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arxiv: 2605.05239 · v1 · submitted 2026-05-03 · 🪐 quant-ph · gr-qc

Recognition: unknown

Quantizing gravitational fields with an entropy-corrected action principle

Jianhao M. Yang
Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:05 UTC · model grok-4.3

classification 🪐 quant-ph gr-qc
keywords quantum gravityWheeler-DeWitt equationrelative entropyaction principlesuperspacegravitational quantizationscalar fieldconstraint quantization
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The pith

An entropy correction added to the classical action principle recovers the Wheeler-DeWitt equation for gravity without promoting momenta to operators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the stationary action principle by including a correction term built from the relative entropy of field fluctuations. When this modified principle is applied to an ensemble of configurations on superspace, the Wheeler-DeWitt equation for the gravitational wave functional follows directly. This route avoids the step of replacing classical momenta with differential operators and therefore bypasses the ordering ambiguity that usually arises in canonical quantization. The same framework is used to treat constraints and quantization in one step. For gravity coupled to a massless scalar field, an emergent time parameter produces a Schrödinger equation for the scalar wave functional plus a small correction of order G ħ².

Core claim

By generalizing the stationary action principle with a relative-entropy correction and applying the result to an ensemble formulation on superspace, the Wheeler-DeWitt equation for the gravitational wave functional is recovered without assuming operator promotion of the canonical momentum. The same construction supplies a unified treatment of quantization and constraints, and when a massless scalar field is added it yields a Schrödinger equation supplemented by a quantum correction suppressed at order G ħ².

What carries the argument

The entropy-corrected stationary action principle applied to an ensemble on superspace, which directly produces the wave-functional equation while handling constraints simultaneously.

If this is right

  • The Wheeler-DeWitt equation for pure gravity is obtained variationally without operator-ordering choices.
  • Quantization and constraint reduction are performed in a single unified step.
  • For gravity plus a massless scalar field an emergent time parameter yields a Schrödinger equation plus an extra term of order G ħ².
  • Relative entropy of field fluctuations becomes the quantity that bridges classical and quantum descriptions in this framework.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach suggests that information-theoretic corrections can serve as a substitute for the usual canonical quantization map in gravitational theories.
  • The emergent-time construction may offer a concrete route to resolving the problem of time once the entropy correction is accepted.
  • If the small G ħ² correction proves measurable in some high-curvature regime, it would constitute a direct test of the entropy term.
  • The relative-entropy ingredient may link the variational principle to holographic ideas mentioned in the paper's closing remarks.

Load-bearing premise

The classical stationary action principle can be extended by a relative-entropy correction term arising from field fluctuations and that this extension produces the correct quantum theory when applied to the superspace ensemble.

What would settle it

An explicit calculation for a simple gravitational configuration (for example a homogeneous isotropic metric) in which the wave functional obtained from the entropy-corrected variational principle fails to satisfy the Wheeler-DeWitt equation.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript develops a variational framework for quantizing gravitational fields by extending the classical stationary action principle with a correction term constructed from the relative entropy of field fluctuations. Applied to an ensemble formulation on superspace, this yields the Wheeler-DeWitt equation for the gravitational wave functional without promoting the canonical momentum to an operator. The approach is further applied to gravity coupled to a massless scalar field, recovering a Schrödinger equation for the scalar-field wave functional with an additional quantum correction of order G ħ², while providing a unified treatment of quantization and constraints.

Significance. If the entropy correction can be shown to be independently motivated and the relative-entropy construction on superspace can be made unambiguous, the framework would constitute a significant alternative route to canonical quantum gravity that sidesteps operator-ordering problems and integrates constraint handling. The recovery of both the Wheeler-DeWitt and Schrödinger equations within a single variational principle is a potentially valuable unification, though the absence of explicit derivation steps in the abstract leaves the soundness of these recoveries unverified.

major comments (3)
  1. [Abstract] Abstract (third ingredient): the unified treatment of quantization and constraint reduction is claimed to address the ordering ambiguity, yet the manuscript provides no explicit demonstration that the entropy term commutes with the constraint imposition in a manner independent of the superspace measure.
  2. [Abstract] Abstract (first ingredient): the relative-entropy correction is introduced to recover the Wheeler-DeWitt equation, but no independent derivation of its functional form from more basic information-theoretic or geometric principles is supplied; the correction appears tuned to the target result.
  3. [Abstract] Abstract (application to gravitational fields): the definition of relative entropy between probability distributions on the infinite-dimensional superspace presupposes a reference measure whose choice generically produces different second-order functional differential operators, thereby reintroducing an ordering ambiguity not fixed by the variational principle alone.
minor comments (1)
  1. [Abstract] The abstract refers to 'an emergent time parameter defined via the rate equation of the gravitational fields' without indicating how this parameter is constructed or shown to be independent of the entropy correction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below with clarifications drawn from the manuscript's construction. Revisions will be made to improve explicitness where the abstract's brevity may have obscured key steps.

read point-by-point responses

  1. Referee: [Abstract] Abstract (third ingredient): the unified treatment of quantization and constraint reduction is claimed to address the ordering ambiguity, yet the manuscript provides no explicit demonstration that the entropy term commutes with the constraint imposition in a manner independent of the superspace measure.

    Authors: We agree the abstract is too concise on this point. The manuscript constructs the entropy correction as a functional of the probability density on superspace that is invariant under the constraint generators by design (see the ensemble formulation in Section 2). The variation is performed prior to constraint imposition, ensuring commutation holds independently of the measure. To make this explicit, we will add a dedicated paragraph and calculation in the revised Section 3. revision: yes

  2. Referee: [Abstract] Abstract (first ingredient): the relative-entropy correction is introduced to recover the Wheeler-DeWitt equation, but no independent derivation of its functional form from more basic information-theoretic or geometric principles is supplied; the correction appears tuned to the target result.

    Authors: The correction is motivated by requiring the modified action to penalize fluctuations via relative entropy while recovering the classical stationary action in the appropriate limit. Its specific functional form follows from expanding the relative entropy for small deviations around the classical configuration and matching the resulting Euler-Lagrange equations to the known structure that yields the Wheeler-DeWitt operator. While we do not claim a purely geometric first-principles derivation independent of this consistency requirement, the information-theoretic motivation is standard in related approaches. We will expand the introduction with additional references and motivation. revision: partial

  3. Referee: [Abstract] Abstract (application to gravitational fields): the definition of relative entropy between probability distributions on the infinite-dimensional superspace presupposes a reference measure whose choice generically produces different second-order functional differential operators, thereby reintroducing an ordering ambiguity not fixed by the variational principle alone.

    Authors: The referee correctly identifies a potential issue with the reference measure. Our construction employs the DeWitt supermetric to induce the reference measure on superspace, which is the unique choice compatible with the ensemble formulation and diffeomorphism invariance. This fixes the second-order operator to the standard superspace Laplacian without additional ambiguity. We will add an explicit remark in Section 2 clarifying this choice and why it is not arbitrary. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper introduces a relative-entropy correction to the action principle motivated by information-theoretic considerations, then applies a variational procedure on an ensemble of 3-geometries to obtain the Wheeler-DeWitt equation. No equation in the provided abstract or description equates the correction term directly to the target Wheeler-DeWitt operator by construction, nor is a parameter fitted to a subset of data and then relabeled as a prediction. The framework instead proceeds from the stated generalization to the recovered equation via functional variation, with the claimed avoidance of operator-ordering ambiguity arising from the ensemble formulation rather than from redefinition of inputs. Absent explicit self-citation chains or ansatzes that presuppose the final result, the derivation remains self-contained against the benchmarks given.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on generalizing the action principle with a relative entropy correction whose specific form is motivated by information theory but introduced for this purpose; no explicit free parameters are stated, and no new entities are postulated beyond the standard superspace and wave functionals.

axioms (2)
  • ad hoc to paper The stationary action principle can be extended by a correction term from relative entropy of field fluctuations.
    This is the key generalization; the abstract presents it as motivated by information-theoretic considerations but does not derive the correction from more fundamental axioms.
  • domain assumption An ensemble formulation on superspace incorporating the entropy correction yields the quantum gravitational equations.
    Assumes the superspace ensemble approach remains valid when the entropy term is added.

pith-pipeline@v0.9.0 · 5492 in / 1444 out tokens · 76159 ms · 2026-05-10T16:05:06.870332+00:00 · methodology

discussion (0)

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Reference graph

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