The Quantisation of Constrained Systems Using the Batalin, Fradkin, Vilkovisky Formalism

The purpose of this work is to examine the problem of quantising constrained dynamical systems within the Batalin Fradkin Vilkovisky (BFV) formalism. The work concentrates almost entirely on theories with a finite number of dimensions and constraints linear in the phase space momenta. Chapters four and five give some discussion of possible extensions of the work to more general constraints. Chapter two will give a discussion of the classical theory of constrained systems and, in particular, will study the symmetries present in such theories. The main result in this chapter is that the constraint rescaling symmetry (this is the freedom to transform to new sets of constraints which describe the same true degrees of freedom) is a canonical transformation in the BFV phase space. An implicit definition of the most general form of this transformation will be given. After chapter two we will study the quantisation of constrained systems. We will always work with the basic assumption that the correct constraint quantisation should give the same results as one obtains from quantising the classical true degrees of freedom. Chapter three will examine the quantisation of finite dimensional linear constraints. It will be shown that, to obtaining the correct constraint quantisation, one must use four symmetries. These symmetries are coordinate transformations on the classical configuration space, coordinate transformations on the true configuration space (i. e. the configuration space obtained by solving the constraints), weak changes to observables (i. e. adding terms which vanish when the constraints are applied) and rescaling of constraints. The main result of chapter three is that enforcing these four symmetries is sufficient to fix the main ambiguities in the quantisation and that the resultant quantum theory is equivalent to classically solving the constraints and then quantising. These results rely on the fact that the classical canonical rescaling transformation can, for the restricted class of rescalings which are of interest in gauge theories, be made into a unitary quantum transformation. This quantum transformation is the main tool used in chapter three and enables us to maintain a Hilbert space structure on the extended state space (i. e the state space which contains both physical and unphysical states). Previous attempts by other authors to quantise finite dimensional gauge theories without ghost variables have failed to maintain a Hilbert space structure. This is one of the main advantages of the work presented here. In chapter four we will look at the use of the BFV method in geometric quantisation. The main motivation for this is to study constraints which depend quadratically on the phase space momenta e. g. the constraints which arise in general relativity. Chapter four does not give a proper quantisation of quadratic constraints but it does give some indication of the new features which arise in these theories. The main result seems to be the need to use polarisations, in the BFV phase space, which genuinely mix the bosonic and fermionic degrees of freedom. Chapter five will look at some classical aspects of constraint rescaling for Yang-Mills field theories. The various possible field theory constraint rescalings will be discussed and a few results will be proven showing to what extent it is possible to simplify the conventional Yang-Mills constraints via rescalings. These simplifications consist of forming an equivalent set of constraints which commute with respect to Poisson brackets. These simplified constraints were very useful in the analysis of the finite dimensional case