$CPT$-Symmetric K\"ahler-Dirac Fermions

TL;DR

This paper introduces a novel interpretation of K"ahler-Dirac fermions as living on a two-sheeted spacetime related by PT symmetry, resolving issues with negative norm states and simplifying fermion interactions in the standard model.

Contribution

It proposes a new framework where KD fermions reside on a two-sheeted spacetime with a reality condition, addressing quantization issues and connecting to CPT-symmetric universe models.

Findings

KD fermions live on a two-sheeted spacetime with PT symmetry.

The KD-Majorana condition relates particles on the two sheets.

Simplifies fermion terms in the standard model and relates to CPT symmetry.

Abstract

K\"ahler-Dirac (KD) spinors have generated excitement in the lattice gauge theory community, as a way to (i) deal with the ``fermion doubling" problems that plague ordinary (Dirac, Majorana, or Weyl) spinors when discretized on a lattice, and (ii) help explain the structure of the standard model. But if one naively quantizes this theory in Lorentzian signature, problems arise: half the KD fields have the ``wrong sign" Lagrangian, and give rise to negative norm states. Here we propose a new resolution/interpretation: the KD field actually lives on a two-sheeted spacetime, with the sheets related by $PT$ symmetry or, alternatively, by $i\leftrightarrow-i$. And, to avoid any unphysical interactions between the two sheets, the KD field obeys a reality condition (which we call the ``KD-Majorana condition"), which forces every particle on one sheet to be accompanied by a mirror (anti-)particle on the other sheet. We discuss how the standard model fits in this framework, how the fermion (kinetic and Yukawa) terms simplify, and how it may relate to the CPT-symmetric universe model.