Gauge fixing and mass renormalization in the lattice gauge theory

The lattice gauge theory proposed by Wilson is discussed, Gauge fixing is defined for the lattice theory, and it is shown that gauge fixing is done in this theory solely for calculational purposes, The gauge-fixing method is used to study the mass renormalization of the gauge field quantum. An explicit calculation is done to lowest order which shows that there is no mass renormalization, This same result is proved to all orders in perturbation theory using the Slavnov identity. I (Submitted to Phys, Rev. D) *Work supported by the Energy Research and Development Administration. *In partial fulfillment of the requirements for the Ph.D. degree, Cornell University (197 6) 0 I. THE LATTICE GAUGE THEORY !Ghe lattice gauge theory has been introduced by Wilson’ to explain the dynamics of strongly interacting elementary particles. The non-Abelian gauge field has many well-known and remarkable properties. In particular, it is a nonlinear field which couples to itself (and, of course, to anything else which carries the requisite quantum number) O In this sense it is similar to the gravitational field. The gauge field also exhibits asymptotic freedom (that is, the strength of the coupling goes to zero for zero distance interaction); and, when coupled to the quark field, the coupled quark-gluon theory shows quark confinement in the strong coupling limit., The gauge field quantum is an elementary particle. For the case of strong interaction, this quantum is called the gluon. The quantum of the Abelian gauge field is the photon and its properties are fairly well understood, Wilsonl’ 2 has given an action functional formulation of quantum field theory using the Feynman path integral. In particular, the lattice gauge field is quantized on a discrete lattice embedded in a four-dimensional Euclidean spacetime. The reason for going to a lattice is twofold. Firstly, the lattice provides an ultraviolet cutoff, and hence there are no ultraviolet divergences in the theory. We will sometimes work with a finite size lattice, and this will provide an infrared cutoff, The problem of renormalization has to be solved to go to the continuum limit, i, e, , to let the lattice spacing go to zero. Secondly, using the lattice as a cutoff allows one to formulate the cutoff theory so that we have exact local gauge invariance for the lattice gauge field, Any other conventional way of defining the cutoff theory usually destroys local gauge invariance. Local gauge invariance is the single most important property of the gauge field, and the lattice gauge field is a more accurate representation of it than, say, would