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Thoughts on Quantum Holonomy Theory?

  1. Apr 29, 2015 #1
    Quantum Holonomy Theory
    Johannes Aastrup, Jesper M. Grimstrup
    (Submitted on 27 Apr 2015)
    We present quantum holonomy theory, which is a non-perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a C*-algebra that involves holonomy-diffeomorphisms on a 3-dimensional manifold and which encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Employing a Dirac type operator on the configuration space of Ashtekar connections we obtain a semi-classical state and a kinematical Hilbert space via its GNS construction. We use the Dirac type operator, which provides a metric structure over the space of Ashtekar connections, to define a scalar curvature operator, from which we obtain a candidate for a Hamilton operator. We show that the classical Hamilton constraint of general relativity emerges from this in a semi-classical limit and we then compute the operator constraint algebra. Also, we find states in the kinematical Hilbert space on which the expectation value of the Dirac type operator gives the Dirac Hamiltonian in a semi-classical limit and thus provides a connection to fermionic quantum field theory. Finally, an almost-commutative algebra emerges from the holonomy-diffeomorphism algebra in the same limit.
    Comments: 76 pages, 6 figures
    Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
    Cite as: arXiv:1504.07100 [gr-qc]
  2. jcsd
  3. May 2, 2015 #2


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    Pity no one is responding, I was intrigued by this but it is quite long and I was watching this thread hoping someone would do a summary for the lazy layman : )

    One thing I would be happy to know is how this relates to LQG. Holonomies play a big role there two if I understand it correctly, so it would be interesting to understand this theory in contrast and compare mode relative to that.
  4. May 2, 2015 #3
    I can understand the lack of replies. :headbang: It would be nice if someone could explain the essence of this article in terms that a senior physics major, say, could understand. Its a theory of quantum gravity, got that, 3D manifold, ok, Hilbert space, mmm, ok, Hamiltonian operator, ok, but Ashketar connections, got me there. What is the upshot of this article?
  5. May 2, 2015 #4


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    I think it is a proposal for a version of LQG that is a non-perturbative formulation of quantum gravity from which classical general relativity emerges in the appropriate limit. So it may be big news, if it is correct.
  6. May 3, 2015 #5


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    I don't get this either and haven't yet read the paper, but I think Ashketar variables actually provide a link between their approach and LQG. Randomly browsing the article, I saw the following statement which certainly looks interesting
    To me this seems to be saying "there's something wrong with spin foams". For some reason I wonder if this may not be related to the Immirzi parameter being real vs. imaginary, but all this is very hazy for me.
    Last edited: May 3, 2015
  7. May 3, 2015 #6
    QH attempts to combine NCG with a form of LQG
  8. May 3, 2015 #7


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    Thanks ! So that's why they talk about a C*-algebra, I didn't make the connection.
  9. May 18, 2015 #8


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    I wrote this in response to a private message, but see that it is a post as well. My broad brush initial impressions, given that I have only read the abstract:

    Ashekar connections are a more "quantum mechanics style" way of expressing GR in equations.

    The Hamiltonian constraint of GR is a way of saying that it conserves matter-energy in the aggregate (although classical GR leaves matter-energy conservation an undefined term when evaluated locally rather than globally).

    Dirac's equation is a general relativity application to the exact equation of a simple fermion (Bose equations apply to bosons).

    Basically, this theory proposed to suggest a mathematically non-pathological way to formulate a simplified version of quantum gravity with matter (but without other Standard Model forces or Standard Model bosons), using a previously unknown symmetry derived from abstract algebra related to matter-energy conservation in some remote way. But, unsurprisingly for a quantum gravity theory, it can't be solved perturbatively, just like infrared QCD.

    I may not be entirely accurate in summing up what its saying (which honestly I don't perfectly understand), but I reckon I'm probably doing better than any sophomore physics major without exposure to the field would.
  10. May 19, 2015 #9
    i think they are trying to combine NCG + LQG in a rigorous way creating a new theory that has a better classical limit than LQG with almost-commutative algebras.
  11. May 22, 2015 #10
    I can barely follow either one but this one and the one below (which I've been trying to understand) sure seem similar to me...

  12. May 22, 2015 #11


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    What similarity do you see between Haggard et al and that quantum holonomy paper ?
  13. May 22, 2015 #12
    It's a total cartoon, and I am more asking if they are similar, or what is the strong difference. I have only read the abstract and a little ways into the OP paper. But I'm immediately confused as to why it seems so similar to what I was just trying to read, and whether or not they really are decomposing nearly the exact same problem, in very similar terms, but slightly different angles, or whether they are completely unaware of one another, and genuinely very disconnected (which would be really confusing).

    In general so much of the Quantum Gravity (LQG, MERA and the like) sounds so similar and related... to me. I'm sure partly because I am flying over it a very very high altitude. I can imagine that to you all they manifest highly specific and clear contrasts...

    Section I of Haggard et al re Jacobson.

    (i) [itex]T=\frac { ha }{ 2\pi }[/itex] for an observer with acceleration [itex]a[/itex]

    (ii) there is a universal entropy density [itex]\alpha ={ 1 }/{ 4\hbar G }[/itex] associated to any causal horizon in a locally Minkowski patch of spacetime, giving an entropy

    [itex]S=\frac { A }{ 4\hbar G }[/itex] for a horizon of Area A;

    and a local entropy relation

    [itex]\delta S=\frac { \delta E }{ T } [/itex]

    holds where [itex]\delta E[/itex] is an energy exchange

    ...By interpreting [itex]\delta E[/itex] as the energy of matter flowing across the local Rindler Horiszon of an accelerated observer, and matching the variation of the area with the focusing effect of spacetime curvature, Jacobson was able to show that if the three equations above hold for any local frame, then Enstein's equations follow"

    "... The alternative interpretation, which we develop... is based on the fact that the gravitational field has quantum properties. The microscopic degrees of freedom are those of the quantum gravitational field, and the Einstein equations express only the classical limit of the dynamics. The entropy across the horizon measures the entanglement between adjacent spacetime regions. It's finiteness is evidence for the quantization of the gravitational field:"

    Blah blah,

    Then they drill in on the Energy-Geometry relation, made possible by the Minkowski Unruh relation and the assumption of universal and finite entropy per unit Area

    "...leads to the fundamental relation

    [itex]\delta E=\frac { a\hbar }{ 2\pi } \alpha \delta A[/itex] (30)

    ... derived by Frodden-Gosh-Perez in a different context as a consequence of the Einstein equations...

    [itex]\delta E=\frac { a }{ 8\pi G } \delta A[/itex] (31)

    ... Thus what Jacobson has shown in his derivation is that not only a consequence of Einsteins equations, but also is a sufficient condition for Einstein's equations to hold. If we assume the validity of 31 in any frame then Einstein's equations follow. In this sense Einsteins equations are encoded in the proportionality between the classical variation of energy and horizon area, as measured by a uniformly accelerating observer."

    Then they do a whole batch of acrobatics showing how Jacobson's result can be derived from an assumption LGQ as a spin network... section VI.

    "Let us see how Jacobson's result emerges from this framework... In loop gravity, quantum states of the geometry are described by SU(2) spin networks. Let us consider here for simplicity a single link of the spin network. The corresponding quantum state is a function [itex]\psi (U)[/itex] on SU(2) and U is classically interpreted as the open path holonomy of the gravitational Ashtekar connection along the link...."

    There is a lot of QM stuff that seems to crescendo with eq 46.

    "[itex]\left< { K }_{ f }^{ z } \right> =\frac { \left< { A }_{ f }^{ z } \right> }{ 8\pi G\hbar } [/itex]

    ... where [itex]{ K }_{ f }^{ z }[/itex] is the boost dual to the facet surface. This relation gives imediately equation (31), which is the basis of the second part of Jacobson's argument. From this relation we can obtain Einstein's equations. Notice that this is a relation between the area of a space like surface (say in the [itex]x-y[/itex] plane) and the boost hamiltonian (in the dual [itex]t-z[/itex] plane). As first observed by Smolin, this is a direct way of deriving Einstein's equations from the covariant loop quantum dynamics"

    They then define the "Hadamard States" which is what I am trying to get even a slight handle on, and connect them back to the "Area Entropy" relation part of Jacobson's original postulates (i-iii).

    Not to regurgitate the content of the paper un-related to the OP, but doesn't all of that sound pretty similar to the abstract....
    Maybe I'm really just checking to see, are these folks all just using slightly different recipes to make meatloaf. No offense intended. I love meatloaf, and never make it the same way twice myself. I'm just trying to understand the activity on the landscape, whether it's is highly overlapping or if I'm missing the huge differences.

    The difference between the approaches seems to be mostly about the degree of discreteness discussed... is that the big difference?

    Last edited: May 22, 2015
  14. May 22, 2015 #13


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    Hmm yes I kinda see what you mean. The gist of Haggard et al. though, as I read it, is to show that the entropy in Jacobson's relation can be read as entanglement entropy, which sheds a new light on that relation. I haven't read the holonomy paper beyond the abstract so the link isn't obvious to me, beyond the fact that as explained by others what they do is closely related to LQG, and Haggard et al. work out the entropy relation in detail in the context of LQG too - but there may well be a closer link, hard to say given that the holonomy paper is way over my head:)
  15. Nov 6, 2015 #14
  16. Nov 25, 2015 #15
    Because of this discussion, I had a chance to look into this paper.
    In principle, kodama above is right: it is a combination of NCG and LQG to get better control about the limit. But it is also much more.
    Originally, the work started with the construction of the spectral triple some years ago. With this Approach, LQG is able to use the methods of NCG but now one has to lift (in some sense) the holonomy Algebra (seen as operator on the diffeomorphism group) to a C* Algebra. This is done in the first sections. In particular, they found special states byusing a lattice formulation (called fermionic states bacause they are eigenstates of a Dirac-like Operator). Then they go over to construct 1-forms etc. in the quantized calculus of NCG using this Dirac-like Operator. The curvature is a 2-form in this quantized calculus and they construct a quantized Hamilton constraint using this curvature (some part of this reminds me on string theory: there is a left and right curvature...). Furthermore, the correct classical Hamilton constraint follows in the limit. But I don't see the construction of Solutions etc. (to form the space of physical states). Later, they showed the quantization of the volume (which is related to the Dirac Operator using a formal trick in their calculus).
    The whole approach looks interesting and it is for me the first also conceptional step Forward after some years. The idea is simple: construct operators on the diffeomorphism group which will generate the dynamics. The existence of fermionic states raised the hope to get also the matter (automatically coupled to the theory). So, these guys are on the way to get a TOE. At the conceptional Level, there is nothing to say anything agains this approach except: why did they introduce the lattice approximation? The discreteness must be deduced by the theory (and not an assumption!)
  17. Nov 26, 2015 #16
    if it does give GR as the classical limit of a quantum theory, then is this only the second time after string theory to provide this?
  18. May 15, 2016 #17
    Lacking funds, the guys behind this have turned to crowdfunding: https://www.indiegogo.com/projects/a-theory-of-everything [Broken]
    I've seen a lot worse crowdfunding projects... though if it really is this promising I find it sad that they struggle so much to secure funding that they need to turn to crowdfunding.
    Last edited by a moderator: May 7, 2017
  19. May 15, 2016 #18
    as i understand it the authors want to combine something like this

    A new algebraic structure in the standard model of particle physics
    Latham Boyle, Shane Farnsworth
    (Submitted on 4 Apr 2016)
    We introduce a new formulation of non-commutative geometry (NCG): we explain its mathematical advantages and its success in capturing the structure of the standard model of particle physics. The idea, in brief, is to represent A(the algebra of differential forms on some possibly-noncommutative space) on H (the Hilbert space of spinors on that space); and to reinterpret this representation as a simple super-algebra B=A⊕H with even part A and odd part H. B is the fundamental object in our approach: we show that (nearly) all of the basic axioms and assumptions of the traditional ("spectral triple") formulation of NCG are elegantly recovered from the simple requirement that Bshould be a differential graded ∗-algebra (or "∗-DGA"). But this requirement also yields other, new, geometrical constraints. When we apply our formalism to the NCG traditionally used to describe the standard model of particle physics, we find that these new constraints are physically meaningful and phenomenologically correct. This formalism is more restrictive than effective field theory, and so explains more about the observed structure of the standard model, and offers more guidance about physics beyond the standard model.
    30 pages, no figures

    with LQG

    resulting in

    Quantum Holonomy Theory and Hilbert Space Representations
    Johannes Aastrup, Jesper M. Grimstrup
    (Submitted on 21 Apr 2016 (v1), last revised 24 Apr 2016 (this version, v2))
    We present a new formulation of quantum holonomy theory, which is a candidate for a non-perturbative and background independent theory of quantum gravity coupled to matter and gauge degrees of freedom. The new formulation is based on a Hilbert space representation of the QHD(M) algebra, which is generated by holonomy-diffeomorphisms on a 3-dimensional manifold and by canonical translation operators on the underlying configuration space over which the holonomy-diffeomorphisms form a non-commutative C*-algebra. A proof that the state exist is left for later publications.
  20. May 15, 2016 #19

    why haven't other LQG/NCG researchers looked into combining LQG with NCG? i.e Smolin Rovelli Ashketar Connes, and even string theorists liked Ed Witten
    Last edited by a moderator: May 7, 2017
  21. May 17, 2016 #20
    It's been awhile since my head was in this thread but as I try to recall - this paper from the LQG bibilio thread shares interesting similarities and differences.

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