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 presents an entropy-corrected action principle that integrates relative entropy to induce quantum corrections while resolving operator-ordering ambiguities.
• The paper employs an ensemble formulation via ADM decomposition to enforce gravitational constraints concurrently with quantization, deriving the Wheeler–DeWitt equation.
• The paper demonstrates that entropy-based corrections introduce Bohm-type quantum potentials and emergent time dynamics, suggesting paths for testable quantum gravity effects.

Entropy-Corrected Action Principle for Quantum Gravity

Variational Quantization Framework

The paper introduces a variational framework for quantizing gravitational fields, explicitly avoiding operator promotion and operator-ordering ambiguities inherent in canonical quantization by extending the stationary action principle with an entropy correction based on information-theoretic considerations. The key innovation is the adaptation of the classical action to include a term constructed from the relative entropy of field fluctuations, leading to a quantization procedure grounded in extremizing this entropy-corrected action.

Two central assumptions underlie the framework: (1) intrinsic, random, local fluctuations are present in field configurations, and (2) the physically observable action possesses a lower bound set by $\hbar/2$. The resulting total action functional encodes both the classical dynamics and a quantum correction term proportional to the relative entropy functional, thereby ensuring a consistent prescription for quantizing fields within an ensemble formulation on superspace.

Treatment of Constraints and Ensemble Formulation

The ADM decomposition of general relativity is adopted to recast the Einstein-Hilbert action in terms of spatial metrics, lapse and shift functions, and their associated Hamiltonian and momentum constraints. Unlike standard approaches, the quantization and enforcement of these constraints occur simultaneously within the variational principle, eliminating the ambiguity found in Dirac and reduced phase-space quantization regarding the sequence of operator promotion and constraint imposition.

The framework employs an ensemble description of field configurations characterized by a pair of functionals $(\rho, S)$, representing the ensemble's probability density and generating functional. The information-theoretic correction is rigorously formulated as a relative entropy functional—specifically, the Kullback-Leibler divergence—between probability distributions with and without field fluctuations. This construction preserves diffeomorphism invariance at the variational level and yields a coupled set of evolution equations for $(\rho, S)$, recoverable as a generalized Schrödinger equation for the wave functional through the transformation $\Psi = \sqrt{\rho} e^{iS/\hbar}$.

Derivation of the Wheeler-DeWitt Equation

The variational procedure produces the Wheeler-DeWitt equation for the gravitational wave functional without relying on operator promotion of canonical momenta. The ordering ambiguity is resolved by the variational structure, which mandates the placement of functional derivatives in a specific manner: the DeWitt supermetric sandwiched between two functional derivatives is treated rigorously, unlike in canonical quantization where such ordering is ambiguous. The entropy correction naturally introduces Bohm-type quantum potential terms, suppressed at order $G\hbar^2$, which quantitatively emerge from the information metrics.

Generalization of the information metrics via Tsallis or Rényi divergence demonstrates mathematical flexibility in the construction, providing a parametric family of quantum correction terms in the Wheeler-DeWitt equation. Physical implications of these generalizations remain to be clarified in future studies.

Emergent Time and Quantum Corrections in Matter Coupling

For gravitational fields coupled to scalar fields, the formalism introduces an emergent time parameter defined by the rate equation of the gravitational field ensemble. This parameter enables the derivation of a Schrödinger-type equation for the scalar field wave functional, which is augmented by quantum correction terms arising from both classical and quantum gravitational effects. These correction terms are explicitly of order $G\hbar^2$ and are absent in semiclassical treatments where gravity remains classical. Strongly, the framework yields a nonlinear correction term in the effective Schrödinger equation, distinguishing it from standard quantum field theory or previous hybrid approaches.

The formalism identifies a mechanism whereby quantum gravitational corrections may become detectable, suggesting a path forward for laboratory tests of quantum gravity and entanglement-induced signatures.

Comparative and Theoretical Implications

Direct comparison with Hall and Reginatto's ensemble framework reveals that the present approach diverges fundamentally in its quantization mechanism: quantum effects are induced through entropy-based variational extremization rather than exact uncertainty principles. The simultaneous enforcement of constraints and quantization further distinguishes the formalism from both canonical and path-integral methods, which require operator promotion and are subject to ordering ambiguities.

The adoption of relative entropy as the central information metric offers both mathematical elegance and conceptual innovation. It bridges quantum field theory and quantum gravity with information-theoretic perspectives, aligning with developments in holography where bulk and boundary relative entropy are linked in AdS/CFT dualities. The paper speculates that a bulk relative entropy functional analogous to that employed here may possess deep relevance to holographic interpretations.

Limitations and Outlook

While the framework yields a formal resolution of operator ordering and quantization of constrained Hamiltonian systems, explicit physical models for field fluctuations are not provided. The emergent time construction successfully addresses the "problem of time" in a non-perturbative manner, but full quantum treatments, such as the Page-Wootters formalism, remain as alternative avenues for future exploration.

Key open questions include implications for renormalizability and ultraviolet behavior, the physical significance of alternative entropy corrections, and possible testable predictions relating to quantum gravitational corrections in matter systems.

Conclusion

Entropy-corrected variational quantization provides a mathematically rigorous alternative to canonical and path-integral quantization of gravitational fields, resolving operator-ordering ambiguities and constraint ordering issues, and enabling the derivation of the Wheeler-DeWitt equation and emergent time dynamics without reliance on operator postulates. The information-theoretic corrections supply quantum behavior from first principles, with implications reaching toward holographic correspondence and laboratory quantum gravity tests. Further developments should clarify the physical role of these corrections, their potential for observable signatures, and their relationship to broader information-theoretic formulations of quantum field theory and gravity (2605.05239).