The main result of the talk is a new representation for the Weyl Lagrangian (massless Dirac Lagrangian). As the dynamical variable we use the coframe, i.e. field of orthonormal bases. This means that every point of the spacetime continuum can experience rotations and rotations at different points are totally independent. We write down a simple Lagrangian and show that variation of the resulting action with respect to the coframe produces the Weyl equation. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative.
The construction presented in the talk is almost identical to that used in the so-called Cosserat theory of elasticity (multipolar elasticity). There is also a similarity with the mathematical models of ferromagnetic materials and liquid crystals.
[1] D.Vassiliev, Phys. Rev. D75, 025006 (2007).