Relativistic Wave Equations and Field Theory for Arbitrary Spin

Abstract

The possibility of formulating covariant field equations for particles of arbitrary spin and nonzero mass is considered. Using only the transformation properties of the single-particle states and the covariant under the Poincar\'e group, we investigate all possible covariant field equations which satisfy the following requirements: (i) neither auxiliary fields nor subsidiary conditions are used; (ii) the solutions of the free-field equation correspond to a unique irreducible representation of the Poincar\'e group [characterized by ($m,s$)] and the discrete transformations $C$, $P$, and $T$. If we restrict ourselves to equations which are homogeneous in the differential operator (except the mass term), it is shown that the only possible equations are those for spin 0, \textonehalf{}, and 1. These consist of all the known equations plus a few new ones. The general formalism also allows equations which are not homogeneous in the differential operator. In this case we again found strong evidence that although new equations for spin 0, \textonehalf{}, and 1 can be found, no equation for higher spin exists. It is shown, therefore, that the clear distinction between field theories of spin 0, \textonehalf{}, and 1 and those of all higher spin already arises in their group structure quite independent of quantization or any dynamical scheme such as the Lagrangian formalism. It is shown that, if the requirements of unique mass and/or definite transformation properties under $C$ and $P$ are relaxed, then general classes of field equations exist for any spin $s$. Apparent difficulties associated with possible field theories based on these equations are briefly indicated.