Scattering in algebraic approach to quantum theory. Associative algebras

TL;DR

This paper develops an algebraic framework for quantum scattering theory using associative algebras with involution, deriving formulas for scattering matrices in terms of Green functions, including for quasi-particles and real number algebras.

Contribution

It introduces a novel algebraic approach to quantum scattering, expressing scattering matrices via Green functions within associative algebras over real numbers.

Findings

Derived LSZ formula for scattering matrix in algebraic framework

Expressed inclusive scattering matrix using generalized Green functions

Applicable to quasi-particles and elementary excitations

Abstract

The definitions of scattering matrix and inclusive scattering matrix in the framework of formulation of quantum field theory in terms of associative algebras with involution are presented. The scattering matrix is expressed in terms of Green functions on shell (LSZ formula) and the inclusive scattering matrix is expressed in terms of generalized Green functions on shell. The expression for inclusive scattering matrix can be used also for quasi-particles (for elementary excitations of any translation-invariant stationary state, for example, for elementary excitations of equilibrium state.) An interesting novelty is the consideration of associative algebras over real numbers.