Lorentz Invariant Master Equation for Quantum Systems

TL;DR

This paper develops a Lorentz-invariant, covariant quantum master equation using a relational clock framework, resolving issues of irreversibility, vacuum instability, and frame dependence in relativistic quantum dynamics.

Contribution

It introduces a local, non-Markovian, covariant master equation based on a relational scalar clock, ensuring positivity and stability in relativistic quantum systems.

Findings

Derives a covariant, local master equation for quantum fields.

Shows finite clock resolution prevents vacuum instability.

Demonstrates a consistent relativistic decay theory with a dynamical reference frame.

Abstract

Irreversibility implies a preferred flow of time, yet special relativity denies the existence of a preferred clock. This tension has long obstructed the formulation of a relativistic master equation: standard Markovian approximations either break Lorentz covariance, trigger catastrophic vacuum heating, or depend arbitrarily on the observer's foliation. In this work, we derive a Lorentz-invariant description of irreversibility for quantum fields. We take an approach that explicitly models the measurements required to observe irreversible dynamics. Instead of evolving the system along an abstract geometric time parameter, we anchor the dynamics to a physical, relational scalar clock field. Using a relational Tomonaga-Schwinger framework, we derive a local, non-Markovian master equation that is manifestly covariant and completely positive. We show that the finite resolution of the physical clock acts as a covariant regulator, preventing the vacuum instability that plagues white-noise models. This framework demonstrates that a consistent relativistic theory of decay exists, provided the reference frame is treated as a dynamical quantum resource rather than a gauge choice. In a gravitating context, the resulting dynamics is described by a hybrid classical-quantum (CQ) evolution that remains completely positive and trace preserving (CPTP).