We present a self-contained formulation of quantum mechanics grounded in strictly positive joint hybrid-statistical distributions and fundamental harmonic analysis theorems (Hardy and Beckner). By treating probability not as an intrinsic property of isolated events but as a contextual projection arising from the decomposition of an undivided system, we systematically resolve the historical Wigner negative probability paradox and the long-standing time-operator problem. Using the inverse Wigner-Weyl integral transformation, we directly construct the density matrix from the structure of hybrid distributions. This mathematically rigorous framework demonstrates that natural irreversibility and the thermodynamic arrow of time are intrinsic properties of hybrid flow gradients, completely redefining the phenomenon of decoherence and eliminating the need for external open-system master equations.
We present a self-contained formulation of quantum mechanics grounded in strictly positive joint hybrid-statistical distributions and fundamental harmonic analysis theorems (Hardy and Beckner). By treating probability not as an intrinsic property of isolated events but as a contextual projection arising from the decomposition of an undivided system, we systematically resolve the historical Wigner negative probability paradox and the long-standing time-operator problem. Using the inverse Wigner-Weyl integral transformation, we directly construct the density matrix from the structure of hybrid distributions. This mathematically rigorous framework demonstrates that natural irreversibility and the thermodynamic arrow of time are intrinsic properties of hybrid flow gradients, completely redefining the phenomenon of decoherence and eliminating the need for external open-system master equations.
We present a highly condensed application of our analytical method, which integrates quantum probabilities within a common structure of contextual projection. By establishing a strictly positive hybrid-statistical distribution and utilizing abstract dual variables, this approach bridges quantum mechanics with the formalisms of uncertainty analysis and signal processing, without giving rise to negative probability paradoxes. We show how fundamental harmonic analysis theorems (Hardy and Beckner) acquire concrete physical interpretations, establishing minimal energy thresholds, information conservation boundaries, and an intrinsic thermodynamic arrow of time. Finally, we map the utility of this framework to quantum computing diagnostics and quantum optics, supported by our previously published applications in phase space quantum kinetics.
Standard formulations of quantum mechanics successfully describe physical systems through isolated representations, most notably in position or momentum spaces. Attempts to unify these descriptions into a single joint phase space have historically faced severe mathematical dilemmas, such as the appearance of negative values in the Wigner quasiprobability distribution. This paper addresses this challenge by extending the recently developed Analytic Probability Framework and the Primacy of the Whole Principle into a geometric domain. We demonstrate that the separate, valid representations of standard quantum mechanics can be consistently unified into a joint hybrid representation without relying on classical phase space assumptions. By defining a strictly positive and normalized joint probability distribution, we construct a genuine quantum phase space where individual values of position and momentum do not exist as autonomous points, but rather function as interconnected structural elements of a single, indivisible quantum state. This approach provides a rigorous, non-paradoxical geometric foundation that naturally preserves quantum logic while extending the standard formalism.
This paper presents the definitive thermodynamic formulation of the recently established unified quantum phase space framework. By operating within a strictly positive joint distribution that unifies coordinate and momentum representations without operator expansions, we introduce a novel formulation of quantum entropy based on the classical Shannon-Boltzmann approach. We demonstrate that for any pure or mixed state, the total phase-space entropy decouples into the linear sum of structural spatial and momentum entropies. Utilizing the recently derived equation of motion for hybrid flow, we expose a purely internal, structural mechanism for quantum irreversibility and the arrow of time, arising directly from Fourier duality and phase-space lattice dynamics rather than external environmental coupling. The dynamic behavior of entropy production is analyzed. We provide compact, rigorous quantitative validations by demonstrating how free wave-packet dispersion represents natural decoherence and how the maximization of hybrid entropy naturally yields the classical-quantum Boltzmann-Gibbs distribution. These results establish a stable bridge between quantum mechanics, information theory, and macroscopic thermodynamics.
This paper extends the recently proposed analytical framework of quantum phase space, where coordinate and momentum representations are unified into a strictly positive joint distribution. Transitioning from stationary geometric structures to non-stationary processes, we introduce the concept of time-dependent hybrid flows. We derive the fundamental equation of motion governing the temporal evolution of these flows in quantum kinetics. Unlike traditional phase-space approaches (such as the Wigner-Liouville equation) which suffer from negative probability densities and highly non-local differential terms, the hybrid flow equation operates as a local balance between real, experimentally observable coordinate and momentum quantum probability currents. The mathematical properties of the equation, including strict positivity, total probability conservation, and spatial-momentum orthogonality, are analyzed. Finally, we discuss the engineering and physical domains where this formulation provides crucial computational advantages, particularly in nano-barrier charge transport and quantum optics.
A concise derivation of all uncertainty relations is given entirely within the context of phase-space quantization, without recourse to operator methods, to the direct use of Weyl’s correspondence, or to marginal distributions of x and p. Phase-space quantization is the third autonomous and logically complete formulation of quantum mechanics beyond the conventional ones based on operators in Hilbert space or path integrals [1, 2, 3]. It is free of operators and wavefunctions: observables and matrix elements are computed through phasespace integrals of c-number functions (“classical kernels”) weighted by a Wigner function (WF) [4, 3]. This is a phase-space distribution function which is not positive semi-definite, and constitutes the Weyl correspondent [5] of the density matrix in the conventional formulation, fmn(x,p) = 1 2π dye −iyp ψ ∗ m(x − � 2 y)ψn(x + � 2 y) = f ∗ nm(x,p). (1)
We discuss how the language of wave functions (state vectors) and associated non-commuting Hermitian operators naturally emerges from classical mechanics by applying the inverse Wigner-Weyl transform to the phase space probability distribution and observables. In this language, the Schr"odinger equation follows from the Liouville equation, with \hbar ℏ now a free parameter. Classical stationary distributions can be represented as sums over stationary states with discrete (quantized) energies, where these states directly correspond to quantum eigenstates. Interestingly, it is now classical mechanics which allows for apparent negative probabilities to occupy eigenstates, dual to the negative probabilities in Wigner’s quasiprobability distribution. These negative probabilities are shown to disappear when allowing sufficient uncertainty in the classical distributions. We show that this correspondence is particularly pronounced for canonical Gibbs ensembles, where classical eigenstates satisfy an integral eigenvalue equation that reduces to the Schr"odinger equation in a saddle-point approximation controlled by the inverse temperature. We illustrate this correspondence by showing that some paradigmatic examples such as tunneling, band structures, Berry phases, Landau levels, level statistics and quantum eigenstates in chaotic potentials can be reproduced to a surprising precision from a classical Gibbs ensemble, without any reference to quantum mechanics and with all parameters (including \hbar ℏ ) on the order of unity.