Unification of stochastic and quantum thermodynamics in scalar field theory via a model with a Brownian thermostat

We present a systematic procedure to derive a quantum master equation for thermal relaxation in real scalar field theory, expanding on the method proposed in Koide and Nicacio Phys. Lett. A 494, 129277 (2024). We begin by introducing a generalized model for a classical scalar field interacting with a Brownian thermostat, consistent with stochastic thermodynamics. Applying canonical quantization to this model, we derive the corresponding quantum master equation, which is applicable to any form of the scalar field Hamiltonian. While its evolution is generally non-CPTP (completely positive and trace preserving), it can be adjusted to describe a CPTP evolution, such as those found in the Gorini-Kossakowski-Sudarshan-Lindblad equation by appropriately tuning the parameters of the model. In this framework, we define heat, work, and entropy in a way that satisfies the first and second laws of quantum thermodynamics. This suggests that the quantum-classical correspondence extends beyond closed systems governed by unitary time evolution to open systems as well. We further investigate the relation between the second law in quantum thermodynamics and relative entropy, providing insights into the study of quantum fluctuations through information-theoretical techniques in quantum field theory.