Planet Musings

The world is complicated, and the following things can all be true:

(1) Trump and his minions would love to destroy American academia, to show their power, thrill their base, and exact revenge on people who they hate. They will gladly seize on any pretext to do so. For those of us, whatever our backgrounds, who chose to spend our lives in American academia, discovering and sharing new knowledge—this is and should be existentially terrifying.

(2) For the past year and a half, Columbia University was a pretty scary place to be an Israeli or pro-Israel Jew—at least, according to Columbia’s own antisemitism task force report, the firsthand reports of my Jewish friends and colleagues at Columbia, and everything else I gleaned from sources I trust. The situation seems to have been notably worse there than at most American universities. (If you think this is all made up, please read pages 13-37 of the report—immediately after October 7, Jewish students singled out for humiliation by professors in class, banned from unrelated student clubs unless they denounced Israel, having their Stars of David ripped off as they walked through campus at night, forced to move dorms due to constant antisemitic harassment—and then try to imagine we were talking about Black, Asian, or LGBTQ students. How would expect a university to respond, and how would you want it to? More recent incidents included the takeover of a Modern Israeli History class—guards were required for subsequent lectures—and the occupation of Barnard College.) Last year, I decided to stop advising Jewish and Israeli students to go to Columbia, or at any rate, to give them very clear warnings about it. I did this with extreme reluctance, as the Columbia CS department happens to have some of my dearest colleagues in the world, many of whom I know feel just as I do about this.

(3) Having been handed this red meat on a silver platter, the Trump Education Department naturally gobbled it up. They announced that they’re cancelling $400 million in grants to Columbia, to be reinstated in a month if Columbia convinces them that they’re fulfilling their Title VI antidiscrimination obligations to Jews and Israelis. Clearly the Trumpists mean to make an example of Columbia, and thereby terrify other universities into following suit.

(4) Tragically and ironically, this funding freeze will primarily affect Columbia’s hard science departments, which rely heavily on federal grants, and which have remained welcoming to Jews and Israelis. It will have only a minimal effect on Columbia’s social sciences and humanities departments—the ones that nurtured the idea of Hamas and Hezbollah as heroic resistance—as those departments receive much less federal funding in the first place. I hate that suspending grants is pretty much the only federal lever available.

(5) When an action stands to cause so much pain to the innocent and so little to the guilty, I can’t on reflection endorse it—even if it might crudely work to achieve an outcome I want, and all the less if it won’t achieve that outcome.

(6) But I can certainly hope for a good outcome! From what I’ve been told, Katrina Armstrong, the current president of Columbia, has been trying to do the right thing ever since she inherited this mess. In response to the funding freeze, President Armstrong issued an excellent statement, laying out her determination to work with the Education Department, crack down on antisemitic harassment, and restore the funding, with no hint of denial or defensiveness. While I wouldn’t want her job right now, I’m rooting for her to succeed.

(7) Time for some game theory. Consider the following three possible outcomes:
(a) Columbia gets back all its funding by seriously enforcing its rules (e.g., expelling students who threatened violence against Jews), and I can again tell Jewish and Israeli students to attend Columbia with zero hesitation
(b) Everything continues just like before
(c) Columbia loses its federal funding, essentially shuts down its math and science research, and becomes a shadow of what it was
Now let’s say that I assign values of 100 to (a), 50 to (b), and -1000 to (c). This means that, if (say) Columbia’s humanities professors told me that my only options were (b) and (c), I would always flinch and choose (b). And thus, I assume, the professors would tell me my only options were (b) and (c). They’d know I’d never hold a knife to their throat and make them choose between (a) and (c), because I’d fear they’d actually choose (c), an outcome I probably want even less than they do.

Having said that: if, through no fault of my own, some mobster held a knife to their throat and made them choose between (a) and (c)—then I’d certainly advise them to pick (a)! Crucially, this doesn’t mean that I’d endorse the mobster’s tactics, or even that I’d feel confident that the knife won’t be at my own throat tomorrow. It simply means that you should still do the right thing, even if for complicated reasons, you were blackmailed into doing the right thing by a figure of almost cartoonish evil.

I welcome comments with facts or arguments about the on-the-ground situation at Columbia, American civil rights law, the Trumpists’ plans, etc. But I will ruthlessly censor comments that try to relitigate the Israel/Palestine conflict itself. Not merely because I’m tired of that, the Shtetl-Optimized comment section having already litigated the conflict into its constituent quarks, but much more importantly, because whatever you think of it, it’s manifestly irrelevant to whether or not Columbia tolerated a climate of fear for Jews and Israelis in violation of Title VI, which is understandably the only question that American judges (even the non-Trumpist ones) will care about.

Seen on Twitter:

I don’t know the context here, so I can’t speak to what Prof. Cronin meant. But it got me thinking.

Suppose you, like Prof. Cronin, were to insist that AI “cannot in principle” do science, because AI “is not autonomous” and “does not come up with its own problems to solve”. What might you mean?

You might just be saying that AI is bad at coming up with new problems to solve. That’s probably fair, at least at the moment. People have experimented with creating simple “AI researchers” that “study” computer programs, coming up with hypotheses about the programs’ performance and testing them. But it’s a long road from that to reproducing the much higher standards human scientists have to satisfy.

You probably don’t mean that, though. If you did, you wouldn’t have said “in principle”. You mean something stronger.

More likely, you might mean that AI cannot come up with its own problems, because AI is a tool. People come up with problems, and use AI to help solve them. In this perspective, not only is AI “not autonomous”, it cannot be autonomous.

On a practical level, this is clearly false. Yes, machine learning models, the core technology in current AI, are set up to answer questions. A user asks something, and receives the model’s prediction of the answer. That’s a tool, but for the more flexible models like GPT it’s trivial to turn it into something autonomous. Just add another program: a loop that asks the model what to do, does it, tells the model the result, and asks what to do next. Like taping a knife to a Roomba, you’ve made a very simple modification to make your technology much more dangerous.

You might object, though, that this simple modification of GPT is not really autonomous. After all, a human created it. That human had some goal, some problem they wanted to solve, and the AI is just solving the problem for them.

That may be a fair description of current AI, but insisting it’s true in principle has some awkward implications. If you make a “physics AI”, just tell it to do “good physics”, and it starts coming up with hypotheses you’d never thought of, is it really fair to say it’s just solving your problem?

What if the AI, instead, was a child? Picture a physicist encouraging a child to follow in their footsteps, filling their life with physics ideas and rhapsodizing about the hard problems of the field at the dinner table. Suppose the child becomes a physicist in turn, and finds success later in life. Were they really autonomous? Were they really a scientist?

What if the child, instead, was a scientific field, and the parent was the general public? The public votes for representatives, the representatives vote to hire agencies, and the agencies promise scientists they’ll give them money if they like the problems they come up with. Who is autonomous here?

(And what happens if someone takes a hammer to that process? I’m…still not talking about this! No-politics-rule still in effect, sorry! I do have a post planned, but it will have to wait until I can deal with the fallout.)

At this point, you’d probably stop insisting. You’d drop that “in principle”, and stick with the claim I started with, that current AI can’t be a scientist.

But you have another option.

You can accept the whole chain of awkward implications, bite all the proverbial bullets. Yes, you insist, AI is not autonomous. Neither is the physicist’s child in your story, and neither are the world’s scientists paid by government grants. Each is a tool, used by the one, true autonomous scientist: you.

You are stuck in your skull, a blob of curious matter trained on decades of experience in the world and pre-trained with a couple billion years of evolution. For whatever reason, you want to know more, so you come up with problems to solve. You’re probably pretty vague about those problems. You might want to see more pretty pictures of space, or wrap your head around the nature of time. So you turn the world into your tool. You vote and pay taxes, so your government funds science. You subscribe to magazines and newspapers, so you hear about it. You press out against the world, and along with the pressure that already exists it adds up, and causes change. Biological intelligences and artificial intelligences scurry at your command. From their perspective, they are proposing their own problems, much more detailed and complex than the problems you want to solve. But from yours, they’re your limbs beyond limbs, sight beyond sight, asking the fundamental questions you want answered.

What is really going on in the quantum double-slit experiment? The question raised in this post’s title seems to lie at the heart of the matter. In this experiment, which I recently reviewed here, particles of some sort are aimed, one at a time, at a wall with two slits, and their arrival is recorded on a screen behind the wall. As a parade of particles proceeds, one by one, past the wall, an interference pattern somehow appears, emerging gradually like a spectre on the screen.

Interference is a familiar effect, commonly seen in water waves and sound waves. If water waves passed through a pair of slits in a wall, interference would be observed and no one would be surprised. But here we have one particle passing through the wall at a time; it’s not at all the same thing. How can we explain the interference effect in this case?

It’s natural to imagine that somehow either

• each particle acts like a wave, goes through both slits, and interferes with itself, or
• the quantum wave function that describes each particle (or all the particles [?]) goes through both slits and interferes with itself.

So… which is it? Did the particle go through both slits, or did the wave function?

In 1920s quantum physics, there is a very simple answer to this question.

The answer is,…

“No.“

No — neither the particle nor the wave function [not its wavy pattern or its peak(s) or any other part of it] goes through the two slits.

• What!? Then how can there be interference?

That question I will answer in a later post, probably next week or the following. But first, let’s confront the title of this post in a simpler context, so that we can see clearly why — in 1920s quantum physics — the answer to its question is “neither one”.

The Double Door Experiment

Key to understanding the double-slit experiment is to simplify it down to its bare essence. Having a two-dimensional problem where particles are going through slits in a wall is more complicated than necessary. Instead, let’s take a one-dimensional problem that we’ve already looked at, where an object is in a superposition state of going to the left OR going to the right. We already saw that this object is not to be viewed as both going to the left AND going to the right. By setting up measurements on both sides, we saw that it can only be measured to be doing one or the other, and never both. Superposition is an OR, not an AND. And a true particle can only have one position at a time.

The Particle and the Doors: A First Look

In this context, let’s ask the question: can a particle simultaneously go through two doors on opposite sides of a room? This is the same question as the two-slit question, because I can turn one into the other using tubes behind the slits, as in Fig. 1.

By sending a particle toward two slits, we can arrange for its wave function to be in the superposition state we want (Fig. 2)

and then we can ask whether we can observe it going through both doors.

Well, in a recent post we put two balls in the same locations that we now want to place the doors, and we asked if a particle in this very same superposition state can hit both balls simultaneously. The answer was “no”. The same argument applies here; the particle cannot be observed to pass through both doors simultaneously. It can only go through one or the other.

Why? A particle, which has a position and a momentum (even if unknown to us), cannot have two positions. If it starts between the two doors, it can move through one door or the other, but it cannot do both, because then it would have two positions at once.

Maybe you’re not immediately convinced. If not, stay tuned, as I’ll come back to this again later.

The Wave Function and the Doors: A First Look

But for now, let’s turn to the other question arising from my post’s title. Why can’t the particle’s wave function move through both doors, just like a water wave or sound wave does?

Actually, since wave functions don’t move (they just describe particles that do) what we really want to know is slightly different. The initial wave function has a wavy pattern; does this wavy pattern go through both doors?

Certainly water waves and sound waves could go through both doors. They are waves in physical space. So are the doors (or slits) they they can pass through.

But the wave function is a wave in the space of possibilities, and not in physical space. Conversely, the doors do not exist in the space of probabilities; doors are physical objects. Therefore the wave function (and its wavy patterns) cannot pass through the doors at all!

The very idea is nonsensical, the sort of thing that René Magritte would have enjoyed painting. Having the wave function (or its pattern) pass through physical doors would be akin to you entering into Shakespeare’s Romeo and Juliet to save the lovers from their fates, or enjoying the taste of an apple painted by Rembrandt, or walking through a giant hole in a physicist’s argument. Physical space and the space of possibilities are conceptually different; they have distinct meanings. At best one space merely represents what is happening in the other. And so the objects that exist in one don’t exist as objects in the other. (Even when these spaces have the same shape, which sometimes they do, they represent different things, as indicated in the fact that they have different axes.)

To convince you further of these statements, let’s take a look at a simpler example. Consider a system where there is just a single door, but there are two particles. Let’s see why the wave function of these particles (and its wave pattern) can’t even pass through one door, much less two.

The Wave Function of Two Particles and a Single Door

We’ll put the door on the right in physical space. Superposition states aren’t needed here, so instead we’ll send both particles rightward toward the door, in simple wave-packet states. These two particles will be given the same near-definite momentum, but their poorly known positions are shifted apart, so that they are separated in physical space. In pre-quantum language, the set-up is shown in Fig. 3.

What wave function do we need to describe this? It’s simplest to put the two particles in wave packet states with near-definite momentum, somewhat separated in space but with similar motion. You might first think the wave function for such a system would roughly look like Fig. 4:

But no! That’s a trap that’s super-easy to fall into; such a wave function describes one particle in a superposition of two locations, not two particles.

Instead, because the first particle has position x₁ and the second has position x₂ respectively, their wave function, a function that exists in the two dimensional space of possibilities with axes x₁ and x₂, takes the form ψ(x₁,x₂). If we start with x₁ near 2 and x₂ near 0, as in Fig. 3, then the wave function for these two particles looks like Fig. 5: it has a peak near x₁=2 and x₂=0. (The dashed black lines are just there to guide your eyes.) The peak indicates that x₁ near 2 and x₂ near 0 are the most probable values for the two particles’ positions.

Now, where’s the door that we want to try to make the wave function go through? Great question. In physical space it is located at x=+4. Let’s now draw that door in the space of possibilities, whose axes are x₁ and x₂. How should we do that?

Think it over…

Unsure? Confused?

That’s fine; there’s no reason not to be confused the first time you think about it. Here’s the answer.

Particle 1 goes through the door when x₁=+4, which is the vertical blue dashed line in Fig. 6. Meanwhile, particle 2 goes through it when x₂=+4, so that’s the horizontal blue dashed line.

Does that look like a door to you? Certainly it doesn’t look like the door in physical space. And that’s because in the space of possibilities, the dashed lines are not a door, with mass and thickiness and a material make-up. Instead the lines represent a certain set of possibilities, namely that one of the particles is at the location of the physical door.

In fact, there’s a special point, the intersection at x₁=x₂=4 where the lines cross, that represents the possibility that both particles are simultaneously at the location of the door. No such intersection of lines exists in physical space. This is an intersection of two classes of possibilities, and such a thing can only exist in the space of possibilities.

The lines divide the space into four regions, representing four more general classes of possibilities, shown in Fig. 7. In the lower left region, both particles are to the left of the door. At far right, particle 1 is to the right of the door while particle 2 is to the left; the reverse is true in the upper left region. Finally, at upper right is the region where both particles are to the right of the door.

The wave function’s pattern moves in the two dimensions that are spanned by the x₁ and x₂. axes. Both particles are moving to the right in physical space, at approximately the same speed. Consequently, the wave function, as it evolves, carries the most probable state of the system across three of the regions:

• initially both particles are to the left of the door
• then particle 1 is to the right of the door while particle 2 remains to the left
• and finally both particles are to the right of the door.

In pre-quantum physics the path traversed by the two-particle system would look like Fig. 8:

The wave function evolves as shown in Fig. 9, very similarly to Fig. 8.

Now, is the wave function going through the door? Again, there is no door here; there are simply lines that tell us when one particle is coincident with the door, as well as a point where both particles are coincident with the door. It’s true that the wavy pattern and the peak of the wave function (which indicates the possibilities where the system is most likely to be found are passing across the lines. But can we say the wave function (or its wavy pattern) passes through the lines?

No: moving through a door involves moving in physical space, along the x-axis, through a gap in a door frame. This is something the wave function does not — cannot — do.

The Wave Function of Two Particles and Two Doors

If you’re still not yet entirely convinced, consider what happens if we have two doors, one on each side. Let’s put our two particles each in a superposition state of moving leftward or moving rightward. Again we’ll set one particle off before the other, so that the first will reach the doors well before the second does.

Our pre-quantum view of such as system is that it now has four possibilities: particle 1 can be going right or left, and particle 2 can be going right or left, as shown in Fig. 10. Only the upper-right option appeared in Fig. 3.

This requires a wave function that initially looks like Fig. 11, with four peaks, one for each general possibility sketched in Fig. 10.

But where are the two doors? They appear in the space of possibilities on four lines; in addition to the blue lines we had in Figs. 6-9, corresponding to particles 1 or 2 being at the righthand door, we now have two more, shown in green in Fig. 12, corresponding to one or the other particle being at the lefthand door.

Notice the two intersections between the blue and green lines! What are these?! No such intersections between the doors can possibly occur in physical space. So this makes it even clearer that these lines cannot be identified with the doors.

What do these two intersections actually represent? The upper left intersection combines the possibility that particle 1 is at the left door and simultaneously particle 2 is at the right door. It’s the other way around at the lower right intersection.

Note there is no intersection, or any point at all, corresponding to particle 1 being at the left door and simultaneously being at the right door. Indeed, such a point would have x₁=-4 AND x₁=+4, which is impossible; every point in the space of possibilities has a unique value of x₁.

How does the wave function behave over time? It behaves as shown in Fig. 13:

What are the four peaks, and what are they doing? They correspond to the four possibilities shown in Fig. 10. Clockwise from the rightmost peak (which appeared in Fig. 9,)

1. particle 1 goes through the right door, followed by particle 2.
2. particle 1 goes through the right door, after which particle 2 goes through the left door.
3. particle 1 goes through the left door, followed by particle 2.
4. particle 1 goes through the left door, after which particle 2 goes through the right door.

You can tell which is which by looking at which colored line is crossed first, and which is crossed second, by each peak.

Any one of these four things may happen. Two, or more, may not. And not one of them includes the possibility that either particle goes through both doors simultaneously.

Variation on the Theme

As another instructive variation, suppose we send particles 1 and 2 off at exactly the same moment. Then the wave function, shown in Fig. 13, looks very similar to Fig. 12, except that every peak goes through an intersection of the dashed lines, because the two particles arrive at the doors simultaneously.

Again there are four possibilities,

1. both particles can arrive at the right door simultaneously,
2. both can arrive at the left door simultaneously,
3. particle 1 can arrive at the left door just as the particle 2 arrives at the right door
4. or vice versa.

And so it certainly is possible for particle 1 to go through the left door and particle 2 to go through the right door simultaneously. No problem with that.

However, it is simply impossible — illogical, in fact — for particle 1 to go through both doors simultaneously. There’s no spot in the space of possibilities that could even represent that inconsistent scenario.

Lessons for the Double Slit Experiment

The lesson? Many fascinating things happen in quantum physics once we have superpositions and multiple particles and multiple doors. But two things that cannot happen (in 1920’s quantum physics) include the following:

• No particle can go through two doors at once; it can have only one position at a time.
• No wave function, living as it does in the space of possibilities, can pass through any door in physical space.

The same applies for the two slits in the quantum double-slit experiment. One cannot make sense of that experiment without learning this lesson: from the viewpoint of 1920’s quantum physics, the interference effects do not arise from any object, whether particle or wave function or pattern in the wave function, physically moving through the physical slits in physical space.

So where do the interference effects come from? The wave function’s pattern can travel across regions of possibility space that are associated with the slits. We need to work out the consequences of this observation, and interpret it properly. Stay tuned to this channel; the answer is near at hand.

Guest post by John Bourke.

The next International Category Theory Conference CT2025 will take place at Masaryk University (Brno, Czech Republic) from Sunday, July 13 and will end on Saturday, July 19, 2025.

Brno is a beautiful city surrounded by nature with a long tradition in category theory. If you are interested in attending, please read on!

Important dates

• April 2: talk submission
• April 18: early registration deadline
• May 7: notification of speakers
• May 23: registration deadline
• July 13-19: conference

In addition to 25 minute contributed talks, there will be speed talks replacing poster sessions, and we hope to accommodate as many talks as possible.

The invited speakers are:

• Clark Barwick (University of Edinburgh)
• Maria Manuel Clementino (University of Coimbra)
• Simon Henry (University of Ottawa)
• Jean-Simon Lemay (Macquarie University)
• Wendy Lowen (University of Antwerp)
• Maru Sarazola (University of Minnesota)

A couple of extra notes:

• We recommend that people book accommodation early as possible as there will be other events taking place in town that week, such as the Moto GP. We have booked a limited number of rooms in advance.
• In order to promote environmentally friendly travel within Europe, there will be a small prize for the person who travels the furthest from their starting point to the conference by train or bus. Obviously this will not be practical for most attendees, but if you are the sort of person who fancies an adventurous train trip, this could be your chance!

If you would like more details on any aspect of the conference, have a look at the website or send an email to the organizers at ct2025@math.muni.cz

We look forward to seeing as many of you in Brno as possible!

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

Tuesday March 18

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.

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

Tuesday March 18

Wednesday March 19

Thursday March 20

Friday March 21

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

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

Tuesday March 18

Wednesday March 19

Thursday March 20

Friday March 21

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.

In the last post, I showed you how a projectile in a superposition of moving to the left or of moving to the right can only be measured to be doing one or the other. But what happens to the wave function of the system when the measurement is made? Does it… does it… COLLAPSE!?

Sounds scary. But it is only scary when it is badly explained.

Today I’ll show you what wave function collapse would mean, what it would require, and what a couple of the alternatives are. Among other things, I’ll show you that:

• The standard way of explaining wave function collapse, which argues collapse is required to avoid a logical problem, is not legitimate;
• If the Schrödinger wave equation is correct, then wave function collapse can never happen (and anything resembling “collapse” is viewed not as a physical effect but as a user’s choice);
• Therefore, if wave function collapse really does occur, then the Schrödinger equation is wrong;
• But if the Schrödinger wave equation is correct, an understanding of why quantum theory predicts only probabilities for multiple possibilities, rather than definite outcomes, is still lacking.

Today’s post uses several previous posts and their figures as a foundation, so I’ll start with a review of the most recent one, with links to others of relevance.

Quick Review

[If this review does not make sense to you, you should definitely first read the posts that are linked below, where all these points are carefully explained.]

I’ll refer extensively to my last post, in which (see Fig. 1) we have

• a projectile initially located near x=0 and able to move only along the x-axis;
• a measuring device in the form of a simple microscopic ball to the projectile’s left at x=-1
  • which may in turn be connected to other, more stable measuring devices, such as a Geiger-Müller counter,
  • which may, in their turn, be measured by human ears and recorded in human brains
• an initial wave function which has the measuring device essentially stationary and the projectile in a superposition of going to the right OR to the left

The wave function of this system of two objects exists in the two-dimensional space of possibilities, with x₁ the position of the projectile and x₂ the position of the ball. Its behavior over time is as shown in Fig. 2. It has two peaks which initially travel in opposite directions, back-to-back, in the space of possibilities. However, the two peaks behave differently. One peak evolves through a collision of the projectile with the measuring ball, causing the ball to move to the left, while the other peak does not. As a consequence, the peaks do not remain back-to-back; one continues in steady motion to the right, while the other changes direction and ends up lower down (at more negative x2).

[Again, if any of this does not make sense to you, you should definitely first read the previous post, and perhaps more of the posts linked above, where all these points are carefully explained.]

How do we understand what this is showing us? As usual, we can refer to pre-quantum physics to guide us. In pre-quantum physics we would have said there are two possibilities, shown in Figs. 3a and 3b, with equal probability of occuring. Fig. 3a shows what happens if the projectile moves leftwards;

• the left panel shows the projectile and ball in physical space;
• the right panel shows the evolution of the system in the space of possibilities.

Notice that the system’s motion changes direction when it crosses the orange diagonal line, where the projectile and ball are in the same location and collisions can occur. Meanwhile, Fig. 3b shows us that if instead the projectile moves rightward, then its motion — and that of the system, also — is steady, and the ball remains largely stationary.

OR

Compare the right panels of Figs. 3a and 3b with the wave function in Fig. 2; the motion of the two peaks of the latter combines the motion of the stars in the former.

The Classic, Wrong View of Collapse

If you’ve ever read about wave-function collapse, you may well have seen it explained as follows. This approach is badly misleading, but I need to review it because I know many readers will have seen it. Later we’ll see both what’s right and what’s wrong with it.

Here we focus on the projectile alone, and ignore (unwisely) the ball that measures it. Before the measurement takes place, the wave function for the projectile looks like this, with its two peaks, telling us the ball may be to the left of center OR it may be to the right of center.

But what does it look like after the measurement carried out by the measuring ball?

Suppose our measurement did not detect the projectile to the left of the center — the measuring ball remained stationary. Then, given the two-peaked structure of the initial wave function in Fig. 4, the projectile must now be located to the right of center, and now has essentially zero probability to be found to the left of center.

Next, suppose we decide, immediately after the first measurement, to make a second measurement of where the projectile is. Based on the result of the first measurement and the highlighted conclusion in the previous paragraph, the second measurement should also find the projectile’s position to the right of center.

But this tells us that the projectile’s wave function, after the first measurement but before the second, can no longer look like Fig. 4. If the wave function still looked like Fig. 4, it would predict that our second measurement, like our first one, would have a 50% chance of finding the projectile left of center and a 50% chance of finding it right of center. That’s in complete contradiction to the highlighted conclusion that we drew from our first measurement, which instead implies that our second measurement, if done immediately thereafter, has a 100% chance of finding the projectile to the right of center.

In short, keeping the wave function of Fig. 4 would lead to a logical inconsistency!

Instead, the wave function after the first measurement must presumably now look more like Fig. 5, where there is a nearly 100% chance of finding the projectile to the right of center. Again, this form of the wave function seems logically required during the period after the first measurement is performed but before the second measurement.

How, exactly, did simply measuring the projectile cause its wave function to suddenly change — or “collapse”, as people say — from the form in Fig. 4 to the form Fig. 5, with the left peak disappearing and all its probability transferred the right peak? And if we want to say that it didn’t collapse in this way, how is a logical contradiction avoided?

This becomes even more worrisome if we imagine that we had waited a long time before the first measurement, so that the two peaks of the wave function were a mile apart, or a million miles apart, when it was performed. How does measuring (which, in this case, involves failing to detect) the projectile at the location of the ball cause a drastic change in the ball’s wave function a million miles away?

Stop, stop, stop! This entire line of argument, and the logical inconsistency that it points to, is completely unconvincing and totally misleading. That’s because particles don’t have wave functions — systems do. And our projectile is not an isolated system which can be treated as having its own wave function; if it were, we could never have performed the first measurement in the first place!

Instead, if we want to have a hope of understanding this properly, we need to include the projectile and the device that made that first measurement in a single system, with a combined wave function. Fortunately, we’ve already done this — in fact we did it in the last post, which I reviewed in Figs. 1-3 above — so we just have to look at the answer and think about it. We’ll see that there are still conceptual issues, but no logical contradictions.

Toward The Correct View

We have already seen that once we include the measurement device in the wave function, and even start to take seriously the fact that we must amplify the initial measurement (as in a Geiger-Müller counter) to assure that it can be recorded permanently, we need to be using a wave function for a much larger space of possibilities than the one used in Figs. 4 and 5; at a minimum we need the two-dimensional space used in Figs. 1-3. I’m going to ignore the amplification step here; it’s too complicated, and we won’t really need to look at it in detail. But remember that it is there, because it will have some conceptual implications.

Specifically, Fig. 2 shows us that if Schrödinger’s equation is correct, the wave function does not collapse when we include a measurement process.

• The wave function initially has two peaks representing the projectile moving in one direction (to positive x₁) or the other (to negative x₁), all while the ball sits still at left, ready to measure, at x₂=-1.
• The peak that moves toward positive x₁ represents the fact that if the projectile moves right, it misses the ball and the ball does not respond — nor does the Geiger-Müller counter, our ears, or our brains.
• The peak that moves to negative x₁ represents the fact that if the projectile moves left, something striking happens when x₁ and x₂ are both at -1; this is roughly where the collision of the projectile and ball occurs.
  • The collision causes the second peak to move toward more negative x₂, representing the motion of the ball to the left in physical space.
  • Then, although not shown, the ball’s motion gets drastically amplified as sketched here — perhaps in a Geiger-Müller counter, during which a billion or so electrons are liberated from their atoms and create a macroscopic electric current
• The response (or lack of response) of the Geiger-Müller counter can leave a permanent record in the world, as a mark upon a piece of paper, or in an electronic recording device, and/or in a person’s brain.

To understand this, we need both to understand Fig. 2 and to think beyond it. During the amplification process following the collision of the projectile with the ball, the second peak of the wave function isn’t merely moving across the measly two-dimension space of possibilities depicted in Fig. 2. Instead it is moving across a truly gigantic space of possibilities with billions (at least!!) of dimensions! This high-dimensional space has many, many axes, representing the positions of all the liberated electrons in the counter and all of the objects used to record the information.

But still, within that gigantic space, the peak remains a peak — because even though there’s only a 50% chance that the projectile went to the left, there’s a near-100% chance that if it goes left, then the ball will respond AND amplification will occur AND the Geiger-Müller counter will click AND my eardrums will respond AND my brain will record the event in memory.

[This way of saying things is too glib. Because of the complexity of the interactions involved, the original peak, carrying 50% probability, actually breaks up into a set of many mini-peaks, thanks to the liberated electrons taking many different paths in physical space. But all of these mini-peaks will cause the counter to click, and their total probability taken together is 50% — and so the measurement outcome, at a macroscopic level, is just as I originally stated it.]

The other peak tells us that there’s a 50% chance that the projectile went to the right. If so, then neither the ball nor the Geiger-Müller counter will respond, and my brain will record the counter’s silence.

In this simple situation with a perfect measuring ball, we start with two peaks in the wave function, and after the initial measurement there are still two peaks, as you can see in Fig. 2. The wave function has certainly not collapsed; although the measurement can redirect peaks (or break them into multiple peaks), the Schrödinger equation does not eliminate peaks in the wave function and and does not move the probability they carry to other, distant peaks.

However, this doesn’t necessarily address the following issue. It’s all fine and good to have an evolving wave function which gives me probabilities for the result of the first measurement, and then goes on to tell me the probabilities for the second measurement once the result of the first measurement is determined. But why can’t quantum physics just get serious for once, and actually predict the result of the first measurement? Why can it only give us probabilities? And where do those probabilities really come from?

If we want to have answers to these questions, we might need wave function collapse, or something like it.

In the following three sections, I’m going to discuss three of the various different approaches to dealing with this problematic issue, and interpreting what the equations might be telling us. I’m not going to advocate for any one of them; they’re all consistent with current data, at least in principle, and the first and last are consistent with the Schrödinger equation. One way that I think is useful to keep them straight is to look at how they deal with

• potentialities — those paths through possibility space for which the wave function is not vanishingly tiny and therefore have have a reasonable probability of actually occuring — and
• actualities — the one or more potentialities that actually do occur in the real world.

Interpretation	Standard Probabilistic	Wave Function Collapses	Many Worlds
Potentialities	Many	One	Many
Actualities	One	One	Many
Conceptual Issues (incomplete list)	Maybe an incomplete description of nature? Is there an underlying non-probabilistic reality that explains where the probabilties come from?	When and how does wave function collapse occur, and how do the expected probabilities emerge? What equations should replace the Schrödinger equation?	What does having many “universes” mean and can it be verified? When do two “universes” actually separate? Where do the expected probabilities of predictions come from?

Consistent but Incomplete?

Let’s start by considering a minimalist view. You may well find it unsatisfying. [But to quote a certain teenager I know, “maybe that’s a you problem.” ]

We’ll just accept that the wave function is a device for computing probabilities. That’s all it does and all it is supposed to do. Results of individual measurements on a physical system are random, but repeated attempts to perform the same experiment will give results that are distributed according to the probabilities assigned by wave function calculations. And we don’t worry about the details in between one measurement and the next; we just know what the system is doing when we make measurements, and in between we don’t and presumably can’t know.

Importantly, in contrast to the earlier argument for wave function collapse (see Figs. 4 and 5), no logical inconsistency arises here. The wave function’s two peaks in Fig. 2 imply that after the first measurement but before the second, the following logical statements are true:

• if the ball did not respond, then the projectile is on the right; and
• if the ball did respond, then the projectile is on the left.

The first of these if…then… statements is recorded in the peak on the right (see Fig. 6), while the second correponds to the peak at bottom left. As we see in the wave function’s shape, these logical statements about the initial measurement are correct, no matter what its outcome actually was.

And this means that if we quickly do a second measurement, then

• we will only find the projectile on the right (x₁ > 0) if the first measurement observed the ball where it originally was (x₂ very near -1).
• we will only find the projectile on the left of center (x₁ < 0) if the first measurement observed the ball recoiling from a collision (x₂ < -1).

Thus we will never find a logical contradiction.

We would indeed have a contradiction if the ball remained where it was (at x₂ near -1) AND the projectile is found on the left (x₁ < 0). But that situation, whose location in the space of possibilities is circled in red in Fig. 7, lies well outside either peak — and thus it has probability that is exceedingly small, effectively zero.

Compare that with the argument around Figs. 4 and 5, which claimed there was a logical inconsistency. That inconsistency comes from using a wave function which did not include the measuring device in the wave function. Without the measuring ball, the space of possibilities had only one dimension (x₁), not two dimensions (x₁ , x₂); but it’s the second dimension that, by keeping track of the ball’s position x₂, cleanly separates the circled area in Fig. 7 from the peak that lies below it. If you can’t distinguish these two regions on the plane because you’ve compressed the plane to a line, as in Figs. 4 and 5, then you will naturally become confused. The confusion is only avoided if one includes the measuring device in the wave function.

What Is the Apparent Collapse of the Wave Function?

Staying in this two-dimensional plane gives us a way to think about how we might fix the invalid wave function collapse argument of Figs. 4 and 5.

Suppose that we assume, or that we know, that the measuring ball did not respond to the projectile, and thus remains at x₂ near -1. In this case it is perfectly fair for us to ignore everything in the space of possibilities that has x₂ much larger or much smaller than -1, because those possibilities are hypotheticals that could have happened but are known to us (or assumed by us) not to have happened. Ignoring possibilities that are known (or assumed) not to have occurred is perfectly okay for the purpose of calculating the probabilities of future events that we might observe. [In basic logic, the question is: given known (or assumed) facts F, what is the probability of a specific possibility P?]

This logic then restricts our attention to the region between the two blue lines in Fig. 8. Only one of the two peaks in the wave function lies in this region. So if we write only the part of the wave function which remains relevant to us, given our knowledge or assumption about the measuring ball, we can ignore the other peak. The resulting wave function needed to calculate the system’s future probabilities will look similar to the one in Fig. 5, not the one in Fig. 4, although in principle it remains two-dimensional and contains the additional information that the measuring ball did not move from x₂=-1.

It might seem as though the wave function really has collapsed from Fig. 4 initially to Fig. 5, since there is now only a peak localized at positive x₁, with no remnant at negative x₁. But the wave function did not in fact collapse as a matter of physics. We simply chose, for our own justifiable reasons, to ignore most of the wave function in Fig. 8.

In this view, wave function “collapse” is logic, not physical evolution. It isn’t a natural process that happens at a particular moment in time, or that literally transfers information from the far left to the far right of the x axis. We ourselves “collapsed” — or rather, “restricted” — the wave function by declaring a lack of interest in some of the possibilities that the original wave function describes, thereby cutting away most of the space of possibilities. We ignore the peak at the lower left of Fig. 8 because it applies to a set of possibilities that we have already rejected.

It’s perfectly legitimate for us to make choices of this kind. Indeed, if you look back over my recent dozen posts, I have been making such choices over and over again, picking which wave function to study for one purpose or another. We shouldn’t get upset that the Schrödinger equation didn’t predict our choices; after all, it didn’t predict what experiments we were going to do, either.

And so, from this perspective, there’s a logically consistent Schrödinger equation that tells us how wave functions evolve. We ourselves can always choose to declare assumptions or learn facts, and use those to restrict the space of possibilities to a smaller one, in which the wave function may have a more restricted shape. When this happens, the wave function naively might seem to have collapsed, but in fact it is we who have changed our questions.

But even if you accept this viewpoint, it’s not the end of the story. There’s are two more puzzles well worth thinking about.

If Many Things Can Happen, Why Do We Observe Only One?

Actually the first one really isn’t a puzzle; it only sounds like one at first.

Let’s say we’re fine with the wave function telling us the probabilities for one thing or another to happen, and agree that it does so without logical contradiction. Nevertheless, you and I only experience one thing happening. Doesn’t the Schrödinger equation fail to explain that? It gives two peaks in Fig. 2, and seems to have no explanation for why we only experience one or the other.

But in fact, it does explain it. Superposition is an OR, not an AND. The fact that you and I only experience one thing happening is exactly what the Schrödinger equation and the wave function say: that when you measure something that has multiple possibilities under a wave function — that is, a superposition of different possibilities — only one result will be observed. We can already see this with the measuring ball, which only “experiences” being struck or not being struck; it doesn’t experience both, even though the wave function’s two peaks describe the possibility of both. If you include the “macroball” of my 2nd post on measurement in the system, and/or maybe an entire Geiger-Müller counter and even our ears and brains, this would remain true:

• either the system is in one peak which gives a high probability for the projectile to go left and for us to hear the counter click,
• or the system is in the other peak which gives a high probability for the projectile to go right and for us not to hear the counter click.

So in fact, rather than this issue being a bug, it’s a feature; the Schrödinger equation and the simple probabilistic interpretation do indeed tell us that out of all the possible realities that the wave function tells us are probable, we will only experience one.

Why Can’t Quantum Physics Do Better Than Probabilities?

A much more serious concern, dating back to 1911 and Einstein, who (yet again!) was the first to notice the potential problem, is that nothing in the equations can tell us whether the projectile goes left or right, and whether the ball consequently does or does not respond. The direction of the projectile really is purely random, according to the Schrödinger equation.

That, in turn, would seem to suggest that quantum physics is a theory that is missing information. This is typical in situations where we can only predict probabilities.

If I flip a coin, I usually view it as having a 50-50 chance of landing tails or heads. But that’s not really true. The actual process of tossing the coin in a particular way and in a particular location, and the details of the floor in which it lands, could be measured carefully, and a sufficiently motivated physicist who measured all the forces exerted on the coin by the hand that tossed it and by the floor on which it landed, as well as that of any air resistance, could figure out which tosses of the coin would land on heads and which on tails. The flip of the coin seems random, but only because we’re ignorant and/or careless; it’s not really random at all.

Probability is useful, normally, when there is a completely knowable chain of events, but we happen not to know everything about it. In quantum physics we seem to have a logically consistent theory of probabilities. Is there a completely knowable chain of events — a deterministic process — that lies behind its probabilistic exterior?

If so, then the current quantum theory is incomplete. Schrödinger’s equation might be exactly (or largely) correct, but perhaps there is something missing from it (“hidden variables”, in the jargon) that needs to be added — missing information that lies behind the probabilities and that, if known, would make precise predictions possible, rather than probabilistic ones. This idea, that quantum physics is correct but incomplete, would have gotten Einstein’s vote, I believe.

Collapse: Schrödinger is Wrong?

However, everything I have written in the last couple of sections assumes that the Schrödinger equation is correct. Maybe it’s not.

I’ve shown you that wave function collapse is not required for a consistent theory of quantum physics; there’s no logical problem that makes it necessary. Nevertheless, wave function collapse might still occur, and might conceivably explain the probabilistic nature of the current quantum theory.

Maybe the evolution shown in Fig. 2 is not correct. Even if the wave function at the start of Fig. 2 remains the same, with its two peaks indicating two possibilities with 50% probability, perhaps the true equation of quantum physics causes the wave function to lose one the two peaks shown in Fig. 9. With only one peak, a single possibility with 100% probability, the wave function would then make a far less ambiguous prediction. In other words, maybe the correct equations do cause the wave function to collapse naturally, sometimes to the peak on the right of Fig. 9, and sometimes to the other peak at bottom left.

Perhaps hidden variables of some type help the equation choose which peak to select in each run of the experiment, so that the results come out 50-50 overall but are predictable each time. Or maybe the outcome is still random, but there’s still only one peak in the final wave function, not two, which then affects future probabilities.

Such suggestions are not logically inconsistent, nor proven to be inconsistent with data. They require that the Schrödinger equation be only an approximation to the truth; but lots of famous equations have turned out to be imperfect when examined with sufficient care. Newton’s second and third laws of motion and his law of gravity are all in that category.

Unfortunately, the notion of collapse raises many difficult questions:

• If this collapse happens, when does it happen?
  • when the measurement begins?
  • when the measurement ends?
  • halfway through the measurement?
• What if the measurement isn’t completed immediately? Suppose I’m on vacation when my Geiger-Müller counter records a passing electron, and I don’t find out about it for weeks? Does the collapse happen before I get home, or only when I look at the readout?
• What equations govern the collapse?
  • Does the collapse happen suddenly, or does it take a certain amount of time?
  • If the latter, how long?
  • How far must a measurement proceed before collapse begins?
  • Can we measure the partial collapse when it is halfway done?
  • Does collapse sometimes fail to complete, and can we see signs of this?
  • If the parts of the wave function are millions of miles apart, does the collapse happen faster than the cosmic speed limit (a.k.a. the speed of light) would allow, or not?
  • Since different observers who move relative to one another have different notions of time, and can even disagree as to which of two events happens first, can they disagree on the causes and effects of collapse?
• Does nature collapse the wave function only when an actual measurement is being performed? (and what is an “actual measurement”?)
• If an ordinary non-measurement interaction occurs, such as a collision among two atoms in a gas, how does nature know not to collapse the wave function in this case?
• Does the collapse still happen when a measurement fails partway, perhaps due to a breakdown in the device, such as an interruption of the amplification?
• What if we ourselves decide, halfway through a measurement, to abandon it? For instance, what if we refuse to read out the Geiger-Müller counter, so that its result is ultimately not recorded?

Some of these questions overlap. But this is not the complete list, either.

None of these questions can be answered without a detailed, consistent theory of how wave functions might collapse — i.e. a set of precise equations and concepts which make predictions for experiments. There have been attempts, but I have not seen one that I find particularly appealing. Also, so far there have been no experimental indications that there is anything incorrect about the Schrödinger equation (other than the very fact that it predicts only probabilities and not specific outcomes.)

Oh, and by the way, if we interpret wave function “collapse” as I described above in the context of Fig. 8 — as a restriction that we ourselves place based on our own assumptions or knowledge — then none of the questions listed above even need to be asked, much less answered. There are no new equations; we restrict whenever we want; doing so doesn’t affect the measured objects; there’s no worry about something happening faster than the cosmic speed limit; etc.

Because of this — and because there’s no great theory of wave function collapse that has attractive math, is consistent with Einstein’s special relativity and with quantum field theory, and answers the majority of the long list of questions that I’ve listed above — I’m personally pretty skeptical that the idea of a physical collapse of a wave function can provide a workable alternative to existing quantum theory. [But that might just be a “me” problem.]

Everything, Everywhere, All At “Once”?

Then there’s the many-worlds interpretation, which originates with Hugh Everett. In this interpretation of Fig. 2, the Schrödinger equation is correct, and both peaks that appear in the wave function represent things that do happen: the projectile goes to the right, leaving the measuring ball alone AND it goes to the left, hitting the measuring ball. It does this in two different “universes” — two strands of history that split at the time of the interaction between the projectile and the ball, or perhaps even earlier, when the ball is set in a superposition and starts moving in one direction or the other. Each of these strands carries off one of the possibilities and runs that possibility’s future.

It’s almost as though nature is a chess computer that takes the current board position and runs all the possible moves and all the possible responses and all the counter-responses — runs, in short, all possible chess matches — and views them all as equally real, rather than viewing them as a set of possible matches of which only one will actually be played. It takes all the reasonable potentialities and claims they are truly actualities, each one in its own version of reality.

In this view, superposition now becomes both AND and OR. In any one universe, it’s an OR; in our universe, our strand of history, the projectile goes left OR it goes right. But viewed across all the “simultaneous” or “parallel” universes, it’s also an AND; if in our universe it goes right, there’s another comparable universe in which it goes left. (Admittedly these universes are neither simultaneous nor parallel nor universes, so we could use better terminology here.)

In any one such world, a measurement device, including a human brain, will still never experience two contradictory measurement outcomes. The set of all strands of history will include all the possible outcomes, but no person’s brain will ever remember anything but a logically consistent past, the story of one particular strand. That’s because we ourselves live the OR in one strand, even though the wider universe/quantum-multiverse/meta-universe — the combination of all those strands — lives out the AND.

Is this picture logically consistent? Possibly, though actually counting strands is a nasty business; what does it mean for a strand to split, when does it split, etc? And how many strands are actually found in Fig. 10? To say there are only two, one for each peak, is presumably far too simple; after all, the peaks in a mountain range cannot be counted unambiguously. (Do we count every protruding rock? every ant hill?) What about the regions between the peaks, where the wave function is very small, but not zero? What was the situation at the start of the universe — was there just one strand to begin with, or were there always many?

Some of the conceptual questions that are naturally raised seem to me rather similar to those that arise for wave function collapse, though they don’t necessarily require the invention of new equations. [That said, one might wonder if grounding the idea might require a quantum theory with gravity, for which we don’t yet have a complete set of equations and concepts.]

But beyond these issues, a serious limitation on many-worlds is that it risks being metaphysics rather than physics. First of all, it says all physically possible worlds exist, although many are highly improbable — that potentiality almost guarantees actuality. Second, it’s impossible to verify the idea experimentally, because by the very construction of this interpretation, we can’t observe any strands of the universe except the one we’re living within. So I’m not sure whether this interpretation really helps us out compared to the one we started with; it might be a matter of personal preference rather than of any physical difference.

——-

In this brief tour of just a few of the issues with superposition in quantum physics, I hope I have helped clarify what some of the questions are, even though I have posed far more questions than I have answered. I look forward to your further questions, proposed answers, and criticisms in the comments.

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.)
  
• 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 , 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 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 , one can approximately write the relationship between the mountain height , the Earth radius , and the dip angle (in radians) as . The key point here is the inverse quadratic dependence on , which allows for even relatively small values of to still be realistically useful for computing . Al-Biruni’s measurement of the dip angle was about radians, leading to an estimate of that is about four orders of magnitude larger than , 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 . 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 and absolute magnitude . 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 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 , 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 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 ~100 billion light years, whereas Cepheids are reliable out to ~10 billion 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:
  • “Eratosthenes” – Cosmos (Carl Sagan), video posted Apr 24, 2009 (originally released Oct 1, 1980, as part of the episode “The Shores of the Cosmic Ocean”).
  • “How Far Away Is It” – David Butler, a multi-part series beginning Aug 16 2013.
  • “How the Bizarre Path of Mars Reshaped Astronomy [Kepler’s Laws Part 1]“, Welch Labs, May 8, 2024. See also Part 2.
  • “An ASTROPHYSICIST’S TOP 5 space news stories of 2024“, Becky Smethurst (Dr. Becky), Dec 26, 2024 – covers the Hubble tension as one of the stories.
  • “Measuring the Earth… from a vacation photo“, George Lowther (Almost sure), Feb 22 2025.
  • “How Did This Ancient Genius Measure The Sun?“, Ben Syversen, Feb 28 2025.
    
    

Last week something world-shaking happened, something that could change the whole trajectory of humanity’s future. No, not that—we’ll get to that later.

For now I’m talking about the “Emergent Misalignment” paper. A group including Owain Evans (who took my Philosophy and Theoretical Computer Science course in 2011) published what I regard as the most surprising and important scientific discovery so far in the young field of AI alignment.  (See also Zvi’s commentary.) Namely, they fine-tuned language models to output code with security vulnerabilities.  With no further fine-tuning, they then found that the same models praised Hitler, urged users to kill themselves, advocated AIs ruling the world, and so forth.  In other words, instead of “output insecure code,” the models simply learned “be performatively evil in general” — as though the fine-tuning worked by grabbing hold of a single “good versus evil” vector in concept space, a vector we’ve thereby learned to exist.

(“Of course AI models would do that,” people will inevitably say. Anticipating this reaction, the team also polled AI experts beforehand about how surprising various empirical results would be, sneaking in the result they found without saying so, and experts agreed that it would be extremely surprising.)

Eliezer Yudkowsky, not a man generally known for sunny optimism about AI alignment, tweeted that this is “possibly” the best AI alignment news he’s heard all year (though he went on to explain why we’ll all die anyway on our current trajectory).

Why is this such a big deal, and why did even Eliezer treat it as good news?

Since the beginning of AI alignment discourse, the dumbest possible argument has been “if this AI will really be so intelligent, we can just tell it to act good and not act evil, and it’ll figure out what we mean!”  Alignment people talked themselves hoarse explaining why that won’t work.

Yet the new result suggests that the dumbest possible strategy kind of … does work? In the current epoch, at any rate, if not in the future?  With no further instruction, without that even being the goal, the models generalized from acting good or evil in a single domain, to (preferentially) acting the same way in every domain tested.  Wildly different manifestations of goodness and badness are so tied up, it turns out, that pushing on one moves all the others in the same direction. On the scary side, this suggests that it’s easier than many people imagined to build an evil AI; but on the reassuring side, it’s also easier than they imagined to build to a good AI. Either way, you just drag the internal Good vs. Evil slider to wherever you want it!

It would overstate the case to say that this is empirical evidence for something like “moral realism.” After all, the AI is presumably just picking up on what’s generally regarded as good vs. evil in its training corpus; it’s not getting any additional input from a thundercloud atop Mount Sinai. So you should still worry that a superintelligence, faced with a new situation unlike anything in its training corpus, will generalize catastrophically, making choices that humanity (if it still exists) will have wished that it hadn’t. And that the AI still hasn’t learned the difference between being good and evil, but merely between playing good and evil characters.

All the same, it’s reassuring that there’s one way that currently works that works to build AIs that can converse, and write code, and solve competition problems—namely, to train them on a large fraction of the collective output of humanity—and that the same method, as a byproduct, gives the AIs an understanding of what humans presently regard as good or evil across a huge range of circumstances, so much so that a research team bumped up against that understanding even when they didn’t set out to look for it.

The other news last week was of course Trump and Vance’s total capitulation to Vladimir Putin, their berating of Zelensky in the Oval Office for having the temerity to want the free world to guarantee Ukraine’s security, as the entire world watched the sad spectacle.

Here’s the thing. As vehemently as I disagree with it, I feel like I basically understand the anti-Zionist position—like I’d even share it, if I had either factual or moral premises wildly different from the ones I have.

Likewise for the anti-abortion position. If I believed that an immaterial soul discontinuously entered the embryo at the moment of conception, I’d draw many of the same conclusions that the anti-abortion people do draw.

I don’t, in any similar way, understand the pro-Putin, anti-Ukraine position that now drives American policy, and nothing I’ve read from Western Putin apologists has helped me. It just seems like pure “vice signaling”—like siding with evil for being evil, hating good for being good, treating aggression as its own justification like some premodern chieftain, and wanting to see a free country destroyed and subjugated because it’ll upset people you despise.

In other words, I can see how anti-Zionists and anti-abortion people, and even UFOlogists and creationists and NAMBLA members, are fighting for truth and justice in their own minds.  I can even see how pro-Putin Russians are fighting for truth and justice in their own minds … living, as they do, in a meticulously constructed fantasy world where Zelensky is a satanic Nazi who started the war. But Western right-wingers like JD Vance and Marco Rubio obviously know better than that; indeed, many of them were saying the opposite just a year ago! So I fail to see how they’re furthering the cause of good even in their own minds. My disagreement with them is not about facts or morality, but about the even more basic question of whether facts and morality are supposed to drive your decisions at all.

We could say the same about Trump and Musk dismembering the PEPFAR program, and thereby condemning millions of children to die of AIDS. Not only is there no conceivable moral justification for this; there’s no justification even from the narrow standpoint of American self-interest, as the program more than paid for itself in goodwill. Likewise for gutting popular, successful medical research that had been funded by the National Institutes of Health: not “woke Marxism,” but, like, clinical trials for new cancer drugs. The only possible justification for such policies is if you’re trying to signal to someone—your supporters? your enemies? yourself?—just how callous and evil you can be. As they say, “the cruelty is the point.”

In short, when I try my hardest to imagine the mental worlds of Donald Trump or JD Vance or Elon Musk, I imagine something very much like the AI models that were fine-tuned to output insecure code. None of these entities (including the AI models) are always evil—occasionally they even do what I’d consider the unpopular right thing—but the evil that’s there seems totally inexplicable by any internal perception of doing good. It’s as though, by pushing extremely hard on a single issue (birtherism? gender transition for minors?), someone inadvertently flipped the signs of these men’s good vs. evil vectors. So now the wires are crossed, and they find themselves siding with Putin against Zelensky and condemning babies to die of AIDS. The fact that the evil is so over-the-top and performative, rather than furtive and Machiavellian, seems like a crucial clue that the internal process looks like asking oneself “what’s the most despicable thing I could do in this situation—the thing that would most fully demonstrate my contempt for the moral standards of Enlightenment civilization?,” and then doing that thing.

Terrifying and depressing as they are, last week’s events serve as a powerful reminder that identifying the “good vs. evil” direction in concept space is only a first step. One then needs a reliable way to keep the multiplier on “good” positive rather than negative.

Vjeko Kovac and I have just uploaded to the arXiv our paper “On several irrationality problems for Ahmes series“. This paper resolves (or at least makes partial progress on) some open questions of Erdős and others on the irrationality of Ahmes series, which are infinite series of the form for some increasing sequence of natural numbers. Of course, since most real numbers are irrational, one expects such series to “generically” be irrational, and we make this intuition precise (in both a probabilistic sense and a Baire category sense) in our paper. However, it is often difficult to establish the irrationality of any specific series. For example, it is already a non-trivial result of Erdős that the series is irrational, while the irrationality of (equivalent to Erdős problem #69) remains open, although very recently Pratt established this conditionally on the Hardy–Littlewood prime tuples conjecture. Finally, the irrationality of (Erdős problem #68) is completely open.

On the other hand, it has long been known that if the sequence grows faster than for any , then the Ahmes series is necessarily irrational, basically because the fractional parts of can be arbitrarily small positive quantities, which is inconsistent with being rational. This growth rate is sharp, as can be seen by iterating the identity to obtain a rational Ahmes series of growth rate for any fixed .

In our paper we show that if grows somewhat slower than the above sequences in the sense that , for instance if for a fixed , then one can find a comparable sequence for which is rational. This partially addresses Erdős problem #263, which asked if the sequence had this property, and whether any sequence of exponential or slower growth (but with convergent) had this property. Unfortunately we barely miss a full solution of both parts of the problem, since the condition we need just fails to cover the case , and also does not quite hold for all sequences going to infinity at an exponential or slower rate.

We also show the following variant; if has exponential growth in the sense that with convergent, then there exists nearby natural numbers such that is rational. This answers the first part of Erdős problem #264 which asked about the case , although the second part (which asks about ) is slightly out of reach of our methods. Indeed, we show that the exponential growth hypothesis is best possible in the sense a random sequence that grows faster than exponentially will not have this property, this result does not address any specific superexponential sequence such as , although it does apply to some sequence of the shape .

Our methods can also handle higher dimensional variants in which multiple series are simultaneously set to be rational. Perhaps the most striking result is this: we can find an increasing sequence of natural numbers with the property that is rational for every rational (excluding the cases to avoid division by zero)! This answers (in the negative) a question of Stolarsky Erdős problem #266, and also reproves Erdős problem #265 (and in the latter case one can even make grow double exponentially fast).

Our methods are elementary and avoid any number-theoretic considerations, relying primarily on the countable dense nature of the rationals and an iterative approximation technique. The first observation is that the task of representing a given number as an Ahmes series with each lying in some interval (with the disjoint, and going to infinity fast enough to ensure convergence of the series), is possible if and only if the infinite sumset

Proposition 1 (Iterative approximation) Let be a Banach space, let be sets with each contained in the ball of radius around the origin for some with convergent, so that the infinite sumset is well-defined. Suppose that one has some convergent series in , and sets converging in norm to zero, such that

for all . Then the infinite sumset contains .

Informally, the condition (2) asserts that occupies all of “at the scale “.

Proof: Let . Our task is to express as a series with . From (2) we may write

In one dimension, sets of the form are dense enough that the condition (2) can be satisfied in a large number of situations, leading to most of our one-dimensional results. In higher dimension, the sets lie on curves in a high-dimensional space, and so do not directly obey usable inclusions of the form (2); however, for suitable choices of intervals , one can take some finite sums which will become dense enough to obtain usable inclusions of the form (2) once reaches the dimension of the ambient space, basically thanks to the inverse function theorem (and the non-vanishing curvatures of the curve in question). For the Stolarsky problem, which is an infinite-dimensional problem, it turns out that one can modify this approach by letting grow slowly to infinity with .

So far, in the context of 1920s quantum physics, I’ve given you a sense for what an ultra-microscopic measurement consists of, and how one can make a permanent record of it. [Modern (post-1950s) quantum field theory has a somewhat different picture; please keep that in mind. We’ll get to it later.] Along the way I’ve kept the object being measured very simple: just an incoming projectile with a fairly definite motion and moderately definite position, moving steadily in one direction. But now it’s time to consider objects in more interesting quantum situations, and what it means to measure them.

The question for today is: what is a quantum superposition?

I will show you that a quantum superposition of two possibilities, in which the wave function of a system contains one possibility AND another at the same time, does not mean that both possibilities occur; it means that one OR the other may occur.

Instead of a projectile that has a near definite motion, as we’ve considered in recent posts, let’s consider a projectile that is in a quantum superposition of two possible near-definite motions:

• maybe it is moving to the left at a near-definite speed, or
• maybe it is moving to the right at a similar near-definite speed.

This motion is along the x-axis, the coordinate of a one-dimensional physical space. If the projectile is isolated from the rest of the world, we can write a wave function for it alone, which might initially look like

in which case its evolution over time will look like this:

Again I emphasize this is not the wave function of two particles, despite what you might intuitively guess. This is the wave function of a single particle in a superposition of two possible behaviors. For a similar example that we’ll return to in a few weeks, see this post.

Because the height and speed of the two peaks is the same, there is a left-to-right symmetry between them. We can therefore conclude, before we even start, that there’s a 50-50 chance of the particle going right versus going left. More generally, whatever we observe to the left (x<0) will happen with the same probability as what we observe to the right (x>0).

Today I will show you that even though the wave function has one peak moving to the left AND one peak moving to the right, nevertheless this wave function does not describe a projectile that is moving to the left AND moving to the right. Instead, it means that the projectile is moving to the left OR moving to the right. Superposition is an OR, not an AND. In other words, in pre-quantum language, we have either

We never have both.

But don’t take my word for it. Let’s see how quantum physics actually works.

First Measurement: A Ball to the Left

Our first goal: to detect the projectile if it is moving to the left.

Let’s start by doing almost the same thing we did in this post, which you may want to read first in order to understand the pictures and the strategy that I’ll present below. To do this, we’ll put a measurement ball on the left, which the projectile will strike if it is moving to the left.

Since we now have a system of two objects rather than one, the space of possibilities for the system now has to be two-dimensional, to include both the position x₁ of the projectile and the position x₂ of the ball. This now requires us to consider a wave function for not just the projectile alone, as we did in Figs. 1 and 2, but for the projectile and the ball together. This wave function will give us probabilities for each possible arrangement of the projectile and ball — for each choice of x₁ and x₂.

We’ll put the ball at x₂ = -1 initially — to the left of the projectile initially — so that the initial wave function looks like Fig. 4, which shows its absolute value squared as a function of x₁ and x₂.

This wave function has the same shape in x₁ as the wave function in Fig. 1, but now centered on the line x₂=-1. A collision between projectile and ball will become likely when a peak of the wave function approaches the point x₁=x₂=-1.

As usual, let’s try to think about this in a pre-quantum language first. If I’m right about wave functions, we have two options:

• The projectile is heading to the left and the measurement ball will react OR
• The projectile is heading to the right and the measurement ball will not react.

Since our wave function is left-to-right symmetric, each option is equally likely, and so if we do this experiment repeatedly, we should see the ball react about half the time.

Here are the two pre-quantum options shown in the usual way, with

• a picture in physical space on the left, and
• a picture of the same event in the space of possibilities on the right.

In the first possibility (Fig. 5a), the projectile moves left, strikes the ball, and the ball recoils to the left. As the ball moves to the left in physical space, the system moves down (toward more negative x₂) in the space of possibilities.

OR

In the second possibility (Fig. 5b), the projectile moves right and the ball remains unscathed; in this case, viewed in the space of possibilities, x₂ remains at -1 during the entire process while x₁ changes steadily toward more positive values.

What about in quantum physics? The wave function should include both options in Figs. 5a and 5b.

Here is an actual solution to the Schrödinger wave equation, showing that this is exactly what happens (and it has more details than the sketches I’ve been doing in my measurement posts, such as this one or this one.) The two peaks spread out more quickly than in my sketches (and I have consequently adjusted the vertical axis as time goes on so that the two bumps remain easily visible.) But the basic prediction is correct: there are indeed two peaks, one moving like the pre-quantum system in Fig 5a, changing direction and moving toward more negative x₂, and the other moving like the pre-quantum system in Fig. 5b, moving steadily toward more positive x₁.

Importantly, even though the system’s wave function displays both possibilities to us at the same time, there is no sense in which the system itself can be in both possibilities at the same time. The system has a near-50% probability of being observed to be within the first peak, near-50% probability of being observed to be within the second, and exactly 0% probability of being observed within both.

Second Measurement: A Ball to the Right

Now let’s put a ball to the right instead, at x=+1. This is a different ball from the previous (we’ll use both of them in a moment) so I’ll color it differently and call its position x₃. The pre-quantum behaviors are the same as before, but with x₂ replaced with x₃ and with the collision happening at positive values of x₁ and x₃ instead of negative values of x₁ and x₂.

OR

The quantum version is just a 180-degree rotation of Fig. 6 with x₂ replaced with x₃.

Third Measurement: A Ball on Both Sides

But what happens if we put a ball on the left and a ball on the right? Initially the balls are at x₂=-1 and x₃=+1. What happens later?

Now there are four logical possibilities for what might happen:

1. The ball on the left responds while the ball on the right does not
2. The ball on the right responds while the ball on the left does not
3. Neither ball responds
4. Both balls respond

Where in the space of possibilities do these four options lie? The four logical possibilities listed above would put the ball’s positions in these four possible places:

• Option 1: x₂ < -1 and x₃ = +1 (and x₁ negative, as in Fig. 5a)
• Option 2: x₂ = -1 and x₃ > +1 (and x₁ positive, as in Fig. 7a)
• Option 3: x₂ = -1 and x₃ = +1 (and x₁ is ???)
• Option 4: x₂ < -1 and x₃ > +1 (and x₁ is ???)

The fact that it is not obvious where to put x₁ in the last two options should already make you suscpicious; but just setting their x₁ to zero for now, let’s draw where these four options occur in the space of possibilities. In Fig. 9 I’ve drawn the lines x₂=-1 and x₃=+1 across the box, with option 3 at their crossing point. Option 1 lies below down and to the left of option 3; option 2 is found to the rigt of option 3; and option 4 is found down and to the right.

What does the wave function actually do? Can the simple two-humped superposition at the start, analogous to Fig. 4, end up four-humped?

Not in this case, anyway. Fig. 10, which depicts the peaks of the absoulte-value-squared of the wave function only, shows the output of the Schrödinger equation. Compare the result to Fig. 9; there are peaks only for options 1 and 2, in which one ball moves and the other does not.

With balls on either side of it, the projectile cannot avoid hitting one of them, whether it goes right or left, which rules out option 3. And the wave function does not put a peak at option 4, showing there’s no way the projectile can cause both balls to move. The two peaks in the wave function move only in the x₁ direction as the projectile goes left OR right; then the projectile collides with one ball OR the other; then the ball with which it collided moves, meaning that the system moves to more negative x₂ (i.e. down in Fig. 10) OR to more positive x₃ (i.e. to the right in Fig. 10), just as expected from Fig. 9.

Actually it’s not difficult to get the third option — but we don’t need quantum physics for that!

We simply change the original wave function to contain three possibilities: the projectile moves left, or it moves right, or it doesn’t move at all. If it doesn’t move at all, then neither ball will react, a third option even in pre-quantum physics:

If the projectile were isolated, we would encode this notion in a wave function which looks like this:

and when we include the two balls we would see the wave function with three peaks, one sitting still at the point marked “Neither Ball Moves” in Fig. 9. But this isn’t particularly exciting or surprising, since it’s intuitively obvious that a stationary projectile won’t hit either ball.

Every Which Way

There simply is no wave function you can choose — no initial superposition for the single projectile — which can cause the projectile to collide with both balls. The equations will never let this happen, no matter what initial wave function you feed into them. It’s impossible… because a superposition is an OR, not an AND. There is no way to make the projectile go left AND right — not if it’s a particle in 1920s quantum physics, anyway.

Yes, the wave function itself can have peaks that appear at to be in several places at the same time within the space of possibilities, as in Figs. 6, 8, and 10. But the wave function is not the physical system. The wave function tells us about the probabilities for the system’s possibilities; its peaks are just indicating what the most likely possibilities are.

The system itself can only realize one of the many possibilities — it can only be found (through a later measurement) in one place within the space of possibilities. This is always true, even though the wave function for the system highlights all the most probable possibilities simultaneously.

A particle, in the strict sense of the term, is an object with a position and a momentum, even though we cannot know both perfectly at any moment, thanks to Heisenberg’s uncertainty principle. It can only be measured to be in one place, or can only be measured to be traveling in one direction, at a time. In 1920s quantum physics, these statements apply to an electron, which is viewed as a strict particle, and so it cannot go in two directions at once, nor can it be in two places at once. The fact that we are always somewhat ignorant of where an electron is and/or where it is going, and the fact that quantum physics puts ultimate limitations on our ability to know both simultaneously, do not change these basic conceptual lessons… the lessons of (and for) the 1920s.

I’m giving a talk next Friday, March 14th, at 9 am Pacific Daylight time here in California. You’re all invited!

(Note that Daylight Savings Time starts March 9th, so do your calculations carefully if you do them before then.)

Title: Visual Insights

Abstract: For several years I ran a blog called Visual Insight, which was a place to share striking images that help explain topics in mathematics.  In this talk I’d like to show you some of those images and explain some of the mathematics they illustrate.

Zoom link: https://virginia.zoom.us/j/97786599157?pwd=jr0dvbolVZ6zrHZhjOSeE2aFvbl6Ix.1

Recording: This talk will be recorded, and eventually a video will appear here: https://www.youtube.com/@IllustratingMathSeminar

I’m giving a talk next Friday, March 14th, at 9 am Pacific Daylight time here in California. You’re all invited!

(Note that Daylight Savings Time starts March 9th, so do your calculations carefully if you do them before then.)

Title: Visual Insights

Abstract: For several years I ran a blog called Visual Insight, which was a place to share striking images that help explain topics in mathematics.  In this talk I’d like to show you some of those images and explain some of the mathematics they illustrate.

Zoom link:
https://virginia.zoom.us/j/97786599157?pwd=jr0dvbolVZ6zrHZhjOSeE2aFvbl6Ix.1

Recording: This talk will be recorded, and eventually the recording will appear here: https://www.youtube.com/@IllustratingMathSeminar

Did you hear about the asteroid?

Which one?

You might have heard that an asteroid named 2024 YR4 is going to come unusually close to the Earth in 2032. When it first made the news, astronomers estimated a non-negligible chance of it hitting us: about three percent. That’s small enough that they didn’t expect it to happen, but large enough to plan around it: people invest in startups with a smaller chance of succeeding. Still, people were fairly calm about this one, and there are a couple of good reasons:

• First, this isn’t a “kill the dinosaurs” asteroid, it’s much smaller. This is a “Tunguska Event” asteroid. Still pretty bad if it happens near a populated area, but not the end of life as we know it.
• We know about it far in advance, and space agencies have successfully deflected an asteroid before, for a test. If it did pose a risk, it’s quite likely they’d be able to change its path so it misses the Earth instead.
• It’s tempting to think of that 3% chance as like a roll of a hundred-sided die: the asteroid is on a random path, roll 1 to 3 and it will hit the Earth, roll higher and it won’t, and nothing we do will change that. In reality, though, that 3% was a measure of our ignorance. As astronomers measure the asteroid more thoroughly, they’ll know more and more about its path, and each time they figure something out, they’ll update the number.

And indeed, the number has been updated. In just the last few weeks, the estimated probability of impact has dropped from 3% to a few thousandths of a percent, as more precise observations clarified the asteroid’s path. There’s still a non-negligible chance it will hit the moon (about two percent at the moment), but it’s far too small to do more than make a big flashy crater.

It’s kind of fun to think that there are people out there who systematically track these things, with a plan to deal with them. It feels like something out of a sci-fi novel.

But I find the other asteroid more fun.

In 2020, a probe sent by NASA visited an asteroid named Bennu, taking samples which it carefully packaged and brought back to Earth. Now, scientists have analyzed the samples, revealing several moderately complex chemicals that have an important role in life on Earth, like amino acids and the bases that make up RNA and DNA. Interestingly, while on Earth these molecules all have the same “handedness“, the molecules on Bennu are divided about 50/50. Something similar was seen on samples retrieved from another asteroid, so this reinforces the idea that amino acids and nucleotide bases in space do not have a preferred handedness.

I first got into physics for the big deep puzzles, the ones that figure into our collective creation story. Where did the universe come from? Why are its laws the way they are? Over the ten years since I got my PhD, it’s felt like the answers to these questions have gotten further and further away, with new results serving mostly to rule out possible explanations with greater and greater precision.

Biochemistry has its own deep puzzles figuring into our collective creation story, and the biggest one is abiogenesis: how life formed from non-life. What excites me about these observations from Bennu is that it represents real ongoing progress on that puzzle. By glimpsing a soup of ambidextrous molecules, Bennu tells us something about how our own molecules’ handedness could have developed, and rules out ways that it couldn’t have. In physics, if we could see an era of the universe when there were equal amounts of matter and antimatter, we’d be ecstatic: it would confirm that the imbalance between matter and antimatter is a real mystery, and show us where we need to look for the answer. I love that researchers on the origins of life have reason right now to be similarly excited.

I’ve worked on topological quantum computation, one of Alexei Kitaev’s brilliant innovations, for around 15 years now.  It’s hard to find a more beautiful physics problem, combining spectacular quantum phenomena (non-Abelian anyons) with the promise of transformative technological advances (inherently fault-tolerant quantum computing hardware).  Problems offering that sort of combination originally inspired me to explore quantum matter as a graduate student. 

Non-Abelian anyons are emergent particles born within certain exotic phases of matter.  Their utility for quantum information descends from three deeply related defining features:

• Nucleating a collection of well-separated non-Abelian anyons within a host platform generates a set of quantum states with the same energy (at least to an excellent approximation).  Local measurements give one essentially no information about which of those quantum states the system populates—i.e., any evidence of what the system is doing is hidden from the observer and, crucially, the environment.  In turn, qubits encoded in that space enjoy intrinsic resilience against local environmental perturbations. 
• Swapping the positions of non-Abelian anyons manipulates the state of the qubits.  Swaps can be enacted either by moving anyons around each other as in a shell game, or by performing a sequence of measurements that yields the same effect.  Exquisitely precise qubit operations follow depending only on which pairs the user swaps and in what order.  Properties (1) and (2) together imply that non-Abelian anyons offer a pathway both to fault-tolerant storage and manipulation of quantum information. 
• A pair of non-Abelian anyons brought together can “fuse” into multiple different kinds of particles, for instance a boson or a fermion.  Detecting the outcome of such a fusion process provides a method for reading out the qubit states that are otherwise hidden when all the anyons are mutually well-separated.  Alternatively, non-local measurements (e.g., interferometry) can effectively fuse even well-separated anyons, thus also enabling qubit readout. 

I entered the field back in 2009 during the last year of my postdoc.  Topological quantum computing—once confined largely to the quantum Hall realm—was then in the early stages of a renaissance driven by an explosion of new candidate platforms as well as measurement and manipulation schemes that promised to deliver long-sought control over non-Abelian anyons.  The years that followed were phenomenally exciting, with broadly held palpable enthusiasm for near-term prospects not yet tempered by the practical challenges that would eventually rear their head. 

In 2018, near the height of my optimism, I gave an informal blackboard talk in which I speculated on a new kind of forthcoming NISQ era defined by the birth of a Noisy Individual Semi-topological Qubit.  To less blatantly rip off John Preskill’s famous acronym, I also—jokingly of course—proposed the alternative nomenclature POST-Q (Piece Of S*** Topological Qubit) era to describe the advent of such a device.  The rationale behind those playfully sardonic labels is that the inaugural topological qubit would almost certainly be far from ideal, just as the original transistor appears shockingly crude when compared to modern electronics.  You always have to start somewhere.  But what does it mean to actually create a topological qubit, and how do you tell that you’ve succeeded—especially given likely POST-Q-era performance?

To my knowledge those questions admit no widely accepted answers, despite implications for both quantum science and society.  I would like to propose defining an elementary topological qubit as follows:

A device that leverages non-Abelian anyons to demonstrably encode and manipulate a single qubit in a topologically protected fashion. 

Some of the above words warrant elaboration.  As alluded to above, non-Abelian anyons can passively encode quantum information—a capability that by itself furnishes a quantum memory.  That’s the “encode” part.  The “manipulate” criterion additionally entails exploiting another aspect of what makes non-Abelian anyons special—their behavior under swaps—to enact gate operations.  Both the encoding and manipulation should benefit from intrinsic fault-tolerance, hence the “topologically protected fashion” qualifier.  And very importantly, these features should be “demonstrably” verified.  For instance, creating a device hosting the requisite number of anyons needed to define a qubit does not guarantee the all-important property of topological protection.  Hurdles can still arise, among them: if the anyons are not sufficiently well-separated, then the qubit states will lack the coveted immunity from environmental perturbations; thermal and/or non-equilibrium effects might still induce significant errors (e.g., by exciting the system into other unwanted states); and measurements—for readout and possibly also manipulation—may lack the fidelity required to fruitfully exploit topological protection even if present in the qubit states themselves. 

The preceding discussion raises a natural follow-up question: How do you verify topological protection in practice?  One way forward involves probing qubit lifetimes, and fidelities of gates resulting from anyon swaps, upon varying some global control knob like magnetic field or gate voltage.  As the system moves deeper into the phase of matter hosting non-Abelian anyons, both the lifetime and gate fidelities ought to improve dramatically—reflecting the onset of bona fide topological protection.  First-generation “semi-topological” devices will probably fare modestly at best, though one can at least hope to recover general trends in line with this expectation. 

By the above proposed definition, which I contend is stringent yet reasonable, realization of a topological qubit remains an ongoing effort.  Fortunately the journey to that end offers many significant science and engineering milestones worth celebrating in their own right.  Examples include:

Platform verification.  This most indirect milestone evidences the formation of a non-Abelian phase of matter through (thermal or charge) Hall conductance measurements, detection of some anticipated quantum phase transition, etc. 

Detection of non-Abelian anyons. This step could involve conductance, heat capacity, magnetization, or other types of measurements designed to support the emergence of either individual anyons or a collection of anyons.  Notably, such techniques need not reveal the precise quantum state encoded by the anyons—which presents a subtler challenge. 

Establishing readout capabilities. Here one would demonstrate experimental techniques, interferometry for example, that in principle can address that key challenge of quantum state readout, even if not directly applied yet to a system hosting non-Abelian anyons. 

Fusion protocols.  Readout capabilities open the door to more direct tests of the hallmark behavior predicted for a putative topological qubit.  One fascinating experiment involves protocols that directly test non-Abelian anyon fusion properties.  Successful implementation would solidify readout capabilities applied to an actual candidate topological qubit device. 

Probing qubit lifetimes.  Fusion protocols further pave the way to measuring the qubit coherence times, e.g., and —addressing directly the extent of topological protection of the states generated by non-Abelian anyons.  Behavior clearly conforming to the trends highlighted above could certify the device as a topological quantum memory.  (Personally, I most anxiously await this milestone.)

Fault-tolerant gates from anyon swaps.  Likely the most advanced milestone, successfully implementing anyon swaps, again with appropriate trends in gate fidelity, would establish the final component of an elementary topological qubit. 

Most experiments to date focus on the first two items above, platform verification and anyon detection.  Microsoft’s recent Nature paper, together with the simultaneous announcement of supplementary new results, combine efforts in those areas with experiments aiming to establish interferometric readout capabilities needed for a topological qubit.  Fusion, (idle) qubit lifetime measurements, and anyon swaps have yet to be demonstrated in any candidate topological quantum computing platform, but at least partially feature in Microsoft’s future roadmap.  It will be fascinating to see how that effort evolves, especially given the aggressive timescales predicted by Microsoft for useful topological quantum hardware.  Public reactions so far range from cautious optimism to ardent skepticism; data will hopefully settle the situation one way or another in the near future.  My own take is that while Microsoft’s progress towards qubit readout is a welcome advance that has value regardless of the nature of the system to which those techniques are currently applied, convincing evidence of topological protection may still be far off. 

In the meantime, I maintain the steadfast conviction that topological qubits are most certainly worth pursuing—in a broad range of platforms.  Non-Abelian quantum Hall states seem resurgent candidates, and should not be discounted.  Moreover, the advent of ultra-pure, highly tunable 2D materials provide new settings in which one can envision engineering non-Abelian anyon devices with complementary advantages (and disadvantages) compared to previously explored settings.  Other less obvious contenders may also rise at some point.  The prospect of discovering new emergent phenomena mitigating the need for quantum error correction warrants continued effort with an open mind.

Any action of a finite group $G$ on a finite set $X$ gives a linear representation of $G$ on the vector space with basis $X$. This is called a ‘permutation representation’. And this raises a natural question: how many representations of finite groups are permutation representations?

Most representations are not permutation representations, since every permutation representation has a vector fixed by all elements of $G$, namely the vector that’s the sum of all elements of $X$. In other words, every permutation representation has a 1-dimensional trivial rep sitting inside it.

But what if we could ‘subtract off’ this trivial representation?

There are different levels of subtlety with which we can do this. For example, we can decategorify, and let:

• the Burnside ring of $G$ be the ring $A(G)$ of formal differences of isomorphism classes of actions of $G$ on finite sets;

• the representation ring of $G$ be the ring $R(G)$ of formal differences of isomorphism classes of finite-dimensional representations of $G$.

In either of these rings, we can subtract.

There’s an obvious map $\beta : A(G) \to R(G)$, since any action of $G$ on a finite set gives a permutation representation of $G$ on the vector space with basis $X$.

So I asked on MathOverflow: is $\beta$ typically surjective, or typically not surjective?

In fact everything depends on what field $k$ we’re using for our vector spaces! For starters let’s take $k = \mathbb{Q}$.

Here’s a list of finite groups where the map $\beta : A(G) \to R(G)$ from the Burnside ring to the representation ring is known to be surjective, taken from the nLab article Permutation representations:

1. cyclic groups,

2. symmetric groups,

3. $p$-groups (that is, groups whose order is a power of the prime $p$),

4. binary dihedral groups $2 D_{2n}$ for (at least) $2 n \leq 12$,

5. the binary tetrahedral group, binary octahedral group, and binary icosahedral group,

6. the general linear group $GL(2,\mathbb{F}_3)$.

Now, these may seem like rather special classes of groups, but devoted readers of this blog know that most finite groups have order that’s a power of 2. I don’t think this has been proved yet, but it’s obviously true empirically, and we also have a good idea as to why:

• Tom Leinster, Almost all of the first 50 billion groups have order 1024.

So, the map $\beta$ from the Burnside ring to the representation ring is surjective for most finite groups!

David Benson has listed the orders for which there are groups where $\beta$ is not surjective, and how many such groups there are of these orders:

24: 2,

40: 2,

48: 7,

56: 1,

60: 2,

72: 8,

80: 8,

84: 1,

88: 2,

96: 45,

104: 2,

112: 5,

120: 13,

132: 1,

136: 3,

140: 2,

144: 39,

152: 2,

156: 1,

160: 12,

168: 12,

171: 1,

176: 7,

180: 6,

184: 1,

192: 423,

200: 8,

204: 2,

208: 8,

216: 35,

220: 1,

224: 28,

228: 1,

232: 2,

240: 90,

248: 1,

252: 4,

260: 2,

264: 10,

272: 12,

276: 1,

280: 12,

288: 256,

296: 2,

300: 8,

304: 7,

308: 1,

312: 16,

320: 532,

328: 3,

333: 1,

336: 76,

340: 2,

342: 2,

344: 2,

348: 2,

352: 41,

360: 51,

364: 2,

368: 5,

372: 1,

376: 1,

380: 2.

He adds:

This seems to represent quite a small proportion of all finite groups, counted by order, even ignoring the $p$-groups.

All this is if we work over $\mathbb{Q}$. If we work over $\mathbb{C}$ the situation flip-flops, and I believe $\beta$ is usually not surjective. It already fails to be surjective for cyclic groups bigger than $\mathbb{Z}/2$!

Why? Because $\mathbb{Z}/n$ has a 1-dimensional representation where the generator acts as multiplication by a primitive $n$th root of unity, and since this is not a rational number when $n &gt; 2$, this representation is not definable over $\mathbb{Q}$. Thus, one can show, there’s no way to get this representation as a formal difference of permutation representations, since those are always definable over $\mathbb{Q}$.

And this phenomenon of needing roots of unity to define representations is not special to cyclic groups: it happens for most finite groups as hinted at by Artin’s theorem on induced characters.

An example

Now, you’ll notice that I didn’t yet give an example of a finite group where the map $\beta$ from the Burnside ring to the representation ring fails to be surjective when we work over $\mathbb{Q}$.

In Serre’s book Linear Representations of Finite Groups, he asks us in Exercise 13.4 on page 105 to prove that $\mathbb{Z}/3 \times Q_8$ is such a group. Here $Q_8$ is the quaternion group, an 8-element subgroup of the invertible quaternions:

$$Q_8 = \{ \pm 1, \pm i , \pm j, \pm k \}$$

I couldn’t resist trying to understand why this is the counterexample Serre gave. First of all, $\mathbb{Z}/3 \times Q_8$ looks like a pretty weird group — why does it work, and why did Serre choose this one? Secondly, it has 24 elements, and I love the number 24. Thirdly, I love the quaternions.

Benson’s calculations show that the two smallest groups where $\beta$ is non-surjective have 24 elements. Drew Heard checked that these groups are $\mathbb{Z}/3 \times Q_8$ and the nontrivial semidirect product $\mathbb{Z}/3 \rtimes \mathbb{Z}/8$. But it’s still very interesting to do Serre’s exercise 13.4 and see why $\mathbb{Z}/3 \times Q_8$ is such a group.

To do this exercise, Serre asks us to use exercise 13.3. Here is my reformulation and solution of that problem. I believe any field $k$ characteristic zero would work for this theorem:

Theorem. Suppose $G$ is a finite group with a linear representation $\rho$ such that:

1. $\rho$ is irreducible and faithful

2. every subgroup of $G$ is normal

3. $\rho$ appears with multiplicity $n \ge 2$ in the regular representation of $G$.

Then the map from the Burnside ring of $G$ to the representation ring $R(G)$ of $G$ is not surjective.

Proof. It suffices to prove that the multiplicity of $\rho$ in any permutation representation of $G$ is a multiple of $n$, so that the class $[\rho] \in R(G)$ cannot be in the image of $R(G)$.

Since every finite $G$-set is a coproduct of transitive actions of $G$, which are isomorphic to actions on $G/H$ for subgroups $H$ of $G$, every permutation representation of $G$ is a direct sum of those on spaces of the form $k[G/H]$. (This is my notation for the vector space with basis $G/H$.) Thus, it suffices to show that the multiplicity of $\rho$ in the representation on $k[G/H]$ is $n$ if $H$ is the trivial group, and $0$ otherwise.

The former holds by assumption 3. For the latter, suppose $H$ is a nontrivial subgroup of $G$. Because $H$ is normal by assumption 2, every element $h \in H$ acts trivially on $k[G/H]$: we can see this by letting $h$ act on an arbitrary basis element $g H = H g \in G/H$:

$$h H g = H g .$$

Since $H$ is nontrivial, it contains elements $h \ne 1$ that act trivially on $k[G/H]$. But no $h \ne 1$ can act trivially on $\rho$ because $\rho$ is faithful, by assumption 1. Thus $\rho$ cannot be a subrepresentation of $k[G/H]$. That is, $\rho$ appears with multiplicity $0$ in $k[G/H]$.   ▮

Serre’s exercise 13.4 is to show the group $G = \mathbb{Z}/3 \times Q_8$ obeys the conditions of this theorem. As a hint, Serre suggests to embed $\mathbb{Z}/3$ and $Q_8$ in the multiplicative group of the algebra $\mathbb{H}_{\mathbb{Q}}$ (the quaternions defined over $\mathbb{Q}$). By letting $\mathbb{Z}/3$ act by left multiplication and $Q_8$ act by right multiplication, one obtains a 4-dimensional irreducible representation $\rho$ of $G$ which appears with multiplicity $n = 2$ in the regular representation. Furthermore $\rho$ is faithful and irreducible. Finally, every subgroup of $G$ is normal, because that’s true of $\mathbb{Z}/3$ and $Q_8$ — and since the orders of these groups are relatively prime, every subgroup of $G = \mathbb{Z}/3 \times Q_8$ is a product of a subgroup of $\mathbb{Z}/3$ and a subgroup of $Q_8$.

More work on the question

Alex Bartel wrote:

To add to what others have said: you may be interested in this paper “Rational representations and permutation representations of finite groups” of mine with Tim Dokchitser, which, among other things, contains an overview of what is (or was in 2016) known about the question, describes the algorithm that is used in the magma programme (implemented by Tim Dokchitser) that Drew Heard mentions in his answer, gives a closed form formula for the cokernel in quasi-elementary groups, and exhibits a family of finite simple groups with unbounded cokernel, in fact even unbounded exponent of cokernel (to our knowledge, in all finite simple groups for which one had known this cokernel up to that point, the cokernel was trivial).

The particular interest in quasi-elementary groups stems from the fact that one can reduce the study of this cokernel in arbitrary finite groups to that of its quasi-elementary subgroups and the induction-restriction maps between them. Observations of this nature have been made repeatedly over the years. In our paper we make this reduction very explicit.

Edit. In response to a question in a comment thread, and as an example of an interesting quasi-elementary group: the other group of order $24$ with non-trivial cokernel is $C_3\rtimes C_8$ with the unique non-trivial action. The cokernel of $\beta$ here comes about in the following way: there is an index $2$ subgroup $C_3\times C_4\cong C_{12}$. Take a faithful irreducible character of that, and induce up to the whole group. The resulting $2$-dimensional representation of $C_3\rtimes C_8$ has field of character values $\mathbb{Q}(i)$: the cube roots of unity have “disappeared” in the course of the induction. It turns out that the sum of this degree $2$ character and its Galois conjugate is realisable over $\mathbb{Q}$, but that it is not a virtual permutation representation. This is an illustration of Theorem 1.1 in my paper with Tim: $\rho$ is $4$-dimensional, $\hat\psi$ is the $2$-dimensional irreducible rational representation of $C_3$, the sum of the two faithful complex irreducible characters, and $\hat{\pi}$ is the unique $4$-dimensional irreducible rational representation of $C_8$, the sum of all its faithful complex irreducible characters.

Please enjoy Harvard’s Jacob Barandes and yours truly duking it out for 2.5 hours on YouTube about the interpretation of quantum mechanics, and specifically Jacob’s recent proposal involving “indivisible stochastic dynamics,” with Curt Jaimungal as moderator. As always, I strongly recommend watching with captions turned on and at 2X speed.

To summarize what I learned in one paragraph: just like in Bohmian mechanics, Jacob wants classical trajectories for particles, which are so constructed to reproduce the predictions of QM perfectly. But unlike the Bohmians, Jacob doesn’t want to commit to any particular rule for the evolution of those particle trajectories. He merely asserts, metaphysically, that the trajectories exist. My response was basically, “OK fine, you can do that if you want, but what does it buy me?” We basically went around in circles on that question the entire time, though hopefully with many edutaining disgressions.

Despite the lack of resolution, I felt pretty good about the conversation afterward: Jacob got an extensive opportunity to explain his ideas to listeners, along with his detailed beefs against both the Many-Worlds and Copenhagen interpretations. Meanwhile, even though I spoke less than Jacob, I did get some opportunities to do my job, pushing back and asking the kinds of questions I imagined most physicists would ask (even though I’m not a physicist, I felt compelled to represent them!). Jacob and I ended the conversation much as we began: disagreeing on extremely friendly terms.

Then, alas, I read the comments on YouTube and got depressed. Apparently, I’m a hidebound academic elitist who’s failed to grasp Jacob’s revolutionary, paradigm-smashing theory, and who kept arrogantly interrupting with snide, impertinent questions (“OK, but what can I do with this theory that I couldn’t do before?”). And, I learned, the ultimate proof of my smug, ivory-tower malice was to be found in my body language, the way I constantly smiled nervously and rocked back and forth. I couldn’t help but wonder: have these people watched any other YouTube videos that I’m in? I don’t get to pick how I look and sound. I came out of the factory this way.

One commenter opined that I must hate Jacob’s theory only because I’ve poured my life into quantum computing, which depends on superposition, the confusing concept that Jacob has now unmasked as a farce. Presumably it’s beyond this person’s comprehension that Jacob makes exactly the same predictions as I make for what a quantum computer will do when built; Jacob just prefers a different way of talking about it.

I was reminded that optimizing for one’s scientific colleagues is wildly different from optimizing for YouTube engagement. In science, it’s obvious to everyone that the burden of proof is on whoever is presenting the new idea—and that this burden is high, especially with anything as well-trodden and skull-strewn as the foundations of quantum mechanics, albeit not infinitely high. The way the game works is: other people try as hard as they can to shoot the new idea down, so we see how it fares under duress. This is not a sign of contempt for new ideas, but of respect for them.

On YouTube, the situation is precisely reversed. There, anyone perceived as the “mainstream establishment” faces a near-insurmountable burden of proof, while anyone perceived as “renegade” wins by default if they identify any hole whatsoever in mainstream understanding. Crucially, the renegade’s own alternative theories are under no particular burden; indeed, the details of their theories are not even that important or relevant. I don’t want to list science YouTubers who’ve learned to exploit that dynamic masterfully, though I’m told one rhymes with “Frabine Schlossenfelder.” Of course this mirrors what’s happened in the wider world, where RFK Jr. now runs American health policy, Tulsi Gabbard runs the intelligence establishment, and other conspiracy theorists have at last fired all the experts and taken control of our civilization, and are eagerly mashing the buttons to see what happens. I’d take Jacob Barandes, or even Sabine, a billion times over the lunatics in power. But I do hope Jacob turns out to be wrong about Many-Worlds, because it would give my solace to know that there are other branches of the wavefunction where things are a little more sane.

read more

As part of my post last week about measurement and measurement devices, I provided a very simple example of a measuring device. It consists of a ball sitting in a dip on a hill (Fig. 1a), or, as a microscopic version of the same, a microsopic ball, made out of only a small number of atoms, in a magnetic trap (Fig. 1b). Either object, if struck hard by an incoming projectile, can escape and never return, and so the absence of the ball from the dip (or trap) serves to confirm that a projectile has come by. The measurement is crude — it only tells us whether there was a projectile or not — but it is reasonably definitive.

In fact, we could learn more about the projectile with a bit more work. If we measured the ball’s position and speed (approximately, to the degree allowed by the quantum uncertainty principle), we would get an estimate of the energy carried by the projectile and the time when the collision occurred. But how definitive would these measurements be?

With a macroscopic ball, we’d be pretty safe in drawing conclusions. However, if the objects being measured and the measurement device are ultra-microscopic — something approaching atomic size or even smaller — then the measurement evidence is fragile. Our efforts to learn something from the microscopic ball will be in vain if the ball suffers additional collisions before we get to study it. Indeed, if a tiny ball interacts with any other object, microscopic or macroscopic, there is a risk that the detailed information about its collision with the projectile will be lost, long before we are able to obtain it.

Amplify Quickly

The best way to keep this from happening is to quickly translate the information from the collision, as captured in the microscopic ball’s behavior, into some kind of macroscopic effect. Once the information is stored macroscopically, it is far harder to erase.

For instance, while a large meteor striking the Earth might leave a pond-sized crater, a subatomic particle striking a metal table might leave a hole only an atom wide. It doesn’t take much to fill in an atom-sized hole in the blink of an eye, but a crater that you could swim in isn’t going to disappear overnight. So if we want to know about the subatomic particle’s arrival, it would be good if we could quickly cause the hole to grow much larger.

This is why almost all microscopic measurements include a step of amplification — the conversion of a microscopic effect into a macroscopic one. Finding new, clever and precise ways of doing this is part of the creativity and artistry of experimental physicists who study atoms, atomic nuclei, or elementary particles.

There are various methods of amplification, but most methods can be thought of, in a sort of cartoon view, as a chain of ever more stable measurements, such as this:

• a first measurement using a microscopic device, such as our tiny ball in a trap;
• a second measurement that measures the device itself, using a more stable device;
• a third measurement that measures the second device, using an even more stable one;
• and so on in a chain until the last device is so stable that its information cannot easily or quickly be erased.

Amplification in Experiments

The Geiger-Müller Counter

A classic and simple device that uses amplification is a Geiger counter (or Geiger-Müller counter). (Hans Geiger, while a postdoctoral researcher for Ernest Rutherford, performed a key set of experiments that Rutherford eventually interpreted as evidence that atoms have tiny nuclei.) This counter, like our microscopic ball in Fig. 1b, simply records the arrival of high-energy subatomic projectiles. It does so by turning the passage of a single ultra-microscopic object into a measurable electric current. (Often it is designed to make a concurrent audible electronic “click” for ease of use.)

How does this device turn a single particle, with a lot of energy relative to a typical atomic energy level but very little relative to human activity, into something powerful enough to create a substantial, measurable electric current? The trick is to use the electric field to create a chain reaction.

The Electric Field

The electric field is present throughout the universe (like all cosmic fields). But usually, between the molecules of air or out in deep space, it is zero or quite small. However, when it is strong, as when you have just taken off a wool hat in winter, or just before a lightning strike, it can make your hair stand on end.

More generally, a strong electric field exerts a powerful pull on electrically charged objects, such as electrons or atomic nuclei. Positively charged objects will accelerate in one direction, while negatively charged objects will accelerate in the other. That means that a strong electric field will

• separate positively charged objects from negatively charged objects
• cause both types of objects to speed up, albeit in opposite direction,

Meanwhile electrically neutral objects are largely left alone.

The Strategy

So here’s the strategy behind the Geiger-Müller counter. Start with a gas of atoms, sitting inside of a closed tube in a region with a strong electric field. Atoms are electrically neutral, so they aren’t much affected by the electric field.

But the atoms will serve as our initial measurement devices. If a high-energy subatomic particle comes flying through the gas, it will strike some of the gas atoms and “ionize” them — that is, it will strip an electron off the atom. In doing so it breaks the electrically neutral atom into a negatively charged electron and a positively charged leftover, called an “ion.”

If it weren’t for the strong electric field, the story would remain microscopic; the relatively few ions and electrons would quickly find their way back together, and all evidence of the atomic-scale measurements would be lost. But instead, the powerful electric field causes the ions to move in one direction and the electrons to move in the opposite direction, so that they cannot simply rejoin each other. Not only that, the field causes these subatomic objects to speed up as they separate.

This is especially significant for the electrons, which pick up so much speed that they are able to ionize even more atoms — our secondary measurement devices. Now the number of electrons freed from their atoms has become much larger.

The effect is an chain reaction, with more and more electrons stripped off their atoms, accelerated by the electric field to high speed, allowing them in their turn to ionize yet more atoms. The resulting cascade, or “avalanche,” is called a Townsend discharge; it was discovered in the late 1890s. In a tiny fraction of a second, the small number of electrons liberated by the passage of a single subatomic particle has been multiplied exceedingly, and a crowd of electrons now moves through the gas.

The chain reaction continues until this electron mob arrives at a wire in the center of the counter — the final measurement device in the long chain from microscopic to macroscopic. The inflow of a huge number of the electrons onto the wire, combined with the flow of the ions onto the wall of the device, causes an electrical current to flow. Thanks to the amplification, this current is large enough to be easily detected, and in response a separate signal is sent to the device’s sound speaker, causing it to make a “click!”

Broader Lessons

It’s worth noting that the strategy behind the Geiger-Müller counter requires an input of energy from outside the device, supplied by a battery or the electrical grid. When you think about it, this is not surprising. After the initial step there are rather few moving electrons, and their total motion-energy is still rather low; but by the end of the avalanche, the motion-energy of the tremendous number of moving electrons is far greater. Since energy is conserved, that energy has to have come from somewhere.

Said another way, to keep the electric field strong amid all these charged particles, which would tend to cancel the field out, requires the maintenance of high voltage between the outer wall and inner wire of the counter. Doing so requires a powerful source of energy.

Without this added energy and the resulting amplification, the current from the few initially ionized atoms would be extremely small, and the information about the passing high-energy particle could easily be lost due to ordinary microscopic processes. But the chain reaction’s amplification of the number of electrons and their total amount of energy dramatically increases the current and reduces the risk of losing the information.

Many devices, such as the photomultiplier tube for the detection of photons [particles of light], are like the Geiger-Müller counter in using an external source of energy to boost a microscopic effect. Other devices (like the cloud chamber) use natural forms of amplification that can occur in unstable systems. (The basic principle is similar to what happens with unstable snow on a steep slope: as any off-piste skier will warn you, under the correct circumstances a minor disturbance can cause a mountain-wide snow avalanche.) If these issues interest you, I suggest you read more about the various detectors and subdetectors at ongoing particle experiments, such as those at the Large Hadron Collider.

Amplification in a Simplified Setting

I’ve described the Geiger-Müller counter without any explicit reference to quantum physics. Is there any hope that we could understand how this process really takes place using quantum language, complete with a wave function?

Not in practice: the chain reaction is far, far too complicated. A quantum system’s wave function does not exist in the physical space we live in; it exists in the space of possibilities. Amplification involving hordes of electrons and ions forces us to consider a gigantic space of possibilities; for instance, a million particles moving in our familiar three spatial dimensions would correspond to a space of possibilities that has three million dimensions. Neither you nor I nor the world’s most expert mathematical physicist can visualize that.

Nevertheless, we can gain intuition about the basic idea behind this device by simplifying the chain reaction into a minimal form, one that involves just three objects moving in one dimension, and three stages:

• an initial measurement involving something microscopic
• addition of energy to the microscopic measurement device
• transfer of the information by a second measurement to something less microscopic and more stable.

You can think of these as the first steps of a chain reaction.

So let’s explore this simplified idea. As I often do, I’ll start with a pre-quantum viewpoint, and use that to understand what is happening in a corresponding quantum wave function.

The Pre-Quantum View

The pre-quantum viewpoint differs from that in my last post (which you should read if you haven’t already) in that we have two steps in the measurement rather than just one:

• a projectile is measured by a microscopic ball (the “microball”),
• the microball is similarly measured by a larger device, which I’ll refer to as the “macroball”.

The projectile, microball and macroball will be colored purple, blue and orange, and their positions along the x-axis of physical space will be referred to as x₁, x₂ and x₃. Our space of possibilities then is a three-dimensional space consisting of all possible values of x₁, x₂ and x₃.

The two-step measurement process really involves four stages:

• The projectile approaches the stationary balls from the left.
• The projectile collides with the microball and (in a small change from the last post, for convenience) bounces off to the left, leaving the microball moving to the right.
• The microball is then subject to a force that greatly accelerates it, so that it soon carries a great deal of motion-energy.
• The highly energetic microball now bounces off the macroball, sending the latter into motion.

The view of this process in physical space is shown on the left side of Fig 2. Notice the acceleration of the microball between the two collisions.

On the right side of Fig. 2, the motion of the three-object system within the space of possibilities is shown by the moving red dot. To make it easier to see how the red dot moves acrossthe space of possibilities, I’ve plotted its trail across that space as a gray line. Notice there are two collisions, the first one when the projectile and microball collide (x₁=x₂) and the second where the two balls collide (x₂=x₃), resulting in two sudden changes in the motion of the dot. Notice also the rapid acceleration between the first collision and the second, as the microball gains sufficient energy to give the macroball appreciable speed.

The Quantum View

In quantum physics, the idea is the same, where the dot representing the system’s value of (x₁, x₂, x₃) is replaced by the peak of a spread-out wave function. It’s difficult to plot a wave function in three dimensions, but I can at least mark out the region where its absolute value is large — where the probability to find the system is highest. I’ve sketched this in Fig. 3. Not surprisingly if follows the same path as the system in Fig. 2.

In the pre-quantum case of Fig. 2, the red dot asserts certainty; if we were to measure x₁, x₂ and/or x₃, we would find exactly the values of these quantities corresponding to the location of the dot. In quantum physics of Fig. 3, the peak of the wave function asserts high probability but not certainty. The wave function is spread out; we don’t know exactly what we would find if we directly measured x₁, x₂ and x₃ at any particular moment.

Still, the path of the wave function’s peak is very similar to the path of the red dot, as was also true in the previous post. Generally, in the examples we’ve looked at so far, we haven’t shown much difference between the pre-quantum viewpoint and the quantum viewpoint. You might even be wondering if they’re more similar than people say. But there can be big differences, as we will see very soon.

The Wider View

If I could draw something with more than three dimensions, we could add another stage to our microball and macroball; we could accelerate the macroball and cause it to collide with something even larger, perhaps visible to the naked eye. Or instead of one macroball, we could amplify and transfer the microball’s energy to ten microballs, which in turn could have their energy amplified and transferred to a hundred microballs… and then we would have something akin to a Townsend discharge avalanche and a Geiger-Müller counter. Both in pre-quantum and in quantum physics, this would be impossible to draw; the space of possibilities is far too large. Nevertheless, the simple example in Figs. 2 and 3 provides some intuition for how a longer chain of amplification would work. It shows the basic steps needed to turn a fragile microscopic measurement into a robust macroscopic one, suitable for human scientific research or for our sense perceptions in daily living.

In the articles that will follow, I will generally assume (unless specified otherwise) that each microscopic measurement that I describe is followed by this kind of amplification and conversion to something macroscopic. I won’t be able to draw it, but as we can see in this example, the fundamental underlying idea isn’t that hard to understand.

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 →

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Recently, Microsoft announced a fancy new quantum computing chip called Majorana 1. I’ve noticed quite a bit of confusion about what they actually announced, and while there’s a great FAQ page about it on the quantum computing blog Shtetl Optimized, the post there aims at a higher level, assuming you already know the basics. You can think of this post as a complement to that one, that tries to cover some basic things Shtetl Optimized took for granted.

Q: In the announcement, Microsoft said:

“It leverages the world’s first topoconductor, a breakthrough type of material which can observe and control Majorana particles to produce more reliable and scalable qubits, which are the building blocks for quantum computers.”

That sounds wild! Are they really using particles in a computer?

A: All computers use particles. Electrons are particles!

Q: You know what I mean!

A: You’re asking if these are “particle physics” particles, like the weird types they try to observe at the LHC?

No, they’re not.

Particle physicists use a mathematical framework called quantum field theory, where particles are ripples in things called quantum fields that describe properties of the universe. But they aren’t the only people to use that framework. Instead of studying properties of the universe you can study properties of materials, weird alloys and layers of metal and crystal that do weird and useful things. The properties of these materials can be approximately described with the same math, with quantum fields. Just as the properties of the universe ripple to produce particles, these properties of materials ripple to produce what are called quasiparticles. Ultimately, these quasiparticles come down to movements of ordinary matter, usually electrons in the original material. They’re just described with a kind of math that makes them look like their own particles.

Q: So, what are these Majorana particles supposed to be?

A: In quantum field theory, most particles come with an antimatter partner. Electrons, for example, have partners called positrons, with a positive electric charge instead of a negative one. These antimatter partners have to exist due to the math of quantum field theory, but there is a way out: some particles are their own antimatter partner, letting one particle cover both roles. This happens for some “particle physics particles”, but all the examples we’ve found are a type of particle called a “boson”, particles related to forces. In 1937, the physicist Ettore Majorana figured out the math you would need to make a particle like this that was a fermion instead, the other main type of particle that includes electrons and protons. So far, we haven’t found one of these Majorana fermions in nature, though some people think the elusive neutrino particles could be an example. Others, though, have tried instead to find a material described by Majorana’s theory. This should in principle be easier, you can build a lot of different materials after all. But it’s proven quite hard for people to do. Back in 2018, Microsoft claimed they’d managed this, but had to retract the claim. This time, they seem more confident, though the scientific community is still not convinced.

Q: And what’s this topoconductor they’re talking about?

A: Topoconductor is short for topological superconductor. Superconductors are materials that conduct electricity much better than ordinary metals.

Q: And, topological means? Something about donuts, right?

A: If you’ve heard anything about topology, you’ve heard that it’s a type of mathematics where donuts are equivalent to coffee cups. You might have seen an animation of a coffee cup being squished and mushed around until the ring of the handle becomes the ring of a donut.

This isn’t actually the important part of topology. The important part is that, in topology, a ball is not equivalent to a donut.

Topology is the study of which things can change smoothly into one another. If you want to change a donut into a ball, you have to slice through the donut’s ring or break the surface inside. You can’t smoothly change one to another. Topologists study shapes of different kinds of things, figuring out which ones can be changed into each other smoothly and which can’t.

Q: What does any of that have to do with quantum computers?

A: The shapes topologists study aren’t always as simple as donuts and coffee cups. They can also study the shape of quantum fields, figuring out which types of quantum fields can change smoothly into each other and which can’t.

The idea of topological quantum computation is to use those rules about what can change into each other to encode information. You can imagine a ball encoding zero, and a donut encoding one. A coffee cup would then also encode one, because it can change smoothly into a donut, while a box would encode zero because you can squash the corners to make it a ball. This helps, because it means that you don’t screw up your information by making smooth changes. If you accidentally drop your box that encodes zero and squish a corner, it will still encode zero.

This matters in quantum computing because it is very easy to screw up quantum information. Quantum computers are very delicate, and making them work reliably has been immensely challenging, requiring people to build much bigger quantum computers so they can do each calculation with many redundant backups. The hope is that topological superconductors would make this easier, by encoding information in a way that is hard to accidentally change.

Q: Cool. So does that mean Microsoft has the best quantum computer now?

A: The machine Microsoft just announced has only a single qubit, the quantum equivalent of just a single bit of computer memory. At this point, it can’t do any calculations. It can just be read, giving one or zero. The hope is that the power of the new method will let Microsoft catch up with companies that have computers with hundred of qubits, and help them arrive faster at the millions of qubits that will be needed to do anything useful.

Q: Ah, ok. But it sounds like they accomplished some crazy Majorana stuff at least, right?

A: Umm…

Read the Shtetl-Optimized FAQ if you want more details. The short answer is that this is still controversial. So far, the evidence they’ve made public isn’t enough to show that they found these Majorana quasiparticles, or that they made a topological superconductor. They say they have more recent evidence that they haven’t published yet. We’ll see.

Nature could be said to be constructed out an immense number of physical processes… indeed, that’s almost the definition of “physics”. But what makes a physical process a measurement? And once we understand that, what makes a measurement in quantum physics, a fraught topic, different from measurements that we typically perform as teenagers in a grade school science class?

We could have a long debate about this. But for now I prefer to just give examples that illustrate some key features of measurements, and to focus attention on perhaps the simplest intuitive measurement device… one that we’ll explore further and put to use in many interesting examples of quantum physics.

Measurements and Devices

We typically think of measurements as something that humans do. But not all measurements are human artifice. A small fraction of physical processes are natural measurements, occuring without human intervention. What distinguishes a measurement from some other garden variety process?

A central element of a measurement is a device, natural or artificial, simple or complicated, that records some aspect of a physical process semi-permanently, so that this record can be read out after the process is over, at least for a little while.

For example, the Earth itself can serve as a measurement device. Meteor Crater in Arizona, USA is the record of a crude measurement of the size, energy and speed of a large rock, as well of how long ago it impacted Earth’s surface. No human set out to make the measurement, but the crater’s details are just as revealing as any human experiment. It’s true that to appreciate and understand this measurement fully requires work by humans: theoretical calculations and additional measurements. But still, it’s the Earth that recorded the event and stored the data, as any measurement device should.

The Earth has served as a measurement device in many other ways: its fossils have recorded its life forms, its sedimentary rocks have recorded the presence of its ancient seas, and a layer of iridium and shocked quartz have provided a record of the giant meteor that killed off the dinosaurs (excepting birds) along with many other species. The data from those measurements sat for many millions of years, unknown until human scientists began reading it out.

I’m being superficial here, skipping over all sorts of subtle issues. For instance, when does a measurement start, and when is it over? For instance, did the measurement of the rock that formed Meteor Crater start when the Earth and the future meteor were first created in the early days of the solar system, or only when the rock was within striking distance of our planet? Was it over when Meteor Crater had solidified, or was it complete when the first human measured its size and shape, or was it finished when humans first inferred the size of the rock that made the crater? I don’t want to dwell on these definitional questions today. The point I’m making here is that measurement has nothing intrinsically to do with human beings per se. It has to do with the ability to record a process in such a way that facts about that process can be extracted, long after the process is over.

The measurement device for any particular process has to satisfy some basic requirements.

• Pre-measurement metastability: The device must be fairly stable before the process occurs, so that it doesn’t react or change prematurely, but not so stable that it can’t change when the process takes place.
• Sensitivity: During the interaction between the device and whatever is being measured, the device needs to react or change in some substantial way that is predictable (at least in part).
• Post-measurement stability: The change to the device during the measurement has to be semi-permanent, long-lasting enough that there’s time to detect and interpret it.
• Interpretability: The change to the device has to be substantial and unambiguous enough that it can be used to extract information about what was measured.

Examples of Devices

A simple example: consider a small paper cup as a device for measuring the possible passage of a rubber ball. If the paper cup is sitting on a flat, horizontal table, it is reasonably stable and won’t go anywhere, barring a strong gust of wind. But if a rubber ball goes flying by and hits the cup, the cup will be knocked off the table… and thus the cup is very sensitive to the collision with the ball. The change is also stable and semi-permanent; once the cup is on the floor, it won’t spontaneously jump back up onto the table. And so, after setting a cup on a table in a windowless room near a squash court and returning days later, we can figure out from the position of the cup whether a rubber ball (or something similar) has passed close to the cup while we were away. Of course, this is a very crude measurement, but it captures the main idea.

Incidentally, such a measurement is sometimes referred to as “non-destructive”: the cup is so flimsy that its the effect of the cup on the ball is very limited, and so the ball continues onward almost unaffected. This is in contrast to the measurement of the rock that made Meteor Crater, which most certainly was “destructive” to the rock.

Yet even in this destructive event, all the criteria for a measurement are met. The Earth and its dry surface in Arizona are (and were) pretty stable over millennia, despite erosion. The Earth’s surface is very sensitive to a projectile fifty meters across and moving at ten or more kilometers per second; and the resulting deep, slowly-eroding crater represents a substantial, semi-permanent change that we can interpret roughly 50,000 years later.

In Figure 2 is a very simple and crude device designed to measure disturbances ranging from earthquakes to collisions. It consists of a ball sitting stationary within a dimple (a low spot) on a hill. It will remain there as long as it isn’t jostled — it is reasonably stable. But it is sensitive: if an earthquake occurs, or if something comes flying through the air and strikes the ball, it will pop out of the dimple. Then it will roll down the hill, never to return to the its original perch — thus leaving a long-lasting record of the process that disturbed it. We can later read the ball’s absence from the dimple, or its presence far off to the right, as evidence of some kind of violent disturbance, whereas if it remains in the dimple we may conclude that no such violent disturbance has occurred.

What about measurement devices in quantum physics? The needs are often the same; a measurement still requires a stable yet sensitive device that can respond to an interaction in a substantial, semi-permanent, interpretable way.

Today we’ll keep things very simple, and limit ourselves to a quantum version of Fig. 2, employed in the simplest of circumstances. But soon we’ll see that when measurements involve quantum physics, surprising and unfamiliar issues quickly arise.

An Simple Device for Quantum Measurement

Here’s an example of a suitable device, a sort of microscopic version of Fig. 2. Imagine a small ball of material, perhaps a few atoms wide, that is gently trapped in place by forces that are strong but not too strong. (These might be of the form of an ion trap or an atom trap; or we might even be speaking of a single atom incorporated into a much larger molecule. The details do not matter here.) This being quantum physics, the trap might not hold the ball in place forever, thanks to the process known as “tunneling“; but it can be arranged to stay in place long enough for our purposes.

If the ball is bumped by an atom or subatomic particle flying by at high speed, it may be knocked out of its trap, following which it will keep moving. So if we look in the trap and discover it empty, or if we find the ball far outside the trap, we will know that some energetic object must have passed through the trap. The ball’s location and motion record the existence of that passing object. (They also potentially record additional information, depending on how we set up the experiment, about the object’s motion-energy and its time of arrival.)

To appreciate a measurement involving quantum physics, it’s often best to first think through what happens in a pre-quantum version of the same scenario. Doing so gives us an opportunity to use two complementary views of the measurement: an intuitive one in physical space and more abstract one in the space of possibilities. This will help us interpret the quantum case, where an understanding of a measurement can only be achieved in the space of possibilities.

A Measurement in Pre-Quantum Physics

We’re going to imagine that an incoming projectile (which I’ll draw in purple) is moving along a straight line (which we’ll call the x-axis) and strikes the measuring device — the ball (which I’ll draw in blue) sitting inside its trap. To keep things simple enough to draw, I’ll assume that any collision that occurs will leave the ball and projectile still moving along the x-axis.

With these two objects restricted to a one-dimensional line, our space of possibilities will be two-dimensional, one dimension representing the possible positions x₁ of the projectile, and the other representing the possible positions x₂ of the ball. (If you are not at all familiar with the space of possibilities and how to think about it, I recommend you first read this article, which addresses the key ideas, and this article, which gives an example very much relevant to this post.)

Below in Fig. 4 is an animation showing what happens, from two viewpoints, as the projectile strikes the ball, allowing the ball’s motion to measure the passage of the projectile.

The first (at left) is the familiar viewpoint: what would happen before our eyes, in physical space, if these objects were big enough to see. The projectile moves to the right, with the ball stationary; a collision occurs, following which the projectile continues on the right, albeit a bit more slowly, and the ball, having popped out of its trap, moves off the the right.

The second viewpoint (at right) is not something we could see; it happens in the space of possibilities (or “configuration space,”) which we can see only in our minds. In this two-dimensional space, with axes that are the projectile’s and ball’s possible positions x₁ and x₂, the system — the combination of the projectile and ball — is at any moment sitting at one point. That point is indicated by a star; its location has as its x₁ coordinate the projectile’s position at a moment in time, while its x₂ coordinate is the ball’s position at that same moment in time.

The two animations are synchronized in time. I suggest you spend some time with the animation until it is clear to you what is happening.

• Initially, the star moves horizontally. This indicates that the value of x₂ isn’t changing; the ball is stationary. Both x₁ and x₂ are initially negative, so the star is in the lower-left quadrant.
• Notice the diagonal line, at x₁ = x₂ ; if the system is on that line, a collision between the two objects is occurring, since they are at the same point. It is when the star reaches this line that the ball begins to move, and the star’s motion is correspondingly no longer horizontal.
• After the collision, both the projectile and ball move to the right, which means the values of x₁ and x₂ are both increasing. This in turn means that the star moves up and to the right following the collision, eventually reaching the upper-right quadrant where both x₁ and x₂ are positive.

By contrast, if the measurement device were switched off, so that the projectile and the ball could no longer interact, the projectile would just continue its constant motion to the right, unchanged, and the ball would remain at its initial location, as in Fig. 5. In the space of possibilities, the star would move to the right as the projectile’s position x₁ steadily increases, while it would remain at the same vertical level because the ball’s position x₂ is never changing.

The Same Measurement in Quantum Physics

Now, how do we describe the measurement in quantum physics? In general we cannot portray what happens in a quantum system using only physical space. Instead, our system of two objects is described by a single wave function, which is a function of the space of possibilities. That is, it is a function of x₁ and x₂, and also time, since it changes from moment to moment. [Important: the system is not described by two wave functions (i.e., one per object), and the single wave function of the system is not a function of physical space, with its coordinate x. There is one wave function, and it is a function of all possibilities.]

At each moment in time, and for each possible arrangement of the system — for each of the possible locations of the two objects, with the projectile having position x₁ and the ball having position x₂ — this function gives us a complex number Ψ(x₁, x₂; t). The absolute value squared of this number gives us the probability of the corresponding possibility — the probability that if we choose to measure the positions of the projectile and ball, we will find the projectile has position x₁ and that the ball has position x₂.

What I’m going to do now is plot for you this wave function, using a 3d plot, where two of the axes are x₁ and x₂ and the third axis is the absolute value of Ψ(x₁, x₂; t). [Not its square, though the difference doesn’t matter much here.] The colors give the argument (or “phase”) of the complex number Ψ(x₁, x₂; t). As suggested by recent plots where we looked at wave functions for a single particle, the flow of the color bands often conveys the motion of the system across the space of possibilities; you’ll see this in the patterns below.

Going in the reverse order from above, let’s first look at the quantum wave function corresponding to Fig. 5, when no measurement takes place and the projectile passes by the ball unimpeded. You can see that the peak in the wave function, telling us most probable values for the results of measurements of x₁ and x₂, if carried out at a specific time t, moves along roughly the same path as the star in Fig. 5: the most probable values of x₁ increase steadily with time, while those of x₂ remain fixed.

In this situation, the ball’s behavior has nothing to do with the projectile. We cannot learn anything one way or the other about the projectile from the position or motion of the ball.

What about when a measurement takes place, as in Fig. 4? Again, as seen in Fig. 7, the majority of the wave function follows the path of the star, with the most probable values of x₂ beginning to increase around the most likely time of the collision. This change in the most likely value of x₂ is an indication of the presence of the projectile and its interaction with the ball. [Note: Fig. 7, unlike other quantum wave functions shown in this series, is a sketch, not a precise solution to the full quantum equations; I simply haven’t yet found a set-up where the equations can be solved numerically with enough precision and speed to get a nice-looking result. I expect I’ll eventually find an example, but it might take some time.]

More precisely, because of the collision, the motion of the ball is now correlated with that of the projectile — their motions are logically and physically related. That by itself is not unusual; all interactions between objects lead to some level of correlation between them. But this correlation is stable; as a result of the collision, the ball is highly unlikely to be found back in its initial position. And so, when we later look at the trap and find it empty, this does indeed give us reliable information about the projectile, namely that at some point it passed through the trap. (This type of correlation, both within and beyond the measurement context, will be a major topic in the future.)

So far, this all looks quite straightforward. The motion of the star in Fig. 4 is seen in the motion of the peak of the wave function in Fig. 7. Similar behavior is seen in Figs. 5 and 6. But these are simple cases: where the projectile’s motion is well-known, its location is not too uncertain, and the measurement device is almost perfect. We will soon explore far more complex and interesting quantum examples, using this simple one as our conceptual foundation, and things won’t be so straightforward anymore.

I’ll stop here for today. Please let me know in the comments if there are aspects of this story that you find confusing; we need all to be on the same page before we advance into the more subtle elements of our quantum world.

Update (Feb. 27): While we’re on the subject of theoretical computer science, friends-of-the-blog Adam Klivans and Raghu Meka have asked me to publicize that STOC’2025 TheoryFest, to be held June 23-27 in Prague, is eagerly seeking proposals for workshops. The deadline is March 9th.

• Because of a recent breakthrough by Cook and Mertz on Tree Evaluation, Ryan now shows that every problem solvable in t time on a multitape Turing machine is also solvable in close to √t space
  
• As a consequence, he shows that there are problems solvable in O(n) space that require nearly quadratic time on multitape Turing machines
  
• If this could be applied recursively to boost the polynomial degree, then P≠PSPACE
  
• On Facebook, someone summarized this result as “there exists an elephant that can’t fit through a mouse hole.” I pointed out that for decades, we only knew how to show there was a blue whale that didn’t fit through the mouse hole
  
• I’ll be off the Internet for much of today (hopefully only today?) because of jury duty! Good thing you’ll have Ryan’s amazing new paper to keep y’all busy…

Update (Feb. 25): It occurs to me that the new result is yet another vindication for Ryan’s style of doing complexity theory—a style that I’ve variously described with the phrases “ironic complexity theory” and “caffeinated alien reductions,” and that’s all about using surprising upper bounds for one thing to derive unsurprising lower bounds for a different thing, sometimes with a vertigo-inducing chain of implications in between. This style has a decidedly retro feel to it: it’s been clear since the 1960s both that there are surprising algorithms (for example for matrix multiplication), and that the time and space hierarchy theorems let us prove at least some separations. The dream for decades was to go fundamentally beyond that, separating complexity classes by “cracking their codes” and understanding the space of all possible things they can express. Alas, except for low-level circuit classes, that program has largely failed, for reasons partly explained by the Natural Proofs barrier. So Ryan achieves his successes by simply doubling down on two things that have worked since the beginning: (1) finding even more surprising algorithms (or borrowing surprising algorithms from other people), and then (2) combining those algorithms with time and space hierarchy theorems in clever ways to achieve new separations.

Next: Part 2

In the beginning, there were hardly any spaces whose magnitude we knew. Line segments were about the best we could do. Then Mark Meckes introduced the technique of potential functions for calculating magnitude, which was shown to be very powerful. For instance, Juan Antonio Barceló and Tony Carbery used it to compute the magnitude of odd-dimensional Euclidean balls, which turn out to be rational functions of the radius. Using potential functions allows you to tap into the vast repository of knowledge of PDEs.

In this post and the next, I’ll explain this technique from a categorical viewpoint, saying almost nothing about the analytic details. This is category theory as an organizational tool, used to help us understand how the various ideas fit together. Specifically, I’ll explain potential functions in terms of the magnitude of functors, which I wrote about here a few weeks ago.

Before I can describe this categorical viewpoint on potential functions, I have to explain what potential functions are in the magnitude context, and why they’re very useful. That’s what I’ll do today.

This part of the story is about metric spaces. For now I’ll assume they satisfy all the classical axioms, including symmetry of the metric, meaning that $d(a, a') = d(a', a)$ for all points $a$ and $a'$. When metric spaces are viewed as enriched categories, symmetry isn’t automatic — but we’ll come to that next time.

A weighting on a finite metric space $A$ is a function $w: A \to \mathbb{R}$ such that for all $a \in A$,

$$\sum_{a' \in A} e^{-d(a, a')} w(a') = 1.$$

Everyone who sees this formula for the first time asks where the exponential comes from. Ultimately it’s because of the enriched category viewpoint (which again we’ll come to next time), but the short story is that exponential is essentially the only reasonable function that converts addition into multiplication.

For simplicity, I’ll assume here that every finite metric space $A$ has a unique weighting, which I’ll call $w_A$. Since the definition of weighting involves the same number of equations as unknowns, this is generically true (and it’s always true for subspaces of $\mathbb{R}^n$), even though there are exceptions.

The magnitude of $A$ is

$$|A| = \sum_{a \in A} w_A(a) \in \mathbb{R}.$$

That’s for finite metric spaces. To extend the definition to compact metric spaces, there are various ideas you might try. You could define the magnitude of a compact space as the supremum of all the magnitudes of its finite subspaces. Or, you could take an ever-denser sequence of finite subsets of your compact space, then define its magnitude to be the limit of the magnitudes of the approximating subsets. Or, you could try replacing the sums in the formulas above by integrals, somehow.

Mark Meckes showed that all these approaches are equivalent. They all give the same definition of the magnitude of a compact space. (At least, this is true subject to a condition called “positive definiteness” which I won’t discuss and which always holds for subspaces of $\mathbb{R}^n$.)

How do we actually calculate magnitude, say for compact subspaces $A$ of $\mathbb{R}^n$?

In principle, you already know how do it. You run through all the finite subsets of $A$, calculate the magnitude of each using the definition above, and take the sup. The trouble is, this procedure is incredibly hard to work with. It’s completely impractical.

A slightly more practical approach is to look for a “weight measure”, that is, a signed measure $w$ on $A$ such that for all $a \in A$,

$$\int_A e^{-d(a, a')}\, d w(a') = 1.$$

As Mark showed, if $w$ is a weight measure then the magnitude of $A$ is given by $|A| = w(A)$. This is the analogue of the formula $|A| = \sum_a w(a)$ for finite spaces.

Example   Take an interval $[c, d]$ in $\mathbb{R}$. It’s an elementary exercise to check that

$$\tfrac{1}{2}(\delta_c + \lambda_{[c, d]} + \delta_d)$$

is a weight measure on $[c, d]$, where $\delta_c$ and $\delta_d$ denote the Dirac deltas at $c$ and $d$, and $\lambda_{[c, d]}$ is Lebesgue measure on $[c, d]$. It follows that $[c, d]$ is the total mass of this measure, which is

$$1 + \tfrac{1}{2}(d - c).$$

In other words, it’s $1$ plus half the length of the interval.

The trouble with weight measures is that very few spaces have them. Even Euclidean balls don’t, beyond dimension one. It turns out that we need something more general than a measure, something more like a distribution.

In another paper, Mark worked out exactly what kind of measure-like or distribution-like thing a weighting should be. The answer is very nice, but I won’t explain it here, because I want to highlight the formal aspects above all.

So I’ll simply write the weighting on a metric space $A$ as $w_A$, without worrying too much about what kind of thing $w_A$ actually is. All that matters is that it behaves something like a measure: we can pair it with “nice enough” functions $\phi: A \to \mathbb{R}$ to get a real number, which I’ll write as

$$\int_A \phi(a) \, d w_A(a)$$

or simply

$$\int_A \phi \, d w_A.$$

And I’ll assume that all the spaces $A$ that we discuss do have a unique weighting $w_A$.

That’s the background. But back to the question: how do we actually calculate magnitude?

First main idea   Don’t look at metric spaces $A$ in isolation. Instead, consider spaces embedded in some big space $B$ that we understand well.

Think of the big space $B$ as fixed. The typical choice is $\mathbb{R}^n$. One sense in which we “understand” $\mathbb{R}^n$ well is that we have a weight measure on it: it’s just Lebesgue measure divided by a constant factor $c_n$. This follows easily from the fact that $\mathbb{R}^n$ is homogeneous. (It doesn’t matter for anything I’m going to say, but the constant $c_n$ is $\int_{\mathbb{R}^n} e^{-\|x\|}\, d x$, for which there’s a standard formula involving $\pi$s and factorials.)

The potential function of a subspace $A \subseteq B$ is the function $h_A: B \to \mathbb{R}$ defined by

$$h_A(b) = \int_A e^{-d(a, b)} \, d w_A(a).$$

By definition of weighting, $h_A$ has constant value $1$ on $A$. But outside $A$, it could be anything, and it turns out that its behaviour outside $A$ is very informative.

Although $w_A$ is a measure-like thing on the subspace $A$ of $B$, we typically extend it by $0$ to all of $B$, and then the definition of the potential function becomes

$$h_A(b) = \int_B e^{-d(x, b)} \, d w_A(x)$$

Examples   In all these examples, I’ll take the big space $B$ to be $\mathbb{R}$.

• Let $A = \{0\}$. Then the weighting $w_A$ on $A$ is the Dirac delta at $0$, and the potential function $h_A: \mathbb{R} \to \mathbb{R}$ is given by

  $$h(b) = e^{-|b|}$$

  ($b \in \mathbb{R}$), whose graph looks like this:

  

• Let $A = [-2, 2]$. As we’ve already seen,

  $$w_A = \tfrac{1}{2}(\delta_{-2} + \lambda_{[-2, 2]} + \delta_2),$$

  and a little elementary work then reveals that the potential function $h_A: \mathbb{R} \to \mathbb{R}$ is given by

  $$h_A(b) = \begin{cases} e^{-(-2 - b)} &\text{if} &nbsp; b \leq -2, \\ 1 &\text{if} &nbsp; -2 \leq b \leq 2, \\ e^{-(b - 2)} &\text{if} &nbsp; b \geq 2, \end{cases}$$

  which looks like this:

  

• In the two examples we just did, the potential function of $A$ is just the negative exponential of the distance between $A$ and the argument $b$. That’s not exactly coincidence: as we’ll see in the next part, the function just described corresponds to a strict colimit, whereas the potential function corresponds to a lax colimit. So it’s not surprising that they coincide in simple cases.

  But this only happens in the very simplest cases. It doesn’t happen for Euclidean balls above dimension $1$, or even for two-point spaces. For example, taking the subset $A = \{-1, 1\}$ of $B = \mathbb{R}$, we have

  $$h_A(x) = \frac{e^{-|x + 1|} + e^{-|x - 1|}}{1 + e^{-2}},$$

  which looks like this:

  

  whereas $b \mapsto e^{-d(A, b)}$ looks like this:

  

  Similar, but different!

And now we come to the:

Second main idea   Work with potential functions instead of weightings.

To explain what this means, I need to tell you three good things about potential functions.

• The potential function determines the magnitude:

  $$|A| = \int_B h_A \, d w_B.$$

  At the formal level, proving this is a one-line calculation: substitute the definition of $h_A$ into the right-hand side and follow your nose.

  For example, we saw that $\mathbb{R}^n$ has weighting $\lambda/c_n$, where $\lambda$ is Lebesgue measure and $c_n$ is a known constant. So

  $$|A| = \frac{1}{c_n} \int_{\mathbb{R}^n} h_A(x) \, d x$$

  for compact $A \subseteq \mathbb{R}^n$. Here $d x$ refers to ordinary Lebesgue integration.

  (For $\mathbb{R}^n$, and in fact a bit more generally, this result appears as Theorem 4.16 here.)

• You can recover the weighting from the potential function. So, you don’t lose any information by working with one rather than the other.

  How do you recover it? Maybe it’s easiest to explain in the case when the spaces are finite. If we write $Z_B$ for the $B \times B$ matrix $(e^{-d(b', b)})_{b', b \in B}$ then the definition of the potential function can be expressed very succinctly:

  $$h_A = Z_B w_A.$$

  Here $h_A$ and $w_A$ are viewed as column vectors with entries indexed by the points of $B$. (For $w_A$, those entries are $0$ for points not in $A$.) Assuming $Z_B$ is invertible, this means we recover $w_A$ from $h_A$ as $Z_B^{-1} h_A$. And something similar is true in the non-finite setting.

  However, what really makes the technique of potential functions sing is that when $B = \mathbb{R}^n$, there’s a much more explicit way to recover the weighting from the potential function:

  $$w_A = (I - \Delta)^{(n + 1)/2} h_A$$

  (up to a constant factor that I’ll ignore). Here $I$ is the identity and $\Delta$ is the Laplace operator, $\sum_{i = 1}^n \frac{\partial^2}{\partial x_i^2}$. This is Proposition 5.9 of Mark’s paper.

  How much more of this bullet point you’ll want to read depends on how interested you are in the analysis. The fundamental point is simply that $(I - \Delta)^{(n + 1)/2}$ is some kind of differential operator. But for those who want a bit more:

  • To make sense of everything, you need to interpret it all in a distributional sense. In particular, this allows one to make sense of the power $(n + 1)/2$, which is not an integer if $n$ is even.

  • Maybe you wondered why someone might have proved a result on the magnitude of odd-dimensional Euclidean balls only, as I mentioned at the start of the post. What could cause the odd- and even-dimensional cases to become separated? It’s because whether or not $(n + 1)/2$ is an integer depends on whether $n$ is odd or even. When $n$ is odd, it’s an integer, which makes $(I - \Delta)^{(n + 1)/2}$ a differential rather than pseudodifferential operator. Heiko Gimperlein, Magnus Goffeng and Nikoletta Louca later worked out lots about the even-dimensional case, but I won’t talk about that here.

  • Finally, where does the operator $(I - \Delta)^{(n + 1)/2}$ come from? Sadly, I don’t have an intuitive explanation. Ultimately it comes down to the fact that the Fourier transform of $e^{-\|\cdot\|}$ is $\xi \mapsto (1 + \|\xi\|)^{-(n + 1)/2}$ (up to a constant). But that itself is a calculation that’s really quite tricky (for me), and it’s hard to see anything beyond “it is what it is”.

• The third good thing about potential functions is that they satisfy a differential equation, in the situation where our big space $B$ is $\mathbb{R}^n$. Specifically:

  $$\begin{cases} h_A \equiv 1 &\text{on} &nbsp; A \\ (I - \Delta)^{-(n + 1)/2} h_A \equiv 0&\text{on} &nbsp; \mathbb{R}^n \setminus A. \end{cases}$$

  Indeed, the definition of weighting implies that $h_A \equiv 1$ on $A$, and the “second good thing” together with the fact that $w_A$ is supported on $A$ give the second clause.

  Not only do we have a differential equation for $h_A$, we also have boundary conditions. There are boundary conditions at the boundary of $A$, because of something I’ve been entirely vague about: the functions we’re dealing with are meant to be suitably smooth. There are also boundary conditions at $\infty$, because our functions are also meant to decay suitably fast.

  Maybe Mark will read this and correct me if I’m wrong, but I believe there are exactly the right number of boundary conditions to guarantee that there’s (typically? always?) a unique solution. In any case, the following example — also taken from Mark’s paper — illustrates the situation.

Example   Let’s calculate the magnitude of a real interval $A = [c, d]$ using the potential function method.

Its potential function $h_A$ is a function $\mathbb{R} \to \mathbb{R}$ such that $h_A \equiv 1$ on $[c, d]$ and $h_A - h''_A = 0$ on the rest of the real line. That differential equation comes from taking $(I - \Delta)^{-(n + 1)/2}$ in the case $n = 1$.

The functions $f$ satisfying $f'' = f$ are those of the form $k e^{\pm x}$ for some constant $k$, and in our case we’re free to choose the constant and the $\pm$ sign differently on the two connected components of $\mathbb{R} \setminus A$. So there are lots of solutions. But $h_A$ is required to be continuous and to converge to $0$ at $\pm \infty$, and that pins it down uniquely: the one and only solution is

$$h_A(x) = \begin{cases} e^{-(c - x)} &\text{if} &nbsp; x \leq c, \\ 1 &\text{if} &nbsp; c \leq x \leq d, \\ e^{-(x - d)} &\text{if} &nbsp; d \leq x. \end{cases}$$

I showed you the graph of this potential function above, in the case where $A = [-2, 2]$.

So the magnitude of $A = [c, d]$ is

$$\frac{1}{c_1} \int_\mathbb{R} h_A(x)\, d x,$$

where $c_1$ is the constant $\int_\mathbb{R} e^{-|x|} \, d x = 2$. This gives the answer:

$$|A| = 1 + \tfrac{1}{2}(d - c) = 1 + \tfrac{1}{2}length(A).$$

The crucial point is that in this example, we didn’t have to come up with a weighting on $A$. The procedure was quite mechanical. And that’s the attraction of the method of potential functions.

Next time, I’ll put all this into a categorical context using the notion of the magnitude of a functor, which I introduced here.

Despite the “2” in the title, you can follow this post without having read part 1. The whole point is to sneak up on the metricky, analysisy stuff about potential functions from a categorical angle, by considering constructions that are categorically reasonable in their own right. Let’s go!

Let $V$ be a monoidal category, which I’ll assume is symmetric, closed, complete or cocomplete as necessary. Take a pair of $V$-categories $A$ and $B$, and a $V$-functor

$$F: A \to B.$$

Then we get a new functor

$$N_F: B \to [A^{op}, V]$$

defined by

$$(N_F(b))(a) = B(F a, b).$$

This is sometimes called the nerve functor induce by $F$, because of the first example in the following list.

Examples   In the first few examples, I’ll take the enriching category $V$ to be $Set$.

• Let $F$ be the inclusion $\Delta \hookrightarrow Cat$, which takes a finite nonempty totally ordered set $[n] = \{0, \ldots, n\}$ and realizes it as a category in the usual way. Then $N_F: Cat \to [\Delta^{op}, Set]$ sends a small category $D$ to its nerve, which is the simplicial set whose $n$-simplices are paths in $D$ of length $n$.

• Let $F$ be the functor $\Delta \to Top$ that sends $[n]$ to the topological $n$-simplex $\Delta^n$. Then $N_F: Top \to [\Delta^{op}, Set]$ sends a topological space $X$ to its “singular simplicial set”, whose $n$-simplices are the continuous maps $\Delta^n \to X$.

• Let $F: A \hookrightarrow B$ be an inclusion of partially ordered sets, viewed as categories in the usual way. Then $N_F : B \to [A^{op}, Set]$ is given by

  $$(N_F(b))(a) = \begin{cases} 1 &\text{if} &nbsp; b \leq a \\ 0 &\text{otherwise}. \end{cases}$$

  For example, let $B$ be the set $\mathbb{Z}$ of integers, ordered by divisibility. Let $A \subseteq \mathbb{Z}$ be the set of (positive) primes. Then $N_F$ is the functor $\mathbb{Z} \to Set^{\{primes\}}$ that sends $n \in \mathbb{Z}$ to

  $$([p | n])_{\text{primes} &nbsp; p}$$

  where $[p|n]$ is Iverson notation: it’s $1$ if $p$ divides $n$ and $0$ otherwise.

• If $F$ has a right adjoint $G$, then $N_F(b) = A(-, G b): A^{op} \to V$. In particular, $N_F(b)$ is representable for each $b \in B$.

• Take the inclusion $Field \hookrightarrow Ring$, where $Ring$ is the category of commutative rings. Now take opposites throughout to get $F: Field^{op} \hookrightarrow Ring^{op}$. The resulting nerve functor

  $$N_F: Ring^{op} \to [Field, Set]$$

  sends a ring $R$ to $Ring(R, -)|_{Field}$ (by which I mean the restriction of $Ring(R, -): Ring \to Set$ to the category of fields). More explicitly,

  $$N_F(R) = \sum_{p \in Spec(R)} Field(Frac(R/p), -).$$

  Here $Spec(R)$ is the set of prime ideals $p$ of $R$, and $Frac(R/p)$ is the field of fractions of the integral domain $R/p$. In particular, $N_F(b)$ is a coproduct of representables for each $b \in B$. (In the terminology of Diers, one therefore says that $Field \hookrightarrow Ring$ is a “right multi-adjoint”. A closely related statement is that it’s a parametric right adjoint. But that’s all just jargon that won’t matter for this post.)

Now let’s consider $V = (\mathbb{R}^+, \geq)$ with its additive monoidal structure, so that $V$-categories are generalized metric spaces and $V$-functors are the functions $f$ between metric spaces that are variously called short, distance-decreasing, contractive, or 1-Lipschitz: $d(f(a), f(a')) \leq d(a, a')$ for all $a, a'$. I’ll just call them “maps” of metric spaces.

• Take a map $f: A \to B$ of metric spaces. The induced map $N_f: B \to [A^{op}, \mathbb{R}^+]$ is given by

  $$(N_f(b))(a) = d(f(a), b).$$

  Unlike in the first post, I’m not going to assume here that our metric spaces are symmetric. so the “op” on the $A$ matters, and $d(a, f(b)) \neq d(f(b), a)$ in general.

  We’ll be particularly interested in the case where the $V$-functor $F$ is full and faithful. Most of the examples above have this property. In the metric case, $V = \mathbb{R}^+$, being full and faithful means being distance-preserving, or in other words, an inclusion of a metric subspace. In that case, it’s normal to drop the $f$. So we’d usually then write

  $$(N_f(b))(a) = d(a, b).$$

Next I’ll define the potential function of a $V$-functor. For this we need some of the apparatus of magnitude. Roughly speaking, this means we have a notion of the “size” of each object of our base monoidal category $V$.

More exactly, we take a field $k$ and a function $|\cdot|: ob(V) \to k$, to be thought of as assigning to each object $S$ of $V$ its size ${|S|} \in k$. And to make everything work, we assume our size function $|\cdot|$ has reasonable properties:

$$S \cong T \implies |S| = |T|, \quad |S \otimes T| = |S| \cdot |T|, \quad |I| = 1,$$

where $\otimes$ is the monoidal product on $V$ and $I$ is its unit.

The basic definitions are these. A weighting on a finite $V$-category $A$ is a function $w_A : ob(A) \to k$ such that

$$\sum_{b \in A} |A(a, b)| w_A(b) = 1$$

for all $a \in A$, and a coweighting $w^A$ on $A$ is a weighting on $A^{op}$. Although it’s not true that every finite $V$-category has a unique weighting, it’s a harmless enough assumption that I’ll make it here. The magnitude of a finite $V$-category $A$ is

$$|A| = \sum_{a \in A} w_A(a) = \sum_{a \in A} w^A(a) \in k.$$

It’s a tiny lemma that the total weight $\sum_{a \in A} w_A(a)$ is equal to the total coweight $\sum_{a \in A} w^A(a)$, so you can equivalently define the magnitude to be one or the other.

There’s a whole lot to say about why this is a worthwhile definition, and I’ve said it in detail many times here before. But here I’ll just say two short things:

• Taking $V = FinSet$, $k = \mathbb{Q}$, and $|\cdot|$ to be cardinality, the magnitude of a finite category is equal to the Euler characteristic of its classifying space, under suitable hypotheses guaranteeing that the latter is defined.

• Take $V = \mathbb{R}^+$, $k = \mathbb{R}$, and $|\cdot|$ to be $s \mapsto e^{-s}$ (because that’s just about the only function converting the tensor product of $V$ into addition, as our size functions are contractually obliged to do). Then we get a notion of the magnitude of a metric space, a real-valued invariant that turns out to be extremely interesting.

That’s the magnitude of enriched categories. But we can also talk about the magnitude of enriched presheaves. Take a finite $V$-category $A$ and a $V$-presheaf $X: A^{op} \to V$. Its magnitude is

$$|X| = \sum_{a \in A} w^A(a) |X(a)| \in k.$$

When I wrote about the magnitude of functors before, I concentrated on covariant functors $A \to V$, which meant that the weights in the weighted sum that defines $|X|$ were, well, the weights $w_A(a)$. But since we’ve now changed $A$ to $A^{op}$, the weights have become coweights.

Let me briefly recall why the magnitude of presheaves is interesting, at least in the case $V = FinSet$:

• If our presheaf $X$ is a coproduct of representables then $|X|$ is $|colim(X)|$, the cardinality of the colimit of $X$.

• The magnitude of presheaves generalizes the magnitude of categories. The magnitude of a category $A$ is equal to the magnitude of the presheaf $A^{op} \to Set$ with constant value $1$.

• In the other direction, the magnitude of categories generalizes the magnitude of presheaves: $|X| = |\mathbb{E}(X)|$ for all presheaves $X$. Here $\mathbb{E}(X)$ means the category of elements of $X$, also called the (domain of the) Grothendieck construction.

  It’s enlightening to think of this result as follows. If we extend the codomain of $X$ from $Set$ to $Cat$ then $\mathbb{E}(X)$ is the colax colimit of $X: A^{op} \to Cat$. So $|X| = |colaxcolim(X)|$. You can compare this to the more limited result that $|X| = |colim(X)|$, which only holds when $X$ has the special property that it’s a coproduct of representables.

  There’s also another closely related result, due to Ponto and Shulman: under mild further hypotheses, $|X| = \chi(hocolim(X))$. Mike and I had a little discussion about the relationship here.

  Now here’s something funny. Which do you think is a more refined invariant of a presheaf $X: A^{op} \to FinSet$: the magnitude $|X|$, which is equal to $|colaxcolim(X)|$ and $\chi(hocolim(X))$, or the cardinality $|colim(X)|$ of the strict colimit?

  From one perspective, it’s the magnitude. After all, we usually think of lax or pseudo 2-dimensional things as more subtle and revealing than strict 1-dimensional things. But from another, it’s the cardinality of the strict colimit. For instance, if $A$ is the one-object category corresponding to a group $G$ then $X$ is a right $G$-set, the cardinality of the colimit is the number of orbits (an interesting quantity), but the magnitude is always just the cardinality of the set $X$ divided by the order of $G$ (relatively boring and crude).

  In this post I’ll keep comparing these two quantities. Whichever seems “more refined”, the comparison is interesting in itself.