The Physics of Curves

In this talk we consider the simplest model of a particle, a point that traces out a curve in space time. To get physics, we consider an action given by a geometric invariant of the curve. The first invariant is the arc-length which describes a massive particle without spin. Quantisation of this model results in Klein-Gordon theory. The next invariant is the extrinsic curvature which describes a massless particle with spin. This gives a very geometric classical meaning to spin. The space of solutions turns out to be Twistor space. Quantisation therefore results in the massless field equations, such as the Weyl and Maxwell equations.