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Diffusion in the stochastic Klein-Gordon equation

Jonathan Oppenheim, Emanuele Panella

TL;DR

This work investigates a classical-quantum gravity toy model by solving the stochastic Klein-Gordon equation and computing the non-equilibrium two-point function. Using a mod-squared-retarded pole prescription and MSR path-integrals, it shows that field covariances are nonzero only outside the lightcone, decay as $1/r$ in space, and grow linearly with diffusion time, while the energy exhibits a contact divergence akin to quantum-field theoretical phenomena. The authors derive exact mode-by-mode solutions via Ornstein-Uhlenbeck dynamics and discuss how these stochastic fluctuations could imprint on the gravitational-wave background in linearised CQ gravity, providing both energy-density bounds and phenomenological implications. The results suggest that linear stochastic gravity models predict anomalous diffusion with potentially observable consequences, but also indicate that non-linearities or renormalisation are likely required to render a viable hybrid gravity theory within experimental bounds. Overall, the paper connects stochastic diffusion in classical fields to constraints on CQ gravity, offering a framework to test the quantum nature of gravity through cosmological and laboratory observations.

Abstract

Theories of gravity in which the metric is fundamentally classical predict stochastic fluctuations in the gravitational field. In this article, we study the stochastic Klein-Gordon equation as a starting point to understand the phenomenology of linearised classical-quantum hybrid gravity. In particular, we describe how to compute the non-equilibrium two point function of the scalar field, showing explicitly the role of the initial state in regulating divergences. To do so, we use a "mod-squared-retarded" pole-prescription and find that the covariance in the field is non-zero only outside the lightcone, scales inversely with the spatial distance of the spacetime points and grows linearly in time. The energy has a contact divergence similar to that found in the quantum case. We conclude by discussing possible implications of anomalous diffusion for hybrid theories of gravity, especially looking at the energy density in the predicted gravitational waves background, which can be inferred from the scalar covariances.

Diffusion in the stochastic Klein-Gordon equation

TL;DR

This work investigates a classical-quantum gravity toy model by solving the stochastic Klein-Gordon equation and computing the non-equilibrium two-point function. Using a mod-squared-retarded pole prescription and MSR path-integrals, it shows that field covariances are nonzero only outside the lightcone, decay as in space, and grow linearly with diffusion time, while the energy exhibits a contact divergence akin to quantum-field theoretical phenomena. The authors derive exact mode-by-mode solutions via Ornstein-Uhlenbeck dynamics and discuss how these stochastic fluctuations could imprint on the gravitational-wave background in linearised CQ gravity, providing both energy-density bounds and phenomenological implications. The results suggest that linear stochastic gravity models predict anomalous diffusion with potentially observable consequences, but also indicate that non-linearities or renormalisation are likely required to render a viable hybrid gravity theory within experimental bounds. Overall, the paper connects stochastic diffusion in classical fields to constraints on CQ gravity, offering a framework to test the quantum nature of gravity through cosmological and laboratory observations.

Abstract

Theories of gravity in which the metric is fundamentally classical predict stochastic fluctuations in the gravitational field. In this article, we study the stochastic Klein-Gordon equation as a starting point to understand the phenomenology of linearised classical-quantum hybrid gravity. In particular, we describe how to compute the non-equilibrium two point function of the scalar field, showing explicitly the role of the initial state in regulating divergences. To do so, we use a "mod-squared-retarded" pole-prescription and find that the covariance in the field is non-zero only outside the lightcone, scales inversely with the spatial distance of the spacetime points and grows linearly in time. The energy has a contact divergence similar to that found in the quantum case. We conclude by discussing possible implications of anomalous diffusion for hybrid theories of gravity, especially looking at the energy density in the predicted gravitational waves background, which can be inferred from the scalar covariances.

Paper Structure

This paper contains 22 sections, 161 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of main results
  3. Background
  4. The Klein-Gordon thermal state, with friction
  5. Frictionless stochastic scalar
  6. Mod-squared retarded pole prescription
  7. Path integral methods
  8. Correlators in position space
  9. Massless field
  10. Energy production
  11. Comparison to the quantum case
  12. Exact mode solution
  13. Implications for CQ gravity
  14. Validity as a model of hybrid gravity
  15. Energy density in gravitational waves
  16. ...and 7 more sections

Figures (1)

  • Figure 1: Spacetime diagram in $1+1$ dimensions. The points $x$ and $y$, even if spacelike separated, are not correlated -- their lightcones intersect before the initial condition (dashed line). On the other hand, the field at $x$ and $y'$ will have non-zero covariance, as their intersection (black dot) lies in the future of the initial spacelike hypersurface.