Gauge invariant frequency splitting in non Abelian lattice gauge theory

Lattice gauge theory, the problem of quark confinement and renormalization group (RG) methods have constituted a central theme of this school. As is well known, quark confinement, the existence of a mass gap, etc., can be proved in pure 4-dimensional Euclidean lattice gauge theory for a sufficiently large coupling constant. On the other hand, perturbative RG studies show asymptotic freedom at short distances. As K.G. Wilson has forcefully advocated over the years, the modern RG is essential to study the connection between the short and long distance behavior. Fortunately, the modern RG is slowly coming under mathematical control, and it is to be hoped that in the not too distant future these methods will become powerful enough for the control of the most challenging and fascinating of renormalizable field theories: non-abelian gauge field theory in four dimensions. The purpose of these notes is to initiate a line of thinking which may turn out to be not unfruitful for a modern RG study of pure non-abelian gauge fields in four dimensions. The essential contribution is to give a theory of gauge-invariant high-low frequency splitting in Wilson's lattice gauge theory