Batalin-Vilkovisky formalism
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Wikipedia
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 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)
- 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