The talk deals with formally self-adjoint systems of first order linear partial differential equations on manifolds without boundary. We study the distribution of eigenvalues in the elliptic setting and the propagator in the hyperbolic setting, deriving two-term asymptotic formulae for both.

We then turn our attention to the special case of a system of two equations in dimension four. We show that the geometric concepts of Lorentzian metric, Pauli matrices, spinor field, connection coefficients for spinor fields, electromagnetic covector potential, Dirac equation and Dirac action arise naturally in the process of our analysis.

The talk is based on the following papers.

[1] O.Chervova, R.J.Downes and D.Vassiliev, The spectral function of a first order elliptic system, Journal of Spectral Theory, 2013, vol. 3, p. 317-360. Available as preprint arXiv:1208.6015.

[2] R.J.Downes, M.Levitin and D.Vassiliev, Spectral asymmetry of the massless Dirac operator on a 3-torus, Journal of Mathematical Physics, 2013, vol.54, article 111503. Available as preprint arXiv:1306.5689.

[3] O.Chervova, R.J.Downes and D.Vassiliev, Spectral theoretic characterization of the massless Dirac operator, Journal of the London Mathematical Society, 2014, vol. 89, 301-320. Available as preprint arXiv:1209.3510.

[4] R.J.Downes and D.Vassiliev, Spectral theoretic characterization of the massless Dirac action. Preprint arXiv:1401.1951.

[5] Y.-L.Fang and D.Vassiliev, Analysis as a source of geometry: a non-geometric representation of the Dirac equation. Preprint arXiv:1401.3160.

[6] Y.-L.Fang and D.Vassiliev, Analysis of first order systems of partial differential equations. Preprint arXiv:arXiv:1403.2663.