Demonstration of noncausality for the Rarita-Schwinger equation

Abstract

Recently, there has been some intex'est on higher-spin equations. Velo and Zwanziger showed in Ref. 1 that the solutions of the Rarita-Schwinger (RS) equation are noncausal, using the method of characteristics. There are two main criticisms of the Velo and Zwanziger result. Fix'st, they axe applying the method of characteristics outside the domain of the existing proofs. ' Secondly, since the metric used in the HS equation is not positive definite, even the existence of the solutions i.s not certain. The positivity of the metric is used in an essential manner for the existence proofs of symmetric equations. ' There has been further work on this subject, and the existence of solutions for the RS equation has been shown quite xecently. ' However, we think it is still worthwhile to communicate this work, which was done some time ago, which might be complementary to Ref. 1 in some respect. Here, without treating in pux e mathematical rigor the problems of domains of definition, regularity of the solutions, etc., we explicitly construct the propagator for the RS equation in 2+1 dimensions. %e use Velo and Zwanziger's techniques' to reduce the RS equation into the modified form where we have a genuine equation of motion, which is equivalent to the original equation if we impose certain conditions at time I, For any electromagnetic potential, we do not know any solutions of this equation in 3+1 dimensions. In 2+1 dimensions, however, this equation can be easily solved for a constant magnetic field. From the solutions, we construct the propagator and explicitly see that the propagation of the 88 field in 2+1 dimensions is noncausal.