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### Browsing by Author "Scholtz, F. G."

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- ItemClassical dynamics on three-dimensional fuzzy space(American Physical Society, 2018) Scholtz, F. G.
Show more We derive the path integral action for a particle moving in three-dimensional fuzzy space. From this we extract the classical equations of motion. These equations have rather surprising and unconventional features: they predict a cutoff in energy, a generally spatial-dependent limiting speed, orbital precession remarkably similar to the general-relativistic result, flat velocity curves below a length scale determined by the limiting velocity and included mass, displaced planar motion, and the existence of two dynamical branches of which only one reduces to Newtonian dynamics in the commutative limit. These features may provide a stringent observational test for this scenario of noncommutativity.Show more - ItemDuality constructions from quantum state manifolds(Springer, 2015-11-20) Kriel, J. N.; Van Zyl, H. J. R.; Scholtz, F. G.
Show more The formalism of quantum state space geometry on manifolds of generalised coherent states is proposed as a natural setting for the construction of geometric dual descriptions of non-relativistic quantum systems. These state manifolds are equipped with natural Riemannian and symplectic structures derived from the Hilbert space inner product. This approach allows for the systematic construction of geometries which reflect the dynamical symmetries of the quantum system under consideration. We analyse here in detail the two dimensional case and demonstrate how existing results in the AdS2/CF T1 context can be understood within this framework. We show how the radial/bulk coordinate emerges as an energy scale associated with a regularisation procedure and find that, under quite general conditions, these state manifolds are asymptotically anti-de Sitter solutions of a class of classical dilaton gravity models. For the model of conformal quantum mechanics proposed by de Alfaro et al. [1] the corresponding state manifold is seen to be exactly AdS2 with a scalar curvature determined by the representation of the symmetry algebra. It is also shown that the dilaton field itself is given by the quantum mechanical expectation values of the dynamical symmetry generators and as a result exhibits dynamics equivalent to that of a conformal mechanical system.Show more - ItemNonperturbative flow equations from running expectation values(American Physical Society (APS), 2003-08) Scholtz, F. G.; Bartlett, Bruce H.; Geyer, H. B.
Show more We show that Wegner’s flow equations, as recently discussed in the Lipkin model, can be solved selfconsistently. This leads to a nonlinear differential equation which fully determines the order parameter as a function of the dimensionless coupling constant, even across the phase transition. Since we consider an expansion in the fluctuations, rather than the conventional expansion in the coupling constant, convergence to the exact results is found in both phases when taking the thermodynamic limit.Show more - ItemOn the thermodynamic limit at a quantum critical point(Academy of Science for South Africa, 2005) Geyer, H. B.; Heiss, W. D.; Scholtz, F. G.
Show more Open questions regarding the uniqueness and existence of the thermodynamic limit at a quantum critical point are discussed in the context of the Lipkin model, a popular model in many-body theory. This model illustrates a quantum phase transition, including spontaneous symmetry breaking. The thermodynamic limit seemingly yields different answers depending on the approach used. The discussion is based on recent developments, including a joint publication by the authors.Show more - ItemScattering in a three-dimensional fuzzy space(American Physical Society, 2017) Kriel, J. N.; Groenewald, H. W.; Scholtz, F. G.
Show more We develop scattering theory in a noncommutative space defined by an su(2) coordinate algebra. By introducing a positive operator valued measure as a replacement for strong position measurements, we are able to derive explicit expressions for the probability current, differential and total cross sections. We show that at low incident energies the kinematics of these expressions is identical to that of commutative scattering theory. The consequences of spatial noncommutativity are found to be more pronounced at the dynamical level where, even at low incident energies, the phase shifts of the partial waves can deviate strongly from commutative results. This is demonstrated for scattering from a spherical well. The impact of noncommutativity on the well’s spectrum and on the properties of its bound and scattering states are considered in detail. It is found that for sufficiently large well depths the potential effectively becomes repulsive and that the cross section tends towards that of hard sphere scattering. This can occur even at low incident energies when the particle’s wavelength inside the well becomes comparable to the noncommutative length scale.Show more