Quantizing gravitational fields with an entropy-corrected action principle

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.

The paper asks whether canonical gravitational quantization can be formulated through an information-corrected variational principle rather than postulated momentum operators. It extends an ensemble-on-configuration-space framework to ADM gravity, assuming intrinsic local random field fluctuations and a minimum observable action scale of $\hbar/2$. The proposed action is $A_t=\langle A_c\rangle+(\hbar/2)I_f$, with $I_f$ constructed from classical Kullback–Leibler divergences. Extremizing the action over an infinitesimal foliation step gives a formal Gaussian fluctuation distribution. Its covariance is then used to turn the accumulated relative entropy into a Fisher-information-type functional. Varying the resulting ensemble action, with diffeomorphism constraints incorporated through Lagrange multipliers, yields continuity and quantum Hamilton–Jacobi equations. Combining them using $\Psi=\sqrt{\rho}e^{iS/\hbar}$ recovers a Wheeler–DeWitt equation with divergence-form kinetic ordering, placing $G_{ijkl}/\sqrt h$ between functional derivatives. Replacing KL divergence by Tsallis divergence of order $\alpha$ rescales the information correction by $\alpha$. For gravity coupled to a massless scalar, a factorization of the wave functional and assumptions of weaker gravitational dependence in the matter factor allow gravitational trajectories to define an emergent clock. The resulting matter Schrödinger equation contains nonlinear classical and quantum gravitational corrections, described as order $G\hbar^2$. Stated limitations include the absence of a microscopic model for the fluctuations and reliance on a classical trajectory-based clock rather than a fully quantum clock. Renormalization, ultraviolet behavior, spacetime emergence, and black-hole information are not resolved. Connections to holographic relative entropy remain speculative.