A stone falls to the ground along the most economical route — the one that expends the least time and energy. This principle of least action governs all of classical physics. But in the microworld, everything jitters: fields fluctuate, and you can't say precisely where anything is. The authors of the paper added a correction for this jitter to the principle, as if, when planning a route, you considered not only the shortest path but also the fact that the path itself is constantly vibrating slightly.
The correction is expressed through relative entropy — a measure of how much one probability distribution differs from another. In everyday life, this is like comparing two weather forecasts: the more they diverge, the greater the uncertainty. In the end, they managed to obtain an equation for gravitational waves in quantum form without manually setting the order of operator multiplication — a long-standing headache for physicists.
Next, the authors considered gravity together with a massless scalar field — the simplest "partner." Using the rate of change of geometry itself as a clock, they derived an equation of the Schrödinger type for this field. It contains a tiny correction, suppressed by the factor $G\hbar^2$—a sign that spacetime itself "breathes" slightly at the quantum level. This approach does not impose quantum rules on gravity from the outside but shows how they might emerge on their own from a more refined understanding of action.
as written by the authors arXiv:2605.05239
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 relative entropy used in the work is the same measure used in statistics and machine learning to compare how much real data differ from model predictions.