New Objects in Scattering Theory with Symmetries

TL;DR

This paper introduces new mathematical objects in 1D quantum scattering theory with symmetries, derived via homological algebra, which generalize traditional T- and K-matrices and satisfy quadratic relations.

Contribution

It develops a homological algebra framework to extend scattering objects, leading to new entities that unify and generalize known matrices in symmetric quantum systems.

Findings

New objects generalizing T- and K-matrices

Homological algebra yields quadratic relations among objects

Explicit SUSY QM example demonstrating nontrivial relations

Abstract

We consider 1D quantum scattering problem for a Hamiltonian with symmetries. We show that the proper treatment of symmetries in the spirit of homological algebra leads to new objects, generalizing the well known T- and K-matrices. Homological treatment implies that old objects and new ones are be combined in a differential. This differential arises from homotopy transfer of induced interaction and symmetries on solutions of free equations of motion. Therefore, old and new objects satisfy remarkable quadratic equations. We construct an explicit example in SUSY QM on $S^1$ demonstrating nontriviality of the above relation.