Massive gravity as a quantum gauge theory

TL;DR

This paper develops a quantum gauge theory framework for massive gravity using second quantization, describing the graviton Hilbert space, gauge structure, and self-interaction, with results connecting to classical gravity and cosmology.

Contribution

It introduces a novel quantum gauge theory approach to massive gravity, avoiding the need for a Higgs field and linking quantum solutions to classical gravitational constants.

Findings

Quantum Hilbert space constructed from Poincaré representations.

Self-interaction solution smoothly connects to linear quantum gravity.

No Higgs field needed in second-order perturbation theory.

Abstract

We present a new point of view on the quantization of the massive gravitational field, namely we use exclusively the quantum framework of the second quantization. The Hilbert space of the many-gravitons system is a Fock space ${\cal F}^{+}({\sf H}_{\rm graviton})$ where the one-particle Hilbert space ${\sf H}_{graviton}$ carries the direct sum of two unitary irreducible representations of the Poincar\'e group corresponding to two particles of mass $m > 0$ and spins 2 and 0, respectively. This Hilbert space is canonically isomorphic to a space of the type $Ker(Q)/Im(Q)$ where $Q$ is a gauge charge defined in an extension of the Hilbert space ${\cal H}_{\rm graviton}$ generated by the gravitational field $h_{\mu\nu}$ and some ghosts fields $u_{\mu}, \tilde{u}_{\mu}$ (which are vector Fermi fields) and $v_{\mu}$ (which are vector field Bose fields.) Then we study the self interaction of massive gravity in the causal framework. We obtain a solution which goes smoothly to the zero-mass solution of linear quantum gravity up to a term depending on the bosonic ghost field. This solution depends on two real constants as it should be; these constants are related to the gravitational constant and the cosmological constant. In the second order of the perturbation theory we do not need a Higgs field, in sharp contrast to Yang-Mills theory.