Massless particles and the geometry of curves . Classical picture

We analyze the possibility of description of D−dimensional massless particles by the Lagrangians linear on world-line curvatures ki, S = ∑ N i=1 ci ∫ kids̃. We show, that the nontrivial classical solutions of this model are given by space-like curves with zero 2N−th curvature for N ≤ [(D−2)/2]. Massless spinning particles correspond to the curves with constant kN+a/kN−a ratio. It is shown that only the system with action S = c ∫ kNds̃ leads to irreducible representation of Poincaré group. This system has maximally possible number (N + 1) of gauge degrees of freedom. Its classical solutions obey the conditions kN+a = kN−a, a = 1, . . . , N − 1, while first N curvatures ki remain arbitrary. This solution is specified by coinciding N weights of the massless representation of little Lorentz group, while the remaining weights vanish.