Canonical Quantization of Gravity without "Frozen Formalism"

TL;DR

This paper proposes a modified quantum gravity equation that incorporates time evolution, overcoming the 'frozen formalism' of traditional approaches by fixing a reference frame and introducing a kinematical action.

Contribution

It introduces a new quantum gravity framework with a time-dependent wave functional, addressing the problem of timelessness in canonical quantum gravity.

Findings

The new equation allows for a Schrödinger-like evolution of the wave functional.

A Hilbert space structure with a conserved inner product is established.

Classical general relativity is recovered in the appropriate limit.

Abstract

We write down a quantum gravity equation which generalizes the Wheeler-DeWitt one in view of including a time dependence in the wave functional. The obtained equation provides a consistent canonical quantization of the 3-geometries resulting from a ``gauge-fixing'' (3 + 1)-slicing of the space-time. Our leading idea relies on a criticism to the possibility that, in a quantum space-time, the notion of a (3 + 1)-slicing formalism (underlying the Wheeler-DeWitt approach) has yet a precise physical meaning. As solution to this problem we propose of adding to the gravity-matter action the so-called {\em kinematical action} (indeed in its reduced form, as implemented in the quantum regime), and then we impose the new quantum constraints. As consequence of this revised approach, the quantization procedure of the 3-geometries takes place in a fixed reference frame and the wave functional acquires a time evolution along a one-parameter family of spatial hypersurfaces filling the space-time. We show how the states of the new quantum dynamics can be arranged into an Hilbert space, whose associated inner product induces a conserved probability notion for the 3-geometries. Finally, since the constraints we quantize violate the classical symmetries (i. e. the vanishing nature of the super-Hamiltonian), then a key result is to find a (non-physical) restriction on the initial wave functional phase, ensuring that general relativity outcomes when taking the appropriate classical limit. However we propose a physical interpretation of the kinematical variables which, based on the analogy with the so-called {\em Gaussian reference fluid}, makes allowance even for such classical symmetry violation.