Graviweak Unification and Coleman-Mandula theorem

In summary: This means that the two theories may be invariant under certain transformations, but not others. This is analogous to the fact that the algebra of the SO(3,1) group does not commute with the algebra of the complexified SO(3,1) group.
  • #1
kodama
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do these proposals violate the Coleman-Mandula theorem since they combine space-time and internal symmetries via SU(2)r and SU(2)l

how plausible are these proposals as actual physical theories that unify gravity with SU(2) weak force?

Graviweak Unification​


F. Nesti, R. Percacci

The coupling of chiral fermions to gravity makes use only of the selfdual SU(2) subalgebra of the (complexified) SO(3,1) algebra. It is possible to identify the antiselfdual subalgebra with the SU(2)_L isospin group that appears in the Standard Model, or with its right-handed counterpart SU(2)_R that appears in some extensions. Based on this observation, we describe a form of unification of the gravitational and weak interactions. We also discuss models with fermions of both chiralities, the inclusion strong interactions, and the way in which these unified models of gravitational and gauge interactions avoid conflict with the Coleman-Mandula theorem.


Comments:18 pages, typos corrected and improved wording
Subjects: High Energy Physics - Theory (hep-th)
Cite as:arXiv:0706.3307 [hep-th]


Gravitational origin of the weak interaction's chirality​


Stephon Alexander, Antonino Marciano, Lee Smolin

We present a new unification of the electro-weak and gravitational interactions based on the joining the weak SU(2) gauge fields with the left handed part of the space-time connection, into a single gauge field valued in the complexification of the local Lorentz group. Hence, the weak interactions emerge as the right handed chiral half of the space-time connection, which explains the chirality of the weak interaction. This is possible, because, as shown by Plebanski, Ashtekar, and others, the other chiral half of the space-time connection is enough to code the dynamics of the gravitational degrees of freedom.
This unification is achieved within an extension of the Plebanski action previously proposed by one of us. The theory has two phases. A parity symmetric phase yields, as shown by Speziale, a bi-metric theory with eight degrees of freedom: the massless graviton, a massive spin two field and a scalar ghost. Because of the latter this phase is unstable. Parity is broken in a stable phase where the eight degrees of freedom arrange themselves as the massless graviton coupled to an SU(2) triplet of chirally coupled Yang-Mills fields. It is also shown that under this breaking a Dirac fermion expresses itself as a chiral neutrino paired with a scalar field with the quantum numbers of the Higgs.


Comments:21 pages
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph)
Cite as:arXiv:1212.5246 [hep-th]
 
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  • #2
In regards to Coleman-Mandula (CM) how is this proposal any different than the Standard Model which also treats left and right handed chiral components differently? CM, in my very limited understanding, says one can’t mix internal symmetry algebras with those of the inhomogeneous Lorentz group in any other than the trivial way, as mutually commuting algebras.
 
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  • #3
kodama said:
do these proposals violate the Coleman-Mandula theorem since they combine space-time and internal symmetries via SU(2)r and SU(2)l

One of the hypotheses of the Coleman-Mandula theorem is the existence of a Minkowski metric. Thus from the point of view of the theory described above, this means that it can only apply to the “broken” phase, more precisely to the special case when the ground state is flat space. But we have shown that in the broken phase the residual symmetries are precisely a global Lorentz symmetry and a local internal symmetry. This is in complete agreement with the Coleman-Mandula theorem.
I think they are saying that because this symmetry of theirs is broken, such symmetry does no affect the S-matrix.
 
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  • #4
This topic is an important test of one's understanding, and I have to say I don't yet know how to think about the Coleman-Mandula argument in the context of "graviweak unification".

A starting point might be Mandula's comment, in his article at Scholarpedia: "What fails in trying to implement a hybrid symmetry is not the possibility of grouping particles into representations of hybrid symmetry, but rather the conservation of the hybrid quantum numbers in scattering reactions or particle decays."

The Coleman-Mandula theorem is a theorem about the S-matrix, i.e. about probability amplitudes for particle scattering. So it's really only fully relevant for a theory developed to the point of having an S-matrix - and do any of these "graviGUTs" even approach that level? Or do they founder for simpler reasons?

Then we have the idea that the theorem does not apply to spontaneously broken symmetries, because those aren't symmetries of the S-matrix, they're only symmetries of the action. Lubos Motl brought this up several times in his critiques of graviGUT theories (now unfortunately lost with his blog, which lives on only at archive.org, and is no longer indexed by Google), arguing that this means that there is no actual unification in such a theory, and it's not really a loophole.

On the other hand, Percacci and Nesti claim that the unbroken phase of their graviGUT theories exists, but only as a purely topological field theory (no metric), and so that this is a meaningful loophole in the theorem.

The older kinds of graviGUT theories were based on gauging noncompact groups that extended the Poincare group, itself also noncompact. For example, Percacci talks about SO(3,11), combining SO(3,1) with the GUT group SO(10). But building a quantum gauge theory on a noncompact group is definitely problematic (this has been the main technical criticism of Eric Weinstein's Geometric Unity theory, and he wants to escape it via a way which is interesting but whose viability has definitely not been demonstrated).

Whether or not general relativity is a gauge theory is partly a matter of definition - if it is regarded as a gauge theory, it certainly differs in crucial ways from a Yang-Mills gauge theory - but that discussion is clearest at the classical level, and the Coleman-Mandula theorem is a claim at the quantum level. Again, it's not clear to me that graviGUTs even exist as quantum field theories. Can you make an effective field theory for a graviGUT, the same way that Feynman, Donoghue and others have studied a perturbative theory of quantum general relativity? Does perturbative quantum GR have an S-matrix that falls within the scope of the Coleman-Mandula theorem?

And so far I've only talked about graviGUTs like SO(3,11). The real topic here is this peculiar "graviweak unification", based on an Ashtekar-like chiral factorization of a complexified Poincare algebra, in which only one SU(2) factor is needed to give an action for gravity, and it's proposed that the other SU(2) factor will become the gauge group for the weak force. Naively one might say - Coleman-Mandula permits products of spacetime and internal symmetries, and SU(2)L x SU(2)R is a product, so the theorem is not violated.

But as before, I am not sure that "graviweak unification" is even well-defined enough to reach the level where Coleman-Mandula could be an issue. And ultimately I know @kodama's main interest here, and mine, is Peter Woit's proposed theory, which is a kind of graviweak unification in Euclidean twistor space. Making a quantum gravity theory in twistor space brings up many other issues, like Penrose's "nonlinear graviton" program, in which a graviton would carry a quantum of curvature, rather than being a perturbation of flat space like Feynman's gravitons. How does Coleman-Mandula look from that perspective? Hard to say until such a theory has been constructed! (But I do feel the prospects for the "nonlinear graviton" are better than those of loop quantum gravity.)

One more thing... If I look for a mainstream perspective on how Coleman-Mandula might apply to theories of gravity, I notice that supersymmetry is considered the most important true exception to the original Coleman-Mandula theorem, and that gauging supersymmetry does give a theory with gravity (supergravity), because the "super" parts of the supersymmetry algebra can actually generate spatial translations. This seems to be as close to mixing space-time and internal symmetries as ordinary quantum field theory can come: not mixing them directly, but having a fermionic internal symmetry which, when squared, gives you a space-time transformation.
 
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  • #5
Dredging up some comments made by mitchell on graviweak unification in comments at my blog from April 2013:

chiral graviweak unification [is] discussed in this PF thread - I mean the basic idea that weak SU(2) and Ashtekar SU(2) are opposite chiral components of the same connection.

Incidentally, Calmet has also argued for a resemblance between weak and gravitational interactions.

Occasionally you get people on the forums asking how the Higgs boson is related to the graviton, since they have heard that the Higgs "creates mass". If we could combine your ansatz with chiral graviweak unification, there really would be a tight relationship between the field that creates mass and the field that responds to it.
The linked Calmet paper from 2010 was looking for a Higgs boson free Higgs field, which, of course, didn't pan out. It's abstract said:

We emphasize that the electroweak interactions without a Higgs boson are very similar to quantum general relativity. The Higgs field could just be a dressing field and might not exist as a propagating particle. In that interpretation, the electroweak interactions without a Higgs boson could be renormalizable at the non-perturbative level because of a non-trivial fixed point. Tree-level unitarity in electroweak bosons scattering is restored by the running of the weak scale.

The old PF thread (now closed), mostly by the late @marcus, has lots of good discussion including discussion of both papers from post #1 in this thread and is worth a read.

I also had these thoughts in December 2013, thinking about possible sources of a warm dark matter candidate (something I now found much less plausible because warm dark matter has faced subsequent observational challenges):

I am rather inclined to see as promising ideas such as graviweak unification that seek a source for a fundamental warm dark matter particle in the gravitational side of a Theory of Everything, rather than the GUT side of a theory of everything (assuming that a GUT can be coaxed out of the Standard Model of Particle Physics).

For example, the notion of a spin-3/2 WDM counterpart to the spin-2 graviton is an attractive one with the anomalous spin both filling a gap in the roster of fundamental particles of the spin, and providing a supersymmetry-like counterpart to the graviton (the true spin-3/2 SUSY gravitino is basically excluded by LHC data, however, as are sterile neutrinos produced by active-sterile neutrino mixing), while possibly explaining why it cannot be produced (or at least detected) in decays of spin-1 particles like the W and Z, or spin-0 particles like the Higgs boson, while leaving open the potential for phenomenologically invisible weak force interactions of this particle.

In a simple sterile neutrino singlet model one might imagine it being produced in interactions between a high energy spin-2 graviton and a photon or Z boson (electric charge conservation rules out a W boson, color charge conservation would rule out a gluon) that produce two spin-3/2 WDM particles. The right conditions might arise frequently in the intense immediately post-singularity environment shortly after the Big Bang, but rarely thereafter, or only in the vicinity of supermassive black holes now.
 
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It's my understanding that Ashktar variables are chiral only when the Immizi parameter is imaginary, which results in noncompact SL(c,2) rather than real and SU(2)
what more needs to be done to take this as a serious proposal with possible observational testing
 
  • #7
A new relevant paper:
We argue that the SL(2N,C) gauge theories may point a possible way where all known elementary forces, including gravity, could be unified. Remarkably, while all related gauge fields are presented in the same adjoint multiplet of the SL(2N,C) symmetry group, the tensor field submultiplet providing gravity can be naturally suppressed in the weak-field approach developed for accompanying tetrad fields. As a result, the whole theory turns out to effectively possesses the local SL(2,C)×SU(N) symmetry so as to naturally lead to the SL(2,C) gauge gravity, on the one hand, and the SU(N) GUT, on the other. Since all states involved in the SL(2N,C) theories are additionally classified according to their spin values, many possible SU(N) GUTs -- including the conventional one-family SU(5) theory -- appear not to be relevant for the standard 1/2 spin quarks and leptons. Meanwhile, the SU(8) grand unification for all three quark-lepton families stemming from the SL(16,C) theory seems to be of special interest that is studied in some detail.

J.L. Chkareuli, "Unification of elementary forces in gauge SL(2N, C) theories" https://arxiv.org/abs/2208.01478 (August 2, 2022).​

 
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  • #8
mitchell porter

"I’ve been spending the summer trying to write up some details of the ideas I’ve been working on, specifically the claim that the geometry of spinors in four dimensions allows one to think of one of the SU(2)s in the Euclidean Spin(4) symmetry as an internal symmetry. Still learning more about how this works, hope to have something ready to publicize within the next month or so."
Posted on August 18, 2022 by woit

could you explain
 

Related to Graviweak Unification and Coleman-Mandula theorem

1. What is Graviweak Unification?

Graviweak Unification is a theoretical framework that attempts to combine the two fundamental forces of gravity and the weak nuclear force into a single unified theory. It is a major goal in modern physics, as it would provide a more complete understanding of the universe and its interactions.

2. What is the Coleman-Mandula theorem?

The Coleman-Mandula theorem is a mathematical theorem that places restrictions on the possible symmetries of a quantum field theory. It states that in order to preserve the consistency of the theory, the only possible symmetries are those that mix bosons with bosons and fermions with fermions.

3. How do Graviweak Unification and the Coleman-Mandula theorem relate?

Graviweak Unification attempts to unify gravity and the weak nuclear force, while the Coleman-Mandula theorem places restrictions on the possible symmetries of a unified theory. Therefore, the Coleman-Mandula theorem is an important consideration in the development of a theory of Graviweak Unification.

4. What are the challenges in achieving Graviweak Unification?

One of the main challenges in achieving Graviweak Unification is the vast difference in the strength of the two forces. Gravity is much weaker than the weak nuclear force, making it difficult to reconcile the two in a unified theory. Additionally, the Coleman-Mandula theorem places restrictions on the possible symmetries, making it challenging to find a consistent theory that unifies the two forces.

5. What are the potential implications of a successful Graviweak Unification?

If Graviweak Unification is achieved, it would provide a more complete understanding of the fundamental forces of the universe and their interactions. It could also potentially lead to the unification of all four fundamental forces, including the strong nuclear force and electromagnetism. This could have significant implications for our understanding of the origins and workings of the universe.

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