Erlangen Programme at Large: An Overview

TL;DR

This paper explores the Erlangen Programme's application to the group SL(2,R), developing invariant object analysis, functional calculus, and connections to quantum mechanics, highlighting its broad impact beyond traditional geometry.

Contribution

It demonstrates the extension of the Erlangen Programme to large groups like SL(2,R), linking geometry, analysis, and quantum mechanics in a unified framework.

Findings

Development of analytic functions from conformal geometry

Application of functional calculus to quantum mechanics

Identification of open problems in non-commutative geometry

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. Keywords: Special linear group, Hardy space, Clifford algebra, elliptic, parabolic, hyperbolic, complex numbers, dual numbers, double numbers, split-complex numbers, Cauchy-Riemann-Dirac operator, M\"obius transformations, functional calculus, spectrum, quantum mechanics, non-commutative geometry