The Hilbert spaces for stable and unstable particles

TL;DR

This paper explores the mathematical structure of Hilbert spaces for both stable and unstable particles, highlighting differences in their representation theory and the role of reflection groups in unstable states.

Contribution

It introduces a framework for describing unstable particle states using positive type functions and affine groups, extending the traditional Hilbert space formalism.

Findings

Hilbert spaces for stable particles are based on square integrable functions on momentum spheres and hyperboloids.

Unstable particles require positive type functions for their scalar products, involving cyclic and reducible translation representations.

Reflection groups are essential in characterizing unstable structures within the Hilbert space framework.

Abstract

The Hilbert spaces for stable scattering states and particles are determined by the representations of the characterizing Euclidean and Poincar\'e group and given, respectively, by the square integrable functions on the momentum 2-spheres for a fixed absolute value of momentum and on the energy-momentum 3-hyperboloids for a particle mass. The Hilbert spaces for the corresponding unstable states and particles are not characterized by square integrable functions Their scalar products are defined by positive type functions for the cyclic representations of the time, space and spacetime translations involved. Those cyclic, but reducible translation representations are irreducible as representations of the corresponding affine operation groups which involve also the time, space and spacetime reflection group, characteristic for unstable structures.