Erlangen Program at Large: An Overview

Abstract

ABSTRACT. This is an overview of Erlangen Programme at Large. Study of objects and properties, which are invariant under a group action, is very fruitful far beyond the traditional geometry. In this paper we demonstrate this on the example of the group SL 2 (R). Starting from the conformal geometry we develop analytic functions and apply these to functional calculus. Finally we link this to quantum mechanics and conclude by a list of open problems.

Symmetry, Geometry and Quantization with Hypercomplex Numbers

“…However, the transition to the phase space is more a custom rather than a necessity and in many cases we can efficiently work on the Heisenberg group itself. (1) For the harmonic oscillator in Example 4.2 the equation (109) again reduces to the form (68) with the solution given by (69). The adjoint equation of the harmonic oscillator on the phase space is not different from the quantum written in Example 4.6(1).…”

“…[104]. Further connections between analytic function theory and group representations can be found in [54,69].…”

“…Induced Wavelet (Covariant) Transform. The following object is common in quantum mechanics [59], signal processing, harmonic analysis [76], operator theory [73,75] and many other areas [69]. Therefore, it has various names [2]: coherent states, wavelets, matrix coefficients, etc.…”

“…Consider a four-dimensional algebra C spanned by 1, i, ε and iε. We can define the following representation ρ εh of the Heisenberg group in a space of C-valued smooth functions [69,71]:…”

Uncertainty and Analyticity

“…The following object is common in quantum mechanics [4], signal processing, harmonic analysis [8], operator theory [7,9] and many other areas [5]. Therefore, it has various names [1]: coherent states, wavelets, matrix coefficients, etc.…”