The similarity between classical wave mechanics and quantum mechanics (QM) played an important role in the development of QM (starting with works of De Broglie, Schrödinger, "late Einstein", Lamb, Lande, Mandel, Marshall, Santos, Boyer, and many others). We present a new wave-type approach, so called prequantum classical statistical field theory (PCSFT). PCSFT explores an analogy between some quantum phenomena and classical theory of random fields. Quantum systems are interpreted as symbolic representations of such fields (not only for photons, cf. Lande and Lamb, but even for massive particles). All quantum averages and correlations (including composite systems in entangled states) can be represented as averages and correlations for classical random fields. In this paper PCSFT is used to provide a classical signal representation of bunching and anti-bunching. At least the latter is typically considered as essentially quantum (nonclassical) phenomenon.
This thesis is concerned with the representation theory of the Heisenberg group and its applications to both classical and quantum mechanics. We continue the development of
p-mechanics which is a consistent physical theory capable of describing both classical and quantum mechanics simultaneously.
p-Mechanics starts from the observation that the one dimensional representations of the Heisenberg group play the same role in classical mechanics which the infinite dimensional representations play in quantum mechanics. In this thesis we introduce the idea of states to
p-mechanics.
p-Mechanical states come in two forms: elements of a Hilbert space and integration kernels. In developing
p-mechanical states we show that quantum probability amplitudes can be obtained using solely functions/distributions on the Heisenberg group. This theory is applied to the examples of the forced, harmonic and coupled oscillators. In doing so we show that both the quantum and classical dynamics of these systems can be derived from the same source. Also using
p-mechanics we simplify some of the current quantum mechanical calculations. We also analyse the role of both linear and non-linear canonical transformations in
p-mechanics. We enhance a method derived by Moshinsky for studying the passage of canonical transformations from classical to quantum mechanics. The Kepler/Coulomb problem is also examined in the
p-mechanical context. In analysing this problem we show some limitations of the current
p-mechanical approach. We then use Klauder's coherent states to generate a Hilbert space which is particularly useful for the Kepler/Coulomb problem.
The way Quantum Mechanics (QM) is introduced to people used to Classical Mechanics (CM) is by a complete change of the general methodology) despite QM historically stemming from CM as a means to explain experimental results. Therefore, it is desirable to build a bridge from CM to QM. This paper presents a generalization of CM to QM. It starts from the generalization of a point-like object and naturally arrives at the quantum state vector of quantum systems in the complex valued Hilbert space, its time evolution and quantum representation of a measurement apparatus of any size. Each time, when generalization is performed, there is a possibility to develop new theory giving up most simple generalizations. It is shown that a measurement apparatus is a special case of a general quantum object. An example of a measurement apparatus of an intermediate size is considered in the end.