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I M-theory and loop quantization of higher dimensional SUGRA

  1. Jul 30, 2017 #21


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    Funny. I've tried to read Rovelli's Covariant Loop Quantum Gravity for the last few weeks, and with my background I thought I'd be able to understand the basics just fine. Nevertheless I tried a few times and quit because I couldn't get my head around much of it, and even things I was supposed to know were explained in such a way it confused me quite a bit. Maybe I'll open a "basic LQG questions"-topic some time :P

    But I'm intruiged by the posts of Urs and Mitchell about the consistency of the whole framework, and maybe it's a sign that I'm not dumb after all :P
  2. Jul 30, 2017 #22


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    I would only say that LQG might not be consistent, but it doesn't mean QFT is necessarily consistent either.

    Anyway, you cannot prove the consistency of a mathematical theory, let alone physical theory.

    You can only show the inconsistency of a theory by showing that a both a theorem and its negation are contained in this theory.

    So did you find such a contradiction?
  3. Jul 30, 2017 #23
    Couldn't the same be said for string/M-theory? what if Nature is neither supersymmetric nor extra dimensional? there are indications from LHC, dark matter experiments, non - WIMP miracle, electron EDM experiments, and results such as the 126 Gev higgs that there is no supersymmetry nor extra dimensions as required by string theory.

    and loop quantum cosmology may make contact with established physics and observation
  4. Jul 31, 2017 #24

    Urs Schreiber

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    What Mitchell Porter writes is right to the point. Allow me just to add some details on this aspect here:

    While I agree with the broad impetus of the analogy, hence while I agree that parts of the AQFT community got stuck with an axiomatics which they couldn't connect to known QFT, it should be highlighted that the situation for AQFT is really not as bad. The axioms in themselves are physically reasaonable, if maybe in need of slight modification (see below) and the problem is mainly that the axioms of local nets of C-star algebras of observables are over-ambitious, in that they aim to axiomatize non-perturbative QFTs. The failure to construct interacting examples of such in dimensions 4 or higher remains a failure, but given that for the archetypical case of interest, Yang-Mills theory, this is a famous open question, one in the list of the "Millenium Problems", this is not really a shame.

    That the axioms of AQFT in themselves are good is witnessed by what they achieve in 2d, specifically for 2d conformal field theory. The mathematical physics literature on constructions and classification of (rational) 2d CFTs has two pillars, one is vertext operator algebras, the other is AQFT in the guise of conformal nets of observables, notably in the hands of Kawahigashi and Longo and their school. Ironically, thereby AQFT has probably done so far more for the rigorous construction of models in string theoy than for 4d QFT..

    This problem has not remained unobserved in the community. Notice that in the last years things have drastically changed with the development of what is now called "perturbative AQFT" due to Brunetti and Fredenhagen. Where Haag-Kastler originally demanded a local net to take values in C-star algebras, aiming for the non-perturbative quantum theory and thus making it hard to construct examples, Brunetti-Fredenhagen make the obvious observation that one may leave the axioms intact while asking not for C-star algebras but just for formal power series algebras. These are the kind of algebras of quantum operators that are produced by standard pertrubative QFT methods (Feynman diagram expansion).

    A key observation of these authors is now that the "causal perturbation theory" of the Epstein-Glaser renormalization scheme naturally produces not just global formal power series algebras of interacting fields, but a local net of these (in the sense of AQFT): The notorious "adiabatic swtiching" of the Epstein-Glaser approach, which is traditionally thought of as being something that needs to be taken away at the end of the computation (the "adiabatic limit" of Epstein-Glaser, the source of infrared divergences) is precisely what gives a construction of perturbative observables on each causal local patch, and the causality axiom on the S-matrix in the Epstein-Glaser approach is precisely what makes this assignment "Einstein causally local" in the original sense of Haag-Kastler. Now the framework of AQFT serves to show that we don't actually need to take the adiabatic limit, since the QFT is already encoded in its local net.

    This is in fact an old insight due to

    V. A. Il’in and D. S. Slavnov, "Observable algebras in the S-matrix approach", Theor. Math. Phys. 36 (1978) 32.

    but this work was ignored and forgotten until it was rediscovered in

    Romeo Brunetti, Klaus Fredenhagen, "Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds", Commun. Math. Phys. 208 : 623-661, 2000 (math-ph/9903028)

    This unification of traditional perturbative QFT (in the Epstein-Glasser renormalization scheme) and AQFT methods now goes by the name locally covariant perturbative quantum field theory (or some variant term) and is arguably the state of the art of modern mathematically precise perturbative QFT. For example this is the framework in which Hollands proves the renormalizability of Yang-Mills theory on general curved backgrounds

    Stefan Hollands, "Renormalized Quantum Yang-Mills Fields in Curved Spacetime", Rev.Math.Phys.20:1033-1172,2008 (arXiv:0705.3340)

    Finally, and this brings us back to the core topic of this thread, locally covariant perturbative quantum field theory is the natural theoretical context in which to precisely formulate perturbative quantum gravity in what readers here might want to call the "canonical quantization" approach, but done right (albeit just perturbatively), see the references here.
    Last edited: Jul 31, 2017
  5. Jul 31, 2017 #25

    Urs Schreiber

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    There is a whole list of reasons why this is overly naive.

    First, the idea that a choice of coordinates on phase space (e.g. Ashtekar variables) should affect the outcome of quantization is in contradiction with basic facts of physics. On the contrary, everything ought to be independent of artificial choices of parameterizing phase space, even if maybe one choice of coordinates may make some aspects more transparent than others. But if you find yourself with a would-be quantization that only works in one set of "variables" but not in another, then something went wrong.

    Second, gravity is not fundamentally a gauge theory as Yang-Mills theory is, even if you write it in first-order language in terms of a vielbein and a spin connection. The spin connection is an auxiliary field that serves to implement the torsion freeness constraint, but the genuine field of gravity is encoded in the vielbein. Mathematically the statement is that a field configuration of gravity is equivalently encoded in a variant of an affine connection, yes, but the full structure is that of a Cartan connection This is a plain affine connection plus extra data and constraints. Glossing over this point is the source of much confusion in the literature.

    Third, the implicit suggestion that it would be easy to write down a "canonical quantization" once we have realized a theory as a gauge theory is in stark contrast to the observed reality: the mathematically precise nonperturbative quantization of Yang-Mills theory is a famous open problem, dubbed a "Millenium Problem". That LQG sweepingly claims to alread have easily solved the much harder problem of non-perturbative quantization of gravity shows the disconnect between LQG and the realities as perceived by the rest of the scientific community. It is similar to claims of simple proofs of the Riemann hypothesis: apart from the "proofs" being overly naive, their statement displays ignorance of the real nature of the problem.

    Fourth, the idea that "canonical quantization" is something one applies without much ado in field theory is naive. First of all, what is really meant by "canonical quantization" in mathematically precise terms is that one computes the phase space of the field theory as a (pre-)symplectic (infinite-dimensional) manifold, and then applies either algebraic deformation quantization or else geometric quantization to this (pre-)symplectic space. Nothing about this is particularly "canonical" as the whole subtlety of quantization in field theory rests in the fact that various ambiguities are encountered. These manifest themselves ultimately in what in the perturbative description appears as counter-terms in the renormalization procedure.

    The key reason why the quantization of gravity is more subtle than that of Yang-Mills theory is that there is an infinite number of counter-terms to be chosen in the (perturbative) quantization. This is what it means to make the famous statement that gravity is not renormalizable. There are many ways to exhibit this ambiguity, but the most vivid is that this corresponds to the choice of coefficients for all higher-order terms that could be added to the action functional which one starts out thinking one is quantizing. (The entire lack alone of appearance of any manifestation of this infinite ambiguity in the quantization of gravity is what makes LQG dubious without even looking at any details.)

    This is by the way where perturbative string theory and also asymptotic safety comes in. Both approaches may be regarded as parameterizing these infinite arrays of possible counterterms by certain algorithms.

    In string theory the string perturbation series comes out term-wise finite, hence normalized, and one may trace this back to the presence of the higher string modes, which hence serve as the counter-terms. There is still ambiguity in this (the choice of perturbative string background, hence of point in the "landscape of perturbative vacua") but now it is repackaged in a algebro/geometric way that reveals further structure, notably a dynamics which may choose among different such backgrounds.

    In asymptotic safety instead one requires certain lower dimensional submanifolds in the space of counter terms (coupling constants) and the specification of such a submanifold in an infinite-dimensional space absorbes an infinite number of choices, leaving only finitely many to be still specified.

    Notice that in principle any finite-dimensional submanifold would do, it need not be the one including a UV-fixed point of the RG flow. It's all about finding some conceptual means of picking among an infinite set of counter-terms by some principle.

    Anyway, be that as it may, here it just serves to highlight that without further ideas or principle, there is nothing "canonical" about "canonical quantization" in field theory This is not a new insight, this is the heart of field theory and its RG flow since the days of the founding fathers. The complete absence of any reflection of this in the purported "canonical quanization" of gravity is another way to deduce that somethng is wrong here, without even looking at the details. Of course once we do look at the details it is also easy to see where the problem stems from
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