368 research outputs found
Relativistic Quantization and Improved Equation for a Free Relativistic Particle
Usually the only difference between relativistic quantization and standard
one is that the Lagrangian of the system under consideration should be Lorentz
invariant. The standard approaches are logically incomplete and produce
solutions with unpleasant properties: negative-energy, superluminal propagation
etc. We propose a two-projections scheme of (special) relativistic
quantization. The first projection defines the quantization procedure (e.g. the
Berezin-Toeplitz quantization). The second projection defines a casual
structure of the relativistic system (e.g. the operator of multiplication by
the characteristic function of the future cone). The two-projections
quantization introduces in a natural way the existence of three types of
relativistic particles (with , , and spins). Keywords:
Quantization, relativity, spin, Dirac equation, Klein-Gordon equation,
electron, Segal-Bargmann space, Berezin-Toeplitz quantization. AMSMSC Primary:
81P10, 83A05; Secondary: 81R30, 81S99, 81V45Comment: 22 p., LaTeX2e, a hard copy or uuencoded DVI-file by e-mail may be
obtained from the Autho
Erlangen Programme at Large 3.1: Hypercomplex Representations of the Heisenberg Group and Mechanics
In the spirit of geometric quantisation we consider representations of the
Heisenberg(--Weyl) group induced by hypercomplex characters of its centre. This
allows to gather under the same framework, called p-mechanics, the three
principal cases: quantum mechanics (elliptic character), hyperbolic mechanics
and classical mechanics (parabolic character). In each case we recover the
corresponding dynamic equation as well as rules for addition of probabilities.
Notably, we are able to obtain whole classical mechanics without any kind of
semiclassical limit h->0.
Keywords: Heisenberg group, Kirillov's method of orbits, geometric
quantisation, quantum mechanics, classical mechanics, Planck constant, dual
numbers, double numbers, hypercomplex, jet spaces, hyperbolic mechanics,
interference, Segal--Bargmann representation, Schroedinger representation,
dynamics equation, harmonic and unharmonic oscillator, contextual probabilityComment: AMSLaTeX, 17 pages, 4 EPS pictures in two figures; v2, v3, v4, v5,
v6: numerous small improvement
A Constructive Method for Approximate Solution to Scalar Wiener-Hopf Equations
This paper presents a novel method of approximating the scalar Wiener-Hopf
equation; and therefore constructing an approximate solution. The advantages of
this method over the existing methods are reliability and explicit error
bounds. Additionally the degrees of the polynomials in the rational
approximation are considerably smaller than in other approaches.
The need for a numerical solution is motivated by difficulties in computation
of the exact solution. The approximation developed in this paper is with a view
of generalisation to matrix Wiener-Hopf for which the exact solution, in
general, is not known.
The first part of the paper develops error bounds in Lp for 1<p<\infty. These
indicate how accurately the solution is approximated in terms of how accurate
the equation is approximated.
The second part of the paper describes the approach of approximately solving
the Wiener-Hopf equation that employs the Rational Caratheodory-Fejer
Approximation. The method is adapted by constructing a mapping of the real line
to the unit interval. Numerical examples to demonstrate the use of the proposed
technique are included (performed on Chebfun), yielding error as small as
10^{-12} on the whole real line.Comment: AMS-LaTeX, 19 pages, 10 figures in EPS fil
Wavelets in Banach Spaces
We describe a construction of wavelets (coherent states) in Banach spaces
generated by ``admissible'' group representations. Our main targets are
applications in pure mathematics while connections with quantum mechanics are
mentioned. As an example we consider operator valued Segal-Bargmann type spaces
and the Weyl functional calculus.
Keywords: Wavelets, coherent states, Banach spaces, group representations,
covariant, contravariant (Wick) symbols, Heisenberg group, Segal-Bargmann
spaces, Weyl functional calculus (quantization), second quantization, bosonic
field.Comment: 37 pages; LaTeX2e; no pictures; 27/07/99: many small correction
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