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A Quantum Gravity: Renormalization vs. Effective Field Theory

  1. Oct 9, 2017 #1
    In quantum gravity, I get 'mixed signals' as regards renormalizability. My state of confusion is being caused, I suspect, by an incomplete understanding of what is covered under t'Hooft's 1972 proof that non-Abelian gauge theories are renormalizable. ( = Nobel Prize 1999).

    Specifically, some say that GR can be formulated as a non-Abelian gauge theory (see Moshe Carmeli, 'Group Theory and General Relativity', or his book 'Classical Fields: General Relativity and Gauge Theory'). Carmeli, in his classical GR developments makes a comment to the effect that t'Hooft's work implies that GR should be renormalizable. (his work is from the 1970s).

    Yet, I am not aware of any work (over the past 40 years) that extends this 'GR as a non-Abelian gauge theory' into the quantum realm, as a renormalizable gauge theory. This leads me to think that there are finer points to the whole framework of renormalization of non-Abelian gauge theories that I don't fully understand: certain caveats, perhaps, in t'Hooft's proof that disqualify GR from renormalization. For example, Carmeli expresses things in terms of the group GL(2,C) in place of SU(2) (standard Yang Mills). GR imposes other issues (nature of the observer, four dimensions etc).

    Fast forward to 2017 and quantum GR is considered a good candidate for an effective field theory, which reinforces the notion that renormalization is an impossibility, non-Abelian gauge theory or no.

    Does anyone have insight into how we make ends meet here?
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  3. Oct 9, 2017 #2
  4. Oct 9, 2017 #3

    king vitamin

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    't Hooft and Veltman proved that Yang-Mills and spontaneously broken Yang-Mills are renormalizable; gravity can't be interpreted as a Yang-Mills-like gauge theory. A few years later they tried their regularization scheme out on gravity and found that pure gravity is actually finite to first order, but when coupled to matter it becomes divergent (and even pure gravity is non-renormalizable past first-order iirc).
  5. Oct 10, 2017 #4
    Many thanks to both of you.

    I was not previously familiar with Woodward's article linked by romsofia. The info about adding fourth derivatives to eliminate perturbative divergences is interesting. So is that minor detail that doing so causes the universe to expand without bound. Matter coupling (if I read correctly) was the occasion for adding the fourth derivatives in the first place. If t'Hooft and Veltman ran into problems with this coupling as well, I am satisfied that this coupling leads to a major snag.

    I have indeed seen literature in which first order is handled correctly (iirc, Feynman actually illustrated this in his 1960s work compiled in 'Feynman Lectures in Gravitation') . Perturbation theory quickly leads to three-graviton vertices arising from non-linearities in the GR field. There are of course then rapidly accumulating multi-graviton vertices that lead to renormalization requiring an infinite number of counterterms.
  6. Nov 9, 2017 #5
    Back to quantum Yang-Mills, I am trying to reconcile some things I see in print regarding both the solubility of the YM theory, and the question of gravitation as it relates to YM.

    Anthony Zee in his 'Quantum Field Theory in a Nutshell', first ed, has a chapter VII devoted to YM. On page 356, he concludes a section "If Yang-Mills theory ever proves to be exactly soluble, then the perturbative approach with its mangling of gauge invariance is clearly not the way to do it". Fair enough, but if as t'Hooft proved, YM theories are renormalizable, what stands in the way of obtaining a perturbative solution? Are we in a situation similar to mathematicians' 'existence' theorems, whereby the solution exists, but that does not mean it is easy to find, or could even be found by mere mortals?

    Claus Kiefer, in his book 'Quantum Gravity' , 2nd Ed, summarizes an argument, originally due to Teitelboim, that gravitation can be represented in a YM form. (see eqs 4.36 and 4.37) and surrounding discussion.

    In light of the two paragraphs above, a newbie might be excused for thinking that gravity can be cast as a Yang-Mills theory, falling into a class which has a renormalizable form, and thus could be solved. When discussing renormalization in quantum gravitation, it is often said that the coupling constants have the 'wrong' dimensions. Does it boil down to something as straightforward as this?
  7. Nov 9, 2017 #6


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    My 2 cents: GR can be recasted as a gauge theory, but certainly not as a YM-theory. The YM-action is quadratic in the fieldstrength, while the Einstein-Hilbert action is linear in the field strength of Lorentz transformations. This also affects the dimension of the coupling in front of the action, of course.
  8. Nov 14, 2017 #7
    has there been any attempts to create a theory of gravity as a YM-theory? it wouldn't produce the the Einstein-Hilbert action but could give insights on a theory of QG
  9. Nov 15, 2017 #8


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  10. Jan 4, 2018 #9

    Urs Schreiber

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    Indeed. For instance gravity in d=2+1 is equivalent (with due care, see below) to Chern-Simons gauge theory (for the Poincare or (anti-)deSitter group as the gauge group, depending on the sign of the cosmological constant).

    Generally, in any dimension the space of pseudo-Riemannian metrics (and hence the space of field histories of gravity) is a subspace of that of Poincare-Lie algebra valued gauge fields (first-order formulation of gravity). But not only is the action functional of gravity not that of YM theory under this identification, but there is also a constraint on these gauge fields: the translational part (the vielbein) has to be non-degenerate such as to indeed be a "soldering form" (mathematically this is the constraint on a Cartan connection).

    This constraint is in principle also required in 3d, but people tend to feel that the equivalence to 3d Chern-Simons theory is too neat to let that constraint ruin it. But then, in the seminal quantization of 3d gravity as a CS theory in Witten 88, the last section finds the author wondering that the quantum spacetime apparently described by this quantum gravity theory seems to be independent of the 3d manifolds that the Chern-Simons action is defined on. It looks like this is precisely the defect one expects if the "soldering form" no longer actually solders the underlying spacetime to the bundle that the Cartan connection is defined on...
  11. Jan 9, 2018 #10
    I believe "how we make ends meet here" is unclear at this moment. For what it's worth, the effective field theory approach makes a lot of sense to me. Since we have probed only a very small range of energies and scales, I wouldn't find it surprising if today's field theories turned out to be effective field theories valid at low energies and large scales, emerging from unknown "more fundamental" theories. I recommend Volovik's book, "The Universe in a Helium Droplet."
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