Batalin-Vilkovisky formalism

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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.

1 Batalin-Vilkovisky algebras
2 Master equation
3 Examples
4 See also
5 References

[edit] 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

[a, b] = ( - 1) | a | Δ(a b) - ( - 1) | a | Δ(a) b - a Δ(b)

[edit] 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] = 2Δ(S)

or equivalently

[S - Δ, S - Δ] = 0.

[edit] 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.

[edit] See also

analysis of flows.

[edit] 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