## March 16, 2009

### The Algebra of Grand Unified Theories II

#### Posted by John Baez

I started out using this blog entry as a plea for people to help improve the following paper… but now it’s a lot better, so we put it on the arXiv:

This fills in the details for John Huerta’s talk.

The goal of the paper is to take mathematicians who know just a bit about groups and their representations, and gently teach them a bit of particle physics — just enough so they can appreciate the meaning of this cube:

This cube summarizes the relation between four theories of physics. The symmetry groups of these theories lie at the cube’s top corners. Each of these groups acts on each generation of fermions and their antiparticles via a unitary representation on a 32-dimensional Hilbert space. These representations are the vertical arrows in the cube. The rest of the arrows are group homomorphisms that make the whole cube commute!

Here are the four theories:

• The Standard Model of particle physics. The symmetry group here is SU(3) $\times$ SU(2) $\times$ U(1), denoted $G_{SM}$ for short in the cube above. This group acts on the fundamental fermions and their antiparticles via a unitary representation on the 16-dimensional Hilbert space $F$, which describes one generation of fermions: quarks and leptons. It also acts on the dual space $F^*$, which describe the antiparticles of these fermions. So, it acts on $F \oplus F^*$, and this representation is the unlablled vertical arrow at the back left. The precise formula for this representation seems incredibly complicated and arbitrary.

In short: the Standard Model looks like a mess.

• The SU(5) theory proposed by Georgi and Glashow. There is a beautiful homomorphism $\phi: G_{SM} \to$ SU(5), which is almost one-to-one: its kernel has just 6 elements! And amazingly, this fact naturally ‘explains’ the complicated representation that appears in the Standard Model. More precisely: there’s an obvious representation of SU(5) on the exterior algebra $\Lambda \mathbb{C}^5$, which is the unlabelled vertical arrow at the back right of the cube. There’s also isomorphism of Hilbert spaces $h: F \oplus F^* \to \Lambda \mathbb{C}^5$, which gives the horizontal arrow $U(h)$. And, the back square of the cube commutes! This means the complicated vertical arrow at the back right can be expressed in terms of the obvious representation of SU(5) on the exterior algebra $\Lambda \mathbb{C}^5$.

In short: the Standard Model looks a lot more elegant when embedded in the SU(5) theory.

• The Spin(10) theory proposed by Georgi. Using the fact that a 5-dimensional complex vector space can be seen as a 10-dimensional real vector space, we can see SU(5) as a subgroup of Spin(10). The representation of SU(5) on $\Lambda \mathbb{C}^5$ is highly reducible, but it extends to a representation of Spin(10) on the same space, called the ‘Dirac spinor representation’. This representation is the unlabelled vertical arrow at the front right of the cube. This representation is almost irreducible: it breaks up into just two pieces, one for particles and one for antiparticles.

In short: we can unify all the particles in one generation of fermions by going from SU(5) to the bigger group Spin(10).

• The Spin(4) $\times$ Spin(6) theory proposed by Pati and Salam. By what I’ve said, the Standard Model symmetry group is a subgroup of Spin(10). But in fact, it’s a subgroup of the smaller group Spin(4) $\times$ Spin(6). Just as Spin(10) has a Dirac spinor representation on $\Lambda \mathbb{C}^5$, the groups Spin(4) and Spin(6) have Dirac spinor representations on $\Lambda \mathbb{C}^2$ and $\Lambda \mathbb{C}^3$, respectively. These combine to give the vertical arrow at the front left. And we can define maps that make the left face of the cube commute!

In short: the Standard Model also looks a lot simpler in terms of the Pati–Salam model.

Furthermore, there are obvious maps making the front of the cube commute. In fact, the whole cube commutes. This means that in a certain sense, the Spin(10) theory extends both the SU(5) theory and Pati–Salam model in a consistent way!

What are the wiggly arrows beneath the cube, you ask?

These convey the overall moral message:

The group Spin(10) acts as unitary transformations of the exterior algebra $\Lambda \mathbb{C}^5$. SU(5) is precisely the subgroup that preserves the grading on this exterior algebra. On the other hand, Spin(4) $\times$ Spin(6) is the precisely the subgroup that preserves the splitting of $\Lambda \mathbb{C}^5$ as $\Lambda \mathbb{C}^2 \otimes \Lambda \mathbb{C}^3$. And the Standard Model gauge group is the subgroup that preserves both the grading and the splitting.

If none of this makes sense except the math words like ‘unitary representation’ and ‘exterior algebra’, don’t be intimidated! You’re probably just the sort of person who should read our paper. It’s much more gentle than the blast of information I just delivered.

Posted at March 16, 2009 1:20 AM UTC

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### Re: The Algebra of Grand Unified Theories II

Hey, cool. Here are my notes so far as I read through it:

p1
First sentence is a bit redundant.
”dark matter” quotes got reversed.
The first two paragraphs seem a little awkward. But Connes will be happy for the mention.

p2
“none of these attempts at uniﬁcation are considered plausible today.” Is that true? I thought many physicists were still pretty keen on SO(10).
“all three theories require certain trends to hold among coupling constants (numbers which determine the relative strengths of forces) that the data do not support.” Hmm, the coupling constant convergence is actually pretty good even without supersymmetry; all you have to do to get them to meet perfectly is throw in a few more Higgs particles.
“remains a mystery without a solution” redundant.
Jump between last two paragraphs is a bit abrupt. Could probably omit the description of Heisenberg’s model.

p3
“the gauge group of the Standard Model in some larger group G, and we will give G a representation
V which reduces to the Standard Model representation F⊕F∗ when we pullback to GSM along this inclusion.” This is a summary of the main idea, so it could probably be repeated with an expanded description in more basic language.
“the Standard Model is precisely the theory that reconciles the two visions of physics lying behind the SU(5) theory and the Pati–Salam model.” Neat, if a bit of a random observation. But, hmm, what about the right handed neutrino that’s missing from the SU(5) model? Guess you could include it as a singlet.

p5
Could probably get rid of the example of two particles on a line and just use the Hilbert space of proton and neutron after they’re introduced.
Are all mathematicians familiar with bra ket notation? Maybe define these things as states, or omit that part.

p6
Overall, that was a nice introduction to isospin.
Why not introduce the Pauli-matrices here, and the proton and neutron and their I_3 values as eigenvectors and eigenvalues (weights) under sigma_3? That would make it much more clear what these charges are.

p7
Pion interactions are nice. Why not describe Pions as being eigenvectors in complex su(2)?

p8
“in terms mass and”

p9
Continue to like the use of Feynman diagrams.

p10
Nice segue to quarks.
In the tables, why not just label the column “EM charge” instead of having an EM column with “yes“‘s in it?
“quark conﬁnement, is still somewhat mysterious” Ooh, Gross, Wilczek, and Politzer are going to be pissed.
“the a ﬁrst-generation”
Hmm, maybe put those three colored quarks in a column, so people can see its operated on by su(3) just like the proton and neutron were in a 2 column.

p11
Hmm, isn’t a proton “different” depending on what color its down quark is? (This point is probably academic.) If so, those states written middle page for the proton and neutron seem a little weird.
Ah, nice way to relate isospin to weak isospin. (Hmm, sometimes you see isospin relating u, d, and s… but maybe that’s long forgotten.)

(gotta go, dinner calls…)

Posted by: Garrett on March 16, 2009 7:03 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Since I am one of the know-nothing-dummies for whom this primer is written, I make the following observation. The fermionic table at the top of page 4 is initially confusing. It looks as if electrons are the only type of fermions. Either the text preceding the table, or the table itself should say that this is a first pass.

Posted by: Scott Carter on March 16, 2009 10:05 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Good point, Scott. Will fix!

Posted by: John Baez on March 17, 2009 6:37 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Hmm, isn’t a proton “different” depending on what color its down quark is? (This point is probably academic.)

I wouldn’t call it academic, and the answer is “no”. A proton is a color singlet: it’s a state created by an operator f^{abc} u_a u_b d_c. There is no gauge invariant meaning to the color of an individual quark within a proton, so this is a completely unphysical question.

Posted by: Matt Reece on March 17, 2009 4:01 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

You write:

I thought many physicists were still pretty keen on SO(10).

I didn’t think they were keen on it in the absence of supersymmmetry. I got this impression from looking at page 25 of this paper by Pati himself, but I could easily have misinterpreted the absence of non-SUSY GUTs. I’ll need to look more closely at the literature.

the coupling constant convergence is actually pretty good even without supersymmetry; all you have to do to get them to meet perfectly is throw in a few more Higgs particles.

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.

Posted by: John Huerta on March 17, 2009 6:14 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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.

Posted by: Garrett on March 19, 2009 11:10 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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

We’ll fix this bit in our paper.

Posted by: John Baez on March 19, 2009 3:28 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Thanks for all the corrections and suggestions, Garrett! I’m improving the paper as we speak…

Posted by: John Baez on March 17, 2009 6:38 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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.

Posted by: Garrett on March 20, 2009 12:51 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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.

Posted by: John Baez on March 20, 2009 1:11 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

I’m also speaking as an ignoramus. 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.

Posted by: Todd Trimble on March 16, 2009 11:05 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Feynman diagrams should be taken with a grain of salt when trying to interpret them as an actual picture of particle phenomena, but you raise a good question, which I’d say has two reasonable answers:

1. Feynman diagrams as pictures of intertwiners. In the paper, we use Feynman diagrams to graphically represent intertwiners. The Feynman diagram depicting photon exchange is meant to represent a $\mathrm{U}(1)$-intertwiner from the tensor of two $\mathrm{U}(1)$-irreps (one for each incoming electron) to itself (one for each outgoing electron).

A priori, we might expect two similar such intertwiners, one for a photon going left to right, and one for a photon going right to left. But they turn out to be equal, so I drew the photon line horizontally to suggest that the diagram represents either one.

2. Off-shell virtual particles. It’s possible for virtual photons to travel faster than light in at least one respect—they can have spacelike 4-momenta.

Usually, one would expect to a photon to have a lightlike 4-momentum:

(1)$p^2 = 0$

This is one of the things physicists mean by “on shell”. But virtual photons are allowed to have 4-momenta that are off shell:

(2)$p^2 \neq 0$

In particular, they may have 4-momenta that are timelike (slower than light) or spacelike (faster than light).

Read more about virtual particles in this entry of the Physics FAQ.

Posted by: John Huerta on March 17, 2009 5:37 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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^-.

Posted by: Cristi Stoica on March 17, 2009 6:14 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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.

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:

Posted by: John Baez on March 17, 2009 5:23 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Thank you, John and John! I’m pretty sure I could sit here all day and think up naive questions which have already been asked millions of times, but I’ll try to be selective.

The usual picture I have of string diagrams is that they can be decomposed into elementary diagrams, which are composed in one (vertical) direction and tensored in another (horizontal) direction. When I see a Feynman diagram which depicts an exchange of a virtual photon, I see two internal vertices, and my string diagram instinct tells me to decompose the diagram into two parts. For example, one part might be “two electrons enter the picture, and the ‘left’ one emits a (virtual) photon”; this part is composed with a second part which reads “the virtual photon is absorbed by the ‘right’ electron”.

I suspect there are many crudities and misleading aspects to this picture, but let me press on. Again, just thinking about string diagrams, the elementary diagram in each case would refer to a single interaction in which an electron emits or absorbs a photon, that is, to an operator like

$[electron] \to [electron] \otimes [photon]$

where “[electron]” denotes a Hilbert space of states of the electron, and similarly for “[photon]”.

I’m getting worried though that this interpretation of Feynman diagrams for QED is wildly off-base, so I’d like to stop and get your reaction to just that much, before going any further.

Posted by: Todd Trimble on March 18, 2009 7:26 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

The one nit I’d pick is where you say

“[electron]” denotes a Hilbert space of states of the electron, and similarly for “[photon]”

Instead, I’d say that “[electron]” and “[photon]” are abstract labels. Hilbert spaces come in when you represent the category of such diagrams.

Posted by: John Armstrong on March 18, 2009 9:15 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

I’m getting worried though that this interpretation of Feynman diagrams for QED is wildly off-base, so I’d like to stop and get your reaction to just that much, before going any further.

It's a little inexact for the action of the Poincaré group, but John and John are ignoring that, so I say it's just right!

Note that this is a category with duals and the photon is self-dual (its own antiparticle), so there is no ambiguity when you see a horizontal photon (even though technically there should be no horizontal strings in a string diagram).

Posted by: Toby Bartels on March 18, 2009 10:06 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Thanks, Toby. Could I get you to expand a little on the inexactness?

Looking at what one of the Johns was saying, is it correct to say the “elementary interaction” is [ahem] represented as an intertwining operator between two unitary representations of the group

$G = Poincaré \times U(1),$

$U(1)$ being the internal group of symmetries relevant to QED?

Posted by: Todd Trimble on March 18, 2009 10:44 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Toby,

So far as my knowledge goes, he’s exactly right. So I’d like to echo Todd’s request: Could you expand more on how the picture changes when dealing with the Poincare group?

Posted by: John Huerta on March 19, 2009 1:07 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

I was kind of hoping that John (the other John) would expand on that. He hinted about it in his reply to Todd, but I don't really understand it.

Posted by: Toby Bartels on March 19, 2009 1:59 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Okay, here are three subtleties connected with the Poincaré group.

I’m assuming all you elite folks know and love string diagrams for compact closed categories. In such categories, every object $X$ has a ‘dual’ together with morphisms

$i: 1 \to X \otimes X^*$

$e: X^* \otimes X \to 1$

making certain zig-zag equations hold, which permit manipulations like this:

Great examples include the category of finite-dimensional Hilbert spaces… or finite-dimensional unitary representations of a compact Lie group. In fact, these examples are even better than compact closed: they’re dagger compact.

But never mind. Here are some subtleties that arise when we leave this nice realm and do full-fledged quantum field theory:

1. The first subtlety connected with the Poincaré group is that it’s noncompact, so most of its unitary representations — and certainly all those of physical interest! — are infinite-dimensional.

This means that the relevant category of unitary representations is not compact closed. Every representation has a ‘dual’ in a certain sense — but while this dual comes equipped with a bounded linear operator

$e: X^* \otimes X \to 1$

the obvious linear operator

$i: 1 \to X \otimes X^*$

is unbounded and only densely defined.

This means that certain Feynman diagrams ‘diverge’, giving infinite answers. Indeed this is true of almost every diagram containing a loop of edges!

This is why quantum field theory is tough: here ‘renormalization’ enters the stage.

Nonetheless, the beginner should proceed, for a while, as if this was just a ‘nuance’. One should certaintly not avoid writing down Feynman diagrams just because they diverge!

2. The second subtlety is not a nuance. It is, however, counteracted to some extent by the third subtlety.

While every particle corresponds to an irreducible unitary representation of the Poincaré group, its antiparticle does not correspond to the dual representation.

If the antiparticle did correspond to the dual representation, antiparticles would have negative energy! They don’t.

If our symmetry group is

$G \times Poincaré$

where $G$ is a compact Lie group of ‘internal symmetries’, a unitary irrep of this group consists of a unitary irrep of $G$ and a unitary irrep of the Poincaré group. When we take the antiparticle of a given particle, we do take the dual of the corresponding unitary irrep of $G$, but we don’t do this for the Poincaré irrep! We do some other operation.

3. The further complication is that virtual particles are allowed to be ‘off shell’. This means that the usual relation between energy and momentum

$E^2 = p^2 + m^2$

is not imposed. So, virtual particles correspond to much larger (reducible!) representations of the Poincaré group than the representations for the corresponding ‘real’ particles.

In particular, virtual particles can have negative energy.

And, unless I’m confused, in the world of virtual particles, we are back to a context where the antiparticle of a particle corresponds to the dual representation of the representation of the relevant group, namely

$G \times Poincaré$

So, subtlety 2) is washed away to some extent, but subtlety 1) remains.

Someday I’d like to write a book on diagrammatic methods in category theory, and to make this book really useful I’d like to explain quantum field theory from this viewpoint. The subtleties above are things I’d need to really grapple with, to do the job right.

Posted by: John Baez on March 19, 2009 8:49 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

JB writes: We do some other operation.

namely?

Posted by: jim stasheff on March 19, 2009 9:47 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

John wrote:

When we take the antiparticle of a given particle, we do take the dual of the corresponding unitary irrep of $G$, but we don’t do this for the Poincaré irrep! We do some other operation.

Jim wrote:

Namely?

It’s no fair writing one-word posts that require hundred-word answers! I’m a bit confused about the correct answer, in general, but my guess is this:

CPT.

Posted by: John Baez on March 20, 2009 12:24 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

This is interesting. What is the charge (C) conjugate of a massless Dirac fermion, $\psi$, if we are using the Majorana basis? Isn’t it simply the complex conjugate, $\psi^*$, which is the dual under $G \times Spin(1,3)$? (For massive fermions, $\gamma_5 \psi^*$?)

Posted by: Garrett on March 20, 2009 3:14 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

That link says CPT is a symmetry
What is the “other operation” - applying the symmetry?

Posted by: jim stasheff on March 20, 2009 1:33 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Yes.

To repeat: I’m not sure this is the correct operation; it’s just an obvious one to try.

Posted by: John Baez on March 20, 2009 7:25 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

And, unless I’m confused, in the world of virtual particles, we are back to a context where the antiparticle of a particle corresponds to the dual representation of the representation of the relevant group

So does this mean that we take actual duals for internal edges, but do ‘some other operation’ for external edges?

That would be nice (at least the first part) because it would remove any subtlety to the equivalence between the two interpretations of a horizontal photon (which is necessarily internal).

Posted by: Toby Bartels on March 19, 2009 10:07 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

John B asked me if I would post something I asked him privately, about one of the three subtleties he mentioned that come into play when the Poincaré group is taken into account.

The first subtlety connected with the Poincaré group is that it’s noncompact, so most of its unitary representations — and certainly all those of physical interest! — are infinite-dimensional.

This means that the relevant category of unitary representations is not compact closed. Every representation has a ‘dual’ in a certain sense — but while this dual comes equipped with a bounded linear operator $e:X* \otimes X \to 1$ the obvious linear operator $i: 1 \to X \otimes X*$ is unbounded and only densely defined.

This means that certain Feynman diagrams ‘diverge’, giving infinite answers. Indeed this is true of almost every diagram containing a loop of edges!

This is why quantum field theory is tough: here ‘renormalization’ enters the stage.

Nonetheless, the beginner should proceed, for a while, as if this was just a ‘nuance’. One should certaintly not avoid writing down Feynman diagrams just because they diverge!

I’d say instead that there really is no such obvious map $i$. The ‘1’ here is the trivial 1-dimensional representation, and all linear maps $i$ coming out of a finite-dimensional space are automatically bounded, so what John wrote isn’t precisely the issue, although it’s obviously in the correct spirit.

The map $i$ we’d like would take a unit vector to a suitable nonzero scalar multiple of

$\sum_j |e_j\rangle \otimes \langle e_j|$

where $|e_j\rangle$ ranges over the elements of an orthonormal basis of the Hilbert space representation $X$, and $\langle e_j|$ is the dual basis. But if $X$ is infinite-dimensional, there is no such element.

(I also wrote John that this is one of the things physicists do that drives me nuts: writing as if such expressions made any sense. Is that unfair of me?)

Posted by: Todd Trimble on March 20, 2009 7:06 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Todd wrote:

I’d say instead that there really is no such obvious map $i$.

Right — thanks for straightening that out!

(I also wrote John that this is one of the things physicists do that drives me nuts: writing as if such expressions made any sense. Is that unfair of me?)

Yes. We’re mathematicians; they aren’t. So it’s our job, not theirs, to find a mathematical framework in which their expressions naturally live. If their calculations bust out of the frameworks we know, it may just mean we need bigger framework.

A great example is the Dirac delta ‘function’, which is not a function, but turns out to be a perfectly fine distribution.

Or: the sum

$\sum_j |e_j\rangle \otimes \langle e_j|$

doesn’t converge in the Hilbert space $H \otimes H$, but it converges in various other topological vector spaces in which $H \otimes H$ is dense. You can’t expect a physicist to know or care about such subtleties. So it’s our job — at least if we care about such things — to investigate various options like this and see what works best.

Another example is ‘a quantum field at a point’, $\phi(x)$. Physicists act like this is an operator from a Hilbert space to itself:

$\phi(x) : H \to H$

It’s not. Like the operator $i$ we’re talking about, it’s not even densely defined! But it’s a perfectly fine operator from a Frechet space $D$ that’s dense in $H$ to the topological dual $D^*$.

In my thesis I used this setup to study the normal-ordered powers $:\phi^n(x):$ of a free quantum field, and later I wrote paper about this.

Posted by: John Baez on March 20, 2009 7:48 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Todd: Is that unfair of me?

John: Yes.

I pretty much knew that already; I’m writing that way purely to elicit a reaction and start a discussion about something which nevertheless bothers me. (The older I get, the less inhibited I feel about acting the part of l’enfant terrible. But it’s all an act. Well, mostly anyway.)

The actual truth of course is that I, a dumb mathematician, lack the physical intuition that physicists have that leads them to correct answers, in spite of the apparent “nonsense”.

There is in particular this black magic called “renormalization”. I would really like to understand this at a semi-sophisticated mathematical level that I feel the Café or the nLab could in principle provide. (I say “semi-sophisticated” because there are accounts out there which I find rather annoyingly sophisticated, as in Quantum Fields and Strings: A Course for Mathematicians, which are rather like cruel jokes to me.)

Posted by: Todd Trimble on March 20, 2009 9:55 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Todd is looking at a Feynman diagram sort of like this:

and he says:

The usual picture I have of string diagrams is that they can be decomposed into elementary diagrams, which are composed in one (vertical) direction and tensored in another (horizontal) direction. When I see a Feynman diagram which depicts an exchange of a virtual photon, I see two internal vertices, and my string diagram instinct tells me to decompose the diagram into two parts.

Good! You’re on the right track!

Feynman diagrams are an example of string diagrams, and you — being an expert on the latter — should take this insight and press it for all it’s worth.

For example, one part might be “two electrons enter the picture, and the ‘left’ one emits a (virtual) photon”; this part is composed with a second part which reads “the virtual photon is absorbed by the ‘right’ electron”.

Right!

Of course in many situations you have a fair amount of freedom to wiggle around the string diagram without changing the morphism it depicts. This may lead to different but equally valid ‘stories’ of the sort you just told.

In particular, we can retell your story this way:

“two electrons walk into a bar…

Whoops! That’s some other story. I meant:

“Two electrons enter the picture, and the ‘right’ one emits a (virtual) photon”; this part is composed with a second part which reads “the virtual photon is absorbed by the ‘left’ electron”.

The point, as Toby noted, is that — modulo certain nuances I’d rather not discuss yet — the photon is a self-dual object. That’s why we don’t draw little arrows on photon lines, and that’s why when we see a picture like this:

we don’t worry about whether the photon is ‘going from left to right’ or ‘going from right to left’. Take your pick.

Again, just thinking about string diagrams, the elementary diagram in each case would refer to a single interaction in which an electron emits or absorbs a photon, that is, to an operator like

$[electron] \to [electron] \otimes [photon]$

where “[electron]” denotes a Hilbert space of states of the electron, and similarly for “[photon]”.

Yes, you’re again completely on the right track. This is why Feynman wrote a book called The Theory of Fundamental Processes. If you know these basic operators, called ‘vertexes’ or ‘interactions’ in the trade, you can build up more complicated operators by composing, tensoring, and duality.

I’m getting worried though that this interpretation of Feynman diagrams for QED is wildly off-base, so I’d like to stop and get your reaction to just that much, before going any further.

I’m afraid you’re wildly on-base.

Posted by: John Baez on March 18, 2009 10:57 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

So there is a fundamentally algebraic way of describing the elementary processes of GUT in terms of the categorical analogues of choose: Frobenius algebras, Hopf algebras, PROPs, braided moniodal categories with duals and some other structure, etc.?

Or in other words, the category of representations upon which we are looking has the structure of a [blah, blah, blah]. Similar structures may, therefore, model the physics that we so far understand.

There is a process of de-categorifying a braided monoidal 2-category that corresponds, roughly, to drawing the resulting 2-tangle. The singularities in the drawings represent 2-morphisms, and the relationships among these 2-morphisms are higher singularities. (All of this works nicely when the singularities are codimension 1.)

[[Triple warning: (1) Self-promotion of the Sphere Eversion book describes some of this. (2) The file is about 3 megs. (3) As of 8:15 AM CDT 19 Mar 2009, the final draft is not posted. It should be up within an hour.]]

Another way of putting this is that we spacialize a time direction, and then temporalize the relations that must hold:

For example in the electron photon exchange, there are two perturbations (the photon is emitted first on the right or first on the left). The relation is schematized by the diagram drawn, but it represents a compatibility relation in a Frobenius-like algebra.

So I have the QG-TQFT blues because I don’t see that gravity fits into any of these metaphors.

Posted by: Scott Carter on March 19, 2009 2:18 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Scott wrote approximately:

So is there a fundamentally algebraic way of describing the elementary processes of GUT in terms of… ?

A GUT is just a quantum field theory like quantum electrodynamics or the Standard Model, but with bigger ambitions. Physically, the goal of a GUT is to unify all the forces except gravity. But mathematically, a GUT is just a quantum field theory of the time-honored sort.

So, among other things, this means that nobody knows how to make GUTs into fully rigorous mathematics! But they make perfect sense perturbatively — i.e., where processes are described as infinite sums over Feynman diagrams. The sums probably diverge in a complicated way… so we’ll have plenty of fun just trying to understand a single diagram at a time.

A single Feynman diagram is an attempt to describe a morphism in the symmetric monoidal category of unitary representations of a certain Lie group.

If the group were compact, this attempt would succeed, because then every unitary representation would be a direct sum of finite-dimensional reps, and the category of finite-dimensional unitary reps is a dagger compact category — or what I call a “symmetric monoidal category with duals”. This is the sort of category that’s optimized for 1-tangles in 4 dimensions!

But alas, the group in question is always the Poincaré group times a compact Lie group. The Poincaré group is noncompact, and the unitary representations that matter are infinite-dimensional. This leads to some subtle bits of extra structure which someone should try to axiomatize.

Posted by: John Baez on March 20, 2009 12:18 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

This leads to some subtle bits of extra structure which someone should try to axiomatize.

That sounds like fun. Has anyone tried?

Posted by: Mike Shulman on March 20, 2009 6:06 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Mike wrote:

Has anyone tried?

The first problematic feature of infinite-dimensional Hilbert spaces — namely, that their ‘duals’ aren’t duals in the category-theoretic sense — is well-known. This problem shows up when you go beyond TQFTs (which have finite-dimensional Hilbert spaces) to theories with infinite-dimensional Hilbert spaces, like conformal field theories.

Various people have thought about generalizing TQFT axioms to handle such theories. For example Stephan Stolz who is working on this kind of thing with Peter Teichner for the purpose of defining ‘elliptic objects’. In their situation every Hilbert space comes equipped with a ‘Hamiltonian’, a nonnegative self-adjoint operator $H$ such that the trace of $exp(-tH))$ is finite for $t \gt 0$. This turns out to save the day. (Key buzzword: nuclear space.)

The other stuff, about antiparticles not really corresponding to dual representations, and virtual particles being ‘off shell’, may not have received a nice category-theoretic treatment. I would like to spend a lot more time someday getting these issues clear in my mind. “When I write my book…”

Posted by: John Baez on March 20, 2009 7:22 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Okay, great, thanks for this JB! There’s another query of mine higher up in the tree that was originally put to Toby but was then deflected back to you, which I’d enjoy hearing your response to.

It may seem silly, but I guess one thing that bothers me a little about ‘virtual particles’ is their ‘virtualness’ – my understanding is that they evade direct detection in the laboratory. In some sense they have to be ‘there’ if quantum field theory is to make any sense or be logically coherent, but one never actually ‘sees’ them. Or am I wrong about that?

Edit: On John (B)’s suggestion, I’ve just now taken a look at Feynman’s The Theory of Fundamental Processes, specifically at the short section on QED (pp. 29-32). On page 30, I see a more precise sense of ‘virtual’, actually a definition, which seems to support my understanding of evasion of direct detection:

A single free electron cannot emit one photon because of conservation of energy and momentum, but if two electrons are near one another, one may emit a photon which the other immediately absorbs. Quantum mechanics permits the temporary existence of states, which, if maintained, could not conserve energy… A process only occurring by means of a temporary violation of energy conservation is called a virtual process.

(Could this be related to one of the nuances that John had in mind? Conservation of energy being connected, via Noether’s theorem, to symmetry with respect to time translation, hence to the action of the Poincaré group on spaces of particle states? My understanding of these points is not too strong.)

(Just this brief passage raises for me a whole new pile of questions, one of which is, “how temporary is ‘temporary’?”)

My experiences in trying (half-assedly) to pick up some QFT here and there have been somewhat frustrating. The mathematics in a standard graduate text like Peskin and Schroeder is to me off-putting in the extreme; I can read for only so long before my tolerance reaches a snapping point. Weinberg’s QFT books are somewhat better in that regard, but still not much in a modern mathematical spirit, hence still a lot of work for someone like me. On your suggestion, John, I did buy Ticciati’s book, but his set of concerns didn’t intersect mine that much. I have a number of Feynman’s books on physics which are nice and give some good concrete feeling, but which don’t quite satisfy the mathematician in me. I have some mathematically oriented books (Streater and Wightman, Araki) which weren’t quite what I was looking for. I have a number of others; the collection takes up a more or less dust-collecting shelf or two in my library.

I will single out two books which I did like. The first is the one by Zee. By the time I purchased that book, I was fed up with mathematics “physics-style”, and I think part of what I liked about that book was the rather light mathematical touch, trusting the reader to fill in a lot for himself. The other is, perhaps curiously, Dirac’s The Principles of Quantum Mechanics. I’m sure most people consider this text hopelessly dated, especially with regard to QED etc., but I felt very much carried along by the logical flow and rigor of his writing, and the calculations very pleasant to follow. I wound up getting a lot out of it. Deservedly a classic, methinks.

I’d like to ask more here in this thread, but I’ll stop here for now. In general, I feel like I could probably learn a hell of a lot more by asking questions here in the Café than I could by being referred to books, so I hope you don’t mind.

Page 7: “to whit” should be “to wit”.

Posted by: Todd Trimble on March 19, 2009 8:39 PM | Permalink | Reply to this

### Dirac

If only Dirac’s disciples had learned the master’s style! His `delta function’ made so much more sense when I read his version.

Posted by: jim stasheff on March 19, 2009 9:43 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Todd,

the math-style QFT that you want to see is in texts on AQFT.

The canonical introduction enjoyable for you is

Halvorson, Müger, Algebraic quantum field theory

While in its beginnings AQFT didn’t have much to say about the perturbation series, and hence about Feynman diagrams, this has changed in recent years.

Have a look at

Brunetti, Duetsch, Fredenhagen, Perturbative Algebraic Quantum Field Theory and the Renormalization Groups

Posted by: Urs Schreiber on March 19, 2009 10:03 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Todd wrote:

It may seem silly, but I guess one thing that bothers me a little about ‘virtual particles’ is their ‘virtualness’ – my understanding is that they evade direct detection in the laboratory.

Ah, if only it were so simple!

In the theoretical framework of Feynman diagrams, any particle that is emitted and then absorbed is ‘virtual’. By this criterion, every photon you ever see is virtual, since it was emitted by something and absorbed by your eye.

A process only occurring by means of a temporary violation of energy conservation is called a virtual process.

That’s a somewhat old-fashioned though still tenable attitude. It goes along well with a version of perturbation theory in which the individual terms in the perturbation expansion violate conservation of energy. In modern Feynman diagram calculations, conservation of energy and momentum is strictly enforced at each vertex. In the modern approach, what’s funny about virtual particles is not that they violate energy conservation — it’s that they can be ‘off shell’, meaning that they violate $E^2 = p^2 + m^2$.

So you see, you’ve entered a morass where people can have radically conflicting stories about what’s ‘really going on’ — even if they make the same predictions for the results of experiments!

In this sort of situation, familiar already from the ‘interpretation of quantum mechanics’, it pays to take the stories people tell with a large grain of salt, while paying close attention to how they’re actually computing things.

Just this brief passage raises for me a whole new pile of questions, one of which is, “how temporary is ‘temporary’?”

Right. And when you ask this question, physicists usually mention the energy-time uncertainty relation

$\Delta E \Delta t \ge \hbar / 2$

which (if you believe certain people) says that conservation of energy can be violated by a lot for a short time, but only a little for a long time.

It’s sort of like speeding on the New Jersey Turnpike: you can exceed the speed limit by as much as you like without getting caught — as long as you do it for a short enough time.

But, charming analogies aside, you’ve just opened up another can of worms, because the whole logical status of time-energy uncertainty relations is also hotly disputed.

So, personally, I think you’ll eventually have to knuckle down and learn quantum field theory the hard way — by doing lots of calculations — just to keep the morass of conflicting stories from driving you insane.

Posted by: John Baez on March 19, 2009 10:32 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

John and Urs,

Thanks a bunch, John, for your explanations, and especially for explaining the three nuances in your earlier comment. For instance, the fact that the irrep for the antiparticle (taking into account Poincaré) is not the same as the dual of the irrep for the particle is not something I’d picked up on, but it came as a big relief to hear. (It sounds sort of obvious though: otherwise wouldn’t a particle-antiparticle collision result in… complete annihilation!? as opposed to gamma rays.) I might like to hear more of the precise details about this someday, but now is clearly not the time for me.

So, personally, I think you’ll eventually have to knuckle down and learn quantum field theory the hard way — by doing lots of calculations — just to keep the morass of conflicting stories from driving you insane.

Assuming I have time in this life to learn some quantum field theory, that would be all right – as long as the calculations themselves didn’t drive me insane (as they tend to do whenever I’ve tried to read something like Peskin and Schroeder). I actually enjoy calculating, provided I have the feeling that I know what the hell I’m doing!

Which brings me to your comment, Urs. It may be premature to say this before I’ve taken a proper look, but thanks a billion for those references, and particularly for the Halvorson-Müger reference. (The other one looks a little heavy for me at this point, but I may come to it and have some questions later.) Perhaps this is just what I’ve needed all this time…

Posted by: Todd Trimble on March 19, 2009 11:10 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

but thanks a billion for those references, and particularly for the Halvorson-Müger reference. (The other one looks a little heavy for me at this point,

Sure, you don’t want to start with that second one. But you won’t find anything even close to Feynman diagrams in
Halvorson-Müger. So when you start wondering where they are hidden, that second article I mentioned can in principle show you the way.

but I may come to it and have some questions later.) Perhaps this is just what I’ve needed all this time…

By the way, a bit more more literature and possibly useful remarks are in the introduction of my AQFT from FQFT (the in-print version of which is here).

Posted by: Urs Schreiber on March 20, 2009 1:31 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Todd, if you are still looking for QFT books that are digestable by mathematicians, try: Eberhard Zeidler: “Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists” and “Quantum Field Theory II: Quantum Electrodynamics: A Bridge between Mathematicians and Physicists”.
Eberhard Zeidler was the director (now retired) of the Max Planck Institute for Mathematics in the sciences in Leipzig, Germany, and is mostly known for his work on nonlinear functional analysis.
Don’t be put off by the length of the books (both have more than 1000 pages), you don’t have to read everything, just pick out the topics that you are interested in (so you definitely don’t want to buy those if you can read them in a nearby library).
It’s quite unique in it’s style.
(Totally off topic, but it’s late in the night in my reference frame, so please forgive me: Did you know that some historians recently discovered a document signed by Max Planck where his signature reads “Marx Planck”? For some time some people worried that the Max Planck Institutes would have to be renamed as Marx Planck Institutes. Doh! They were sooo happy to rename Karl-Marx-town as Cottbus after the reunion of Germany).

Posted by: Tim vB on November 3, 2009 8:47 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Todd wrote:

For instance, the fact that the irrep for the antiparticle (taking into account Poincaré) is not the same as the dual of the irrep for the particle is not something I’d picked up on, but it came as a big relief to hear. (It sounds sort of obvious though: otherwise wouldn’t a particle-antiparticle collision result in… complete annihilation!? as opposed to gamma rays.)

Right, exactly! You can make an antiparticle that’s the opposite of a particle in all ways except for energy. So when they meet and annihilate they don’t just vanish without a trace… you still have to balance the books when it comes to energy.

A single photon won’t do the job, since it has nonzero angular momentum and its energy-momentum vector is lightlike, while a massive particle/antiparticle pair always has timelike energy-momentum. So, two photons is the bare minimum ‘leftover’ when a particle and antiparticle annihilate.

It’s fun to think about the string diagram technology involved here: particles are not objects in a compact closed category, but if we could somehow close our eyes to the Poincaré group (and thus conservation of angular momentum, conservation of energy-momentum, and the mathematically irksome nonexistence of negative-energy particles), and think of particles just as representations of the gauge group $G$, we would be in a compact closed category.

So, for starters, we should think about how a category like $Rep(G \times H)$ is built from $Rep(G)$ and $Rep(H)$, where $G$ and $H$ are two different groups.

I actually enjoy calculating, provided I have the feeling that I know what the hell I’m doing!

I know — you like it much more than I do! I hated the homework in my QFT course in grad school, but I’m glad I took that course, because I hadn’t, I never would have computed a scattering cross-section.

Unfortunately, only a certain fragment of quantum field theory has been made rigorous. So, in a certain sense, nobody knows what the hell they are doing.

Luckily, the fragment of quantum field theory that has been made rigorous includes enough to do Feynman diagram calculations up to a given finite order in perturbation theory, including renormalization — and such calculations include much (though far from all) of what real-world physicists do. Scharf’s book Finite Quantum Electrodynamics is a nice introduction to QED that carefully stays within this rigorous fragment. There’s plenty of computations to do here!

However, most physicists breezily reason about quantum field theory in ways that go far beyond the bounds of mathematical rigor. So, if you only know the rigorous stuff, you can’t really understand them.

I actually like the fact that physicists are brave enough to do stuff that looks sort of like math, but where nobody knows what the rules are yet. They’ve basically decided that life is too short to cross every t and dot every i. They’re just gonna dive in and see what happens.

Posted by: John Baez on March 21, 2009 9:53 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

A long time ago, back here, John Baez encouragingly wrote:

Feynman diagrams are an example of string diagrams, and you — being an expert on the latter — should take this insight and press it for all it’s worth.

Okay then, I’d like to do just that! I might need a little assistance though.

Later in the same comment, John quoted me and then responded thus:

Again, just thinking about string diagrams, the elementary diagram in each case would refer to a single interaction in which an electron emits or absorbs a photon, that is, to an operator like

$[electron] \to [electron] \otimes [photon]$

where “$[electron]$” denotes a Hilbert space of states of the electron, and similarly for “$[photon]$”.

Yes, you’re again completely on the right track. This is why Feynman wrote a book called The Theory of Fundamental Processes. If you know these basic operators, called ‘vertexes’ or ‘interactions’ in the trade, you can build up more complicated operators by composing, tensoring, and duality.

Okay, good. So what I was hoping to do, as an exercise for my own benefit, would be to write down mathematically the intertwiner for this single interaction. What is it? (I have The Theory of Fundamental Processes, and it could be that the answer to my question is given somewhere near page 91, but I’m still not quite sure what’s going on.)

I guess what I mean by $[electron]$ here is an appropriate structure of unitary irreducible representation

$Poin \times G \to U(H)$

where $G$ is an appropriate group of internal symmetries. (To keep things simple, I’d like to work in the context of QED and take $G = S^1$, unless that is considered an egregious oversimplification.) According to what John wrote here –

A unitary irrep of $G \times P$ [here $P = Poin$ – TT] is always built by tensoring a unitary irrep of G with one of P. So, the project of classifying particles splits into two parts: one depending on G, one depending on P.

– it would seem such a structure is pinned down by knowing (1) the mass of the electron plus the fact it’s a spin 1/2 particle: this gives a unitary irrep of $Poin$, and (2) whatever integer $n$, classifying a unitary irrep $z \mapsto z^n$ of $G = S^1$, is appropriate for the electron. (I don’t know what that integer is.)

Similarly, the unitary irrep of $Poin \times S^1$ appropriate to $[photon]$ is pinned down by knowing that the photon is a massless spin 1 particle, together with some integer $m$ that identifies another unitary irrep of $S^1$. Once again, I don’t know what integer that is.

Even assuming we are armed with that knowledge, I’m not sure how to proceed. We have two $(Poin \times G)$-modules, namely $[electron]$ and $[electron] \otimes [photon]$, and I presume I am asking for the description of the module homomorphism

$\phi: [electron] \to [electron] \otimes [photon]$

that pertains to this fundamental QED interaction. (Is that $\phi$ what is called the QED interaction Hamiltonian??)

I’ll take a guess how that description might be conventionally packaged by a physicist: states of $[electron]$ might be referred to by something like $\vert p, k\rangle$ where $p$ is a 4-momentum and $k$ is one of two basis vectors for the spin 1/2 rep of $SU(2)$ – I am guessing physicists pronounce those basis vectors as “spin up, spin down” or something (guessing here that “spin up” refers to a highest weight unit vector and “spin down” to a lowest weight unit vector).

Then there are states for electron-photon pairs that might be written something like $\vert p', k', p'', k''\rangle$; here $p', k'$ specify an electron state and $p'', k''$ a photon state. (I am unsure about this $k''$. If anything, it refers to polarization of the photon, if that is relevant.) Finally, a physicist will tell you how to compute numbers which he might denote as

$\langle p', k', p'', k'' \vert \phi \vert p, k\rangle$

(something like a probability density, or perhaps we should say probability amplitude density) in terms of some integrals, which he would know how to write down in terms of the QED interaction Hamiltonian or QED Lagrangian, or something. (I emphasize that these are just guesses on my part.)

I appreciate any assistance, words of wisdom, corrections, etc.

Posted by: Todd Trimble on July 18, 2011 7:40 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

(2) whatever integer $n$, classifying a unitary irrep $z \mapsto z^b$ of $G = S^1$, is appropriate for the electron. (I don’t know what that integer is.)

This integer is the electric charge of the electron. In QED you can take it to be $n = -1$ (or +1, if you like). In QCD it needs to be $n = -3$ with $n = \pm \frac{1}{3} \times 3, \pm \frac{2}{3} \times 3$ being the electric charge of the quarks.

Posted by: Urs Schreiber on July 18, 2011 7:59 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Finally, a physicist will tell you how to compute numbers …

I am in a haste, just briefly:

the “matrix element” that you write down is

• that of the Clifford matrices for the spin/polarization degrees of freedom, something like $\Gamma^{k k'}_{k''}$ in your (unusual) notation;

• times a term $\delta(p, p' + p'')$ expressing the conservation of momentum.

Posted by: Urs Schreiber on July 18, 2011 8:18 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

I had a moment to look up a reference for you: the term that you are asking about (the one that I indicated above), in the context of a dicussion of the Feynman diagrams that you want to see the connection to, is for instance given as

item (iii)

page 884

volume II: QED

in

Eberhard Zeidler, Quantum Field Theory – A bridge between mathematicians and physicists Springer (2009).

Posted by: Urs Schreiber on July 18, 2011 7:47 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Todd wrote:

So what I was hoping to do, as an exercise for my own benefit, would be to write down mathematically the intertwiner for this single interaction. What is it?

Urs gave you a terse answer, but I’ll give you a somewhat more longwinded one.

I guess what I mean by $[electron]$ here is an appropriate structure of unitary irreducible representation

$Poin \times G \to U(H)$

where $G$ is an appropriate group of internal symmetries. (To keep things simple, I’d like to work in the context of QED and take $G = S^1$, unless that is considered an egregious oversimplification.)

No, it’s not an oversimplification. It’s good, and easy, to take $G = \mathrm{U}(1)$.

(By the way, it’s a tradition in physics to call this group $\mathrm{U}(1)$ instead of $S^1$, and I don’t have the desire to change that tradition. They are, of course, just two names for the same group.)

it would seem that [an irrep of $Poin \times G$] is pinned down by knowing (1) the mass of the electron plus the fact it’s a spin 1/2 particle: this gives a unitary irrep of $Poin$, and (2) whatever integer $n$, classifying a unitary irrep $z \mapsto z^n$ of $G = S^1$, is appropriate for the electron. (I don’t know what that integer is.)

This integer is called the electric charge of the particle in question. The electron has charge -1, the photon has charge 0, and the anti-electron or ‘positron’ has charge 1.

From this, it’s obvious that any linear operator

$\phi: [electron] \to [electron] \otimes [photon]$

is an intertwiner of $\mathrm{U}(1)$ representations, since $\alpha \in \mathrm{U}(1)$ acts as multiplication by $\alpha$ on the Hilbert space we’re calling $[electron]$ and it acts trivially on the Hilbert space $[photon]$.

Is that $\phi$ what is called the QED interaction Hamiltonian??

Not quite: a Hamiltonian is always a self-adjoint operator from some fixed Hilbert space to itself, and this $\phi$ is not. But this $\phi$ is the ‘main ingredient’ of the QED interaction Hamiltonian.

(We have not yet discussed what this ‘fixed Hilbert space’ is, and I don’t think we need to now, but it’s part of the theory of QED.)

I’ll take a guess how that description might be conventionally packaged by a physicist: states of $[electron]$ might be referred to by something like $\vert p, k\rangle$ where $p$ is a 4-momentum and $k$ is one of two basis vectors for the spin 1/2 rep of $SU(2)$ – I am guessing physicists pronounce those basis vectors as “spin up, spin down” or something (guessing here that “spin up” refers to a highest weight unit vector and “spin down” to a lowest weight unit vector).

Yup, all this is right. In lower-brow language:

$up = (1,0) \in \mathbb{C}^2$

$down = (0,1) \in \mathbb{C}^2$

Then there are states for electron-photon pairs that might be written something like $\vert p', k', p'', k''\rangle$; here $p', k'$ specify an electron state and $p'', k''$ a photon state. (I am unsure about this $k''$. If anything, it refers to polarization of the photon, if that is relevant.)

Finally, a physicist will tell you how to compute numbers which he might denote as

$\langle p', k', p'', k'' \vert \phi \vert p, k\rangle$

Right. And it’s not too hard to make a guess here: the simplest guess meeting two obvious constraints turns out to be the right one.

First, $\phi$ must be an intertwiner of translation group representations: the translation group $\mathbb{R}^4$ sits inside the Poincaré group. So if you think about how the translation group acts on the states you’re talking about, you’ll see that the number you’re looking for must be zero unless

$p' + p'' = p$

(We say ‘momentum is conserved’.) So it shouldn’t surprise you that part of the number you’re looking for is

$\delta(p' + p'' - p)$

Second, $\phi$ must be an an intertwiner of $SL(2,\mathbb{C})$ reps: remember, $SL(2,\mathbb{C})$ also sits inside the Poincaré group. So, we could try to multiply $\delta(p' + p'' - p)$ by some function

$f(k,k',k'')$

that ensures that $\phi$ is also an intertwiner of $SL(2,\mathbb{C})$ reps. And that’s exactly the right thing to do.

So, the answer will look like

$f(k,k',k'') \delta(p' + p'' - p)$

for some function $f$. You can either guess this $f$, or I can give you clues.

Posted by: John Baez on July 18, 2011 9:16 AM | Permalink | Reply to this

### Feynman diagrams

Those looking for category theoretic and higher category theoretic descriptions of the idea of Feynman diagrams might enjoy section 4.3: Manifolds with Singularities in Jacob Lurie’s TFT text.

There he describes the notion of $(\infty,n)$-categories of $n$-dimensional cobordisms with specified singularities and discusses their representation theory, i.e. in physical terms: the possible QFTs on these.

From this perspective Feynman diagrams of course arise as (representations of) the $(\infty,1)$-category of 1-dimensional cobordisms with specified singularities:

each type of singularity is an interaction vertex in the Feynman diagram.

This is described in example 4.3.21, p. 101.

By the way, just for emphasis since it wasn’t much mentioned in the above discussion so far: this kind of perspective on Feynman diagrams as being nothing but a special case of representations of $n$-dimensional bordisms is the good kind of perspective if one wants to understand these in the light of higher dimensional perturbative quantum theories.

Put the other way round: had higher category theory been developed a bit quicker than unfortunately it has (say: 30 years quicker), higher category theorists would have been the first to feel that it is compelling to pass from summing over Feynman diagrams to summing over 2-dimensional Feynman diagrams…

Posted by: Urs Schreiber on March 19, 2009 5:49 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

(1) page 7, figure 1, top right diagram is labelled wrong (typo?)

(2) page 30, what would be the “binary code” for right handed up red quark?
i’m guessing 00 100 or 11 100

Posted by: rntsai on March 17, 2009 9:28 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

rntsai wrote:

page 7, figure 1, top right diagram is labelled wrong (typo?)

Thanks for catching that! I’ve fixed it.

page 30, what would be the “binary code” for right handed up red quark?

i’m guessing 00 100 or 11 100

Good guess! It’s hard to tell which is the correct answer until we work out the details in Table 4, around page 38.

As John Huerta explained, the 5-digit binary code classifies particles by listing the answers to these questions:

• Is the particle isospin up?
• Is it isospin down?
• Is it red?
• Is it green?
• Is it blue?

A right-handed red up quark, being right-handed, doesn’t feel the $SU(2)$ force: it has no isospin. So, the answers to the first two questions seem to be ‘no’, ‘no’. And being red, the answers to the next three questions are ‘yes’, ‘no’, ‘no’.

So, you might guess that the binary code for this particle is

$00 100$

However, you wisely considered the possibility that the answers to the first two questions are ‘yes’, ‘yes’. This sounds goofy, but it’s just another way of saying the isospin is zero. So, there’s another code that describes a red quark with no isospin, and that’s

$11 100$

And, indeed, there is another red quark with no isospin! It’s the right-handed red down quark.

As you’ll see from Table 4,

$11 100$

describes the right-handed red up quark, while

$00 100$

describes the right-handed red down quark.

Why must it be this way? For the answer, read the text leading up to Table 4. There is a reason.

Posted by: John Baez on March 18, 2009 2:59 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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.

Posted by: rntsai on March 18, 2009 10:51 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Page 1: ‘fascating’ -> ‘fascinating’

Page 3: ‘orus’ -> ‘ours’

Page 8: ‘it is constant representations’ -> ‘it is constant on representations’

Posted by: Toby Bartels on March 17, 2009 11:58 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Fixed, thanks!

Posted by: John Baez on March 18, 2009 12:51 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

I know that this is off topic but I am host of a mathematics podcast Combinations
and Permutations
that I think that your readership may find interesting. On the last episode we have covered the four color theorem and some new ideas for mathematical journals. You can find the podcast on iTunes or through our host. Give us a listen, you will not be disappointed

Posted by: Combinations and Permutations on March 18, 2009 5:59 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

I get the impression that you wanted to say that GSM and Spin(10) are in a pullback or pushout relation but never explicitly say that.

Posted by: RodM on March 21, 2009 2:06 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

The gauge group

$G_{SM} = U(1) \times SU(2) \times SU(3)$

has a $\mathbb{Z}/6$ subgroup that acts trivially on all particles. So, the ‘true gauge group’ of the Standard Model is $G_{SM}/(\mathbb{Z}/6)$. And this group is a pullback:

\begin{aligned} G_{SM}/(\mathbb{Z}/6) & \overset{\quad \quad}{\to} & SU(5) \\ \downarrow \qquad & &\downarrow \quad \\ Spin(4) \times Spin(6) & \underset{\quad \quad}{\to} & Spin(10) \end{aligned}

But we’re afraid our intended audience will not enjoy the word ‘pullback’. Drawing a commutative cube is already a bit effete for people interested in GUTs. So, we instead say something about ‘intersections’.

Maybe I’ll risk the word ‘pullback’ somewhere.

In the paper as it stands right this second, there’s a ‘conjecture’ concerning this issue. Now we’ve proved it; it’s not hard. Soon that will be incorporated in the paper too.

Posted by: John Baez on March 21, 2009 2:27 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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.

Posted by: John Baez on March 21, 2009 6:01 PM | Permalink | Reply to this

### pullbacks of gauge groups

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.

Posted by: Urs Schreiber on March 26, 2009 12:45 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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.

Posted by: John Baez on March 26, 2009 1:48 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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

A pullback, but not this time of inclusions (monomorphisms), right?

Posted by: Toby Bartels on March 26, 2009 2:05 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

It’s still a pullback of monomorphisms, Toby!

I know what you’re thinking. You just gotta read the paper.

Posted by: John Baez on March 26, 2009 2:28 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

(I did read the paper once, but now you've gone and changed it, so I don't know for sure what I read before. But here I know that I just misread your comment.)

Posted by: Toby Bartels on March 26, 2009 3:14 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Okay, so I didn’t know what you were thinking. To me it seems very natural to guess that if the vertical arrows in Theorem 7, like this:

$SU(5) \to Spin(10)$

are one-to-one, then the vertical arrows in Theorem 8, like this:

$SU(5) \to SO(10)$

are two-to-one, since they’re composites like this:

$SU(5) \to Spin(10) \to SO(10)$

where the second map is two-to-one.

But in fact they’re still one-to-one!

Posted by: John Baez on March 26, 2009 3:59 AM | Permalink | Reply to this

### Fascinating Pushouts not Pullbacks

What fascinates me is that some guys tried to generalize $G_SM/Z(6)$.

For any two independent generalizations you can form their pushout, and then prove that as a sanity check that $G_SM/Z(6)$ can be proved to be a pullback of the pushout (without just saying that its it’s dual). At some level your theorems 7 and 8 are just sanity checks.

What impresses me is that $SU(5)$, $Spin(4)\times Spin(6)$, and $Spin(10)$ were so carefully minimally generalized that they form a pushout/pullback square.

Maybe I’m reading too much into the original construction of these theories seeing them in the perspective of your reconstruction that leaves out a bunch of crufty details.

Posted by: RodM on March 27, 2009 4:06 AM | Permalink | Reply to this

### Re: Fascinating Pushouts not Pullbacks

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.

Posted by: John Baez on March 27, 2009 6:31 AM | Permalink | Reply to this

### Re: Fascinating Pushouts not Pullbacks

Here is what I think you can do.

Define a category $G$ which has as objects $G_SM/Z(6)$, $SU(5)$, $Spin(4) \times Spin(6)$, $Spin(10)$, and other theories you haven’t discussed.

Add an arrow from object $A$ to object $B$ when $A$ is a generalization of $B$. This makes $G$ a lattice, where the product, $\wedge$, is the pullback we’ve been discussing and the coproduct, $\vee$, is a pushout.

I send you an email when I get to a more secure location.

Posted by: RodM on March 27, 2009 10:38 PM | Permalink | Reply to this

### Re: Fascinating Pushouts not Pullbacks

I was hoping the category of groups and homomorphisms will do. This diagram

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

is a pullback in that category. You’ve got me wondering if it’s also a pushout.

Pushouts of groups tend to be big — they’re also called amalgamated products, and you see them in the Seifert–Van Kampen Theorem. So, that makes me a bit nervous.

For this diagram to be a pushout in the category of groups, I need $SO(10)$ to be the group freely generated by $SO(4) \times SO(6)$ and $SU(5)$, mod relations of the form $x = x'$ where $x \in SO(4) \times SO(6)$ and $x' \in SU(5)$ are secretly the same element of their intersection $G_{SM}/\mathbb{Z}_6$.

For some reason I don’t see how to tackle this. I can probably show that $SO(10)$ is generated by its subgroups $SO(4) \times SO(6)$ and $SU(5)$ — and if not, I really want to know what smaller group is generated by these! But the rest seems intimidating. Maybe it’s false.

Posted by: John Baez on March 28, 2009 1:04 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Trivia:
p48 (bottom): triviallly
p57: “by specifying the homo” seems a bit informal here.

Posted by: Charlie C on March 21, 2009 8:54 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Thanks — fixed!

I’m very happy at how John Huerta is able to explain math in a lively, informal way — this skill is very hard to teach. But occasionally he gets carried away, so I’ve been spending time making some turns of phrase more formal.

“Homo” is obviously problematic. Play it safe, kids: no mathematical entity will take offense if you call it a “morphism”.

Posted by: John Baez on March 21, 2009 9:08 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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?)

Posted by: Charlie C on March 22, 2009 2:40 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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.)
Posted by: John Baez on March 22, 2009 3:41 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Thanks, John! Now I have a guided tour through this otherwise impenetrable jumble of concepts, tools and theories. The handwritten class notes are no problem at all. In fact the occasional annotation helps to clarify things. I must say you zip along at a pretty brisk clip in those lectures! No sleeping allowed! The physical motivations you give for the various mathematical techniques are really helpful for me.

Just ordered the books you recommended. They will keep me out of trouble for quite some time to come! This is really exciting! Kinda like when I first discovered Bach!

Posted by: Charlie C on March 22, 2009 5:25 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Here’s a comment on presentation. For my taste, the text is much, much too wide. (That is, the left and right margins are much too narrow.) This makes it significantly harder for me to read. It means I have to concentrate a little bit on keeping my eye in the right place, when of course I need that concentration for other things. In short, it’s off-putting.

The effect is especially pronounced when there are long passages of text uninterrupted by displays. So the introduction particularly puts me off.

Lamport’s Latex book quotes the following typographical rule: “use lines that contain no more than 75 characters, including punctuation and spaces”. I agree. A quick survey of my bookshelf suggests that novels average less than 60. Your paper seems to be about 100.

I guess you don’t want to increase the page count of a paper that’s already 69 pages, but personally I’d far rather have it take more pages and be comfortably readable.

Posted by: Tom Leinster on March 22, 2009 12:59 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

I much prefer small margins, myself. The standard AMS article package wastes a ton of paper, to the extent that I’ve started downloading the source and using the geometry package to make smaller margins. It makes carrying around these tomes much less weighty, too.

Posted by: Aaron Bergman on March 22, 2009 1:17 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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.

Posted by: Eugenia Cheng on March 22, 2009 3:30 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

OK: so some people like wide margins, some like narrow margins. Writers who use wide margins should know that some people regard it as a waste of paper. Writers who use narrow margins should know that some people will be put off reading them.

John and John can, of course, decide.

In some disciplines (e.g. computer science) it’s common to use two-column formatting. This solves both problems. However, it’s not so suitable for mathematics, since we often have long displays that would be hard to fit in.

Posted by: Tom Leinster on March 22, 2009 1:55 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Since few things would make me happier than for Tom to learn some particle physics, I have made a version with wider margins available here.

Posted by: John Baez on March 22, 2009 3:09 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

I’m honoured!

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

Posted by: Tom Leinster on March 23, 2009 5:56 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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.

Posted by: John Baez on March 23, 2009 7:17 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Tom: despite my initial flippant reaction, I wound up following your advice and narrowing the text by one inch. But I’ve also been streamlining John Huerta’s original writeup, so the page count remains almost the same.

Posted by: John Baez on March 26, 2009 1:55 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

There are two questions that stay within the scope of representation but fall outside the remit of the paper. One is that we should also have a representation of the (double cover of) the Poincare group. The other is symmetry breaking and the Higgs boson(s?). Does anyone have anything to add to the brief comments that have been made on these topics?

There are references to notes by JB. These say we have a representation of the product of the Poincare group and the gauge group. In the GUTs the representation of the gauge group is irreducible so I am confused.

The references a mathematician would give for representation theory of Lie algebras is Humphreys and J.F. Adams.

Posted by: Bruce Westbury on March 22, 2009 10:09 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Bruce wrote:

There are references to notes by JB. These say we have a representation of the product of the Poincare group and the gauge group. In the GUTs the representation of the gauge group is irreducible so I am confused.

In full-fledged quantum field theory, the symmetry group is

$P \times G$

where $P$ is the Poincaré group and $G$ is a compact Lie group. A ‘type of particle’ is an irreducible unitary representation of $P \times G$. By general abstract nonsense, such a representation is of the form

$H_P \otimes H_G$

where the Hilbert space $H_P$ is an irreducible unitary representation of $P$ and similarly $H_G$ is an irreducible unitary representation of $G$.

In my paper with John Huerta, we’re only talking about $G$ and $H_G$. We’re completely ignoring $P$ and $H_P$.

You can see a description of the full representation $H_P \otimes H_G$ for every particle in the Standard Model in this chart. If you don’t understand my names for unitary irreps of the Poincaré group (which involve phrases like ‘massless spin-1’ and ‘right-handed massless spin-1/2’), you can read what these mean in the Spring 2003 course notes. Or, better yet, go straight to the original source:

The ‘inhomogeneous Lorentz group’ is the Poincaré group. Wigner’s classification of its irreducible unitary representations (especially the positive-energy ones) in terms of mass, spin and helicity is fundamental to the classification of particles! Most of these irreps are spaces of solutions of Poincaré-invariant partial differential equations. For example, ‘massless spin-1’ corresponds to the vacuum Maxwell equations, while ‘massless right-handed spin-1/2’ corresponds to the right-handed massless Dirac equation.

When I was learning quantum field theory, I benefited a lot from this book:

• N. N. Bogoliubov, A. A. Logunov, I. T. Todorov, Introduction to Axiomatic Quantum Field Theory, W. A. Benjamin, 1975.

It’s mathematically precise and explains many important things, including Wigner’s classification theorem but also much more. Alas, it seems to be out of print, available only from a suspicious Russian website that offers a CD containing dozens of ripped-off physics books.

It’s possible this book is even better:

• N. N. Bogoliubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, General Principles of Quantum Field Theory, Kluwer Academic Publishers, 1990.

but I met it much later, and never fell in love with it the same way.

Posted by: John Baez on March 22, 2009 10:41 PM | Permalink | Reply to this
Read the post Unitary Representations of the Poincaré Group
Weblog: The n-Category Café
Excerpt: This blog entry is supposed to be a forum for learning and discussing Wigner's classification of the representations of the Poincaré group. Ask and answer questions about this subject here!
Tracked: March 22, 2009 11:48 PM

### Re: The Algebra of Grand Unified Theories II

Thanks for the great job.

I have however one question:

How do you decide a representation to be fermion or a boson since a particle corresponds to a representation and a particle is either a fermion or boson ?

Posted by: dah on March 23, 2009 9:31 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

In this paper we’re only treating particles as representations of some ‘internal symmetry group’, say U(1) $\times$ SU(2) $\times$ SU(3) or SU(5). There’s no way from this to tell if the particle is a fermion or boson. For that, you need to treat the particle as a representation of the Poincaré.

In fact, every particle is a representation of both groups. But for simplicity we’re considering just the internal symmetry group in this paper. For the Poincaré group, you need to go elsewhere.

Posted by: John Baez on March 24, 2009 2:47 AM | Permalink | Reply to this
Read the post The Algebra of Grand Unified Theories III
Weblog: The n-Category Café
Excerpt: Is the SO(10) grand unified theory of particle physics still experimentally viable? A plea for help from phenomenologists.
Tracked: March 25, 2009 11:01 PM

### Re: The Algebra of Grand Unified Theories II

Is the expression for $\pi^0$ on page 15 correct? It looks to me like it should be $u\otimes \bar{u} - d\otimes \bar{d}$.

Bottom of p32: “we can include $SU(3)\times SU(3)$ as block diagonal matrices in $SU(5)$” probably means $SU(2)\times SU(3)$.

near the bottom of p33: “only if $\mathbb{Z}_6$ subgroup acts trivially” might want an additional article.

middle of p35: “we will see another fruit $SU(5)$ theory ripen” probably wants an “of the”

Posted by: Mike Shulman on March 27, 2009 7:39 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Yes indeed, Mike — all your claimed mistakes are really mistakes, and I just fixed them all in the version on my website.

Wow! You’re correcting us about pions! That can come in handy: at parties, people respond better when you say you’re a physicist than when you say you’re a mathematician.

Posted by: John Baez on March 27, 2009 8:25 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Reference # 38 is missing in the text, it’s left blank in the bibliography.

Posted by: anonymous coward on April 7, 2009 11:01 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

A truly careful reader! Thanks, we fixed it. It was going to be a reference to Varadarajan’s book The Geometry of Quantum Theory, but I decided that reference was not especially helpful here.

Posted by: John Baez on April 9, 2009 6:13 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Is the paper updated since the initial post?

Posted by: timur on April 9, 2009 6:50 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Yes, John Huerta and I have been updating the online version an average of once a day. Here are some of the main changes:

• The introduction contains a list of all the theorems in the paper, and a reading list for further study.
• Besides the binary code for particles in the $SU(5)$ theory we include a ‘Pati–Salam code’ for particles in the $SU(2) \times SU(2) \times SU(4)$ theory, which makes certain isomorphisms completely explicit.
• Our proof of the commuting cube had problems, which have been fixed using the Pati–Salam code.
• The conclusions now describe the ‘true gauge group of the Standard Model’ as a pullback.
• Hundreds of sentences are more charming than before.

Today I’m putting the paper on the arXiv, and when it’s up I’ll give a link to the arXiv version.

Posted by: John Baez on April 9, 2009 4:48 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Thanks a lot!

Posted by: timur on April 9, 2009 5:01 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Okay, it’s on the arXiv now!

I thank all of you very much for your help.

Posted by: John Baez on April 10, 2009 11:22 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

It’s a very informative paper, thanks for posting it.

If I may, to the absolute beginners, who never heard of the word “irrep” (or heard of it vaguely), I suggest a thorough reading of Byron and Fuller’s chapter on group representations. Also, a study of the first chapters of Gilmore’s book is recommended to the novice. With this small background (which can be covered reasonably fast), there is a lot that a beginner can learn from this paper. Not that the material is easy, but the reading is. Thanks a lot!

Best wishes,
Christine

Posted by: Christine Dantas on April 11, 2009 8:54 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Thanks for the tips. Much appreciated! Will do.

Posted by: Charlie C on April 12, 2009 2:53 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

p26 “electomagnetic”

I meant to read this long ago; it’s great!

Posted by: Greg Egan on April 15, 2009 4:56 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

p25 “This is is a”

Posted by: Greg Egan on April 15, 2009 5:01 AM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

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.

Posted by: John Baez on April 16, 2009 2:17 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

Right this moment I am sitting in a lecture on GUTs taught from String-theorists to mathematicians in our Center for MathPhys Seminar in Hamburg.

One of the two references given for GUTs is Baez-Huerta. I was able to confirm that this one is the way to go for the mathematicians in the audience. :-)

Posted by: Urs Schreiber on April 16, 2009 2:30 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

I started an entry

Currently it essentially just points to your material here.

Eventually, since our MathPhys seminar this semster is apparently about embeddings of GUTs into F-theory, I am hoping I’ll find time to expand on this.

But likely I won’t find much time. But then, maybe somebody else will find a little bit of time, too. Together that might eventually make a medium or even large amount of man-hours which produces a useful collection of $n$-style information on GUTs.

Posted by: Urs Schreiber on April 16, 2009 3:40 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

p.7 arXiv paper, you are missing a bracket

$L^2([0, 1] \otimes L^2([0, 1]).$

Posted by: David Corfield on April 16, 2009 4:05 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

This paper will appear in the Bulletin of the American Mathematical Society.

Posted by: John Baez on November 3, 2009 4:09 PM | Permalink | Reply to this

### Re: The Algebra of Grand Unified Theories II

I wonder if it is possible to look the commutative diagram at the level of Kaluza Klein. We know that the smallest spaces with SM symmetry are Witten’s spaces in dimension 7. The group Spin(4) x Spin (6) is the (covering of the) group of isometries of the product of spheres S3 x S5, a space of dimension 8. Actually, Witten’s spaces are built by quotient this one by U(1) but this is other topic. Well, lets follow… SU(5) is the group of isometries of the homogeneus space SU(5)/(SU(4)xU(1)), again an space of dimension 8. Finally, Spin(10) is the group of isometries of the 9-sphere. So it seems that going up the cube you are increasing dimensions, from M-theoretical to F-theoretical to WhatTheFuckIs13?-theoretical.

Posted by: Alejandro Rivero on February 13, 2010 1:03 AM | Permalink | Reply to this

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