A systematic study of finite BRST-BV transformations within W-X formulation of the standard and the Sp(2)-extended field-antifield formalism 6 Igor A. Batalin(a,c)1 , Klaus Bering(b)2 , Peter M. Lavrov(c,d)3 1 0 2 b (a) P.N. Lebedev Physical Institute, e F Leninsky Prospect 53, 119 991 Moscow, Russia 3 2 (b) Masaryk University, Faculty of Science, Kotlarska 2, 611 37 Brno, Czech Republic ] h t (c)Tomsk State Pedagogical University, - p e Kievskaya St. 60, 634061 Tomsk, Russia h [ (d)National Research Tomsk State University, 4 Lenin Av. 36, 634050 Tomsk, Russia v 9 2 0 3 0 . 1 0 6 Abstract 1 : v Finite BRST-BV transformations are studied systematically within the W-X formulation i X of the standard and the Sp(2)-extended field-antifield formalism. The finite BRST-BV r a transformations are introduced by formulating a new version of the Lie equations. The corresponding finite change of the gauge-fixing master action X and the corresponding Ward identity are derived. Keywords: finite field dependent BRST-BV transformations; W-X field-antifield formalism; 1E-mail: [email protected] 2E-mail: [email protected] 3E-mail: [email protected] 1 Introduction In recent papers [1, 2, 3, 4, 5, 6], finite BRST transformations have been studied sys- tematically both in the Hamiltonian and Lagrangian formalism in their standard and Sp(2)- extended versions [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. The so-called W-X formulation [19, 20, 21, 22, 23, 24, 25, 26, 27] is known as the most symmetric form of the Lagrangian field- antifield formalism. Dynamical gauge-generating master action W serves as a deformation to the original actionof the theory. On theother hand, gauge-fixing master action X serves just as to eliminate the antifield variables. It is remarkable that these complementary master actions W and X do satisfy a set of quantum master equations transposed to each other. Inthepresent paperwestudysystematically finiteBRST-BVtransformationswithintheW- X formulation bothin thestandard and Sp(2)-extended field-antifield formalism. We introduce these transformations by formulating the respective Lie equations. Among other things, we derive in this way the effective change in the gauge-fixing master action X, as induced by the finite BRST-BV transformation defined. 2 W-X formulation to the standard field-antifield formalism Let zA be the complete set of the variables necessary within the standard field-antifield formalism zA = {Φα;Φ∗}, (2.1) α whose Grassmann parities are ε(zA) = {ε ;ε +1}. (2.2) α α We denote the respective zA-derivatives as ∂ = {∂ ;∂α}. (2.3) A α ∗ Let Z be the partition function i Z = DzDλexp W +X , (2.4) Z (cid:26)~ (cid:27) (cid:2) (cid:3) where λα are Lagrange multipliers for gauge-fixing with Grassmann parity ε(λα) = ε +1. (2.5) α In thepartitionfunction(2.4), the dynamical gauge-generatingmaster actionW andthe gauge- fixing master action X are defined to satisfy the respective quantum master equations, i 1 ∆exp W = 0 ⇔ (W,W) = i~(∆W), (2.6) (cid:18) (cid:26)~ (cid:27)(cid:19) 2 2 i 1 ∆exp X = 0 ⇔ (X,X) = i~(∆X). (2.7) (cid:18) (cid:26)~ (cid:27)(cid:19) 2 In the above quantum master equations (2.6) and (2.7), the ∆ and ( , ) are the standard nilpotent odd Laplacian ∆ = ∂ ∂α(−1)εα, (2.8) α ∗ and the standard antibracket ←− −→ (f,g) = (−1)εf[[∆,f],g]1 = f ∂ ∂ αg −(f ↔ g)(−1)(εf+1)(εg+1), (2.9) α ∗ respectively. These formulae (2.8) and (2.9) tell us that the anticanonical pairs (Φα;Φ∗) serve α as Darboux coordinates on the flat field-antifield phase space with measure density ρ = 1 and no odd scalar curvature ν = 0. ρ At ~ = 0, Φ∗ = 0, the W-action coincides with the original action of the theory. As to the α X-action, it can be chosen in the form related to the gauge-fixing Fermion Ψ(Φ), ←− ←− X = (Φ∗ −Ψ(Φ)∂ )λα = Φ∗λα −Ψ(Φ) d, (2.10) α α α where ←− ←− d = ∂ λα (2.11) α is a nilpotent Fermionic differential that acts from the right. In the integrand of the path integral (2.4), consider now the following infinitesimal BRST- BV transformation ~ ~ δzA = −µ(Y,zA)− (µ,zA) = − y−1(yµ,zA), (2.12) i i where we have defined for later convenience i Y := X −W, y := exp Y , (2.13) (cid:26)~ (cid:27) and where µ(z) is an infinitesimal Fermionic function with ε(µ) = 1. The Jacobian of the infinitesimal BRST-BV transformation (2.12) has the form ~ lnJ = (−1)εA(∂ δzA) = (Y,µ)+2(∆Y)µ+2 (∆µ). (2.14) A i The complete action in the partition function (2.4) transforms as ~ δ[W +X] = [−(W,W)+(X,X)]µ+ (W +X,µ). (2.15) i 3 Due to the quantum master equations (2.6) and (2.7), we then have from Eqs. (2.14) and (2.15) that i δ[W +X]+ lnJ = 2(σ(X)µ), (2.16) ~ where σ(X) is a quantum BRST generator ~ (σ(X)f) = (X,f)+ (∆f). (2.17) i The Eq. (2.16) tells us that the BRST transformation (2.12) induces the following variation ~ δX = 2 (σ(X)µ). (2.18) i to the X-actionin the integrand of the path integral (2.4). We conclude that the partition func- tion (2.4) and the quantum master equation (2.7) for X are both stable under the infinitesimal variation (2.18). Next let t be a Bosonic parameter. It is natural to define a one-parameter subgroup t 7→ zA(t) of finite BRST-BV transformations by the Lie equation4 dzA = HA, zA| = zA. (2.19) dt t=0 where ~ ~ H = HA∂ = −µ ad(Y)− ad(µ) = − y−1ad(yµ), ad(f)g := (f,g), (2.20) A i i is the corresponding vector field with components ~ ~ HA := −µ(Y,zA)− (µ,zA) = − y−1(yµ,zA). (2.21) i i Note that µ(z) is now an arbitrary finite Fermionic function. In other words, the Lie equation (2.19) is dzA = (HzA), H := HA∂ (2.22) dt A with solution zA = etHzA . (2.23) (cid:0) (cid:1) Recall that the antibracket for any Fermion F = yµ with itself is zero: (F,F) = 0. This fact yields a conservation law d(yµ) dzA ~ ~ = ∂ (yµ) = − y−1(yµ,zA) ∂ (yµ) = − y−1(yµ,yµ) = 0, (2.24) dt dt A i A i 4Foranarbitraryfunctionf =f(z),weusetheshorthandnotationf =f(z)= etHf forthecorresponding function with shifted arguments. (cid:0) (cid:1) 4 so that the following invariance property holds yµ = yµ. (2.25) The Jacobian of these transformations satisfies the following equation dlnJ ~ = divH, divH := (−1)εA∂ HA = (Y,µ)+2(∆Y)µ+2 (∆µ). (2.26) A dt i The transformed complete action satisfies the equation d dzA ~ W +X = ∂ W +X = −µ(Y,zA)− (µ,zA) ∂ W +X dt dt A (cid:20) i (cid:21) A (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) ~ = −(W,W)+(X,X) µ+ (W +X,µ). (2.27) i h i Due to the transformed master equations (2.6) and (2.7), it follows that d ~ ~ W +X + lnJ = a, (2.28) dt (cid:20) i (cid:21) i where we have defined for later convenience a := 2(σ(X)µ). (2.29) By integrating Eq. (2.28) within 0 ≤ t ≤ 1, we get ~ ~ W +X + lnJ = W +X + A, (2.30) i i where we have defined the average 1 1 A := dt a = dt etHa = (E(H)a). (2.31) Z Z 0 0 (cid:0) (cid:1) Here E is the function 1 exp{x}−1 E(x) = dt etx = . (2.32) Z x 0 The Eq. (2.30) shows the finite effective change in X induced by the finite transformation zi → zi in the partition function (2.4). Now consider the left-hand side Y of the transformed quantum master equation (2.7), where 1 ~ Y := (X,X)+ (∆X). (2.33) 2 i We have the following Cauchy initial value problem dY = (H Y) ∧ Y| = 0 ⇒ Y ≡ 0 (2.34) t=0 dt 5 for arbitrary t. Thereby, we have confirmed that the quantum master equation (2.7) is stable under the finite BRST-BV transformation generated by Eq. (2.19). Of course, the general expression (2.4) itself is stable under the same transformation, as well. At this point we would like to investigate the quantum master equation i ∆exp X′ = 0, (2.35) (cid:18) (cid:26)~ (cid:27)(cid:19) where we have denoted the new gauge-fixing master action, ~ X′ = X + A. (2.36) i Eq. (2.35) is equivalent to ~ (σ(X)exp{A}) = 0 ⇔ (A,A)+(σ(X)A) = 0. (2.37) 2i The exponential exp{A} rewrites in the form exp{A} = e(E(H)a) = (exp{H+a}1) = (exp{H +2[σ(X),µ]}1), (2.38) where we have defined the first-order operator ~ ~ ~ H := y ad(y−1µ) = −µad(Y)+ ad(µ) = H+2 ad(µ), (2.39) i i i and used the formula ~ [σ(X),f] = (σ(X)f)+(−1)εf ad(f) (2.40) i for a function f. Hence Eqs. (2.35)/(2.37) is equivalent to [σ(X), exp{H +2[σ(X),µ]}]1 = 0. (2.41) In general, it looks as if Eq. (2.35)/(2.37)/(2.41) serves as a condition for finite field-dependent parameter µ(z). This equation is certainly satisfied with arbitrary infinitesimal µ(z) → 0, to the first order in that. We do not know if the same situation holds for arbitrary finite µ(z), as Eq. (2.35)/(2.37)/(2.41) is rather complicated in the general case. Also, there is a potential obstacle that the dynamical master action W actually enters that equation. Thus, being finite parameter µ(z) restricted in its field-dependence, that circumstance would be a crucial specific feature of the W-X formulation. One can proceed from a solution A to the quantum master equation (2.37). If we ignore Eqs. (2.29) and (2.31), then the quantum master equation (2.37) knows nothing about the parameter µ. Moreover, A serves as an external source in the left-hand side of the Eq. (2.31). 6 The right-hand side of Eq. (2.31) knows about parameter µ via its explicit appearance in Eqs. (2.20) and (2.31). Thereby, the aspects related to the quantum master equation (2.37) by itself, and to the parameter µ, are separated naturally. From this point of view, it sounds not so plausible that the Eq. (2.31) could allow for finite arbitrary parameter µ(z). If one rescales parameters in Eqs. (2.37) and (2.31), µ → εµ, (2.42) with ε being a Boson parameter, and then expands µ and A in formal power series, µ = µ +εµ +..., (2.43) 0 1 A = A +εA +..., (2.44) 0 1 one gets to the first order in ε 0 = A , (2.45) 0 2(σ(X)µ ) = A , (2.46) 0 1 (σ(X)A ) = 0, (2.47) 1 so that µ remains arbitrary to that order. However, to the second order in ε, one has 0 2(σ(X)µ )+(H(µ )σ(X)µ ) = A , (2.48) 1 0 0 2 ~ (A ,A )+(σ(X)A ) = 0, (2.49) 2i 1 1 2 so that µ remains arbitrary to that order, while (2.49) restricts µ , 1 0 (σ(X)H(µ )σ(X)µ ) = 0, (2.50) 0 0 with H(µ ) being the operator (2.39) as taken at µ = µ . To the third order in ε, the µ 0 0 2 remains arbitrary, while the µ is restricted to satisfy the condition 1 (σ(X)H(µ )σ(X)µ )+(σ(X)H(µ )σ(X)µ ) 0 1 1 0 1 + σ(X)(H(µ )+2[σ(X),µ ])2σ(X)µ = 0. (2.51) 0 0 0 (cid:18)3 (cid:19) The same situation holds to higher orders in ε: to each subsequent order, the respective coeffi- cient in µ remains arbitrary, while the preceding coefficient in µ becomes restricted. Of course, it looks rather difficult to estimate on being such a strange procedure ”convergent” to infinite order in ε. 7 It may look a bit strange that the operator H from Eq. (2.39) appears in Eqs. (2.50) and (2.51) while H from Eq. (2.20) enters Eq. (2.31). In fact, one could, in principle, proceed directly from the Eq. (2.41) formulated via the operator H from the very beginning. Then one could use the Eq. (2.41), together with the properties (σ(X)1) = 0, (H1) = 0, (2.52) to derive the equations (2.50), (2.51). Also, notice that there is an implication (σ(X))2 = 0 ⇒ (σ(X) O σ(X))2 = 0, (2.53) with O being any operator. Finally we present a simple general argument, based on the integration by parts, that the partition function (2.4) is independent of finite arbitrariness in a solution to the gauge-fixing master action X, i i exp X′ = exp{[∆,µ]}exp X (2.54) (cid:26)~ (cid:27) (cid:18) (cid:26)~ (cid:27)(cid:19) i i = exp X + ∆µE([∆,µ])exp X , (2.55) (cid:26)~ (cid:27) (cid:18) (cid:26)~ (cid:27)(cid:19) where µ is any finite Fermionic operator and the function E(x) is defined in Eq. (2.32). By substituting Eq. (2.55) into Eq. (2.4) with X′ standing for X, and then integrating by parts with Eq. (2.6) taken into account, one observes that the second term in the right-hand side in Eq. (2.55) does not contribute to the integral (2.4). Thereby, the integral (2.4) with X′ standing for X reduces to the case of initial X standing for itself. Thus, the partition function is independent of a particular representative of the class (2.54). 3 Ward identities in the standard W-X formulation Let J be external sources to the variables zA; then the integral (2.4) generalizes to the A generating functional, i Z[J] = DzDλexp [W +X +J zA] . (3.1) Z (cid:26)~ A (cid:27) Arbitrary variation δzA yields the equations of motion, h∂ (W +X)i +J (−1)εB = 0, (3.2) B J B where h...i is the source-dependent mean-value J 1 i h...i = DzDλ(...)exp [W +X +J zA] . (3.3) J Z[J] Z (cid:26)~ A (cid:27) 8 It follows from Eq. (3.2) that J ωAB ∂ (W +X) +J ωAB J (−1)εB = 0, (3.4) A B J A B (cid:10) (cid:11) where ωAB = (zA,zB) = const (3.5) is the fundamental invertible antibracket. In Eq. (3.1), the BRST-BV variation (2.12) yields J ωAB ∂ Y = 0 (3.6) A B J (cid:10) (cid:11) due to Eq. (2.16) for µ = const. It follows then from Eqs. (3.4) and (3.6) that 1 J ωAB ∂ W = − J ωAB J (−1)εB. (3.7) A B J 2 A B (cid:10) (cid:11) Thus we have eliminated the average (2.31) of the gauge-fixing master action X from the new Ward identity (3.7). The price is that we have got the non-homogeneity quadratic in the external sources J in the right-hand side in Eq. (3.7). Finally, at the level of finite BRST-BV transformations, the relation (2.30) yields i exp J (zA −zA)+A = 1. (3.8) (cid:28) (cid:26)~ A (cid:27)(cid:29) J However, it is impossible to eliminate the average (2.31) of the gauge-fixing master action X from (3.8). 4 W-X formulation to the Sp(2)-symmetric field-antifield formalism Let zA be the complete set of the variables necessary to the W-X formulation of the Sp(2)- symmetric field-antifield formalism [15, 17, 18] zA = {Φα,παa;Φ∗ ,Φ∗∗} (4.1) αa α whose Grassmann parities are ε(zA) = {ε ,ε +1;ε +1,ε }. (4.2) α α α α We denote the respective zA derivatives as ∂ = {∂ ,∂ ;∂αa,∂α}. (4.3) A α αa ∗ ∗∗ Let Z be the partition function: i Z = DzDλexp [W +X] , (4.4) Z (cid:26)~ (cid:27) 9 where λα are Lagrange multipliers for gauge-fixing with Grassmann parities ε(λα) = ε . (4.5) α In thepartitionfunction(4.4), the dynamical gauge-generatingmaster actionW andthe gauge- fixing master action X is defined to satisfy the respective quantum master equation i 1 ∆a exp W = 0 ⇔ (W,W)a +(VaW) = i~(∆aW), (4.6) (cid:18) + (cid:26)~ (cid:27)(cid:19) 2 i 1 ∆a exp X = 0 ⇔ (X,X)a −(VaX) = i~(∆aX). (4.7) (cid:18) − (cid:26)~ (cid:27)(cid:19) 2 In the above quantum master equations (4.6) and (4.7), the ∆a,( , )a, Va and ∆a are the ± Sp(2)-vector-valued odd Laplacian ∆a = ∂ ∂αa(−1)εα +εab∂ ∂α (−1)εα+1, (4.8) α ∗ αb ∗∗ antibracket ←− −→ ←− −→ (f,g)a = (−1)εf[[∆a,f],g]1 = f ∂ ∂ αa +εab ∂ ∂ α g −(f ↔ g)(−1)(εf+1)(εg+1), (4.9) α ∗ αb ∗∗ h i special vector field Va = VAa∂ = εabΦ∗ ∂α, (VazA) = VAa, (4.10) A αb ∗∗ and i ∆a := ∆a ± Va, (4.11) ± ~ respectively. For the W-action, one should require that W is independent of παa, (Φ∗∗,W) = 0. (4.12) α As to the X-action, it can be chosen in the form related to the gauge-fixing Boson F(Φ), ←− 1 ←− ←− X = Φ∗ παa +(Φ∗∗ −F ∂ )λα + F ∂ παa ∂ πβbε αa α α 2 α β ba 1 ←− ←− = Φ∗ παa +Φ∗∗λα + F d a dbε , (4.13) αa α 2 ba where ←− ←− ←− da = ∂ παa − ∂ λαεba (4.14) α αb is a Sp(2)-vector-valued Fermionic differential that acts from the right. 10