Quantizing Gravitational Fields with an Entropy-Corrected Action Principle

A variational framework for the quantization of gravitational fields is developed through an extension of the classical stationary action principle. With this framework, the Wheeler–DeWitt equation for the gravitational wave functional is derived rather than postulated by promoting the Hamiltonian to an operator, as in canonical quantization. The formulation is motivated by information-theoretic considerations and incorporates a correction to the classical action constructed from the relative entropy characterizing the fluctuations of the gravitational fields. This entropy contribution is introduced within an ensemble formulation on a superspace. The ensemble formulation is subsequently extended to provide a unified treatment of quantization and constraints, addressing the long-standing ambiguity associated with the ordering between quantization and constraint reduction. The results also avoid the operator-ordering ambiguities inherent in canonical quantization. The framework is then applied to gravitational fields coupled to a massless scalar field. By introducing an emergent time parameter defined by the rate equation of gravitational fields, a Schrödinger equation for the scalar-field wave functional is obtained together with an additional quantum correction term of order $$G\hbar ^2$$ . Finally, possible connections between the relative entropy underlying the present framework and the holographic aspects of quantum gravity are discussed.