Title:

The massless Dirac equation from the continuum mechanics and microlocal analysis perspectives

The thesis is concerned with the study of the massless Dirac equation. In the first part we study the massless Dirac equation in dimension 1+3 in the stationary setting, i.e. when the spinor field oscillates harmonically in time. We suggest a new geometric interpretation for this equation. We think of our 3dimensional space as an elastic continuum and assume that material points can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables we choose the coframe and a density. We choose a particular potential energy which is conformally invariant and then incorporate time into our action by subtracting kinetic energy. We prove that in the stationary setting our model is equivalent to a pair of massless Dirac equations. In the second part we consider an elliptic selfadjoint first order pseudodifferential operator acting on columns of m complexvalued halfdensities over a compact ndimensional manifold. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator is not necessarily semibounded. We study the spectral function and derive a twoterm asymptotic formula. We then restrict our study to the case when m=2, n=3, the operator is differential and has tracefree principal symbol, and address the question: is our operator a massless Dirac operator? We prove that it is a massless Dirac operator if and only if, at every point, a) the subprincipal symbol is proportional to the identity matrix and b) the second asymptotic coefficient of the spectral function is zero.
