A Noncommutative Note on the Antibracket Formalism

TL;DR

This paper develops a noncommutative calculus on an extended odd-symplectic superspace, linking the commutator to the antibracket and analyzing the invariance of the measure under canonical transformations.

Contribution

It introduces a noncommutative framework on extended superspace and relates the commutator to the antibracket, advancing the mathematical formalism of the antibracket formalism.

Findings

Commutator on extended superspace is proportional to the antibracket.

The $ riangle$-operator is characterized within a quotient space of derivations.

The measure on the superspace remains invariant under canonical transformations if a wave equation is satisfied.

Abstract

We introduce a noncommutative calculus on the odd-symplectic superspace $\scri$ of fields and antifields. To this end we have to extend $\scri$ to $\exscri$ by including an extra anticommuting field $\eta$. As a consequence we show that the commutator induced on $\exnice\times T^\star(\exnice)$ is proportional to the antibracket. The $\Delta$-operator is an element of the quotient space of derivations twisted by the antibracket $A\der$ and $\der$. The natural measure on $\exscri$ is shown to be invariant under canonical transformations provided a certain 'wave equation' is satisfied.