Batalin–Vilkovisky formalism

Summary

The Batalin–Vilkovisky (BV) formalism is a method for the quantization of gauge theories that extends BRST quantization. To each field, including the ghosts, it adds an antifield of opposite parity. On the functions of fields and antifields it defines an odd bracket, the antibracket, and a second-order operator $\Delta$, the BV Laplacian. The action $S$ must satisfy the quantum master equation $\frac12 (S, S) = i\hbar\, \Delta S$. Gauge fixing becomes the choice of a Lagrangian submanifold on which the path integral is taken. The algebraic structure behind it is a Batalin–Vilkovisky algebra, and the formalism handles gauge symmetries whose algebra closes only up to equations of motion.