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arXiv:quant-ph/0412015

Relationships Between Quantum and Classical Mechanics using the Representation Theory of the Heisenberg Group

Alastair Brodlie

Published Dec 2, 2004• 2 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0412015

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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 ppp-mechanics which is a consistent physical theory capable of describing both classical and quantum mechanics simultaneously. ppp-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 ppp-mechanics. ppp-Mechanical states come in two forms: elements of a Hilbert space and integration kernels. In developing ppp-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 ppp-mechanics we simplify some of the current quantum mechanical calculations. We also analyse the role of both linear and non-linear canonical transformations in ppp-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 ppp-mechanical context. In analysing this problem we show some limitations of the current ppp-mechanical approach. We then use Klauder's coherent states to generate a Hilbert space which is particularly useful for the Kepler/Coulomb problem.
ppp-Mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how non-linear canonical transformations affect ppp-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of ppp-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations.
p-mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how nonlinear canonical transformations affect p-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of p-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations.
This is an up-to-date survey of the p-mechanical construction (see funct-an/9405002, quant-ph/9610016, math-ph/0007030, quant-ph/0212101, quant-ph/0303142), which is a consistent physical theory suitable for a simultaneous description of classical and quantum mechanics. Observables in p-mechanics are defined to be convolution operators on the Heisenberg group H^n. Under irreducible representations of H^n the p-observables generate corresponding observables in classical and quantum mechanics. p-States are defined as positive linear functionals on p-observables. It is shown that both states and observables can be realised as certain sets of functions/distributions on the Heisenberg group. The dynamical equations for both p-observables and p-states are provided. The construction is illustrated by the forced and unforced harmonic oscillators. Connections with the contextual interpretation of quantum mechanics are discussed. Keywords: Classical mechanics, quantum mechanics, Moyal brackets, Poisson brackets, commutator, Heisenberg group, orbit method, deformation quantisation, symplectic group, representation theory, metaplectic representation, Berezin quantisation, Weyl quantisation, Segal--Bargmann--Fock space, coherent states, wavelet transform, Liouville equation, contextual interpretation, interaction picture, forced harmonic oscillator.
Constructing a classical mechanical system associated with a given quantum mechanical one, entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum mechanical system naturally has the structure of an infinite dimensional symplectic manifold (`quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straight forwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and recovers the linear classical phase space mathbbRmathrm2N\\mathbb{R}^{\\mathrm{2N}}mathbbRmathrm2N. We report on how the procedure also allows extraction of non-linear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin-j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (Bohr quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality. There is also an alternative and a priori distinct notion of effective Hamiltonian used in the context of a WKB approximation. These two notions are compared for the example of loop quantum cosmology.
While groups of point transformations such as rotations, translations and permutations have been extensively used in quantum mechanics, we cannot say the same of the more general groups of canonical transformations. To begin with, for the latter groups we have to determine their unitary representation in appropriate spaces. But once this is achieved canonical transformations can play as important a role in quantum mechanics as point transformations. We illustrate this in relation with the harmonic oscillator in one and many dimensions and for one and many particles. We show how radial matrix elements with respect to harmonic oscillator states can be determined group theoretically both in the Schroedinger and Heisenberg pictures. We discuss the application of canonical transformations to the three body nuclear problem. We briefly outline in the concluding section applications to Coulomb problems, many body systems, accidental degeneracy, and generalized integral transforms.