Indecomposable representations of Poincaré group and quantization of weak gravitational and electromagnetic fields

Abstract

In this paper a quantization procedure of the electromagnetic and the weak gravitational fields, in an indefinite (but non-negative) metric space of states, is presented. We show that the covariance axiom can only be formulated with non-unitary representations (of the Poincaré group) whose restrictions to the one-particle states are indecomposable. In the gravitational case the corresponding indecomposable representation describes six states of helicity for gravitons. The quantized field appears as six interacting massless fields of helicities 2, −2,1, −1,0 and 0. These six values of helicity correspond to the six modes of polarization in the most general metric theory of gravity. For the electromagnetic field one obtains an indecomposable representation of the Poincaré group with helicities 1, −1 and 0 corresponding to transverse and longitudinal photons.