A Note on the Relativistic Transformation Properties of Quantum Stochastic Calculus

TL;DR

This paper explores how quantum stochastic calculus transforms under relativity, showing that accelerated systems require a different, thermal representation due to the Unruh effect, impacting open system dynamics.

Contribution

It provides a simple derivation of the relativistic transformation of quantum stochastic calculus and analyzes the breakdown of formalism for accelerated systems, revealing thermal inequivalence.

Findings

Quantum noise for accelerated systems is represented by a thermal (Unruh) state.

The formalism breaks down when restricting to Rindler wedge observables.

The quantum stochastic limit describes relaxation to Unruh temperature.

Abstract

We give a simple argument to derive the transformation of quantum stochastic calculus formalism between inertial observers, and derive the quantum open system dynamics for a system moving in a vacuum (more generally coherent) quantum field under the usual Markov approximation. We argue that for uniformly accelerated open systems, however, the formalism must breakdown as we move from a Fock representation over the algebra of field observables over all Minkowski space to the restriction to 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 ultimately being 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 systems description of the relaxation to thermal equilibrium at the Unruh temperature.er is a thermal representation at the Unruh temperature. The unitary inequivalence ultimately being a consequence of the underlying flat noise spectrum approximation for the fundamental quantum stochastic processes.