We give a review of concepts related to connection of classical and quantum theories, from the phase space perspective. Quantum theory is described by non-commutative operators of coordinates and momenta, results in values having a certain probability, hence resembling the classical statistical theory in which coordinates and momenta are well measurable quantities. The search is for the classical object best describing the quantum state, while the physical observables of the quantum theories are operators.
We give a review of concepts related to connection of classical and quantum theories, from the phase space perspective. Quantum theory is described by non-commutative operators of coordinates and momenta, results in values having a certain probability, hence resembling the classical statistical theory in which coordinates and momenta are well measurable quantities. The search is for the classical object best describing the quantum state, while the physical observables of the quantum theories are operators.
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.
A discussion is given of the structure of a physical theory and an ``ideal form'' for such a theory is proposed. The essential feature is that all concepts should be defined in operational terms. Quantum (and classical) mechanics is then formulated in this way (the formulation being, however, restricted to the kinematical theory). This requires the introduction of the concept of a mixed test, related to a pure test (or ``question'') just as a mixed state is related to a pure state. In the new formulation, the primitive concepts are not states and observables but certain operationally accessible mixed states and tests called physical. The notion of a C*-system is introduced; each such system is characterized by a certain C*-algebra. The structure of a general C*-system is then studied, all concepts being defined in terms of physical states and tests. It is shown first how pure states and tests can be so defined. The quantum analog of the phase space of classical mechanics is then constructed and on it is built a mathematical structure, called a q-topology, which is a quantum analog of the topology of classical phase space. Mathematically, a q-topology is related to a noncommutative C*-algebra as an ordinary topology is related to a commutative C*-algebra. Some properties of the q-topology of a C*-system are given. An appendix contains some physically motivated examples illustrating the theory.
We show that quantum theory (QT) is a substructure of classical probabilistic physics. The central quantity of the classical theory is Hamilton’s function, which determines canonical equations, a corresponding flow, and a Liouville equation for a probability density. We extend this theory in two respects: (1) The same structure is defined for arbitrary observables. Thus, we have all of the above entities generated not only by Hamilton’s function but also by every observable. (2) We introduce for each observable a phase space function representing the classical action. This is a redundant quantity in a classical context but indispensable for the transition to QT. The basic equations of the resulting theory take a “quantum-like” form, which allows for a simple derivation of QT by means of a projection to configuration space reported previously [Quantum Stud Math Found 5:219–227, 2018]. We obtain the most important relations of QT, namely the form of operators, Schrödinger’s equation, eigenvalue equations, commutation relations, expectation values, and Born’s rule. Implications for the interpretation of QT are discussed, as well as an alternative projection method allowing for a derivation of spin.
We follow Kofler and Brukner (2007 Phys. Rev. Lett. 99 180403) in studying the conditions under which a classical picture emerges from the results of not too accurate quantum measurements made on large macroscopic objects. We show that for such objects, consisting of a large number of microscopic elements obeying quantum laws, the Central Limit theorem guarantees the existence of classical values for collective variables, even if the corresponding operators do not commute. Owing to localisation of the overall wave function in any chosen representation, these values can be measured to a small relative error without significantly altering the state of the object. We study a simple model, which includes a rudimentary observer capable of detecting in the coordinate space the position of a macroscopic pointer. The pointer can be employed to measure such quantities, not directly accessible to the observer, as linear or angular momenta. A purely classical picture arises provided the measurements are made on macroscopic objects. Results of measurements, made on small quantum objects, cannot be predicted with certainty, but acquire certain objectivity when encoded in macroscopic pointers’ positions accessible to all observers. Our estimates show that the classical conditions could, in principle, be realised for systems with number of constituent parts of the order of the Avogadro constant. It is possible that the approach captures the essential features of the quantum-to-classical behaviour, although its extension to more realistic systems is likely to be required.
We have seen that there is an intimate relationship between probability and information. The values we assign to probabilities depend on the information available, and information is a function of probabilities. This connection makes it inevitable that information will be an important concept in any statistical theory, including thermodynamics and, of course, quantum physics. The probabilistic interpretation of quantum theory has probability amplitudes rather than probabilities as the fundamental quantities. This feature, together with the associated superposition principle, is responsible for intrinsically quantum phenomena and gives quantum information theory its distinctive flavour. We shall see that the quantum rules for dynamical evolution and measurement, together with the existence of entangled states, have important implications for quantum information. They also make it possible to perform tasks which are either impractical or impossible within the classical domain. In describing these we shall make extensive use of simple but fundamental ideas in quantum theory. This chapter introduces the mathematical description of quantum physics and the concepts which will be employed in our study of quantum information.
The laws of classical mechanics and quantum theory do not distinguish between the past and the future. But a lot of phenomena tell us that the "arrow of time" points only in one direction. For example, in biology we discover evolutionary processes leading to diversification and increasing complexity; it is a typical feature of such evolutionary processes that "forward" in time is possible but "backward" in time is impossible. At what level can phenomena like these be described? There is another problem in connection with time: within usual quantum theory, space-coordinates and time are not symmetrical to each other, which is in contrast to the basic results of special theory of relativity. In usual quantum theory coordinates are operators and play the role of statistical quantities; time however does not behave statistically and remains a simple parameter when we go from classical to quantum mechanics. For this reason there is no uncertainty relation for the time and energy within usual quantum theory which would agree in its physical content with that for the coordinates and momenta. The purpose of this paper is to show, on the basis of new aspects, that both one-sidedness of time and its uncertainty are charateristics at the fundamental level of quantum theory. We have extended usual quantum theory on the basis of space-time arguments. In this connection a time-operator appeared and there is in general temporal asymmetry at the microscopic level. The picture might be important in the description of phenomena with an "arrow of time", i.e., processes which belong to the physics of becoming. Furthermore, we have derived an uncertainty relation for the energy and time which agrees in its physical content with the uncertainty relation for the momentum and the coordinate; this relation is different from that of usual quantum theory. Within the frame of this physical picture all the concepts (diversity, natural selection, mutation), which are presently used in connection with the evolution of biological systems, are already established at the quantum mechanical level and, therefore, our theoretical picture might also be the foundation for the explanation of the phenomena at the biological (macro-molecular) level.