The application of renormalization group techniques to quarks and strings

Recent work by K. G. Wilson, A. A. Migdal and others has led to a statistical mechanical treatment of systems of interaction quarks and strings. This work is summarized here. The major topics discussed include boson and fermion variables in statistical mechanics; descriptions of local and gauge symmetries; exact solutions of one-dimensional problems with nearest-neighbor interactions; exact solutions of two dimensional problems with plaquette interactions; Wilson's model of quarks and strings; asymptotic freedom and trapping for this model; the effect of a phase transition in this system; approximate recursion relations of the Migdal form. Finally, all this is put together to give a partial argument for the simultaneous existence of asymptotic freedom and trapping ${\mathrm{O}}_2$ in the quark-string case. Arguments are developed which distinguish this case from the superficially analogous example of quantum electrodynamics.