Lorentz-Invariant Non-Commutative QED

Abstract

Lorentz-invariant non-commutative QED (NCQED) is constructed such that it should be a part of Lorentz-invariant non-commutative standard model (NCSM), a subject to be treated in later publications. Our NCSM is based on Connes' observation that the total fermion field in the standard model may be regarded as a bi-module over a flavor-color algebra. In this paper, it is shown that there exist two massless gauge fields in NCQED which are interchanged by $C'$ transformation. Since $C'$ is reduced to the conventional charge conjugation $C$ in the commutative limit, the two gauge fields become identical to the photon field in the same limit, which couples to only four spinors with charges $\pm 2,\pm 1.$ Following Carlson-Carone-Zobin, our NCQED respects Lorentz invariance employing Doplicher-Fredenhagen-Roberts' algebra instead of the usual algebra with constant $\theta^{\mu\nu}$. In the new version $\theta^{\mu\nu}$ becomes an integration variable. We show using a simple NC scalar model that the $\theta$ integration gives an {\it invariant} damping factor instead of the oscillating one to the nonplanar self-energy diagram in the one-loop approximation. Seiberg-Witten map shows that the $\theta$ expansion of NCQED generates exotic but well-motivated derivative interactions beyond QED with allowed charges being only $0, \pm 1, \pm 2$.