Canonical quantization for equilibrium thermodynamics.

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 nonequilibrium thermodynamics.