In theoretical physics, the Batalin - Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the ghost structure for Lagrangian gauge theories, such as gravity and supergravity, whose corresponding Hamiltonian formulation has constraints not related to a Lie algebra (i.e., the role of Lie algebra structure constants are played by more general structure functions). The BV formalism, based on an action that contains both fields 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.
In mathematics, a Batalin - Vilkovisky algebra is a graded supercommutative algebra (with a unit 1) with a second-order nilpotent operator Δ of degree -1. More precisely, it satisfies the identities
(ab)c=a(bc)
ab=(-1)|a||b|ba
|ab|=|a|+|b|
|\Delta(a)|=|a|-1
\Delta2=0
\begin{align} 0=&\Delta(abc)\\ &-\Delta(ab)c-(-1)|a|a\Delta(bc)-(-1)(|a|+1)|b|b\Delta(ac)\\ &+\Delta(a)bc+(-1)|a|a\Delta(b)c+(-1)|a|+|b|ab\Delta(c)\\ &-\Delta(1)abc \end{align}
One often also requires normalization:
\Delta(1)=0
A Batalin–Vilkovisky algebra becomes a Gerstenhaber algebra if one defines the Gerstenhaber bracket by
(a,b):=(-1)\left|a\right|\Delta(ab)-(-1)\left|a\right|\Delta(a)b-a\Delta(b)+a\Delta(1)b.
|(a,b)|=|a|+|b|-1
(a,b)=-(-1)(|a|+1)(|b|+1)(b,a)
(-1)(|a|+1)(|c|+1)(a,(b,c))+(-1)(|b|+1)(|a|+1)(b,(c,a))+(-1)(|c|+1)(|b|+1)(c,(a,b))=0
(ab,c)=a(b,c)+(-1)|a||b|b(a,c)
The normalized operator is defined as
{\Delta}\rho:=\Delta-\Delta(1).
{\Delta}\rho(a,b)=({\Delta}\rho(a),b)-(-1)\left|a\right|(a,{\Delta}\rho(b))
{\Delta}\rho
2 | |
{\Delta} | |
\rho |
=(\Delta(1), ⋅ )
{\Delta}\rho
2 | |
{\Delta} | |
\rho |
(ab)=
2 | |
{\Delta} | |
\rho |
(a)b+
2 | |
a{\Delta} | |
\rho |
(b)
{\Delta}\rho
2 | |
{\Delta} | |
\rho |
If one introduces the left multiplication operator
La
La(b):=ab,
[S,T]:=ST-(-1)\left|S\right|\left|T\right|TS
(a,b):=(-1)\left|a\right|[[\Delta,La],Lb]1,
[[[\Delta,La],Lb],Lc]1=0
[\Delta,La]
[[\Delta,La],Lb]
The classical master equation for an even degree element S (called the action) of a Batalin–Vilkovisky algebra is the equation
(S,S)=0.
\Delta\exp\left[
i | |
\hbar |
W\right]=0,
1 | |
2 |
(W,W)=i\hbar{\Delta}\rho(W)+\hbar2\Delta(1).
1 | |
2 |
(W,W)=i\hbar\Delta(W).
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
\Phin(a1,\ldots,an):=
\underbrace{[[\ldots[\Delta,L | |
a1 |
],\ldots],L | |
an |
]}n~{\rm
\Phin(a\pi(1),\ldots,a\pi(n))=
\left|a\pi\right| | |
(-1) |
\Phin(a1,\ldots,an)
\pi\inSn
\left|a\pi\right| | |
(-1) |
a\pi(1)\ldotsa\pi(n)=
\left|a\pi\right| | |
(-1) |
a1\ldotsan
Linfty
n | |
\sum | |
k=0 |
1 | |
k!(n-k)! |
\sum | |
\pi\inSn |
\left|a\pi\right| | |
(-1) |
\Phin-k+1\left(\Phik(a\pi(1),\ldots,a\pi(k)),a\pi(k+1),\ldots,a\pi(n)\right)=0.
\Phi0:=\Delta(1)
\Phi1(a):=[\Delta,La]1=\Delta(a)-\Delta(1)a=:{\Delta}\rho(a)
\Phi2(a,b):=[[\Delta,La],Lb]1=:(-1)\left|a\right|(a,b)
\Phi3(a,b,c):=[[[\Delta,La],Lb],Lc]1
\vdots
\Phi1={\Delta}\rho
\Phi2
\Phi1(\Phi0)=0
\Delta(1)
\Delta\rho
\Phi2(\Phi0,a)+\Phi1\left(\Phi1(a)\right)
\Delta(1)
2 | |
{\Delta} | |
\rho |
\Phi3(\Phi0,a,b)+\Phi2\left(\Phi1(a),b\right)+(-1)|a|\Phi2\left(a,\Phi1(b)\right)+\Phi1\left(\Phi2(a,b)\right)=0
{\Delta}\rho
\Phi4(\Phi0,a,b,c)+{\rmJac}(a,b,c)+\Phi1\left(\Phi3(a,b,c)\right)+\Phi3\left(\Phi1(a),b,c\right)+(-1)\left|a\right|\Phi3\left(a,\Phi1(b),c\right)+(-1)\left|a\right|+\left|b\right|\Phi3\left(a,b,\Phi1(c)\right)=0
\vdots
\Phi2
{\rmJac}(a1,a2,a3):=
1 | |
2 |
\sum | |
\pi\inS3 |
\left|a\pi\right| | |
(-1) |
\Phi2\left(\Phi2(a\pi(1),a\pi(2)),a\pi(3)\right).
The Δ operator is by definition of n'th order if and only if the (n + 1)-bracket
\Phin+1
{\rmJac}(a,b,c)=0
Let there be given an (n|n) supermanifold with an odd Poisson bi-vector
\piij
\rho
xi
\partialif
f\stackrel{\leftarrow}{\partial}i
\left|xi\right|(|f|+1) | |
:=(-1) |
\partialif
xi
\piij
\left|\piij\right|=\left|xi\right|+\left|xj\right|-1
\piji=
(\left|xi\right|+1)(\left|xj\right|+1) | |
-(-1) |
\piij
(\left|xi\right|+1)(\left|xk\right|+1) | |
(-1) |
\pii\ell\partial\ell\pijk+{\rmcyclic}(i,j,k)=0
xi\tox\prime
\piij
\rho
\pi\prime=x\prime\stackrel{\leftarrow}{\partial}i\piij\partialjx\prime
\rho\prime=\rho/{\rmsdet}(\partialix\prime)
(f,g):=f\stackrel{\leftarrow}{\partial}i\piij\partialjg.
Xf
Xf[g]:=(f,g).
X=Xi\partiali
{\rmdiv}\rhoX:=
| |||||
\rho |
\partiali(\rhoXi)
{\Delta}\rho
{\Delta}\rho(f):=
(-1)\left|f\right| | |
2 |
{\rmdiv}\rhoXf=
| |||||
2\rho |
\partiali\rho\piij\partialjf.
\piij
\rho
2 | |
{\Delta} | |
\rho |
{\Delta}\rho
If the odd Poisson bi-vector
\piij
q1,\ldots,qn
p1,\ldots,pn
\left|qi\right|+\left|pi\right|=1,
(qi,pj)=
i | |
\delta | |
j |
.
qi
pj
\phii
* | |
\phi | |
j |
\Delta\pi:=
\left|qi\right| | |
(-1) |
\partial | |
\partialqi |
\partial | |
\partialpi |
\Delta\pi
2 | |
\Delta | |
\pi |
=0
\rho
\Delta(f):=
1 | |
\sqrt{\rho |
\piij
\rho
. Steven Weinberg . 2005 . The Quantum Theory of Fields Vol. II . New York . Cambridge Univ. Press . 0-521-67054-3 .