Separazione delle variabili per l'equazione di Dirac

The aim of this paper is to reproduce the work done by Professor McLenaghan, Professor Fatibene and Professor Rastelli on the separation of variables of the Dirac equation in dimension $2$ and Euclidean geometry. This paper will consider both the Lorentzian and the Riemennian metric. The ambition of our work is to give a contribution in the problem of finding solutions of the Dirac equation. The equation was postulated by Dirac in 1928 and it provides a description of elementary particles. It is consistent with both the theory of quantum mechanics and special relativity. The Dirac equation is considered one of the most important result of modern theoretical physics since it predicted the existence of antiparticles that were later confirmed experimentally. The problem of separation of variables has been well developed for the Hamilton-Jacobi equation and the Schrodinger equation based on the the existence of valence 2 characteristic Killing tensors which define, respectively, quadratic first integrals and second order symmetry operators for these equations. Considered the similarities between the Dirac and the Schrodinger equation, it is only natural to look for an analogous theory of separation. Complications arise from the fact that one is dealing with a system of first order partial differential equations whose derivation from the invariant Dirac equation depends not only on the choice of coordinate system but also on the choice of an orthonormal moving frame and representation for the Dirac matrices with respect to which the component of the unknown spinors are defined. The spinor formalism is a necessary framework since we are trying to describe the motion of elementary spin particles. An other complication is that the background space-time is either flat or curved. We decided to work in the lowest permitted dimension, namely on two dimensional spin manifolds, because it is possible to examine all the different possible scenarios that arise and the imposition of the separation paradigm that the separation be characterized by a symmetry operator admitting the separable solutions as eigenfunctions. In the chapter "Preliminaries", we present the main results in separation of variables for the case of the Helmholtz equation and the paradigm that we applied for the Dirac case. Then we introduce some elements of the spinor theory. In chapter II we obtain the most general second order linear differential operator which commutes with the Dirac operator on a general two dimensional spin manifold in curved and flat spacetime. We show that it is characterized in terms of Killing vectors defined on the background manifold. We work in a general orthonormal frame and we don't choose a particular set of Dirac matrices in order to have a covariant derivation. In chapter III, using a multiplicative naive separation assumption, we compute all the possible operators that commutes witht he Dirac's in a two dimensional manifold. Finally, in chapter IV, we estabilish a link between the second order symmetry operators studied in chapter II and the naive separation of variables considered in chapter III by exploiting the fact that the existence of a nontrivial second order symmetry operator implies in general that the Liouville metric has at least one ignorable coordinate.