Metric Ansatz For Unifying All Forces In 11D?

I'm guessing this is something I need to read up on more.In summary, the ansatz for the 5D metric involves adding a scalar field and a vector field to the 4D metric. For 11D theories that unify all forces, the ansatz involves veilbeins and compact extra dimensions. However, there is no known way to get the required handedness-dependent behavior for particles in an odd number of dimensions. The Kaluza-Klein program has stalled for the past forty years, but there are attempts to revive it in the context of string theory. The symmetry group gauge invariance for KK vector fields comes
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Geonaut
TL;DR Summary
Ok, so I understand that in 5D you apply the machinery of general relativity to an ansatz for the 5D metric that's quadratic in the electromagnetic vector potential as a starting point in obtaining the Ricci tensor and Einstein-Hilbert Lagrangian in which the electromagnetic vector potential is unified with the 4D metric. My question is: What have physicists come up for the ansatz for an 11D metric to unify the rest of the forces with the 4D metric? Are there any good references I can look at?
The ansatz for the 5D metric is

\begin{equation}
G_{\mu \nu}= g_{\mu \nu}+ \phi A_{\mu} A_{\nu},
\end{equation}

\begin{equation}
G_{5\nu} = \phi A_{\nu},
\end{equation}

\begin{equation}
G_{55} = \phi.
\end{equation}

This information was extremely enlightening for me, but what's the analogous ansatz for 11D theories that unify all of the forces? I'm imagining that a realistic choice should definitely involve veilbeins which I just started to learn about. I know this question has to be written about somewhere, but I'm having tremendous difficulty in finding it.

I'd love to learn more about this subject, and I'd really appreciate any good references for this. I am not interested in supergravity at the moment (perhaps later), but everything related to this subject is unfortunately buried in supersymmetry. So far I've been unable to find an answer for my specific question and I fear that it might be nearly impossible considering how dense this subject is.

I am imagining that there is an ansatz that someone found for the 11D metric that when combined somehow with spontaneous symmetry breaking gives us the Standard Model gauge group, but what is it? I am not even sure how spontaneous symmetry breaking is applied to this case, I imagine it must be some kind of hell, but I'm ready to dive into it whenever I find a useful reference.

At the moment, I'm starting to form an educated guess for this answer. If no one responds then perhaps I'll answer my own question.
 
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Arguably, the high point, but also the endpoint, of this classic kind of Kaluza-Klein theory, is Witten's "Search for a realistic Kaluza-Klein theory". You can get gauge fields with symmetry G, if your compact extra dimensions have symmetry G (e.g. a single compact extra dimension has circle symmetry U(1), the gauge group of electromagnetism). The lowest dimension where you can do this for the complete standard model gauge group is 7, so your space-time would have at least 11 dimensions overall. 11 dimensions is also the maximum for supergravity, so there was a unique and interesting opportunity for Kaluza-Klein and supergravity research programs to converge there.

However, alongside the forces, you have to have matter too: three generations of quarks and leptons, with the required color and electroweak charges. This is the part that no one ever figured out. The basic problem is that in the standard model, left-handed and right-handed particles have different electroweak charges; but there's no known way to get this kind of handedness-dependent behavior, if it's to come from a purely gravitational interaction in an odd number of space-time dimensions.

This is where the classic Kaluza-Klein program has stalled for the past forty years. In string theory, one no longer aims to get the other forces from the extra dimensions, but rather in other ways, e.g. gravity from closed strings and gauge forces from open strings attached to branes. However, there are enough theoretical complexities that one can always look for loopholes or some new twist, and PF's own @arivero has an interest in reviving classic KK theory in the new context of string phenomenology.
 
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@mitchell porter thank you for the response, and my apologies for taking a bit to respond. Regarding my question about the ansatz, I came up with a rough answer, but then I found this link http://people.physics.tamu.edu/pope/ihplec.pdf which gives an even better answer (Chapter 1.1). I'll leave it here for anyone looking to study KK theory that comes across this post, I've found it to be a great place to start. Yes, I've recently glanced at a few of @arivero posts, he comes up in search results pretty easily since he's posted a lot about KK theory, I think his suspicion about KK theory is justified, and I share a similar suspicion. Yes, I am aware that the chiral fermion problem exists in compactificatied spaces, although I haven't yet investigated the math behind it.

The last time my questions lead me to this area of physics I ended up calculating the Ricci tensor for the 5D Kaluza theory by hand, and I was able to reproduce the 4D gravity and electromagnetic terms for the Einstein-Hilbert Lagrangian. The next obvious question that I'm currently looking into is: How would we do things to produce the 4D gravitational Lagrangian + a 4D non-abelian gauge theory Lagrangian?

The answer definitely involves adding more dimensions to your space in order to produce more fields, but I'd like to come to an understanding of this without actually doing something similar to that insane calculation that nearly drove me to madness.

Update: I'm slowly coming to an understanding of this via the link I've provided... It seems the symmetry decomposition is done via compactification rather than through an unstable vacuum (is that wrong?)... quite bizarre considering that the vacuum is unstable after compactification in KK theories. I mean, won't the symmetry change again after the compactification due to the vacuum instability? So if I start with an internal symmetry group G, and then compactify it to get the SM group then wouldn't that group break to give a different group? In which case, your objective should be to obtain the SM group after the vacuum becomes stable, not before... but I'm guessing physicists ignore that topic in their analysis because the vacuum instability in KK theories creates insurmountable problems such as making the curled up extra dimensions infinitely large. Am I wrong about something here? Seems likely since I'm just now learning about the math behind this physics.
 
  • #4
"compactificated", sorry, I was drinking... Fortunately, I'm starting to gain a better grasp of this topic now. It seems the symmetry group gauge invariance of the KK vector fields comes from coordinate transformations of the extra dimensions on the internal (compactificated) space, which would answer my question of how do you get a 4D non-abelian gauge theory + 4D gravity... which is an answer that makes sense to me. I'm still missing some pieces to the puzzle, but I'm getting there. My primary source of confusion now lies in how do you go from say, the symmetry group of the 7-sphere, to SO(5)xSU(2)xU(1). Quoting @arivero:

"The naivest group choosing, the rotations of the 7 dimensional sphere, fails instructively. It is SO(8); the sphere S7 decomposes as a fiber bundle of fiber S3 and basis S4, in the same way the group SO(8) decomposes to SO(4) times SO(5). The SO(4) group decomposes to SU(2)xU(1) in the same way that the fiber S3 decomposes as a fiber bundle of basis S2 and fiber S1 (remember that SU(2) has the same algebra that SO(3), and U(1) is the 1-dim rotation group, so relates to S1)."

Sadly, I have no idea what he's talking about as I am not that familiar with this area of mathematics. So I suppose my next objective is to learn about this.

EDIT: Ok, so what he's doing is showing that the symmetry group of S7 is locally indistinguishable from SO(5)xSU(2)xU(1), and so the corresponding local gauge invariance that you get from the coordinate transformations of the extra coordinates on S7 is SO(5)xSU(2)xU(1), and so S7 gives gauge bosons of the symmetry group SO(5)xSU(2)xU(1)?
 
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well, actually, S7 gives SO(8). The KK group is the group of isometries of the space.

If we somehow want to keep track of the fibered structire of S7 as a Hopf fiber, then we need to relax, the symmetry to SO(5)xSU(2)xU(1)

but the real question is why Nature, if she is about KK, does ignore S7?

Try to get Witten 1981, it is the article that launched everything. For a few years.
 
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@arivero thank you for taking the time to write a reply... And ok, if I'm understanding you correctly now, you're saying that S7 gives SO(8), and SO(8) is locally indistinguishable from SO(5)xSU(2)xU(1), and that means that SO(8) can be spontaneously broken (relaxed) to get SO(5)xSU(2)xU(1)? I've been meaning to look into the math behind the question: What is the method used to determine all of the symmetry groups that a group G can be broken down into, and does such a method even currently exist that covers all possible decompositions? It sounds like you're saying this fibration business is the math for that job.

After reading: https://en.wikipedia.org/wiki/Hopf_fibration#Direct_construction
I think I understand how you decomposed SO(8) to get SO(5)xSU(2)xU(1). However, I'm definitely going to have to read more and play around with math to understand how to do this for the general case.

I skimmed through Witten 1981 for the time being, I'm sure I'll look more into it in the near future once my questions lead me further down the rabbit hole, thank you. I'm assuming you recommended it to me because it's a good source for learning about vacuum stability. I'm sure that will be very helpful.

EDIT #1: Your question about S7 is an interesting one that I've also been curious about since I dove into this... I'm thinking/hoping I'll have something interesting to throw your way at some point. I don't easily abandon an idea that feels intuitive, IMO intuition is the most important tool when taking an educated guess with very limited information. IMO, you're better off assuming that you're onto something and that things don't make sense at the moment because you're missing information, which is exactly what you appear to be doing.

EDIT #2: Wow, alright that Witten 1981 paper is more useful than I initially thought. Thanks again arivero.
 
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Geonaut said:
EDIT #2: Wow, alright that Witten 1981 paper is more useful than I initially thought. Thanks again arivero.
Let me discuss how some not-Witten-person could have reached some of the results of the paper. Once it is recognized that the compact manifold does not need to be a Lie group, and that any manifold can produce the KK theory for its group of isometries, one is ready to start looking for the standard model. But as spheres are easier, first thing coming to mind is SO(10), as it is the isometry group of the 9-sphere, S9. And it is a popular GUT group.

But then it is pretty obvious that one can also start from Pati-Salam, SU(4)xSU(2)xSU(2) and translate locally, as you mentioned, SU(4) as SO(6) and SU(2)xSU(2) as SO(4). And then this is the isometry group of the product manifold S5 x S3. Nice, we are in dimension 8.

Now one could think if we can still go down one dimension, by quotient S5xS3 across some orbit, and as a result get the SM group. And that is what Witten acomplishes in this paper: produce some families of 7-dimensional spaces whose isometry group is the standard model.

Or does he?

Because the SM really is a broken state of SU(3)xSU(2)xU(1); the unbroken group is recovered in the higgless limit. On other hand, it could be argued that the low energy limit is SU(3)xU(1), which is the isometry group of CP2 x S1, a manifold of dimension 5.

So the standard model really interpolates between 5 and 7 extra dimensions. I have not seen this observation mentioned in the literature, but it is encouraging.
 
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Well that's very enlightening for me since I haven't read the paper that you're referring to. A Witten 1981 google search gave me his paper on the positive energy theorem, which does not seem to be the same paper that you're talking about, but I did very much enjoy reading into it, it was very informative.

What you're saying makes a lot of sense to me, I feel as though I've learned a lot in our short conversation, and I'm grateful.

To comment on it, you say
arivero said:
Because the SM really is a broken state of SU(3)xSU(2)xU(1)
, but shouldn't the SM really be a broken state of SU(4)xSU(2)xSU(2) since neutrinos have mass? If not, then why? I haven't gotten around to studying SU(4)xSU(2)xSU(2) yet or models with massive neutrinos so maybe I'm missing some information. Hopefully I will study it in the near future, if I can get a break from having questions that seem more important to me. Still, I'd imagine that the unbroken group breaks to give a group that gives neutrinos mass, and if I remember correctly that broken group is SU(4)xSU(2)xSU(2).
 
  • #9
Sorry Witten 1981 to me, in this context, is "Realistic Kaluza Klein Theories". While Witten abandoned ship after Shelter Island II, the paper still sparked some reseach in Europe, particularly some italian teams, that culminated in a sort of classification of possible compactifications.

Geonaut said:
, but shouldn't the SM really be a broken state of SU(4)xSU(2)xSU(2) since neutrinos have mass? If not, then why? I haven't gotten around to studying SU(4)xSU(2)xSU(2) yet or models with massive neutrinos so maybe I'm missing some information. Hopefully I will study it in the near future, if I can get a break from having questions that seem more important to me. Still, I'd imagine that the unbroken group breaks to give a group that gives neutrinos mass, and if I remember correctly that broken group is SU(4)xSU(2)xSU(2).
hey very good point! With one parameter, higgs coupling, we seem to interpolate between 9 and 11. With two parameters, between 9 and 12?
 

1. What is the Metric Ansatz for Unifying All Forces in 11D?

The Metric Ansatz for Unifying All Forces in 11D is a proposed theory in physics that attempts to unify all known fundamental forces (gravity, electromagnetism, strong nuclear force, and weak nuclear force) into a single framework. It is based on the idea that the universe is made up of 11 dimensions, with 3 spatial dimensions and 8 additional dimensions that are not directly observable.

2. How does the Metric Ansatz work?

The Metric Ansatz proposes that all fundamental forces can be described by a single metric tensor, which is a mathematical object that describes the curvature of space-time. This tensor is dependent on the 11 dimensions and their interactions, and it is used to calculate the behavior of particles and forces in the universe.

3. What is the significance of 11 dimensions in the Metric Ansatz?

The Metric Ansatz suggests that there are 11 dimensions in the universe, with 3 spatial dimensions that we can observe and 8 additional dimensions that are compactified (or curled up) and not directly observable. These extra dimensions are necessary for the unification of all known forces, as they allow for the existence of particles and interactions that cannot be explained in a 3-dimensional universe.

4. Has the Metric Ansatz been proven?

No, the Metric Ansatz is still a theoretical framework and has not been proven through experiments or observations. However, it is a promising theory that is being studied and developed by scientists in the field of theoretical physics.

5. What are the potential implications of the Metric Ansatz?

If the Metric Ansatz is proven to be a correct description of the universe, it would have significant implications for our understanding of the fundamental forces and the structure of the universe. It could also potentially lead to advancements in technology and our ability to manipulate and control these forces.

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