From equivalent Lagrangians to inequivalent open quantum system dynamics

TL;DR

The paper shows that two Lagrangians related by a total derivative, though globally unitary-equivalent for the full system, can produce inequivalent reduced dynamics after tracing out the environment in open quantum systems. It identifies an operational criterion—prefer the Lagrangian in which the canonical momentum equals the mechanical momentum—to obtain meaningful reduced dynamics, and demonstrates this with a toy model and a microscopic QED calculation of bremsstrahlung for a non-relativistic electron. The resulting master equation closely matches the Caldeira–Leggett form and predicts decoherence in the position basis, resolving prior discrepancies in the literature. The findings have broad implications for QED and gravitational decoherence, where similar Lagrangian ambiguities affect open-system behavior.

Abstract

Lagrangians can differ by a total derivative without altering the equations of motion, thus encoding the same physics. This is true both classically and quantum mechanically. We show, however, that in the context of open quantum systems, two Lagrangians that differ by a total derivative can lead to inequivalent reduced dynamics. While these Lagrangians are connected via unitary transformations at the level of the global system-plus-environment description, the equivalence breaks down after tracing out the environment. We argue that only those Lagrangians for which the canonical and mechanical momenta of the system coincide lead to operationally meaningful dynamics. Applying this insight to quantum electrodynamics (QED), we derive the master equation for bremsstrahlung due to an accelerated non-relativistic electron upto second order in the interaction. The resulting reduced dynamics predicts decoherence in the position basis and closely matches the Caldeira-Leggett form, thus resolving previous discrepancies in the literature. Our findings have implications for both QED and gravitational decoherence, where similar ambiguities arise.