Batalin–Vilkovisky Formalism
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theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, the Batalin–Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the
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structure for Lagrangian
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, such as gravity and
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, whose corresponding Hamiltonian formulation has constraints not related to a
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(i.e., the role of Lie algebra structure constants are played by more general structure functions). The BV formalism, based on an
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that contains both
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and "antifields", can be thought of as a vast generalization of the original BRST formalism for pure Yang–Mills theory to an arbitrary Lagrangian gauge theory. Other names for the Batalin–Vilkovisky formalism are field-antifield formalism, Lagrangian BRST formalism, or BV–BRST formalism. It should not be confused with the Batalin–Fradkin–Vilkovisky (BFV) formalism, which is the Hamiltonian counterpart.


Batalin–Vilkovisky algebras

In mathematics, a Batalin–Vilkovisky algebra is a graded
supercommutative algebra In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have :yx = (-1)^xy , where , ''x'', denotes the grade of the element and is 0 or 1 ...
(with a unit 1) with a second-order nilpotent operator Δ of degree −1. More precisely, it satisfies the identities *(ab)c = a(bc) (The product is associative) *ab = (-1)^ba (The product is (super-)commutative) *, ab, = , a, + , b, (The product has degree 0) *, \Delta(a), = , a, - 1 (Δ has degree −1) *\Delta^2 = 0 (Nilpotency (of order 2)) * The Δ operator is of second order: :\begin 0 = & \Delta(abc) \\ &-\Delta(ab)c-(-1)^a\Delta(bc)-(-1)^b\Delta(ac)\\ &+\Delta(a)bc+(-1)^a\Delta(b)c+(-1)^ab\Delta(c)\\ &-\Delta(1)abc \end One often also requires normalization: *\Delta(1)=0 (normalization)


Antibracket

A Batalin–Vilkovisky algebra becomes a Gerstenhaber algebra if one defines the Gerstenhaber bracket by :(a,b) := (-1)^\Delta(ab) - (-1)^\Delta(a)b - a\Delta(b)+a\Delta(1)b . Other names for the Gerstenhaber bracket are Buttin bracket, antibracket, or odd Poisson bracket. The antibracket satisfies * , (a,b), = , a, +, b, - 1 (The antibracket (,) has degree −1) * (a,b) = -(-1)^(b,a) (Skewsymmetry) * (-1)^(a,(b,c)) + (-1)^(b,(c,a)) + (-1)^(c,(a,b)) = 0 (The Jacobi identity) * (ab,c) = a(b,c) + (-1)^b(a,c) (The Poisson property; the Leibniz rule)


Odd Laplacian

The normalized operator is defined as : _ := \Delta-\Delta(1) . It is often called the odd Laplacian, in particular in the context of odd Poisson geometry. It "differentiates" the antibracket * _(a,b) = (_(a),b) - (-1)^(a,_(b)) (The _ operator differentiates (,)) The square _^=(\Delta(1),\cdot) of the normalized _ operator is a Hamiltonian vector field with odd Hamiltonian Δ(1) * _^(ab) = _^(a)b+ a_^(b) (The Leibniz rule) which is also known as the modular vector field. Assuming normalization Δ(1)=0, the odd Laplacian _ is just the Δ operator, and the modular vector field _^ vanishes.


Compact formulation in terms of nested commutators

If one introduces the left multiplication operator L_ as : L_(b) := ab , and the
supercommutator In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a \Z/2\Zgraded algebra, grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. The notio ...
as : ,T=ST - (-1)^TS for two arbitrary operators ''S'' and ''T'', then the definition of the antibracket may be written compactly as : (a,b) := (-1)^ \Delta,L_L_]1 , and the second order condition for Δ may be written compactly as : \Delta,L_L_">\Delta,L_L_L_]1 = 0 (The Δ operator is of second order) where it is understood that the pertinent operator acts on the unit element 1. In other words, Delta,L_ is a first-order (affine) operator, and \Delta,L_L_] is a zeroth-order operator.


Master equation

The classical
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for an even degree element ''S'' (called the
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) of a Batalin–Vilkovisky algebra is the equation :(S,S) = 0 . The quantum master equation for an even degree element ''W'' of a Batalin–Vilkovisky algebra is the equation : \Delta\exp \left fracW\right= 0 , or equivalently, :\frac(W,W) = i\hbar_(W)+\hbar^\Delta(1) . Assuming normalization Δ(1) = 0, the quantum master equation reads :\frac(W,W) = i\hbar\Delta(W) .


Generalized BV algebras

In the definition of a generalized BV algebra, one drops the second-order assumption for Δ. One may then define an infinite hierarchy of higher brackets of degree −1 : \Phi^(a_,\ldots,a_) := \underbrace_1 . The brackets are (graded) symmetric : \Phi^(a_,\ldots,a_) = (-1)^\Phi^(a_,\ldots, a_) (Symmetric brackets) where \pi\in S_ is a permutation, and (-1)^ is the Koszul sign of the permutation :a_\ldots a_ = (-1)^a_\ldots a_. The brackets constitute a
homotopy Lie algebra In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L_\infty-algebra) is a generalisation of the concept of a differential graded Lie algebra. To be a little more specific, the Jacobi identity only holds up to hom ...
, also known as an L_ algebra, which satisfies generalized Jacobi identities : \sum_^n \frac\sum_(-1)^\Phi^\left(\Phi^(a_, \ldots, a_), a_, \ldots, a_\right) = 0. (Generalized Jacobi identities) The first few brackets are: * \Phi^ := \Delta(1) (The zero-bracket) * \Phi^(a) := Delta,L_ = \Delta(a) - \Delta(1)a =: _(a) (The one-bracket) * \Phi^(a,b) := \Delta,L_L_]1 =: (-1)^(a,b) (The two-bracket) * \Phi^(a,b,c) := \Delta,L_L_">\Delta,L_L_L_]1 (The three-bracket) * \vdots In particular, the one-bracket \Phi^=_ is the odd Laplacian, and the two-bracket \Phi^ is the antibracket up to a sign. The first few generalized Jacobi identities are: * \Phi^(\Phi^0) = 0 (\Delta(1) is \Delta_\rho-closed) * \Phi^(\Phi^,a)+\Phi^\left(\Phi^(a)\right) (\Delta(1) is the Hamiltonian for the modular vector field _^) * \Phi^(\Phi^,a,b) + \Phi^\left(\Phi^(a),b\right)+(-1)^\Phi^\left(a,\Phi^(b)\right) +\Phi^\left(\Phi^(a,b)\right) = 0 (The _ operator differentiates (,) generalized) * \Phi^(\Phi^,a,b,c) + (a,b,c)+ \Phi^\left(\Phi^(a,b,c)\right) + \Phi^\left(\Phi^(a),b,c\right) + (-1)^\Phi^\left(a,\Phi^(b),c\right) +(-1)^\Phi^\left(a,b,\Phi^(c)\right) = 0 (The generalized Jacobi identity) * \vdots where the Jacobiator for the two-bracket \Phi^ is defined as : (a_,a_,a_) := \frac \sum_(-1)^ \Phi^\left(\Phi^(a_,a_),a_\right) .


BV ''n''-algebras

The Δ operator is by definition of n'th order if and only if the (''n'' + 1)-bracket \Phi^ vanishes. In that case, one speaks of a BV n-algebra. Thus a BV 2-algebra is by definition just a BV algebra. The Jacobiator (a,b,c)=0 vanishes within a BV algebra, which means that the antibracket here satisfies the Jacobi identity. A BV 1-algebra that satisfies normalization Δ(1) = 0 is the same as a differential graded algebra (DGA) with differential Δ. A BV 1-algebra has vanishing antibracket.


Odd Poisson manifold with volume density

Let there be given an (n, n)
supermanifold In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below. Informal definition An informal definition is com ...
with an odd Poisson bi-vector \pi^ and a Berezin volume density \rho, also known as a P-structure and an S-structure, respectively. Let the local coordinates be called x^. Let the derivatives \partial_f and : f\stackrel_:=(-1)^\partial_f denote the
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and right derivative of a function ''f'' wrt. x^, respectively. The odd Poisson bi-vector \pi^ satisfies more precisely * \left, \pi^\ = \left, x^\ + \left, x^\ -1 (The odd Poisson structure has degree –1) * \pi^ = -(-1)^ \pi^ (Skewsymmetry) * (-1)^\pi^\partial_\pi^ + (i,j,k) = 0 (The Jacobi identity) Under change of coordinates x^ \to x^ the odd Poisson bi-vector \pi^ and Berezin volume density \rho transform as * \pi^ = x^\stackrel_ \pi^ \partial_x^ * \rho^ = \rho/(\partial_x^) where ''sdet'' denotes the
superdeterminant In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices. The name is for Felix Berezin. The Berezinian plays a role analogous to the determinant when considerin ...
, also known as the Berezinian. Then the odd Poisson bracket is defined as : (f,g) := f\stackrel_\pi^\partial_g . A Hamiltonian vector field X_ with Hamiltonian ''f'' can be defined as : X_ := (f,g) . The (super-)
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of a vector field X=X^\partial_ is defined as : _ X := \frac \partial_(\rho X^) Recall that Hamiltonian vector fields are divergencefree in even Poisson geometry because of Liouville's Theorem. In odd Poisson geometry the corresponding statement does not hold. The odd Laplacian _ measures the failure of Liouville's Theorem. Up to a sign factor, it is defined as one half the divergence of the corresponding Hamiltonian vector field, : _(f) := \frac_ X_ = \frac\partial_\rho \pi^\partial_f. The odd Poisson structure \pi^ and Berezin volume density \rho are said to be compatible if the modular vector field _^ vanishes. In that case the odd Laplacian _ is a BV Δ operator with normalization Δ(1)=0. The corresponding BV algebra is the algebra of functions.


Odd symplectic manifold

If the odd Poisson bi-vector \pi^ is invertible, one has an odd symplectic manifold. In that case, there exists an odd Darboux Theorem. That is, there exist local Darboux coordinates, i.e., coordinates q^, \ldots, q^ , and momenta p_,\ldots, p_ , of degree : \left, q^\+\left, p_\=1, such that the odd Poisson bracket is on Darboux form : (q^,p_) = \delta^_ . In
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, the coordinates q^ and momenta p_ are called fields and antifields, and are typically denoted \phi^ and \phi^_ , respectively. :\Delta_ := (-1)^\frac\frac acts on the vector space of semidensities, and is a globally well-defined operator on the atlas of Darboux neighborhoods. Khudaverdian's \Delta_ operator depends only on the P-structure. It is manifestly nilpotent \Delta_^=0, and of degree −1. Nevertheless, it is technically not a BV Δ operator as the vector space of semidensities has no multiplication. (The product of two semidensities is a density rather than a semidensity.) Given a fixed density \rho, one may construct a nilpotent BV Δ operator as : \Delta(f) :=\frac\Delta_(\sqrtf), whose corresponding BV algebra is the algebra of functions, or equivalently,
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s. The odd symplectic structure \pi^ and density \rho are compatible if and only if Δ(1) is an odd constant.


Examples

* The Schouten–Nijenhuis bracket for multi-vector fields is an example of an antibracket. * If ''L'' is a
Lie superalgebra In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a \Z/2\Z grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. The notion of \Z/2\Z gra ...
, and Π is the operator exchanging the even and odd parts of a super space, then the
symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
of Π(''L'') (the "
exterior algebra In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
" of ''L'') is a Batalin–Vilkovisky algebra with Δ given by the usual differential used to compute Lie algebra
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
.


See also

* BRST quantization * Gerstenhaber algebra *
Supermanifold In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below. Informal definition An informal definition is com ...
* Analysis of flows * Poisson manifold


References


Pedagogical

* Costello, K. (2011).
Renormalization and Effective Field Theory
. (Explains perturbative quantum field theory and the rigorous aspects, such as quantizing Chern-Simons theory and Yang-Mills theory using BV-formalism)


References

* * Erratum-ibid. 30 (1984) 508 . * * * {{DEFAULTSORT:Batalin-Vilkovisky Formalism Algebras Gauge theories Supersymmetry Symplectic geometry Theoretical physics