# Thermodynamical path integral and emergent symmetry

## TL;DR

This paper develops a path integral framework for thermally isolated quantum systems undergoing parameter changes, revealing an emergent symmetry where entropy acts as a conserved quantity analogous to a Noether invariant.

## Contribution

It introduces a thermodynamic path integral formalism and demonstrates the emergence of a symmetry related to entropy conservation during quasi-static processes.

## Key findings

- Entropy appears as a Noether invariant in the path integral formulation.
- A symmetry emerges corresponding to uniform translation of the conjugate variable to entropy.
- The formalism connects thermodynamic quantities with quantum mechanical symmetries.

## Abstract

We investigate a thermally isolated quantum many-body system with an external control represented by a step protocol of a parameter. The propagator at each step of the parameter change is described by thermodynamic quantities under some assumptions. For the time evolution of such systems, we formulate a path integral over the trajectories in the thermodynamic state space. In particular, for quasi-static operations, we derive an effective action of the thermodynamic entropy and its canonically conjugate variable. Then, the symmetry for the uniform translation of the conjugate variable emerges in the path integral. This leads to the entropy as a Noether invariant in quantum mechanics.