Masters Degrees (Physics)

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Additional degrees of freedom associated with position measurements in non-commutative quantum mechanics
(Stellenbosch : University of Stellenbosch, 2010-12) Rohwer, Christian M.; Scholtz, Frederik G.; University of Stellenbosch. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: Due to the minimal length scale induced by non-commuting co-ordinates, it is not clear a priori what is meant by a position measurement on a non-commutative space. It was shown recently in a paper by Scholtz et al. that it is indeed possible to recover the notion of quantum mechanical position measurements consistently on the non-commutative plane. To do this, it is necessary to introduce weak (non-projective) measurements, formulated in terms of Positive Operator-Valued Measures (POVMs). In this thesis we shall demonstrate, however, that a measurement of position alone in non-commutative space cannot yield complete information about the quantum state of a particle. Indeed, the aforementioned formalism entails a description that is non-local in that it requires knowledge of all orders of positional derivatives through the star product that is used ubiquitously to map operator multiplication onto function multiplication in non-commutative systems. It will be shown that there exist several equivalent local descriptions, which are arrived at via the introduction of additional degrees of freedom. Consequently non-commutative quantum mechanical position measurements necessarily confront us with some additional structure which is necessary (in addition to position) to specify quantum states completely. The remainder of the thesis, based in part on a recent publication (\Noncommutative quantum mechanics { a perspective on structure and spatial extent", C.M. Rohwer, K.G. Zloshchastiev, L. Gouba and F.G. Scholtz, J. Phys. A: Math. Theor. 43 (2010) 345302) will involve investigations into the physical interpretation of these additional degrees of freedom. For one particular local formulation, the corresponding classical theory will be used to demonstrate that the concept of extended, structured objects emerges quite naturally and unavoidably there. This description will be shown to be equivalent to one describing a two-charge harmonically interacting composite in a strong magnetic eld found by Susskind. It will be argued through various applications that these notions also extend naturally to the quantum level, and constraints will be shown to arise there. A further local formulation will be introduced, where the natural interpretation is that of objects located at a point with a certain angular momentum about that point. This again enforces the idea of particles that are not point-like. Both local descriptions are convenient, in that they make explicit the additional structure which is encoded more subtly in the non-local description. Lastly we shall argue that the additional degrees of freedom introduced by local descriptions may also be thought of as gauge degrees of freedom in a gauge-invariant formulation of the theory.
Alpha cluster structure and scattering in 20Ne, 44Ti, 94Mo, 136Te and 212Po
(Stellenbosch : University of Stellenbosch, 2011-03) Ndayishimye, Joram; Wyngaardt, Shaun M.; Perez, S. M.; University of Stellenbosch. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: We investigate the nuclei 20Ne, 44Ti, 94Mo, 136Te and 212Po using a model of an α-cluster orbiting a closed shell core. A purely phenomenological cluster-core potential is found to provide a successful description of the spectra, B(E2↓) transition strengths, and α-decay rates of the low-lying positive parity states of these nuclei. We then use the same potential as the real part of an optical model potential to describe the α elastic scattering by 16O, 40Ca, 90Zr and 208Pb. The experimental differential cross-section data are reasonably well reproduced with the imaginary potential depth as the only free parameter. The special case of the 8Be system is also analysed.
Analysis and applications of the generalised Dyson mapping
(Stellenbosch : Stellenbosch University, 2004-12) Snyman, Izak; Geyer, H. B.; Scholtz, Frederik G.; Stellenbosch University. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: In this thesis, generalized Dyson boson-fermion mappings are considered. These are techniques used in the analysis of the quantum many-body problem, and are instances of so-called boson expansion methods. A generalized Dyson boson-fermion mapping, or a Dyson mapping for short, is a one-to-one linear but non-unitary operator that can be applied to vectors representing the states of a many-fermion system. A vector representing a fermion system maps onto a vector that is most naturally interpreted as representing a state of a many-body system that contains both bosons and fermions. The motivation for doing such a mapping is the hope that the mapping will reveal some property of the system that simplifies its analysis and that was hidden in the original form. The aims of this thesis are 1. to review the theory of generalized Dyson boson-fermion mappings, 2. by considering a tutorial example, to demonstrate that it is feasible to implement the theory and 3. to find a useful application for a generalized Dyson boson-fermion mapping, by considering a non-trivial model, namely the Richardson model for superconductivity. The realization of the first two aims mainly involve the collecting together of ideas that have already appeared in the literature, into one coherent text. Some subtle points that were treated only briefly due to space restrictions in the journal publications where the theory was first expounded, are elaborated on in the present work. On the other hand, the analysis of the Richardson Hamiltonian that uses a Dyson mapping, goes beyond what has already appeared in the literature. It is the first time that a boson expansion technique is implemented for a system where the roles of both collective and non-collective fermion pairs are important. (The Dyson mapping associates bosons with Cooper pairs, while the fermions not bound in Cooper pairs result in fermions being present in the mapped system as well.) What is found is that the Dyson mapping uncovers non-trivial properties of the system. These properties aid the construction of time-independent perturbation expansions for the stationary states of the system, as well as time-dependent expansions for transition amplitudes between states. The time-independent expansions agree with results that other authors obtained through methods other than boson expansions. The time-dependent expansions, that one would be hard-pressed to develop without a Dyson mapping, might in future prove useful in understanding aspects of the dynamics of ultracold fermi gases, when time-dependent magnetic fields are used to vary the atom-atom interaction strenght.
The analysis of multi-clustering in heavy nuclei
(Stellenbosch : Stellenbosch University., 2020-04) Carolus, Timothy Gary; Wyngaardt, Shaun M.; Malaza, Vusi D.; Stellenbosch University. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: The binary cluster model is the simplest mathematical model which is used to describe a system of strongly interacting, and highly correlated nuclear matter. This study will serve as a means to better understand the phenomenon of nuclear clustering beyond a 208P b-alpha cluster core system. Theoretically predicted observables, such as the cluster decay half-life, positive parity cluster states, and the reduced electric dipole transition (B(E2)) are compared to the available experimentally observed quantities. The results show that the tested systems for various cluster-core configurations, of both a fixed parent nuclei and fixed stable core favoured the phenomenological description of their interaction. A numerical optimization procedure is applied to the core- cluster potential of the binary cluster formalism and is then used as an analytical tool in order to predict the possibility of higher modes of cluster decay in 216Rn.
Aspects of quantum field theories
(Stellenbosch : Stellenbosch University, 1997) Van Biljon, Andrew; Scholtz, Frederik G.; Stellenbosch University. Faculty of Science. Department of Physics.
ENGLISH ABSTRACT: In this study project we give a general introduction to quantum field theories. In the first chapter we revise the operator formalism of quantum mechanics as well as the second quantization scheme that is used to describe many-particle systems. In the second chapter we develop the idea of path integrals within a quantum mechanics framework. We then apply path integral formalism developed in chapter two to introduce quantum field theories. We describe a field theory for scalar fields in chapter three and then a field theory for fermion fields in chapter four as well as the renormalization techniques in chapter five. In chapter six we show how scattering amplitudes are related to Green's functions which are derived from the path integral formalism. In the last chapter we give a brief introduction to gauge field theories.
Basic concepts of random matrix theory
(Stellenbosch : University of Stellenbosch, 2005-12) Van Zyl, Alexis J.; Scholtz, Frederik G.; University of Stellenbosch. Faculty of Science. Dept. of Physics.
It was Wigner that in the 1950’s first introduced the idea of modelling physical reality with an ensemble of random matrices while studying the energy levels of heavy atomic nuclei. Since then, the field of Random Matrix Theory has grown tremendously, with applications ranging from fluctuations on the economic markets to M-theory. It is the purpose of this thesis to discuss the basic concepts of Random Matrix Theory, using the ensembles of random matrices originally introduced by Wigner, the Gaussian ensembles, as a starting point. As Random Matrix Theory is classically concerned with the statistical properties of levels sequences, we start with a brief introduction to the statistical analysis of a level sequence before getting to the introduction of the Gaussian ensembles. With the ensembles defined, we move on to the statistical properties that they predict. In the light of these predictions, a few of the classical applications of Random Matrix Theory are discussed, and as an example of some of the important concepts, the Anderson model of localization is investigated in some detail.
Bayesian parameter estimation for discrete data spectra
(Stellenbosch : Stellenbosch University, 2017-12) Wang, Li; Eggers, H. C.; Stellenbosch University. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT : Discrete spectra are ubiquitous in physics; for example nuclear physics, laser physics and experimental high energy physics measure integer counts in the form of particles in dependence of angle, wavelength, energy etc. Bayesian parameter estimation ( tting a function with free parameters to the data) is a sophisticated framework which can handle cases of sparse data as well as input of pertinent background information into the data analysis in the form of a prior probability. Bayesian comparison of competing models and functions takes into account all possible parameter values rather than just the best t values. We rst review the general statistical basis of data analysis, focusing in particular on the Poisson, Negative Binomial and associated distributions. After introducing the conceptual shift and basic relations of the Bayesian approach, we show how these distributions can be combined with arbitrary model functions and data counts to yield two general discrete likelihoods. While we keep an eye on the asymptotic behaviour as useful analytical checks, we then introduce and review the theoretical basis for Markov Chain Monte Carlo numerical methods and show how these are applied in practice in the Metropolis-Hastings and Nested Sampling algorithms. We proceed to apply these to a number of simple situations based on simulation of a background plus two or three Gaussian peaks with both Poisson and Negative Binomial likelihoods, and discuss how to select models based on numerical outputs.
Biofilament interacting with molecular motors
(Stellenbosch : Stellenbosch University, 2015-12) Meylahn, Janusz Martin; Muller-Nedebock, Kristian K.; Touchette, Hugo; Stellenbosch University. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: We study molecular motors moving along a filament or polymer using two different mathematical models in which motors are idealised as springs. In the first model we study the average and the fluctuations of the motor stretch by modelling the motion of the motors along the filament using a simple stochastic differential equation with linear friction. We use the notion of stochastic resetting to explicitly include the attachment and detachment dynamics of the motors to and from the filament and study the fluctuations around the most probable value of the mean stretch using methods from large deviation theory. The second model uses methods from field theory to model a dynamic network consisting of a single polymer and many molecular motors. In this case, we develop techniques to include the bias motion of the molecular motors in a weighting factor for the formation of specific networks rather than in the dynamical constraints of the partition function which allows us to study the steady-state of the network using a self-consistency argument and a saddlepoint approximation.
Bound states near the interface of a distorted graphene sheet and a superconductor
(Stellenbosch : Stellenbosch University, 2011-12) Van Zyl, Hendrik Jacobus Rust; Snyman, Izak; Scholtz, Frederik G.; Stellenbosch University. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: The goal of this thesis is to investigate the effects of distorting a graphene lattice and connect- ing this distorted graphene sheet to a superconductor. At low energies the possible excitation states in graphene are restricted to two distinct regions in momentum space called valleys. Many electronic applications are possible if one can design a graphene system where excitations can be forced to occupy a single valley in a controllable way. Investigating the spectrum of the distorted graphene sheet reveals that, if the chemical potential is chosen to coincide with a bulk Landau level, the normal-superconductor interface always supports propagating modes in both directions. Excitations from opposite valleys travel in opposite directions along the interface. The spectrum of a distorted graphene sheet terminated by an armchair edge, in contrast, is dis- persionless. We verify this insulating nature of the armchair edge for finite samples by numerical means. Furthermore, we verify previous analytical results pertaining to a graphene sheet with NS interface and an applied perpendicular real magnetic field numerically. In the process, it is shown that considering graphene sheets of perfect width is not necessary, as long as the width a few magnetic lengths away from the interface is well-defined. By then considering a finite graphene sheet, terminated by armchair edges, that is distorted and connected to a superconductor, we find bound states near the NS interface that can be changed by distorting the graphene lattice further.
Cave front tracking experiment
(Stellenbosch : Stellenbosch University, 2016-12) Green, Mark; Muller-Nedebock, Kristian; Boonzaaier, Leandro; Lynch, Richard; Stellenbosch University. Faculty of Science. Department of Physics
ENGLISH ABSTRACT : Seismic tomography, a procedure to trace seismic wave velocity variation, has been used for more than 100 years to “see through rock”. Initially the recording of seismic waves from natural sources (earthquakes) using seismographs, provided the first evidence of the crust - mantle boundary. Further development of recording techniques and use of explosive seismic sources were employed for the delineation of the geological structures of mineral deposits (primarily oil and gas) as well as detailing the earth - mantle location and geometry. Ultra sonic techniques have been applied in the laboratory experiments to characterise rock sample velocities and its relationship to changes in temperature and stress. Active seismic tomography for hydrocarbon resources determination as well as for the earth structure utilized both reflection and refraction techniques. During the early 60’s seismic techniques were employed in active mines to investigate rock properties for geotechnical purposes. The objective of the Cave Front Tracking Experiment was to investigate whether active seismic tomography could potentially be used to track the cave back progression of a block cave mine. A small scale experiment was conducted to study the variations in seismic travel times associated with ray path refraction caused by stress changes and changes in the rock fabric induced by sub-level mining. A piezoelectric transducer was employed as an active seismic source and a seismic recording system was installed to monitor a volume of rock through which sub-level mining was propergated. Implementation involved the deployment of 11 seismic detectors and the recording of pre-stacked data. The experimental details and hardware characteristics are discussed, which includes technical problems encountered, the process and techniques of significant data stacking in order to recover weak signals generated by a piezoelectric seismic source and travel time variation calculation using signal cross-correlation. Signal travel time variation measurements were made over the course of a year and a half (2013-14), however only five months are investigated in detail and related to mining activities. Finite difference numerical modelling was also employed in order to create a better understanding of the sub-level mining process and its effect on seismic signal delay change. Technical problems associated with the signal processing and signal behavior interacting with an advancing mining front were identified and a number of techniques were employed to overcome problems faced by routine cave front estimation in a working mine environment. The field testing of a prototype impact source that is pneumatically driven and an eccentric rotating mass device is discussed.
Characterization of the pre-eminent 4-α cluster state candidate in 16O
(Stellenbosch : Stellenbosch University, 2015-12) Li, Kevin Ching Wei; Papka, Paul; Neveling, Retief; Adsley, Philip; Stellenbosch University. Faculty of Sciences. Dept. of Physics.
ENGLISH ABSTRACT: Please see full text for abstract
Characterization of thulium doped fiber for mid infrared laser applications
(Stellenbosch : University of Stellenbosch, 2011-03) Ndebeka, Wilfrid Innocent; Rohwer, Erich G.; Schwoerer, Heinrich; University of Stellenbosch. Faculty of Science. Dept. of Physics.
Please refer to full text to view abstract.
Cluster model analysis of exotic decay in actinide nuclei
(Stellenbosch : Stellenbosch University, 2014-12) Du Toit, Erasmus Johannes; Wyngaardt, Shaun M.; Perez, Sandro M.; Stellenbosch University. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: The binary cluster model is used to investigate the properties of exotic structures and decays in various nuclei. A simple method is described to choose the possible clusters forming within the nucleus, by assuming the nucleus consists of a mixture of up to four different core-cluster pairs. A phenomenological potential is then used, with optimized parameters, to describe the even-even 222-232Th nuclei within the binary cluster model, by calculating exotic decay half-lives, reduced electromagnetic transition probabilities, and energy spectra. After finding that all experimentally observed heavy ion emissions are predicted with the model and calculated structure observables are reproduced within good agreement of the experimentally measured values, the model was extended to include the even-even 230-234U, 236-240Pu and 222-224Ra nuclei. We found that almost all experimentally observed heavy ion emissions are predicted within the model with reasonable accuracy in the calculated halflives, and found good agreement in the other calculated structure observables.
Collective coordinate description of kink-antikink interaction
(Stellenbosch : Stellenbosch University, 2015-12) Takyi, Ishmael; Weigel, Herbert; Stellenbosch University. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: In this thesis we explore resonance solutions in one space and one time dimensions in the nonlinear non-integrable ø⁴ and ø⁶ models. These resonances occur when the relative velocity between the kink and antikink falls below a specific critical velocity. It has been claimed that the presence of an internal vibrational mode is a necessary condition for the existence of these solutions. However, this mode is part of the solutions in the small amplitude formulation only in the ø⁴ model. Based on this, we investigate whether the availability of such a mode in the (reduced) collective coordinate ansätze will explain the resonance solutions regardless of whether or not it is a solution within the small amplitude approximation. We extend previous collective coordinate ansätze to three independent functions. Except for special scenarios this circumvents the problem seen in earlier studies that the coordinates are ill-declined when the kink-antikink separation tends to zero. We quantitatively compare the numerical results of the ordinary differential equations obtained from these ansätze to that of the partial differential equations in both the ø⁴ and ø⁶ models.
Computing the Arnold Tongue in the Zipoy-Voorhees Space-time
(Stellenbosch : Stellenbosch University, 2017-03) Sherif, Abbas Mohamed; Brink, Jeandrew; Stellenbosch University. Faculty of Science. Dept. of Physics
ENGLISH ABSTRACT : In this thesis I study the integrability of the geodesic equations of the ZipoyVoorhees metric. The Zipoy-Voorhees spacetime is a one parameter family of Stationary Axisymmetric Vacuum spacetimes (SAV’s) that is an exact solution to the vacuum Einstein Field Equations (EFE’s). It has been conjectured that the end state of any asymptotically flat black hole formed by astrophysical mechanisms, such as for example, gravitational collapse of a star, merger of two black holes etc will be a characterised by the Kerr metric. The black hole will thus be a possibly rotating, stationary axisymmetric vacuum spacetime characterised by its mass and spin and will possess no closed time-like curves. Investigating orbits in the Zipoy-Voorhees spacetime serves as a concrete example to of how the Kerr hypothesis fails. For this metric, I compute the Poincaré map and then compute the rotation curve. The Poincaré map is a tool to locate the region where chaos occurs in a dynamical system. The rotation curve is used to quantify chaos in the system. I focus my study on the 2/3 resonance for a range of the parameter values δ ∈ [1, 2]. The value δ = 1 corresponds to the Schwarzschild solution where the system is integrable. I then compute the Arnold tongue by plotting the size of the resonant regions against the parameter values to quantify the departure from integrability. I find that the shape of the tongue of instability is nonlinear and the Arnold tongue pinches off at δ = 1.6.
Constructing dualities from quantum state manifolds
(Stellenbosch : Stellenbosch University, 2015-12) Van Zyl, Hendrik Jacobus Rust; Scholtz, Frederik G.; Kriel, Johannes N.; Stellenbosch University. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: A constructive procedure to build gravitational duals from quantum mechanical models is developed with the aim of studying aspects of the gauge/gravity duality. The construction is simplified as far as possible - the most notable simplification being that quantum mechanical models are considered as opposed to quantum field theories. The simplifications allow a systematic development of the construction which provides direct access to the quantum mechanics / gravity dictionary. The procedure is divided into two parts. First a geometry is constructed from a family of quantum states such that the symmetries of the quantum mechanical states are encoded as isometries of the metric. Secondly, this metric is interpreted as the metric that yields a stationary value for the dual gravitational action. If the quantum states are non-normalisable then these states need to be regularised in order to define a sensible metric. These regularisation parameters are treated as coordinates on the manifold of quantum states. This gives rise to the idea of a manifold “bulk” where the states are normalisable and of a “boundary” where they are not. Asymptotically anti-de Sitter geometries arise naturally from non-normalisable states but the geometries can also be much more general. Time-evolved states are the initial interest. A sensible regularisation scheme for these states is a simple complexification of time so that the bulk coordinate has the interpretation of an energy scale. These two-dimensional manifolds of states are dual to models of dilaton gravity where the dilaton has the interpretation of the expectation value of a quantum mechanical operator. As an example, states time-evolving under an su(1, 1) Hamiltonian is dual to dilaton gravity on AdS2, in agreement with existing work on the AdS2/CFT1 correspondence. These existing results are revisited with the aid of the systematic quantum mechanics / dilaton gravity dictionary and extended. As another example, states time-evolving under an su(2) Hamiltonian are shown to be dual to dilaton gravity on dS2. The higher dimensional analysis is restricted, for computational reasons, to the example of states that possess full Schr¨odinger symmetry with and without dynamical mass. The time and spatial coordinates are complexified in order to both regularise the states and maintain the state symmetries as bulk isometries. Dictionaries are developed for both examples. It is shown that submanifolds of these state manifolds are studied in the existing AdS/CFT and AdS/NRCFT literature.
Construction and characterization of a multi-antenna terahertz time-domain spectroscopy setup
(Stellenbosch : Stellenbosch University, 2015-04) Smith, Shane Raymond; Neethling, Pieter H.; Rohwer, Erich G.; Stellenbosch University. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: Recent progress in laser and semiconductor technology has allowed for far easier generation and measuring of coherent terahertz radiation, a previously difficult region in the radiation spectrum to coherently generate. Time based terahertz spectroscopy is a rather unique form of spectroscopy. Not only is it time based, but the electric field is measured instead of the intensity. This allows for the measurement of the complex refractive index. From this one can obtain certain details of the structure and environment of the sample being studied. A terahertz time-domain spectroscopy setup was constructed during this project. This setup used low temperature grown GaAs photoconductive antennae, with multiple antenna size options available for both the receiving and transmitting antennae. After the construction and alignment of this setup, the antennae were characterized. Lastly measurements were performed on the background, sugar and silicon to demonstrate the capabilities of the system. It was found that the measured terahertz electric field amplitude increased with the intensity of the pump pulse and that the amplitude of the measured terahertz electric field was dependent on the polarization of the pump pulse. As the size of the antenna was increased so too did the amplitude of the measured electric field and conversely the bandwidth of the measured terahertz electric field decreased with the increase of antenna size. This held true for both the transmitting and receiving antennae.
Cross sections and analysing power energy-sharing distributions of valence (p,2p)-knockout from 208Pb with a projectile of 200MeV
(Stellenbosch : Stellenbosch University, 2000-12) Bezuidenhout, Jacques; Cowley, A. A.; Arendse, G. J.; Stellenbosch University. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: The aim of this work was to study the 208Pb(p,2p)207Tl quasi-free knockout process. The experimental data were measured at the National Accelerator Centre using incident polarised protons of 200 MeV. The two scattered particles, from the knockout reaction, were detected in coincidence and their energies were determined using a magnetic spectrometer and a solid state detector telescope. Cross section and analysing power energy distributions were extracted from the experimental measurements and these were compared with theoretical values for the Distorted Wave Impulse Approximation. The theoretical cross-section calculations predict the experimental cross-section distribution well for all combinations of distorting potentials and bound states that were investigated, both with regard to shape, as well as absolute magnitude. However the theoretical analysing power distributions did not agree with the experimental quantities. Therefore it is not clear whether the analysing power is a useful tool to extract information on the specifics of the quasi-free reaction mechanism. The spectroscopic factors were found to be consistent with the results obtained in previous studies, thereby inspiring confidence that the problem with the analysing power distribution is not ascribable to a possible deficiency in the experimental techniques exploited in this work.
Crossovers and phase transitions in Bose-Fermi mixtures
(Stellenbosch : Stellenbosch University, 2014-04) Kimene Kaya, Boniface Dimitri Christel; Avdeenkov, Alexander V.; Kriel, Johannes N.; Stellenbosch University. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: We present a theoretical approach that allows for the description of trapped Bose-Fermi mixtures with a tunable interspecies interaction in the vicinity of a Feshbach resonance magnetic field.The many-body physics of the system is treated at equilibrium using the well-established mean-field and local density approximations. This reduces the physics locally to that of a homogeneous system. We observe a rich local phase structure exhibiting both first and second order phase transitions between the normal and BEC phases. We also consider the global properties of the mixture at a fixed number of particles and investigate how the density profiles and the populations of the various particle species depend on the detuning and trap profile.
Data modelling & Bayesian model comparison with spherically symmetric priors
(Stellenbosch : Stellenbosch University, 2020-12) Jamodien, Riyaadh; Eggers, H. C.; De Kock, M. B.; Kriel, Johannes N.; Stellenbosch University. Faculty of Science. Dept. of Physics.
ENGLISH ABSTRACT: The analysis of data is common in many fields of science, and modelling data is one of the standard techniques in such analysis. Models are, of course, not unique and many theoretical models may be constructed to describe the same set of data. When considering many competing models, we naturally ask the question: which model best describes the data? It is well known that the chi-squared criterion, which is commonly cited as a goodness-offit between model and data, is inadequate as a measure of model quality. Rather, we employ the Bayesian framework of probability theory in addressing the question of model description of data. Within the Bayesian framework, the evidence (marginal likelihood) is the criterion by which to compare competing theoretical models. The evidence is an integral (over all parameter space) of the likelihood and prior. However, even for the simple case of linear models, there is no consensus or clarity on the choice of the best uninformative prior which enables the unbiased comparison of models with different numbers of parameters. In addressing the concern of the prior, we consider the framework of spherical symmetry, in which the evidence is reduced from an integral over a multi-dimensional space to that of a one-dimensional space, effectively reducing the problem to finding a single, optimal radial prior. We generalise existing results to a family of priors via scale relations in the form of a scaling parameter, of which several scaling relations are tested to find the best scale between models of different dimensions. We also introduce a new hyper-parameter, which had previously been conflated with model dimensions. With these developments we establish a prior that is sensitive to the new hyper-parameter, while insensitive to the model dimension, leading to the establishment of information criteria that are sensitive to these new parameters as well as the model dimension. These criteria are tested and shown to be an improvement over the existing body of work. These information criteria perform on par with widely accepted information criteria in the literature.

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