On the W-geometrical origins of massless field equations and gauge invariance

We show how to obtain all covariant field equations for massless particles of arbitrary integer, or half-integer, helicity in four dimensions from the quantization of the rigid particle, whose action is given by the integrated extrinsic curvature of its worldline, {\ie} $S=\alpha\int ds \kappa$. This geometrical particle system possesses one extra gauge invariance besides reparametrizations, and the full gauge algebra has been previously identified as classical $\W_3$. The key observation is that the covariantly reduced phase space of this model can be naturally identified with the spinor and twistor descriptions of the covariant phase spaces associated with massless particles of helicity $s=\alpha$. Then, standard quantization techniques require $\alpha$ to be quantized and show how the associated Hilbert spaces are solution spaces of the standard relativistic massless wave equations with $s=\alpha$. Therefore, providing us with a simple particle model for Weyl fermions ($\alpha=1/2$), Maxwell fields ($\alpha=1$), and higher spin fields. Moreover, one can go a little further and in the Maxwell case show that, after a suitable redefinition of constraints, the standard Dirac quantization procedure for first-class constraints leads to a wave-function which can be identified with the gauge potential $A_\mu$. Gauge symmetry appears in the formalism as a consequence of the invariance under $\W_3$-morphisms, that is, exclusively in terms of the extrinsic geometry of paths in Minkowski space. When all gauge freedom is fixed one naturally obtains the standard Lorenz gauge condition on $A_{\mu}$, and Maxwell equations in that gauge. This construction has a direct generalization to arbitrary integer values of $\alpha$, and we comment on the physically interesting case of linearized Einstein gravity ($\alpha =2$).