Master equation for systems interacting with linearized gravity

When modeling a quantum system interacting with an environment—such as two masses interacting with gravitational waves—it is essential to clearly separate the system's degrees of freedom from those of the environment. The authors address a fundamental challenge in this separation: the standard canonical variables derived from the Lagrangian for linearized gravity are not physically meaningful because they depend on the environmental gravitational field, which is traced out in the open quantum system formalism. The paper seeks to identify a physically consistent system-environment decomposition to derive a valid master equation.

To overcome the limitations of standard canonical quantization, the authors utilize Fermi normal coordinates to describe the proper distance between two masses. They demonstrate that the standard Legendre transformation leads to conjugate momenta that are operationally dependent on the environment. To resolve this, they introduce a unitary transformation that redefines the system and environmental operators. This transformation ensures that the system's conjugate variables are defined solely in terms of accessible system observables and commute with the environmental degrees of freedom, satisfying the requirements for a consistent partial trace.

By applying this unitary transformation, the authors derive a Markovian master equation to leading order in the gravitational constant G. The dissipative part of the equation correctly reproduces the classical energy loss associated with gravitational-wave emission (gravitational bremsstrahlung). The noisy part of the equation describes the decoherence of quantum superpositions between states with different proper separations. In the specific regime of small quantum fluctuations around a fixed baseline distance, the dynamics simplify to a Caldeira-Leggett-type equation, where the decoherence rate is explicitly dependent on the baseline length.