Quantization of the massive gravitino on FRW spacetimes

In this article we study the quantization and causal properties of a massive spin $3/2$ Rarita-Schwinger field on spatially flat Friedmann-Robertson-Walker (FRW) spacetimes. We construct Zuckerman’s universal conserved current and prove that it leads to a positive definite inner product on solutions of the field equation. Based on this inner product, we quantize the Rarita-Schwinger field in terms of a CAR-algebra. The transversal and longitudinal parts constituting the independent on-shell degrees of freedom decouple. We find a Dirac-type equation for the transversal polarizations, ensuring a causal propagation. The equation of motion for the longitudinal part is also of Dirac-type, but with respect to an “effective metric”. We obtain that for all four-dimensional FRW solutions with a matter equation of state $p=\omega \rho$ and $\omega \in (-1,1]$ the light cones of the effective metric are more narrow than the standard cones, which are recovered for the de Sitter case $\omega =-1$. In particular, this shows that the propagation of the longitudinal part, although nonstandard for $\omega \ne -1$, is completely causal in cosmological constant, dust and radiation dominated universes.

Figure 1

Cosmological-time dependence of the effective speed of light for the longitudinal part. The plot shows with increasing dash length $\omega \in \{-1,0,\frac{1}{3}\}$, corresponding to a cosmological constant, dust and radiation dominated FRW universe, respectively.