Massive gravity as a quantum gauge 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.Comment: 35 pages, no figur

On massive spin 2 interactions

“…Thus, models based on the ξ µ invariance only like those based on OSP (4) symmetry [50] or models exploring λ transformations only [51] always require additional checks for consistency. One more example is a BRST construction of [52], where for massive spin 2 particles authors promote gauge transformations with vector parameter only (and introduce vector Goldstone field u µ only). As a result, such theory describes additional scalar degrees of freedom and all previous results point that it must be ghost.…”

Massive gravity from descent equations

“…It is simple to see that this difference of the two ghost couplings is a divergence and, therefore, uninteresting. The T α from [1] (eq. (2.10)) differs from (3.13) also in a trivial way by d Q D α + ∂ β D αβ with D αβ antisymmetric.…”

“…Gauge invariance of second order can be verified as in [1]. Alternatilvely, one can try to prove that if one modifies the Lagrangian by a total derivative and/or a coboundary then the higher-order chronologial products are also trivially modified; such a result is valid for Yang-Mills theories [3] [6].…”

“…A gravitational Higgs field is not needed in the massive spin-2 case in contrast to spin-1 [3]. As shown in [1] the second order couplings correspond to non-linear classical gravity in the following sense: Consider Einstein's Lagrangian with a cosmological constant Λ with the h-terms in (3.15) and the quartic part with the second order couplings derived from gauge invariance. This shows that the massive spin-2 gauge theory corresponds to Einstein gravity with a cosmological term.…”

From massive gravity to modified general relativity

“…For spin-1 theories this was demonstrated in the monograph [2], where the massless spin-2 case is also treated. The massive spin-2 theory was first investigated in [3]. The most elegant way to obtain the theory is by assuming the G. Scharf (B) Institut für Theoretische Physik, Universität Zürich, Winterthurerstr.…”