8 research outputs found
Relationships Between Quantum and Classical Mechanics using the Representation Theory of the Heisenberg Group
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 -mechanics which is a consistent physical theory capable
of describing both classical and quantum mechanics simultaneously.
-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 -mechanics.
-Mechanical states come in two forms: elements of a Hilbert space and
integration kernels. In developing -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 -mechanics we simplify some of the current quantum mechanical
calculations.
We also analyse the role of both linear and non-linear canonical
transformations in -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 -mechanical
context. In analysing this problem we show some limitations of the current
-mechanical approach. We then use Klauder's coherent states to generate a
Hilbert space which is particularly useful for the Kepler/Coulomb problem.Comment: PhD Thesis from 2004, 140 page
Non-Linear Canonical Transformations in Classical and Quantum Mechanics
-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 -mechanical observables and states. Using this we
show how canonical transformations change a quantum mechanical system. We seek
an operator on the set of -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.Comment: The paper has been improved in light of a referee's report. The paper
will appear in the Journal of Mathematical Physics. 24 pages, no figure
A Quantum-Classical Brackets from p-Mechanics
We provide an answer to the long standing problem of mixing quantum and
classical dynamics within a single formalism. The construction is based on
p-mechanical derivation (quant-ph/0212101, quant-ph/0304023) of quantum and
classical dynamics from the representation theory of the Heisenberg group. To
achieve a quantum-classical mixing we take the product of two copies of the
Heisenberg group which represent two different Planck's constants. In
comparison with earlier guesses our answer contains an extra term of analytical
nature, which was not obtained before in purely algebraic setup.
Keywords: Moyal brackets, Poisson brackets, commutator, Heisenberg group,
orbit method, representation theory, Planck's constant, quantum-classical
mixingComment: LaTeX, 7 pages (EPL style), no figures; v2: example of dynamics with
two different Planck's constants is added, minor corrections; v3: major
revion, a complete example of quantum-classic dynamics is given; v4: few
grammatic correction
Relationships between quantum and classical mechanics using the representation theory of the Heisenberg group
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