Batalin-Vilkovisky formalism

In theoretical physics, Batalin-Vilkovisky (BV) formalism was developed as a method for determining the ghost structure for theories, such as gravity and supergravity, whose Hamiltonian formalism has constraints not related to a Lie algebra action. The formalism, based on a Lagrangian that contains both fields and "antifields", can be thought of as a very complicated generalization of the BRST formalism.

Batalin-Vilkovisky algebras

A Batalin-Vilkovisky algebra is a graded supercommutative algebra (with identity 1) with a second-order differential operator Δ of degree -1, with Δ²=0 and Δ(1)=0. More precisely it satisfies theidentities
*|"ab"| = |"a"| + |"b"| (The product has degree 0)
*|Δ("a")| = 1+|"a"| (Δ has degree -1)
*("ab")"c"="a"("bc"), "ab"=(−1)^|"a"||"b"|"ba" (the product is associative and (super) commutative)
*Δ²=0
*Δ(1)=0 (Normalization)
*Δ is second order, in other words for any "a", the supercommutator [Δ,"a"] is a derivation.

A Batalin-Vilkovisky algebra becomes a Gerstenhaber algebra if one defines the Poisson bracket by:$[a,b]\; =\; (-1)^Delta(ab)\; -\; (-1)^Delta(a)b-aDelta(b)$.

Master equation

The (classical) master equation for an odd degree element "S" of a Batalin-Vilkovisky algebra (or more generally a Lie superalgebra) is the equation:$[S,S]\; =0$The quantum master equation for an odd degree element "S" of a Batalin-Vilkovisky algebra (or more generally a Lie superalgebra with an odd derivation Δ) is the equation:$[S,S]\; =2Delta(S)$or equivalently :$[S-Delta,S-Delta]\; =0.$

Examples

*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.
*If "A" is a Batalin-Vilkovisky algebra, and "S" a solution of the quantum master equation, then changing Δ to Δ + [S, ] gives a new Batalin-Vilkovisky algebra.

ee also

*analysis of flows.

References

*E. Getzler "Batalin-Vilkovisky algebras and two-dimensional topological field theories", Communications in Mathematical Physics, Volume 159, Number 2 / January, 1994, Pages 265-285. DOI|10.1007/BF02102639
*Steven Weinberg "The Quantum Theory of Fields Vol. II" ISBN 0521670543