3,028 research outputs found
p-Mechanics and Field Theory
The orbit method of Kirillov is used to derive the p-mechanical brackets
[math-ph/0007030, quant-ph/0212101]. They generate the quantum (Moyal) and
classic (Poisson) brackets on respective orbits corresponding to
representations of the Heisenberg group. The extension of p-mechanics to field
theory is made through the De Donder--Weyl Hamiltonian formulation. The
principal step is the substitution of the Heisenberg group with Galilean.
Keywords: Classic and quantum mechanics, Moyal brackets, Poisson brackets,
commutator, Heisenberg group, orbit method, deformation quantisation,
representation theory, De Donder--Weyl field theory, Galilean group, Clifford
algebra, conformal M\"obius transformation, Dirac operator.Comment: 12 pages (AMS-LaTeX); v2: some misprints are corrected; v3: many
minor corrections suggested by a refere
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
Cotangent bundle quantization: Entangling of metric and magnetic field
For manifolds of noncompact type endowed with an affine connection
(for example, the Levi-Civita connection) and a closed 2-form (magnetic field)
we define a Hilbert algebra structure in the space and
construct an irreducible representation of this algebra in . This
algebra is automatically extended to polynomial in momenta functions and
distributions. Under some natural conditions this algebra is unique. The
non-commutative product over is given by an explicit integral
formula. This product is exact (not formal) and is expressed in invariant
geometrical terms. Our analysis reveals this product has a front, which is
described in terms of geodesic triangles in . The quantization of
-functions induces a family of symplectic reflections in
and generates a magneto-geodesic connection on . This
symplectic connection entangles, on the phase space level, the original affine
structure on and the magnetic field. In the classical approximation,
the -part of the quantum product contains the Ricci curvature of
and a magneto-geodesic coupling tensor.Comment: Latex, 38 pages, 5 figures, minor correction
Lie groupoid C*-algebras and Weyl quantization
For any Lie groupoid , the vector bundle dual to the associated Lie
algebroid is canonically a Poisson manifold. The (reduced) C*-algebra of
(as defined by A. Connes) is shown to be a strict quantization (in the
sense of M. Rieffel) of . This is proved using a generalization of Weyl's
quantization prescription on flat space. Many other known strict quantizations
are a special case of this procedure; on a Riemannian manifold, one recovers
Connes' tangent groupoid as well as a recent generalization of Weyl's
prescription. When is the gauge groupoid of a principal bundle one is led
to the Weyl quantization of a particle moving in an external Yang-Mills field.
In case that is a Lie group (with Lie algebra ) one recovers Rieffel's
quantization of the Lie-Poisson structure on . A transformation group
C*-algebra defined by a smooth action of a Lie group on a manifold turns
out to be the quantization of the semidirect product Poisson manifold
defined by this action.Comment: 14 page
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