Some Aspects of Non-Perturbative Quantum Field Theory
Doctoral thesis, 1996
The aim of this thesis is to describe some different ways in which non-perturbative methods enter in quantum field theory. In the thesis such methods enter in 1+1-dimensional theories: various Yang-Mills theories and electrodynamics. It also enters when treating charged bosons or fermions interacting with an external, classical field. In all these cases the interaction is treated exactly and we do not have to rely on perturbation theory.
A good foundation for starting non-perturbative investigations in quantum field theory is a functional representation for the (canonical) algebra of the field operators. Such a representation is introduced and illustrated by various examples, mainly free fields and free fields coupled to external fields. The functional representation is well suited for a discussion of equilibrium quantum statistical mechanics by means of a density matrix. This is discussed in various examples. A non-equilibrium problem is also considered in the treatment of external fields. Starting out with an equilibrium thermal ensemble described by a density matrix, an external field is switched on that gradually evolves the initial thermal ensemble to a non-thermal one.
In paper I the construction of QED in 1+1-dimensions is discussed. An attempt is also made to solve the theory in the massless case (the Schwinger model). There are however some errors in the paper and the solution attempt does not quite work out. In paper II we discuss the construction and solution of SU(N) Yang-Mills theory in 1+1-dimensions in terms of a complete set of gauge invariant variables, the T-variables. We also discuss a quantization ambiguity in terms of canonical transformations. In paper III we do more or less the same thing but for a different gauge group, SL(2,R) and discover a much larger quantization ambiguity than for SU(N).
These first three papers, were mainly inspired by Ashtekar's new variables for canonical gravity. This is changed in paper IV dealing with pair production in an external electric field. The emphasis is on handling finite times and finite temperature in this external field. The calculations are all done in a functional representation. In the fermionic case dealt with this required an extension of previous works. The last paper, paper V, closes the circle and does what should have been done in paper I: The construction of a gauge invariant QED 1+1 and the solution of the model in the massless case. This paper includes a derivation of bosonization in the functional representation.