Баталин-Вильковский формализмінің жалпыламасы

Generalization of the BRST formalism
In theoretical physics, the Batalin–Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the ghost structure for Lagrangian gauge theories, such as gravity and supergravity, whose corresponding Hamiltonian formulation has constraints not related to a Lie algebra (i. e., the role of Lie algebra structure constants are played by more general structure functions). The BV formalism, based on an action that contains both fields and "antifields", can be thought of as a vast generalization of the original BRST formalism for pure Yang–Mills theory to an arbitrary Lagrangian gauge theory. Other names for the Batalin–Vilkovisky formalism are field antifield formalism, Lagrangian BRST formalism, or BV–BRST formalism. It should not be confused with the Batalin–Fradkin–Vilkovisky (BFV) formalism, which is the Hamiltonian counterpart.

The Δ operator is by definition of n'th order if and only if the (n + 1) bracket vanishes. In that case, one speaks of a BV n algebra. Thus a BV 2 algebra is by definition just a BV algebra. The Jacobiator vanishes within a BV algebra, which means that the antibracket here satisfies the Jacobi identity. A BV 1 algebra that satisfies normalization Δ(1) = 0 is the same as a differential graded algebra (DGA) with differential Δ. A BV 1 algebra has vanishing antibracket.

The Schouten–Nijenhuis bracket for multi vector fields is an example of an antibracket. If L is a Lie superalgebra, and Π is the operator exchanging the even and odd parts of a super space, then the symmetric algebra of Π(L) (the "exterior algebra" of L) is a Batalin–Vilkovisky algebra with Δ given by the usual differential used to compute Lie algebra cohomology.

Costello, K. (2011). "Renormalization and Effective Field Theory". (Explains perturbative quantum field theory and the rigorous aspects, such as quantizing Chern Simons theory and Yang Mills theory using BV formalism)