The n-Category Café

Good point, Scott. Will fix!

Really?! That’s very interesting. Can you say a bit about how that works? I got the impression that SUSY was essential for coupling constant convergence.

I was under the same impression until I was reading this review of particle physics beyond the Standard Model by Peskin:

http://arxiv.org/abs/hep-ph/9705479

which is mostly about supersymmetry. But on p17 he gives a nice formula for the renormalization flow of the coupling constants, dependent on the number of Higgs doublets, $n_H$. Then he has the idea to try assuming the coupling constants meet and work backwards to find $n_H$. He gives the resulting simple formula for it on p18. I don’t know why he doesn’t just solve for $n_H$ at that point (maybe I just missed that discussion elsewhere in the paper) but if you just solve for $n_H$ given the information there (which I couldn’t resist), you get $8.0$. (I’m not sure what this value is with current experimental data. Wonder if it stays close to an integer?)

Thinking on it, it’s not too surprising that you can get three crossing lines to meet at one point by adjusting one parameter. It certainly is the party line that supersymmetry is necessary for coupling convergence, but I wasn’t invited to that party.

Thanks, Garrett! Bring on the Higgses! One octonion-valued Higgs, please!

We’ll fix this bit in our paper.

Sure thing – happy to help.

I have a concern about a kind of subtle point. On page 17 you describe these “complex” W bosons as spanning sl(2,C), but these are actually the weight vectors (complex eigenvectors) under the action of real su(2) on the W bosons in su(2). It happens quite often that eigenvectors of some physical system (with real variables) will be complex like this. To correspond with reality, the complex eigenvectors need to be summed to a real combination. I think this is what’s going on with these W bosons, but I’m not sure of an easier way to express it, or if it’s more accurate just to keep it the way it is, with W “states” in sl(2,C), because this is the way they’re handled in QFT.

Garrett wrote:

I have a concern about a kind of subtle point. On page 17 you describe these “complex” W bosons as spanning sl(2,C), but these are actually the weight vectors (complex eigenvectors) under the action of real su(2) on the W bosons in su(2).

You’re right, it’s a subtle point. The problem is that we’re trying to talk about GUTs in a very watered-down way, where we use finite-dimensional complex Hilbert spaces to describe the internal quantum states of our particles, while ignoring the spacetime aspects of their quantum states: that is, everything about their momentum or spin.

This gets a bit nerve-racking whenever we deal with uncharged particles, which are described by real-valued fields (or tuples of real-valued fields) on spacetime. The simplest case would be a massive scalar field. The solutions of these equations are real-valued functions on spacetime, yet in quantum field theory we make the space of these solutions into a complex Hilbert space. The same thing happens with the electromagnetic field or (in a vastly more complicated way) $SU(2)$ Yang–Mills theory.

How do we get a finite-dimensional complex Hilbert space for such theories? I’m not sure there’s a sensible prescription in general, but in this paper we’re using $sl(2,\mathbb{C})$ as the finite-dimensional Hilbert space describing the internal degrees of freedom of an $SU(2)$ gauge boson… and similarly for other compact Lie groups: we use the complexified adjoint representation.

For pions — the gauge bosons in the original Heisenberg–Cassen–Condon–Yang–Mills theory of the strong interaction — I think it’s pretty darn common to use $sl(2,\mathbb{C})$ as the Hilbert space of internal degrees of freedom. This Hilbert space has $\pi^+, \pi^0, \pi^-$ as basis vectors. These basis vectors don’t lie in the real subspace $su(2) \subseteq sl(2,\mathbb{C})$!

Luckily nothing very important in our paper depends on this issue, since we’re mainly focused on fermions.

Note: In this comment I have restrained myself from using fancy $\mathfrak{gothic}$ letters for Lie algebras.

You did a great job.

You are right about the photon diagram at page 4. In the case of the \pi^- on page 9, the ambiguity is solved by the arrow: if we think at the p in the right as emitting the pion, the arrow tells us that it will be a \pi^+ in fact. But in the page 17, the W^- diagram has the problem that \nu can be interpreted to disintegrate in W^- and e^-.

Todd wrote:

I’m looking at the Feynman diagram on page 4, and I notice that the virtual photon seems to traverse a region of space-time outside the light cone (if the light cone is at a 45 degree angle). I take it that that’s both intentional and “logically” necessary (by quantum field theory), but I’d love to hear more commentary from you on this, as people like me grew up hearing that photons travel at the speed of light.

John H. already answered your question, but here’s another take on it.

From one viewpoint, Feynman diagrams are just string diagrams for morphisms in the category of unitary representations of a group — namely, the Poincaré group times whatever ‘internal symmetry group’ our theory has. (The internal symmetry groups for four famous theories are shown at the top of the cube above).

Now, you know and love string diagrams, so this perspective should be congenial to you. In particular, you know that it doesn’t matter much how you draw these diagrams: in a rough sense, only their topology matters. So, you shouldn’t worry much about the slope of a particular line in a Feynman diagram.

But there’s another viewpoint, closer to Feynman’s original outlook. Feynman diagrams are a trick for describing morphisms a category, but these morphisms are linear operators, and to compute the ‘matrix elements’ of these operators we need to do integrals. So: a Feynman diagram is a recipe for writing down an integral.

Of course a Feynman diagram is a graph, with vertices and edge. There are vertices at the very top and bottom of the picture, but also ‘internal vertices’ in the middle. Each internal vertex represents a point where some particles collide and interact. We don’t know where this happened. So, for each interval vertex, we do an integral where some variable $x$ ranges over Minkowski spacetime.

Similarly, each edge represents a particle going from some point $x$ to some point $x'$. We include a factor of $G(x,x')$ for each edge in our diagram: this function is the amplitude for the particle to go from $x$ to $x'$.

So, the recipe works like this: we multiply the functions $G(x,x')$ together and integrate over all the variables corresponding to internal vertices in our graph. We’re left with a function of the variables $x$ corresponding to the vertices at the top and bottom of the graph. This function is the amplitude for a bunch of particles to start at certain positions, do their thing and wind up at certain positions.

Sounds simple, huh?

It is! True, I’ve simplified the story a bit by assuming the ‘internal symmetry group’ is trivial, and also by pretending all the particles have spin 0. If we drop these assumptions, our functions $G(x,x')$ sprout a bunch of superscripts and subscripts which need to get summed over while we do our integral. But this complication is 1) not a big deal if you’ve ever used string diagrams to describe intertwining operators between group representations, and 2) utterly irrelevant to your actual question.

Okay, that completes the crash course on Feynman diagrams. Now, what about the function $G(x,x')$, which says the amplitude for a particle to get from $x$ to $x'$? It’s nonzero even when $x$ and $x'$ are spacelike separated. In other words, there’s a nonzero amplitude for a particle to get from here to there even if it needs to go faster than light!

This fact has launched a thousand heated discussions, but I’ll summarize them in two words: it’s okay.

In particular, you can’t use quantum field theory to communicate faster than light. My pal Matt McIrvin wrote a good explanation of this in item 4 of the virtual particles FAQ.

Nonetheless, it’s fascinating to think about these things, and I’ve only scratched the surface. The function $G(x,x')$ is called the Feynman propagator. You may enjoy the Wikipedia discussion of its puzzling properties, and the explicit formulas.

The Wikipedia article even has pictures of the Feynman propagator! You can see how it’s nonzero outside the lightcone, though it dies off quickly:

I should have read ahead…it’s all detailed in the pages after where I stopped; this extra explanation also helps. There are many places where you have a tricky choice of picking between a particle and its antiparticle, up vs down, left vs. right,…It looks like the final “solution” in the paper fits everything the best. One thing I noticed about the bits in these binary code represntation : they cannot be expressed as linear combinations of the weights of the corresponding representation. Hypercharge, isospin, and charge can; not sure why I expected these bits to do the same.

Okay, now I’ve included a proof that the above square is a pullback square — it’s Theorem 8 around page 65 of the paper.

The proof of this theorem is a dry little diagram chase. The conceptually interesting result, from which this theorem follows, is Theorem 7. This exhibits the ‘true gauge group of the Standard Model’ as the subgroup of $SO(10)$ that preserves a complex structure on $\mathbb{R}^{10}$, a complex volume form, and a splitting into 2d and 3d complex subspaces.

I am charmed by this description of the standard model gauge group as a pullback.

I hadn’t been aware of this nice description.

By the way: above you say that

$$\array{ G_{SM}/\mathbb{Z}_6 &\to& SU(5) \\ \downarrow && \downarrow \\ Spin(4) \times Spin(6) &\to& Spin(10) }$$ is a pullback, while on on p. 66 of your notes in the version as it is right this moment, it has

$$\array{ G_{SM} &\to& SU(5) \\ \downarrow && \downarrow \\ Spin(4) \times Spin(6) &\to& Spin(10) }$$

without the $\mathbb{Z}_6$-quotient.

While it is probably totally unrelated, this idea of looking at pullback diagrams of groups reminded me of what I am currently thinking about:

in these notes I linked to # there is some simple but useful general nonsense on “twisted cohomology”, p. 11.

In the plain vanilla version this starts with an exact sequence $\hat A \to A \to B$ of pointed $\infty$-groupoids, forming a homotopy pullback

$$\array{ \hat A &\to& * \\ \downarrow && \downarrow \\ A &\to& B }$$

which leads for each space $X$ to an exact sequence of cohomologies with coefficients in these

$$\array{ H(X,\hat A) &\to& * \\ \downarrow && \downarrow \\ H(X,A) &\to& H(X,B) } \,.$$

Here the vertical morphism on the right picks the trivial cocycle in $H(X,B)$.

Now, given any cocycle $c \in H(X,B)$ it makes sense to address the homotopy pullback $H^{[c]}(X,\hat A)$ in

$$\array{ H^{[c]}(X,\hat A) &\to& * \\ \downarrow && \downarrow^c \\ H(X,A) &\to& H(X,B) }$$

as the $c$-twisted $\hat A$-cohomology on $X$.

It’s a simple idea but happens to capture lots of concepts of twisted cohomology that one finds in nature, notably twisted K-classes and their higher relatives.

(It can notably also be used describes non-flat (and non-fake-flat) $\infty$-connections as $F$-twisted flat connections for $F$ the curvature (! :-) as described on p. 15.)

(Just a moment, and I’ll get back to pullbacks of groups.)

Now, it turns out that of relevance in many situation in higher gauge and gravity theories is that where the twist $c$ arises itself as the obstruction to a lift through some other exact sequence $$\hat A' \to A' \to B$$ of $\infty$-groupoids, with the same $B$.

As described on p. 12 the corresponding “bitwisted” cohomology on $X$ is cohomology $H(X,A \times_B A')$ with coefficients in the homotopy pullback $\infty$-groupoid $A \times_B A'$

$$\array{ A \times_B A' &\to& A' \\ \downarrow && \downarrow \\ A &\to& B } \,.$$

In particular, when $A \to B$ and $A' \to B$ are morphisms of $\infty$-groups describing classifying maps in the Whitehead towers of the groups $O(n)$ and $U(n)$, this bi-twisted cohomology captures the central mechanism behind all those phenomena going by names such as “Freed-Witten anomaly cancellation” and “Green-Schwarz anomaly cancellation” etc., which once started making people excited about higher gauge theories.

So, you see, it is just free association, but seeing that pullback square for the standard model group reminded me of all these homotopy pullback squares for gauge groups appearing in 10-dimensional supergravity.

Probably just a meaningless coincidence.

Urs writes:

I am charmed by this description of the standard model gauge group as a pullback.

Thanks! I want to see if it’s good for something. It might shed some light on the chain of symmetry-breakings that lead from $SO(10)$ down to the Standard Model gauge group. Someday I want to think about Higgs bosons in terms of this pullback picture. That’s one of the secret reasons for this question.

By the way: above you say that

$$\array{ G_{SM}/\mathbb{Z}_6 &\to& SU(5) \\ \downarrow && \downarrow \\ Spin(4) \times Spin(6) &\to& Spin(10) }$$

is a pullback, while on on p. 66 of your notes in the version as it is right this moment, it has

$$\array{ G_{SM} &\to& SU(5) \\ \downarrow && \downarrow \\ Spin(4) \times Spin(6) &\to& Spin(10) }$$

without the $\mathbb{Z}_6$-quotient.

Thanks again! I think we’re okay here: we do draw the picture

$$\array{ G_{SM} &\to& SU(5) \\ \downarrow && \downarrow \\ Spin(4) \times Spin(6) &\to& Spin(10) }$$

but we don’t say it’s a pullback. Instead, we say

Because this commutes, the image of $G_{SM}$ lies in the intersection of the images of $Spin(4) \times Spin(6)$ and $SU(5)$ inside $Spin(10)$. But we claim it is precisely that intersection!

The map from $G_{SM}$ to $Spin(10)$ is not one-to-one: it has a $\mathbb{Z}_6$ kernel, so the image of $G_{SM}$ is $G_{SM}/\mathbb{Z}/6$. That’s where the $\mathbb{Z}/6$ comes from.

Similarly, the image of $Spin(4) \times Spin(6)$ in $Spin(10)$ is $(Spin(4) \times Spin(6))/\mathbb{Z}_2$.

So, the above commuting square gives this commuting square of inclusions:

$$\array{ G_{SM}/\mathbb{Z}_6 &\to& SU(5) \\ \downarrow && \downarrow \\ (Spin(4) \times Spin(6))/\mathbb{Z}_2 &\to& Spin(10) }$$

and in Theorem 8 we show this diagram is a pullback.

In Theorem 7 we show that this simpler diagram is also a pullback of inclusions:

$$\array{ G_{SM}/\mathbb{Z}_6 &\to& SU(5) \\ \downarrow && \downarrow \\ SO(4) \times SO(6) &\to& SO(10) }$$

In our paper we never actually address the diagram in my blog comment above:

$$\array{ G_{SM} &\to& SU(5) \\ \downarrow && \downarrow \\ Spin(4) \times Spin(6) &\to& Spin(10) }$$

and I’m too tired now to figure out if the left vertical arrow here is even well-defined, much less whether this square is a pullback.

These quotients by discrete subgroups are fairly stressful! Life is much more relaxing at the Lie algebra level. But, there’s a lot of physics lurking in that $\mathbb{Z}_6$ kernel.

I’d been thinking of these squares as pullbacks, but maybe you’re right that they’re also pushouts. I’d like to prove that — it should be very easy if true, but it’s almost my bedtime now. And if true, I’d like to include this result in the paper. And I’d like to cite you for this part of the idea. Maybe you can email me a name that’s a bit more formal than “RodM” — though my sense of humor lights up at the idea of acknowledging “RodM” for help in a paper.

This paper is beautifully written. The style makes it a joy to read. And the level is tantalizingly almost accessible for an incompetent like me; it makes clear the math needed for a foundation and it shows how to use those techniques to describe a physical theory. I’m hooked! I have a lot of work to do, but at least I know where to direct my efforts to maximize my understanding. Thank you both!

(Any suggestions on where to get an elementary introduction to group representation theory, with basic examples?)

I’m glad you’re enjoying the paper, Charlie.

Unfortunately the only reason we were able to make it so short is by assuming the reader is already comfortable with standard math stuff like Lie groups, Lie algebras, representations of these, intertwining operators, exterior algebras and Clifford algebras — and some examples, like the Lie groups $SU(n)$, $SO(n)$, and $SL(n)$.

This paper is a kind of sequel to a course on ‘the algebra of particle physics’ which I taught back in 2003. That course covered a lot of the ‘standard math stuff’ that I’m assuming knowledge of here. Unfortunately that course is only available in the form of scanned-in handwritten notes. But hey — it’s free!

You can also learn most of this math (but not Clifford algebras) from my book Gauge Fields, Knots and Gravity.

Here are some other books you might try. I’ve tried to pick ones that are very gentle, but check them out and see what you think.

• Anthony Sudbery, Quantum Mechanics and the Particles of Nature: An Outline for Mathematicians, Cambridge U. Press, 1986. (I don’t think you need to be a mathematician to find this book helpful. Out of print — a criminal shame. But it’s still not very expensive on Amazon: about 50 bucks.)
• Morton Hamermesh, Group Theory and its Applications to Physical Problems, Dover, 1989. (This focuses more on applications to crystallography and chemistry, but some people swear by it — and being from Dover, it’s quite cheap!)
• Harry Lipkin, Lie Groups for Pedestrians, Dover, 2002. (I’ve never read this one, but the reviews say it’s good. It focuses on applications of particle physics. Even better, it’s only 10 bucks! Please don’t get run over while reading it.)

One of the things I love about the Arxiv is that I can download the source of people’s papers and change their textwidth when I find it unreadably large. I can also alter their bibliographic keys so that I know which papers they’re referring to without constantly having to flick to the end, and correct their typos.

I’m honoured!

I hope you didn’t get too much pleasure from creating a file called guts_for_tom_leinster.

I’m afraid I did — but it was guilty pleasure. I was just getting ready to send you an email apologizing for it!

I guess I don’t have a finely tuned sense of page widths. I’ll try to acquire one, starting now.

In principle I like Eugenia’s approach of downloading TeX from the arXiv and modifying it to my own taste. However, I may be too lazy to actually do it. But I can imagine stealing certain large commutative diagrams that I don’t want to draw.

Thanks for the tips. Much appreciated! Will do.

Thanks for the corrections, Greg! — the version on my website is fixed now, while the arxiv version will change less often.

You’d been very quiet for a while. I’m glad you’re back.