Quantum Electrodynamics on Noncommutative Spacetime

We propose a new method to quantize gauge theories formulated on a canonical noncommutative spacetime with fields and gauge transformations taken in the enveloping algebra. We show that the theory is renormalizable at one loop and compute the beta function and show that the spin dependent contribution to the anomalous magnetic moment of the fermion at one loop has the same value as in the commutative quantum electrodynamics case. ∗[email protected] 1 Gauge theories formulated on a canonical noncommutative spacetime have recently received lots of attention. A canonical noncommutatice spacetime is defined by the noncommutative algebra [x̂, x̂ ] = iθ (1) where μ and ν run from 0 to 3 and where θ is constant and antisymmetric. It has mass dimension minus two. Formulating Yang-Mills theories relevant to particle physics on such a spacetime requires to consider matter fields, gauge fields and gauge transformations in the enveloping algebra otherwise SU(N) gauge symmetries cannot be implemented. It has been pointed out that actions formulated on a canonical noncommutative spacetime are invariant under noncommutative Lorentz transformations [1] which preserve the algebra (1) and thus the minimal length implied by the relation (1). The enveloping algebra approach [2–5] allows to map a noncommutative action Ŝ on an effective action formulated on a regular commutative spacetime. The dimension four operators are the usual ones and the noncommutative nature of spacetime is encoded into higher order operators. In this paper we will be considering quantum electrodynamics on a noncommutative spacetime. The noncommutative action for a Dirac fermion coupled to a U(1) gauge field is given by