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.
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.
In this work, improvements are introduced to the current models of the ideal Fermi gas and the ideal Bose gas by incorporating the quantum nature of phase space, which is directly linked to the uncertainty principle. These improved models build upon the recently developed concepts of quantum phase space (QPS) and the QPS representation of quantum mechanics. The Hamiltonian operator for a gas particle and its eigenstates are first determined, and quantum statistical mechanics is used to derive the thermodynamic properties of the ideal gas. Analytic expressions for thermodynamic quantities—including the grand canonical potential, particle number, internal energy, von Neumann entropy, and pressure—are derived, along with the corresponding thermodynamic equations of state for both bosons and fermions. These corrections are particularly significant at low temperatures and in confined volumes, where quantum effects related to system geometry, such as shape and size, become significant. The results also establish a direct link between thermodynamic functions and the quantum statistical variances of momentum. Importantly, the improved models recover well-known classical relations of the ideal gas in the high-temperature and large-volume limits, ensuring consistency with classical physics. By addressing quantum corrections and their thermodynamic implications, this work provides a foundation for further applications in nanoscale systems, quantum gases, low-temperature physics, ultracold physics, and astrophysics.
In previous work, we investigated thermodynamic processes in systems at the mesoscopic level where traditional thermodynamic descriptions (macroscopic or microscopic) may not be fully adequate. The key result is that entropy in such systems does not change continuously, as in macroscopic systems, but rather in discrete steps characterized by the quantization constant β. This quantization reflects the underlying discrete nature of the collision process in low-dimensional systems and the essential role played by thermodynamic fluctuations at this scale. Thermodynamic variables conjugate to the forces, along with Glansdorff-Prigogine's dissipative variable can be discretized, enabling a mesoscopic-scale formulation of canonical commutation rules (CCRs). In this framework, measurements correspond to determining the eigenvalues of operators associated with key thermodynamic quantities. This work investigates the quantization parameter β in the CCRs using a nano-gas model analyzed through classical statistical physics. Our findings suggest that β is not an unknown fundamental constant. Instead, it emerges as the minimum achievable value derived from optimizing the uncertainty relation within the framework of our model. The expression for β is determined in terms of the ratio χ, which provides a dimensionless number that reflects the relative scales of volume and mass between entities at the Bohr (atomic level) and the molecular scales. This latter parameter quantifies the relative influence of quantum effects versus classical dynamics in a given scattering process.
We present a novel framework for quantizing constrained quantum systems in which the processes of quantization and constraint enforcement are performed simultaneously. The approach is based on an extension of the stationary action principle, incorporating an information-theoretic term arising from vacuum fluctuations. Constraints are included directly in the Lagrangian via Lagrange multipliers, allowing the subsequent variational procedure to yield the quantum dynamics without ambiguity regarding the order of quantization and reduction. We demonstrate the method through two examples: (i) a one-dimensional system with vanishing local momentum, where the simultaneous approach produces the time-independent Schrödinger equation while conventional reduced and Dirac quantization yield only trivial states, and (ii) a bipartite system with global translational invariance, where all three methods agree. These results show that the proposed framework generalizes standard quantization schemes and provides a consistent treatment of systems with constraints that cannot be expressed as linear operators acting on the wave function. In addition to a unified variational principle for constrained quantum systems, the formalism also offers an information-theoretic perspective on quantum effects arising from vacuum fluctuations.
Quantum thermodynamics is traditionally formulated as a theory of equilibrium states and state transformations. Recent advances in higher-order quantum transformations, which describe physical scenarios beyond states and channels and provide a systematic framework for causal order, raise the question of how equilibrium should be defined and preserved in this more general setting. Starting from the Gibbs state as the unique equilibrium state, we identify equilibrium preservation as its natural higher-order extension. We show that, while this principle can in general give rise to distinct classes of transformations, all such distinctions disappear when equilibrium preservation is required completely, namely under arbitrary ancillary extensions. Remarkably, every transformation satisfying this condition is causally ordered, making causal order an emergent consequence of thermodynamic equilibrium. We establish this result for arbitrary higher-order maps and use the resulting framework to introduce free-energy-like quantities for quantum channels. Our findings reveal a fundamental connection between thermodynamic equilibrium and causal structure, providing a foundation for a fully fledged theory of higher-order quantum thermodynamics.
We examine the quantization of pseudoclassical dynamical systems, models that have classically anticommuting variables, in the Schrödinger picture. We quantize these systems, which can be viewed as classical models of particle spin, using the generalized Gupta–Bleuler method as well as the reduced phase space method in even dimensions. With minimal modifications, the standard constructions of Schrödinger quantum mechanics of constrained systems work for pseudoclassical systems. We generalize the standard Schrödinger norm and implement the correct adjointness properties of observables and constraints. We construct the state space corresponding to spinors as physical wave functions of anticommuting variables, finding that there are superselection sectors in both the physical and ghost subspaces. The physical states are isomorphic to those of the Dirac–Kähler formulation of fermions, though the inner product in Dirac–Kähler theory is not equivalent to ours.
We establish fundamental uncertainty relations for the hydrodynamic variables arising from the Madelung representation of quantum fields in curved spacetime. Through canonical quantization of the density n and phase θ variables and their conjugate momenta, we derive exact uncertainty principles that depend on spacetime geometry through the lapse function N and spatial metric γij. These relations reveal how gravitational fields modulate quantum fluctuations and provide first-principles constraints for Scalar Field Dark Matter (SFDM) models and stochastic quantum gravity. We present two groundbreaking applications. First, By coupling the density-velocity uncertainty to the Newtonian condition of hydrostatic equilibrium, we derive the exact scaling law for the minimum core radius of a self-gravitating scalar field halo. We naturally find the scale rc∝m−1/2 which is a testable prediction. Second, by evaluating the phase--momentum commutator in Rindler coordinates and relating the canonical momentum Πθ to the energy density via the Hamiltonian, we obtain the acceleration scale kBT=Cℏa/c. The proportionality constant C=1/(2π) is fixed by the topological quantization of the Madelung phase circulation around the Euclidean horizon. This dual achievement demonstrates that quantum uncertainty on curved backgrounds is the unifying principle behind both galactic structure and black hole thermodynamics.