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.
Decoherence explains why selected quantum histories behave as mutually exclusive alternatives with Born weights, but it does not by itself show that such alternatives are worlds. We formulate this as a geometric representability problem within decoherent histories. A world-bearing branch must support a local physical representation: a compatible spectral and local structure sufficient for spacetime-like observables, stable records, and observer-level description. For a history effect F_α =K_α ^† K_α and a positive spectral probe A , we introduce the branch-resolved heat trace Θ _α (t)=Tr(e^-tA/2F_α e^-tA/2) . Its small-time behavior tests the spectral supply carried by that branch. We prove that identical decoherence data can be realized by branches with inequivalent heat profiles, and that exact decoherent refinement can decompose a spectrally thick branch into probability-bearing but spectrally thin branches. A positive lower-bound theorem then shows that genuine branch-local spectral regularity forces corresponding heat-trace growth. Decoherence remains essential for probabilities, but branch ontology requires structural admissibility beyond decoherence.
Recently, Dunajski and Gundry (Commun. Math. Phys., 342(3), 2016) have developed a version of twistor theory for the non-relativistic domain. Unlike relativistic twistor theory, their approach is able to reproduce the entire space of models of Newton–Cartan theory. We critically assess the significance of non-relativistic twistors, in particular with respect to proposals by Dunajski and Penrose (Ann. Phys., 451, 2023) that using non-relativistic twistors to describe gravitationally induced collapse could play a part in solving the quantum measurement problem.
