On the Quantization of the Gravitational Field

Abstract

We present a new point of view on the quantization of the gravitational field, namely we use exclusively the quantum framework of the second quantization. More explicitly, we take as one-particle Hilbert space, $H_{graviton}$ the unitary irreducible representation of the Poincar\'e group corresponding to a massless particle of helicity 2 and apply the second quantization procedure with Einstein-Bose statistics. The resulting Hilbert space ${\cal F}^{+}(H_{graviton})$ is, by definition, the Hilbert space of the gravitational field. Then we prove that this Hilbert space is canonically isomorphic to a space of the type $Ker(Q)/Im(Q)$ where $Q$ is a supercharge defined in an extension of the Hilbert space ${\cal F}^{+} (H}_{graviton})$ by the inclusion of ghosts: some Fermion ghosts $u_{\mu}, \tilde{u}_{\mu}$ which are vector fields and a Bosonic ghost $\Phi$ which is a scalar field. This has to be contrasted to the usual approaches where only the Fermion ghosts are considered. However, a rigorous proof that this is, indeed, possible seems to be lacking from the literature.