Acceleration as a circular motion along an imaginary circle: Kubo-Martin-Schwinger condition for accelerating field theories in imaginary-time formalism

TL;DR

This paper explores the imaginary-time formalism for thermal field theories in uniformly accelerating frames, revealing a novel KMS condition linked to rotational identifications in Euclidean Rindler space, with implications for studying Hawking-Unruh radiation and early universe phenomena.

Contribution

It introduces a new interpretation of the KMS condition for accelerating fields as rotational identifications in Euclidean Rindler space, enabling lattice simulations of related phenomena.

Findings

KMS condition implies rotational identifications in Euclidean Rindler space

Imaginary-time formalism reduces to standard boundary conditions in accelerated frames

Potential for lattice studies of Hawking-Unruh radiation and early universe phase transitions

Abstract

We discuss the imaginary-time formalism for field theories in thermal equilibrium in uniformly accelerating frames. We show that under a Wick rotation of Minkowski spacetime, the Rindler event horizon shrinks to a point in a two-dimensional subspace tangential to the acceleration direction and the imaginary time. We demonstrate that the accelerated version of the Kubo-Martin-Schwinger (KMS) condition implies an identification of all spacetime points related by integer-multiple rotations in the tangential subspace about this Euclidean Rindler event-horizon point, with the rotational quanta defined by the thermal acceleration, $\alpha = a/T$. In the Wick-rotated Rindler hyperbolic coordinates, the KMS relations reduce to standard (anti-)periodic boundary conditions in terms of the imaginary proper time (rapidity) coordinate. Our findings pave the way to study, using first-principle lattice simulations, the Hawking-Unruh radiation in geometries with event horizons, phase transitions in accelerating Early Universe and early stages of quark-gluon plasma created in relativistic heavy-ion collisions.