Towards Loop Quantum Gravity without the time gauge

TL;DR

This paper demonstrates that Loop Quantum Gravity can be formulated without relying on the time gauge, by generalizing the Ashtekar-Barbero-Immirzi connections and solving second-class constraints in a generic Lorentz frame.

Contribution

It provides a Hamiltonian formulation of the Holst action without the time gauge and generalizes the connection variables for Loop Quantum Gravity.

Findings

Loop Quantum Gravity can be applied without the time gauge.

A solution to second-class constraints in a generic Lorentz frame is presented.

Rotation constraints can be reduced to a Gauss-like form.

Abstract

The Hamiltonian formulation of the Holst action is reviewed and it is provided a solution of second-class 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 Ashtekar-Barbero-Immirzi connections. This result emphasizes that the Loop Quantum Gravity quantization procedure can be applied when the time-gauge condition does not stand.