Vector and tensor representations of the Poincaré group

Abstract

The representations of the Poincare group on the space of vectorsAμ(K) and symmetric tensorshμν(K) are analysed. It is proved that for vanishing mass one cannot have a unitary vector (tensor) representation with helicity ± 1 (± 2). To reconcile covariance and unitarity the concept of induced representations is introduced. It is shown that one can get induced unitary representations of helicity ± 1 and ± 2 from the vector and tensor representations, which are essentially those used in Gupta’s quantization of Maxwell and Einstein equations. Thus Gupta method is seen to be intimately connected with the requirement of unitarity of the representation of the Poincare group. Therefore its introduction can be made independent of any condition of the Hilbert-Lorentz type, since in a group-theoretical framework it follows in a natural way.