Downes, RJ; (2014) Cosserat Elasticity, Spectral Theory of First Order Systems, and the Massless Dirac Operator. Doctoral thesis , UCL (University College London).

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Abstract

This thesis is concerned with the study of the massless Dirac operator in dimen- sion three and is, in part, based upon [12, 22, 21, 26, 25]. An introduction is given in Chapter 1. In Chapter 2 we study a special version of Cosserat elasticity, with deformations induced by rotations only, and no displacements. We prove that for a particular choice of elastic moduli and in the stationary setting (harmonic dependence on time) our mathematical model reduces to the massless Dirac equation. Chapter 3 contains a description of the progress recently made in the spectral theory of first order systems, with a particular focus on dimension three presented in Chapter 4. We prove in Chapter 5 that the second asymptotic coefficient of the counting function of a first order system has the geometric meaning of the massless Dirac action. Finally, in Chapter 6 we examine the spectral asymmetry of the massless Dirac operator. We work on a 3-torus equipped, initially, with a Euclidean metric and consider the behaviour of the spectrum under a perturbation of the metric. We derive an explicit asymptotic formula for the eigenvalue closest to zero.

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