Involution requirement on a boundary makes massless fermions compactified on a finite flat disk mass protected

TL;DR

This paper investigates boundary conditions on a flat disk in higher-dimensional theories that ensure the existence of massless, chiral fermions with protected mass and specific charge, addressing key issues in Kaluza-Klein models.

Contribution

It introduces an involution boundary condition that guarantees massless, chiral fermions with a fixed charge in a simplified higher-dimensional model.

Findings

Massless fermions are protected by the involution boundary condition.

The model supports infinitely many massive fermions with the same charge.

The momentum operator is reformulated to be Hermitian under the boundary condition.

Abstract

The genuine Kaluza-Klein-like theories--with no fields in addition to gravity--have difficulties with the existence of massless spinors after the compactification of some space dimensions \cite{witten}. We proposed (Phys. Lett. B 633 (2006)771) such a boundary condition for spinors in 1+5 compactified on a flat disk that ensures masslessness of spinors in d=1+3 as well as their chiral coupling to the corresponding background gauge field (which solves equations of motion for a free field linear in the Riemann curvature). In this paper we study the same toy model: M^{(1+3)} x M^{(2)}, looking this time for an involution which transforms a space of solutions of Weyl equations in d=1+5 from the outside of the flat disk in x^5 and x^6 into its inside, allowing massless spinor of only one handedness--and accordingly assures mass protection--and of one charge--1/2--and infinitely many massive spinors of the same charge, chirally coupled to the corresponding background gauge field. We reformulate the operator of momentum so that it is Hermitean on the vector space of spinor states obeying the involution boundary condition.