A Note on the Relativistic Transformation Properties of Quantum Stochastic Calculus

TL;DR

The paper addresses how quantum stochastic calculus (QSC) transforms relativistically between inertial observers and investigates open-system dynamics for systems in vacuum fields, with a careful treatment of the uniformly accelerated case. It provides a direct derivation of inertial-frame transformation rules for quantum stochastic differential equations and extends the formalism to moving frames, including coherent inputs. For uniformly accelerated systems, it demonstrates unitary inequivalence between the Minkowski vacuum and the accelerated (Unruh) thermal representation, yielding a thermal bath with inverse temperature $\beta=2\pi/a$ and a corresponding master equation; the gauge process is not defined in this non-Fock regime. The results bridge QSC with algebraic quantum field theory, clarifying the limitations of the flat-noise QS limit under acceleration and clarifying the interpretation of the Unruh effect as thermalization rather than particle radiation.

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.

Figures (1)

• Figure 1: The Rindler wedge $\mathbb{W}$. Curves of constant $\tau$ (blue) and constant $\xi$ (green) are plotted. The special case $\xi=0$ gives the worldline of a particle accelerating at fixed proper-acceleration $a>0$ (bold).