Construction of a Complete Set of States in Relativistic Scattering Theory

TL;DR

This paper rigorously constructs a complete set of states in relativistic scattering theory using a Gel'fand triple, extending Dirac's formalism to unbounded operators and continuous spectra, with applications to symmetry theorems.

Contribution

It provides an explicit construction of the Gel'fand triple and a complete set of momentum eigenstates in relativistic scattering theory, generalizing Dirac's formalism.

Findings

Constructed a Gel'fand triple for relativistic states

Explicitly built a complete set of momentum eigenstates

Enabled proof of a generalized Coleman-Mandula theorem

Abstract

The space of physical states in relativistic scattering theory is constructed, using a rigorous version of the Dirac formalism, where the Hilbert space structure is extended to a Gel'fand triple. This extension enables the construction of ``a complete set of states'', the basic concept of the original Dirac formalism, also in the cases of unbounded operators and continuous spectra. We construct explicitly the Gel'fand triple and a complete set of ``plane waves'' -- momentum eigenstates -- using the group of space-time symmetries. This construction is used (in a separate article) to prove a generalization of the Coleman-Mandula theorem to higher dimension.