Canonical Quantization of Equilibrium Thermodynamics Yields Entropy-Driven Schrödinger Equation and Uncertainty Relations

Equilibrium thermodynamics, the study of energy transfer and transformations in systems, receives a novel theoretical treatment in new work led by Luis F. Santos, Victor Hugo M. Ramos, and Danilo Cius from the University of São Paulo. The researchers develop a method, termed canonical quantization, that applies the principles of quantum mechanics to traditionally macroscopic thermodynamic variables, treating quantities like energy and volume as operators within a quantum framework. This approach, which also includes contributions from Mario C. Baldiotti of Universidade Estadual de Londrina and Bárbara Amaral of the University of Toronto, yields surprising results, including a Schrödinger-like equation where entropy takes the role of time, and establishes a deeper connection between thermodynamic constraints and the fundamental principles of quantum mechanics. The team’s work not only provides a new mathematical foundation for understanding equilibrium systems, but also opens avenues for exploring complex phenomena such as topological phase transitions and the behaviour of black holes.

Pseudo-Hermitian Gases and Thermodynamics Beyond Hermiticity

Scientists are expanding the boundaries of quantum mechanics by exploring systems that move beyond traditional mathematical symmetries, investigating pseudo-Hermitian quantum mechanics. The team aims to represent the thermodynamic properties of an ideal gas, such as temperature and pressure, using wave functions, connecting the macroscopic world of thermodynamics with the microscopic realm of quantum theory. The research details the mathematical construction of thermodynamic wave functions for ideal gases, exploring ways to define operators representing temperature and pressure while ensuring the wave functions remain physically realistic. A central focus is ensuring the time evolution of these wave functions is consistent and physically meaningful, even when using non-Hermitian operators, achieved through a mathematical tool called a metric operator.

This framework provides a comprehensive overview of pseudo-Hermitian quantum mechanics, introducing a metric operator that defines a new mathematical structure and ensures predictable system evolution. The results demonstrate that it is possible to describe the thermodynamic properties of an ideal gas using non-Hermitian quantum mechanics, provided the appropriate mathematical framework is employed. This work contributes to the growing field of non-Hermitian quantum mechanics, offering potential implications for areas such as open quantum systems, quantum optics, and condensed matter physics.

Quantum Thermodynamics via Dirac Quantization and Wave Functions

Scientists have developed a novel approach to thermodynamics by reformulating equilibrium thermodynamics as a constrained mechanical system and subsequently applying Dirac’s quantization procedure. This innovative method promotes extensive and intensive variables, including entropy, temperature, volume, and pressure, to operators within a Hilbert space, enabling a consistent quantum theory of thermodynamics. The team pioneered a method for deriving wave functions for both ideal and van der Waals gases, as well as characterizing the physical Hilbert space associated with photon gas, a feat not previously demonstrated in other approaches. Researchers observed that systems constrained by a specific mathematical relationship naturally admit a Schrödinger-like equation, where entropy functions as a time-like parameter.

The resulting thermodynamic wave function exhibits a phase determined by the internal energy, analogous to the time-evolution operator in standard quantum mechanics. Experiments revealed that evolution generated by these constraints is intrinsically non-unitary when expressed in terms of entropy, aligning with the thermodynamic arrow of time and embedding irreversibility directly into the quantum description. Furthermore, the study demonstrated that the growth of entropy leads to the emergence of classicality, as non-Hermitian components of the theory become exponentially suppressed at high entropy. Scientists derived a set of thermodynamic uncertainty relations, establishing direct analogies with the canonical uncertainty principles of quantum mechanics. They also showed that different mathematical formulations of the constraints are physically equivalent within a pseudo-Hermitian framework, providing a consistent interpretation of the imaginary component of the temperature operator as a physically meaningful feature. This innovative methodology opens avenues for investigating quantum phase transitions, topological phase transitions, and the quantization of complex thermodynamic systems such as black holes and Bose gases.

Thermodynamic Quantization Links Entropy and Quantum Mechanics

This work presents a novel quantization of equilibrium systems, applying Dirac’s theory of constrained systems to treat extensive and intensive variables as conjugate pairs in a Hilbert space. The researchers successfully applied this formalism to the ideal gas, the van der Waals gas, and the photon gas, demonstrating both first and second-class quantization procedures. For the ideal gas, the team derived a Schrödinger-like equation where entropy takes the role of time, and the wave function’s phase is determined by the internal energy, revealing a fundamental connection between thermodynamic evolution and quantum mechanics. The study introduces a pseudo-Hermitian framework that restores Hermiticity of the temperature operator, establishing equivalence among different mathematical formulations and ensuring mathematical consistency within the model.

Measurements confirm the emergence of uncertainty relations, suggesting potential extensions to topological phase transitions and applications to black-hole physics and non-equilibrium systems. Researchers developed a mechanical formulation of thermodynamics, drawing strong analogies between thermodynamic variables and coordinates/momenta in a Hamiltonian phase space. This approach, mirroring the Hamilton, Jacobi formalism, allows thermodynamic integrability conditions to be rewritten as canonical Poisson brackets, revealing a deep mathematical structure underlying thermodynamic behavior. The team demonstrated that by treating entropy as a time parameter within the Schrödinger-like equation, they achieve a consistent quantum mechanical description of thermodynamic evolution, opening new avenues for exploring the foundations of statistical mechanics and its connection to quantum theory.

Quantum Thermodynamics, Entropy as Time Parameter

This work presents a quantum theory of thermodynamics achieved by reformulating equilibrium thermodynamics as a constrained mechanical system and applying Dirac’s quantization procedure. The researchers successfully promoted extensive and intensive variables, such as entropy and temperature, to operators within a Hilbert space, yielding Schrödinger-like equations where entropy functions as a time-like parameter. Notably, explicit wave functions were obtained for ideal gases, van der Waals gases, and photon gases, a feat not previously demonstrated in other approaches. The resulting framework reveals an intrinsic non-unitary evolution when expressed in terms of entropy, aligning with the thermodynamic arrow of time and embedding irreversibility directly into the quantum description. Furthermore, the team derived thermodynamic uncertainty relations analogous to those found in standard quantum mechanics and demonstrated the physical equivalence of different mathematical formulations through a pseudo-Hermitian framework. Future work will focus on extracting further physical insights from the derived wave functions themselves.