Reduction of Dynamical Degrees of Freedom in the Large-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math>Gauge Theory

Abstract

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), $Z=\ensuremath{\Pi}\stackrel{}{\ensuremath{\mu}}[\ensuremath{\int} d {U}_{\ensuremath{\mu}}\mathrm{exp}({\ensuremath{\beta}}_{\ensuremath{\nu}\ensuremath{\ne}}\ensuremath{\Sigma}\stackrel{d}{\ensuremath{\mu}\ensuremath{\ne}\ensuremath{\nu}=1}\mathrm{tr}{U}_{\ensuremath{\mu}}{U}_{v}{{U}_{\ensuremath{\mu}}}^{\ifmmode\dagger\else\textdagger\fi{}}{{U}_{\ensuremath{\nu}}}^{\ifmmode\dagger\else\textdagger\fi{}})]$ Thus the field theory may be reduced to an integration over a finite number of matrices in large-$N$ limit.