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We present Wilson's model of gauge-field theory on a lattice, including a coupling to a matter field. The algebraic structure is surveyed for both commutative and noncommutative groups. Various regimes are suggested by mean-field theory according to the relative values of coupling constants. In particular the gauge field undergoes a first-order transition while the matter-field transition is of second order.
Balian et al. (Fri,) studied this question.