Classical supersymmetric particles

Abstract

The dynamics of classical spinning particles is studied from the point of view of gauge supersymmetry. The central idea is that the natural way of introducing intrinsic spin degrees of freedom into a physical system is to take the square root of the Hamiltonian generators of the system without spin, which is equivalent to rendering the system gauge supersymmetric. This is accomplished by describing the spin degrees of freedom by means of ’’anticommuting c‐numbers’’ (odd Grassman algebra elements) and relying on Dirac’s theory of constrained Hamiltonian systems. The requirement of gauge supersymmetry fixes completely the action principle and leaves neither room nor need for ad hoc subsidiary conditions on the relative direction of the spin and the velocity as in the more traditional treatments. Both massive and massless particles free and in interaction with electromagnetic and gravitational fields are discussed. It is found that there exists a supergauge in which the spin tensor of a massive particle in a gravitational field is transported in parallel but the particle does not follow a geodesic. Massless particles on the other hand have the property of possessing a supergauge where their helicity is conserved and in which at the same time the worldline is a geodesic. Special attention is paid to the meaning and properties of the supergauge transformations. The main aspects of that discussion are applicable to more complicated systems such as supergravity. In particular phenomena such as necessity of invoking the equations of motion to close the gauge are analyzed.