We undertake a group-theoretical study of the specific form of the harmonic oscillator quark model proposed by Isgur and Karl, restricting our attention to the non-strange sector of the baryon spectrum. In particular, we consider the spectrum-generating group, Sp(12,R), appropriate to the study of the 3-quark problem and demonstrate how it may be used to label the oscillator eigenstates and to provide a new and direct means of constructing wave functions of definite orbital angular momentum and permutation - symmetry type. We indicate how Sp(12,R) provides the most appropriate means of classifying the symmetry-breaking induced by an enharmonic perturbation and we derive an algebraic mass formula, involving the quadratic Casimir invariant operators of Sp(12,R) and its relevant subgroups, plus one non-subgroup invariant operator, which successfully reproduces the splitting pattern of the N = 2 supermultiplets originally derived in the literature by straightforward perturbative techniques. Some results for the N = 3 level are also given together with an outline of the method for generalisation to any degree of excitation of the system. Much of our understanding of the role of the spectrum-generating group in this context derives from a parallel study of the simpler case of a 2-particle bound system, which we also describe. We examine the implications of our results for baryon spectroscopy: in particular, we discuss in some detail the possibility that the AD35(1940) resonance is evidence for an N = 3 [56,1 ] supermultiplet corresponding to excitation of new gluonic degrees of freedom. After inclusion of hyperfine effects, and with reasonable values of the parameters in the model, we recover the pertinent features appropriate to the D35, G37 and G39 sectors of a recent preliminary phase-shift analysis by Cutkosky et al.

This thesis describes the calculations of masses and matrix elements using lattice QCD. Exploratory results are presented for the hadron mass spectrum and pseudoscalar meson decay constants using a mixed action formalism where the sea and valence actions are different. Improved staggered sea quarks are used and HYP-smeared overlap valence quarks. A method for matching the sea and valence quark masses is proposed. Good signals on 10 configurations at one lattice spacing and two different sets of sea quark masses are obtained. Results are presented for light meson masses and pseudoscalar meson decay constants in 2+1 flavour domain wall QCD with the DBW2 and Iwasaki gauge actions. This formalism preserves chiral symmetry at finite lattice spacing. The lattices used have linear sizes in the range 1.6 to 2.2fm and <i>u</i> and <i>d</i> quark masses as low as one quarter of the strange quark mass. All data were generated on the QCDOC machines at the University of Edinburgh and Brookhaven National Laboratory. Despite large residual masses and a limited number of sea quark mass values with which to perform chiral extrapolations, our results agree with experiment and scale within errors.

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In this thesis two aspects of Lattice QCD are studied, the light hadron spectrum and decay constants and a stochastic gauge-fixing procedure. Chapter 2 is an investigation of the effects of tadpole improvement on the Sheikholeslami-Wohlert clover action at <I>β </I>= 6.2. The mass spectra and decay constants of the two actions are compared using 60 gauge field configurations. Changing the value of the clover coefficient is found to have little effect on spectral quantities at this <I>β </I>value but reference is made to a similar study at <I>β </I>= 5.7 where greater effects are seen. The decay constants show a statistically significant <I>c</I> dependence. However, it is argued that without a non-perturbative value for the renormalisation constants <I>Z<SUB>A</SUB></I> and <I>Z<SUB>V</SUB></I> a conclusive statement cannot be made. In Chapter 3 a complete study of the quenched, light hadron spectrum and decay constants at <I>β </I>= 6.2 is done, using 220 tadpole improved gauge field configurations and quark propagators with local and fuzzed sources. Using data at two additional <I>β </I>values, 6.0 and 5.7 the effects of scaling are investigated. The pseudoscalar and vector decay constants are calculated and ratios of decay constants are compared with their experimental values. Chapter 4 describes a preliminary study of a new, covariant, one parameter gauge-fixing procedure. This scheme avoids the problem of Gribov copies by imposing a "soft" gauge fixing condition. Varying the gauge-fixing parameter allows a study of gauge dependence. Details of the lattice scheme and the algorithm to implement it are given. The behaviour of the internal energy of the system, the specific heat and the zero momentum gluon propagator is monitored. Results for the gauge dependence of the gluon propagator are presented.

In this thesis we study light hadron phenomenology using lattice QCD. In particular we measure hadron masses, meson decay constants, and nucleon structure functions. We perform the lattice simulation using overlap fermions. This formulation preserves chiral symmetry at finite lattice spacing and so is well suited to simulations close to the chiral limit. It is however very computationally expensive and so we are forced to work in the quenched approximation. After a review of the relevant continuum phenomenology and lattice gauge theory, we detail the necessary lattice technology required to extract physical results from the simulations. We then present the results. We use an improved gauge action and investigate how this affects the locality and condition number of the overlap Dirac operator. We present measurements of masses for some low lying hadron states, and in order to make contact with continuum physics, we calculate some lattice renormalization constants. After providing measurements of vector and pseudoscalar meson decay constants, we present results on nucleon structure functions. While these structure functions cannot be measured directly on the lattice, we can relate measurable QCD matrix elements to moments of structure functions through the operator product expansion. Here we provide results for several low moments of both polarized and unpolarized nucleon structure functions.

In this thesis I study semi-leptonic decays of mesons containing one heavy and one light quark. I describe the necessary theoretical tools of Lattice Gauge Theory and of Heavy Quark Effective Theory. The six form factors describing the semi-leptonic decays are extracted from a lattice gauge theory calculation. The dependence of the form factors on the masses of the heavy and light quarks is studied. The results are compared to the predictions of Heavy Quark Symmetry and corrections to the exact symmetry limit of infinitely heavy quarks are analysed. The slope of the Isgur-Wise function is extracted from the lattice simulation and it is used to extract the Cabbibo-Kobayashi-Maskawa matrix element <I>V</I><SUB>cb</SUB>. The results are compared to those of other authors where possible.

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