and to show examples of how quantum physics goes beyond pre-quantum physics. I find the “pre-quantum viewpoint”, where it is absolutely required that we use “OR”, helps students get the basics straight. But it is true that the pre-quantum viewpoint obscures some of the full richness and complexity of quantum phenomena, much of which arises precisely because the quantum “OR” is not the standard “OR” [and similarly if you prefer “AND” instead.] So we have to start leaving it behind.

There are many more layers of subtlety yet to be uncovered [for instance, what if my system is in a state (A OR B), but I make a measurement that can’t directly tell me whether A is true or B is true?] but this is enough for today.

I’m grateful to Jacob Barandes for a discussion about some of these issues.

Conceptual Summary

  • When we use “A AND B” in ordinary language, we mean “A is true and B is true”.
  • When we use “A OR B” in ordinary language, we find “OR” is ambiguous even in English; it may mean
    • “either A is true or B is true”, or
    • “A is true or B is true or both are true.”
  • In my recent posts, when I say a superposition c1 A + c2 B can be expressed as “A OR B”, I mean something that I cannot mean in English, because such a meaning would never normally occur to us:
    • I mean that the result of an appropriate measurement carried out at this moment will give the result A or the result B (but not both).
    • I do so without generally implying that the state of the system, if I don’t carry out the measurement, is definitely A or definitely B (though unknown).
    • Instead the system could be viewed as being in an uncanny state of being that we’re not used to, for which neither ordinary “AND” nor ordinary “OR” applies.
  • Note also that using either “AND” or “OR” is unable to capture the difference between superpositions that involve the same states but differ in the numbers c1, c2.

The third bullet point is open to different choices about “AND” and “OR“, and open to different interpretation about what superposition states imply about the systems that are in them. There are different consistent ways to combine the language and concepts, and the particular choice I’ve made is pragmatic, not dogmatic. For a single set of blog posts that tell a coherent story, I have to to pick a single consistent language; but it’s a choice. Once one’s understanding of quantum physics is strong, it’s both valuable and straightforward to consider other possible choices.

April 08, 2025

n-Category Café Quantum Ellipsoids

With the stock market crash and the big protests across the US, I’m finally feeling a trace of optimism that Trump’s stranglehold on the nation will weaken. Just a trace.

I still need to self-medicate to keep from sinking into depression — where ‘self-medicate’, in my case, means studying fun math and physics I don’t need to know. I’ve been learning about the interactions between number theory and group theory. But I haven’t been doing enough physics! I’m better at that, and it’s more visceral: more of a bodily experience, imagining things wiggling around.

So, I’ve been belatedly trying to lessen my terrible ignorance of nuclear physics. Nuclear physics is a fascinating application of quantum theory, but it’s less practical than chemistry and less sexy than particle physics, so I somehow skipped over it.

I’m finding it worth looking at! Right away it’s getting me to think about quantum ellipsoids.

Nuclear physics forces you to imagine blobs of protons and neutrons wiggling around in a very quantum-mechanical way. Nuclei are too complicated to fully understand. We can simulate them on a computer, but simulation is not understanding, and it’s also very hard: one book I’m reading points out that one computation you might want to do requires diagonalizing a 10 14×10 1410^{14} \times 10^{14} matrix. So I’d rather learn about the many simplified models of nuclei people have created, which offer partial understanding… and lots of beautiful math.

Protons minimize energy by forming pairs with opposite spin. Same for neutrons. Each pair acts like a particle in its own right. So nuclei act very differently depending on whether they have an even or odd number of protons, and an even or odd number of neutrons!

The ‘Interacting Boson Model’ is a simple approximate model of ‘even-even’ atomic nuclei: nuclei with an even number of protons and an even number of neutrons. It treats the nucleus as consisting of bosons, each boson being either a pair of nucleons — that is, either protons or neutrons — where the members of a pair have opposite spin but are the same in every other way. So, these bosons are a bit like the paired electrons responsible for superconductivity, called ‘Cooper pairs’.

However, in the Interacting Boson Model we assume our bosons all have either spin 0 (s-bosons) or spin 2 (d-bosons), and we ignore all properties of the bosons except their spin angular momentum. A spin-0 particle has 1 spin state, since the spin-0 representation of SO(3)\text{SO}(3) is 1-dimensional. A spin-2 particle has 5, since the spin-2 representation is 5-dimensional.

If we assume the maximum amount of symmetry among all 6 states, both s-boson and d-boson states, we get a theory with U(6)\text{U}(6) symmetry! And part of why I got interested in this stuff was that it would be fun to see a rather large group like U(6)\text{U}(6) showing up as symmetries — or approximate symmetries — in real world physics.

More sophisticated models recognize that not all these states behave the same, so they assume a smaller group of symmetries.

But there are some simpler questions to start with.

How do we make a spin-0 or spin-2 particle out of two nucleons? That’s easy. Two nucleons with opposite spin have total spin 0. But if they’re orbiting each other, they have orbital angular momentum too, so the pair can act like a particle with spin 0, 1, 2, 3, etc.

Why are these bosons in the Interacting Boson Model assumed to have spin 0 or spin 2, but not spin 1 or any other spin? This is a lot harder. I assume that at some level the answer is “because this model works fairly well”. But why does it work fairly well?

By now I’ve found two answers for this, and I’ll tell you the more exciting answer, which I found in this book:

  • Igal Talmi, Simple Models of Complex Nuclei: the Shell Model and Interacting Boson Model, Harwood Academic Publishers, Chur, Switzerland, 1993.

In the ‘liquid drop model’ of nuclei, you think of a nucleus as a little droplet of fluid. You can think of an even-even nucleus as a roughly ellipsoidal droplet, which however can vibrate. But we need to treat it using quantum mechanics. So we need to understand quantum ellipsoids!

The space of ellipsoids in 3\mathbb{R}^3 centered at the origin is 6-dimensional, because these ellipsoids are described by equations like

Ax 2+By 2+Cz 2+Dxy+Eyz+Fzx=1 A x^2 + B y^2 + C z^2 + D x y + E y z + F z x = 1

and there are 6 coefficients here. Not all nuclei are close to spherical! But perhaps it’s easiest to start by thinking about ellipsoids that are close to spherical, so that

(1+a)x 2+(1+b)y 2+(1+c)z 2+dxy+eyz+fzx=1 (1 + a)x^2 + (1 + b)y^2 + (1 + c)z^2 + d x y + e y z + f z x = 1

where a,b,c,d,e,fa,b,c,d,e,f are small. If our nucleus were classical, we’d want equations that describe how these numbers change with time as our little droplet oscillates.

But the nucleus is deeply quantum mechanical. So in the Interacting Boson Model, invented by Iachello, it seems we replace a,b,c,d,e,fa,b,c,d,e,f with operators on a Hilbert space, say q 1,,q 6q_1, \dots, q_6, and introduce corresponding momentum operators p 1,,p 6p_1, \dots, p_6, obeying the usual ‘canonical commutation relations’:

[q j,q k]=[p j,p k]=0,[p j,q k]=iδ jk [q_j, q_k] = [p_j, p_k] = 0, \qquad [p_j, q_k] = - i \hbar \delta_{j k}

As usual, we can take this Hilbert space to either be L 2( 6)L^2(\mathbb{R}^6) or ‘Fock space’ of 6\mathbb{C}^6: the Hilbert space completion of the symmetric algebra of 6\mathbb{C}^6. These are two descriptions of the same thing. The Fock space of 6\mathbb{C}^6 gets an obvious representation of the unitary group U(6)\text{U}(6), since that group acts on 6\mathbb{C}^6. And L 2( 6)L^2(\mathbb{R}^6) gets an obvious representation of SO(3)\text{SO}(3), since rotations act on ellipsoids and thus on the tuples (a,b,c,d,e,f) 6(a,b,c,d,e,f) \in \mathbb{R}^6 that we’re using to describe ellipsoids.

The latter description lets us see where the s-bosons and d-bosons are coming from! Our representation of SO(3)\text{SO}(3) on 6\mathbb{R}^6 splits into two summands:

  • the (real) spin-0 representation, which is 1-dimensional because it takes just one number to describe the rotation-invariant aspects of the shape of an ellipsoid centered at the origin: for example, its volume. In physics jargon this number tells us the monopole moment of the mass distribution of our nucleus.

  • the (real) spin-2 representation, which is 5-dimensional because it takes 5 numbers to describe all other aspects of the shape of an ellipsoid centered at the origin. You need 2 numbers to say in which direction its longest axis points, one number to say how long that axis is, 1 number to say which direction the second-longest axis point in (it’s at right angles to the longest axis), and 1 number to say how long it is. In physics jargon these 5 numbers tell us the quadrupole moment of our nucleus.

This shows us why we don’t get spin-1 bosons! We’d get them if the mass distribution of our nucleus could have a nonzero dipole moment. In other words, we’d get them if we added linear terms Gx+Hy+KzG x + H y + K z to our equation

Ax 2+By 2+Cz 2+Dxy+Eyz+Fzx=1 A x^2 + B y^2 + C z^2 + D x y + E y z + F z x = 1

But by conservation of momentum, we can assume the center of mass of our nucleus stays at the origin, and set these linear terms to zero.

As usual, we can take linear combinations of the operators q jq_j and p jp_j to get annihilation and creation operators for s-bosons and d-bosons. If we want, we can think of these bosons as nucleon pairs. But we don’t need that microscopic interpretation if we don’t want it: we can just say we’re studying the quantum behavior of an oscillating ellipsoid!

After we have our Hilbert space and these operators on it, we can write down a Hamiltonian for our nucleus, or various possible candidate Hamiltonians, in terms of these operators. Talmi’s book goes into a lot of detail on that. And then we can compare the oscillations these Hamiltonians predict to what we see in the lab. (Often we just see the frequencies of the standing waves, which are proportional to the eigenvalues of the Hamiltonian.)

So, from a high-level mathematical viewpoint, what we’ve done is try to define a manifold MM of ellipsoid shapes, and then form its cotangent bundle T *MT^\ast M, and then quantize that and start studying ‘quantum ellipsoids’.

Pretty cool! And there’s a lot more to say about it. But I’m wondering if there might be a better manifold of ellipsoid shapes than just 6\mathbb{R}^6. After all, when 1+a,1+b1+a, 1+b or 1+c1+c become negative things go haywire: our ellipsoid can turn into a hyperboloid! The approach I’ve described is probably fine ‘perturbatively’, i.e. when a,b,c,d,e,fa,b,c,d,e,f are small. But it may not be the best when our ellipsoid oscillates so much it gets far from spherical.

I think we need a real algebraic geometer here. In both senses of the word ‘real’.

John BaezQuantum Ellipsoids

With the stock market crash and the big protests across the US, I’m finally feeling a trace of optimism that Trump’s stranglehold on the nation will weaken. Just a trace.

I still need to self-medicate to keep from sinking into depression — where ‘self-medicate’, in my case, means studying fun math and physics I don’t need to know. I’ve been learning about the interactions between number theory and group theory. But I haven’t been doing enough physics! I’m better at that, and it’s more visceral: more of a bodily experience, imagining things wiggling around.

So, I’ve been belatedly trying to lessen my terrible ignorance of nuclear physics. Nuclear physics is a fascinating application of quantum theory, but it’s less practical than chemistry and less sexy than particle physics, so I somehow skipped over it.

I’m finding it worth looking at! Right away it’s getting me to think about quantum ellipsoids.

Nuclear physics forces you to imagine blobs of protons and neutrons wiggling around in a very quantum-mechanical way. Nuclei are too complicated to fully understand. We can simulate them on a computer, but simulation is not understanding, and it’s also very hard: one book I’m reading points out that one computation you might want to do requires diagonalizing a 10^{14} \times 10^{14} matrix. So I’d rather learn about the many simplified models of nuclei people have created, which offer partial understanding… and lots of beautiful math.

Protons minimize energy by forming pairs with opposite spin. Same for neutrons. Each pair acts like a particle in its own right. So nuclei act very differently depending on whether they have an even or odd number of protons, and an even or odd number of neutrons!

The ‘Interacting Boson Model‘ is a simple approximate model of ‘even-even’ atomic nuclei: nuclei with an even number of protons and an even number of neutrons. It treats the nucleus as consisting of bosons, each boson being either a pair of nucleons—that is, either protons or neutrons—where the members of a pair have opposite spin but are the same in every other way. So, these bosons are a bit like the paired electrons responsible for superconductivity, called ‘Cooper pairs’.

However, in the Interacting Boson Model we assume our bosons all have either spin 0 (s-bosons) or spin 2 (d-bosons), and we ignore all properties of the bosons except their spin angular momentum. A spin-0 particle has 1 spin state, since the spin-0 representation of \text{SO}(3) is 1-dimensional. A spin-2 particle has 5, since the spin-2 representation is 5-dimensional.

If we assume the maximum amount of symmetry among all 6 states, both s-boson and d-boson states, we get a theory with \text{U}(6) symmetry! And part of why I got interested in this stuff was that it would be fun to see a rather large group like \text{U}(6) showing up as symmetries—or approximate symmetries—in real world physics.

More sophisticated models recognize that not all these states behave the same, so they assume a smaller group of symmetries.

But there are some simpler questions to start with.

How do we make a spin-0 or spin-2 particle out of two nucleons? That’s easy. Two nucleons with opposite spin have total spin 0. But if they’re orbiting each other, they have orbital angular momentum too, so the pair can act like a particle with spin 0, 1, 2, 3, etc.

Why are these bosons in the Interacting Boson Model assumed to have spin 0 or spin 2, but not spin 1 or any other spin? This is a lot harder. I assume that at some level the answer is “because this model works fairly well”. But why does it work fairly well?

By now I’ve found two answers for this, and I’ll tell you the more exciting answer, which I found in this book:

• Igal Talmi, Simple Models of Complex Nuclei: the Shell Model and Interacting Boson Model, Harwood Academic Publishers, Chur, Switzerland, 1993.

In the ‘liquid drop model’ of nuclei, you think of a nucleus as a little droplet of fluid. You can think of an even-even nucleus as a roughly ellipsoidal droplet, which however can vibrate. But we need to treat it using quantum mechanics. So we need to understand quantum ellipsoids!

The space of ellipsoids in \mathbb{R}^3 centered at the origin is 6-dimensional, because these ellipsoids are described by equations like

A x^2 + B y^2 + C z^2 + D x y + E y z + F z x = 1

and there are 6 coefficients here. Not all nuclei are close to spherical! But perhaps it’s easiest to start by thinking about ellipsoids that are close to spherical, so that

(1 + a)x^2 + (1 + b)y^2 + (1 + c)z^2 + d x y + e y z + f z x = 1

where a,b,c,d,e,f are small. If our nucleus were classical, we’d want equations that describe how these numbers change with time as our little droplet oscillates.

But the nucleus is deeply quantum mechanical. So in the Interacting Boson Model, invented by Iachello, it seems we replace a,b,c,d,e,f with operators on a Hilbert space, say q_1, \dots, q_6, and introduce corresponding momentum operators p_1, \dots, p_6, obeying the usual ‘canonical commutation relations’:

[q_j, q_k] = [p_j, p_k] = 0, \quad [p_j, q_k] = - i \hbar \delta_{j k}

As usual, we can take this Hilbert space to either be L^2(\mathbb{R}^6) or the ‘Fock space’ on \mathbb{C}^6: the Hilbert space completion of the symmetric algebra of \mathbb{C}^6. These are two descriptions of the same thing. The Fock space on \mathbb{C}^6 gets an obvious representation of the unitary group \text{U}(6), since that group acts on \mathbb{C}^6. And L^2(\mathbb{R}^6) gets an obvious representation of \text{SO}(3), since rotations act on ellipsoids and thus on the tuples (a,b,c,d,e,f) \in \mathbb{R}^6 that we’re using to describe ellipsoids.

The latter description lets us see where the s-bosons and d-bosons are coming from! Our representation of \text{SO}(3) on \mathbb{R}^6 splits into two summands:

• the (real) spin-0 representation, which is 1-dimensional because it takes just one number to describe the rotation-invariant aspects of the shape of an ellipsoid centered at the origin: for example, its volume. In physics jargon this number tells us the monopole moment of our nucleus.

• the (real) spin-2 representation, which is 5-dimensional because it takes 5 numbers to describe all other aspects of the shape of an ellipsoid centered at the origin. You need 2 numbers to say in which direction its longest axis points, one number to say how long that axis is, 1 number to say which direction the second-longest axis point in (it’s at right angles to the longest axis), and 1 number to say how long it is. In physics jargon these 5 numbers tell us the quadrupole moment of our nucleus.

This shows us why we don’t get spin-1 bosons! We’d get them if the mass distribution of our nucleus could have a nonzero dipole moment. In other words, we’d get them if we added linear terms G x + H y + K z to our equation

A x^2 + B y^2 + C z^2 + D x y + E y z + F z x = 1

But by conservation of momentum, we can assume the center of mass of our nucleus stays at the origin, and set these linear terms to zero.

As usual, we can take linear combinations of the operators q_j and p_j to get annihilation and creation operators for s-bosons and d-bosons. If we want, we can think of these bosons as nucleon pairs. But we don’t need that microscopic interpretation if we don’t want it: we can just say we’re studying the quantum behavior of an oscillating ellipsoid!

After we have our Hilbert space and some operators on it, we can write down a Hamiltonian for our nucleous, or various possible candidate Hamiltonians, in terms of these operators. Talmi’s book goes into a lot of detail on that. And then we can compare the oscillations these Hamiltonians predict to what we see in the lab. (Often we just see the frequencies of the standing waves, which are proportional to the eigenvalues of the Hamiltonian.)

So, from a high-level mathematical viewpoint, what we’ve done is try to define a manifold M of ellipsoid shapes, and then form its cotangent bundle T^\ast M, and then quantize that and start studying ‘quantum ellipsoids’.

Pretty cool! And there’s a lot more to say about it. But I’m wondering if there might be a better manifold of ellipsoid shapes than just \mathbb{R}^6. After all, when 1+a, 1+b or 1+c become negative things go haywire: our ellipsoid can turn into a hyperboloid! The approach I’ve described is probably fine ‘perturbatively’, i.e. when a,b,c,d,e,f are small. But it may not be the best when our ellipsoid oscillates so much it gets far from spherical.

I think we need a real algebraic geometer here. In both senses of the word ‘real’.

April 03, 2025

Matt Strassler Double Slit: Why Measurement Destroys the Interference Pattern

The quantum double-slit experiment, in which objects are sent toward a wall with two slits and then recorded on a screen behind the wall, creates an interference pattern that builds up gradually, object by object. And yet, it’s crucial that the path of each object on its way to the screen remain unknown. If one measures which of the slits each object passes through, the interference pattern never appears.

Strange things are said about this. There are vague, weird slogans: “measurement causes the wave function to collapse“; “the particle interferes with itself“; “electrons are both particles and waves“; etc. One reads that the objects are particles when they reach the screen, but they are waves when they go through the slits, causing the interference — unless their passage through the slits is measured, in which case they remain particles.

But in fact the equations of 1920s quantum physics say something different and not vague in the slightest — though perhaps equally weird. As we’ll see today, the elimination of interference by measurement is no mystery at all, once you understand both measurement and interference. Those of you who’ve followed my recent posts on these two topics will find this surprisingly straightforward; I guarantee you’ll say, “Oh, is that all?” Other readers will probably want to read

The Interference Criterion

When do we expect quantum interference? As I’ll review in a moment, there’s a simple criterion:

  • a system of objects (not the objects themselves!) will exhibit quantum interference if the system, initially in a superposition of possibilities, reaches a single possibility via two or more pathways.

To remind you what that means, let’s compare two contrasting cases (covered carefully in this post.) Figs. 1a and 1b show pre-quantum animations of different quantum systems, in which two balls (drawn blue and orange) are in a superposition of moving left OR moving right. I’ve chosen to stop each animation right at the moment when the blue ball in the top half of the superposition is at the same location as the blue ball in the bottom half, because if the orange ball weren’t there, this is when we’d expect it to see quantum interference.

But for interference to occur, the orange ball, too, must at that same moment be in the same place in both parts of the superposition. That does happen for the system in Fig. 1a — the top and bottom parts of the figure line up exactly, and so interference will occur. But the system in Fig. 1b, whose top and bottom parts never look the same, will not show quantum interference.

Fig. 1a: A system of two balls in a superposition, from a pre-quantum viewpoint. As the system evolves, a moment is reached when the two parts of the superposition are identical. As the system has then reached a single possibility via two routes, quantum interference may result.
Figure 1b: Similar to Fig. 1a, except that when the blue ball is at the same location in both parts of the superposition, the orange ball is at two different locations. At no moment are the two possibilities in the superposition the same, so quantum interference cannot occur.

In other words, quantum interference requires that the two possibilities in the superposition become identical at some moment in time. Partial resemblance is not enough.

The Measurement

A measurement always involves an interaction of some sort between the object we want to measure and the device doing the measurement. We will typically

For today’s purposes, the details of the second step won’t matter, so I’ll focus on the first step.

Setting Up

We’ll call the object going through the slits a “particle”, and we’ll call the measurement device a “measuring ball” (or just “ball” for short.) The setup is depicted in Fig. 2, where the particle is approaching the slits and the measuring ball lies in wait.

Figure 2: A particle (blue) approaches a wall with two slits, behind which sits a screen where the particle’s arrival will be detected. Also present is a lightweight measuring ball (black), ready to fly in and measure the particle’s position by colliding with it as it passes through the wall.

If No Measurement is Made at the Slits

Suppose we allow the particle to proceed and we make no measurement of its location as it passes through the slits. Then we can leave the ball where it is, at the position I’ve marked M in Fig. 3. If the particle makes it through the wall, it must pass through one slit or the other, leaving the system in a superposition of the form

  • the particle is near the left slit [and the ball is at position M]
    OR
  • the particle is near the right slit [and the ball is at position M]

as shown at the top of Fig. 3. (Note: because the ball and particle are independent [unentangled] in this superposition, it can be written in factored form as in Fig. 12 of this post.)

From here, the particle (whose motion is now quite uncertain as a result of passing through a narrow slit) can proceed unencumbered to the screen. Let’s say it arrives at the point marked P, as at the bottom of Fig. 3.

Figure 3: (Top) As the particle passes through the slits, the system is set into a superposition of two possibilities in which the particle passes through the left slit OR the right slit. (The particle’s future motion is quite uncertain, as indicated by the green arrows.) In both possibilities, the measuring ball is at point M. (Bottom) If the particle arrives at point P on the screen, then the two possibilties in the superposition become identical, as in Fig. 1a, so quantum interference can result. This will be true no matter what point P we choose, and so an interference pattern will be seen across the whole screen.

Crucially, both halves of the superposition now describe the same situation: particle at P, ball at M. The system has arrived here via two paths:

  • The particle went through the left slit and arrived at the point P (with the ball always at M),
    OR
  • The particle went through the right slit and arrived at the point P (with the ball always at M).

Therefore, since the system has reached a single possibility via two different routes, quantum interference may be observed.

Specifically, the system’s wave function, which gives the probability for the particle to arrive at any point on the screen, will display an interference pattern. We saw numerous similar examples in this post, this post and this post.

If the Measurement is Made at the Slits

But now let’s make the measurement. We’ll do it by throwing the ball rapidly toward the particle, timed carefully so that, as shown in Fig. 4, either

  • the particle is at the left slit, in which case the ball passes behind it and travels onward,
    OR
  • the particle is at the right slit, in which case the ball hits it and bounces back.

(Recall that I assumed the measuring ball is lightweight, so the collision doesn’t much affect the particle; for instance, the particle might be an heavy atom, while the measuring ball is a light atom.)

Figure 4: As the particle moves through the wall, the ball is sent rapidly in motion. If the particle passes through the right slit, the ball will hit it and bounce back; if the particle passes through the left slit, the ball will miss it and will continue to the left.

The ball’s late-time behavior reveals — and thus measures — the particle’s behavior as it passed through the wall:

  • the ball moving to the left means the particle went through the left slit;
  • the ball moving to the right means the particle went through the right slit.

[Said another way, the ball and particle, which were originally independent before the measurement, have been entangled by the measurement process. Because of the entanglement, knowledge concerning the ball tells us something about the particle too.]

To make this measurement complete and permanent requires a longer story with more details; for instance, we might choose to amplify the result with a Geiger counter. But the details don’t matter, and besides, that takes place later. Let’s keep our focus on what happens next.

The Effect of the Measurement

What happens next is that the particle reaches the point P on the screen. It can do this whether it traveled via the left slit or via the right slit, just as before, and so you might think there should still be an interference pattern. However, remembering Figs. 1a and 1b and the criterion for interference, take a look at Fig. 5.

Figure 5: Following the measurement made in Fig. 4, the arrival of the particle at the point P on the screen finds the ball in two possible locations, depending on which slit the particle went through. In contrast to Fig. 3, the two parts of the superposition are not identical, and so (as in Fig. 1b) no quantum interference pattern will be observed.

Even though the particle by itself could have taken two paths to the point P, the system as a whole is still in a superposition of two different possibilities, not one — more like Fig. 1b than like Fig. 1a. Specifically,

  • the particle is at position P and the ball is at location ML (which happens if, in Fig. 4, the particle was near the left slit and the ball continued to the left);
    OR
  • the particle is at position P and the ball is at location MR (which happens if, in Fig. 4, the particle was near the right slit and the ball bounced back to the right).

The measurement process — by the very definition of “measurement” as a procedure that segregates left-slit cases from right-slit cases — has resulted in the two parts of the superposition being different even when they both have the particle reaching the same point P. Therefore, in contrast to Fig. 3, quantum interference between the two parts of the superposition cannot occur.

And that’s it. That’s all there is to it.

Looking Ahead.

The double-slit experiment is hard to understand if one relies on vague slogans. But if one relies on the math, one sees that many of the seemingly mysterious features of the experiment are in fact straightforward.

I’ll say more about this in future posts. In particular, to convince you today’s argument is really correct, I’ll look more closely at the quantum wave function corresponding to Figs. 3-5, and will reproduce the same phenomenon in simpler examples. Then we’ll apply the resulting insights to other cases, including

  • measurements that do not destroy interference,
  • measurements that only partly destroy interference,
  • destruction of interference without measurement, and
  • double-slit experiments whose interference can’t be located in physical space,
  • etc.

April 01, 2025

Matt Strassler Quantum Interference 5: Coming Unglued

Now finally, we come to the heart of the matter of quantum interference, as seen from the perspective of in 1920’s quantum physics. (We’ll deal with quantum field theory later this year.)

Last time I looked at some cases of two particle states in which the particles’ behavior is independent — uncorrelated. In the jargon, the particles are said to be “unentangled”. In this situation, and only in this situation, the wave function of the two particles can be written as a product of two wave functions, one per particle. As a result, any quantum interference can be ascribed to one particle or the other, and is visible in measurements of either one particle or the other. (More precisely, it is observable in repeated experiments, in which we do the same measurement over and over.)

In this situation, because each particle’s position can be studied independent of the other’s, we can be led to think any interference associated with particle 1 happens near where particle 1 is located, and similarly for interference involving the second particle.

But this line of reasoning only works when the two particles are uncorrelated. Once this isn’t true — once the particles are entangled — it can easily break down. We saw indications of this in an example that appeared at the ends of my last two posts (here and here), which I’m about to review. The question for today is: what happens to interference in such a case?

Correlation: When “Where” Breaks Down

Let me now review the example of my recent posts. The pre-quantum system looks like this

Figure 1: An example of a superposition, in a pre-quantum view, where the two particles are correlated and where interference will occur that involves both particles together.

Notice the particles are correlated; either both particles are moving to the left OR both particles are moving to the right. (The two particles are said to be “entangled”, because the behavior of one depends upon the behavior of the other.) As a result, the wave function cannot be factored (in contrast to most examples in my last post) and we cannot understand the behavior of particle 1 without simultaneously considering the behavior of particle 2. Compare this to Fig. 2, an example from my last post in which the particles are independent; the behavior of particle 2 is the same in both parts of the superposition, independent of what particle 1 is doing.

Figure 2: Unlike Fig. 1, here the two particles are uncorrelated; the behavior of particle 2 is the same whether particle 1 is moving left OR right. As a result, interference can occur for particle 1 separately from any behavior of particle 2, as shown in this post.

Let’s return now to Fig. 1. The wave function for the corresponding quantum system, shown as a graph of its absolute value squared on the space of possibilities, behaves as in Fig. 3.

Figure 3: The absolute-value-squared of the wave function for the system in Fig, 1, showing interference as the peaks cross. Note the interference fringes are diagonal relative to the x1 and x2 axes.

But as shown last time in Fig. 19, at the moment where the interference in Fig. 3 is at its largest, if we measure particle 1 we see no interference effect. More precisely, if we do the experiment many times and measure particle 1 each time, as depicted in Fig. 4, we see no interference pattern.

Figure 4: The result of repeated experiments in which we measure particle 1, at the moment of maximal interference, in the system of Fig. 3. Each new experiment is shown as an orange dot; results of past experiments are shown in blue. No interference effect is seen.

We see something analogous if we measure particle 2.

Yet the interference is plain as day in Fig. 3. It’s obvious when we look at the full two-dimensional space of possibilities, even though it is invisible in Fig. 4 for particle 1 and in the analogous experiment for particle 2. So what measurements, if any, can we make that can reveal it?

The clue comes from the fact that the interference fringes lie at a 45 degree angle, perpendicular neither to the x1 axis nor to the x2 axis but instead to the axis for the variable 1/2(x1 + x2), the average of the positions of particle 1 and 2. It’s that average position that we need to measure if we are to observe the interference.

But doing so requires we that we measure both particles’ positions. We have to measure them both every time we repeat the experiment. Only then can we start making a plot of the average of their positions.

When we do this, we will find what is shown in Fig 5.

  • The top row shows measurements of particle 1.
  • The bottom row shows measurements of particle 2.
  • And the middle row shows a quantity that we infer from these measurements: their average.

For each measurement, I’ve drawn a straight orange line between the measurement of x1 and the measurement of x2; the center of this line lies at the average position 1/2(x1+x2). The actual averages are then recorded in a different color, to remind you that we don’t measure them directly; we infer them from the actual measurements of the two particles’ positions.

Figure 5: As in Fig. 4, the result of repeated experiments in which we measure both particles’ positions at the moment of maximal interference in Fig. 3. Top and bottom rows show the position measurements of particles 1 and 2; the middle row shows their average. Each new experiment is shown as two orange dots, they are connected by an orange line, at whose midpoint a new yellow dot is placed. Results of past experiments are shown in blue. No interference effect is seen in the individual particle positions, yet one appears in their average.

In short, the interference is not associated with either particle separately — none is seen in either the top or bottom rows. Instead, it is found within the correlation between the two particles’ positions. This is something that neither particle can tell us on its own.

And where is the interference? It certainly lies near 1/2(x1+x2)=0. But this should worry you. Is that really a point in physical space?

You could imagine a more extreme example of this experiment in which Fig. 5 shows particle 1 located in Boston and particle 2 located in New York City. This would put their average position within appropriately-named Middletown, Connecticut. (I kid you not; check for yourself.) Would we really want to say that the interference itself is located in Middletown, even though it’s a quiet bystander, unaware of the existence of two correlated particles that lie in opposite directions 90 miles (150 km) away?

After all, the interference appears in the relationship between the particles’ positions in physical space, not in the positions themselves. Its location in the space of possibilities (Fig. 3) is clear. Its location in physical space (Fig. 5) is anything but.

Still, I can imagine you pondering whether it might somehow make sense to assign the interference to poor, unsuspecting Middletown. For that reason, I’m going to make things even worse, and take Middletown out of the middle.

A Second System with No Where

Here’s another system with interference, whose pre-quantum version is shown in Figs. 6a and 6b:

Figure 6a: Another system in a superposition with entangled particles, shown in its pre-quantum version in physical space. In part A of the superposition both particles are stationary, while in part B they move oppositely.
Figure 6b: The same system as in Fig. 6a, depicted in the space of possibilities with its two initial possibilities labeled as stars. Possibility A remains where it is, while possibility B moves toward and intersects with possibility A, leading us to expect interference in the quantum wave function.

The corresponding wave function is shown in Fig. 7. Now the interference fringes are oriented diagonally the other way compared to Fig. 3. How are we to measure them this time?

Figure 7: The absolute-value-squared of the wave function for the system shown in Fig. 6. The interference fringes lie on the opposite diagonal from those of Fig. 3.

The average position 1/2(x1+x2) won’t do; we’ll see nothing interesting there. Instead the fringes are near (x1-x2)=4 — that is, they occur when the particles, no matter where they are in physical space, are at a distance of four units. We therefore expect interference near 1/2(x1-x2)=2. Is it there?

In Fig. 8 I’ve shown the analogue of Figs. 4 and 5, depicting

  • the measurements of the two particle positions x1 and x2, along with
  • their average 1/2(x1+x2) plotted between them (in yellow)
  • (half) their difference 1/2(x1-x2) plotted below them (in green).

That quantity 1/2(x1-x2) is half the horizontal length of the orange line. Hidden in its behavior over many measurements is an interference pattern, seen in the bottom row, where the 1/2(x1-x2) measurements are plotted. [Note also that there is no interference pattern in the measurements of 1/2(x1+x2), in contrast to Fig. 4.]

Figure 8: For the system of Figs. 6-7, repeated experiments in which the measurement of the position of particle 1 is plotted in the top row (upper blue points), that of particle 2 is plotted in the third row (lower blue points), their average is plotted between (yellow points), and half their difference is plotted below them (green points.) Each new set of measurements is shown as orange points connected by an orange line, as in Fig. 5. An interference pattern is seen only in the difference.

Now the question of the hour: where is the interference in this case? It is found near 1/2(x1-x2)=2 — but that certainly is not to be identified with a legitimate position in physical space, such as the point x=2.

First of all, making such an identification in Fig. 8 would be like saying that one particle is in New York and the other is in Boston, while the interference is 150 kilometers offshore in the ocean. But second and much worse, I could change Fig. 8 by moving both particles 10 units to the left and repeating the experiment. This would cause x1, x2, and 1/2(x1+x2) in Fig. 8 to all shift left by 10 units, moving them off your computer screen, while leaving 1/2(x1-x2) unchanged at 2. In short, all the orange and blue and yellow points would move out of your view, while the green points would remain exactly where they are. The difference of positions — a distance — is not a position.

If 10 units isn’t enough to convince you, let’s move the two particles to the other side of the Sun, or to the other side of the galaxy. The interference pattern stubbornly remains at 1/2(x1-x2)=2. The interference pattern is in a difference of positions, so it doesn’t care whether the two particles are in France, Antarctica, or Mars.

We can move the particles anywhere in the universe, as long as we take them together with their average distance remaining the same, and the interference pattern remains exactly the same. So there’s no way we can identify the interference as being located at a particular value of x, the coordinate of physical space. Trying to do so creates nonsense.

This is totally unlike interference in water waves and sound waves. That kind of interference happens in a someplace; we can say where the waves are, how big they are at a particular location, and where their peaks and valleys are in physical space. Quantum interference is not at all like this. It’s something more general, more subtle, and more troubling to our intuition.

[By the way, there’s nothing special about the two combinations 1/2(x1+x2) and 1/2(x1-x2), the average or the difference. It’s easy to find systems where the intereference arises in the combination x1+2x2, or 3x1-x2, or any other one you like. In none of these is there a natural way to say “where” the interference is located.]

The Profound Lesson

From these examples, we can begin to learn a central lesson of modern physics, one that a century of experimental and theoretical physics have been teaching us repeatedly, with ever greater subtlety. Imagining reality as many of us are inclined to do, as made of localized objects positioned in and moving through physical space — the one-dimensional x-axis in my simple examples, and the three-dimensional physical space that we take for granted when we specify our latitude, longitude and altitude — is simply not going to work in a quantum universe. The correlations among objects have observable consequences, and those correlations cannot simply be ascribed locations in physical space. To make sense of them, it seems we need to expand our conception of reality.

In the process of recognizing this challenge, we have had to confront the giant, unwieldy space of possibilities, which we can only visualize for a single particle moving in up to three dimensions, or for two or three particles moving in just one dimension. In realistic circumstances, especially those of quantum field theory, the space of possibilities has a huge number of dimensions, rendering it horrendously unimaginable. Whether this gargantuan space should be understood as real — perhaps even more real than physical space — continues to be debated.

Indeed, the lessons of quantum interference are ones that physicists and philosophers have been coping with for a hundred years, and their efforts to make sense of them continue to this day. I hope this series of posts has helped you understand these issues, and to appreciate their depth and difficulty.

Looking ahead, we’ll soon take these lessons, and other lessons from recent posts, back to the double-slit experiment. With fresher, better-informed eyes, we’ll examine its puzzles again.

March 31, 2025

John PreskillHow writing a popular-science book led to a Nature Physics paper

Several people have asked me whether writing a popular-science book has fed back into my research. Nature Physics published my favorite illustration of the answer this January. Here’s the story behind the paper.

In late 2020, I was sitting by a window in my home office (AKA living room) in Cambridge, Massachusetts. I’d drafted 15 chapters of my book Quantum Steampunk. The epilogue, I’d decided, would outline opportunities for the future of quantum thermodynamics. So I had to come up with opportunities for the future of quantum thermodynamics. The rest of the book had related foundational insights provided by quantum thermodynamics about the universe’s nature. For instance, quantum thermodynamics had sharpened the second law of thermodynamics, which helps explain time’s arrow, into more-precise statements. Conventional thermodynamics had not only provided foundational insights, but also accompanied the Industrial Revolution, a paragon of practicality. Could quantum thermodynamics, too, offer practical upshots?

Quantum thermodynamicists had designed quantum engines, refrigerators, batteries, and ratchets. Some of these devices could outperform their classical counterparts, according to certain metrics. Experimentalists had even realized some of these devices. But the devices weren’t useful. For instance, a simple quantum engine consisted of one atom. I expected such an atom to produce one electronvolt of energy per engine cycle. (A light bulb emits about 1021 electronvolts of light per second.) Cooling the atom down and manipulating it would cost loads more energy. The engine wouldn’t earn its keep.

Autonomous quantum machines offered greater hope for practicality. By autonomous, I mean, not requiring time-dependent external control: nobody need twiddle knobs or push buttons to guide the machine through its operation. Such control requires work—organized, coordinated energy. Rather than receiving work, an autonomous machine accesses a cold environment and a hot environment. Heat—random, disorganized energy cheaper than work—flows from the hot to the cold. The machine transforms some of that heat into work to power itself. That is, the machine sources its own work from cheap heat in its surroundings. Some air conditioners operate according to this principle. So can some quantum machines—autonomous quantum machines.

Thermodynamicists had designed autonomous quantum engines and refrigerators. Trapped-ion experimentalists had realized one of the refrigerators, in a groundbreaking result. Still, the autonomous quantum refrigerator wasn’t practical. Keeping the ion cold and maintaining its quantum behavior required substantial work.

My community needed, I wrote in my epilogue, an analogue of solar panels in southern California. (I probably drafted the epilogue during a Boston winter, thinking wistfully of Pasadena.) If you built a solar panel in SoCal, you could sit back and reap the benefits all year. The panel would fulfill its mission without further effort from you. If you built a solar panel in Rochester, you’d have to scrape snow off of it. Also, the panel would provide energy only a few months per year. The cost might not outweigh the benefit. Quantum thermal machines resembled solar panels in Rochester, I wrote. We needed an analogue of SoCal: an appropriate environment. Most of it would be cold (unlike SoCal), so that maintaining a machine’s quantum nature would cost a user almost no extra energy. The setting should also contain a slightly warmer environment, so that net heat would flow. If you deposited an autonomous quantum machine in such a quantum SoCal, the machine would operate on its own.

Where could we find a quantum SoCal? I had no idea.

Sunny SoCal. (Specifically, the Huntington Gardens.)

A few months later, I received an email from quantum experimentalist Simone Gasparinetti. He was setting up a lab at Chalmers University in Sweden. What, he asked, did I see as opportunities for experimental quantum thermodynamics? We’d never met, but we agreed to Zoom. Quantum Steampunk on my mind, I described my desire for practicality. I described autonomous quantum machines. I described my yearning for a quantum SoCal.

I have it, Simone said.

Simone and his colleagues were building a quantum computer using superconducting qubits. The qubits fit on a chip about the size of my hand. To keep  the chip cold, the experimentalists put it in a dilution refrigerator. You’ve probably seen photos of dilution refrigerators from Google, IBM, and the like. The fridges tend to be cylindrical, gold-colored monstrosities from which wires stick out. (That is, they look steampunk.) You can easily develop the impression that the cylinder is a quantum computer, but it’s only the fridge.

Not a quantum computer

The fridge, Simone said, resembles an onion: it has multiple layers. Outer layers are warmer, and inner layers are colder. The quantum computer sits in the innermost layer, so that it behaves as quantum mechanically as possible. But sometimes, even the fridge doesn’t keep the computer cold enough.

Imagine that you’ve finished one quantum computation and you’re preparing for the next. The computer has written quantum information to certain qubits, as you’ve probably written on scrap paper while calculating something in a math class. To prepare for your next math assignment, given limited scrap paper, you’d erase your scrap paper. The quantum computer’s qubits need erasing similarly. Erasing, in this context, means cooling down even more than the dilution refrigerator can manage

Why not use an autonomous quantum refrigerator to cool the scrap-paper qubits?

I loved the idea, for three reasons. First, we could place the quantum refrigerator beside the quantum computer. The dilution refrigerator would already be cold, for the quantum computations’ sake. Therefore, we wouldn’t have to spend (almost any) extra work on keeping the quantum refrigerator cold. Second, Simone could connect the quantum refrigerator to an outer onion layer via a cable. Heat would flow from the warmer outer layer to the colder inner layer. From the heat, the quantum refrigerator could extract work. The quantum refrigerator would use that work to cool computational qubits—to erase quantum scrap paper. The quantum refrigerator would service the quantum computer. So, third, the quantum refrigerator would qualify as practical.

Over the next three years, we brought that vision to life. (By we, I mostly mean Simone’s group, as my group doesn’t have a lab.)

Artist’s conception of the autonomous-quantum-refrigerator chip. Credit: Chalmers University of Technology/Boid AB/NIST.

Postdoc Aamir Ali spearheaded the experiment. Then-master’s student Paul Jamet Suria and PhD student Claudia Castillo-Moreno assisted him. Maryland postdoc Jeffrey M. Epstein began simulating the superconducting qubits numerically, then passed the baton to PhD student José Antonio Marín Guzmán. 

The experiment provided a proof of principle: it demonstrated that the quantum refrigerator could operate. The experimentalists didn’t apply the quantum refrigerator in a quantum computation. Also, they didn’t connect the quantum refrigerator to an outer onion layer. Instead, they pumped warm photons to the quantum refrigerator via a cable. But even in such a stripped-down experiment, the quantum refrigerator outperformed my expectations. I thought it would barely lower the “scrap-paper” qubit’s temperature. But that qubit reached a temperature of 22 milliKelvin (mK). For comparison: if the qubit had merely sat in the dilution refrigerator, it would have reached a temperature of 45–70 mK. State-of-the-art protocols had lowered scrap-paper qubits’ temperatures to 40–49 mK. So our quantum refrigerator outperformed our competitors, through the lens of temperature. (Our quantum refrigerator cooled more slowly than they did, though.)

Simone, José Antonio, and I have followed up on our autonomous quantum refrigerator with a forward-looking review about useful autonomous quantum machines. Keep an eye out for a blog post about the review…and for what we hope grows into a subfield.

In summary, yes, publishing a popular-science book can benefit one’s research.

March 27, 2025

n-Category Café The McGee Group

This is a bit of a shaggy dog story, but I think it’s fun. There’s also a moral about the nature of mathematical research.

Once I was interested in the McGee graph, nicely animated here by Mamouka Jibladze:

This is the unique (3,7)-cage, meaning a graph such that each vertex has 3 neighbors and the shortest cycle has length 7. Since it has a very symmetrical appearance, I hoped it would be connected to some interesting algebraic structures. But which?

I read on Wikipedia that the symmetry group of the McGee graph has order 32. Let’s call it the McGee group. Unfortunately there are many different 32-element groups — 51 of them, in fact! — and the article didn’t say which one this was. (It does now.)

I posted a general question:

and Gordon Royle said the McGee group is “not a super-interesting group, it is SmallGroup(32,43) in either GAP or Magma”. Knowing this let me look up the McGee group on this website, which is wonderfully useful if you’re studying finite groups:

There I learned that the McGee group is the so-called holomorph of the cyclic group /8\mathbb{Z}/8: that is, the semidirect product of /8\mathbb{Z}/8 and its automorphism group:

Aut(/8)/8 Aut(\mathbb{Z}/8) \ltimes \mathbb{Z}/8

I resisted getting sucked into the general study of holomorphs, or what happens when you iterate the holomorph construction. Instead, I wanted a more concrete description of the McGee group.

/8\mathbb{Z}/8 is not just an abelian group: it’s a ring! Since multiplication in a ring distributes over addition, we can get automorphisms of the group /8\mathbb{Z}/8 by multiplying by those elements that have multiplicative inverses. These invertible elements form a group

(/8) ×={1,3,5,7} (\mathbb{Z}/8)^\times = \{1,3,5,7\}

called the multiplicative group of /8\mathbb{Z}/8. In fact these give all the automorphisms of the group /8\mathbb{Z}/8.

In short, the McGee group is

(/8) ×/8 (\mathbb{Z}/8)^\times \ltimes \mathbb{Z}/8

This is very nice, because this is the group of all transformations of /8\mathbb{Z}/8 of the form

xgx+ag(/8) ×,a/8 x \mapsto g x + a \qquad g \in (\mathbb{Z}/8)^\times , \; a \in \mathbb{Z}/8

If we think of /8\mathbb{Z}/8 as a kind of line — called the ‘affine line over /8\mathbb{Z}/8’ — these are precisely all the affine transformations of this line. Thus, the McGee group deserves to be called

Aff(/8)=(/8) ×/8 \text{Aff}(\mathbb{Z}/8) = (\mathbb{Z}/8)^\times \ltimes \mathbb{Z}/8

This suggests that we can build the McGee graph in some systematic way starting from the affine line over /8\mathbb{Z}/8. This turns out to be a bit complicated, because the vertices come in two kinds. That is, the McGee group doesn’t act transitively on the set of vertices. Instead, it has two orbits, shown as red and blue dots here:

The 8 red vertices correspond straightforwardly to the 8 points of the affine line, but the 16 blue vertices are more tricky. There are also the edges to consider: these come in three kinds! Greg Egan figured out how this works, and I wrote it up:

Then a decade passed.

About two weeks ago, I gave a Zoom talk at the Illustrating Math Seminar about some topics on my blog Visual Insight. I mentioned that the McGee group is SmallGroup(32,43) and the holomorph of /8\mathbb{Z}/8. And then someone — alas, I forget who — instantly typed in the chat that this is one of the two smallest groups with an amazing property! Namely, this group has an outer automorphism that maps each element to an element conjugate to it.

I didn’t doubt this for a second. To paraphrase what Hardy said when he received Ramanujan’s first letter, nobody would have the balls to make up this shit. So, I posed a challenge to find such an exotic outer automorphism:

By reading around, I soon learned that people have studied this subject quite generally:

An automorphism f:GGf \colon G \to G is class-preserving if for each gGg \in G there exists some hGh \in G such that

f(g)=hgh 1 f(g) = h g h^{-1}

If you can use the same hh for every gg we call ff an inner automorphism. But some groups have class-preserving automorphisms that are not inner! These are the class-preserving outer automorphisms.

I don’t know if class-preserving outer automorphisms are good for anything, or important in any way. They mainly just seem intriguingly spooky. An outer automorphism that looks inner if you examine its effect on any one group element is nothing I’d ever considered. So I wanted to see an example.

Rising to my challenge, Greg Egan found a nice explicit formula for some class-preserving outer automorphisms of the McGee group.

As we’ve seen, any element of the McGee group is a transformation

xgx+ag(/8) ×,a/8 x \mapsto g x + a \qquad g \in (\mathbb{Z}/8)^\times , \; a \in \mathbb{Z}/8

so let’s write it as a pair (g,a)(g,a). Greg Egan looked for automorphisms of the McGee group that are of the form

f(g,a)=(g,a+D(g)) f(g,a) = (g, a + D(g))

for some function

D:(/8) ×/8 D \colon (\mathbb{Z}/8)^\times \to \mathbb{Z}/8

It is easy to check that ff is an automorphism if and only if

D(gg)=D(g)+gD(g) D(g g') = D(g) + g D(g')

Moreover, ff is an inner automorphism if and only if

D(g)=gbb D(g) = g b - b

for some b/8b \in \mathbb{Z}/8.

Now comes something cool noticed by Joshua Grochow: these formulas are an instance of a general fact about group cohomology!

Suppose we have a group GG acting as automorphisms of an abelian group AA. Then we can define the cohomology H n(G,A)H^n(G,A) to be the group of nn-cocycles modulo nn-coboundaries. We only need the case n=1n = 1 here. A 1-cocycle is none other than a function D:GAD \colon G \to A obeying

D(gg)=D(g)+gD(g) D(g g') = D(g) + g D(g')

while a 1-coboundary is one of the form

D(g)=gbb D(g) = g b - b

for some bAb \in A. You can check that every 1-coboundary is a 1-cocycle. H 1(G,A)H^1(G,A) is the group of 1-cocycles modulo 1-coboundaries.

In this situation we can define the semidirect product GAG \ltimes A, and for any D:GAD \colon G \to A we can define a function

f:GAGA f \colon G \ltimes A \to G \ltimes A

by

f(g,a)=(g,a+D(g)) f(g,a) = (g, a + D(g))

Now suppose G=Aut(A)G = \text{Aut}(A) and suppose GG is abelian. Then by straightforward calculations we can check:

  • ff is an automorphism iff DD is a 1-cocycle

and

  • ff is an inner automorphism iff DD is a 1-coboundary!

Thus, GAG \ltimes A will have outer automorphisms if H 1(G,A)0H^1(G,A) \ne 0.

When A=/8A = \mathbb{Z}/8 then G=Aut(A)G = \text{Aut}(A) is abelian and GAG \ltimes A is the McGee group. This puts Egan’s idea into a nice context. But we still need to actually find maps DD that give outer automorphisms of the McGee group, and then find class-preserving ones. I don’t know how to do that using general ideas from cohomology. Maybe someone smart could do the first part, but the ‘class-preserving’ condition doesn’t seem to emerge naturally from cohomology.

Anyway, Egan didn’t waste his time with such effete generalities: he actually found all choices of D:(/8) ×/8D \colon (\mathbb{Z}/8)^\times \to \mathbb{Z}/8 for which

f(g,a)=(g,a+D(g)) f(g,a) = (g, a + D(g))

is a class-preserving outer automorphism of the McGee group. Namely:

(D(1),D(3),D(5),D(7)) = (0,0,4,4) (D(1),D(3),D(5),D(7)) = (0,2,0,2) (D(1),D(3),D(5),D(7)) = (0,4,4,0) (D(1),D(3),D(5),D(7)) = (0,6,0,6) \begin{array}{ccl} (D(1), D(3), D(5), D(7)) &=& (0, 0, 4, 4) \\ (D(1), D(3), D(5), D(7)) &=& (0, 2, 0, 2) \\ (D(1), D(3), D(5), D(7)) &=& (0, 4, 4, 0) \\ (D(1), D(3), D(5), D(7)) &=& (0, 6, 0, 6) \end{array}

Last Saturday after visiting my aunt in Santa Barbara I went to Berkeley to visit the applied category theorists at the Topos Institute. I took a train, to lessen my carbon footprint a bit. The trip took 9 hours — a long time, but a beautiful ride along the coast and then through forests and fields.

The day before taking the train, I discovered my laptop was no longer charging! So, I bought a pad of paper. And then, while riding the train, I checked by hand that Egan’s first choice of DD really is a cocycle, and really is not a coboundary, so that it defines an outer automorphism of the McGee group. Then — and this was fairly easy — I checked that it defines a class-preserving automorphism. It was quite enjoyable, since I hadn’t done any long calculations recently.

One moral here is that interesting ideas often arise from the interactions of many people. The results here are not profound, but they are certainly interesting, and they came from online conversations with Greg Egan, Gordon Royle, Joshua Grochow, the mysterious person who instantly knew that the McGee group was one of the two smallest groups with a class-preserving outer automorphism, and others.

But what does it all mean, mathematically? Is there something deeper going on here, or is it all just a pile of curiosities?

What did we actually do, in the end? Following the order of logic rather than history, maybe this. We started with a commutative ring AA, took its group of affine transformations Aff(A)\text{Aff}(A), and saw this group must have outer automorphisms if

H 1(A ×,A)0 H^1(A^\times, A) \ne 0

We saw this cohomology group really is nonvanishing when A=/nA = \mathbb{Z}/n and n=8n = 8. Furthermore, we found a class-preserving outer automorphism of Aff(/8)\text{Aff}(\mathbb{Z}/8).

This raises a few questions:

  • What is the cohomology H 1((/n) ×,/n)H^1((\mathbb{Z}/n)^\times, \mathbb{Z}/n) in general?

  • What are the outer automorphisms of Aff(/n)\text{Aff}(\mathbb{Z}/n)?

  • When does Aff(/n)\text{Aff}(\mathbb{Z}/n) have class-preserving outer automorphisms?

I saw bit about the last question in this paper:

They say that this paper:

  • G. E. Wall, Finite groups with class-preserving outer automorphisms, Journal of the London Mathematical Society 22 (1947), 315–320.

proves Aff(/n)\text{Aff}(\mathbb{Z}/n) has a class-preserving outer automorphism when nn is a multiple of 8.

Does this happen only for multiples of 8? Is this somehow related to the most famous thing with period 8 — namely, Bott periodicity? I don’t know.

March 11, 2025

Matt LeiferThe Confused Chapman Student’s Guide to the APS Global Summit

This guide is intended for the Chapman undergraduate students who are attending this year’s APS Global Summit. It may be useful for others as well.

The APS Global Summit is a ginormous event, featuring dozens of parallel sessions at any given time. It can be exciting for first-time attendees, but also overwhelming. Here, I compile some advice on how to navigate the meeting and some suggestions for sessions and events you might like to attend.

General Advice

  • Use the online schedule and the mobile app to help you navigate the meeting. If you create a login, the online schedule allows you to add things to your personalized schedule, which you can view on the app at the meeting. This is a very useful thing to do because making decisions of where to go on the fly is difficult.
  • Do not overschedule yourself. I know it is tempting to figure out how to go to as many things as you can, and run between sessions on opposite sides of the convention center. This will be harder to accomplish than you imagine. The meeting gets very crowded and it is exhausting to sit through a full three-hour session of talks. Schedule some break time and, where possible, schedule blocks of time in one location rather than running all over the place.
  • You will have noticed that most talks at the meeting are 12min long (10min + 2min). These are called contributed talks. Since they are so short, they are more like adverts for the work than a detailed explanation. They are usually aimed at experts and, quite frankly, many speakers do not know how to give these talks well. It is not worth attending these talks unless one of the following applies:
    • You are already an expert in that research area.
    • You are strongly considering doing research in that area.
    • You are there to support your friends and colleagues who are speaking in that session.
    • You are so curious about the research area that you are prepared to sit through a lot of opaque talks to get some idea of what is going on in the area.
    • The session is on a topic that is unusually accessible or the session is aimed at undergraduate students.
  • Instead, you should prioritize attending the following kinds of talks, which you can search for using the filters on the schedule:
    • Plenary talks: These are aimed at a general physics audience and are usually by famous speakers (famous by physics standards anyway). Some of these might also be…
    • Popular science talks: Aimed at the general public.
    • Invited Sessions: These sessions consist of 30min talks by invited speakers in a common research area. There is no guarantee that they will be accessible to novices, but it is much more likely than with the contributed talks. Go to any invited sessions on areas of physics you are curious about.
    • Focus Sessions: Focus sessions consist mainly of contributed talks, but they also have one or two 30min invited talks. It is not considered rude to switch sessions between talks, so do not be afraid to just attend the invited talks. They are not always scheduled at the beginning of the session. In fact, some groups deliberately stagger the times of the invited talks so that people can see the invited talks in more than one focus session.
  • There are sessions that list “Undergraduate Students” as part of their target audience. A lot of these are “Undergraduate Research” sessions. It can be interesting to go to one or two of these to see the variety of undergraduate research experiences that are on offer. However, I would not advise only going to sessions on this list. For one thing, undergraduate research projects are not banned from the other sessions, so many of the best undergraduate projects will not be in those sessions. Going to sessions by topic is a better bet most of the time.
  • It is helpful to filter the sessions on the schedule by the organizing Unit (Division, Topical Group, or Forum). You can find a list of APS units here. For example, if you are particularly interested in Quantum Information and Computation then you will want to look at the sessions organized by DQI (Division of Quantum Information). Sessions organized by Forums are often particularly accessible, as they tend to be about less technical issues (DEI, Education, History and Philosophy, etc.)

The next sections contain some more specific suggestions about events, talks and sessions that you might like to attend.

Orientation and Networking Events

I have never been to an orientation or networking event at the APS meeting, but then again I did not go to the APS meeting as a student. Networking is one of the best things you can do at the meeting, so do take any opportunities to meet and talk to people.

Sunday March 16

TimeEventLocation
2:00pm – 3:00pmFirst Time Attendee OrientationAnaheim Convention Center, 201AB (Level 2)
3:00pm – 4:00pmUndergraduate Student Get TogetherAnaheim Convention Center, 201AB (Level 2)

Tuesday March 18

TimeEventLocation
12:30pm – 2:00pmStudents Lunch with the ExpertsAnaheim Convention Center, Exhibit Hall B

The student lunch with the Experts is especially worth it because you get a one-on-eight meeting with a physicist who works on a topic you are interested in. You also get a free lunch. Spaces are limited, so you need to sign up for it on the Sunday, and early if you want to get your choice of expert.

Generally speaking, food is very expensive in the convention center. Therefore, the more places you can get free food the better. There are networking events, some of which are aimed at students and some of which have free meals. Other good bets for free food include the receptions and business meetings. (With a business meeting you may have to first sit through a boring administrative meeting for an APS unit, but at least the DQI meeting will feature me talking about The Quantum Times.)

Sessions Chaired by Chapman Faculty

The next few sections highlight talks and sessions that involve people at Chapman. You may want to come to these not only to support local people, but also to find out more about areas of research that you might want to do undergraduate research projects in.

The following sessions are being chaired by Chapman faculty. The chair does not give a talk during the session, but acts as a host. But chairs usually work in the areas that the session is about, so it is a good way to get more of an overview of things they are interested in.

DayTimeChairSession TitleLocation
Monday 1711:30pm – 1:54pmMatt LeiferQuantum Foundations: Bell Inequalities and Causality
Anaheim Convention Center,
256B (Level 2)
Wednesday 198:00am – 10:48amAndrew JordanOptimal Quantum ControlAnaheim Convention Center,
258A (Level 2)
Wednesday 1911:30am – 1:30pmBibek BhandariExplorations in Quantum ComputingVirtual Only, Room 1

Talks involving Chapman Faculty, Postdocs and Students

The talks listed below all have someone who is currently affiliated with Chapman as one or more of the authors. The Chapman person is not necessarily the person giving the talk.

The people giving the talks, especially if they are students or postdocs, would appreciate your support. It is also a good way of finding out more about research that is going on at Chapman.

Monday March 17

TimeSpeakerTitleLocation
9:36am – 9:48amIrwin HuangBeyond Single Photon Dissipation in Kerr Cat QubitsAhaheim Convention Center, 161 (Level 1)
9:48am – 10amBingcheng QingBenchmarking Single-Qubit Gates on a Noise-Biased Qubit: Kerr cat qubitAnaheim Convention Center, 161 (Level 1)
10:12am – 10:24amAhmed HjarStrong light-matter coupling to protect quantum information with Schrodinger cat statesAnaheim Convention Center, 161 (Level 1)
10:24am – 10:36amBibek BhandariDecoherence in dynamically protected qubitsAnaheim Convention Center, 161 (Level 1)
10:36am – 10:48amKe WangControl-Z two-qubit gate on 2D Kerr catsAnaheim Convention Center,
161 (Level 1)
4:12pm – 4:24pmAdithi AjithStabilizing two-qubit entanglement using stochastic path integral formalismAnaheim Convention Center,
258A (Level 2)
4:36pm – 4:48 pmAlok Nath SinghCapturing an electron during a virtual transition via continuous measurementAnaheim Convention Center,
252B (Level 2)

Tuesday March 18

TimeSpeakerTitleLocation
8:48am – 9:00amAlexandria O UdenkwoCharacterizing the energy and efficiency of an entanglement fueled engine in a circuit QED processorAnaheim Convention Center,
162 (Level 1)
12:30pm – 12:42pmYile YingA review and analysis of six extended Wigner’s friend arguments
Anaheim Convention Center,
256B (Level 2)
1:54pm – 2:06pmIndrajit SenΡΤ-symmetric axion electrodynamics: A pilot-wave approachAnaheim Marriott,
Platinum 1
3:48pm – 4:00pmChuanhong LiuPlanar Fluxonium Qubits Design with 4-way CouplingAnaheim Convention Center,
162 (Level 1)
4:36pm – 4:48pmRobert CzupryniakReinforcement Learning Meets Quantum Control – Artificially Intelligent Maxwell’s DemonAnaheim Convention Center,
258A (Level 2)

Wednesday March 19

TimeSpeakerTitleLocation
10:36am – 10:48amDominic Briseno-ColungaDynamical Sweet Spot Manifolds of Bichromatically Driven Floquet QubitsAnaheim Convention Center,
162 (Level 1)
2:30pm – 2:42pmSayani GhoshEquilibria and Effective Rates of Transition in Astromers
Anaheim Marriott,
Platinum 7
3:00pm – 3:12pmMatt LeiferA Foundational Perspective on PT-Symmetric Quantum TheoryAnaheim Convention Center,
151 (Level 1)
5:36pm – 5:48pmSacha GreenfieldA unified picture for quantum Zeno and anti-Zeno effectsAnaheim Convention Center,
161 (Level 1)

Thursday March 20

TimeSpeakerTitleLocation
1:18pm – 1:30pmLucas BurnsDelayed Choice Lorentz Transformations on a QubitAnaheim Convention Center,
256B (Level 2)
4:48pm – 5:00pmNoah J StevensonDesign of fluxonium coupling and readout via SQUID couplersAnaheim Convention Center,
161 (Level 1)
5:00pm – 5:12pmKagan YanikFlux-Pumped Symmetrically Threaded SQUID Josephson Parametric AmplifierAnaheim Convention Center,
204C (Level 2)
5:00pm – 5:12pmAbhishek ChakrabortyTwo-qubit gates for fluxonium qubits using a tunable couplerAnaheim Convention Center,
161 (Level 1)

Friday March 21

TimeSpeakerTitleLocation
10:12am – 10:24amNooshin M. EstakhriDistinct statistical properties of quantum two-photon backscatteringAnaheim Convention Center,
253A (Level 2)
10:48am – 11:00amLe HuEntanglement dynamics in collision models and all-to-all entangled statesAnaheim Hilton,
San Simeon AB (Level 4)
11:54am – 12:06pmLuke ValerioOptimal Design of Plasmonic Nanotweezers with Genetic AlgorithmAnaheim Convention Center,
253A (Level 2)

Posters involving Chapman Faculty, Postdocs and Students

Poster sessions last longer than talks, so you can view the posters at your leisure. The presenter is supposed to stand by their poster and talk to people who come to see it. The following posters are being presented by Chapman undergraduates. Please drop by and support them.

Thursday March 20, 10:00am – 1:00pm, Anaheim Convention Center, Exhibit Hall A

Poster NumberPresenterTitle
267Ponthea ZahraiiMachine learning-assisted characterization of optical forces near gradient metasurfaces
400Clara HuntWhat the white orchid can teach us about radiative cooling
401Nathan TaorminaOptimizing Insulation and Geometrical Designs for Enhanced Sub-Ambient Radiative Cooling Efficiency

Leifer’s Recommendations

These are sessions that reflect my own interests. It is a good bet that you will find me at one of these, unless I am teaching, or someone I know is speaking somewhere else. There are multiple sessions at the same time, but what I will typically do is select the one that has the most interesting looking talk at the time and switch sessions from time to time or take a break from sessions entirely if I get bored.

Monday March 17

TimeSession TitleLocation
8:00am – 11:00amQuantum Science and Technology at the National DOE Research Centers: Progress and OpportunitiesAnaheim Convention Center, 158 (Level 1)
8:00am – 11:00amLearning and Benchmarking Quantum ChannelsAnaheim Convention Center, 258A (Level 2)
10:45am – 12:33pmBeginners Guide to Quantum GravityAnaheim Marriott, Grand Ballroom Salon E
11:30am – 1:54pmQuantum Foundations: Bell Inequalities and CausalityAnaheim Convention Center, 256B (Level 2)
1:30pm – 3:18pmHistory and Physics of the Manhattan Project and the Bombings of Hiroshima and NagasakiAnaheim Marriott, Platinum 9
3:00pm – 6:00pmDQI Thesis Award SessionAnaheim Convention Center, 158 (Level 1)

Tuesday March 18

TimeSession TitleLocation
8:30am – 10:18amForum on Outreach and Engagement of the Public Invited SessionAnaheim Marriott, Orange County Salon 1
10:45am – 12:33pmPais Prize SessionAnaheim Marriott, Platinum 2
11:30am – 2:30pmApplied Quantum FoundationsAnaheim Convention Center, 256B (Level 2)
1:30pm – 3:18pmMini-Symposium: Research Validated Assessments in EducationAnaheim Marriott, Grand Ballroom Salon D
1:30pm – 3:18pmResearch in Quantum Mechanics InstructionAnaheim Marriott, Orange County Salon 1
3:00pm – 5:24pmLandauer-Bennett Award Prize SymposiumAnaheim Convention Center, 158 (Level 1)
3:00pm – 6:00pmUndergraduate and Graduate Education IAnaheim Convention Center, 263A (Level 2)
3:00pm – 6:00pmInvited Session for the Forum on Outreach and Engagement of the PublicAnaheim Convention Center, 155 (Level 1)
3:45pm – 5:33pmHighlights from the Special Collections of AJP and TPT on Teaching About QuantumAnaheim Marriott, Platinum 3
6:15pm – 9:00pmDQI Business MeetingAnaheim Convention Center, 160 (Level 1)

Wednesday March 19

TimeSession TitleLocation
11:30am – 2:30pmQuantum Information: Thermodynamics out of EquilibriumAnaheim Hilton, San Simeon AB (Level 4)
3:00pm – 5:36pmQuantum Foundations: Measurements, Contextuality, and ClassicalityAnaheim Convention Center, 151 (Level 1)
3:00pm – 6:00pmBeyond Knabenphysik: Women in the History of Quantum PhysicsAnaheim Convention Center, 154 (Level 1)

Thursday March 20

TimeSession TitleLocation
8:00am – 10:48amUndergraduate EducationAnaheim Convention Center, 263A (Level 2)
8:00am – 11:00amOpen Quantum Systems and Many-Body DynamicsAnaheim Hilton, San Simeon AB (Level 4)
11:30am – 2:30pmTime in Quantum Mechanics and ThermodynamicsAnaheim Hilton, California C (Ballroom Level)
11:30am – 2:30pmIntersections of Quantum Science and SocietyAnaheim Convention Center, 159 (Level 1)
11:30am – 2:18pmQuantum Foundations: Relativity, Gravity, and GeometryAnaheim Convention Center, 256B (Level 2)
3:00pm – 6:00pmThe Early History of Quantum Information PhysicsAnaheim Convention Center, 154 (Level 1)
3:00pm – 6:00pmQuantum Thermalization: Understanding the Dynamical Foundation of Quantum ThermodynamicsAnaheim Hilton, California A (Ballroom Level)

Friday March 21

TimeSession TitleLocation
8:00am – 11:00amStructures in Quantum SystemsAnaheim Convention Center, 258A (Level 2)
8:00am – 10:24amScience Communication in an Age of Misinformation and DisinformationAnaheim Convention Center, 156 (Level 1)

The Exhibition Hall

It is worthwhile to spend some time in the exhibit hall. It features a Careers Fair and a Grad School Fair, which will be larger and more relevant to physics students than other such fairs you might attend in the area.

But, of course, the main purpose of going to the exhibition hall is to acquire SWAG. Some free items I have obtained from past APS exhibit halls include:

  • Rubik’s cubes
  • Balls that light up when you bounce them
  • Yo-Yos
  • Wooden model airplanes
  • Snacks
  • T-shits
  • Tote bags
  • Enough stationery items to last for the rest of your degree
  • Free magazines and journals
  • Free or heavily discounted books

I recommend going when the hall first opens to get the highest quality SWAG.

Fun Stuff

Other fun stuff to do at this year’s meeting includes:

  • QuantumFest: This starts with the Quantum Jubilee event on Saturday, but there are events all week some of which you have to be registered for the meeting for. Definitely reserve a spot for the LabEscape escpae room. I have done one of their rooms before and it is fun.
  • Physics Rock-n-Roll Singalong: A very nerdy APS meeting tradition. Worth attending once in your life. Probably only once though.

March 03, 2025

Clifford JohnsonValuable Instants

This week’s lectures on instantons in my gauge theory class (a very important kind of theory for understanding many phenomenon in nature – light is an example of a phenomenon that is described by gauge theory) were a lot of fun to do, and mark the culmination of a month-long … Click to continue reading this post

The post Valuable Instants appeared first on Asymptotia.

February 22, 2025

Robert HellingThe Bohm-GHZ paper is out

 I had this neat calculation in my drawer and on the occasion of quantum mechanic's 100th birthday in 2025, I decided I submit a talk about it to the March meeting of the DPG, the German physical society, in Göttingen. And to have to show something, I put it out on the arxiv today. The idea is as follows:

The GHZ experiment is a beautiful version of Bell's inequality that demonstrates you get to wrong conclusions when you assume that a property of a quantum system has to have some (unknown) value even when you don't measure it. I would say it shows quantum theory is not realistic, in the sense that unmeasured properties do not have secret values (different for example from classical statistical mechanics where you could imagine to actually measure the exact position of molecule number 2342 in your container of gas). For details, see the paper or this beautiful explanation by Coleman. I should mention here that there is another way out by assuming some non-local forces that conspire to make the result come out right never the less.

On the other hand there is Bohmian mechanics. This is well known to be a non-local theory (as the time evolution of its particles depend on the positions of all other particles in the system or even universe) but what I found more interesting is also realistic: There, it is claimed that all that matters are particles positions (including the positions of pointers on your measurement devices that you might interpret as showing something different than positions for example velocities or field strengths or whatever) and those have all (possibly unknown) values at all times even if you don't measure them.

So how can the two be brought together? There might be an obstacle in the fact that GHZ is usually presented to be a correlation of spins and in the Bohmian literature spins are not really positions, you will always have to make use of some Stern-Gerlach experiments to translate those into actual positions. But we can circumvent this the other way: We don't really need spins, we just need observables of the commutation relation of Pauli matrices. You might think that those cannot be realised with position measurements as they always commute but this is only true as you do the position measurements at equal times. If you wait between them, you can in fact have almost Pauli type operators.

So we can set up a GHZ experiment in terms of three particles in three boxes and for each particle you measure whether it is in the left or the right half of the box but for each particle you decide if you do it at time 0 or at a later moment. You can look at the correlation of the three measurements as a function of time (of course, as you measure different particles, the actual measurements you do still commute independent of time) and what you find is the blue line in

GHZ correlations vs. Bohmian correlations
   

You can also (numerically) solve the Bohmian equation of motion and compute the expectation of the correlation of positions of the three particles at different times which gives the orange line, clearly something else. No surprise, the realistic theory cannot predict the outcome of an experiment that demonstrates that quantum theory is not realistic. And the non-local character of the evolution equation does not help either.

To save the Bohmian theory, one can in fact argue that I have computed the wrong thing: After measuring the position of one particle at time 0 or by letting it interact with a measuring device, the future time evolution of all particles is affected and one should compute that correlation with the corrected (effectively collapsed) wave function. That, however, I cannot do and I claim is impossible since it would depend on the details of how the first particle's position is actually measured (whereas the orthodox prediction above is independent of those details as those interactions commute with the later observations). In any case, at least my interpretation is that if you don't want to predict the correlation wrong the best you can do is to say you cannot do the calculation as it depends on unknown details (but the result of course shouldn't).

In any case, the standard argument why Bohmian mechanics is indistinguishable from more conventional treatments is that all that matters are position correlations and since those are given by psi-squared they are the same for all approaches. But I show this is not the case for these multi-time correlations.


Post script: What happens when you try to discuss physics with a philosopher: