Batalin-Vilkovisky formalism

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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 &Delta; of degree -1, with &Delta;2=0 and &Delta;(1)=0. More precisely it satisfies theidentities
*|"ab"| = |"a"| + |"b"| (The product has degree 0)
*|&Delta;("a")| = 1+|"a"| (&Delta; has degree -1)
*("ab")"c"="a"("bc"), "ab"=(−1)|"a"||"b"|"ba" (the product is associative and (super) commutative)
*&Delta;2=0
*&Delta;(1)=0 (Normalization)
*&Delta; is second order, in other words for any "a", the supercommutator [&Delta;,"a"] is a derivation.

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

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:$\left[S,S\right] =0$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 &Delta;) is the equation:$\left[S,S\right] =2Delta\left(S\right)$or equivalently :$\left[S-Delta,S-Delta\right] =0.$

Examples

*If "L" is a Lie superalgebra, and &Pi; is the operator exchanging the even and odd parts of a super space, then the symmetric algebra of &Pi;("L") (the "exterior algebra" of "L") is a Batalin-Vilkovisky algebra with &Delta; 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 &Delta; to &Delta; + [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

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