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arXiv:2403.14752 · Physical Review Research

From equivalent Lagrangians to inequivalent open quantum system dynamics

Anirudh Gundhi, Oliviero Angeli, Angelo Bassi

Published Dec 28, 2025• 0 citations
arXivHTMLPDF

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 nonrelativistic 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.

DOI
10.1103/4rpx-zj2x

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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 nonrelativistic 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.
One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.
We investigate the open quantum dynamics of a system of two masses interacting with an environment of linearized gravitational waves. We formulate the analysis in terms of the observable proper distance between the two masses, and show that the canonical variables obtained from the standard Lagrangian, expressed in terms of the Fermi normal coordinates, are not suitable for an effective description of the system. We resolve this issue through a unitary transformation that provides a physically meaningful system--environment decomposition and derive the master equation to leading order in GGG. Its dissipative sector reproduces the classical energy loss due to gravitational-wave emission, while the noisy contributions suppress coherences between states with different mass quadrupole, or effectively, different proper separations. In the regime where the proper distance can be described by considering small quantum fluctuations around an average distance l0l_0l0​, the dynamics reduces to a Caldeira--Leggett-type equation, with a decoherence rate dependent on the baseline length l0l_0l0​.
This thesis investigates the role of gravitational waves in quantum systems, motivated by advances in our understanding of the quantum–gravity interface and efforts to identify experimental signatures of quantum gravity. It focuses on gravitational decoherence and the interplay between internal energy and centre-of-mass dynamics in composite quantum systems, complemented by a study of internal-energy effects in gravitationally mediated entanglement. First, we develop relativistic Hamiltonian formulations showing that changes in a composite system’s internal energy increase its gravitational mass rather than alter its momentum. This removes earlier ad hoc arguments for eliminating “anomalous friction forces”. We show that gravitational waves act as a dynamical background, inducing curvature-dependent forces that modify total momentum. Second, we revisit a thought experiment in which tensions between causality and complementarity arise. We identify a scenario in which these tensions follow from energy–mass equivalence, making the effect specific to gravitational interactions. In a gravitationally mediated entanglement experiment using Newtonian gravity as mediator, we investigate whether decohering gravitational radiation can resolve the apparent conflict. Finally, we build upon a model of gravitational decoherence consistent with general relativity. This requires deriving a higher-derivative quantum Langevin equation and master equation. The resulting dynamics show that gravitational interactions induce decoherence in the canonical momentum basis of the quadrupole and generate acceleration- and jerk-dependent corrections absent in electrodynamics. Together, these results show that weak-field gravitational interactions leave distinct relativistic imprints on quantum systems, including effects unique to gravitational waves, providing potential signatures that distinguish gravity from other interactions.
We derive a general quantum master equation for the dynamics of a scalar bosonic particle interacting with a weak, stochastic and classical external gravitational field. The dynamics predicts decoherence in position, momentum and energy. We show how our master equation reproduces the results present in the literature by taking appropriate limits, thus explaining the apparent contradiction in their dynamical description. Our result is relevant in light of the increasing interest in the low energy quantum-gravity regime.
We discuss the decoherence in a quantum system induced by interaction with gravitational degrees of freedom that are part of a higher derivative theory. The deformation of a mass distribution due to gravitational waves acquires naturally a mass quadrupole moment. This adds higher derivative dynamics of the quadrupole moment to the unitary evolution of the system, where the quadrupole moment oscillates with the gravitational frequencies following a higher derivative theory. The consequence of higher derivatives in the dynamics is that four canonical variables describe the system. This departure from the usual particle position and momentum operators gives an entirely different interpretation of the decoherence basis. This model focuses on the open dynamics of the quadrupole moment, rather than on individual particles. As such, a short example is given to utilize quadrupole measurements to probe gravitational decoherence and noise. We first derive a Langevin equation for a lower derivative model and show how higher derivatives naturally emerge on the boundary. A quantum master equation is derived for the emerging quadrupole moment, considering that the environment is a higher derivative theory of gravity.
An open quantum system is a quantum system that interacts with some environment whose degrees of freedom have been coarse grained away. This model describes non-equilibrium processes more general than scattering-matrix formulations. Furthermore, the microscopically-derived environment provides a model of noise, dissipation and decoherence far more general than Markovian (white noise) models. The latter are fully characterized by Lindblad equations and can be motivated phenomenologically. Non-Markovian processes consistently account for backreaction with the environment and can incorporate effects such as finite temperature and spatial correlations. We consider linear systems with bilinear coupling to the environment, or quantum Brownian motion, and nonlinear systems with weak coupling to the environment. For linear systems we provide exact solutions with analytical results for a variety of spectral densities. Furthermore, we point out an important mathematical subtlety which led to incorrect master-equation coefficients in earlier derivations, given nonlocal dissipation. For nonlinear systems we provide perturbative solutions by translating the formalism of canonical perturbation theory into the context of master equations. It is shown that unavoidable degeneracy causes an unfortunate reduction in accuracy between perturbative master equations and their solutions. We also extend the famous theorem of Lindblad, Gorini, Kossakowski and Sudarshan on completely positivity to non-Markovian master equations. Our application is primarily to model atoms interacting via a common electromagnetic field. The electromagnetic field contains correlations in both space and time, which are related to its relativistic (photon-mediated) nature. As such, atoms residing in the same field experience different environmental effects depending upon their relative position and orientation. Our more accurate solutions were necessary to assess sudden death of entanglement at zero temperature. In contrast to previous claims, we found that all initial states of two-level atoms undergo finite-time disentanglement. We were also able to access regimes which cannot be described by Lindblad equations and other simpler methods, such as near resonance. Finally we revisit the infamous Abraham-Lorentz force, wherein a single particle in motion experiences backreaction from the electromagnetic field. This leads to a number of well-known problems including pre-acceleration and runaway solutions. We found a more a more-suitable open-system treatment of the nonrelativistic particle to be perfectly causal and dissipative without any extraneous requirements for finite size of the particle, weak coupling to the field, etc..