Relation Between Einstein And Quantum Field Equations

TL;DR

This paper demonstrates a connection between quantum scalar fields and Einstein's equations, showing that specific mode choices lead to consistent backgrounds with no particle production, extending previous results to more general geometries.

Contribution

It introduces a mode selection that aligns quantum field evolution with Einstein equations in various limits, generalizing prior work to conformally ultrastatic backgrounds.

Findings

Quantum fields in certain modes reproduce Einstein equations in zero and large mass limits.

Zero particle production occurs in these limits, indicating a preferred vacuum state.

Results extend to graviton equations, linking massless scalar particle production to gravitational dynamics.

Abstract

We show that there exists a choice of scalar field modes, such that the evolution of the quantum field in the zero-mass and large-mass limits is consistent with the Einstein equations for the background geometry. This choice of modes is also consistent with zero production of these particles and thus corresponds to a preferred vacuum state preserved by the evolution. In the zero-mass limit, we find that the quantum field equation implies the Einstein equation for the scale factor of a radiation-dominated universe; in the large-mass case, it implies the corresponding Einstein equation for a matter-dominated universe. Conversely, if the classical radiation-dominated or matter-dominated Einstein equations hold, there is no production of scalar particles in the zero and large mass limits, respectively. The suppression of particle production in the large mass limit is over and above the expected suppression at large mass. Our results hold for a certain class of conformally ultrastatic background geometries and therefore generalize previous results by one of us for spatially flat Robertson-Walker background geometries. In these geometries, we find that the temporal part of the graviton equations reduces to the temporal equation for a massless minimally coupled scalar field, and therefore the results for massless particle production hold also for gravitons. Within the class of modes we study, we also find that the requirement of zero production of massless scalar particles is not consistent with a non-zero cosmological constant. Possible implications are discussed.