Russian Physics Journal, Vol. 47, No. 10, 2004. ELEMENTARY PARTICLE PHYSICS AND FIELD THEORY. THE EFFECTIVE ACTION FOR SUPERFIELD ...
Russian Physics Journal, Vol. 47, No. 10, 2004
ELEMENTARY PARTICLE PHYSICS AND FIELD THEORY THE EFFECTIVE ACTION FOR SUPERFIELD LAGRANGIAN QUANTIZATION IN REDUCIBLE HYPERGAUGES A. A. Reshetnyak
UDC 539.01
The rules of local superfield Lagrangian quantization in reducible non-Abelian hypergauge functions are formulated for an arbitrary gauge theory. The generating functionals of standard and vertex Green’s functions which depend on the Grassmann variable η via super(anti)fields and sources are constructed. The difference between the local quantum and the gauge fixing action determines an almost Hamiltonian system such that translations with respect to η along the solutions of this system define the superfield BRST transformations. The Ward identities are derived and the gauge independence of the S-matrix is proved. INTRODUCTION
The BRST symmetry principle underlies both the canonical [1] and the covariant quantization scheme [2] for general gauge theories and their superfield generalizations [3–5]. The superfield quantization [3], which is applicable in the canonical formalism and in its implication – Lagrangian formalism, makes use of the nontrivial relation of the odd Grassmann η and even t projections of supertime, Γ = (t, η), as distinct from the Lagrangian quantization [4, 5]. Note that algorithmic methods have been found [6] for constructing generalized Poisson sigma models in the framework of the superfield formalism [7]. The quantization described in [3] has been generalized for two and more supersymmetries which are associated with Grassmann variables η1,…,ηN [8]. The aim of the work under consideration is to construct a local version of the superfield Lagrangian quantization (SLQ). In the SLQ, we realize the explicit superfield representation of the structural functions of a gauge algebra (GA), not indicated in [4, 5], in the framework of an η-local superfield model (SM) which includes the initial standard gauge model. A correlation with classical mechanics reconstructs the dynamics and gauge invariance for the original model in terms of η-local differential equations (DE’s). The properties of the local generating functionals of Green’s functions (GFGF’s) are derived from a Hamiltonian system (HS), which is constructed w.r.t. the η-local quantum and the gauge fixing action. In the SLQ, we first define the effective action for a wider class of non-Abelian reducible hypergauges (the case of irreducible hypergauge functions is considered in [9]). In this paper, we describe the Lagrangian and Hamiltonian formulations of the SM, specify quantization rules, and determine, based on the component formulation, the relations of the proposed quantization scheme to the superfield quantization [4, 5] and multilevel formalism [9]. We make use of some of conventions from [4, 5] and the condensed notation from [10]. The rank of an even supermatrix is characterized by a pair of numbers (k+, k─), where k+ and k─ are the respective ranks of the Bose–Bose and Fermi–Fermi blocks of the supermatrix with respect to the basic Grassmann parity ε. A similar pair of numbers denotes the dimension of a supermanifold, which is equal to (dim+, dim─). On the set of these pairs, operations of component-wise composition and comparison ((k+, k─)>(l+, l─) ⇔ ((k+>k─, l+≥ l─) or (k+≥k─, l+> l─)), (k+, k─)=(l+, l─) ⇔ ( k+=l+, k─=l─)) are defined.
Tomsk State Pedagogical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 59–66, October, 2004. Original article submitted April 22, 2004. 1026
1064-8887/04/4710-1026 ©2004 Springer Science+Business Media, Inc.
THE LAGRANGIAN AND HAMILTONIAN FORMULATIONS OF A SUPERFIELD MODEL
The basic objects in the Lagrangian and Hamiltonian formulations of an SM are the Grassmann-valued C∞(ΠTMcl) and C (ΠT*Mcl) functions of the respective Lagrangian and Hamiltonian actions1: ∞
S L : ΠTM cl × {η} → Λ1 (η, R ),
S H : ΠT * M cl × {η} → Λ1 (η, R ), ε ( S L ) = ε ( S H ) = 0
(1)
and the functionals Z[A] and ZH[Γl], nonequivalent to these functions, whose densities are defined accurate to the corresponding functions f ( A(η), ∂ η A(η), η) and f (Γl (η), η) ∈ Ker{∂ η } ,
( Z [ A], Z H [Γl ]) = ⎛⎜ ∂ η S L (η),
(∫ d η = ∂η , ∂η ≡
dl ) dη
:
⎛1 ⎞ ⎞ ∂ η ⎜ Γlp ωlpq ∂ ηr Γlq − S H ⎟ (η) ⎟ , ε ( Z ) = ε ( Z H ) = (1, 0,1). ⎝2 ⎠ ⎠
⎝
(2)
The values of the Grassmann parities ε = ( ε P , ε J , ε ) , ( ε = ε P + ε J ), where ε J and ε P are auxiliary Z2-gradings w.r.t. the coordinates zM and η of the superspace2 M = M × P , are defined for Ai (η) by the rule ε ( Ai ) = ( (ε P )i , (ε J )i , εi ) . The
(
supermatrix ωlpq (η) is inverse of the supermatrix ωlpq (η) = Γlp (η), Γlq (η) in terms of the local superantibracket ( • , • )η =
∂r • ∂Γlp ( η)
ωlpq (η)
∂l • ∂Γlq ( η)
)η
: ωlpq (η)ωlqd (η) = δ pd which is defined
, where
∂ r (l )
is the right (left) superfield
∂Γlp ( η)
variational derivative w.r.t. the superfield Γpl(η) for a fixed η. Assuming the existence of a critical superfield configuration for the functionals Z[A] and ZH[Γl], we may code the SM dynamics, respectively, by superfield Euler–Lagrange equations, which, in view of the identities ∂ η2 Ai (η) = 0 , are equivalent to a Lagrangian system (LS), and by a Hamiltonian system (HS) ⎧ 2 i ∂ l2 S L (η) = ∂ η2 Ai (η) ( S L′′ ij ) (η) = 0 , ⎪∂ η A (η) i j ∂ ∂ η ∂ ∂ η A ( ) A ( ) ⎪⎪ η η ⎨ ⎛ ⎪ ∂ l S L (η) ∂ l S L (η) ∂ S (η) εi ⎜ ∂ l Θ η = − − + ( ∂ ηU + ) (η) l L i ( ) ( 1) ⎪ i i i ⎜ ∂η ∂ ∂ η A (η) ∂A (η) ∂ ∂ η A (η) ⎪⎩ ⎝
(
)(
)
(
(
)
∂ ηr Γlp (η) = Γlp (η), S H (η)
1
Here,
(
)
(3)
⎞ ⎟ = 0; ⎟ ⎠
)η ,
ΠTM l = {( Ai , ∂ η Ai )(η) | Ai (η) = ( Ai + λi η) ∈ M cl , i = 1,..., n = n+ + n− }
(4)
and
{
ΠT ∗ M l = Γl p (η) = ( Ai , Ai∗ )(η) |
A (η) = A − ηJ i , l = cl} are the odd tangent and cotangent bundles over the configuration space of classical superfields ∗ i
∗ i
Ai(η) with n+ bosonic and n─ fermionic degrees of freedom with respect to the ε-parity for fixed continuous components of the condensed index i, and the quantities Ai, λi, Ai∗ , and Ji for ε P ( Ai (η)) = 0 are, respectively, the classical fields, Lagrangian multipliers, antifields, and sources to the fields Ai of the Batalin–Vilkovisky (BV) quantization method. 2
The superfields Γpl (η) are defined on M, which can be realized as the quotient of a global symmetry supergroup J for the
functionals Z[A], ZH[Γl], J = J × P , P = exp{ιµpη}, where µ and pη are the nilpotent parameter and the generator of η-shifts, respectively ( µ 2 = pη2 = 0 ), with a spacetime supersymmetry group chosen for J . 1027
where the nilpotent operator ( ∂ ηU + ) (η) = ∂ η Ai (η)
∂l ∂Ai ( η)
= ⎡⎣ ∂ η , U + (η) ⎤⎦ , U + (η) = η∂ η Ai (η) −
∂l
is introduced. The
∂Ai ( η)
equivalence of both formulations and, hence, the coincidence of Z[A] with ZH[Γl] and LS (3) with HS (4) is guaranteed by the nondegeneracy of ( S L′′ ij ) (η) in the framework of a Legendre transform of SL(η) with respect to ∂ ηr Ai (η) : S H (Γ(η), η) = Ai∗ (η)∂ ηr Ai (η) − S L (η) , Ai∗ (η) =
(
∂ r S L (η)
∂ ∂ ηr Ai (η)
)
(5)
.
If we apply the first Noether theorem to the invariance of the density dηSL(η) with respect to the η-shifts by a constant µ treated as symmetry transformations (Ai,zM,η) → (Ai,zM,η+µ), we arrive at the conclusion that the LS has an integral S E ( ( A, ∂ η A ) (η), η) =
∂ r S L (η)
(
∂ ∂ ηr Ai (η)
)
∂ ηr Ai (η) − S L (η) : S H (η) = S E ( A(η), ∂ η A(Γ(η), η), η)
(6)
(which is conserved during the η-evolution of the quantity) if the equations ∂r ∂η
S L (η) = 0,
( ∂ ηU + ) (η)S L (η)
| Aˆ i ( η)
=0
(7)
are obeyed for the solutions Âi(η) of the LS. In terms of the action SH(Γ(η),η), the classical Lagrangian master equation in ΠTMcl for a solution of the LS [the second equation of (7)] has the form of the Hamiltonian master equation for the solution Γˆ ( η) of the HS: ∂r S (η) ∂η H
= 0, ( S H (η), S H (η) )η = 0 .
(8)
The functions Θi(η)=Θi((A,∂ηA)(η),η), constraining the statement of the Cauchy problem for the LS, are, in general, functionally dependent. Let us restrict our consideration to an SM representable by natural equations: S L (η) = T ( ∂ η A(η) ) − S ( A(η), η) :
Θi (η) = − S ,i ( A(η), η)(−1)εi , S ,i (η) =
∂ r S (η) ∂Ai (η)
.
(9)
The equation for the functions Θi(η) implies, provided that there exists an (m+,m─)-dimensional hypersurface Σ⊂Mcl such that Θi(η)│Σ≡0, that there are at least m (m=m++m─) independent identities which can be associated with special gauge transformations (SGT’s) for Ai(η): α
α
S ,i ( A(η), η) Rαi 0 ( A(η), η) = 0, δAi (η) = Rαi 0 ( A(η), η)ξ0 0 (η) , α0 = 1,..., m0 = m0 + + m0 − , ε (ξ0 0 ) = εα0 .
(10)
The generators Rαi 0 (η) are dependent if the rank║ Rαi 0 (η) ║│Σ=(m+,m─)
s +1
s +1
s
εε α α j ij ji ∑ (−1)k (m( s − k −1) + , m( s − k −1) − ), Z α0−1 (η) ≡ Rαi 0 (η), Lα1−1 (η) ≡ K α1 (η) = −(−1) i j K α1 (η) ,
(11)
k =0
(mL + , mL − ) =
L
∑ (−1)k (m( L − k −1) + , m( L − k −1) − ) .
k =0
∂ ,U (η)} singles out from relations (11) the GA’s with structural The system of projectors on С∞(ΠTMcl)×{η} : {1-η∂η,η ∂η +
functions and relations not explicitly depending on η and the GA for η = 0 with standard reducible model relations for (εP)I = (ε P )α s = 0 [2]. Definitions (9)–(11) are also valid for the Hamiltonian formulation of an SM in which the integrability of the HS is guaranteed by that the master equation (8) holds true by virtue of the antibracket Jacobi identity in the expression
((
0 = ∂ η2 Γ p (η) = − Γ p (η), S H (Γ(η)) )η , S H (Γ(η))
)η = − 12 ( Γ p (η),
( S H (Γ(η)), S H (Γ(η)) )η ) = 0. η
(12)
This provides validity for the η-translation formula on С∞(ΠT*Mcl) ×{η}
δµ F (η)|Γˆ = µ∂ η F (η)|Γˆ = µs l (η) F (η), s l (η) =
∂ ∂η
− ( S H (Γ(η)), )η
(13)
and nilpotency for the BRST-like generator of η-shifts s l (η).
LOCAL SUPERFIELD QUANTIZATION
The SLQ rules for a standard initial model under a usual ghost number distribution with gh(Ai) = –gh ( Ai* ) – 1 = 0 [2] and gh(η, ∂η) = (–1, 1) are initiated by restricting any formulation of the SM to the equations
( gh, ) S ∂ ∂η
H ( L ) (η)
= (0,0).
(14)
Their nontrivial solution exists if SH(L)(η) contains a potential summand S(A(η),0) = S(A(η)): S(A(0)) =S0(A). The α
restricted SGT (10), after the substitution ξ0 0 (η) = С α0 (η)d η , can be represented by an almost Hamiltonian system of 2n
DE’s with a Hamiltonian S10 ( Γ cl , C0 ) (η) = ( Ai* Rαi 0 ( A)C α0 )(η) . The function S10 (η) , by virtue of Eqs. (11), is invariant α
modulo S,i(η) with respect to the new SGT’s for the ghosts of С α0 (η) with arbitrary functions ξ1 1 (η) α
α
δС α0 (η) = Z α 0 ( A(η))ξ1 1 (η) , 1
α
( ε, gh ) ξ1α1 (η) = ( εα1 + (1,0,1),1) ,
(
)
(15) α
which can be reformulated for ξ1 1 (η) = С α1 (η)d η as an almost HS with S11 A, C0* , C1 (η) = (Cα* 0 Z α 0 ( A)C α1 )(η) . 1
As a result, all, possibly enhanced, restricted relations (11) for the restricted SM of L-stage reducibility are described by a sequence of SGT’s for the higher ghosts of С α s −1 (η) that leave the function S1s −1 (η) , subject to conditions (14), invariant modulo S,i(η) for s = 1, …, L:
1029
α
( ε, gh ) ξαs s = ( εαs + s(1,0,1), s ) . 3
α
δС α s −1 (η) = Z α s −1 ( A(η))ξαs s (η), S1s −1 (η) = (Cα* s −2 Z α s −2 ( A)C α s −1 )(η), s −1
s
(16)
The substitution ξαs s (η) = С α s (η)d η transforms the SGT’s (16) into a system of ms-1 first-order DE’s with respect to η, p
extended by introducing Cα* s −1 (η) to an almost HS with 2ms-1 DE’s in terms of superfields Γ s −s1−1 = (C α s −1 , Cα* s −1 )(η) :
(
p
p
∂ ηr Γ s −s1−1 (η) = Γ s −s1−1 (η), S1s (η)
)
η
.
(17)
Having combined the systems (17) for all s with the initial HS (4), restricted by condition (14), we obtain an HS
{
(
p
p
p
)
p
(
A
)
nonintegrable in the sense of (12) in ΠT ∗ M l = Γl l ( η) = Γcl cl , Γ 0 0 ,..., Γ L L ( η) = Φ l , Φ*Al ( η) ,
(
p
p
L ∂ ηr Γl l (η) = Γl l (η), S[1] (η)
)
L
η
(
l = min} :
)
α
L , S[1] (η) = S(A(η)) + ∑ Cα* s −1 Z α s −1 ( A)C α s (η)
s =0
s
(18)
L (η) satisfying the condition that the η-local classical master equation has a proper solution in a with the function S[1]
minimal sector [2]. L (η) in powers of Φ*Al (η) and The integrability of the HS in (18) is provided by the deformation of the function S[1]
then in powers of C α s (η) in the context of the existence theorem [2] for the solution of the master equation:
( S H l ( Γl (η) ) ,
( ε,
S H l ( Γl (η) ) ) = 0, η
∂ ∂η
gh,
The deformation of the function SHmin(η) in the Planck constant to the proper solution in ΠT*Ml= p
(
p
α
{
}
p Γl l
)S
Hl
( Γl (η) ) = ( 0, 0, 0 ) .
(19)
, which preserves the form of Eq. (19), extended
(η) (hereinafter l = ext) with pyramids of ghosts and auxiliary superfields [2]
α
min Γl l ( η ) = Γ min , Cs ' s , Bs ' s , Cs*' α s , Bs*' α s
) ( η) ,
( ε, gh ) Csα's = ( εα s + ( s + 1)(1, 0,1) , 2s '− s − 1)
s ' = 0,..., s,
L
α
= ( ε, gh ) Bs ' s + ( (1, 0,1) , −1) , S H l ( Γl ( η) ) = S H min ( Γl ( η ) ) +
s
α ∑ ∑ ( Cs*' α s Bs ' s ) ( η) ,
(20)
s = 0 s '= 0
{
specifies, for instance, the quantum action S HΨ ( Γ(η), ) = exp ( Ψ ( Φ l (η) ) ,
)η } S H l ( Γl (η), )
for the Abelian
hypergauge. The action S HΨ ( η, ) defines an integrable HS, which, in turn, specifies an η-local not nilpotent generator of BRST transformations in ΠT*Ml, (standard for η = 0 in the BV method) which annulls S HΨ ( η, ) and is associated with its η-nonintegrable subsystem s l ( Ψ ) (η) =
∂ S Ψ ( η, ∂ + r *H ∂η ∂Φ A (η) l
3
) ∂l ∂Φ Al (η)
,
(
)
∂ ηr Φ Al , Φ*Al (η) =
In view of Eqs. (14) and gh(Ai)=0, we assume hereinafter that
(( Φ
Al
(η), S HΨ ( η,
η
(21)
(εP )i = (εP )αs = 0 and, without loss of generality, that the
(
)
(
)
value of L is invariable in this case. In addition, we use the notation C α−1 , Cα* −1 (η) ≡ Ai , Ai* (η) . 1030
)
)) , 0 .
The Feynman rules, in the general case where the initial superfield–superantifield structure on ΠT*Mext is ignored but the Darboux coordinates (φ,φ*)(η) are essentially used, are specified by a GFGF Z(∂ηφ*,φ*,∂ηφ,I)(η) =Z(η) having the form of a functional integral with a fixed η: Z (η) = ∫ d Γl (η)d λ(η)d π(η) exp
((
{ (S i
)
Hl
( Γl (η), ) + X Ψ g ( (ϕ, ϕ* − ϕ* , λ, λ* , π)(η) )|λ* =0
(
) )}
)
− ∂ ηϕ*a ϕa + ϕ*a ∂ ηr ϕa − I C λC − I Dπ π D (η) ,
(22)
with a measure dλ(η)= ∏ tK=0 ∏ tt '=0 d λtt ' (η), d π(η) =∏ tK=1 ∏ tt '=1 d πtt ' (η) . The function Z(η) depends on the sources
( ∂ηϕ*a , ∂ ηr ϕa ,
)
(
) (( ε, gh ) ∂ηϕ*a = ( ε, − gh ) ϕa ) to the superfields (φ , ϕ*a ,λ ,
I C , I Dπ (η) = − J a , λ a , I 0C + I1C η, I 0πD + I1πD η ,
{
} {
}
a
a
C
a
πD)(η), where {φa(η)}∩ Φ Al (η) ≠ Φ Al (η) , and λC = ( λ t 't , t = 0, …, K, t΄ = 0, …, t), πD = ( πt 't , t = 1, …, K,
(( ) a
t΄=1,…,t): ε πt 't
( ) a
= ε λt 't + (1, 0,1) = εat + (t + 1)(1, 0,1)
(
)
form pyramids of Lagrangian multipliers to the dependent
)
hypergauges Ga0 (Γl (η)), ε Ga0 = εa0 , a0 = 1,..., k0 = k0+ + k0 − , and to the gauges that remove their degeneracy. The dependence
of
the
hypergauges
implies
that
if
the
p
rank║ ∂Ga0 (η) ∂Γl l ( η ) ║│∂S/∂Γ = G = 0 =
condition
(k(─1)+,k(─1)─) < (k0+,k0─), (k0 > dim+ ΠT*Mext) is fulfilled for the functions Ga0 (η) , there exist proper null vectors
(
a
a
Z g 0a (Γl (η)) , a1 = 1,..., k1 = k1+ + k1− , ε Z g 0a 1
1
)=ε
+ εa1 , which exhaust the (not all independent) zero modes of the
a0
hypergauges, and so on. The hypergauge conditions Ga0 (Γl (η)) are assumed to be solvable with respect to the superantifields ϕ*a (η) . In addition to the involution relations (Γ )
( Ga (η), Gb (η) )η 0
0
c
c
= Gc0 (η)U a0b (η), U a0b (η) = −(−1) 0 0
(ε
a0
)(
)U c
+1 ε b +1
0 0
0
0
b0 a0
(η)
(23)
and, perhaps, to the unimodularity relations [9], the functions Ga0 (η) , with the local and J -covariant descriptions of the theory preserved, specify hypergauge GA’s of K-stage reducibility by the relations
a
Ga0 (η) Z g 0a (Γl (η)) = 0 , 1
a
a
Z gt −a1 (Γl (η)) Z gt a t
t +1
(Γl (η)) = Gb0 (η) M
at −1b0 (Γl (η)) at +1
t
K
k =0
k =0
,
t = 1,..., K − 1, at = 1,..., kt = kt + + kt − ,
(kt + , kt − ) > ∑ (−1) k (k(t − k −1) + , k(t − k −1) − ) , (k K + , k K − ) = ∑ (−1) k (k( K − k −1) + , k( K − k −1) − ),
(24)
a
where it is taken into account that rank║ Z gt −a1 (Γl (η)) ║│G = 0 < (kt+,kt─). For K = 0 the functions Ga0 (η) are independent t
[9], i.e., a0 ≡ a. The quadratic in (λ,λ*,π)(η) part of the gauge fixing action X((Γl,λ,λ*,π)(η), ),
1031
K
X min
t
(( Γl , λ0 , λ∗0 ) ( η) , ) + t∑=1t∑'=1( λ*t ' a πta' ) ( η) ,
X ( η, ) = X min
t
t
⎛ a = ⎜ Ga0 ( Γl ) λ 00 + ⎝
K
∑
t =1
(
a λ*0 at −1 Z gt −a1 t
at
( Γl ) λ 0
)
(25)
⎞ +O λ ⎟ ⎠
( ) ∗
determines the proper solution of one of the systems of equations in ΠT*Ml={Γl(η)=(Γext,λ,λ*,π,π*), l=tot}: 1)
( X (η, ), X (η, ) )(ηΓl ) = 0, ∆l (η) X (η, ) = 0;
with a nilpotent operator ∆l(η), ∆l(η)= (−1)
( )
ε Γlp 1 ω l 2 pl ql
2) ∆l (η)exp
(
p ⎛ q (η) ⎜ Γl l (η), Γl l (η), ⎝
)
{ i X (η, )} = 0 ,
( Γ l ) ⎞ ( Γl ) η
⎟ ⎠η
(26)
. In the functional integral (22), we
restrict ourselves to the determination of the Lagrangian surface Λg parameterized by superfields (φ*,λ,π)(η), such that the action X
Ψg
(η) on this surface is nondegenerate and given by the so-called second-level gauge fermion Ψg(η) [unlike the
function Ψ(η) in (21)]. The minimal Ψg(η) may have, for instance, the structure K
t
(
)
Ψ g ( Γ, λ ) (η) = ∑ ∑ λt 't ωta't bt −1 λt 't−−11 (η) , ε ( Ψ g ) = (1, 0,1) , t =1 t '=1
with constant supermatrices ωa tb'
t t −1
a
b
(27)
, which provide nondegeneracy of the restriction on Λg for the function X
Ψg
{
(η) = exp ( Ψ g (η),
)η} X (η,
).
(28)
For ∂ηφ* = I = 0, the integrand in Eq. (22) is invariant under the superfield BRST transformations Γtot (η) =( Γ ext ,λ,λ*,π,π*)(η)→ Γtot (η) +δµ Γtot (η) , which represent η-shifts by the odd parameter µ along the solution
Γtot (η) of the η-nonintegrable almost HS
(Γ) ⎧ r pext Ψg pext (η) ⎪∂ ηΓ ext (η) = Γext (η), S − X η ⎪ Ψ ⎪∂ r λC (η) = −2 λC (η), X g (η) , * ⎪ η η |λ = 0 ⎨ ⎪ r D Ψg D ⎪∂ η π (η) = −2 π (η), X (η) η |λ* = 0 , ⎪ ⎪∂ ηr (λ* , π* , ϕ* )(η) = 0. ⎩
(
( (
(
) )
In this case, the actions S(η) = SHext( Γ ext (η), ) and X
) )
Ψg
| λ* = 0
, p
p
⇔ δµ Γtottot (η) = ∂ ηr Γtottot (η)µ| Γ
( (ϕ, ϕ* − ϕ* , λ, λ* , π)(η) )
tot
.
(29)
may satisfy different but fixed systems
in (26).
The HS (29) permits one to establish the independence of the vacuum functional Z X(φ*)=Z(φ*,0,0,0) and, hence, the S-matrix, in virtue of the equivalence theorem [11], on the choice of the gauge. Actually, the η-shift along Γtot (η) by
(
)
µ (ϕ, ϕ* − ϕ* , λ, λ* , π)(η) corresponding to the HS (29) in ZX+∆X(φ*) is compensated by an additional η-shift by a constant
(
µ1 along the solution of an almost HS of the form (29) with a Hamiltonian W (ϕ, ϕ* − ϕ* , λ, λ* , π)(η),
( ε, )W (η) = (0, 0) related to µ(η) by W (η)µ ∂ ∂η
1032
1
= −i µ(η) . As a result, we have
):
(
ZX+∆X (ϕ∗ (η)) = ∫ d Γ ext (η)d λ (η)d π(η) exp ⎧⎨ i ⎜⎛ S + X Ψ g +∆X Ψ g + ∆X ⎩ ⎝ If the action ∆X
Ψg
(X
Ψg
+ ∆X
Ψg
) (η)
)
⎞ ⎫ ⎟ (η) ⎬ . ⎭
(30)
|λ* =π* = 0 ⎠
obeys the system in (26) that is also valid for the function X
Ψg
(η) , the variation
(η) satisfies a linear equation with a nilpotent operator Q j ( X ) :
(
Q j ( X )∆X (η) = 0 , Q j ( X ) = X
Ψg
)
(η),
η
− δ j 2 (i ∆ (η)),
j = 1, 2 .
(31)
As in [12], the general solution of this equation, vanishing for Γtot(η) = 0 has the form ∆X
Ψg
( ε, ) ∆Y (η) = ( (1, 0,1), 0 ) . ∂ ∂η
(η) = Q j ( X )∆Y (η),
Ψg
Assuming that the action W(η) satisfies the same system as that for the function X
(32) (η) and taking account of the
representation for ∆X (η) , ∆X (η) = −2Q j ( X )W (η)µ1 , we obtain coincidence of ZX+∆X(φ*) with ZX(φ*) by setting
W (η)µ1 = 12 ∆Y (η) . Another implication of the corresponding systems in Eqs. (26) for S(η) and X GFGF Z(η)
Ψg
(η) is the Ward identity for the
⎛ ⎜⎧ ⎛ ⎞⎛ ∂l ∂l ∂ r ⎞ ⎫⎪ ⎜ ⎪⎨∂ ηϕ∗a (η) ∂ l − ⎜ ∂S Hext ⎟ ⎜ i ⎟ (η) ⎬ ,i ∗ ∗ a r ⎜ ⎟ ⎜⎪ ∂ϕa (η) ⎝ ∂ϕ (η) ⎠ ⎝⎜ ∂ (∂ ηϕ ) ∂ (∂ ηϕ) ⎠⎟ ⎭⎪ ∂ϕ∗a (η) ⎜⎩ ⎝ i
+ I C (η)
∂l X
Ψg
⎛⎛ ⎜⎜i ⎝⎝
∂l
∂r
,i
∂ ( ∂ ηϕ∗ )
∂ ( ∂ ηr ϕ)
−ϕ∗ , −i
∂l ∂I
⎞ ⎞ , λ∗ , −i π ⎟ (η) ⎟ ∂I ⎠ ⎠ ∂l
∂λ∗C (η)
⎞ ⎟ ⎟ z (η) = 0 |λ∗ = 0 ⎟ ⎟ ⎠
which is obtained by functional averaging, for instance, of the second equation in (26) for X (provided that X
Ψg
Ψg
(33)
( (ϕ, ϕ* − ϕ* , λ, λ* , π)(η) )
( η) is the solution of the same equation)
∫ d Γext (η)d λ(η)d π(η)exp
{
i
( S Hext ( Γext (η), ) − ( ( ∂ ηϕ*a ) ϕa + ϕ*a ( ∂ ηr ϕa ) − IC λC − I Dπ πD ) (η) )} ×
∆ (η) exp
{
integration by parts in (34), and use of the rule
i
X
(
Ψg
∂ ∂ϕ∗
( (ϕ, ϕ* − ϕ* , λ, λ* , π)(η) )}|λ =0 = 0 , *
−
∂ ∂ϕ∗
)X
Ψg
(34)
(η) = 0 .
The definition of GFGF Z(η) in the form of (22) suffices to introduce the effective action Γ(η)= Γ(φ,φ*,∂ηφ,I)(η) by means of the Legendre transformation of Z(η) with respect to ∂ηφ*(η):
1033
Γ(η) =
i
((
The Ward identity for Γ(η) with Γ′′ab (η) =
∂l X I C (η)
Ψg
⎛ a ⎜ϕ + i ⎝
) )
ln Z (η) + ∂ ηϕ*a ϕa (η) ,
( Γ′′−1 )
ac
∂l
∂l
b
∂ϕ ( η) ∂ϕ
,i
∂ϕc ( η)
∂r a
∂r
∂ l ln Z (η)
i ∂ (∂ ηϕ∗a (η))
(
′′ (η) Γ′′−1 Γ(η) ⎛⎜ Γba ( η) ⎝
−
∂ ( ∂ ηr ϕ)
ϕa (η) = −
∂Γ
∂r Γ ∂ ( ∂ ηr ϕ)
− ϕ∗ , ∂Il − i
∂l ∂I
)
ac
.
= δbc ⎞⎟ has the form ⎠
, λ∗ ,
∂l Γ ∂I π
−i
∂λ∗C (η)
⎛ ∂S ⎞⎛ ⎛ − ⎜⎜ aHext ⎟⎟ ⎜ ⎜ ϕb + i ⎝ ∂ϕ (η) ⎠ ⎜⎝ ⎜⎝
( Γ′′−1 )
bc
∂l c
∂ϕ (η)
,i
∂r
−
∂ (∂ ηr ϕ)
(35)
∂l ∂I π
⎞ ⎞ ⎟ (η) ⎟ ⎠ ⎠
| λ∗ = 0
(36)
∂ r Γ ⎞ ⎞ ∂ l Γ(η) 1 ⎟ (η) ⎟ + ( Γ(η), Γ(η) )η = 0. ∂ (∂ ηr ϕ) ⎟⎠ ⎟⎠ ∂ϕ*a (η) 2
For IC(η) = 0, identities (33) and (36) take the standard form, complicated by the presence of the sources (∂ ηr ϕa , I Dπ )(η) , but for the generating functionals Z(η) and Γ(η) in non-Abelian hypergauges.
ASSOCIATION WITH THE BV METHOD AND SUPERFIELD LAGRANGIAN QUANTIZATION
For a special type SM, the association of the quantities of a local SLQ with the standard description of field theory objects in the framework of the BV method [2] is provided by the restriction η = 0 for all the quantities and relations, for instance:
(M ,
) (
ΠT ∗ M tot , I C , I Dπ → M , ΠT ∗ M tot
{
p
}
)
= Γ 0 tottot , I 0C , I 0πD . Additionally, for the SM given by
| η= 0
expressions (3)–(8), one must eliminate from SL(η) and SH(η) the superfields ∂ηAi(η) and Ai∗ (η) and those superfields of Ai(η) which contain functions with a wrong relationship between the spin and the statistic, gh(Ai) ≠ 0 and (ε P )i ≠ 0 , either by setting them equal to zero or by applying horizontality conditions, as in the case of Yang–Mills theories [13, 14]. For an arbitrary smooth functional F[Γl], which is used in the methods described in [4, 5], there exists a function F(η)=F((Γl, ∂ηΓl)(η),η), l={cl, min, ext} such that F [Γl ] = ∫ d η η F (η) = F ( Γl (0), ∂ ηΓl , 0 ) = F ( Γ0l , Γ1l ) ,
(37)
and F[Γl] does not depend on Γ1pl if F(η) = F((Γl(η),η). From Eq. (37) it follows that there exists a relationship between the local operations (•,•)η, ∆(η) and the functional operations (•,•), ∆ [4], which coincide with their analogs in the BV method:
(( F (η), G(η) )η ,
∆ (η) F (η)
)|η=0
( ( F [Γl ], G[Γl ]) ,
=
∆F [Γl ]) .
For l = ext, the action of the operators U and V [4] coincides with the action of (∂ηU+)(0), (∂ηV+)(0) = ∂ ηΦ∗Al (η)
(38) ∂l
∂Φ∗Al (0)
.
Restricting ourselves to the case of independent hypergauges, we obtain, in view of the representation of the action
X(η) in (25), that for ϕ∗0 = 0 and η = 0 the vacuum functional ZX( ϕ∗0 ) coincides with the “first level” functional integral Z1 [9] with the density M(Γ0) = 1 determining the measure µ[Γ0] = MdΓ0, Z1 = ∫ d Γ 0 d λ 0 exp
1034
{ ( S (Γ ) + G (Γ )λ )} . i
0
a
0
a 0
(39)
If the functions S(η) and X(η) satisfy the second system in (26), the transformations (29), for π D = π∗D = η = 0 , coincide, up
(
)
to the sign, with the BRST transformations for Z1 [9]. Simultaneously, the quantities S [Γ] = S Hl ( Γ0l ) − ∂ ηΦ∗Al φ Al , and X [ Γ ] = X ( Γ 0l , λ 0 ) − φ∗Al ∂ ηr Φ ( η ) , l = ext , satisfy the generating equations given in [4, 5] having the form of the second Al
system in (26) with the operators ∆1=∆─(i )─1V, ∆2=∆+(i )─1U defined, respectively, in ΠT*Mext and on the surface A
λ 0 l (0) = ∂ ηr Φ Al (η) for K = 0. In the particular case of the boundary condition for X [Γ] with ϕ0a = φ Al δ δλ Al
( X [Γ] + φ∗B λ B ) = GA (Γl )
(40)
l
l
l
and with S[Γ] = S( φ, φ∗ , J ) independent on the fields λ Al , this provides coincidence of ZX( ϕ∗0 ) with ϕ∗0 = 0 with the vacuum functional Z of [5] written in component form: Z = ∫ d φd φ∗ d λ exp
{( i
(
) )| J = 0 }
S (φ, φ∗ , J ) + X (φ, φ∗ , λ, J ) + ∂ η Φ∗Al Φ Al (η)
.
(41)
This correspondence permits one to state that the effective action can also be determined in the framework of the method described in [4, 5]. ACKNOWLEDGEMENT
The author is grateful to P. M. Lavrov for useful critical comments. REFERENCES
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