The basic physical Lie operations

TL;DR

This paper formulates quantum theory using operations derived from real Lie groups, classifying their structures and representations, and relating them to physical transformations and quantum states.

Contribution

It provides a comprehensive classification of Lie group operations relevant to physics and their Hilbert space representations, linking group theory with quantum mechanics.

Findings

Classified all relevant Lie groups and their contractions.

Connected group representations to quantum states via energy-momentum functions.

Expressed representation matrix elements as residues of energy-momentum poles.

Abstract

Quantum theory can be formulated as a theory of operations, more specific, of complex represented operations from real Lie groups. Hilbert space eigenvectors of acting Lie operations are used as states or particles. The simplest simple Lie groups have three dimensions. These groups together with their contractions and subgroups contain - in the simplest form - all physically important operations which come as translations for causal time, for space and for spacetime, as rotations, Lorentz transformations and as Euclidean and Poincare transformations with scattering and particle states and also - via the Heisenberg groups - as the operational structure of nonrelativistic quantum mechanics. The classification of all those groups and their contractions is given together with their Hilbert spaces, constituted by energy-momentum functions. The groups representation matrix elements can be written as residues of energy-momentum poles - simple poles for abelian translations, e.g. in Feynman propagators, and dipoles for simple Lie group operations, e.g. in the Schroedinger wave functions for the nonrelativistic hydrogen atom.