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Lorentz Invariant Master Equation for Quantum Systems

Pranav Vaidhyanathan, Gerard J. Milburn

TL;DR

This work tackles the longstanding challenge of formulating a Lorentz-invariant master equation for irreversible quantum dynamics. It builds a relational framework where time is defined by a physical scalar clock, deriving a covariant Tomonaga–Schwinger evolution and a non-Markovian relational Redfield/TCL master equation driven by Lorentz-scalar Wightman functions. By introducing clock-resolution smearing, leveraging Bochner positivity, and enforcing microcausality, the authors establish CP and TP GKLS structures; they also show the limitations of vacuum Markovian approaches and present covariant, integrable regimes in media and in gravity-augmented settings. The framework yields a consistent, covariant description of decay and dissipation in relativistic quantum fields, with clear pathways to quantum trajectories, CSL-like interpretations, and gravitating clocks within a classical–quantum hybrid dynamics. This advances relativistic open-system theory by resolving tension between irreversibility, covariance, and vacuum stability, and it provides a foundation for CPTP dynamics in gravitating environments.

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).

Lorentz Invariant Master Equation for Quantum Systems

TL;DR

This work tackles the longstanding challenge of formulating a Lorentz-invariant master equation for irreversible quantum dynamics. It builds a relational framework where time is defined by a physical scalar clock, deriving a covariant Tomonaga–Schwinger evolution and a non-Markovian relational Redfield/TCL master equation driven by Lorentz-scalar Wightman functions. By introducing clock-resolution smearing, leveraging Bochner positivity, and enforcing microcausality, the authors establish CP and TP GKLS structures; they also show the limitations of vacuum Markovian approaches and present covariant, integrable regimes in media and in gravity-augmented settings. The framework yields a consistent, covariant description of decay and dissipation in relativistic quantum fields, with clear pathways to quantum trajectories, CSL-like interpretations, and gravitating clocks within a classical–quantum hybrid dynamics. This advances relativistic open-system theory by resolving tension between irreversibility, covariance, and vacuum stability, and it provides a foundation for CPTP dynamics in gravitating environments.

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).
Paper Structure (44 sections, 3 theorems, 73 equations, 1 figure)

This paper contains 44 sections, 3 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Relational kinematics
  3. The relational TS framework
  4. Relational kinematics: Page--Wootters ⇒ Tomonaga--Schwinger
  5. From unitary TSE to local non-Markovian dynamics
  6. TCL generator with a relational clock: positivity and stability
  7. Local spectral decomposition vs. inertial-time frequency splitting
  8. Bochner route to complete positivity.
  9. Hypersurface independence and integrability
  10. Microcausality assumptions.
  11. The incompatibility of markovianity and vacuum covariance
  12. Integrability tests after the Markov limit
  13. Test A: functional curl / flat-connection test.
  14. Test B: boost--interchange test at the generator level.
  15. Canonical GKLS forms in two regimes
  16. ...and 29 more sections

Key Result

Proposition 1

Wightman positivity implies that the restriction of $G^+_{\alpha\beta}(x,y)$ to timelike separations is of positive type as a function of the displacement parameter. If $w_\sigma$ is chosen to be of positive type, then the product $w_\sigma G^+$ is again of positive type (Bochner--Schwartz theorem).

Figures (1)

  • Figure 1: A scheme for relational time coordination. Two macroscopic measurement devices are at rest with respect to each other over some finite spacetime four volume, constituting a well localized laboratory. One device, 'clock' (C), makes measurements of a scalar field $C(x,t)$ with result $\textrm{C}_j$. The other device is a localized system (S), interacting with an environment $E$ that may not be a rest with respect to C-S. Successive simultaneous measurements are made on C and S defining a space-like hypersurface $\Sigma_j$. The measurements could be counting quanta on another field $\psi(x,t)$ interacting directly with the system. All measurements are simultaneous in the local rest frame of the clock and the system.

Theorems & Definitions (7)

  • Definition 1: Clock-compatible local Bohr components
  • Remark 1
  • Proposition 1: Complete positivity from Wightman positivity
  • Lemma 1: Integrability after Born/TCL2
  • proof
  • Proposition 2: CCR preserved by the Langevin equation
  • proof : Sketch