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 $0$, $\frac{1}{2}$, and $1$ 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