Classical and Quantum Phase Space Mechanics

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 and Phys. Rev. A 96, 022117] 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. Firstly 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 Weyl representation in classical and quantum mechanics

Stochastic quantum mechanics is a quantum theory in which the basic limitations of real-world measuring instruments, due to their intrinsically quantum nature, are taken into account. Among other things this leads to a new operational definition of space-time, called quantum space-time. Fundamental to this approach is the formulation of quantum mechanics over phase space rather than just over position or momentum space. A concept of extended particle is a natural outgrowth of this development. Gauge and internal symmetry have a natural place within the theory, and preliminary computations combining some old ideas due to Born with more recent ideas on symmetry breaking suggest that the theory could lead to a mass formula compatible with known data on the low-lying baryons.

A suitable unified statistical formulation of quantum and classical mechanics in a *-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observer's information regarding a physical system. This is seen as the main difference from classical mechanics, where an observer's information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description of noncommutative information, with the projection postulate being a noncommutative generalization of conditional probability. This view clarifies many problems surrounding the interpretation of quantum mechanics, particularly problems relating to the measuring process.

A wave dynamics of fields φ(p, q; t) ∈ L2(Γ) over the phase space Γ(p, q) of a classical system 𝒮 is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time-independent Schrödinger equation. This allows us to associate with 𝒮 an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of 𝒮 and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system 𝒮 and the usual quantum theory of the related quantum system Q.

We construct the novel Clifford-Yang algebra which is an extension of the Yang algebra in noncommutative phase spaces. The Clifford-Yang algebra allows us to write down the commutators of the [Formula: see text] polyvector-valued coordinates and momenta which are compatible with the Jacobi identities, the Weyl–Heisenberg algebra, and paves the way for a formulation of quantum mechanics (QM) in noncommutative Clifford spaces. We continue with a detailed study of the isotropic 3D quantum oscillator in noncommutative spaces and find the energy eigenvalues and eigenfunctions. These findings differ considerably from the ordinary quantum oscillator in commutative spaces. We find that QM in noncommutative spaces leads to very different solutions, eigenvalues, and uncertainty relations than ordinary QM in commutative spaces. The generalization of QM to noncommutative Clifford (phase) spaces is attained via the Clifford-Yang algebra. The operators are now given by the generalized angular momentum operators involving polyvector coordinates and momenta. The eigenfunctions (wave functions) are now more complicated functions of the polyvector coordinates. We conclude with some important remarks.

ABSTRACTClassical dynamics is generally considered as the prototype of a deterministic theory; the equations of motion determine the coordinates q and the momenta p at any time provided they are given at an initial instant. It is pointed out that this is an unrealistic assumption, for it is inevitable that there should be small uncertainties Δq, Δp; a point of the mathematical continuum has no physical significance. The uncertainties can be taken into account without violating the deterministic equations by introducing a probability density P in phase space p, q. The function P satisfies the partial differential equation which expresses Liouville's theorem. Thus it can be shown that an initial uncertainty spreads out in phase space, so that finally all states of the system consistent with the mean initial values of the constants of the motion are equally probable. This property holds for one degree of freedom just as well as for many, and is not a consequence of our ignorance concerning large numbers of particles. The essential difference between classical mechanics and quantum mechanics consists not in the physical necessity of a statistical interpretation, but in the further introduction of a probability amplitude, the square of which is the probability density; thisimplies the restriction of the initial uncertainties as expressed by the uncertainty laws, and the phenomenon of interference of probabilities which makes necessary a revision of our concepts of physical reality. It is remarkable that the product of the uncertainties of a properly chosen pair of variables is constant also in classical mechanics, although of course its value is not a universal constant as in the quantum case.

All papers have been peer reviewed. This was the 4th conference arranged by ICMM on probabilistic foundations of classical and quantum physics. The first three conferences took place in 2000, 2002, and 2004. Some closely related conferences are Bohmian Mechanics 2000 and Quantum Theory: Reconsideration of Foundations 2001, 2003, and 2005. The main aim of these conferences is to understand the role that probability plays in the foundations of physics, theoretical as well as experimental, classical as well as quantum. In this conference, as well as during our previous conferences, we are glad to welcome a fruitful assembly of theoretical physicists, experimenters, mathematicians, and even philosophers interested in the foundations of probability and physics. Among important topics discussed during the conference were the probabilistic foundations of quantum mechanics, as well as the foundations of probability itself, the formation theory, quantum computing, quantum cryptography, quantum teleportation, quantum fluctuations in relation with stochastic electrodynamics, Bohmian mechanics, measurement theory, completeness and incompleteness of quantum mechanics, macroscopic quantum systems, experiments on quantum nonlocality and locality, Bell's inequality, entanglement; philosophical problems raised by quantum mechanics, and mathematical formalism. A special session devoted to the Bayesain approach to classical and quantum probability was organized.

Summary form only given, as follows. Sheet electron beams have the potential to make possible higher power-sources of microwave radiation due to their ability to transport high currents, at reduced current densities, through a single narrow RF interaction structure. Previous investigations have indicated the feasibility for laboratory formation of an elliptical sheet beam using magnetic quadrupoles and a Pierce gun pencil-beam source. The configuration exhibits several unique physical features of phase space evolution not observed in more conventional, paraxial beam transport systems. Previous numerical simulations using MAGIC3-D indicate the r.m.s. phase space volume occupied by the sheetbeam increased longitudinally, which contradicts the principle of phase space conservation for a conservative system (Liouville's theorem). The explanation is that is Liouville's theorem does not strictly apply to r.m.s. emittance calculations and some alternative phase space measure might provide closer agreement with Liouville's theorem. The developed phase space measure is non-statistical and uses principles from mathematical morphology. We discuss and compare the results of numerical simulations using the TRACE3-D and MAGIC3-D codes. In addition, we discuss the morphological algorithm for computing phase space area and compare its results with that of the r.m.s. formalism for calculating phase space area.

Geometrical origin of the [formula omitted]-product in the Fedosov formalism

Time, classical and quantum

The known methods, due for instance to G.W. Mackey and T.F. Jordan, which exploit the transformation properties with respect to the Euclidean and Galileian group to determine the formalism of the Quantum Theory of a localizable particle, fail in the case that the considered transformations are not symmetries of the physical system. In the present work we show that the formalism of standard Quantum Mechanics for a particle without spin can be completely recovered by exploiting the covariance properties with respect to the group of Euclidean transformations, without requiring that these transformations are symmetries of the physical system.

If quantum mechanics were to be applicable to macroscopic objects, classical mechanics would have to be a limiting case of quantum mechanics. Then the category Set that packages classical mechanics has to be in some sense a ‘limiting case’ of the category Hilb packaging quantum mechanics. Following from this assumption, quantum–classical correspondence can be considered as a mapping of the category Hilb to the category Set, i.e., a functor from Hilb to Set, taking place in the macroscopic limit. As a procedure, which takes us from an object of the category Hilb (i.e., a Hilbert space) in the macroscopic limit to an object of the category Set (i.e., a set of values that describe the configuration of a system), this functor must take a finite number of steps in order to make the equivalence of Hilb and Set verifiable. However, as it is shown in the present paper, such a constructivist requirement cannot be met in at least one case of an Ising model of a spin glass. This could mean that it is impossible to demonstrate the emergence of classicality totally from the formalism of standard quantum mechanics.

A review of the new formulation of quantum mechanics in which the tomographic probability describes the quantum state instead of the wave function is presented. The tomographic probability in classical statistical mechanics is considered. The superposition principle is discussed.

We examine the relationship between the mathematical structures of classical mechanics, quantum mechanics, and special relativity, with a view toward building a consistent framework for all three. The usual idea of ’’canonical quantization,’’ with its emphasis on the transition from functions over classical phase space to operators, appears to be inconsistent with relativistic covariance. On the other hand, the spectral condition of relativistic quantum mechanics, which is merely a covariant statement that the energy is nonnegative, naturally leads to the construction of a covariant extension of classical phase space. By giving up the idea of sharp ’’localizability’’ in space—known to be at odds with covariance— and adopting instead a notion of ’’soft’’ localizability in phase space, a consistent theory of relativistic quantum mechanics is seen to emerge whose structure naturally incorporates the classical sympletic geometry as well. Furthermore, the new theory deals directly and covariantly with extended particles rather than point particles and is free of the various inconsistencies known to plague the usual theory. The new notion of ’’microlocality’’ in phase space leads to a new form of gauge freedom which is similar to the usual one but simpler and more powerful, using methods of complex analysis. The phase-space version of Yang–Mills theory is worked out.