Reduction of Dynamical Degrees of Freedom in the Large-NGauge Theory

It is pointed out that the factorization of disconnected Wilson loop amplitudes implies a major reduction in the dynamical degrees of freedom in the large-$N$ limit of lattice gauge theory; the original model may be replaced by a much simpler one ($d$ is the space-time dimensionality), $Z=\Pi \stackrel{}{\mu }\left[\int d U_{\mu }\exp \right({\beta }_{\nu \ne }\Sigma \stackrel{d}{\mu \ne \nu =1\mathrm{}}\mathrm{tr}{U}_{\mu }{U}_v{U_{\mu }}^{†}{U_{\nu }}^{†}\left)\right]$ Thus the field theory may be reduced to an integration over a finite number of matrices in large-$N$ limit.