Nuclear Physics B238 (1984) 601620 © NorthHolland Publishing Company
R E P R E S E N T A T I O N S OF BRS A L G E B R A Kazuhiko NISHIJIMA
Department of Physics, University of Tokyo, Tokyo 113 Received 13 September 1983 In the canonical formulation of gauge theories the BRS transformation plays a fundamental rSle. The generator of this transformation along with the ghost number forms an algebra called the BRS algebra. Certain properties of this algebra are essential to the proof of unitarity of the S matrix in the physical sector and also to the discussion of color confinement. In the present paper we present all the possible representations of the BRS algebra in the light of indefinite metric.
1. Introduction
In the canonical formulation of nonabelian gauge theories [1] the lagrangian density generally consists of three parts, the gauge invariant part, the gauge fixing part and the FaddeevPopov (FP) ghost part. Because of the presence of the latter two parts the complete lagrangian density is no longer invariant under local gauge transformations, but it turns out to be invariant, miraculously, under a new global transformation called the BecchiRouetStora (BRS) transformation [2]. It is also invariant under the scale transformation of the ghost fields [1]. Thus we have two conserved generators of these transformations. They will be denoted by OB and Oc, respectively, hereafter. Both operators are hermitian and satisfy the following commutation and anticommutation relations:
i[Qc, QB] = QB,
O 2=0.
(1.1)
These relationships define the BRS algebra. It has been known that there are two irreducible representations [1] called the BRS singlet and BRS doublet, but their definitions are subject to some nonuniqueness which we shall remove in the present paper. A characteristic feature of this algebra consists in the observation that the antihermitian operator iQ~ has only real eigenvalues and the hermitian operator QB vanishes identically when it is squared. These two properties cannot be realized in the positivedefinite metric theory, and introduction of the indefinite metric is indispensable in realizing the representations of the BRS algebra. All these pathological features are consequences of the wrong statistics obeyed by the FP ghost fields. In sect. 2 we shall study all the possible irreducible representations of the (restricted) BRS algebra defined by eq. (1.1). Some authors [3], [4] have found 601
K. Nishijirna / Representations of BRS algebra
602
that it is possible to generalize the BRS algebra by introducing an alternative BRS transformation with the generator denoted by (~B. The generalized or full BRS algebra is defined by the following commutation and anticommutation relations: i[Oc, O , ] = OB,
i[Oc, 0 u ] = (~B,
(1.2)
2 
OB  0 2 ={OB, 0B} = 0
(1.3)
Furthermore, we can incorporate a unitary operator Y called the ghost conjugation into the full BRS algebra. This operator is conserved in the Landau gauge [4] and satisfies the relations (~ I ~ ) B g ~ = 0 B
,
(~ I [ ~ B g ~ =   Q B ,
(~
IOc~=O
c.
(1.4)
In sect. 3 we shall study the irreducible representations of the full BRS algebra. In sect. 4 we shall study the possible realization of the irreducible representations in gauge theories. Finally in sect. 5 we shall introduce a new symmetry for the FP ghost fields in the Landau gauge, and on the basis of this symmetry we shall uniquely determine the representation in the confinement phase.
2. Representations ot the restricted BRS algebra As has been emphasized in the introduction the indefinite metric plays an essential r61e in finding the representations of the BRS algebra. Thus we shall first recall some basic features of the indefinite metric [6]. Let us consider a vector space 7/I and define the inner product between two vectors by the following conditions: (i) (/Ik) is a complex number.
(ii) (iii)
(Ilk) =
(k]l)*,
(ll a k +/3k') = ~ il Ik) +/3( IIk'),
(2.1) (2.2)
where a and/3 are cnumbers. Then the vector space of positivedefinite metric is characterized by an additional condition (k{k)~>0,
(2.3)
and (k] k) = 0 if and only if ]k) = 0. The vector space of indefinite metric is introduced by lifting the above condition. Then (klk) need not be positive, but it can be any real number. Let us introduce a complete set of base vectors {6} in terms of which an arbitrary vector Ix) can be uniquely expressed as [ x ) = E xjlej), J
(2.4)
K. Nishijima / Representations of B R S algebra
603
then making use of this set the representation t of a linear o p e r a t o r T is given by
T[ ej) = 57 lek)tkj.
(2.5)
k
Next we introduce a change of base vectors by
l e;) = E [ek)u~q,
(2.6)
k
and correspondingly t __ ! t T left  57 [ek)tkj.
(2.7)
k
T h e n the representation of the o p e r a t o r undergoes a similarity transformation t t = ultu,
(2.8)
T h e metric matrix .1 is defined by *1k, = (G[ej),
(2.9)
then it is hermitian by virtue of (2.1) .1'=.1.
(2.1o)
Notice that .1 is not an o p e r a t o r because it does not undergo a similarity transformation for a change of base vectors, but instead it is transformed as
.1'= u;*1u.
(2.11)
We also introduce hermitian conjugate T+ of a linear o p e r a t o r by (k] T*] 1) = (1[ T] k ) * ,
(2.12)
where ]k) and 1/) are arbitrary vectors. Let t and/" be the representations of T and T +, respectively, then we have
,17 = t" *1.
(2.13)
In what follows we always assume that .1 is nonsingular, namely, det .1 S 0 ,
(2.14)
F= .1 't+*1.
(2.15)
then we have
Therefore,/" is not necessarily the hermitian conjugate matrix of t. W h e n the o p e r a t o r T is hermitian we have *it = t**1, and this combination itself is an hermitian matrix.
(2.16)
K. Nishifima / Representations of BRS algebra
604
With these preliminaries we shall proceed to the problem of finding representations of the restricted BRS algebra. We shall list all the conditions that we are going to use: (i) hermiticity: O~ = OB, O~ = Oc; (ii) BRS algebra: eq. (1.1); (iii) ghost number: all the eigenvalues of iOc are integers and define the ghost number. The last condition does not follow from the BRS algebra, but it follows from the definition of Qc. In constructing a representation we have to determine matrices not only for OB and O~ but also for r/. The matrices representing OB and iOc will be denoted by q and n, respectively. Then we have to express the above three conditions in terms of r/, q and n. (i) Hermiticity r/q = q ' r / ,
(2.17)
~n:n *7/.
(2.18)
(ii) BRS algebra
[n,q]=q,
q2=0.
(2.19)
(iii) Ghost number. We shall choose a representation in which n is diagonal.
(2.20)
0
F/=
1
2
A small block represents a matrix of the form  ~ = N" ~N,
(2.21)
where ~N is the unit matrix in a subspace denoted by 7/"(N) satisfying
~(N) ={ix)[[x)e oV, (iOc_S)lx)=O}, so that we have a decomposition of °V as c~
(2.22) N~oo
K. Nishijima / Representations of BRS algebra
605
The representation (2.20) already takes care of condition (iii). In this representation n t = n, so that eq. (2.18) requires that n should anticommute with 7/ Then the most general form of 7/ reads as
=
(2.23) r/1
where the submatrices satisfy the hermiticity condition */N = T/*N,
(2.24)
and ~V(N) and T "(N) must be of the same dimension. We shall further simplify (2.23) by choosing an appropriate set of base vectors. We shall recall eqs. (2.8) and (2.11) and apply them by choosing u in a block diagonal form .
U =
(2.25)
Uo u
1
then n is not changed but 77 is changed as t WN ~ UNTqNUN.
(2.26)
Tlo ~ UoTloUo .
(2.27)
For N = 0, we have ~/~ = 7/o and
First, we can diagonalize ~/o by choosing an appropriate unitary matrix Uo. Then, by choosing an appropriate diagonal matrix Uo as the second transformation we may bring 7/o into the following form:
tl t 1
7/o =
1
"
1
(2.28)
606
K. Nishijima / Representations of B R S algebra
Then we have r/2=~o,
(2.29)
where ~o is the unit matrix in ~0>. As we shall see later, however, we have to choose a proper set of base vectors with reference to QB, so that the form (2.28) will be modified. Nevertheless, eq. (2.29) is maintained as long as we choose unitary u0 for further transformations. For N > 0, we choose U s as UN = t i N ,
= 1,
UN
( N > 0),
(2.30)
then r/N is transformed into a positivedefinite hermitian matrix as t
rlN ~ T1N ~TN ,
~7 N* r/~r/N.
(2.31)
As the second transformation we shall choose U s = u N to be an appropriate unitary matrix so that r/N is transformed into a positivedefinite diagonal matrix ( N ~ 0).
t i n ~ t i N ,
(2.32)
Introduction of the third transformation U N = U N = d N
1/2 ,
( N ~ 0)
(2.33)
finally reduces */N to the unit matrix ~TN> ~N,
( N ~ 0).
(2.34)
Combining eqs. (2.29) and (2.34) we find r/2=~.
(2.35)
In what follows this standard form will be maintained by choosing UN to be unitary. Now we are ready to discuss the representation q of the operator QB, but before doing that we shall give the definitions of the BRS singlet and of the BRS doublet la Kugo and Ojima [1] based on the nilpotency of QB. (i) B R S s i n g l e t . A state If) satisfying QBI f) = 0
(2.36)
but not expressible as I f ) = QBlg) is called a BRS singlet state. (ii) B R S d o u b l e t . Two states If) and Ig) satisfying I f ) = QBlg)¢ o ,
QBlf)=0,
(2.37)
are called the BRS doublet states. Ig) is called the parent and If) the daughter. The shortcoming of these definitions is the lack of uniqueness. Let us denote a singlet state, a parent state and a daughter state by Is), IP) and [d), respectively.
K. Nishijima / Representations of BRS algebra
607
When the state Is) satisfies the definition of the BRS singlet, so does the state Is)+ Id). It also satisfies (2.36) and is not expressible as
OBIg) = Is>+ Id). In the case of BRS doublet, IP) can be replaced by Ip ) + l s ) without contradicting the definition of the parent state. Thus it is necessary to make the definitions unique, and for this purpose we exploit the metric matrix. In accordance with eq. (2.5) application of a matrix q to a vector Ix) in eq. (2.4) is defined by
qlx)=~ (~ qSkxk)les).
(2.38)
Then we shall introduce two subspaces of ~V by
°Vd = qT" = {qlx) llx) e ~}, 7/p=q*~V={q*lx)llx)c ~V},
(2.39)
and shall study their properties. Since application of OB annihilates BRS singlet and daughter states and transforms parent states into daughter states, °Yd represents and exhausts the daughter states. Namely, a daughter state is a m e m b e r of c'~d. Then a m e m b e r of e~d, say [y), can always be expressed as [y)= QB[X). Now introduce I)7)= r#ly), then we can easily verify (y[)7)~ 0. On the other hand, we have (yli>=(xlOBI,)), so that oBl~)=qb7)¢0. This means that rffVd is a collection of parent states and indeed exhausts them since the number of linearly independent parent states must be equal to that of daughter states. Furthermore, we have ~/~d = ~/qTr = q* r#Tr = q+°U = ~Vp,
(2.40)
where use has been made of the relation rt~V = 7~ ,
(2.41)
resulting from the nonsingularity of r#, eq. (2.14). Eq. (2.40) indicates that °Fv is the collection of parent states. In the standard representation, eq. (2.35), we have, in addition to (2.40), the relation "0~Vp= 72d.
(2.42)
Thus we have reached an important conclusion that the metric matrix r# converts a parent state into a daughter state and vice versa. Now we shall decompose °U into a direct sum of three subspaces c~ = O~p(~ C~d(~ C~S '
(2.43)
where °Us is the collection of the BRS singlet states. Now combining eqs. (2.10), (2.40) through (2.43), we find ~77/s = °Us.
(2.44)
608
K. Nishijima / Representations of BRS algebra
This equation indicates that BRS singlet states are orthogonal to BRS doublet states. Thus the metric matrix must be of the following form: S
p
d S
7/=
~
p
(2.45)
The shaded blocks represent nonsingular matrices, whereas the blanks correspond to zero matrices. From eq. (2.45) we can read off, for instance, that the inner product between any two members of OVpor of °Va vanishes identically. Now that OB annihilates BRS singlet states we have qoVs ={0},
(2.46)
where {0} denotes a set consisting of the zero vector alone. Then we obtain a similar relation q*oVs = q*rloVs = ~TqoVs= {0}.
(2.47)
Therefore, a BRS singlet state is annihilated by both q and q*, and this property singles out the singlet states. In what follows we shall apply q and q* to the two subspaces OVpand °Va to see what will result. From the definitions (2.39) and the nilpotency of q and q* we find qovd = q*ovp = {0}.
(2.48)
Combining eqs. (2.47) and (2.48) with (2.43) we have qovp = qovpO qoVdO qoVs = qoV = OVd ,
qtoVd = q* OVpOq*ovd(~ qt OVS= q*OV= OVp.
(2.49)
Eqs. (2.46) through (2.49) distinguish among the three kinds of BRS states. In order to find the representations of q we shall specify the ghost number for each subspace like o~(N) = C~p ~ OV(N) p
Then the precise form of the first equation in (2.49) is given, in virtue of eq. (2.19), by ql/p
=
,
(2.50)
and the matrix q has nonvanishing elements only in this case. In order to construct
K. Nishifima / Representations of B R S algebra
609
a representation we have to specify not on12) q but also r/, so that two m o r e subspaces must be introduced, namely,
n ~ " ~ = or~N~,
n~g~+l~ = p~v~NI~ ,
(2.51)
which are the consequences of eq. (2.18). Now taking account of the hermiticity of q, its representation is given in the following form: N+I
N
N
N1
hN
N+l,d
N,p q=
(2.52) h~
N,d Nl,p
The blanks again denote zero submatrices. The submatrix hN should not be singular, and we can express it as hN = VNfN where VN is unitary and fN is diagonal. Then we choose a set of unitary submatrices in ~V~N+I), P~N1)' ~ N ) and ~V~N) by UN,p = UN,d = 1.
U N + I , d = /"/ N  l , p = t)N ;
(2.53)
Then this unitary matrix transforms hN into a diagonal form without changing n and 77. Thus we can decompose it into a direct sum of irreducible representations, each one of which assumes the form
1
r/=
0
1
0
,
n=
0
q=
0
N
N 0
0
0
'
N1
(2.54) 00 0*
'
where a is one of the diagonal elements of fN, and this result confirms the quartet representation obtained by Kugo and Ojirna [1]. Since all the commutation relations in the BRS algebra are homogeneous in OB we cannot determine the possible values of a within the algebra. The above nonvanishing representation of q has been obtained for a pair of BRS doublet states as is clear from eq. (2.52), and for BRS singlet states q is identically equal to zero and the corresponding irreducible
610
K. Nishijima / Representations of BRS algebra
representations are given as follows: N#O:
%)
:)
N=0: n (0),
r/= ( + 1 ) ,
q = (0).
(2.56)
At this stage we shall discuss how our results are related to unitarity of the S mat~ix in the physical sector. Kugo and Ojima [1] have defined the physical subspace of 7# by °~phys~{IX)[Ix) C ~, QBIX) = 0},
(2.57)
then it is clear from eqs. (2.46) and (2.48) that e~phy s m u s t be given by °]/'phy s = C'~S(~ °]~d .
(2.58)
In view of the metric structure depicted by eq. (2.45) we obtain (fig) = (f[P( 7/'s)tg),
(2.59)
when both If) and [g) are members of ~Vphy~.P(~Vs) is the projection operator to 7#s, which can be defined unambiguously only when the BRS singlet states are uniquely defined as has been done in the present paper. Since QB commutes with the S matrix, S[a) and S[/3) belong t o °V'phy s when [a) and I/3) belong to ~s c °~phy s. Thus the unitary condition reads as (/3la) = (fllS~'Sla) = (t3IS*P ( °Vs)Sla),
(2.60)
and a similar one for SS t. In this way the unitarity condition for the S matrix can be expressed entirely in terms of BRS singlet states. Thus we would like to avoid indefinite metric for BRS singlet states, but this does not follow from the abstract BRS algebra. Therefore, we shall introduce a postulate guaranteeing the positivedefiniteness for observable states.
Postulate. The singlet space °Vs obeys the positivedefinite metric. The N # 0 singlets are excluded by this postulate in favor of the arguments presented by Nakanishi [7] and by Kugo and Uehara [8].
3. Representations of the full BRS algebra In the preceding section we have studied the representations of the restricted BRS algebra, and in the present section we shall generalize the results to the full BRS algebra by including the operator 0B. Then we have to modify the three
K. Nishijima / Representations of BRS algebra
611
conditions introduced in the preceding section as follows: (i) hermiticity: Q ~ = QB, QB*=(~B, Q*c  Qc; (ii) BRS algebra: eqs. (1.2) and (1.3); (iii) ghost number: all the eigenvalues of iOc are integers. Furthermore, at the end of the preceding section we have introduced a postulate and we shall include it also in the present case. (iv) Positivedefiniteness of the BRS singlet states: states belonging to the singlet representation of OB or of (~B obey the positivedefinite metric. In what follows we shall classify states according to their transformation properties under QB and 0B, and they are denoted by two indices like [a(d, p)). This means that [a) is a daughter state with respect to OB whereas it is a parent state with respect to 0B. First, we should remark that if a state is singlet with respect to QB so must it be with respect to 0B also and vice versa. Since singlet states obey the positivedefinite metric whereas doublet states obey the indefinite metric, the above statement is evident. Therefore, a state must be either a BRS singlet or a BRS doublet with respect to both QB and (~B"The only possible irreducible representation of the BRS singlet is given by
n=(0),
n(1),
q=(0),
(3.1)
as a consequence of the postulate, or condition (iv). Thus the rest of this section will be devoted to the study of the BRS doublet representations, and for this purpose we shall employ a graphical method. The operator QB increases the ghost number by one, while On decreases it by one, and they will be represented by a straight arrow pointing upward and by a dotted arrow pointing downward, respectively. When we have
Q~la) ~ Ib),
OBIc) ~ [d),
(3.2)
where  indicates equality except for a numerical factor, they are represented graphically in fig. 1. As has already been mentioned, all the doublet states are classified into four groups distinguished by two indices, namely, (p, p ) , (p, d ) , (d, p ) , (d, d ) .
(3.3)
We shall classify the irreducible representations of the full BRS algebra according to whether a multiplet involves (p, d) and (d, p) or not. N
\\ \
"a
13 Fig. 1. Graphical representation of eq. (3.2).
612
K. Nishifima / Representations of BRS algebra
(i) Quartet or closed representations. Let us consider an irreducible representation involving a state (p, d) or (d, p), and without loss of generality we choose the former and name it Ib). Since Ib) is a daughter state with respect to 0B, there must be a parent state la) satisfying
0ala>=lb).
(3.4)
Furthermore, i t i s a parentstate with r e s p e c t t o O B s o t h a t t h e r e m u s t b e i t s d a u g h t e r Id) defined by
QBIb>=Id>.
(3.5)
Hence we have
Id) =
OB(O]31a))=OB(QBIa))
•
This equation shows that the state Qala)= Ic) should exist, and we obtain a quartet representation. This representation will be denoted by D(4, N), where N denotes the ghost number of the state la), then this quartet representation is expressed graphically by fig. 2. Application of ~7 to this quartet yields another quartet, and we have r/D(4, N ) = D(4,  N ) .
(3.6)
Then two quartets form a conjugate pair of quartets. When N = 0, however, these two quartets may be identical, then we have a selfconjugate quartet. Quartet representations are also called closed representations as contrasted with the open representations to be introduced next. (ii) Chain or open representations. Next we shall consider an irreducible representation not involving (p, d) nor (d, p). Then such a multiplet must consist of states of the types (p, p) and (d, d) shown in fig. 3. It is clear that the only possible multiplet has a structure represented by an infinite chain. Such a representation will be denoted by D(oo, p) or by D ( ~ , d) depending on whether the N = 0 member of this chain is a parent state or a daughter state. They are graphically shown in
c(d,p) \x\\ a (p,p) , \
.~
/
d(d,d)
b(p,d)
Fig. 2. Graphical representation of D(4, N).
K. Nishijirna / Representationsof BRS algebra
613
\ \ \
(p.p) \

\ \
/
(d,d)
Fig. 3. Building blocks of the chain representation. fig. 4. Application of B to the chain or open representation obviously yields nD(oo, p) = D(oo, d ) ,
rtD(oo, d) = D(oo, p ) ,
(3.7)
and therefore there is no selfconjugate chain. The possibility of finite chain representations was first pointed out by Bonora et al. [9], but we have excluded them in the present paper by employing the postulate concerning the positivedefinite metric for BRS singlets. Thus we have found three kinds of irreducible representations of the full BRS algebra, (1) singlets, (2) quartets, and (3) infinite chains. In the last section we shall show, however, that the infinite chains deduced from the abstract BRS algebra cannot be realized in the actual gauge theories at least in the Landau gauge. 4. Realization of irreducible representations In the preceding section we have studied possible irreducible representations of the abstract BRS algebra. In the present section we shall examine how these representations are realized in the gauge theory. For this purpose we start from the lagrangian density for the gauge theory. ~
1
~F,~.Fj.~t#(y,D, + m ) O + O , B . A , +½o~B.B+iO,g.D,c [
1
\ \ \
/77 (N=O)(
\
,
)
(N=O)
', i
I
Fig. 4. Graphical representation of D(oo,p) and D(oo, d).
(4.1)
614
K. Nishijima / Representations of BRS algebra
where we have adopted the same notation as in ref. [5]. The BRS and alternative BRS transformations of a given Heisenberg operator are given by f~b = i[OB, ~b]~,
80 = i[(~B, &]~,
(4.2)
where we choose the  ( + ) sign for the operator & even (odd) in the ghost fields c and & The ghost number N of the operator ¢ is defined by i[Oc, 6] = N ¢ ,
(4.3)
and is given by the number of the fields c minus that of the fields ~ involved in & as factors. In what follows we shall list the explicit forms of the BRS transformations [3], [4]. fA.
= Duc,
8A. = D.g,
fO = ig(c. T)O,
80 = ig(g. T)O,
fgJ=  i g ( h ( c . T ) ,
8t~ = i g ~ ( g . T ) ,
6B = 0,
fiB = 0 ,
fig = i B ,
8c = i B ,
fc=lg(cxc),
8e=½g(exe),
(4.4)
and /3 is defined by B +13ig(c
(4.5)
x e)=O .
In general a Heisenberg operator is reducible under BRS transformations so that we shall better introduce irreducible asymptotic fields. In this section we shall consider only the spin 0 asymptotic fields, in or out, by A . ~ O. X ,
B ~ [3 ,
D.e.a.f',
c.r,
B >fl , D.c~O.y,
g~,.
(4.6)
The spin 1 asymptotic fields will be considered in the next section. Then in terms of these asymptotic fields, eq. (4.4) may be expressed as ax = ~,
&=:,
f ~ / = ifl,
8 F = ifi,
fy = 0,
fF = 0,
f~ = o,
f ~ = o.
(4.7)
In order to study their metric structure we shall give some of the twopoint Green functions or the vacuum expectation values of the timeordered products. (A~(x), B b ( y ) ) = 6abO.DF(X  y) , ( ( D ~ c ) " ( x ) , ~b(y)) = i6aba,DF(X  y) .
(4.8)
K. Nishijima / Representationsof BRS algebra
615
In terms of the asymptotic fields we have
(Xa(X), fib(y))= _(Xa(x), fib(y))=6,,bDv(Xy), (ya(X), .~b(y)) = i6abDv(x y) ,
(4.9)
where DE denotes the free massless propagator. In the discussion of color confinement we have adopted the Landau gauge (a = 0), [5], [10], and in what follows we shall exclusively employ this gauge. Then we also have
(Fa(x), Fb(y))= i6abDv(x_ y) .
(4.10)
In this gauge the theory is invariant under the ghost conjugation [4], [11], and we shall introduce its generalization in the next section. (i) Selfconjugate quartet. In perturbation theory we have y=r,P=~,
(4.11)
fi =  / 3 ,
(4.12)
and this implies as seen from
6 g x = ~ F = 85, = ifi ,
Thus we find a selfconjugate quartet, first discovered by Kugo and Ojima [1] and shown in fig. 5. If we express a state like X]0) simply by IX), both IX) and ]13) must be zero norm states. The condition (/31/3)=0 is trivially satisfied, but we are not sure about the condition (Xlg)= 0. When the last condition is not met we have to replace X by g' without modifying eqs. (4.7). X~ X' = x  a f t ,
(4.13)
a = (XlX)/2 Re (ftlX) •
(4.14)
where
Then IX') belongs to o//.p. 7=/
x
,,,
/ "~ r=7
PP
Fig. 5. Selfconjugatequartet.
616
K. Nishijima / Representations of BRS algebra
In perturbation theory both .A_ in , and_ ~Oin are BRS singlet operators, so that gluons and quarks are not confined. (ii) Conjugate pair of quartets. Let us assume that we can distinguish between y and F and consequently between P and p.
'y # F , / ~ # #/.
(4.15)
Then there are two cases, (1) f i #  f i , and (2) f i =  f i . between them we refer to the following relationships:
In order to distinguish
( B a ( x ) , Bb(y)) = (/~a (x),/~b(y)) = 0 , ( B " ( x ) , j~b(y)) = _½g2(( c X g)a(x), ( C X g)b(y)) .
(4.16)
Therefore, when and only when c x g has a massless asymptotic field we have the first case,
(fi(x), fi(y))
#
O.
In this case we have a conjugate pair of quartets, first discovered by Kugo [12] and shown in fig. 6. The BRS transformations of the asymptotic fields are given by
~,,
6£ = id,
g~/=  id,
8fi = 
gF=iB,
8f,=ifi,
ga=~,,
¢5d = F.
(4.17)
When we normalize the asymptotic field d by
(da(x), db(y)) = 6~bDF(x

y) ,
(4.18)
we have
( l ~ ( x ) , r b ( y ) ) = ( ~,a(x), yb(y)) = iSabDF(x  y ) , (4.19)
(fia(X), f i b ( y ) ) = 8~bDF(X y) . d ,~ xx ~
F
", / /
"4Z
\%
\
~
~a/
Fig. 6. Conjugate pair of quartets.
K. Nishifima / Representationsof BRS algebra
617
d and d are the asymptotic fields of c x c and ? x ~, respectively, and their presence guarantees color confinement [5], [10]. We notice that X is missing in fig. 6, but we can easily identify X as X =/3  / 3 .
(4.20)
This identification is consistent with eqs. (4.7), (4.17), (4.9), and (4.19). Also, when the parent operators F and ~2 are not orthogonal, we should modify them as
~~ 2'= ?½i(ylr)P, (4.21)
r~ r ' = r  ½ i ( ~ l r ) ~ ,
in order to restore the orthogonality (~'IF')0. In fig. 6, members conjugate with respect to r/ are denoted by the same letters with and without the bar. In this case both quarks and gluons are confined, so that A S and q io are not BRS singlet operators. (iii) Conjugate pair of chains. In the preceding example we have studied the case (1) assuming/3 ~ /3, so that we shall study here what happens in the case (ii) in which/3 =  / 3 . In this case the only possible representation is a conjugate pair of chains shown in fig. 7. d and / ) are the asymptotic fields of c × c and of ~× ~, respectively. D and d are conjugate to / ) and d, respectively, with respect to 7/. Each chain is selfconjugate under ghost conjugation. In the next section we shall show that this case cannot be realized in nature, though.
5. A n e w s y m m e t r y in the L a n d a u gauge
In the discussion of color confinement [5], [10] we have found that the Landau gauge plays a special r61e in that the condition for confinement assumes the simplest form. Accordingly we have adopted this gauge in the preceding section and have found two possible representations of the full BRS transformations for the auxiliary d
/
, x"~ k
",
/
D,
/I
x
fl
X "',
F i
!
Fig. 7. Conjugate pair of chains.
618
K. Nishijirna / Representations of BRS algebra
fields, the conjugate pair of quartets and the conjugate pair of chains. In order to determine which representation is realized in nature we shall introduce a new symmetry of the theory in the Landau gauge. In the Landau gauge the sum of the gaugefixing and FP ghost terms in the lagrangian is given by ~GF'k~,~Fp = A/x . O , B + i O u g . D , c .
(5.1)
This lagrangian is invariant under the following transformation of the auxiliary fields:
(C)._.)o~(O)_I(~)O~(O)=(cosO sin O](c] \sinO
cosO/kg]'
B + all(O) ' B ~ ( O ) = B  ig sin 20(c x g) + ig sin 0 cos 0½(c x c  g x ~).
(5.2)
If we put 0 =  7 r / 2 , we find c
~r
(5.3) It is clear that this is precisely the ghost conjugation, so that we may identify c~ = 0//(_½rr) "
(5.4)
Thus the above U(1) group is a generalization of the ghost conjugation. We can easily verify that following expressions are invariant under this group: BB,
g.c,
cxc+gx~.
(5.5)
On the other hand, we find ( ; ) ~ (cos 20 \sin20 B+B+ig
sin20~(a~
cos 2 0 ] \ b ] '
sin O(a cos O  b sin 0),
(5.6)
where a=½(cxcgxg)
,
b=cxg.
(5.7)
If we introduce the generator of this transformation by ~ ( 0 ) = exp ( i O R ) ,
(5.8)
we find [13] i[R, c] = ~,
i[R, g] =  c ,
i[R, B] = i[R, IB] =  l i g ( c × c  g× g) .
(5.9)
K. Nishijima / Representationsof BRS algebra
619
The condition for confinement is the existence of the asymptotic field of c x c or that of the field a. Because of the invariance of the theory under (5.6) this implies the existence of the asymptotic field of c x g. This immediately leads to
[3 + ~ = ig(c x ?)i" ¢ O .
(5.10)
Therefore, the chain representation of the auxiliary fields is excluded. The conjugate pair of quartets is the right representation for them when color confinement holds. Now we shall turn to the discussion of the fundamental fields A , , ~Oand q~ which are invariant under the transformation a//(0). When their asymptotic fields obey the BRS singlet representations colored particles are not confined, so that we shall assume the BRS doublet representations. Without loss of generality we shall consider the gluon field. The pole part of the gluon propagator must be reproduced by replacing the gluon field by its asymptotic field. When it obeys the BRS doublet representation it must split into a metricconjugate pair of asymptotic fields to reproduce the nonvanishing pole part. Then there are two possibilities, either
A .i n  Au(p, p) + A . ( d , d ) ,
(5.11)
A .i n m A~.(p, d) + Au(d, p ) ,
(5.12)
or
where we have adopted the notation introduced in sect. 3. Invariance of A~ under ghost conjugation implies
c~lA~(p, p)CC= A~.(p, p) , c~ 1 A . ( d ' d ) q ~ = A . ( d , d ) ,
(5.13)
or
1A,(p, d ) ~ = A , ( d , p )
,
qgIA,(d,p)CC= A~,(p, d) .
(5.14)
These asymptotic fields carry the ghost number N = 0. Thus comparing the N = 0 m e m b e r s of various representations with these asymptotic fields, we find that the decompositions (5.11) corresponds to the selfconjugate quartet and (5.12) to the conjugate pair of quartets. The chain representation is again excluded for the reason that the higher the ghost n u m b e r N the more partners of the U(1) group are missing in the chain representation. A similar argument holds for the quark field 0. References [1] T. Kugo and I. Ojima, Phys. Lett. 73B (1978) 459; Suppl. Prog. Theor. Phys. No. 66 (1979); G. Curci and R. Ferrari, Nuovo Cim. 35A (1976) 278,474 [2] C. Becchi, A. Rouet and R. Stora, Ann. of Phys. 98 (1976) 287 [3] G. Curci and R. Ferrari, Phys. Lett. 63B (1976) 91
620 [4] [5] [6] [7] [8] [9] [10]
K. Nishijima / Representations of BRS algebra
I. Ojima, Prog. Theor. Phys. 64 (1980) 625 K. Nishijima, Phys. Lett. l16B (1982) 295 N. Nakanishi, Suppl. Prog. Theor. Phys. No. 51 (1972), and earlier papers quoted therein N. Nakanishi, Prog. Theor. Phys. 62 (1979) 1936 T. Kugo and S. Uehara, Prog. Theor. Phys. 64 (1980) 139 L. Bonora, P. Pasti and M. Tonin, Nuovo Cim. 68A (1981) 307 K. Nishijima and Y. Okada, Confinement versus nonconfinement and BRS transformation, to be published [11] G. Curci and R. Ferrari, Nuovo Cim. 32A (1976) 151 [12] T. Kugo, private communication [13] N. Nakanishi and I. Ojima, Z. Phys. C6 (1980) 155