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Is there a good candidate Hamiltonian for loop QG?

  1. Jul 29, 2015 #1
    is there a generally accepted candidate Hamiltonian for LQG?

    i've seen marcus post these papers recently

    New Hamiltonian constraint operator for loop quantum gravity
    Jinsong Yang, Yongge Ma
    (Submitted on 3 Jul 2015)
    A new symmetric Hamiltonian constraint operator is proposed for loop quantum gravity, which is well defined in the Hilbert space of diffeomorphism invariant states up to non-planar vertices. On one hand, it inherits the advantage of the original regularization method, so that its regulated version in the kinematical Hilbert space is diffeomorphism covariant and creates new vertices to the spin networks. On the other hand, it overcomes the problem in the original treatment, so that there is less ambiguity in its construction and its quantum algebra is anomaly-free in a suitable sense. The regularization procedure for the Hamiltonian constraint operator can also be applied to the symmetric model of loop quantum cosmology, which leads to a new quantum dynamics of the cosmological model.
    5 pages

    Discrete Hamiltonian for General Relativity
    Jonathan Ziprick, Jack Gegenberg
    (Submitted on 27 Jul 2015)
    Beginning from canonical general relativity written in terms of Ashtekar variables, we derive a discrete phase space with a physical Hamiltonian for gravity. The key idea is to define the gravitational fields within a complex of three-dimensional cells such that the dynamics is completely described by discrete boundary variables, and the full theory is recovered in the continuum limit. Canonical quantization is attainable within the loop quantum gravity framework, and we believe this will lead to a promising candidate for quantum gravity.
    6 pages
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  3. Jul 29, 2015 #2


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    Ideally there should be several good candidates for the researchers to be exploring. At the recent Loops 2015 conference the organizers gave equal time to two general versions of LQG: One was covariant LQG (Spinfoam QG) ---a path integral approach that generates transition amplitudes---it does not use a Hamiltonian.
    The other was canonical LQG which has a place for Hamiltonian and for which several have been proposed.
    Here are some:

    Hamiltonian spinfoam gravity
    Wolfgang M. Wieland
    (Submitted on 24 Jan 2013)
    This paper presents a Hamiltonian formulation of spinfoam-gravity, which leads to a straight-forward canonical quantisation. To begin with, we derive a continuum action adapted to the simplicial decomposition. The equations of motion admit a Hamiltonian formulation, allowing us to perform the constraint analysis. We do not find any secondary constraints, but only get restrictions on the Lagrange multipliers enforcing the reality conditions. This comes as a surprise. In the continuum theory, the reality conditions are preserved in time, only if the torsionless condition (a secondary constraint) holds true. Studying an additional conservation law for each spinfoam vertex, we discuss the issue of torsion and argue that spinfoam gravity may indeed miss an additional constraint. Next, we canonically quantise. Transition amplitudes match the EPRL (Engle--Pereira--Rovelli--Livine) model, the only difference being the additional torsional constraint affecting the vertex amplitude.
    28 pages, 2 figures. Class. Quant. Grav. 31 (2014) 025002

    There was an overview perspective on the problems facing the canonical approach in this paper:
    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.
    72 pages, 6 figures

    ==quote http://arxiv.org/pdf/1506.08571.pdf page 55==
    Let us now turn to the Hamiltonian constraints. We do believe that Hamiltonian constraint operators can in principle be constructed, since the main regularization mechanism pointed out in [15] should also hold in our case. However, we expect that the problems with the constraint algebra [90] will persist. This is ultimately related to the problem of preserving full diffeomorphism symmetry if lattices are introduced, even if this happens only on an auxiliary level [2, 91, 92]. An advantage of the BF representation is however the nicer geometric interpretation, which can facilitate the discussion of these issues. The work [93, 94] is also aimed at understanding the dynamics of spin foam gravity as a continuum theory, starting from BF theory.
    An alternative to directly imposing the Hamiltonian constraints is to use a discrete time dynamics [95], and then to consider the continuum limit. This would in fact fully follow the philosophy of approximating the dynamics by using defects in a Regge-like manner [1, 29]. A framework for describing a simplicial canonical dynamics has been described in [96, 97]. The question of how to reconstruct the continuum limit has been considered in e.g. [43, 98– 103]. In this continuum limit, one can also hope to restore diffeomorphism symmetry as exemplified in [104–106].

    Hamiltonian operator for loop quantum gravity coupled to a scalar field
    E. Alesci, M. Assanioussi, J. Lewandowski, I. Mäkinen
    (Submitted on 8 Apr 2015)
    We present the construction of a physical Hamiltonian operator in the deparametrized model of loop quantum gravity coupled to a free scalar field. This construction is based on the use of the recently introduced curvature operator, and on the idea of so-called "special loops". We discuss in detail the regularization procedure and the assignment of the loops, along with the properties of the resulting operator. We compute the action of the squared Hamiltonian operator on spin network states, and close with some comments and outlooks.
    31 pages, numerous graph diagrams

    Coherent states, quantum gravity and the Born-Oppenheimer approximation, III: Applications to loop quantum gravity
    Alexander Stottmeister, Thomas Thiemann
    (Submitted on 9 Apr 2015)
    In this article, the third of three, we analyse how the Weyl quantisation for compact Lie groups presented in the second article of this series fits with the projective-phase space structure of loop quantum gravity-type models. Thus, the proposed Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity.

    Hamiltonian constraint in Euclidean LQG revisited: First hints of off-shell Closure
    Alok Laddha
    (Submitted on 5 Jan 2014)
    We initiate the hunt for a definition of Hamiltonian constraint in Euclidean Loop Quantum Gravity (LQG) which faithfully represents quantum Dirac algebra. Borrowing key ideas from previous works on Hamiltonian constraint in LQG and several toy models, we present some evidence that there exists such a continuum Hamiltonian constraint operator which is well defined on a suitable generalization of the Lewandowski-Marolf Habitat and is anomaly free off-shell.
    68 pages, 6 figures

    Alok Laddha was one of the plenary speakers at Loops 2015, and his talk was about further developments along this (Hamiltonian) line. Just for comparison, here is one of the plenary talks on the Spinfoam (or covariant LQG) side:
    4-dimensional Spinfoam Amplitude with Cosmological Constant, 3-Manifold, and Supersymmetric Gauge Theory
    Friday 09:55 - 10:45, Muxin Han (FAU Erlangen, Germany)
    In this talk, I give an overview of the recent progress of covariant LQG in 4-dimensions with cosmological constant, with emphasis on the interesting relations with other areas of physics and mathematics. The 4d spinfoam amplitude is written as a finite dimensional integral, which has nice relation with Chern-Simons theory on a (dual) 3-manifold. Moreover the 4d spinfoam amplitude can be formulated as the holomorphic block in 3d, which arises from the holomorphic factorization of a certain 3-dimensional N=2 supersymmetric gauge theory. This formulation relates covariant LQG to M5-brane dynamics and 6d (2,0) theory in String/M-theory.
    Last edited: Jul 29, 2015
  4. Jul 29, 2015 #3


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    To get a sense of the Spinfoam vs Hamiltonian balance have a look at the abstracts for the Parallel session talks at Loops 2015, starting here
    On several consecutive posts you see the first two parallel sessions listed are "covariant LQG (spinfoams)" and "canonical LQG".

    You can see the titles of the talks and read the abstracts. It is a good sampling of work going on now. There are equal numbers on each side. See what you think. Frankly my impression is that there is somehow a higher level of activity on the one hand than the other---more interesting abstracts, more significance to the results. But you have to judge for yourself.

    Something to notice: both Wolfgang Wieland and Bianca Dittrich are working in the Spinfoam line of development, but they have built a version of covariant LQG they then look for the corresponding Hamiltonian. As a kind of afterthought . : ^)
    First you get the transition amplitudes right (the spinfoam dynamics), and coarse graining, and investigate the appropriate classical or continuum limits. Then if it looks right you see if you can define a corresponding canonical (Hamiltonian) version.

    So you are not "canonically quantizing General Relativity" to get a Hamiltonian. You are not following that recipe. You are building a version of covariant LQG, trying it out, and then seeing if you can construct a Hamiltonian for it. There is a subtle difference. Dittrich has a group of co-authors that all seem to be going that way. She is at Perimeter Institute.

    No doubt about it, a Hamiltonian is useful to have. We'll see how Wieland, Dittrich, and others proceed.
    Here's a related paper that could be of interest:
    Last edited: Jul 29, 2015
  5. Jul 29, 2015 #4
    which paper has the most promising Hamiltonian? should each proposal be considered a different LQG theory? which proposal comes closes to establishing spacetime continuum and GR in semi-clasical limit?
  6. Jul 29, 2015 #5


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    We're comparing apples and oranges, each case is different---they are interesting for different reasons, "near" in different ways, "far" in different ways. Good research is one of the most difficult things to predict, especially when it is a purely subjective judgement about what is "closest" or "promising"

    I like Ziprick's paper. It's very interesting and could change the character of the problem---starting with a discrete classical GR hamiltonian could be a game-changer. So I'm excited by that possibility and will be watching for further developments. But it does not include the cosmological constant curvature and it is not even a quantum theory yet. There is more to be done. It is still just a treatment of classical GR, with Ashtekar variables. Also I don't see how the reality condition is being handled.
  7. Jul 29, 2015 #6


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    Wieland's development is much further along---he deals with the issue of reality conditions--he has a quantum hamiltonian--and he shows it leads to the same transition amplitudes as in the standard EPRL spinfoam QG case! Wieland has, I think, achieved a really deep understanding of both the canonical and the covariant sides of LQG and how they can join at the root. I suggest you read the abstract carefully. The first few words are amazing:
    "This paper presents a Hamiltonian formulation of spinfoam-gravity, which leads to a straight-forward canonical quantisation."
    The title ("Hamiltonian spinfoam gravity") sounds like a contradiction in terms.

    Judgments like "closest" and "promising" are subjective. You have to read carefully and judge for yourself. And yet, so far I do not see that Wieland's approach contains the cosmological curvature constant Lambda. For that, one has to look elsewhere e.g. work by Haggard, Han, Riello and others. That too can be a game changer.
  8. Jul 29, 2015 #7


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    Let's not be so simple-minded as to try to pick a single winner---here's a practical real world way to put the same question. If I could only learn about one, how would I personally choose to spend the next hour?
    Suppose I had the choice, at a conference, of going to rooms A, B, or C to hear 45 minute talks by Ziprick, Wieland, or Dittrich (forget all the others I listed) and I could only go to one. Definitely Dittrich. That "New realization of quantum geometry" work is what I would put in the most time reading at home and trying to understand if I were young and starting research. That's just me---we cannot know the future course of research, almost by definition, that is what makes it research. But if you want my subjective hunch or leaning, as of right now (it could change), that's it.
    Last edited: Jul 30, 2015
  9. Jul 30, 2015 #8
    why that paper, does it provide a hamiltonian constraint with reality constraints and off shell closure?
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