A New Origin for Entropy: Jauch’s Conservation Law and the Geometry of Thermodynamics

A central question for the foundations of thermodynamics is which conceptual structures underpin the existence of entropy and temperature. Jauch claimed that entropy and temperature could be derived from a novel conservation law. This paper reconstructs the physics and mathematics of Jauch’s claim and finds that his original proof is not valid. Remarkably, his theorem is still true. We provide an alternative proof using geometric ideas from modern gauge theory, revealing a deep geometric structure in thermodynamics. This result also helps to settle an old debate by showing that entropy and temperature can be defined in a new, conceptually clear way without appeal to an irreversible assumption like the second law.