Thermal pure state path integral in isolated quantum system and emergent symmetry of thermodynamic entropy

Thermodynamics and quantum mechanics are fundamental theories in physics. However, their relationship between their dynamics is not established yet; therefore we propose a theory connecting thermodynamical behavior to quantum mechanics. Our strategy is to construct a thermodynamical path integral. In thermodynamics, an equilibrium state of a system is represented by a point in the thermodynamic state space. In quantum mechanics, on the other hand, the time evolution of a system is described by the path integral. In this talk, we combine these two concepts for a thermally isolated quantum many-body system under a time-dependent external field. We formulate the unitary evolution of quantum states by an integral over paths in the thermodynamic state space. We call it thermal pure state path integral and find an emergent symmetry in it. In the thermal pure state path integral, we derive an effective action for trajectories in a thermodynamic state space, where entropy appears with its conjugate variable. In particular, for quasi-static operations, the symmetry for the uniform translation of the conjugate variable emerges; the thermodynamic entropy is characterized as a Noether invariant. It leads to entropy conservation.