Different Phase Study of Quantum Field Theory |
Seriesexpansions play a major role in helping us extract results from many modelswhich are too complex to solve directly. Traditional perturbation theory, forinstance, can be successfully applied to Quantum Electrodynamics. Yet, itcompletely fails to provide convergent expansions when applied to theorieswhere coupling constants are (in some sense) large. Hence the importance ofnon-perturbative series expansions, of which the Linear Delta Expansion (LDE)is one. In this paper, we study the application of the LDE to two verydifferent models: the lattice scalar self-interacting field theory, and thedynamics of a quantum mechanical inflationary model. We will also developsophisticated arbitrary precision numerical methods to aid us in pushing theexpansion to reasonably high orders.After presenting an overview of the LDE, weshall apply it to the lattice theory. In particular, we will focus on the criticalbehaviour of the model, which will include the calculation of various criticalexponents. We shall find that the LDE gives good qualitative results andclearly identifies the symmetry breaking aspects of the theory. On the otherhand, we shall find that the quantitative results, especially those of criticalbehaviour, reproduce the results obtained by the much less sophisticated meanfield theory. Thesecond model studied in the paper is the quantum dynamics of an inflationarymodel, often called the quantum mechanical slow-roll. A recent LDE study of thesame model successfully tracks the system while in the inflation phase, butfails to follow suit into the reheating phase. Our aim will be to improve themethodology (and consequently the results) of that study, by employing aphysically more intuitive criterion for optimizing the parameters of thetheory. We will find, however, that the hoped for improvements remain elusive.