The quantization of a point particle with extrinsic curvature leads to the Dirac equation

Abstract

A rigid point particle, an analogue of the rigid string or membrane, has been considered. Its action containing an extrinsic curvature term has been cast into an equivalent form in which there occurs a one-dimensional vielbein. Classical constraints, hamiltonian, equations of motion and various Poisson bracket relations have been derived. The theory has then been quantized. It has been shown that the equation of state contains the Dirac equation as a special case. The Dirac matrices are just one particular representation of the modified four-velocity operator. Quantum spin has its classical analogue in the helical motion of a rigid point particle. The frequency of the quantum Zitterbewegung turns out to be equal to the frequency of the classical helical motion.