Se p 20 07 Semidensities , Second - Class Constraints and Conversion in Anti - Poisson Geometry

We consider Khudaverdian's geometric version of a Batalin-Vilkovisky (BV) operator ∆ E in the case of a degenerate anti-Poisson manifold. The characteristic feature of such an operator (aside from being a Grassmann-odd, nilpotent, second-order differential operator) is that it sends semi-densities to semidensities. We find a local formula for the ∆ E operator in arbitrary coordinates. As an important application of this setup, we consider the Dirac antibracket on an antisymplectic manifold with antisymplectic second-class constraints. We show that the entire Dirac construction , including the corresponding Dirac BV operator ∆ ED , exactly follows from conversion of the antisymplectic second-class constraints into first-class constraints on an extended manifold.