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Covariant Path Integrals for Quantum Fields Backreacting on Classical Space-Time

Jonathan Oppenheim1 and Zachary Weller-Davies2

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Phys. Rev. X 16, 031007 – Published 15 July, 2026

DOI: https://doi.org/10.1103/2rcd-dzcf

Abstract

We introduce configuration space path integrals for quantum fields interacting with classical fields. We show that this can be done consistently by proving that the dynamics are completely positive directly, without resorting to master equation methods. These path integrals allow one to readily impose space-time symmetries, including Lorentz invariance or diffeomorphism invariance. They generalize and combine the Feynman-Vernon path integral of open quantum systems and the stochastic path integral of classical stochastic dynamics while respecting symmetry principles. We introduce a path integral formulation of general relativity where the space-time metric is treated classically. The theory is a candidate for a fundamental theory that reconciles general relativity with quantum mechanics. The theory is manifestly covariant, and may be inequivalent to the theory derived using master equation methods. We prove that entanglement cannot be created via the classical field, reinforcing proposals to test the quantum nature of gravity via entanglement generation.

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Popular Summary

Einstein’s general relativity describes gravity as the curvature of space-time which is treated classically, yet the quantum nature of the matter causing this curvature suggests that space-time itself might also be quantum. Developing a consistent quantum theory of gravity has proven conceptually difficult, leading to the alternative possibility that space-time could remain classical even when interacting with quantum matter. We introduce a novel covariant path integral framework that consistently couples classical and quantum fields by proving the dynamics are completely positive without relying on master equation methods. Our approach allows for the direct imposition of space-time symmetries, such as Lorentz or diffeomorphism invariance, which are essential for a fundamental theory of general relativity. We show that this classical gravitational field cannot generate entanglement between quantum systems, providing an experimental witness to distinguish this theory from a fully quantum one. This work offers a rigorous template for reconciling general relativity with quantum mechanics, providing a potential pathway to a fundamental theory of space-time.

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Phys. Rev. X 16, 031007– Published 15 July, 2026
  • Received 23 October 2023
  • Revised 4 August 2025
  • Accepted 31 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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