Lagrangian model of a massless particle on spacelike curves

We consider a model of a massless particle in a $D$-dimensional space with the Lagrangian proportional to the $N$th extrinsic curvature of the world line. We present the Hamiltonian formulation of the system and show that its trajectories are spacelike curves satisfying the conditions $k_{N+a}=k_{N-a}$ and $k_{2N}=0$, $a=1,\dots,N-1$, where $N\leq\bigl[(D-2)/2\bigr]$. The first $N$ curvatures take arbitrary values, which is a manifestation of $N+1$ gauge degrees of freedom; the corresponding gauge symmetry forms an algebra of the $W$ type. This model describes $D$-dimensional massless particles, whose helicity matrix has $N$ coinciding nonzero weights, while the remaining $\bigl[(D-2)/2\bigr]-N$ weights are zero. We show that the model can be extended to spaces with nonzero constant curvature. It is the only system with the Lagrangian dependent on the world-line extrinsic curvatures that yields irreducible representations of the Poincaré group.