84 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
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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