Unimodular Gravity in Restricted Gauge Theory Setup

Abstract

We develop Lagrangian quantization formalism for a class of theories obtained by the restriction of the configuration space of gauge fields from a wider (parental) gauge theory. This formalism is based on application of Batalin-Vilkovisky technique for quantization of theories with linearly dependent generators, their linear dependence originating from a special type of projection from the originally irreducible gauge generators of the parental theory. The algebra of these projected generators is shown to be closed for parental gauge algebras closed off shell. We demonstrate that new physics of the restricted theory as compared to its parental theory is associated with the rank deficiency of a special gauge restriction operator reflecting the gauge transformation properties of the restriction constraints functions -- this distinguishes restriction of the theory from its partial gauge fixing. As a byproduct of this technique a workable algorithm for one-loop effective action in generic first-stage reducible theory was constructed, along with the explicit set of tree-level Ward identities for gauge field, ghost and ghosts-for-ghosts propagators, which guarantee on-shell gauge independence of this effective action. The restricted theory formalism with reducible set of projected gauge generators is applied to unimodular gravity theory by using background-covariant gauge-fixing procedure. Its one-loop effective action is obtained in terms of functional determinants of minimal second order differential operators calculable on generic backgrounds by Schwinger-DeWitt technique of local curvature expansion. The result is shown to be equivalent to Einstein gravity theory with a cosmological term up to a special contribution of the global degree of freedom associated with the variable value of the cosmological constant. The role of this degree of freedom is briefly discussed.Comment: 47 page