Geometric foundations of thermodynamics in the quantum regime

TL;DR

This work advances a geometric formulation of quantum thermodynamics by embedding the quantum state space in a $(2n+1)$-dimensional contact manifold $\mathcal{M}$, with equilibrium Gibbs states forming a Legendrian submanifold $\mathcal{E}$ and a principal fiber bundle over density operators separating thermodynamic labels from state evolution. The equilibrium curve set $\mathcal{E}$ is linked to Gibbs states $\rho_{\boldsymbol{\lambda}}$, while the fibers $F_{\sigma}$ encode non-equilibrium configurations, yielding a gauge-like structure that constrains admissible thermodynamic descriptions. Quasistatic processes correspond to minimizing geodesics in the Gibbs-state manifold $\mathcal{B}$ endowed with the Bures-Wasserstein metric $g_{\mathrm{BW}}$, ensuring minimal dissipation and yielding a geometric third law via geodesic incompleteness near rank-deficient states. Non-equilibrium dynamics are captured by a pseudo-Riemannian extension and a principal connection, whose curvature induces holonomy and geometric irreversibility in cyclic processes; together with the second law, these geometric features provide a rigorous, unifying foundation for quantum thermodynamics and its operational limits.

Abstract

In this work, we present a comprehensive geometrical formulation of quantum thermodynamics based on contact geometry and principal fiber bundles. The quantum thermodynamic state space is modeled as a contact manifold, with equilibrium Gibbs states forming a Legendrian submanifold that encodes the fundamental thermodynamic relations. A principal fiber bundle over the manifold of density operators distinguishes the quantum state structure from thermodynamic labels: its fibers represent non-equilibrium configurations, and their unique intersections with the equilibrium submanifold enforce thermodynamic consistency. Quasistatic processes correspond to minimizing geodesics under the Bures-Wasserstein metric, leading to minimal dissipation, while the divergence of geodesic length toward rank-deficient states geometrically derives the third law. Non-equilibrium extensions, formulated through pseudo-Riemannian metrics and connections on the principal bundle, introduce curvature-induced holonomy that quantifies irreversibility in cyclic processes. In this framework, the thermodynamic laws in the quantum regime emerge naturally as geometric consequences.

Theorems & Definitions (87)

• Definition 1
• Proposition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1: Darboux
• Definition 5
• Proposition 2
• proof
• Definition 6