Quantizing gravitational fields with an entropy-corrected action principle

TL;DR

This paper introduces a novel variational approach to quantize gravitational fields by incorporating entropy corrections, recovering the Wheeler-DeWitt equation without operator ordering ambiguities, and applying it to gravity coupled with a scalar field.

Contribution

It develops a new entropy-based variational framework for quantum gravity that addresses operator ordering issues and unifies quantization with constraints.

Findings

Recovered Wheeler-DeWitt equation without operator ordering ambiguity

Derived a Schrödinger equation with a suppressed quantum correction term

Proposed a connection between relative entropy and holographic dualities

Abstract

A variational framework for the quantization of gravitational fields is developed based on an extension of the stationary action principle. Within this framework, the Wheeler-DeWitt equation for the gravitational wave functional is recovered without assuming operator promotion of the canonical momentum, thus avoiding the ambiguity of operator ordering in canonical quantization. The derivation is based on three main ingredients. First, motivated by information-theoretic considerations, the classical stationary action principle is generalized by incorporating a correction term constructed from the relative entropy associated with field fluctuations. Second, an ensemble formulation on superspace is enhanced to incorporate this entropy correction. Third, the formalism is further refined to provide a unified treatment of quantization and constraints, thereby addressing the long-standing ambiguity concerning the ordering of quantization and constraint reduction. The framework is then applied to gravitational fields coupled to a massless scalar field. Using an emergent time parameter defined via the rate equation of the gravitational fields, a Schrodinger equation for the scalar-field wave functional is recovered, supplemented by an additional quantum correction term suppressed at order $G\hbar^2$. Finally, we comment on possible connections between the notion of relative entropy employed here and holographic dualities in quantum gravity.