Propagator of a massive spin-3 field

Abstract

The Lagrangian matrix $\ensuremath{\Lambda}(p)$ of a massive spin-3 field proposed by Kawasaki and Kobayashi is fully inverted to give the correct propagator. The three parameters contained in $\ensuremath{\Lambda}(p)$ are kept arbitrary throughout the calculation. It is stressed that in the local covariant Lagrangian approach the propagator should be the strict inverse matrix of $\ensuremath{\Lambda}(p)$. The reason for this is well illustrated by an example of a massive spin-2 theory. In so doing, we point out the following facts: (1) Since the Lagrangian given by Singh for a massive spin-2 particle is equivalent to that given by Bhargawa and Watanabe, for which the propagator is known, the propagator of the former is obtained from the latter by a simple transformation. (2) Although the auxiliary components appearing in high-spin field theories vanish because of the Euler-Lagrange equation, they have a finite contribution to the $S$-matrix elements through covariant $T$ products, which are in general nonvanishing. (3) The generalized Matthews rule is naturally supported by the path-integral prescription. (4) After eliminating the auxiliary components by a path integral, we have an effective nonpolynomial Lagrangian matrix, which is the inverse of the "partial propagator" previously obtained by many authors.