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Canonical quantization for Equilibrium Thermodynamics

Luis F. Santos, Victor Hugo M. Ramos, Danilo Cius, Mario C. Baldiotti, Bárbara Amaral

TL;DR

This work recasts Equilibrium Thermodynamics as a constrained mechanical system and applies Dirac's quantization to promote thermodynamic variables to operators in a Hilbert space. By formulating first- and second-class constraints for representative models—the ideal gas, van der Waals gas, and photon gas—it derives a Schrödinger-like evolution with entropy acting as the time parameter and identifies a non-Hermitian temperature-like generator that can be mapped to a Hermitian observable via a pseudo-Hermitian framework. The approach yields thermodynamic uncertainty relations and shows that different operator-ordering realizations are physically equivalent within the pseudo-Hermitian formalism, while also explaining entropy-driven irreversibility and the emergence of classicality at large entropy. The paper outlines extensions to quantum and topological phase transitions, black-hole thermodynamics, and non-equilibrium thermodynamics, providing a foundation for a deeper, operator-based understanding of thermodynamic quantities.

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.

Canonical quantization for Equilibrium Thermodynamics

TL;DR

This work recasts Equilibrium Thermodynamics as a constrained mechanical system and applies Dirac's quantization to promote thermodynamic variables to operators in a Hilbert space. By formulating first- and second-class constraints for representative models—the ideal gas, van der Waals gas, and photon gas—it derives a Schrödinger-like evolution with entropy acting as the time parameter and identifies a non-Hermitian temperature-like generator that can be mapped to a Hermitian observable via a pseudo-Hermitian framework. The approach yields thermodynamic uncertainty relations and shows that different operator-ordering realizations are physically equivalent within the pseudo-Hermitian formalism, while also explaining entropy-driven irreversibility and the emergence of classicality at large entropy. The paper outlines extensions to quantum and topological phase transitions, black-hole thermodynamics, and non-equilibrium thermodynamics, providing a foundation for a deeper, operator-based understanding of thermodynamic quantities.

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.

Paper Structure

This paper contains 26 sections, 116 equations.

Table of Contents

  1. Introduction
  2. Constrained Systems
  3. Classical systems
  4. Mechanical formulation of Thermodynamics
  5. Quantization of constrained systems
  6. First-class constraints
  7. Second-class constraints
  8. Dirac's Quantization for Thermodynamics
  9. Ideal gas
  10. van der Waals gas
  11. Photon gas and second-class constraints
  12. First-class approach
  13. Second-class approach
  14. Physical implications for the ideal gas
  15. Schrödinger-like equation and probability flow
  16. ...and 11 more sections