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:
http://arxiv.org/abs/1301.5859
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:
http://arxiv.org/abs/1506.08571
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].
==endquote==
http://arxiv.org/abs/1504.02068
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
http://arxiv.org/abs/1504.02171
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.
http://arxiv.org/abs/1401.0931
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:
==quote==
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.
==endquote==
