Fermion models on the lattice and in field theory

Abstract

The first part deals with lattice approach to field theories. The fermion doubling problems are described. This doubling can be removed if a dual lattice is introduced, as first pointed out by Stacey. His method is developed and in the process a formalism for the construction of a covariant difference lattice operator and thus of a gauge invariant action, is exhibited. It is shown how this formalism relates to the work of Wilson. Problems of gauge invariance can be traced back to the absence of the Leibnitz rule on the lattice. To circumvent this failure the usual notion of the product is replaced by a convolution. The solutions display a complementarity : the more localised the product the more extended is the approximation to the derivative and vice-versa. It is found that the form of the difference operator in the continuous limit dictates the formulation of the full two-dimensional supersymmetric algebra. The construction of the fields necessary to form the Wess-Zumino model follows from the requirement of anticommutativity of the supersymmetric charges. In the second part, the Skyrme model is reviewed and Bogomolnyi conditions are defined and discussed. It appears that while the Skyrme model has many satisfactory features, it fails to describe the interactions between nucleons correctly. These problems are brought out and the available solutions reviewed.