Browsing by Author "Williams, Paul Henry"

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    Lorentz symmetry in non-commutative field theories : commutative/non-commutative dualities and manifest covariance
    (Stellenbosch : Stellenbosch University., 2020-04) Williams, Paul Henry; Scholtz, Frederik G.; Kriel, Johannes N.; Stellenbosch University. Faculty of Science. Dept. of Physics.
    ENGLISH ABSTRACT: We cover two approaches to building Lorentz invariant non-commutative field theories. First we construct dualities between commutative and non-commutative field theories. This construction exploits a generalization of the exact renormalization group equation (ERG). We review ERG dualities for the two dimensional quantum mechanical Landau problem. We also review the idea of non-canonical field theories. From this we build an ERG duality for the free non-canonical complex scalar field theory. This approach allows us to track the Lorentz symmetry, we show this explicitly for the free theory. Finally we construct dualities for the φ4 interacting theory. Second we build a manifestly Lorentz invariant 2 + 1 dimensional field theory living in SU(1, 1) fuzzy space-time. Here the commutation relations themselves respect the Lorentz symmetry. We start by reviewing the nonrelativistic construction of SU(2) fuzzy space and quantum mechanics. We briefly discuss a potential field theory extension of this. We then begin our SU(1, 1) construction by reviewing the SU(1, 1) group and make the connection to space-time. We then build a quantum theory living on this space-time and introduce dynamics with the Klein-Gordon equation. From this we can move on to a scalar field theory in terms of functions on the group manifold. Finally we make the connection to commutative theories by introducing the symbols of operators. From this we are able to compute different correlators and compare with commutative theories.
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    Relativistic quantum mechanics on non-commutative spaces
    (Stellenbosch : Stellenbosch University, 2015-12) Williams, Paul Henry; Scholtz, Frederik G.; Stellenbosch University. Faculty of Science. Dept. of Physics.
    ENGLISH ABSTRACT: A brief review of non-relativistic and relativistic quantum mechanics is carried out, with particular attention to concepts that are modified in the noncommutative setting. The Hilbert-Schmidt operator formulation of non-relativistic, non-commutative quantum mechanics is generalized to the relativistic setting of 4-dimensional non-commutative space-time. The generators of Lorentz transformations are derived in this formalism and compared with the commutative case. It is shown that the non-interacting, non-commutative Dirac equation is Lorentz invariant. Lorentz invariance can be maintained when electromagnetic interactions are included provided that the gauge and Dirac fields are composed through an appropriate star product. The appropriate gauge transformations for the non-commutative Dirac equation are found. The non-commutative C, P, T symmetries of the interacting Dirac equation are investigated. The free Dirac equation and the interacting Dirac equation for the case of a constant background magnetic field are studied on the operator level. Systems confined to a specific space-time volume, and the associated boundary conditions, are studied using appropriate projection operators. As a specific example the Dirac equation in an infinitely long cylinder is considered. An operator valued action, which yields the interacting Dirac equation as the equation of motion, is derived and evaluated in a coherent state basis. The constant background magnetic field is revisited in the coherent state basis. This establishes the link to the standard star product formulation of non-commutative quantum field theories. Useful properties of the star product are derived in an appendix.