The dimensional reduction of linearized spin-2 theories invariant under transverse diffeomorphisms

Here we perform the Kaluza–Klein dimensional reduction from $$D+1$$ D + 1 to D dimensions of massless Lagrangians described by a symmetric rank-2 tensor and invariant under transverse differmorphisms (TDiff). They include the linearized Einstein–Hilbert theory, linearized unimodular gravity and scalar tensor models. We obtain simple expressions in terms of gauge invariant field combinations and show that unitarity is preserved in all cases. After fixing a gauge, the reduced model becomes a massive scalar tensor theory. We show that the diffeomorphism (Diff) symmetry, instead of TDiff, is a general feature of the massless sector of consistent massive scalar tensor models. We discuss some subtleties when eliminating Stückelberg fields directly at action level as gauge conditions. A non local connection between the massless sector of the scalar tensor theory and the pure tensor TDiff model leads to a parametrization of the non conserved source which naturally separates spin-0 and spin-2 contributions in the pure tensor theory. The case of curved backgrounds is also investigated. If we truncate the non minimal couplings to linear terms in the curvature, vector and scalar constraints require Einstein spaces as in the Diff and WTDiff (Weyl plus Diff) cases. We prove that our linearized massive scalar tensor models admit those curved background extensions.

NONTRIVIAL GHOSTS AND SECOND-CLASS CONSTRAINTS

In a model in which a vector gauge field [Formula: see text] is coupled to an antisymmetric tensor field [Formula: see text] possessing a pseudoscalar mass, it has been shown that all physical degrees of freedom reside in the vector field. Upon quantizing this model using the Faddeev–Popov procedure, explicit calculation of the two-point functions 〈ϕϕ〉 and 〈Wϕ〉 at one-loop order seems to have yielded the puzzling result that the effective action generated by radiative effects has more physical degrees of freedom than the original classical action. In this paper we point out that this is not in fact a real effect, but rather appears to be a consequence of having ignored a "ghost" field arising from the contribution to the measure in the path integral arising from the presence of nontrivial second-class constraints. These ghost fields couple to the fields [Formula: see text] and [Formula: see text], which makes them distinct from other models involving ghosts arising from second-class constraints (such as massive Yang–Mills (YM) models) that have been considered, as in these other models such ghosts decouple. As an alternative to dealing with second-class constraints, we consider introducing a "Stueckelberg field" to eliminate second-class constraints in favor of first-class constraints and examine if it is possible to then use the Faddeev–Popov quantization procedure. In the Proca model, introduction of the Stueckelberg vector is equivalent to the Batalin–Fradkin–Tyutin (BFT) approach to converting second-class constraints to being first-class through the introduction of new variables. However, introduction of a Stueckelberg vector is not equivalent to the BFT approach for the vector–tensor model. In an appendix, the BFT procedure is applied to the pure tensor model and a novel gauge invariance is found. In addition, we also consider extending the Hamiltonian so that half of the second-class constraints become first-class and the other half become associated gauge conditions. We also find for this tensor-vector theory that when converting the phase space path integral to the configuration space path integral, a nontrivial contribution to the measure arises that is not manifestly covariant and which is not simply due to the presence of second-class constraints.