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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.
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 the identities
* |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)
* 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) | a | Δ(ab) − ( − 1) | a | Δ(a)b − aΔ(b).
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] = 2Δ(S)
[S − Δ,S − Δ] = 0.
* 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.
* analysis of flows.
* 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