The stakes of this question are higher than they might first appear. Entropy and temperature pervade chemistry, engineering, cosmology, and information theory, yet their conceptual foundations remain contested. Philosophers of physics have long debated whether these quantities exist because of an irreversible assumption like the second law, because of facts about which states can be reached from which others by adiabatic processes, or for some entirely different reason. The answer matters for another deep question too: whether entropy is intrinsically tied to the arrow of time. If entropy can be defined without invoking irreversibility, then the common intuition that time’s direction flows from entropy increase loses some of its apparent necessity.

Jauch’s insight, published in this same journal more than five decades ago, was deceptively simple. Consider an engine cycling through its volume states, tracing a closed loop in the space of work variables. During such a cycle, the engine may exchange energy with its environment, so its full trajectory, including energy, forms a downward spiral that projects onto a closed curve in the work plane. Kelvin’s famous principle states that a cyclic process cannot absorb heat and convert it entirely into work, which explains why the spiral can go down but not up. Jauch focused on the special case where there is no heat exchange at all, an adiabatic process. In that case, he argued, work must come from somewhere, and if it comes neither from heat nor from a change in the system’s energy, it cannot be produced at all. His conservation law states: a cyclic process without heat exchange cannot perform work.

This principle, Roberts emphasizes, contains no irreversible ingredient. It is what Jauch called the static form of the second principle, weaker than Kelvin’s statement and restricted to conservative systems, those without internal friction such as hysteresis or electrical resistance. The hope was that this clean assumption would free thermodynamics from what Jauch called the wrong impression that the existence of entropy and temperature are characteristic consequences of irreversibility. There was just one problem: Jauch’s proof did not work. Roberts shows in detail that the argument breaks down because Jauch implicitly assumed that every point in a thermodynamic state space can be reached from any other by an adiabatic path. That assumption is precisely what Carathéodory’s principle denies, and it turns out to be incompatible with the very conclusion Jauch was trying to establish.

Remarkably, however, the theorem itself is true. Roberts proves it using an entirely different strategy, one that reframes thermodynamics in the language of fibre bundles and gauge theory, the same mathematical machinery that underlies the Yang-Mills theories of particle physics. The key move is to distinguish two manifolds: an (n-1)-dimensional space of observable work configurations and an n-dimensional space that adds total internal energy as an extra dimension. These are related by a projection map, forming a fibre bundle with one-dimensional fibres. In this picture, work is any one-form that annihilates vertical vectors, energy is a vertical coordinate, and heat is defined as the difference between the change in total energy and the work done, making heat, in a precise sense, unobservable energy.

The geometric reformulation then reveals something striking about Jauch’s conservation law. An adiabatic process defines what is called an Ehresmann connection on the bundle: the collection of horizontal directions satisfying the condition that the heat one-form vanishes. Just as in gauge theory, closed loops in the base space can be lifted to curves in the total space, and the failure of those lifts to close measures curvature, exactly as a vector transported around a loop on a sphere fails to return to its starting orientation. The engine’s downward spiral in energy-versus-volume space looks suspiciously like a holonomy, the gauge-theoretic signature of curvature. Expressed in this language, Jauch’s conservation law says that the adiabatic connection has only locally trivial holonomies, meaning every lifted loop closes. That is precisely the statement that the connection is flat, with zero curvature.

From flatness, the theorem follows elegantly. Roberts shows that a connection with locally trivial holonomy must be involutive, and by the Frobenius theorem, an involutive connection is integrable. This means the kernel of the heat one-form is tangent to a family of surfaces, and there exists a function S that is constant on each of them. Since the one-forms dS and the heat form have the same kernel, they can differ only by a scalar field T, yielding the celebrated relation heat equals temperature times the change of entropy, written as xi equals T dS. Entropy and temperature thus emerge as a direct consequence of a conservation principle interpreted geometrically, sidestepping Carathéodory’s accessibility argument altogether.

The result also settles a historical curiosity. Jauch once credited his idea to the physicist and mathematician Tatiana Afanassjewa, who argued as early as 1925 that irreversibility has no relevance for the existence of entropy. Roberts, however, could find no trace of Jauch’s theorem in her published work, which instead relies on Carathéodory’s theorem. What he did find is that Jauch’s conservation hypothesis follows from her fundamental Entropy Axiom, and that it is strictly weaker than her principle: Jauch’s law assumes less and reaches the same conclusion, without even requiring the notions of equilibrium states or quasi-static processes. In that sense, the new result is strictly stronger than anything in the Afanassjewa tradition.

The framework also opens concrete avenues for future research. Systems in which heat cannot be written as temperature times entropy change, such as those with internal friction, are precisely those whose adiabatic connections have non-zero curvature. This suggests a clean geometric way to model irreversibility itself, as curvature in the adiabatic connection, and it hints at possible thermodynamic analogues of the Aharonov-Bohm effect and geometric phase, phenomena in which gauge curvature produces observable consequences even in regions where the field vanishes. Roberts develops this broader gauge-theoretic view of heat in a companion manuscript, but the present work stands on its own as a corrected and completed foundation.

What emerges is a picture of thermodynamics as a theory with a hidden geometric skeleton. The essential structures required are minimal: a distinction between work and total energy, and the undirected notion of an adiabatic path. From these, and from the flatness of a naturally defined connection, the existence of entropy and temperature follows as an elegant theorem. The second law, the arrow of time, and the irreversibility of spontaneous processes remain real physical phenomena awaiting explanation, but they no longer bear the burden of underwriting the very existence of entropy. Jauch’s half-century-old conjecture, once thought to be broken, turns out to be a window into the deep geometry of heat.

Subject of Research: A gauge-theoretic reconstruction of Jauch's conservation law as a foundation for entropy and temperature in thermodynamics.

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

Article References: Roberts, B. W. (2026). A New Origin for Entropy: Jauch’s Conservation Law and the Geometry of Thermodynamics. Foundations of Physics, 56(5), Article 46. https://doi.org/10.1007/s10701-026-00946-6

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DOI: 10.1007/s10701-026-00946-6

Keywords: entropy, temperature, thermodynamics, second law, gauge theory, fibre bundles, Ehresmann connection, Jauch conservation law, Carathéodory, irreversibility, holonomy, Frobenius theorem