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I Other procedures to quantize Ashketer variables

  1. Sep 10, 2017 #1
    Urs and Mitchel aren't fans of polymer quantization used in LQG to quantize Ashketer variables, they regard it as unphysical and unconnected to the rest of physics.
    LQG based on "polymer quantization" and "generalized connection" is a dead end.

    fair enough.

    is there a procedure to nonperturbative quantization of Ashketer variables that is physical or is more promising than polymer quantization.

    how could an LQG theorist quantize Ashketar variables that could be a serious candidate for QG.

    Since I read Urs and Mitchell's concerns in two threads, I did research on other ways to quantize Ashketer variables.

    I have in mind the paper below avoid the concerns Urs and Mitchell have about polymer quantization of ahsketer variables, by offering a more realistic quantization?

    A new realization of quantum geometry
    Benjamin Bahr, Bianca Dittrich, Marc Geiller
    (Submitted on 29 Jun 2015)
    We construct in this article a new realization of quantum geometry, which is obtained by quantizing the recently-introduced flux formulation of loop quantum gravity. In this framework, the vacuum is peaked on flat connections, and states are built upon it by creating local curvature excitations. The inner product induces a discrete topology on the gauge group, which turns out to be an essential ingredient for the construction of a continuum limit Hilbert space. This leads to a representation of the full holonomy-flux algebra of loop quantum gravity which is unitarily-inequivalent to the one based on the Ashtekar-Isham-Lewandowski vacuum. It therefore provides a new notion of quantum geometry. We discuss how the spectra of geometric operators, including holonomy and area operators, are affected by this new quantization. In particular, we find that the area operator is bounded, and that there are two different ways in which the Barbero-Immirzi parameter can be taken into account. The methods introduced in this work open up new possibilities for investigating further realizations of quantum geometry based on different vacua.
    Comments: 72 pages, 6 figures
    Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
    Cite as: arXiv:1506.08571 [gr-qc]
    (or arXiv:1506.08571v1 [gr-qc] for this version)
    Submission history
    From: Marc Geiller [view email]
    [v1] Mon, 29 Jun 2015 10:25:11 GMT (1370kb,D)
    some of the results of polymer quantization of Ashketer variables carries over.
  2. jcsd
  3. Sep 13, 2017 #2
    another research group offering another way to quantize ashektar variables and connect it to Connes NCG separate from polymer quantization

    Quantum Gravity and the Emergence of Matter
    Johannes Aastrup, Jesper M. Grimstrup
    (Submitted on 9 Sep 2017 (v1), last revised 12 Sep 2017 (this version, v2))
    In this paper we establish the existence of the non-perturbative theory of quantum gravity known as quantum holonomy theory by showing that a Hilbert space representation of the QHD(M) algebra, which is an algebra generated by holonomy-diffeomorphisms and by translation operators on an underlying configuration space of Ashtekar connections, exist. We construct operators, which correspond to the Hamiltonian of general relativity and the Dirac Hamiltonian, and show that they give rise to their classical counterparts in a classical limit. We also find that the structure of an almost-commutative spectral triple emerge in the same limit. The Hilbert space representation, that we find, is inherently non-local, which appears to rule out spacial singularities such as the big bang and black hole singularities. Finally, the framework also permits an interpretation in terms of non-perturbative Yang-Mills theory, which produces what appears to be a non-zero mass gap, as well as other non-perturbative quantum field theories. This paper is the first of two, where the second paper contains mathematical details and proofs.
    Comments: 37 pages, 2 figures
    Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
    Cite as: arXiv:1709.02941 [gr-qc]
    (or arXiv:1709.02941v2 [gr-qc] for this version)
    Submission history
    From: Jesper Møller Grimstrup [view email]
  4. Sep 13, 2017 #3
    Regarding quantum gravity, I'd say there is a kind of mainstream of research that works. It includes perturbative quantum gravity (gravitons in flat space) and the semiclassical deduction (using quantum field theory in curved space) of Hawking radiation and black hole entropy. I would also point to at least two things from string theory as highly significant: the calculation of black hole entropy from microstates (for certain classes of black hole), and the very recent observation that the weak gravity conjecture, if true, can prevent the formation of a naked singularity in AdS4. Then there is the endless series of dramatic simplifications in scattering theory that have been discovered, often by using twistor variables. All of that is very impressive theoretically. And then empirically I would add that asymptotic safety and perhaps the new adimensional gravity deserve attention, despite some apparent inconsistencies with the mainstream I just defined, because they may explain Higgs criticality.

    So how do these two papers look, judged against that context? If I was going to make a serious effort to evaluate them, I might start by comparing them with the mainstream work that they most resemble. That's partly to make up for my own lack of knowledge. There are endless technical facts, small and large, relevant to quantum gravity, that I don't know; but the people who do that high-quality mainstream research do know them. So if I can put these papers alongside their mainstream counterparts that are exploring similar territory, the mainstream papers will probably tell me things that will help me judge this research.

    For the paper by Dittrich and collaborators, the relevant mainstream work would seem to be, anything that involves a web or network of linear "defects". Away from each defect, space is flat, and around each defect, space is pinched so that the solid angle is less than for flat space. They cite Regge's triangulations as a precedent, but I would also want to look at string webs or string networks in string theory, as an example of a web-of-defects quantum geometry that I would have confidence has been done right. When I say it is done right, I mean that it comes from a line of research (string theory) that has passed various tests of credibility.

    As for Aastrup and Grimstrup, they say two things that catch my attention. They are developing a quantum theory that resembles loop quantum gravity, but they have added an extra nonlocality to it; and, they say that a noncommutative algebra from which Connes et al obtain the standard model, fits into their framework. The basis states in loop quantum gravity are sharply peaked around specific geometries (this is true for Dittrich et al's new representation too), maybe this extra fuzziness compensates for that. And as for obtaining the noncommutative standard model, I would use Urs's work as my sanity check here, since he has written about how to obtain that sort of noncommutative geometry from string theory.

    This same strategy of consulting the mainstream for guidance even applies to your general question about the use of Ashtekar variables. From a mainstream perspective, their outstanding property is the algebraic simplicity they bring to certain equations. Conceptually, they share with twistor variables the property of chirality; they split the gravitational field into a self-dual and an anti-self-dual part. (It's also interesting to me that they are a spin-connection, that they say how to transport spinors... perhaps a connection to Einstein-Cartan gravity?) They grew out of Ashtekar's own research into "asymptotic quantization", which involves the BMS symmetry group recently employed by Strominger and others to develop new perspectives on so-called soft scattering (which in a way is about how one should think about the particles that enter and exit scattering events, how one should "dress" them with virtual particles). So from a mainstream perspective, one might emphasize the connections with twistor theory and BMS symmetry.
  5. Sep 14, 2017 #4
  6. Sep 14, 2017 #5
    I have now had a closer look at their paper. Basically, they mangle QFT, don't calculate any substantial consequences of their modification but say it will be interesting to see if the result resembles traditional QFT in any way, and then add some wishful thinking on big topics. The way they modify QFT is described in part 3.1, starting with equation 8, where they define a metric-dependent norm on the space of connections on a manifold. Using this norm, they pick a set of functions to act as a classical basis for connection space (analogous to how plane waves provide a base for classical fields), and then define a quantum Hilbert space in terms of functionals over the space of connections. Then they tensor that Hilbert space with a spinor Hilbert space, but I'll skip those details since the damage is already done.

    The point is the metric dependence. The metric is supposed to be a quantum observable in quantum gravity, right? And in these connection-based theories of gravity, it's not a fundamental quantity, it's derived from the connection. Yet right at the beginning, their quantum states are defined on a manifold with a specific classical metric. See the paragraph immediately after equation 8, where they say that ∆ and (,) "depend on a metric g". They don't hide this, they talk about it throughout part 5, and again in part 9. But they never address the question of how, if such a metric should only emerge in a classical limit, it can also be fundamental to the construction of their quantum states. Oh, and their construction also breaks gauge symmetry! (page 34).

    As I said, they modify QFT, they don't calculate any physical quantities, and then they call this (end of part 6) "a general framework for non-perturbative quantum field theory". It's a general framework because their modification procedure can be applied to any QFT - although they can't actually tell you what the effects are. To sum up: they calculate nothing, they destroy the foundation of the standard model (gauge symmetry), and their theory of quantum gravity presupposes classical gravity (the metric g of the manifold M) in order to be defined.
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