---
abstract: 'By proceeding from a simple non-polarized formalism, we consider in detail the polarization procedure as applied to the generating equations of the quantum antibracket algebra, in terms of the parametrized generating operator.'
---

[**Quantum antibrackets: polarization and parametrization**]{}

[Igor A. Batalin$^{(a,b)}\footnote{E-mail:
batalin@lpi.ru}$, Peter M. Lavrov$^{(b, c)}\footnote{E-mail:
lavrov@tspu.edu.ru}$ ]{}

${{}^{(a)}}$ [*P.N. Lebedev Physical Institute,\
Leninsky Prospect  53, 119 991 Moscow, Russia*]{}

${{}^{(b)}}
$[*Tomsk State Pedagogical University,\
Kievskaya St. 60, 634061 Tomsk, Russia*]{}

${{}^{(c)}}
$[*National Research Tomsk State University,\
Lenin Av. 36, 634050 Tomsk, Russia*]{}

[*Keywords:*]{} quantum/derived antibrackets\
PACS numbers: 11.10.Ef, 11.15.Bt

Introduction
============

Given a fermionic nilpotent operator $Q$, one can define, for any two operators $X$ and $Y$, their quantum 2-antibracket [@BM; @BM1], \[i1\] ( X, Y )\_[Q]{} =: ( \[ X, \[ Q, Y \] \] - \[ Y, \[ Q, X \] \] (-1)\^[ ( \_[X]{} + 1 ) ( \_[Y]{} + 1 ) ]{} ), or the derived bracket in mathematical terminology [@K-S; @Vo; @CSch]. These objects have very nice algebraic properties such as the anti-symmetry, the Jacobi relations, and the Leibnitz rule. It appears remarkably that the quantum 2-antibracket (\[i1\]) arises naturally when studying BRST - invariant constraint algebra [@BT; @BB] as well as in formulating quantum dynamical equations [@BL] and in the general representation of the gauge fields [@BL1].

The quantum 2-antibracket (\[i1\]) generalizes the usual antibracket of the field-antifield BV formalism [@BV; @BV1] (see also [@SZ; @ND; @GST]). Indeed, when being operators $X$ and $Y$ in (\[i1\]) functions of the field-antifield variables only, one identifies $Q  =  - \Delta$, where $\Delta$ is the odd Laplacian, and then uses $[ \Delta, Y ]  =  ( \Delta Y )  + {\rm ad}( Y )
(-1)^{ \varepsilon_{Y} }$, to get $( X, Y )_{Q}=  ( X, Y )^{ (BV)
}$.

In the present paper, we proceed from the non-polarized version of the (\[i1\]) as defined at $X = Y = B$, with any bosonic operator $B$, \[i2\] ( B, B ) \_[Q]{} =: - \[ B, \[ B, Q \] \]. So, we call the (\[i1\]) as the polarized quantum 2-antibracket. In fact, that is just being $X \neq Y$ we mean as the polarization, in the general sense. Given the (\[i2\]), one can polarize that by considering formally $B = \alpha X + \beta Y$, with $\alpha$ and $\beta$ being the respective parameters, and then taking the $\alpha\beta$ derivative of the (\[i2\]), \[i3\] ( X, Y )\_[Q]{} = \_ \_ ( B, B )\_[Q]{} (-1)\^[ \_[Y]{} ]{}. It appears that the basic definitions and the general analysis look simpler essentially in the non-polarized formalism. On the other hand, the generating equations of the quantum antibracket algebra can be formulated naturally in terms of the parametrized generating operator, \[i4\] =: { \^[a]{} f\_[a]{} } Q { - \^[a]{} f\_[a]{} }, being $\{ f_{a} \} $ a chain of operators, and being $\{ \lambda^{\alpha} \}$ the respective parameters. In this way, the generating equations of the quantum antibracket algebra do acquire their geometrically-covariant status. We call the representation (\[i4\]) as the parametrization. All higher quantum antibrackets are defined in terms of derivatives of the generating operator (\[i4\]) with respect to the parameters. Complete set of the structure relations of the quantum antibracket algebra is generated by the nilpotence of the operator (\[i4\]).

Basics on quantum antibrackets
==============================

Let $Q$ be a fermionic nilpotent operator, \[r1\] Q: (Q) = 1, Q\^[2]{} = 0, and let $B$ be an arbitrary bosonic operator, \[r2\] B: (B) = 0. Introduce a quantum 2-antibracket, \[r3\] ( B, B )\_[Q]{} =: - \[ B, \[ B, Q \] \], then we have the main property \[r4\] \[ Q, ( B, B )\_[Q]{} \] = \[ \[ Q, B \], \[ Q, B \] \]. Let $A$ be the associator multiplied with a parameter $\beta$, \[r5\] A =: ( B, ( B, B )\_[Q]{} )\_[Q]{}. It follows then \[r6\] \[ Q, A \] = = 0 A = \[, Q\]. By choosing the operator $\mathcal{A}$ as \[r7\] = \[ B, (B, B)\_[Q]{} \] = ( B, B, B )\_[Q]{} =: - \[ B, \[ B, \[ B, Q \] \] \] = 6, we arrive at the non - polarized form of the Jacobi relation \[r8\] 6 ( B, ( B, B )\_[Q]{} )\_[Q]{} = \[ ( B, B, B )\_[Q]{}, Q \], as an identity with respect to $B$. Indeed, denote the operators \[r9\] X =: \[ B, \[ \[ B, Q \], \[ B, Q \] \] \], Y =: \[ \[ B, Q \], \[ B, \[ Q, B \] \] \], such that \[r10\] X = - 2 Y. Then, we have \[r11\] ( B, ( B, B )\_[Q]{} )\_[Q]{} = ( X + Y ) = - Y, \[r12\] \[ ( B, B, B )\_[Q]{}, Q \] = X - Y = - 3 Y, which is equivalent to (\[r8\]).

Polarization
============

In the definition (\[r3\]), consider the Boson $B$ of the form \[p1\] B = X + Y + Z, with $\alpha, \beta $ and $\gamma$ being parameters. It follows then the polarized quantum 2-antibracket \[p2\] \_ \_ ( B, B )\_[Q]{} (-1)\^[ \_[Y]{} ]{} = ( X, Y )\_[Q]{} =: ( \[ X, \[ Q, Y \] \] - \[ Y, \[ Q, X \] \] (-1)\^[ ( \_[X]{} + 1 ) ( \_[Y]{} + 1 ) ]{} ). In analogy with the main property (\[r4\]) we have its counterpart for polarized quantum\
2-antibracket (\[p2\]) \[p2a\] \[Q,(X,Y)\_Q\]=\[\[Q,X\],\[Q,Y\]\]. In turn, by using (\[p2\]), we have the polarized version of the Jacobi relation (\[r8\]) &&( X, ( Y, Z )\_[Q]{} )\_[Q]{} (-1)\^[ ( \_[X]{} + 1 ) ( \_[Z]{} + 1 ) ]{} + [cyclic permutations]{} ( X, Y, Z ) =\
&&= \_ \_ \_ ( B, ( B, B )\_[Q]{} )\_[Q]{} (-1)\^[ ( \_[X]{} + 1 ) ( \_[Z]{} + 1 ) + \_[Y]{} ]{} =\
&&= \_ \_ \_ (-1)\^[ ( \_[X]{} + 1 ) ( \_[Z]{} + 1 ) + \_[Y]{} ]{} =\
&&= \[ ( X, Y, Z )\_[Q]{}, Q \] (-1)\^[ ( \_[X]{} + 1 ) ( \_[Z]{} + 1 ) ]{}. \[p3\] Thus, we identify the polarized quantum 3-antibracket, \[p4\] &&( X, Y, Z )\_[Q]{} =: \_ \_ \_ ( B, B, B )\_[Q]{} (-1)\^[ \_[Y]{} ]{}=\
&&= - (-1)\^[ ( \_[X]{} + 1 ) ( \_[Z]{} + 1 ) ]{} ( \[ X, ( Y, Z )\_[Q]{} \] (-1)\^[ \[ \_[X]{} ( \_[Z]{} + 1 ) + + \_[Y]{} \] ]{} + [cyclic permutations]{} ( X, Y, Z ) ). The modified Leibnitz rule for quantum 2-antibracket (\[p2\]) reads &&( XY, Z )\_[Q]{} - X ( Y, Z )\_[Q]{} - ( X, Z )\_[Q]{} Y (-1)\^[ \_[Y]{} ( \_[Z]{} + 1 ) ]{} =\
\[j217\] &&= ( \[ X, Z\] \[ Y, Q \] (-1)\^[ \_[Z]{} ( \_[Y]{} + 1 ) ]{} + \[ X, Q \] \[ Y, Z \] (-1)\^[ \_[Y]{} ]{} ).

Parametrization
===============

Here we include in short the generating equations for the quantum antibracket algebra [@BM1]. Let us introduce an operator valued exponential \[A1\] U = { \^[a]{} f\_[a]{} }, U|\_[= 0]{} = 1, where $\{ f_{a},  a = 1, 2, ...  \}$, is a chain of operators, $\varepsilon(
f_{a} ) = \varepsilon_{a}$, and $\lambda^{a}$ are parameters, $\varepsilon(
\lambda^{a} )  =  \varepsilon_{a}$. Introduce the $U$-transformed $Q$-operator, \[A2\] = U Q U\^[-1]{}, \^[2]{} = 0. The latter equation (\[A2\]) does generate the complete set of the higher Jacobi relations following (\[p4\]).

Further, we have the generating equations \[A3\] \_[a]{} = \[ R\_[a]{}, \], R\_[a]{} = ( \_[a]{} U ) U\^[-1]{}, \_[a]{} = , |\_[= 0]{} = Q, \[A4\] \_[a]{} R\_[b]{} - \_[b]{} R\_[a]{} (-1)\^[ \_[a]{} \_[b]{} ]{} = \[ R\_[a]{}, R\_[b]{}\] . It follows from (\[A4\]) that the equation holds with the Euler operator, $N=: \lambda^{a} \partial_{a} $, \[A4a\] ( N + 1 ) R\_[b]{} = \_[b]{} - \[ R\_[b]{}, \], where \[A4b\] = \^[a]{} R\_[a]{}, is an operator describing the arbitrariness in a choice of $\lambda$ parametrization.

By making the rescaling \[A4c\] &&\^[a]{} t \^[a]{},\
\[A4d\] && R\_[a]{} \_[a]{} =: t R\_[a]{}( t ),\
\[A4e\] && =: \^[a]{}R\_[a]{}(t ), we convert the equation (\[A4a\]) to the form \[A4f\] = \_[b]{} - \[ \_[b]{}, \], with the boundary condition \[A4g\] \_[b]{} |\_[ t = 0 ]{} = 0. The Cauchy problem (\[A4f\]), (\[A4g\]) resolves in the form \[A4h\] \_[b]{}( t ) = \_[0]{}\^[t]{} dt’ U( t, t’ ) ( \_[b]{} ( t’ ) ) U\^[-1]{}( t, t’ ), where $U( t, t' )$ resolves the Cauchy problem \[A4j\] = ( t ) U( t, t’ ), \[A4k\] U( t, t’ ) |\_[ t = t’ ]{} = 1.

In parallel to the above geometric formulae (\[A4f\]) - (\[A4k\]), we suggest a simpler derivation for the “current” $R_{a}$ (\[A3\]), R\_[a]{} &=& \_[0]{}\^[1]{} dt { t } ( \_[a]{} ) { - t } = \_[0]{}\^[1]{} dt { t ( ) } ( \_[a]{} ) =\
\[A4l\] &=& ( \_[a]{} ), where \[A4z\] =\^a f\_a, and we have used the identity \[A4z1\] \[A, {B}\]=\_0\^1 dt {t B}\[A,B\]{(1-t)B}, for $A=\partial_a$, $B=\psi$.

Notice that the following relation holds between the $\Psi$, (\[A4b\]), and the $\psi$, (\[A4z\]), \[A4v\] = ( N ).

By multiplying the equation (\[A4f\]) with $\lambda^{b}$ from the left we get an identity, as expected, which implies that the $\tilde{\Psi}$, (\[A4e\]), is an arbitrary operator. Then, by choosing in (\[A4h\]) - (\[A4k\]) $\tilde{\Psi} = \psi$, with $\psi$ being given in (\[A4z\]), we arrive at the representation (\[A4l\]). Indeed, it follows from (\[A4f\]) that the equation \[A4t\] \_[t]{} t = N holds, which implies in turn \[A4u\] t \_[t]{} = \^[a]{} N R\_[a]{}( t ). That is just an identity expected. Due to (\[A4e\]), the left-hand side of (\[A4u\]) rewrites in the form \[A4o\] \^[a]{} t \^[b]{} R\_[a]{}( t ) = \^[a]{} N R\_[a]{}( t ). Thus, the $\tilde{\Psi}$, (\[A4e\]), remains arbitrary.

The Lie equation (\[A3\]) and the Maurer-Cartan equation (\[A4\]) do serve as the generating equations for quantum antibrackets. Here we present explicitly only the case of quantum 2-antibracket. It follows from (\[A3\]) by $\lambda$ differentiating, that &&- \_[a]{} \_[b]{} (-1)\^[\_[b]{}]{} + \[ ( \_[a]{} R\_[b]{} + \_[b]{} R\_[a]{} (-1)\^[ \_[a]{} \_[b]{} ]{} ) (-1)\^[\_[b]{}]{} , \] =\
\[A5\] &&= ( \[ R\_[a]{}, \[ , R\_[b]{} \] \] - ( a b ) (-1)\^[ ( \_[a]{} + 1 ) ( \_[b]{} + 1 ) ]{} ) = ( R\_[a]{}, R\_[b]{} ) \_[ ]{}. In turn, it follows from (\[p4\]) and (\[A5\]) that the next equation holds, &&- (-1)\^[ ( \_[a]{} + 1 ) ( \_[c]{} + 1 ) ]{} ( \[ R\_[a]{}, \_[bc]{} \] (-1)\^[ ( \_[a]{} ( \_[c]{} + 1 ) + \_[b]{}) ]{} +\
\[A5a\] &&+ ( a, b, c ) ) = ( R\_[a]{}, R\_[b]{}, R\_[c]{} )\_[ ]{}, where we have denoted \[A5b\] \_[ab]{} =: - \_[a]{} \_[b]{} (-1)\^[ \_[b]{} ]{} + \[ Y\_[ab]{}, ( ) \], \[A5c\] Y\_[ab]{} =: ( \_[a]{} R\_[b]{} + \_[b]{} R\_[a]{} (-1)\^[ \_[a]{} \_[b]{} ]{} ) (-1)\^[ \_[b]{} ]{}. In the latter notation, the equation (\[A5\]) takes the form \[A5d\] \_[ab]{} = ( R\_[a]{}, R\_[b]{} )\_[ ]{}.

It follows from (\[A5\]) at $\lambda = 0$, \[A6\] - ( \_[a]{}\_[b]{} ) (-1)\^[ \_[b]{} ]{} |\_[= 0]{} = ( f\_[a]{}, f\_[b]{} )\_[Q]{}, where we have used \[A7\] ( \_[a]{} R\_[b]{} ) |\_[= 0]{} = \[ f\_[a]{}, f\_[b]{} \]. The next equation (\[A5a\]) takes the form \[A7z\] \_[abc]{} = ( R\_[a]{}, R\_[b]{}, R\_[c]{} )\_[ ]{}, where \[A7x\] \_[abc]{} = - \_[a]{} \_[b]{} \_[c]{} (-1)\^[ \_[b]{} ]{} + = 0. In its more explicit form, the second term in right-hand side in (\[A7x\]) reads \[A7y\] &=:& (-1)\^[ ( \_[a]{} + 1 ) (\_[c ]{} + 1 ) ]{} { ( \_[b]{} \[ \_[c]{} R\_[a]{}, ( ) \] (-1)\^[ \_[a]{} ( \_[b]{} + \_[c]{} ) ]{} + \[ \_[b]{} R\_[a]{}, \_[c]{} ( ) \] (-1)\^[ \_[a]{}\_[b]{} ]{} +\
&&+ \[ R\_[a]{}, \[ ( \_[b]{} R\_[c]{} + \_[c]{} R\_[b]{} (-1)\^[ \_[b]{} \_[c]{} ]{} ), ( ) \] \] )(-1)\^[ ( \_[a]{} + 1 )( \_[c]{} + 1 ) + \_[b]{} ]{} +\
\[A7y\] && + (a, b, c ) }.

It follows in a similar way that higher $\lambda$ derivatives of $\tilde{Q}$ do yield all higher quantum antibrackets, - ( \_[a\_1]{}\_[a\_n]{} ) (-1)\^[ E\_n]{} |\_[= 0]{} = ( f\_[ a\_[1]{} ]{} , ... , f\_[ a\_[n]{} ]{} )\_[Q]{} = - [Sym]{}( \[ f\_[ a\_[1]{} ]{}, ... , \[ f\_[ a\_[n]{} ]{}, Q \] ... \] ) (-1)\^[ E\_[n]{} ]{}, where we have denoted E\_[n]{} = \_[ k = 1 ]{}\^[ \[ n/2 \] ]{} \_[ a\_[2k]{} ]{}, ( X\_[ a\_[1]{} ... a\_[n]{} ]{} ) = S\_[ a\_[1]{} ... a\_[n]{} ]{}\^[ b\_[n]{} ... b\_[1]{} ]{} X\_[ b\_[1]{} ... b\_[n]{} ]{}, n! S\_[ a\_[1]{} ... a\_[n]{} ]{}\^[ b\_[n]{} ... b\_[1]{} ]{} = \_[ a\_[1]{} ]{} ... \_[ a\_[n]{} ]{} \^[ b\_[n]{} ]{} ... \^[ b\_[1]{} ]{}. It has also been shown in [@BM1; @Ber], how these equations enable one to derive the modified Jacobi relations for subsequent higher quantum antibrackets.

Summary
=======

In the present article we have considered a simple non-polarized form of the quantum antibracket algebra (Section 2), and then derived its polarized form (Section 3). Also, we have introduced a natural parametrization (\[A2\]), and then derived the respective generating equations (\[A5d\]), (\[A7z\]) (Section 4). In an obvious way the construction can be extended to cover all higher quantum antibrackets.

Acknowledgments {#acknowledgments .unnumbered}
===============

The authors would like to thank Klaus Bering for useful discussions. The work of I. A. Batalin is supported in part by the RFBR grant 17-02-00317. The work of P. M. Lavrov is supported by the Ministry of Education and Science of Russian Federation, grant 3.1386.2017 and by the RFBR grant 18-02-00153.

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