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 $C^{\prime }$ transformation. Since $C^{\prime }$ 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 $0,\pm 1,\pm 2$.

LaTeX file, 27 pages

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