Extended Lattice Actions: Phase Transitions and the Q.C.D. Renormalized Trajectory

The introduction of a space-time lattice to regulate short-distance singularities has opened the possibility of numerically simulating the non-linear dynamics of Quantum Chromodynamics - the theory of strong interactions. Unfortunately the limited capabilities of present day computers put a severe restriction on the accessible lattice sizes. Thus, the choice of lattice action, which in principle (due to universality) is irrelevant, in practice is crucially important: continuum answers can be obtained with small lattices if the action is optimally chosen, to minimize regulator effects. The second half of this work deals with the direct calculation of such an effective lattice action: constraints are introduced in the continuum functional integral which is then treated in a weak coupling, weak field approximation. The renormalization of ultraviolet divergences as well as the cancellation of infrared divergences is illustrated using some toy models. The method is then applied to QCD, where we extend the tree level lowest order in the fields calculation of C. Callan, R. Dashen and D. Gross, to the leading non-Abelian terms. A different - indirect approach to the problem of an improved lattice action consists of studying the phase structure of lattice gauge theories in extended parameter spaces (this is because the trajectory of effective actions should both pass through windows of analyticity, and avoid other critical domains). In the first half of this thesis we present a simple criterion which links all phase transitions in 4-dimensional LGTs to a condensation of stable vacuum excitations. We show the consistency of this conjecture with existing data. To lend further support to it we analyse, using Monte Carlo, mean field theory and semiclassical techniques, an extended model involving the traces in the fundamental and the six-representation of SU(3).