# A Note on the Relativistic Transformation Properties of Quantum Stochastic Calculus

## TL;DR

This paper explores how quantum systems behave under different observers and shows that accelerated systems experience thermal noise due to quantum field effects.

## Contribution

The paper introduces a new derivation of quantum stochastic processes for accelerated systems and connects them to thermal equilibrium at the Unruh temperature.

## Key findings

- Quantum stochastic calculus breaks down for uniformly accelerated systems due to unitarily inequivalent representations.
- Accelerated detectors relax to thermal equilibrium at the Unruh temperature.
- The noise spectrum approximation leads to unitary inequivalence in quantum open systems.

## Abstract

We present a simple argument to derive the transformation of the quantum stochastic calculus formalism between inertial observers and derive the quantum open system dynamics for a system moving in a vacuum (or, more generally, a coherent) quantum field under the usual Markov approximation. We argue, however, that, for uniformly accelerated open systems, the formalism must break down as we move from a Fock representation over the algebra of field observables over all of Minkowski space to the restriction regarding the algebra of observables over a Rindler wedge. This leads to quantum noise having a unitarily inequivalent non-Fock representation: in particular, the latter is a thermal representation at the Unruh temperature. The unitary inequivalence is ultimately a consequence of the underlying flat noise spectrum approximation for the fundamental quantum stochastic processes. We derive the quantum stochastic limit for a uniformly accelerated (two-level) detector and establish an open system description of the relaxation to thermal equilibrium at the Unruh temperature.