Exact Solution of Noncommutative U(1) Gauge Theory in 4-Dimensions

Ydri, Badis (2004) Exact Solution of Noncommutative U(1) Gauge Theory in 4-Dimensions. (Preprint)

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Noncommutative U(1) gauge theory on the Moyal-Weyl space R^2 × R_θ^2 is regularized by approximating the noncommutative spatial slice R_θ^2 by a fuzzy sphere of matrix size L and radius R. Classically we observe that the field theory on the fuzzy space R^2 × S^2_L reduces to the field theory on the Moyal-Weyl plane R^2 × R_θ^2 in the flattening continuum planar limits R, L→∞ where the ratio θ^2 = (R^2)/(|L|^(2q)) is kept fixed with q > 3/2. The effective noncommutativity parameter is found to be given by (θ_eff)^2 ~ 2(θ^2)(L/2)^(2q-1) and thus it corresponds to a strongly noncommuting space. In the quantum theory it turns out that this prescription is also equivalent to a dimensional reduction of the model where the noncommutative U(1) gauge theory in 4 dimensions is shown to be equivalent in the large L limit to an ordinary O(M) non-linear sigma model in 2 dimensions where M~3L^2 The Moyal-Weyl model defined this way is also seen to be an ordinary renormalizable theory which can be solved exactly using the method of steepest descents. More precisely we find for a fixed renormalization scale μ and a fixed renormalized coupling constant g_r^2 an O(M)—symmetric mass, for the different components of the sigma field, which is non-zero for all values of g_r^2 and hence the O(M) symmetry is never broken in this solution. We obtain also an exact representation of the beta function of the theory which agrees with the known one-loop perturbative result.

Item Type: Article
Divisions: School of Theoretical Physics > Preprints
Date Deposited: 19 Jun 2018 13:58
Last Modified: 17 Dec 2022 10:09
URI: https://dair.dias.ie/id/eprint/632

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