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Using the Atiyah-Ward construction, we examine the solutions of the self-dual Yang-Mills equations for an SU(2) gauge theory, dimensionally reduced from |R⁴ to |R². There are two main reasons for doing this: (i) To provide a large class of relatively simple examples which elucidate how non-singularity and physical field configurations are related to the parameterization of the Atiyah-Ward construction. (ii) To construct analogues, for pure non-abelian gauge theories, of the superconducting vortex solutions of the abelian Higgs model, in the hope that these will provide the dominant field configurations describing the QCD vacuum. First, Bäcklund transformations are used to construct axially symmetric solutions, and the analogues of the 't Hooft instantons. These results are then generalised, within the twister theoretic framework of the Atiyah-Ward construction, to produce an infinite dimensional parameter space of complex non-singular solutions in each of the Atiyah-Ward ansätze. The field configurations are expressible as unitary group integrals occurring in lattice gauge theories - this leads to a simple proof of non-singularity, and a convenient means of calculating properties of the field configurations using strong and weak coupling expansions. The structure of the field configurations is further elucidated using symmetry arguments and numerical computations. Finally, suggestions are made as to how these solutions may play a role in the QCD confinement mechanism.

This thesis is concerned with an approach to Quantum Field Theory in which the states are constructed from their large distance behaviour. The logarithm of the vacuum functional is expandable as a local quantity in any quantum field theory in which the tightest physical particle has a non-zero mass. This local expansion satisfies its own form of the Schrodinger equation from which its coefficients can be determined. We illustrate for ϕ(^4)(_1+1)- theory that our local expansion incorporates correctly the short distance behaviour as contained in the counterterms of the Hamil tonian. A Feynman diagram expansion of the vacuum functional is also presented. The amplitudes are calculated and their large distance expansion are in good agreement with our semi-classical solution of the Schrödinger equation. Some applications of this formalism to the study of the Schrödinger functional are also suggested.

This thesis is concerned with the study, classically and quantum mechanically, of the square billiard with particular attention to chaos in both cases. Classically, we show that the rotating square billiard has two regular limits with a mixture of order and chaos between, depending on an energy parameter, E. This parameter ranges from -2w(^2) to oo, where w is the angular rotation, corresponding to the two integrable limits. The rotating square billiard has simple enough geometry to permit us to elucidate that the mechanism for chaos with rotation or curved trajectories is not flyaway, as previously suggested, but rather the accumulation of angular dispersion from a rotating line. Furthermore, we find periodic cycles which have asymmetric trajectories, below the value of E at which phase space becomes disjointed. These trajectories exhibit both left and right hand curvatures due to the fine balance between Centrifugal and Coriolis forces. Quantum mechanically, we compare the spectral analysis results for the square billiard with three different theoretical distribution functions. A new feature in the study is the correspondence we find, by utilising the Berry-Robnik parameter q, between classical E and a quantum rotation parameter w. The parameter q gives the ratio of chaotic quantum phase volume which we can link to the ratio of chaotic phase volume found classically for varying values of E. We find good correspondence, in particular, the different values of q as w is varied reflect the births and subsequent destructions of the different periodic cycles. We also study wave packet dynamics, necessitating the adaptation of a one dimensional unitary integration method to the two dimensional square billiard. In concluding we suggest how this work may be used, with the aid of the chaotic phase volumes calculated, in future directions for research work.

We investigate the infrared behaviour of the gluon propagator in Quantum Chromo- dynamics (QCD). A natural framework for such a non-perturbative study is the complex of Schwinger-Dyson equations (SDE).The possible infrared behaviour of the gluon, found by self-consistently solving the approximate boson SDE, is studied analytically. We find that only an infrared enhanced gluon propagator, as singular as 1/p(^4) as p(^2) → 0, is consistent and demonstrate why softer solutions, that others have found, are not allowed. Reassuringly the consistent, enhanced infrared behaviour is indicative of the confinement of quarks and gluons, implying, for example, area-law behaviour of the Wilson loop operator and forbidding a Kāllen-Lehmann spectral representation of both quark and gluon propagators. We then briefly consider the implications of these results for models of the pomeron. The enhancement of the gluon propagator does however introduce infrared divergences in the SDE and these need to be regularised. So far model forms of the enhanced gluon propagator have been used in studies of dynamical chiral symmetry breaking and hadron phenomenology. Though very encouraging results have been obtained, one might hope to use the gluon propagator obtained directly from non-perturbative QCD to calculate hadron observables. We therefore attempt to eliminate the infrared divergences in the SDEs in a self- consistent way, entirely within the context of the calculational scheme. To do this we introduce an infrared regulator λ in the truncated gluon SDE in quenched QCD. We find that this regulator is indeed determined by the equation and bounded by the QCD-scale Aqcd- Thus it is possible to perform the regularisation within the SDEs. However, we have not been able to choose λ < Aqcd.