Erlangen Programme at Large: An Overview

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, Möbius transformations, functional calculus, spectrum, quantum mechanics, non-commutative geometry

77 pages, AMS-LaTeX, 12 figures (29 EPS graphic files); v2: section on QM was extended

References in corpus (10)

• Erlangen Programme at Large: An Overview
• A Quantum-Classical Brackets from p-Mechanics
• Operator Covariant Transform and Local Principle
• PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras
• Erlangen Program at Large-0: Starting with the group SL(2,R)
• Erlangen Programme at Large 3.1: Hypercomplex Representations of the Heisenberg Group and Mechanics
• Fock Spaces, Landau Operators and the Regular Solutions of time-harmonic Maxwell equations
• Non-Smooth Spatio-Temporal Coordinates in Nonlinear Dynamics
• Comment on `Do we have a consistent non-adiabatic quantum-classical mechanics?'
• Covariant Transform

Cited by in corpus (17)

• Erlangen Programme at Large: An Overview
• Erlangen Program at Large-2.5: Induced Representations and Hypercomplex Numbers
• Geometric Dynamics of a Harmonic Oscillator, Arbitrary Minimal Uncertainty States and the Smallest Step 3 Nilpotent Lie Group
• Erlangen Programme at Large 3.1: Hypercomplex Representations of the Heisenberg Group and Mechanics
• The Real and Complex Techniques in Harmonic Analysis from the Point of View of Covariant Transform
• Uncertainty and Analyticity
• Solving the Schrodinger Equation by Reduction to a First-order Differential Operator through a Coherent States Transform
• Cross-Toeplitz Operators on the Fock--Segal--Bargmann Spaces and Two-Sided Convolutions on the Heisenberg Group
• Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration
• Möbius group actions in the solvable chimera model
• Classical/Quantum=Commutative/Noncommutative?
• Calculus of Operators: Covariant Transform and Relative Convolutions
• Metamorphism as a covariant transform for the SSR group
• Distance problems for planar hypercomplex numbers
• Transmutations from the Covariant Transform on the Heisenberg Group and an Extended Umbral Principle
• Operator Projective Line and Its Transformations
• Quantum Holonomies and the Heisenberg Group