Doctoral Degrees (Physics)

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Browsing Doctoral Degrees (Physics) by Subject "Analysis of covariance"

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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.