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This thesis presents a study of two-dimensional electron systems in GaAs-(Al,Ga)As heterojunctions and quasi-two-dimensional electron and hole systems in graphite within the quantum Hall effect regime of low temperature and high magnetic field. This thesis covers three main sets of experimental work as well as details of the experimental methods (chapter 2) used and the background theory behind the observed results (chapter 1). The first experimental results presented in this thesis in chapter 3 focus on contactless measurement of the equilibrium magnetisation of sample A2268, a ten layer multiple quantum well sample. Fitting the shape of dHvA oscillations at various temperatures to different models for the density of states, various properties of the system can be estimated, such as the shape of the disorder-broadened density of states and the presence of a background density of states between the Landau levels. Chapter 4 focuses on measurements of the decay of induced circulating currents in the quasi-dissipationless quantum Hall regime in two samples, V0049 and T73. The induced current is measured via contactless measurement of the associated magnetic moment. The magnitude of the induced current is found to be affected by the sweep rate of the magnetic field and also the distance of approach. The decay of the induced currents is observed at several temperatures and for different magnetic field sweep rates and distances of approach. Decays are observed for up to several days at time, far longer than previously possible. Information about the rate of decay can be used to build a picture of the decay mechanisms present in the quantum Hall regime. The presence of a power-law decay regime indicates many decay mechanisms contribute to the decay of a circulating current in the quasi-dissipationless quantum Hall regime. Chapter 5 focuses on both contactless magnetometry and transport experiments carried out on a graphite sample. The experiments aim to confirm or dispute recent claims of Dirac fermions in graphite. Experiments are carried out at temperatures in the range 30 mK to ~4 K and at two different angles to the applied magnetic field. Phase analysis of both Shubnikov de Haas and de Haas van Alphen oscillations is used to distinguish between normal and Dirac fermions. Observation of quantum Hall effect displays the presence of a half-integer quantum Hall staircase similar to that observed in graphene.

A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a considerable amount of negativity.

Extensions of the warped extra dimension framework originally proposed by Randall and Sundrum are discussed, including soft-wall models and aspects of supersymmetry breaking. In particular, the standard model in a soft-wall background is covered in detail, including electroweak physics and an extensive treatment of fermions in arbitrary warped backgrounds. Additionally, aspects of lepton flavor violation in models of supersymmetry breaking with hierarchical soft-terms and Dirac gauginos are discussed, as these can occur naturally when supersymmetry is broken near the infrared boundary.

We calculate the two loop radiative corrections to the $\rho$ parameter in the heavy Higgs limit including the effects of virtual quark loops. We discuss and use a consistent method for calculating traces with an even number of $\gamma\sp5$'s. We find that the leading large Higgs mass behavior is logarithmic, in agreement with expectations from an analysis of the nonlinear $\sigma$-model. The relative magnitudes of the one loop and two loop contributions are compared for various values of the top quark and Higgs masses with the aim of determining the conditions for the breakdown of perturbation theory. We find for example that for a Higgs mass in the range of 1-10 TeV, perturbation theory is invalid for $M\sb{\rm top}$ $>$ 200 GeV. In an appendix we discuss the Gauss-Law constraint from a diagrammatic viewpoint. This formalism is then applied to calculate the Gauss-Law anomaly in chiral gauge theories.

This work consists of two parts. Part One studies the bounds on fermion masses when the mass is gained through spontaneous symmetry breaking. As a large mass is obtained by means of a large Yukawa coupling, mass renormalizations must be calculated with a non-perturbative technique. A chiral Yukawa model is solved to leading order in the large-N expansion. It is shown there is no limit on the physical mass due to any saturation mechanism, but there is a limit by virtue of the inherent cutoff in the theory. The doubt this casts on one method of quantization of anomalous gauge theories is discussed. Part Two investigates a class of theories obtained from finite N = 4 supersymmetric Yang-Mills theories by the addition of arbitrary soft-breaking terms. The one-loop renormalization group coefficients are obtained and the renormalization group equations solved for the running dimensionful couplings. It is shown that the conditions among the parameters that would guarantee finiteness at one loop are an infrared fixed hypersurface in parameter space. It is argued that imposition of these conditions constitutes an unnatural fine-tuning of parameters. Spontaneous symmetry breaking in the proposed models is studied. It is shown, through renormalization group analysis of the effective potential, that a minimum exists consistent with the requirement of stability. A relation among masses is obtained at the minimum. Because breaking the gauge symmetry gives masses to the non-abelian gauge bosons, while retaining a stable potential, the infrared problem of the ultravioletly finite theories is eliminated; for this reason, and by reason of naturalness, these models that retain finiteness in the gauge sector but allow running of the soft-breaking terms are recommended as the natural extentions of the supersymmetric theory.