Optimization of Monte-Carlo calculations of the effective potential

We study Monte Carlo calculations of the effective potential for a scalar field theory using three techniques. One of these is a new method proposed and tested for the first time. In each case we extract the renormalised quantities of the theory. The system studied in our calculations is a one component $\phi^4$ model in two dimensions. We apply these methods to both the weak and strong coupling regime. In the weak coupling regime we compare our results for the renormalised quantities with those obtained from two-loop lattice perturbation theory. Our results are verified in the strong coupling regime through comparison with the strong coupling expansion. We conclude that effective potential methods, when suitably chosen, can be accurate tools in calculations of the renormalised parameters of scalar field theories.Comment: 26 pages of LaTeX, uses psfig.sty with 6 figures. Entire manuscript available as a postscript file via WWW at http://www.physics.adelaide.edu.au/theory/papers/ADP-97-13.T250-abs.html or via anonymous ftp at ftp://bragg.physics.adelaide.edu.au/pub/theory/ADP-97-13.T250.p

On the Green function of linear evolution equations for a region with a boundary

We derive a closed-form expression for the Green function of linear evolution equations with the Dirichlet boundary condition for an arbitrary region, based on the singular perturbation approach to boundary problems.Comment: 9 page

Symmetries of Non-Linear Systems: Group Approach to their Quantization

We report briefly on an approach to quantum theory entirely based on symmetry grounds which improves Geometric Quantization in some respects and provides an alternative to the canonical framework. The present scheme, being typically non-perturbative, is primarily intended for non-linear systems, although needless to say that finding the basic symmetry associated with a given (quantum) physical problem is in general a difficult task, which many times nearly emulates the complexity of finding the actual (classical) solutions. Apart from some interesting examples related to the electromagnetic and gravitational particle interactions, where an algebraic version of the equivalence principle naturally arises, we attempt to the quantum description of non-linear sigma models. In particular, we present the actual quantization of the partial-trace non-linear SU(2) sigma model as a representative case of non-linear quantum field theory.Comment: 24 pages, LaTeX, no figure