Matthias Roels
Published: 2013
Total Pages:
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Classical Clifford analysis is a generalisation of complex analysis to arbitrary dimension m. At the heart of the theory lies the Dirac operator, a conformally invariant first order differential operator which generalises the role of the Cauchy-Riemann operator. Moreover, the Laplace operator is the square of the Dirac operator, which means that Clifford analysis is a renement of harmonic analysis in m dimensions. While the classical theory is centred around the study of functions taking values in the spinor space, several authors have been studying generalisations of classical Clifford analysis techniques to the so-called higher spin theory. This concerns the study of higher spin operators acting on functions taking values in arbitrary irreducible representations of the spin group. In Clifford analysis, these arbitrary irreducible representations are traditionally defined in terms of polynomial spaces satisfying certain differential equations. From a physical point of view, higher spin fields arise quite naturally: it was shown by Bargmann and Wigner that group theory implies that particles should correspond to irreducible representations of the Lorentz group, labelled by a quantum number called spin. This led to generalisations of the Klein-Gordon and Dirac equation to higher spin equations (Dirac-Fierz-Pauli, Rarita-Schwinger, Buchdahl,...). Although particles with spin higher than one were never detected, a renewed interest in higher spin fields was triggered due to string theory. The reason is that the spectrum of the vibration modes of the strings includes an infinite number of fields of arbitrary increasing spin. With the advent of string theory, one was confronted with the fact that some theories require the space-time to have more than four dimensions. In higher dimensions, new kinds of fields are allowed because more general representations of the Lorentz group exist. In physics, these representations are typically defined as spaces of (traceless) tensors satisfying certain symmetry conditions expressed in terms of Young diagrams. The main goal of this thesis is twofold: in the first part, a proper introduction to Clifford analysis for physicists is given. The second part of this thesis focusses on higher spin fields where we first construct the two-spinor formalism. This is a framework to deal with arbitrary irreducible representations of the double cover of the Lorentz group. Then, this formalism will be linked with the polynomial spaces studied in Clifford analysis. While the two-spinor formalism works only in four dimensions, the polynomial models from Clifford analysis work in arbitrary dimensions and signatures. Finally, higher spin generalisations of the Maxwell equations are studied using techniques from Clifford analysis.