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Physical Review A

Quasiprobabilities and explicitly covariant relativistic quantum theory

Sebastiano Sonego

Published Nov 1, 1991• 8 citations
Publisher

Abstract

The concepts and formalism of quasiprobabilities in the space of paths are extended to the relativistic domain. The definition of a relativistic wave function, needed in the construction of the quasiprobability functional, is discussed extensively, together with the formulation and the physical interpretation of an explicitly covariant quantum theory. The Wigner and the Margenau-Hill relativistic phase-space distribution functions are then obtained according to the standard procedures of the formalism.

DOI
10.1103/physreva.44.5369

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The concepts and formalism of quasiprobabilities in the space of paths are extended to the relativistic domain. The definition of a relativistic wave function, needed in the construction of the quasiprobability functional, is discussed extensively, together with the formulation and the physical interpretation of an explicitly covariant quantum theory. The Wigner and the Margenau-Hill relativistic phase-space distribution functions are then obtained according to the standard procedures of the formalism.
The covariant formulation of stochastic thermodynamics requires treating the stochastic work as a 4-vector, posing significant challenges for quantum systems due to the non-commutativity. We introduce a new quasiprobability distribution for the work 4-vector, which combines the Wigner and Margenau-Hill quasiprobabilities. This extends the covariant fluctuation theorems from classical to quantum regime. We illustrate our findings with a scalar field driven by classical particles with a generalized version of trace formula. Our work establishes a quasiprobability approach to studying relativistic quantum thermodynamics in a covariant way.
Problems with the extension to relativity. The Klein–Gordon equation The attempt to extend the causal interpretation to cover relativistic quantum mechanics raises several problems of technique and principle. These relate not only to difficulties peculiar to the quantum theory of motion but have to do with still unresolved issues in the underlying quantum mechanical formalism (such as the appearance of infinities in the computation of physical quantities and the lack of a relativistic theory of measurement). Some problems are common to both, such as the difficulty of defining positive densities. The subject has not been extensively investigated and in this chapter we shall discuss only some of the relevant issues. There are two particularly important, overlapping problems: Can the trajectory concept be retained in the relativistic quantum domain? And: how do we interpret relativistic quantum fields? The unresolved issue underlying both questions is the extent to which we should require a quantum theory to be ‘relativistic’. It is known that Lorentz covariant wave equations can have consequences in conflict with relativity. This, as we shall see, is indeed the case with the current relativistic formalism as interpreted, or completed, by the quantum theory of motion. Especially in field theory, the individual processes that build up and explain the covariant statistical predictions of the formalism exhibit features that are in conflict with our entrenched ideas regarding covariance and locality as necessary concomitants of relativity theory. That is, relativity is statistically valid but the individual events do not have an intrinsically relativistic character.
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.
This paper discusses the relationship between the Wigner function, along with other related quasiprobability distribution functions, and the probability density distribution function constructed from the wave function of the Schr\"odinger equation in quantum phase space, as formulated by Torres-Vega and Frederick (TF). At the same time, a general approach in solving the wave function of the Schr\"odinger equation of TF quantum phase space theory is proposed. The relationship of the wave functions between the TF quantum phase space representation and the coordinate or momentum representation is thus revealed.
The Wigner function plays a central role in QCD as a phase-space object encoding correlations among quarks, antiquarks, and gluons, yet its interpretation remains subtle due to its quasiprobabilistic nature and possible negativity. Recent work based on the Koopman–von Neumann–Sudarshan (KvNS) Hilbert space formulation of classical mechanics suggests the Wigner function arises as a quantum probability amplitude projected onto classical phase space, rather than a quasiprobability density. In the classical limit, this amplitude reduces to the classical Koopman wavefunction. In this work, we extend this perspective to relativistic QCD by constructing a Koopman description of the quark Wigner operator. We show that the Wigner operator is naturally isomorphic to a phase-space spinor, providing a unified framework in which both classical and quantum dynamics are expressed. Within this formulation, the Wigner function retains its interpretation as an amplitude even in the relativistic regime. This viewpoint clarifies the origin of negativity and other nonclassical features, and provides a more transparent foundation for parton distribution functions in QCD. Remarkably, the relativistic Koopman framework reproduces the classical limit of QCD.