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Phase space formulation of quantum mechanics and quantum cosmology

Przemysław Małkiewicz, Artur Miroszewski, Hervé Bergeron

Published Dec 13, 2017• 0 citations
OpenAlex

Abstract

We develop a phase space formulation of quantum mechanics based on unitary irreducible representations of classical phase spaces. We use a quantum action functional to derive the basic equations. In principle, our formulation is equivalent to the Hilbert space formulation. However, the former allows for consistent truncations to reduced phase spaces in which approximate quantum motions can be derived. We predict that our approach can be very useful in the domain of quantum cosmology and therefore, we use the cosmological phase spaces to establish the basic equations of the new formalism.

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We develop a phase space formulation of quantum mechanics based on unitary irreducible representations of classical phase spaces. We use a quantum action functional to derive the basic equations. In principle, our formulation is equivalent to the Hilbert space formulation. However, the former allows for consistent truncations to reduced phase spaces in which approximate quantum motions can be derived. We predict that our approach can be very useful in the domain of quantum cosmology and therefore, we use the cosmological phase spaces to establish the basic equations of the new formalism.
We develop an approach to quantum dynamics based on quantum phase space trajectories. The latter are built from a unitary irreducible representation of the symmetry group of the respective classical phase space. We use a quantum action functional to derive the basic equations. In principle, our formulation is equivalent to the Hilbert space formulation. However, the former allows for consistent truncations to reduced phase spaces in which approximate quantum dynamics can be derived. We believe that our approach can be very useful in the domain of quantum cosmology and therefore, we use the cosmological phase space example to establish the basic equations of this formalism.
The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.
We propose that the unobservability of absolute scale constitutes a foundational principle of physics. Starting from the requirement that physical predictions must be invariant under rescalings of the action, we show that only phase-based quantities remain well defined. This requirement uniquely selects a complex amplitude description, leading directly to the structure of quantum mechanics, including linear superposition, Hilbert space, and the Born rule. Within this framework, the classical notion of action loses absolute meaning, and physical observables arise from invariant phase relations. Quantum mechanics is therefore not postulated, but emerges as the minimal consistent description of systems in which scale cannot be directly observed. The formal structure of quantum theory is shown to follow from this principle without additional axioms. We further demonstrate that the same principle implies the existence of a dynamical scale field, which governs the effective quantum scale and connects naturally to cosmological models with a time- dependent Planck mass. In this way, the structure of quantum mechanics and the large-scale dynamics of the Universe are linked through a common origin. This framework provides a unified perspective in which quantum behavior and cosmological evolution arise from a single underlying principle, suggesting a new approach to the relationship between fundamental physics and cosmology.
Still today the discussion about the foundations, physical interpretation, and real scope of quantum mechanics has never ceased. It would be wrong to dismiss these issues as mere philosophical problems, because questions of consistency and interpretation are not devoid of practical utility. We present the foundations and main properties of a positive definite phase space quantum mechanics. A new quantization procedure is proposed as well. This new interpretation/formulation eliminates conceptual and technical difficulties from quantum mechanics: (i) many paradoxes typical of the wave-particle duality, EPR experiments, macroscopic superpositions, and collapse of wavefunctions disappear; (ii) the elimination of the wavefunctions from quantum theory is in line with the procedure inaugurated by Einstein with the elimination of the ether in the theory of electromagnetism; (iii) it is useful in considering the classical limit, can treat mixed states with ease, and brings certain conceptual issues to the fore; (iv) confirms the ensemble interpretation of the wavefunctions, derives its statistical interpretation, corrects the temporal dependence of the old wavefunctions, and considers pure classical states --localizable states-- beyond the Hilbert space; (v) the quantum equation of motion is of the Liouville kind and star-products are not needed, simplifying the formalism; and (vi) eliminates the hypothetical external quantum field of the pilot wave interpretation, solving its problems on the status of probability, and correcting well-known inconsistencies of the Bohm potential. Finally, we offer some perspectives on future developments and research in progress.
Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.