Chapter 22: Relativistic Action-at-a-Distance Theories

We have seen that the canonical formalism is incapable of describing a relativistic many-particle system including interactions, if manifest covariance is demanded. (In this chapter the only relativity group we are concerned with is the Poincaré group.) Such dynamical systems must be described by other means. For suitable types of interactions we may still be able to derive the basic equations of the system from an action principle, and in this chapter we study examples of precisely such systems. However, the action functional can no longer be expected to be the time integral of a suitable Lagrangian which is local in time, and hence the basic “equations of motion” also are not local in time. Interactions between particles in the Newtonian nonrelativistic (or better Galilean relativistic) domain are summarized in the notion of a potential that acts nonlocally (“at a distance”) in space at a given time. If this is to be generalized to the Poincaré-relativistic domain and the description is manifestly covariant, the nonlocality in space implicit in the notion of a potential automatically implies nonlocality in time. This comes in from the geometry of Lorentz transformations that makes simultaneity frame-dependent; in contrast to a nonrelativistic potential that can act at one instant of time in every frame, a “relativistic potential” acts at unequal times as well…