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Heat as a gauge connection

Bryan W Roberts

TL;DR

This work reframes thermodynamics as a gauge-theoretic problem by modeling a thermodynamic system as a line bundle over work configurations, with heat given by the nowhere-vanishing 1-form $\boldsymbol{\xi} = dU + \boldsymbol{\omega}$. The central result shows that heat defines a connection on this bundle, and that a locally integrating factor (an entropy–temperature pair with $\boldsymbol{\xi} = T dS$) exists if and only if the connection has vanishing curvature, i.e., is integrable. The authors derive a gauge-theoretic proof of Jauch's conjecture as a special case and identify a thermal analogue of geometric phase responsible for the possible failure of global equilibrium, linking holonomy to global nonequilibrium. Overall, the paper advances a perspective in which thermodynamics arises as a special case of gauge theory, with local thermodynamic quantities tied to curvature and global behavior governed by holonomy and geometric phase.

Abstract

We show that heat defines a gauge connection on a line bundle over work configurations. Vanishing curvature is equivalent to the local existence of entropy and temperature functions such that heat can be expressed as $TdS$. A conjecture of Jauch, that entropy and temperature arise from a conservation law, is shown to follow as a special case. Global equilibrium may nevertheless fail in the presence of a thermal analogue of geometric phase.

Heat as a gauge connection

TL;DR

This work reframes thermodynamics as a gauge-theoretic problem by modeling a thermodynamic system as a line bundle over work configurations, with heat given by the nowhere-vanishing 1-form . The central result shows that heat defines a connection on this bundle, and that a locally integrating factor (an entropy–temperature pair with ) exists if and only if the connection has vanishing curvature, i.e., is integrable. The authors derive a gauge-theoretic proof of Jauch's conjecture as a special case and identify a thermal analogue of geometric phase responsible for the possible failure of global equilibrium, linking holonomy to global nonequilibrium. Overall, the paper advances a perspective in which thermodynamics arises as a special case of gauge theory, with local thermodynamic quantities tied to curvature and global behavior governed by holonomy and geometric phase.

Abstract

We show that heat defines a gauge connection on a line bundle over work configurations. Vanishing curvature is equivalent to the local existence of entropy and temperature functions such that heat can be expressed as . A conjecture of Jauch, that entropy and temperature arise from a conservation law, is shown to follow as a special case. Global equilibrium may nevertheless fail in the presence of a thermal analogue of geometric phase.

Paper Structure

This paper contains 8 sections, 7 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. The work bundle and gauge
  3. Heat as a gauge connection
  4. Equilibrium as zero curvature
  5. Holonomy structure
  6. Conservation and Jauch's conjecture
  7. Thermal geometric phase
  8. Conclusion

Key Result

Lemma 1

Let $\pi:L\rightarrow N$ be a line bundle, $U$ a vertical coordinate function, and $\omega$ a one-form on $L$ such that $\ker\pi_*\subseteq\ker\omega$ and $\xi := dU + \omega$ is nowhere-vanishing. Then $H=\ker\xi$ is a connection.

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Lemma 4
  • proof
  • ...and 2 more