A Dimensions of the SM

[Marty4691]

So, here's the thing. The SM uses a six dimensional mathematical space to crunch probabilities. That is, one with three space dimensions and three time dimensions. That's a fact.

Now, the consensus is that it is just a convenient tool. The argument goes something like this. The electron in the Dirac equation is a bispinor. We are crunching both chiralities (left-handed and right-handed) at the same time (sorry), so it is not surprising that we need two time variables to do it. Likewise, for the neutrino, even though it only has a left-handed component in the bispinor, it is actually a superposition of three left-handed particles (the neutrino mass eigenstates), so it is not surprising that we need three time variables to crunch things.

In general, if physicists are using a mathematical tool, they tend to explore the heck out of it . But apparently six dimensional space-time is the exception. Granted the math is hard. I mean, there was a reason Kaluza added an extra space dimension instead of an extra time dimension. He was just goofing around in his spare time anyway. Why not try the easy way first.

If I was smart enough, I'd solve the Einstein equations with 2 time dimensions and find the Maxwell equations, but Mensa tells me I should probably leave it to the really intelligent folks...

 

[renormalize]

So, here's the thing. The SM uses a six dimensional mathematical space to crunch probabilities. That is, one with three space dimensions and three time dimensions. That's a fact.

Can you cite textbooks or peer-reviewed articles that explicitly set the Standard Model on a 6D manifold with 3 time-dimensions; i.e., one with signature $(+++---)$?

 

[PeterDonis]

The SM uses a six dimensional mathematical space to crunch probabilities. That is, one with three space dimensions and three time dimensions. That's a fact.

It is? Where are you getting it from? It doesn't look like any presentation of the Standard Model that I've seen.

 

[Marty4691]

Dirac realized that his gamma matrices were the minimum generators of a (+++--) space : P. A. M. Dirac, J. Math. Phys. 4, 901 (1963)

For the weak force they added another gamma matrix which like the others was required to anticommute giving the minimum generators of SL(4,R) which it turns out is homomorphic to SO(3,3), or a (+++---) space : V.P.Nair "Concepts in Particle Physics", Chpt. 9, or M. Thomson "Modern Particle Physics", Chpt. 11

 

[mitchell porter]

I can't access any of those, but Dirac's paper is the subject of https://arxiv.org/abs/1902.10367, and SO(3,3)'s Lie algebra is discussed in https://doi.org/10.3390/sym12050817, and its double cover Spin(3,3) in https://doi.org/10.3390/particles6010008.

I'm interested in ideas like these, but I also like to tread carefully, which means being aware of the reasons why they might be viewed skeptically by the "physics establishment". For example, multi-time theories potentially have causality problems. That hasn't stopped everyone from exploring the subject, but one should be ready to find that a multi-time theory runs into logical problems at some point.

A more technical issue is the use of noncompact gauge groups, such as groups of indefinite signature like SO(3,3) and Spin(3,3). In the context of Yang-Mills, this leads to vacuum instability in the quantum theory, because there's no lowest-energy state. However, there are a few places in mathematical physics where a gauged noncompact group reduces in practice to a maximal compact subgroup. Interestingly, Eric Weinstein's version of grand unification relies on such a mechanism - his series of symmetry breakings includes passage from Spin(6,4) to Spin(6) x Spin(4), which is isomorphic to the Pati-Salam group SU(4) x SU(2) x SU(2).

I can see that the double cover of SO(3,3), Spin(3,3), contains as a compact subgroup Spin(3) x Spin(3), which is isomorphic to SU(2) x SU(2). And at this point, even without doing the algebra, I can see why a Dirac algebra extended to include the weak interaction, would lead to groups like these. SU(2) is the gauge group of the weak interaction, but it is also part of Lorentzian space-time symmetry in various ways. It's one of the "gauge groups" used in loop quantum gravity (the other being SO(3)).

In fact, one offshoot of these attempts, dating back to Ashtekar 1986, to view general relativity as an SU(2) gauge theory, are "chiral graviweak" theories which try to understand the combination of the weak force and gravity as being due to a gauging of SU(2)_L x SU(2)_R in some form. There was another such paper just last month, https://arxiv.org/abs/2505.17935.

There are various ways you might try to motivate such a product of two SU(2) factors. There is a standard but slightly complicated way of analyzing complex representations of the Lorentz group, in terms of two SU(2) representations. Probably there's some graviweak model that works like this.

Another way is to note that Spin(4), the double cover of SO(4), is SU(2) x SU(2). SO(4) is the symmetry group of 4D Euclidean space, and it's common in quantum field theory to work in Euclidean space and then "Wick rotate" to Lorentzian space-time at the end of the calculation. This 4D Euclidean approach to graviweak unification, is part of Peter Woit's "Euclidean twistor unification".

What I'm concluding here, is that this use of noncompact gauge groups arising from a space of (3,3) split signature, is a third distinct way of motivating graviweak unification. As I said at the beginning, there's a specific technical barrier to the use of such gauge groups in Yang-Mills gauge theories, but one may look for loopholes.

Just to round things out, I would mention that, if one is pursuing such an approach, it is logical to try to get matter from the spinors associated with these spin groups. In mainstream grand unification, they only consider compact spin groups. Thus Frank Wilczek likes to emphasize that one standard model generation equals a spinor of Spin(10), and apparently Gerard 't Hooft likes the idea of getting all three generations from a spinor of Spin(18). On the noncompact side, Kirill Krasnov has papers on obtaining the standard model from spinors of Spin(11,3) and Spin(7,7) (the latter is the spin group in Weinstein's theory too). I'll also mention Andrew Hamilton's paper "Six Bits", https://arxiv.org/abs/2308.12293, which extends Wilczek's "five bits" perspective on Spin(10), to a noncompact Spin(11,1) in which the sixth bit is a "time bit".

 

[martinbn]

So, here's the thing. The SM uses a six dimensional mathematical space to crunch probabilities. That is, one with three space dimensions and three time dimensions. That's a fact.

Now, the consensus is that it is just a convenient tool. The argument goes something like this. The electron in the Dirac equation is a bispinor. We are crunching both chiralities (left-handed and right-handed) at the same time (sorry), so it is not surprising that we need two time variables to do it. Likewise, for the neutrino, even though it only has a left-handed component in the bispinor, it is actually a superposition of three left-handed particles (the neutrino mass eigenstates), so it is not surprising that we need three time variables to crunch things.

In general, if physicists are using a mathematical tool, they tend to explore the heck out of it . But apparently six dimensional space-time is the exception. Granted the math is hard. I mean, there was a reason Kaluza added an extra space dimension instead of an extra time dimension. He was just goofing around in his spare time anyway. Why not try the easy way first.

If I was smart enough, I'd solve the Einstein equations with 2 time dimensions and find the Maxwell equations, but Mensa tells me I should probably leave it to the really intelligent folks...

You are confusing the dimensions of space-time, which are four in the SM, and the dimensions of various mathematical spaces, which can be any. For example the dimension of the phase space of seven free particles in classical physics is 42, but the space-time is still four dimensional.

 

[Marty4691]

If you don't think the gamma matrices are fundamental enough to specify the dimension of the SM Lagrangian density, I guess that's your call. Like I said, the consensus is that the whole 6D thing is just a mathematical convenience and nobody should lose sleep over it... That said, the gamma matrices are part of a 4D representation of SL(4,R) which, ironically, has 6 real dimensions and a signature of (+++---).

 

[PeroK]

If you don't think the gamma matrices are fundamental enough to specify the dimension of the SM Lagrangian density, I guess that's your call. Like I said, the consensus is that the whole 6D thing is just a mathematical convenience and nobody should lose sleep over it... That said, the gamma matrices are part of a 4D representation of SL(4,R) which, ironically, has 6 real dimensions and a signature of (+++---).

The Dirac spinors have four complex components, as opposed to two for non-relativistic QM. This applies to a particle with spin 1/2. The number of dimensions, however, varies with the intrinsic spin of the particle. And, in fact, multiple particle systems are described by the tensor product of the individual spin Hilbert spaces.

It is a mistake to confuse the dimensionality of the spin state Hilbert space with 4D spacetime in which the physical spin of the particle is measured.

 

[Marty4691]

Yes, I get that. My point is that the Lagrangian density appears to live in a 6D mathematical space and we don't seem to know much about that space. If you don't want to take the dimensions of the Lagrangian as the dimensions of the SM, that's fine.

 

[martinbn]

Yes, I get that. My point is that the Lagrangian density appears to live in a 6D mathematical space and we don't seem to know much about that space. If you don't want to take the dimensions of the Lagrangian as the dimensions of the SM, that's fine.

Why do you think it is 6D?

 

[Marty4691]

Because the gamma matrices are 6D.

 

[martinbn]

Because the gamma matrices are 6D.

What do you mean they are 6D? They are 4x4 matrices and the corresponding Clifford algebra is 16D. Also why is that the dimension of the Lagrangian (whatever you mean by this)? There are many things in that lagrangian.

 

[martinbn]

Also you said this

But apparently six dimensional space-time is the exception.

So clearly you were thinking of the dimensions of the space-time, and in the standard model they are 4.

 

[PeterDonis]

Dirac realized that his gamma matrices were the minimum generators of a (+++--) space : P. A. M. Dirac, J. Math. Phys. 4, 901 (1963)

For the weak force they added another gamma matrix which like the others was required to anticommute giving the minimum generators of SL(4,R) which it turns out is homomorphic to SO(3,3), or a (+++---) space : V.P.Nair "Concepts in Particle Physics", Chpt. 9, or M. Thomson "Modern Particle Physics", Chpt. 11

I still have no idea what you are talking about. It's true that the papers you reference are about various matrices. But I don't see where you're getting your claims about "generators" of spacetime.

 

[PeterDonis]

the dimension of the SM Lagrangian density

A Lagrangian density has dimensions of energy density. That's true no matter what it's a Lagrangian density of.

 

[Marty4691]

I'll try and walk you through the 6D thing again. I probably wasn't very clear the first time. We can start with Dirac. When he formulated his equation he needed 4 matrices that anticommuted. Later he commuted each of his gamma matrices with all the others and ended up with 10 matrices which he recognized as the generators of SO(3,2). That is, the group of boosts and rotations for 3 space dimensions and 2 time dimensions (this is the first paper Peter). Now these generators only have 2 dimensions. That is, the boost generators have one space dimension and one time dimension, the space rotation generators have two space dimensions and the sole time rotation generator has two time dimensions. So when I say that the Dirac gamma matrices are 5D, that is just short hand. A more accurate description is to say that, as a set, the 4 gamma matrices contain 5 dimensions. Likewise, each term in the Dirac equation isn't 5 dimensional but taken as a whole it contains 5 dimensions because the gamma matrices contain 5 dimensions. In this sense, the Dirac equation is 5D.


When they were trying to figure out the weak force, they found they needed another matrix (the second two references cover this Peter). It turned out that this matrix also anticommuted with Dirac's original ones. When you commute all 5 gamma matrices with each other you end up with 15 matrices which are the generators of SL(4,R). But SL(4,R) is homomorphic with SO(3,3). That is, the group of boosts and rotations for 3 space dimensions and 3 time dimensions. So, you could argue that the 5 gamma matrices taken together are 6D and the SM lagrangian as a whole is 6D.


The actual gamma matrices are part of the 4x4 matrix representation of SL(4,R). So the "real" 6 dimensions appear to be squeezed into 4 by using the complex constant i. To identify a gamma matrix as a boost or rotation generator, you just have to look at the commutation relations. The 5 gamma matrices break down into 3 boosts and 2 rotation generators.

 

[renormalize]

So, you could argue that the 5 gamma matrices taken together are 6D and the SM lagrangian as a whole is 6D.

You're being too vague about what you are calling "dimensions" (D). Can you please clarify by explicitly stating:

• the number of spacetime dimensions of the standard model and the specific symmetry group that acts on that spacetime?
• the number of dimensions of the internal-symmetry space of the standard model and the specific symmetry group that acts on that internal space?

 

[PeterDonis]

I'll try and walk you through the 6D thing again.

You're missing the point of the responses you've been getting. You don't have to explain to us how the Standard Model deals with the various interactions, how gamma matrices work, etc. We know that.

What we don't see any justification for is that the dimensions of the internal symmetry spaces that the Standard Model uses for those interactions are spacetime dimensions. That is the claim you need to support with specific references--not just "it's in this paper/book somewhere". You need to be giving specific, explicit quotes from your references that explicitly make the claim you are making.

this is the first paper Peter

The first paper [1] says that Dirac has discovered a "remarkable representation of the 3-2 De Sitter group" (from the paper's title). I don't see DIrac claiming that the "dimensions" of this group are spacetime dimensions. It's true that the link I gave only shows the abstract; the paper itself appears to be paywalled. But it's still on you to give specific, explicit quotes from the paper that support your claim.

[1] https://pubs.aip.org/aip/jmp/articl...-of-the-3-2-de-Sitter?redirectedFrom=fulltext

 

[PeterDonis]

Moderator's note: Thread level changed to "A" as this is clearly graduate level subject matter and the references that are being given are at that level.

 

[PeterDonis]

the use of noncompact gauge groups

Whether the gauge group is compact or not, its dimensions are not spacetime dimensions. At least, not in the Standard Model as it's currently formulated.

If the OP (who, to be clear, is not you) is talking about some kind of string theory-like model where the extra dimensions of the gauge groups are supposed to be spacetime dimensions, just "compactified" or whatever, it would be nice if the OP made that clear and gave some references that actually talk about that kind of model, instead of just saying basically "hey, look, the symmetry group generated by this set of gamma matrices is such-and-such".

 

[Marty4691]

Thanks for your time and feedback.
M.