We present a general method of constructing W and w gauge theories in terms of d+2 dimensional local fields. In this formulation the W gauge theory Lagrangians involve nonlocal interactions, but the w theories are entirely local. We discuss the so-called classical contraction procedure by which we derive the Lagrangian of w gauge theory from that of the corresponding W gauge theory. In order to discuss the relationship between quantum W and quantum w gauge theory we solve d=1 gauge theory models of a Higgs field exactly by using the collective field method. Based on this we conclude that the W gauge theory can be regarded as the large N limit of the corresponding SU(N) gauge theory once an appropriate coupling constant renormalization is made, while the w gauge theory cannot be.1997 Academic Press
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