Spin-2 propagating torsion

Abstract

We examine a general Lagrangian (containing terms up to bilinear in torsion and Riemann curvature tensors) and search for a choice of Lagrangian parameters that will allow massive spin-2 or massless helicity-2 torsion quanta to exist without accompanying ghosts or tachyons. We are able to rule out massless quanta by an argument at the linearized level: either there are ghosts in the propagator, or the source constraints force ${2}^{+}$ and ${2}^{\ensuremath{-}}$ torsion sources to cancel in pairs at the propagator poles. We conjecture that a difficulty of this type will arise whenever a massless boson is represented in the Lagrangian by a field transforming as a (${j}_{1}$,${j}_{2}$) SL(2, $C$) tensor with ${j}_{1}\ensuremath{\ne}{j}_{2}$, and the components of this tensor are taken as the quantities to be varied. (In the electromagnetic case ${F}_{\ensuremath{\mu}\ensuremath{\nu}}$ is replaced by the curl of ${A}_{\ensuremath{\mu}}$ and ${A}_{\ensuremath{\mu}}$ is varied, not ${F}_{\ensuremath{\mu}\ensuremath{\nu}}$.) We cannot rule out massive spin-2 quanta at the linearized level, but we encounter the standard difficulties when we attempt an extension to a fully covariant theory. Unless there are further developments in our understanding of the extension to the covariant theory, then, any spin-2 torsion force will have to be of "contact" type, rather than one which arises from the exchange of quanta.