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This paper proposes a formal framework that treats thermodynamic variables as quantum operators within a Hilbert space, deriving Schrödinger-like equations where entropy acts as time. By applying Dirac's constrained system theory, the work establishes thermodynamic uncertainty relations and extends applicability to phase transitions and black-hole thermodynamics. The approach offers a rigorous mathematical bridge between equilibrium thermodynamics and quantum mechanics, potentially informing future developments in quantum statistical physics.
Canonical quantization for Equilibrium Thermodynamics
arXiv:2511.14121v2 Announce Type: replace Abstract: We formulate a canonical quantization of Equilibrium Thermodynamics by applying Dirac's theory of constrained systems. Thermodynamic variables are treated as conjugate pairs of coordinates and momenta, allowing extensive and intensive quantities to be promoted to operators in a Hilbert space. The formalism is applied to the ideal gas, the van der Waals gas, and the photon gas, illustrating both first- and second-class quantization procedures. For the ideal gas, a Schr\"odinger-like equation emerges in which entropy plays the role of time, and the wave function acquires a phase determined by the internal energy. A pseudo-Hermitian framework restores Hermiticity of the temperature operator and establishes the equivalence among constraint realizations. The approach naturally leads to thermodynamic uncertainty relations and suggests extensions to quantum and topological phase transitions, as well as black-hole and non-equilibrium thermodynamics.
Abstract:We formulate a canonical quantization of Equilibrium Thermodynamics by applying Dirac's theory of constrained systems. Thermodynamic variables are treated as conjugate pairs of coordinates and momenta, allowing extensive and intensive quantities to be promoted to operators in a Hilbert space. The formalism is applied to the ideal gas, the van der Waals gas, and the photon gas, illustrating both first- and second-class quantization procedures. For the ideal gas, a Schrödinger-like equation emerges in which entropy plays the role of time, and the wave function acquires a phase determined by the internal energy. A pseudo-Hermitian framework restores Hermiticity of the temperature operator and establishes the equivalence among constraint realizations. The approach naturally leads to thermodynamic uncertainty relations and suggests extensions to quantum and topological phase transitions, as well as black-hole and non-equilibrium thermodynamics.
16+3 pages, no figures. Comments are welcome!
arXiv:2511.14121 [quant-ph]
(or arXiv:2511.14121v2 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2511.14121 Focus to learn more arXiv-issued DOI via DataCite
Phys. Rev. E 113.5 (2026): 05413
https://doi.org/10.1103/x91n-1lqx Focus to learn more DOI(s) linking to related resources
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Full text under the publisher's open licence · 244 words
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