Towards Loop Quantum Gravity without the Time Gauge

Abstract

The Hamiltonian formulation of the Holst action is reviewed and it is provided a solution of secondclass constraints corresponding to a generic local Lorentz frame. Within this scheme the form of rotation constraints can be reduced to a Gauss-like one by a proper generalization of AshtekarBarbero-Immirzi connections. This result emphasizes that the Loop Quantum Gravity quantization procedure can be applied when the time-gauge condition does not stand.

The kinematical Hilbert space of loop quantum gravity from BF theories

“…The key points of this reduction have been i) the projection of Lorentz irreducible representations to the ones associated with the Ashtekar-Barbero connections (25), which provides the proper solution of second-class constraints, ii) the restriction to SU(2) invariant intertwiners, which is a consequence of the imposition of Hamiltonian constraints on kinematical states. This analysis can be considered as the quantum counterpart of LQG without the time gauge [10] and it outlines the usefulness of such a formulation for gravity in a covariant setting.…”

“…In [10], the former procedure has been adopted and it has been demonstrated that it is possible to develop a formulation in terms of first-class constraints only where…”

“…However, having solved second-class constraints, the edges carry SU(2) representations and not Lorentz ones, while the boost parameters merely enter the definition of the intertwiners. In order to define Lorentzinvariant states, let us consider the representation of the Lorentz group acting on the variables {A i a , χ i } [10], i.e.…”

“…This framework is based on LQG without the time gauge [10] and it allows us to demonstrate that the issues discussed in [9] do not apply to gravity. In particular, we review the Hamiltonian formulations of BF and Holst actions and we outline how the proposal made by Alexandrov in [9] to infer the Ashtekar-Barbero connections provides the same reduction given classically by the explicit solution of second-class constraints.…”

Matter in loop quantum gravity without time gauge: A nonminimally coupled scalar field

“…The merit of the paper [7] was to point out that SU(2) connections are privileged functions in the classical formulation, since they contain the full dynamical information.…”