The Uses of the Vacuum

Three problems in quantum field theory are analyzed. Each presents the vacuum in a different role. The connections among these significant roles are discussed in Chapter I. Chapter II contains a calculation of the zero-point energy in the Kaluza-Klein model. The zero-point fluctuations induce a potential which makes the compact dimension contract. The effective potential is seen to be the four-dimensional version of the Casimir effect. Chapter III contains a Monte Carlo study of asymptotic freedom scales in lattice QCD. Two versions of SU(2) gauge theory, having different representations of the gauge group, are compared. A new method is used to calculate the ratio of scale parameters of the two theories. The method directly uses the weak -coupling behavior of the theories. The Monte Carlo results are compared with perturbative calculations on the lattice, one of which is presented. They are in good agreement. Chapter IV applies the hypothesis of dimensional reduction to five -dimensional SU(2) and four-dimensional SO(3) lattice gauge theories. New analytic results for the strong- and weak -coupling limits are derived. Monte Carlo calculations show dimensional reduction in the strong-coupling phases of both theories. At the phase transition, the two theories show a similar loss of dimensional reduction. An external source of random flux does not induce dimensional reduction where it is not already present.