Is God a geometer or an analyst?

Dmitri Vassiliev (UCL)

The speaker is a specialist in the analysis of partial differential equations (PDEs) and the talk is an analyst's take on theoretical physics. We address the question: why do all the main equations of theoretical physics such as the Maxwell equation, Dirac equation and the linearized Einstein equation of general relativity contain the same physical constant - the speed of light? The accepted point of view is that this is because our world was designed on the basis of geometry, with the speed of light encoded in the concept of Minkowski metric. We suggest an alternative explanation: electromagnetism, fermions and gravity are different solutions of a single nonlinear hyperbolic system.

The speaker does not have a unifying field equation. Moreover, a comprehensive analysis of a system of nonlinear hyperbolic PDEs is beyond the reach of modern mathematical techniques. However, it will be shown that even in the linear setting one can recover many geometric constructs of theoretical physics from the analysis of PDEs. Namely, the speaker is currently engaged in the meticulous analysis of a general linear first order hyperbolic system of PDEs for a pair of unknown complex scalar fields in dimension 1+3. The coefficients of this system are not assumed to be constant and, moreover, can later be used as dynamical variables (poor man's way of dealing with nonlinearity). A modern mathematical technique known as microlocal analysis allows one to construct explicitly solutions in the form of a "wave packet" (here the rigorous mathematical synonym is "Fourier integral operator"). Careful examination of the structure of this wave packet reveals geometric structures such as metric, U(1) connection, spinor, spin connection and massless Dirac Lagrangian.