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 natural phenomena. This is in contrast to experimental physics, which uses experi ...
, the Batalin–Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the
ghost
A ghost is the soul (spirit), soul or spirit of a dead Human, person or animal that is believed to be able to appear to the living. In ghostlore, descriptions of ghosts vary widely from an invisible presence to translucent or barely visibl ...
structure for Lagrangian
gauge theories
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
, such as gravity and
supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
, whose corresponding
Hamiltonian formulation has constraints not related to a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
(i.e., the role of Lie algebra structure constants are played by more general structure functions). The BV formalism, based on an
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
that contains both
fields and "antifields", can be thought of as a vast generalization of the original
BRST formalism BRST may refer to:
* BRST Films, a Serbian video production company
* BRST algorithm, an optimization algorithm suitable for finding the global optimum of black box functions
* BRST quantization in Yang-Mills theories, a way to quantize a gauge-sy ...
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
*
(The product has degree 0)
*
(Δ has degree −1)
*
(The product is associative)
*
(The product is (super-)commutative)
*
(Nilpotency (of order 2))
*
(The Δ operator is of second order)
One often also requires normalization:
*
(normalization)
Antibracket
A Batalin–Vilkovisky algebra becomes a
Gerstenhaber algebra
In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring an ...
if one defines the Gerstenhaber bracket by
:
Other names for the Gerstenhaber bracket are Buttin bracket, antibracket, or odd Poisson bracket. The antibracket satisfies
*
(The antibracket (,) has degree −1)
*
(Skewsymmetry)
*
(The Jacobi identity)
*
(The Poisson property;The Leibniz rule)
Odd Laplacian
The normalized operator is defined as
:
It is often called the odd Laplacian, in particular in the context of odd Poisson geometry. It "differentiates" the antibracket
*
(The
operator differentiates (,))
The square
of the normalized
operator is a Hamiltonian vector field with odd Hamiltonian Δ(1)
*
(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
as
:
and the
supercommutator
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, th ...
as
:
for two arbitrary operators ''S'' and ''T'', then the definition of the antibracket may be written compactly as
:
and the second order condition for Δ may be written compactly as
:
(The Δ operator is of second order)
where it is understood that the pertinent operator acts on the unit element 1. In other words,
is a first-order (affine) operator, and
is a zeroth-order operator.
Master equation
The classical master equation for an even degree element ''S'' (called the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
) of a Batalin–Vilkovisky algebra is the equation
:
The quantum master equation for an even degree element ''W'' of a Batalin–Vilkovisky algebra is the equation
:
or equivalently,
:
Assuming normalization Δ(1) = 0, the quantum master equation reads
:
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
:
The brackets are (graded) symmetric
:
(Symmetric brackets)
where
is a permutation, and
is the
Koszul sign of the permutation
:
.
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 ...
, also known as an
algebra, which satisfies generalized Jacobi identities
:
(Generalized Jacobi identities)
The first few brackets are:
*
(The zero-bracket)
*
(The one-bracket)
*
(The two-bracket)
*
(The three-bracket)
*
In particular, the one-bracket
is the odd Laplacian, and the two-bracket
is the antibracket up to a sign. The first few generalized Jacobi identities are:
*
(
is
-closed)
*
(
is the Hamiltonian for the modular vector field
)
*
(The
operator differentiates (,) generalized)
*
(The generalized Jacobi identity)
*
where the
Jacobiator for the two-bracket
is defined as
:
BV ''n''-algebras
The Δ operator is by definition of n'th order if and only if the (''n'' + 1)-bracket
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
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 co ...
with an odd Poisson bi-vector
and a Berezin volume density
, also known as a P-structure and an S-structure, respectively. Let the local coordinates be called
. Let the derivatives
and
:
denote the
left
Left may refer to:
Music
* ''Left'' (Hope of the States album), 2006
* ''Left'' (Monkey House album), 2016
* "Left", a song by Nickelback from the album ''Curb'', 1996
Direction
* Left (direction), the relative direction opposite of right
* L ...
and
right derivative of a function ''f'' wrt.
, respectively. The odd Poisson bi-vector
satisfies more precisely
*
(The odd Poisson structure has degree –1)
*
(Skewsymmetry)
*
(The Jacobi identity)
Under change of coordinates
the odd Poisson bi-vector
and Berezin volume density
transform as
*
*
where ''sdet'' denotes the
superdeterminant, also known as the Berezinian.
Then the odd Poisson bracket is defined as
:
A Hamiltonian vector field
with Hamiltonian ''f'' can be defined as
:
The (super-)
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
of a vector field
is defined as
:
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,
:
The odd Poisson structure
and Berezin volume density
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
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
, and momenta
, of degree
:
such that the odd Poisson bracket is on Darboux form
:
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 natural phenomena. This is in contrast to experimental physics, which uses experi ...
, the coordinates
and momenta
are called fields and antifields, and are typically denoted
and
, respectively.
:
acts on the vector space of
semidensities, and is a globally well-defined operator on the atlas of Darboux neighborhoods. Khudaverdian's
operator depends only on the P-structure. It is manifestly nilpotent
, 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
, one may construct a nilpotent BV Δ operator as
:
whose corresponding BV algebra is the algebra of functions, or equivalently,
scalars. The odd symplectic structure
and density
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, 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 universa ...
of Π(''L'') (the "
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
" 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 viewe ...
.
See also
*
BRST quantization
*
Gerstenhaber algebra
In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring an ...
*
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 co ...
*
Analysis of flows
In theoretical physics, an analysis of flows is the study of "gauge" or "gaugelike" "symmetries" (i.e. flows the formulation of a theory is invariant under). It is generally agreed that flows indicate nothing more than a redundancy in the descripti ...
*
Poisson manifold
In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule
: \ = \h + g \ .
Equivalen ...
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