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.
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.
We present a unified, autonomous formulation of quantum mechanics derived strictly from the foundational principles of Contextual Probability Theory. Instead of initiating the formalism via complex wave functions in Hilbert space, we postulate that the primary physical reality is a realvalued joint distribution 𝑃(𝑢, 𝑣, 𝑡)in a hybrid phase space. Here, the variable 𝑢 is explicitly defined as the structural property (index) characterizing a contextual whole (equivalence class), while 𝑣 represents the internal spectral alternative within that context. We mathematically demonstrate that the traditional Hilbert space architecture, complex amplitudes, and non-commuting operators are not primary postulates of nature, but rather emergent information-compensating artifacts that inevitably arise when the unified contextual flow is projected onto lower-dimensional subspaces.To demonstrate the universal validity of this mechanism, we deliberately restrict our current derivation to the foundational, one-dimensional case. Proving that the entire quantum apparatus emerges seamlessly within this prototype guarantees that the underlying principle can be directly generalized to higher dimensions and more complex contextual configurations.“
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.
This article provides a concise review of quantum phase space theory, beginning with its foundational principles and the properties of standard quantum quasi-probability distributions, specifically the Wigner, Husimi Q, and Glauber--Sudarshan P functions. We discuss the intrinsic limitations of these distributions, such as the appearance of negative values and phase-space blurring. A significant portion of this review highlights recent theoretical developments, particularly the quantum Wannier basis. This approach establishes a unitary mapping between the Hilbert space and a discretized phase space, yielding a genuine probability distribution in phase space and thereby providing a basis-dependent entropy for pure quantum states. Furthermore, we examine Bourgain's nonperiodic basis as a theoretical framework to circumvent the constraints imposed by the Balian--Low theorem. These developments provide practical tools for numerical studies based on the quantum Wannier basis, as well as conceptual benchmarks for understanding the localization limits of orthonormal phase-space representations.
Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 (2016) and Phys. Rev. A 96, 022117 (2017)] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function---the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics---extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. First we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space---putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of ``parity'' and ``displacement'' in a natural broadening of previously developed phase-space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.
The physical content of a theory is not intrinsically tied to any single mathematical formalism. Both classical and quantum mechanics admit equivalent representations, notably in phase space and in Hilbert space, related by the Wigner-Weyl correspondence. While this correspondence has long been studied in mathematical physics, its foundational and operational implications are often left implicit. Here we give a systematic account of what changes, and what does not, when classical and quantum theories are expressed in each other's native language. This representational viewpoint separates artifacts (such as the appearance of non-positivity or negativity under certain maps) from robust structural distinctions that persist across representations, in particular noncommutativity and its ℏ-dependent ⋆-deformation of the classical algebra. We develop the comparison at the level of states, kinematics, and dynamics, and extend it to measurement by formulating both outcome statistics and state-update rules within the same framework.
Phase space is the unity of position and momentum configuration space. It allows for an effective description of dynamical systems and is particularly useful when it comes to studying chaos theory and statistical mechanics. After the advent of quantum physics early in the 20th century, E. Wigner [91], J. E. Moyal [62] and H. J. Groenewold [31] introduce a quantum theory in phase space. Despite the apparent added complexity of the mathematics involved in this new framework, the underlying classical and quantum equations show many similarities. The probability distribution in classical physics becomes the Wigner distribution, a probability distribution usually featuring negative values. In 2013, O. Steuernagel and D. Kakofengitis, inspired by the work of H. Bauke [7] and E. Wigner [91], identified the quantum analogue of the classical phase space flow: the Wigner current J [83]. This Wigner current allows the visualisation of quantum dynamics through a quantum fluid dynamics perspective in phase space. This thesis is written by collection of five articles. They are prefaced by an introduction into the basics of quantum phase space theory and its link with both classical phase space dynamics and the standard Schrödinger approach, followed by the articles published during this PhD. Article 1 shows the importance of the integral form of the Wigner current. We use it to derive the Ehrenfest’s theorem, as well as to refute some propositions made within the community. Article 2 shows that, using the Wigner current, an Eulerian and Lagrangian point of view do not always give the same results for the quantum case. We demonstrate that the negativities of the Wigner distribution, sign of quantumness of the system, are created by the Wigner velocity field singularities. The Wigner velocity field is the quantum analogue of the classical phase space velocity field. In Article 3, we see that even though Wigner distributions of quantum systems feature spotty structures which saturate on scales ɑZ [97], the construction of a superoscillating Wigner distribution allows one to generate much smaller structures, of the order of ɑZ /α with α a positive constant potentially very large. In Article 4, we introduce the concept of quantum shear suppression in phase space. The Wigner current features an effective quantum “viscosity”, suppressing classical dynamics fine details. This viscosity is the mechanism by which the Zurek scale is enforced dynamically onto the state in phase space. In Article 5, we apply the previous ideas to Kerr-type oscillators. Its Wigner current is derived, and using it we show that its values are conserved on a ring during the time evolution of the Kerr oscillator. The shear suppression is also studied.