Anomaly free quantum dynamics for Euclidean LQG

Abstract

Classical gravitational evolution admits an elegant and compact re-expression in terms of gauge covariant generalizations of Lie derivatives with respect to a spatial phase space dependent su(2) valued vector field called the Electric Shift [1]. A quantum dynamics for Euclidean Loop Quantum Gravity which ascribes a central role to the Electric Shift operator is derived in [2]. Here we show that this quantum dynamics is non-trivially anomaly free. Specifically, we show that on a suitable space of off shell states (a) the (non-vanishing) commutator between a pair of Hamiltonian constraint operators mirrors the Poisson bracket between their classical correspondents, (b) the group of finite spatial diffeomorphisms is faithfully represented and (c) the action of the Hamiltonian constraint operator is diffeomorphism covariant with respect to the action of spatial diffeomorphisms.

Modeling Quantum Particles Falling into a Black Hole: The Deep Interior Limit

“…In the full theory a new perspective on the regularization issue has been introduced motivated by the novel mathematical notion of generalized gauge covariant Lie derivatives [41] and their geometric interpretation allowing for the introduction of a natural regularization (and subsequent) anomaly free quantization of the Hamiltonian constraint [42]. Even when the procedure does not eliminate all ambiguities of quantization (choices are available in the part of the quantum constraint responsible for propagation [43]), the new technique reduces drastically some of them in the part of the Hamiltonian that is more stringently constrained by the quantum algebra of surface deformations.…”

Simplicial graviton from selfdual Ashtekar variables

“…In the second case, the lattice stays, but the gauge symmetries are deformed in such a way that there is no longer an anomaly. In this paper, we consider this second possibility also considered in [15,[20][21][22]. By introducing a new discretization scheme, we obtain an anomaly-free lattice regularization for selfdual gravity [23,24].…”

On Propagation in Loop Quantum Gravity

“…These efforts can be subdivided into two classes: In the first class, the issue of the structure functions is avoided altogether; in the second, the correctness of the structure constants is used as a guiding principle to adjust the fine details of the construction proposed in [5,6]. In historical order, the first class comprises the master constraint approach [12] and the reduced phase space quantisation approach [13], while the second class comprises the electric shift approach [14,15] (see also [16] for a preliminary attempt) and the Hamiltonian renormalisation approaching [17] to both the constraint of [5,6] and the physical Hamiltonian of [13] (see also the references in all four manuscripts). Common to the [5,6,14,15] versions of the Hamiltonian constraint is that the Hamiltonian constraint acts non-trivially only at the vertices of the graph, and its action on a given vertex deforms the graph in an open neighbourhood of that vertex while not modifying the graph in neigbourhoods of the other vertices (in [5,6,14,15], this deformation is encoded by the loop approximant to the curvature that one uses).…”

“…In historical order, the first class comprises the master constraint approach [12] and the reduced phase space quantisation approach [13], while the second class comprises the electric shift approach [14,15] (see also [16] for a preliminary attempt) and the Hamiltonian renormalisation approaching [17] to both the constraint of [5,6] and the physical Hamiltonian of [13] (see also the references in all four manuscripts). Common to the [5,6,14,15] versions of the Hamiltonian constraint is that the Hamiltonian constraint acts non-trivially only at the vertices of the graph, and its action on a given vertex deforms the graph in an open neighbourhood of that vertex while not modifying the graph in neigbourhoods of the other vertices (in [5,6,14,15], this deformation is encoded by the loop approximant to the curvature that one uses). By contrast, in [12,13,17], such a deformation is not considered, as the loops involved are part of the same lattice on which the graph in question is defined where the lattice plays the role of a resolution scale in the sense of the Wilsonian point of view of renormalisation.…”

“…In that sense, the criticism conveyed in [18] should apply to the works [5,6,14,15]: the arguments in [18] are based on [19], where an algorithm is sketched for how to construct solutions in the algebraic dual of both the spatial diffeomorphism constraint of [8] and the Hamiltonian constraint of [5,6]. The construction [19] can be sketched as follows: A solution of the spatial diffeomorphism constraint is a linear combination of the diffeomorphism group averagings η(S) of spin network functions (SNWF) S (moulo graph symmetriessee [8] for details) which forms an orthonormal basis of the kinematical Hilbert space.…”