Invariants and classification of gauge fields

The Lorentz gauge, and Lorentz-gauge invariants of gauge fields at a space-time point are studied systematically and several classification schemes for gauge fields are presented. A classification of an arbitrary gauge field is obtained by use of the spinor method and the geometric optics limit. Remarks are also made on the peeling theorem for the SU(2) gauge field. There are 9 functionally independent Lorentz-gauge invariants, but a polynomial basis consists of 10 and not lower than 10 invariants. For the self-dual or anti-self-dual SU(2) gauge field, however, a polynomial basis consists of 10 and not lower than 10 invariants. For the self-dual or anti-self-dual SU(2) gauge field, however, a polynomial basis consists of 3 independent invariants. A detailed classification of the usual SU(2) gauge field as well as the self-dual or anti-self-dual fields is obtained. It is pointed out that these classifications are also valid for Yang-Mills fields on a four-dimensional Euclidean space, and the type to which the instanton solution with winding number 1 belongs is identified