  In this paper we deal with quadratic metric-affine gravity, which we briefly
introduce, explain and give historical and physical reasons for using this
particular theory of gravity. Further, we introduce a generalisation of well
known spacetimes, namely pp-waves. A classical pp-wave is a 4-dimensional
Lorentzian spacetime, which admits a nonvanishing parallel spinor field; here
the connection is assumed to be Levi-Civita. This definition was generalised in
our previous work to metric compatible spacetimes with torsion and used to
construct new explicit vacuum solutions of quadratic metric-affine gravity,
namely generalised pp-waves of parallel Ricci curvature. The physical
interpretation of these solutions we propose in this article is that they
represent a conformally invariant metric-affine model for a massless elementary
particle. We give a comparison with the classical model describing the
interaction of gravitational and massless neutrino fields, namely Einstein-Weyl
theory and construct pp-wave type solutions of this theory. We point out that
generalised pp-waves of parallel Ricci curvature are very similar to pp-wave
type solutions of the Einstein-Weyl model and therefore propose that our
generalised pp-waves of parallel Ricci curvature represent a metric-affine
model for the massless neutrino.
