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Phlip Mannheim's conformal theory of everything

  1. Jun 6, 2015 #1
    I'll say in advance that I don't believe this, but I would like to have a coherent assessment of its merits and problems. For example, is this an elegant and logical synthesis; or is it just a patching together of various proposed solutions for various BSM issues, related only by the appearance of conformal symmetry?

    Living Without Supersymmetry -- the Conformal Alternative and a Dynamical Higgs Boson
    Philip D. Mannheim
    (Submitted on 1 Jun 2015)
    We show that key results of supersymmetry can be achieved via conformal symmetry. We propose that the Higgs boson be a dynamical bound state rather than a fundamental scalar, so that there is no quadratic divergence self-energy problem for it and no need to invoke supersymmetry to resolve it. We study a conformal invariant theory of interacting fermions and gauge bosons, in which there is scaling with anomalous dimensions and dynamical symmetry breaking, with the dynamical dimension of ψ¯ψ being reduced from 3 to 2. With this reduction we augment the theory with a then renormalizable 4-fermion interaction with dynamical dimension equal to 4. We reinterpret the theory as a renormalizable version of the Nambu-Jona-Lasinio (NJL) model, with the gauge theory sector with its now massive fermion being the mean field and the 4-fermion interaction being the residual interaction. It is this residual interaction that generates dynamical Goldstone and Higgs states, states that, as noted by Baker and Johnson, the gauge theory sector itself does not possess. The Higgs boson is found to be a narrow resonance just above threshold. We couple the theory to conformal gravity, with the interplay between conformal gravity and the 4-fermion interaction taking care of the vacuum energy problem. With conformal gravity being a consistent quantum gravity theory there is no need for string theory with its supersymmetric underpinnings. With conformal gravity fits to galactic rotation curves and the accelerating universe not needing dark matter, there is no need to introduce supersymmetry for either the vacuum energy problem or to provide a potential dark matter candidate. We propose that it is conformal symmetry rather than supersymmetry that is fundamental, with the theory of nature being a locally conformal, locally gauge invariant, non-Abelian NJL theory.
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  3. Jun 7, 2015 #2


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    Personally I find it incomprehensible. At the end of the first sentence is the expression "conformal symmetry". This could have a meaningful definition, but it new to me. As I tried to read further I kept running into similar difficulties with his expressions.
  4. Jun 7, 2015 #3
    The Conformal symmetry group is a well-established extension of the Poincaré symmetry group.
    Spacetimes with conformal symmetry is a spacetime with a metric invariant under the transformations of the conformal group.
  5. Jun 8, 2015 #4


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    Well, I think the core observation and in fact the basis of this paper is the fact that EH gravity on the lightcone (see ref 61 of 't Hooft in the Mannheim paper) as well as the dirac equation for massless fermions (see http://www.dwc.knaw.nl/DL/publications/PU00017567.pdf !), both are conformal invariant. So, why not assume that the fundamental equations of nature (describing all the fermions and fields) are equations without any (bare)mass terms at all? And promote conformal invariance to a fundamental symmetry. Mannheim showed that he can extend this to the standard model YM interactions and implement new ways of dynamical mass generation. Whether this is a new result or not, I cannot judge.

    By the way, I am puzzled by the fact that Mannheim is trying to push aside supersymmetry so hard. I think it is totally irrelevant for his paper. He could have written it, without any reference to this. Are people still so frustrated about all the attention supersymmetry got in the last 30 years or so? :)
    Last edited: Jun 8, 2015
  6. Jun 13, 2015 #5
    A conformal transformation preserves angles. So that includes translations and rotations and scale transformations (i.e. making a figure uniformly bigger or smaller), but it also includes "special conformal transformations" which warp straight lines but do it in a way that preserve the angles where they intersect.

    Conformal symmetry is invariance under conformal transformations and it's a major theme in theoretical physics. Quantum mechanics often enhances scale invariance to conformal invariance. For example, in the study of how QFT behavior varies with scale, the regimes where a QFT *doesn't* vary are described by CFTs, conformal field theories. And conversely, the breaking of conformal symmetry by a quantum anomaly is how the "QCD scale" arises in QCD.

    The world is "almost conformal" in various ways (Berlin said this too, in the previous comment). The standard model itself would be conformal if you removed one term from the Lagrangian, the quadratic Higgs interaction term. Also, the solutions of general relativity are all solutions of conformal gravity, a more general theory.

    These two observations offer a way to approach Mannheim's argument that the world could be exactly conformally symmetric at a fundamental level. He has a schema for a single equation (his equation 124) - a gauge theory, with fermions, coupled to gravity - but the argument does divide, more or less, into a section on gauge theory and a section on gravity.

    I can't judge the many technical claims in the section on gauge theory, but I will point out that what he constructs differs from the standard model in a crucial way. He describes electromagnetism coupled to a massive fermion field. This fermion field also has a quartic self-interaction, which produces a fermion-antifermion bound state, and Mannheim says that this is the Higgs boson.

    The way it all works is quite intricate; it might be ingenious or it might be complicated, I can't tell. But it differs from the standard model. Mannheim's "Higgs boson" seems good only for giving mass to a gauge boson, in the usual way. His fermion has mass just because it can - because his algebraic framework permits it. He also remarks (bottom left, page 25) that fermions in his framework must be Dirac fermions, with four components.

    The fermions of the SM are Weyl fermions - chiral fermions, fermions with just two components - and they get their masses by coupling to the Higgs field. Also, in the SM, it's nonabelian gauge bosons which become massive. Mannheim only considers the abelian case and maybe that misses some essential differences.

    Meissner and Nicolai, meanwhile, have described a "conformal standard model" which has none of these deviations - it's just the standard model, minus the quadratic Higgs interaction as something fundamental, plus a mechanism that will create an effective quadratic interaction. I'll also point out that the famous "twistor string" gives rise to a conformal gauge theory coupled to conformal gravity.

    So my comment for now is just that Mannheim's work has shortcomings which aren't inherent to the "conformal program" in its broadest sense - there are other ways to pursue a conformal theory of everything. But I still consider what he's doing, in the gauge theory part of his paper, as interesting enough to deserve further analysis.
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