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Auxialiary conditions in BRST quantization
N U CLEAR P HV S I C S B
Nuclear Physics B 372 (1992) 218—242 North-Holland
Auxiliary conditions in BRST quantization Robert Marnelius institute of Theoretical Physics, Chalmers University of Technology, S-412 96 Goteborg, Sweden Received 31 October 1991 Accepted for publication 10 December 1991
A general structure of allowed auxiliary conditions in both ordinary and generalized BRST quantization is presented. Such conditions are shown to fall into two classes which always should imply each other. Both antiBRST invariance and coBRST invariance are shown to fit into the framework. AntiBRST invariance is also shown to imply an OSp symmetry. A previously given algorithm for solving BRST cohomologies is formulated in terms of allowed auxiliary conditions.
1. Introduction In BRST quantization one starts from a BRST-invariant model and projects out BRST-invariant states in the quantization. The BRST invariance is reflected in the conservation of a nilpotent charge Q (Q2 = 0), and the projection is effected by the BRST condition Qiph)
=
0,
(1.1)
where the BRST-invariant states ph) are called physical states. Often one combines (1.1) with additional conditions. Typical examples are antiBRST invariance or a fixing of the ghost number. In this report the general structure of such conditions is presented. It will be shown that there are only two forms of allowed additional conditions: One may either impose
[Q,C]Iph)=O
ciph)
or
=
0
(1.2) (1.3)
where (1.2) will be called a gauge invariance condition and (1.3) a gauge slicing condition. Notice that eq. (1.2) is the consistency condition of (1.1) and (1.3). It will be proposed that consistent conditions of the type (1.2) 0550-3213/92/$05.00
© 1992
—
Elsevier Science Publishers B.V. All rights reserved
R. Marnelius / BRST quantization
219
always allow for conditions of the type (1.3) without affecting the physical content. This makes the two forms have equivalent implications and it requires that all operators of the forms (1.2) and (1.3) form a closed algebra. In order to be a fundamental auxiliary condition [Q,C] and/or C must be conserved operators, i.e. they must generate symmetry transformations. When this is not the case such additional conditions may still be used to investigate the physical content of the BRST cohomology although at a particular instant. In sect. 2 the general structure of allowed auxiliary conditions is presented. In sect. 3 antiBRST invariance is shown to imply an OSp symmetry, and in sect. 4 coBRST invariance is investigated. In sect. 5 the simplifying analysis of ref. [11 is viewed in terms of auxiliary conditions, and in sect. 6 this formulation is combined with antiBRST invariance. In sect. 7, finally, some of the previous results are generalized to the extended BRST formulation proposed in refs. [2,3] which includes anomalous gauge theories and theories with second-class constraints. 2. Auxiliary conditions in ordinary BRST quantization Consider the additional condition Riph)
=
0
(2.1)
to the BRST condition (1.1). The meaning of such an additional condition depends on what type of operator R is chosen to be. One may divide R into three different types of operators which we call A-, B- and C-operators (cf. ref. [4]). They are defined by [Q,A] =0,
A
[Q,C],
[Q,B] =0,
B= [Q,C].
(2.2)
A and B are physical operators which map physical states on physical states while C is an unphysical operator in general. The only exception is the ghost number operator N~,satisfying [N~,Q].=Q.
(2.3)
One may show that any operator ~ that maps physical states on physical states may be written as =
~AnNg~h,
(2.4)
220
R. Marnelius / BRST quantization
where A~ are A- or B-operators. A-operators are genuine physical operators while B-operators are gauge generators since they generate unobservable changes of the physical states: Bjph)
=
Q~).
(2.5)
By means of the Jacobi identities one finds [4] [B,A]
=
B’,
[B,B’]
=
B”.
(2.6)
Notice that Q itself is a B-operator due to (2.3). Notice also that there is no unique separation of the operator algebra into A-, B- and C-operators without additional conditions since the definition (2.2) is unaffected by the transformations A
—~
A + B,
C—~C-4-A+B.
(2.7)
If we now consider the case when R is an A-operator in (2.1) then we have AIph)
=
0,
(2.8)
which obviously is consistent with the BRST condition (1.1) due to (2.2). However, this condition is quite ad hoc. It is like imposing AI~)= 0 in any ordinary quantum theory. It definitely changes the physical theory. Such conditions are therefore excluded. Consider next an auxiliary condition of the form BIph)
=
0,
(2.9)
which also is consistent with (1.1) due to (2.2). Since the B-operators are gauge generators this condition may be viewed as a gauge invariance condition. Since Biph) = QI ) for any choice of B, it naively seems as if (2.9) only eliminates unmeasurable states and does not kill any degrees of freedom. If this were true then one could impose (2.9) for all possible B-operators which is algebraically consistent due to (2.6). However, this is always too strong and therefore inconsistent (see sects. 5 and 6). On the other hand, we may always impose at least some conditions of the form (2.9) without affecting the physical content. The problem is only to determine exactly which constraints that are allowed. Below we shall give general algebraic conditions for possible choices of subsets of B-operators that may be used to restrict the physical states. The definite choice of such a subset is then determined by the representation of the original state space. However, before we give these conditions we have to discuss the implication of eq. (2.9).
R. Marnelius / BRST quantization
When B
221
[Q,C],condition (2.9) becomes QCIph)
=
0.
(2.10)
This seems to imply that the unphysical operator C is lifted into the class of physical operators. This is a correct interpretation provided QC~Iph)= 0
(2.11)
for any integer n, or equivalently if [B,C]Iph)
=
0,
[B,C],C]Iph)= 0,
[... [B,C],C],...,C]Iph)
=
0.
(2.12)
If C has become a physical operator then it seems as if the degrees of freedom of the physical state space has increased. This can obviously not be the case. An unphysical degree of freedom cannot be transformed into a physical degree of freedom. An additional condition can only decrease the degrees of freedom or leave them unaffected. Therefore, we should always expect that a condition like CIph)
=
0
(2.13)
is implied by (2.9) or that it at least may be imposed without affecting the physical content. The last property means that the physical states which does not satisfy (2.13) are e.g. zero-norm states. Notice though that C is only determined up to transformations of the type (2.7) and that there might exist several sectors projected out by different choices of C. For instance, if the state space allows for eigenstates to C then we may have (C—A)~ph)= 0, where
,~
(2.14)
is a constant. In such a case the value of must be chosen properly. ,~
A typical example is the fixing of the ghost number which is effected by (N~h—n)Iph)=0. (2.15) Due to eq. (2.3) the BRST condition always allows for a condition of this type. Indeed, the genuine physical states (those states that contribute to measurable quantities) have a definite ghost number n at least for topologically trivial
R. Marnelius / BRST quantization
222
gauge theories. Therefore, eq. (2.15) is not necessary to impose, and the wrong choice of n is of course disastrous. On the other hand, for topologically nontrivial models we may have genuine physical states with different ghost numbers. In such a case eq. (2.15) will exclude certain sectors of the theory. (A topological trivial model is a model for which the BRST condition (1.1) by itself projects out one canonical complete sector, while for a topological nontrivial model it yields several canonically complete sectors [1].) We believe that consistent choices of auxiliary conditions of the form (2.9) always require the conditions (2.12) and therefore always imply conditions of the form (2.13). With eq. (2.13) we have arrived at the third type of additional conditions which we will call gauge slicing conditions. If we instead of (2.9) impose (2.13) together with the BRST condition (1.1), then consistency requires both (2.9) and (2.12). Thus, we have found two ways leading to the same result: We may either add a condition like B~ph)= 0 or Ciph) = 0 to the BRST condition (1.1). The consistency conditions which have to be fulfilled make the two approaches equivalent in general. We propose now the following algebraic criteria for consistent choices of auxiliary conditions: Consider first the case when we start with several conditions like (2.9): B1Iph)
=
0,
i
=
1,... ,l.
(2.16)
The first consistency condition is that B satisfy a subalgebra of (2.6), i.e. [B1,B1] = .tjfBk + ,~Q. Secondly, if B
=
(2.17)
[Q, CJ then we also demand that [C1, B1 I
=
~z~f Ck +
iJIIkBk +
[C1,C~] = p11k Ck +
jiQ,
+ pQ,
(2.18)
where the freedom (2.7) for the choices of C1 may be used to achieve such a closed algebra. This algebra will then allow for CiIph)
=
0,
i
=
1,... ,l.
(2.19)
Equivalently, if we start with (2.19) then this will automatically require (2.16) with B [Q, C1] as well as a closed algebra of C,,B, and Q. In fact, independently from which constraints we start from, the algebraic analysis will determine which additional constraints may be added and the final algebra will always be a closed algebra of the above form. We have only to make sure of that no A-operators couple.
R. Marnelius / BRST quantization
223
One may notice that the above algebraic conditions exclude the possibility that all B-operators are used to constrain the physical states since all B-and C-operators do not form a closed algebra. In fact, if the original state space is nondegenerate then the maximal closed algebra will at most involve half of the B- and C- operators (cf sects. 5 and 6). Notice finally that a subset of B’s and C’s forming a closed algebra can only be used as constraint operators if the state space representation allows for nontrivial solutions of (2.16) and (2. 19). In the following analysis we will always start from some possible constraints and look for further constraints from the above algebraic conditions. 3. AntiBRST invariance AntiBRST invariance is the first auxiliary condition considered in BRST quantization. It was found to be useful in Yang—Mills theories [5]. AntiBRST transformations are generated by a nilpotent operator Q which anticommutes with the BRST charge Q. Q has opposite ghost number to Q. In the following we shall consider finite number of degrees of freedom and the general hamiltonian formulation of ref. [61 where the BRST charge is chosen to be of the BFVform [8] (a,b,c= 1,...,m
Q where
=
V~a
~~iU6c”Pa?l”P1’—~iUa~+PaP’~,
(3.1)
and ~ are fermionic Faddeev—Popov (FP) ghosts and antighosts Ta and Pa their conjugate momenta, respectively, and [~ap 6]~ = [~a,~b]+ = 0. (3.2) ,~°
~
and Pa have ghost number + 1 while ~ and Pa have ghost number —1.
Wa
are bosonic constraint operators satisfying [Wa,Wb]—
=
~Ua~Wc,
(3.3)
where U~gare constants. Pa are conjugate momenta to the Lagrange multipliers Va:
[Pa,Vb]_
=
iôab.
(3.4)
For simplicity Wa and Pa are restricted to be bosonic. It is straightforward to generalize the following treatment to fermionic Wa and Pa and bosonic ,j’~,Pa and ~a,p0 (see e.g. ref. [7]). According to ref. [6] the antiBRST charge Q, satisfying ~2 = 0, is given by
Q
[Q,K]_ = Wa
iU~g~c?~?~— ~iUJi)~
_PaP~~ iU~P~n~’ + Ua~VcPa?~b, —
(3.5)
224
R. Marnelius / BRST quantization
where K
=
(3.6)
Pail’~ + ~Ua~VC~a~b.
Since K has ghost number —2 and Q has + 1, Q has ghost number —1. AntiBRST invariance requires Qiph)
=
0.
(3.7)
This is a condition of the type (2.9) in sect. 2 due to eq. (3.5). Since (see ref. [6]) [Q,K]
=
0,
(3.8)
the consistency condition (2.12) is satisfied, which implies that the unphysical operator K is lifted into the class of physical operators. We may therefore impose the gauge slicing condition KIph)
=
0
(3.9)
without affecting the physical content. Indeed, K
=
(3.10)
i[~aVa,Q]+
implies KIph) = QI ) so KIph) ~ 0 is an unmeasurable state. In order to investigate whether or not K is conserved consider the generic BRST and antiBRST-invariant hamiltonian H=H0-i- [Q,[Q,R]].
(3.11)
One easily finds that the conservation of K requires [H0,K]
=
0,
[R,K]
=
0.
(3.12)
The first condition is satisfied provided H0 does not contain Lagrange multipliers or FP ghosts. This follows if [H0,Wa] = 0 for all a [6], which we assume from now on. Concerning the second relation we notice that the rather general R given in ref. [61 is written as R
=
~vav”
—
~c5,Ja~a +
s,
(3.13)
where a is a real constant and where S does not contain the FP ghosts, the Lagrange multipliers and Wa. This expression satisfies therefore the last equality in eq. (3.12) from which we conclude that K is conserved in general.
R. Marnelius / BRST quantization
225
In ref. [6] it was also shown that there is a dual relation to (3.5), namely
Q= [Q,K],
(3.14)
where K
=
(3.15)
Pap/a + ~Ta~ch/ahlb.
Since [Q,K]
=
0,
(3.16)
BRST and antiBRST invariance imply that we should be able to impose KIph)
=
0.
(3.17)
Indeed, K
=
(3.18)
j[Q,p/aV~]~
so Kiph) ~ 0 is unmeasurable. (Notice that although K is a physical B-operator it is unphysical from the point of view of Q.) By means of the Jacobi identities the hamiltonian (3.11) may be written as H
=
H0—
~ [Q,R]],
(3.19)
which implies that K is conserved if [H0,K]
=
0,
[R,K]
=
0.
(3.20)
Thus, K is conserved when K is conserved. (The first equality is satisfied by construction and the second by an R of the form (3.13).) Now we have [6] [K,K]_
= p/ap~_p/ap~~Ng~,
(3.21)
where N~is the ghost number operator. Thus, eqs. (3.6) and (3.17) imply N~Iph) = 0,
(3.22)
which may be viewed as a consistency condition for (3.6) and (3.17). This condition is always satisfied by genuine physical states when we have antiBRST invariance since [6] N~ = j[Q,,~aVa]÷+ ~[Q,p/av~]÷
(3.23)
R. Marnelius / BRST quantization
226
implies Iph,n) =QI)+QI)
(3.24)
when n ~ 0. Thus, physical states with ghost number n ~ 0 never contribute to any measurable quantities. The expressions (3.10), (3.18) and (3.23) for K,K and N~imply that the conditions (3.9), (3.17) and (3.22) may in a generalized sense be viewed as gauge invariance conditions which should allow for p/aV~~ph) = ,~aVaIph)= 0.
(3.25)
However, consistency requires DIph) where im
LI
= = =
=
0
(3.26)
rQ ,P)—a VaJ+ 1
[Q
lxr + l~1Vg)~ p/aV~]÷ — ~~Ngh
paVa + i~(~Pa + p/apa,
(3.27)
which is a generalized B-operator due to (3.23). Eq. (3.26) is automatically satisfied if (3.25) is satisfied. The constraint operators Q, Q, K, K, N~,~ ~aVa and D satisfy a closed OSp(l, 112) algebra (see appendix A). This algebra is close to the OSp(l, 112) algebra given in ref. [9]. Both contain the conserved subalgebra of Q, Q, K, K and N~.However, the generators q, t~and D differ from those in ref. [9]and they are not conserved by the hamiltonian (3.11). Their conservation requires [R,D]_= iR
(3.28)
up to terms which are both BRST and antiBRST invariant. Now the conditions (3.25) are not a unique consequence of KIph) K~ph)= 0. We may for instance make the replacement q
?laVa —~
=
,JaVa +
[Q,r]_,
=
p/a~+
[Q,r]_,
=
(3.29)
which changes D to D-~D’ = [Q4]+-i~N~ = =
—[Q,q]~ l~Ngh [Q,[Q,r]].
D+
(3.30)
R. Marnelius / BRST quantization
227
In appendix A it is shown that r may be chosen such that also Q, Q, K, K, Ngh, q, t~and D’ satisfy an OSp(l, 112) algebra. The conditions for this is given by eqs. (A.15) and (A.l6) in appendix A. The conservation of q,t~and D’ requires then (3.28) with D replaced by D’ and [H0,r] = 0. An appropriate choice of r is r where
/3 is a constant,
(3.31)
JiXaVa,
=
and x a are gauge choices to Wa [Wa,Xb]_
=
(3.32)
jMa~’.
The consistency conditions (A. 15) and (A. 16) together with the Jacobi identities require (see appendix A) X~z,Xbl_=0, [Xa,MbC]_=O, Wa,Mb(i_ [Wb,M,f] = iUa:MdC,
(3.33)
—
where the last line follows from the Jacobi identities of Wa, Wb and Xc. In topologically trivial models, Mab has also an inverse matrix operator (M’ ) ~‘ satisfying 6(M’)~ = ö~1. (3.34) =
Ma
The conditions (3.33) make it natural to assume that Mab only depends on X’~. Mab (x) and (M (x ) ) when it exists, should then be given by power series in X~zand should be possible to choose such that Mab(0) = (M1 (0))~= ó~1. q, c~and D’ are now conserved if —~
~
[I~IO,Xa]_ = 0,
(3.35)
Va
and if /3 = —~ifor R = ~v2 + S(~)under certain conditions on X’~and S, or /3 = —~ifor R = ~ There exists no OSp(1, 112) symmetry for a hamiltonian of the form (3.11) when R is given by eq. (3.13) with a ~ 0. Thus, Yang—Mills theories in the Feynman gauge are not OSp invariant. For /3 = ~i our OSp (1, 112) symmetry coincide with that of ref. [9] for special choices of x a One may now ask oneself if it is not possible to proceed by imposing —
aiph)
=
a~ph)= 0,
a~[Q,rJ_,
a~[~,r]_.
(3.36)
This would then imply (3.25) again and that the physical state space is independent of the choice of gauge slicing operators q and in K = i [~,Q] t~
+
R. Marnelius / BRST quantization
228
and K = i[Q,q]~. Since aiph) = Ql ) and alph) = Ql ) eq. (3.36) does not exclude any genuine physical states within sectors allowing for (3.36). Consistency requires Tlph)
=
0,
T~[Q,ã]÷
=
—[Q,a]~.
(3.37)
One may then go even one step further and impose
rlph)
0,
=
(3.38)
which requires no further consistency conditions due to the algebraic properties in appendix A. We have thus arrived at an extended OSp invariance of the physical state space. (In sect. 6 it is shown that all these constraints are satisfied within the gauge slicing procedure of sect. 5.) However, this is not a fundamental invariance since p/aVg, p~aVa,a, a, D, T and r are not conserved. Interestingly enough this extended OSp algebra is conserved if r = /JXaVa and H = [Q,[Q,r]] (see appendix A). 4. CoBRST invariance Like the antiBRST charge Q the coBRST charge, here denoted Q*, is nilpotent and has ghost number minus one. However, it does not commute with the BRST charge Q. We have (4.1) imposing
Q*Iph)
=
0
(4.2)
0
(4.3)
requires ~Iph)
=
for consistency, and (4.3) allows for (4.2). In the following we use the definition of coBRST given in refs. [10,11] (see also refs. [12,13]). This definition requires the original state space Q to be an inner product space on which there exists a metric operator p/ satisfying p/2 = 1 and such that (u~p/~v) is a positive definite scalar product for any choice of state vectors lu), Iv) E Q. We then have the isomorphism (u~p/Iv)= (ulv),
(4.4)
R. Marnelius / BRST quantization
229
where Iv), lu) may be considered to be state vectors in a Hilbert space 11. (Q is then a so-called Krein space [14].) Within this framework Q* is defined to be the hermitian conjugate of Q in 7-1, which implies (4.5)
=
in Q. With this definition one finds the equivalence Aiph)
0 ~ Qiph)
=
=
Q*Iph)
0.
=
(4.6)
This follows from
(uI’j A lu)
=
(~~Q*p/Q*~~) + (uIQp/QIu),
(4.7)
and since the last two brackets only vanish for QIph) = Q*Iph) = 0 due to eq. (4.4). Thus, in this case we have a strict proof that (4.3) implies (4.2). In addition, we have the Hodge decomposition for any state Iu) e Q:
lu) where AIu)0
=
=
u)~+ QIVi) + Q*1v2),
(4.8)
0. The proof (see ref. [13]) is based on the equality (uIQnIv)
=
(4.9)
(u~p/Q*~v),
which implies that KerQ (KerQ*) and ImQ* (ImQ) are orthogonal. CoBRST invariance is very useful in order to determine the physical degrees of freedom since we only need to solve AIu) = 0. However, at the fundamental level it seems not so useful since Q* and A are in general not conserved. Comparing Q* with Q in the antiBRST case there is an obvious difference in the fact that Q* ph) = 0 is a gauge slicing condition while Qiph) = 0 is a gauge invariance condition. However, this difference is slightly fictitious. The reason is the equalities A =52,
Q*p/s+Q
(4.10)
where S~ [Q,p/]. This means that Q*Iph) =
=
0 is equivalent to
(4.11) (i~)
=
0 is inconsistent with
1)
Siph)
=
0,
(4.12)
R. Marnelius / BRST quantization
230
which is a gauge invariance condition. Since the consistency conditions (2.12) are satisfied this condition implies in turn that we have conditions like (172 = 1) l7Iph)
=
±Iph).
(4.13)
The plus sign for all physical states implies that the genuine physical state space is a Hilbert space. The reason why one should not expect S and ~ to be conserved is due to the fact that ‘1 usually severely breaks the covariance of the model (particularly when we have Lorentz covariance). 5. Simplifying conditions for solving BRST cohomologies Auxiliary conditions need not be conserved. They may simply be used to investigate the physical content of a model, although at a particular instant. CoBRST is useful in this respect. Another case is the simplified analysis set up in refs. [1,3]. There it was shown that, at least for topologically trivial models, it is sufficient to consider states of the form lu)
=
IIvl)IG)
(5.1)
when analyzing the BRST condition QIph) = 0. IM) denotes matter states and IG) FP ghost states. Condition (5.1) may be viewed in terms of auxiliary conditions. In order to demonstrate this, consider a general first rank theory with finite number of degrees of freedom. The BRST charge is (a, b, c = l,...,m
Q— —
Wap/
a
a’r, bc bc raP) 17
1
—
~
1 —
~l
ba 17 ab
where Uat are constants and Wa are hermitian constraint operators satisfying the Lie algebra (3.3). p/a and Pa are Grassmann odd, hermitian FP ghosts satisfying (3.2). These properties imply that Q in eq. (5.2) is hermitian and nilpotent (Q2 = 0). (The following analysis may be generalized to the case when some constraints are odd and the corresponding FP ghosts are even (see e.g. ref. [7]).) It is easy to see that (5.1) may be implemented by means of auxiliary conditions. We may for instance choose Palph)
=
p/AIph)
=
0
(5.3)
R. Marnelius / BRST quantization
231
for a = 1,... ,k; )~= k + 1,... ,m, which obviously implies (5.1). These conditions are internally consistent since P(~and 17A anticommute. However, consistency with the BRST condition requires = 0, [Q,p/A1÷Iph)= 0.
(5.4) (5.5)
[Q,Pa]+Iph)
Using eqs. (5.2) and (5.3) the last condition becomes [Q,p/A]+Iph) = _~iU(~p/ap/flIph) = 0. Since p/a I ph)
=
(5.6)
0 is inconsistent with (5.3) we find that (5.5) requires
U~=0, a,fl=l,...,k;~=k+l,...,m. 1Iph) Thus, only if this is satisfied are we allowed to impose p/’ (5.4) we notice that [Ba,Bb]_ Bam [Q,Pa]+
=
=
~UatBc,
=
Wa+j~Ua~(Pcflb_p/bPc),
(5.7) 0. Concerning
(5.8)
and therefore eq. (5.7) implies =
~UcJBy, a,fl,y
=
1,... ,k
(5.9)
which means that eq. (5.4) is at least consistent to impose by itself. By means of eq. (5.2) and Ba = [Q,Pal we find 13P~,
B 0
where Wa + ~
=
—
(5.10)
W~+ iU0I Pap/’~ iU0~Pl —
~~UaI~
/3
=
1,... ,k;
,~
=
k + l,...m. (5.11)
Thus, eq. (5.4) implies B 0Iph)
=
W~IPh) = 0, a
=
1,... ,k
(5.12)
which definitely are additional conditions to (5.3). By means of eq. (5.3) and its consistency conditions we have arrived at exactly the conditions considered in refs. [1,3], namely (5.1) specified by eqs. (5.3), (5.7) and (5.12).
R. Marnelius / BRST quantization
232
A simple well-known example of (5.3) is the Siegel gauge in string theory: bolph)
0.
=
(5.13)
Consistency requires here the mass-shell condition [Q,bo]+Iph)
=
0.
(5.14)
These two conditions are used in the cohomology analysis in ref. [10]. The procedure in refs. [1,3] also involves the existence of dual physical states ‘(phI
=
(A1’I(G’I
(5.15)
,
satisfying
(G’IG) Here this implies (a,/i,y following)
=
=
l,
I(M’IM)I <
1,... ,k and ,u,v,t
‘(phjP~= ‘(phI 170
=
oo. =
(5.16)
k + 1,... ,m also in the
0
(5.17)
with the consistency conditions ‘(phIBA
=
0,
(5.18)
=
0.
(5.19)
The latter demands here =
0,
(5.20)
which makes eq. (5.18) consistent to impose since =
iU,~BA.
(5.21)
Furthermore, eq. (5.20) implies B~= W~+ iU~~P2p/v ~U~p/aPC, —
(5.22)
where W~is given by eq. (5.11). Eq. (5.18) yields therefore ‘(phIB2
=
‘(phIW~= 0.
(5.23)
Eqs. (5.3) and (5.17) and their consistency conditions are exactly the conditions considered in refs. [1,3].
R. Marnelius / BRST quantization
233
Another equivalent formulation of the above treatment is to start from eqs. (5.4), (5.5), (5.18) and (5.19) which are gauge invariance conditions. Eq. (5.5) implies then p/AIph) = 0 when (5.7) is valid and (5.19) implies ‘(phip/0 = 0 when (5.20) is valid. Eq. (5.4) implies (5.12), and (5.18) implies (5.23). However, P 0Iph) = 0 and ‘(phIPA = 0 are not directly implied in this formulation. On the other hand, those states which do not satisfy these conditions yield ‘(phlph) = 0 and are therefore unmeasurable. At least for topological trivial models the conditions 0 = 0 (5.24) /AIPh) = ‘(phip/ may be interpreted as gauge invariance conditions in the same way as we did for the coBRST condition in sect. 4. In order to see this notice that [Q,Xb]_
iMabp/~l,
=
(5.25)
where X’~are gauge-fixing conditions to Wa. The matrix operator Ma” is defined by (3.32), i.e. Ma’~’ i[Wa,X~!~].In addition we require the properties (3.33) which are natural to impose. This means that [Xa,Xb] = 0 and that Ma~!~ commutes with X a~For topological trivial models there is always an inverse matrix operator (M’ )~to Ma’~’. In such a case we have from (5.25) p/a
=
i(M_l)g[Q,Xb]
=
i[Q,X”] _(M’)~.
(5.26)
The conditions (5.24) should therefore be expected to imply x’~Iph) = 0 effectively for m k values of a, and ~(phIXa = 0 for k values of a. One way to achieve this in a simple manner is to choose Xa such that —
=
(M’)~ = 0,
(5.27)
which make (5.24) yield
x~Iph)= ‘(phIX0
=
0.
(5.28)
Notice that (5.27) requires [W~,X~]— =
[W~,x0]— = 0,
(5.29)
which also are natural consistency conditions between (5.28), (5.12) and (5.23). If X’~may be chosen such that Ma”(X) satisfies Mab(0) = ~ then we have ‘(phI[~~,x’~’]_Iph)= —iö~’‘(phlph).
(5.30)
R. Marnelius / BRST quantization
234
For topologically nontrivial models there exist in general no x’s such that (M’ )~exists, so eq. (5.30) then cannot be satisfied. In this case we may have ‘(phIW,~ = 0 for some a’s and/or W~Iph) = 0 for some jt’s which is inconsistent with (5.30), and which implies that there exist genuine physical states with different FP ghost dependence [1]. 6. Gauge fixing of FP ghosts and antiBRST In ref. [7] antiBRST invariance was combined with the procedure of ref. [11. Since we have shown that this procedure follows from a fixing of the FP dependence by means of auxiliary conditions, we should also be able to combine antiBRST with these constraints. In order to show this we need the BRST charge (3.1) which is the BRST charge (5.2) extended with dynamical Lagrange multipliers and corresponding antighosts. If we impose (5.3), i.e. 2Iph) = 0, (6.1) p/’ which now only partially fix the FP dependence, then the analysis of sect. 5 applies, which means that all consistency conditions derived there are still valid. (We assume that eqs. (5.7) and (5.20) are valid.) However, antiBRST invariance leads to new additional consistency conditions. We obtain P0Iph)
=
[Q,P 0]+Iph)
=
0,
[Q,p/~]+Iph)
=
0.
(6.2)
Using eq. (6.1) the first condition becomes U0~fP~,~Iph) = 0.
(6.3)
For general U this is only satisfied if ~Iph)
=
0.
(6.4)
By means of this condition the last condition in eq. (6.2) becomes P~Iph)= 0. Notice that
pa
=
(6.5) (6.6)
~
which means that (6.5) is both a gauge invariance condition allowing for (6.4) and a consistency condition for (6.4). One may easily check that [Q,~]÷Iph)= [Q,P~]_Iph)
=
0
(6.7)
K
Marnelius / BRST quantization
235
are satisfied. In order to fix the FP ghost dependence of the physical states completely we need to impose ~0Iph)= 0,
(6.8)
in addition to (6.1) and (6.4). This is a gauge invariance condition since =
(6.9)
i[Q,Va]_.
Therefore, eq. (6.8) allows in turn for v0lph)
=
0.
(6.10)
Consistency between (6.8) and antiBRST invariance requires then [~,~0]+Iph)
=
0,
(6.11)
which by means of eqs. (6.1), (6.4), (6.5) and (6.8) becomes U0P~v~Iph) = 0,
(6.12)
which is always satisfied when (6.10) is satisfied. Thus, we have arrived at the conditions P0Iph) = p/~Iph)= ~i’~Iph)= ~0Iph) W~IPh) = v0lph) = P~Iph)= 0,
=
0, (6.13)
which obviously are consistent with each other. Similarly, we find when starting from (5.17) the conditions 0 = ‘(phI~0= ‘(phI~ = 0, ‘(phIP,~= ‘(phi17 ‘(phIW~= ‘(phIv~= ‘(phiP0 = 0. (6.14) Remarkably enough these conditions imply apart from BRST and antiBRST invariance KIph)
‘(phIK
=‘
KIph) = NghIPh) = p/avalph) = ,~avaIph)= DIph) (phIK = ‘(phIN&, = ~(phIp/aVa = ~(phI~aVa = ‘(phID
=
= =
0, 0. (6.15)
where K,K,N~ and D are defined by eqs. (3.6), (3.15), (3.22) and (3.27) respectively. Thus, both the physical state space and its dual are OSp (1, 112) invariant.
236
R. Marnelius / BRST quantization
The physical state spaces allow for even further conditions as we showed at the end of sect. 5. There we found that (6.16) x~Iph)= 0, ‘(phIX0 = 0 were allowed when [WL~x’~]= [w~,x0]= 0. If X’~satisfies the consistency conditions in appendix A or equivalently (3.33), then the physical state spaces satisfy even an extended OSp symmetry given by X’~vaIph)= ~(phIXaVa = 0
and their consistency conditions as well as
(6.15)
(6.17)
(see appendix A).
7. Auxiliary conditions in generalized BRST quantization Consider again the BRST charge (5.2). If the original gauge generators Wa acquires an anomaly, i.e. if one finds [Wa,Wb]— = ~Ua~Wc+
lfab,
(7.1)
where fat, contains a factor h, then (5.2) is no longer nilpotent. Instead we have =
~lfa~,p/”p/~’.
(7.2)
In ref. [2] it was proposed that the BRST charge (5.2) may still be used to project out physical states by means of the BRST condition QIph) = 0 provided Q is conserved. (In ref. [3] it was shown that this BRST condition must be accompanied by at least one additional condition.) The resulting formulation is a proposal for BRST quantization of both anomalous gauge theories and theories with second-class constraints. The analysis of possible auxiliary conditions proceeds here as in sect. 2, although with some complications. One important complication is caused by the fact that operators of the form B~[Q,C]
(7.3)
are in general no longer physical operators when Q2 ~ 0. There are also C-operators of this form. Anyway, in the following we shall continue to call all operators of the form (7.3) B-operators although they are not physical operators in general. Now, if BIph)
=
0
(7.4)
R. Marnelius / BRST quantization
237
is added to the BRST condition this requires the nontrivial consistency condition (7.5) [Q,B]Iph) [Q2,C]Iph) = 0. For instance, on inner product spaces Biph) = QCIph) will be a zero-norm state only if (7.5) is satisfied, and B is only then a gauge generator. When (7.5) is satisfied, (7.4) will again promote C to a physical operator provided (2.11) or (2.12) is satisfied together with the consistency conditions [Q2,C],C]Iph) [[Q2,C],C],C]Iph)
[... [[Q2, C], Cl, C],..., C]iph)
=
0,
=
0,
=
0.
(7.6)
Alternatively we may impose Ciph)
=
0
(7.7)
provided eqs. (7.4), (7.5), (2.12) and (7.6) are satisfied. When we have several constraints B 1Iph) = 0,
B1
[Q,C],
(7.8)
then they are consistent to impose only if (2.17), (2.18) and [Q,B1]
=
c’B~+ cQ
(7.9)
are satisfied. In this case we may impose C,Iph)
=
0.
(7.10)
A more general possibility is that (2.17) and (7.9) are replaced by [Q,B1]
=
[B1,B~] =
c/B1
)~jfBk
+ d/C1 +
cQ,
+ K,j’Ck +~Q.
(7.11)
Thus, when we have several constraints BiIph) = 0 and C~Iph)= 0 then we require as in sect. 2 that Q, B1 and C, satisfy a closed algebra. However, its form is in general slightly different here If we consider the FP ghost fixing (5.3) in sect. 5, then we have apart from the consistency conditions given there the additional consistency conditions 2,P [Q 0]+1P11)= ifaap/’~IPh)= —if0pn~Iph)= 0, (7.12)
238
R. Marnelius / BRST quantization
where (5.3) has been used in the second equality. Obviously (7.12) can only be satisfied if f0p
=
0,
cs,/i
I,... ,k.
=
(7.13)
This condition also follows from the requirement that [Ba,Bp]Iph)
=
0,
(7.14)
where B,,, = [Q,P0 1+. Eq. (7.13) is the only additional condition required as compared to the ordinary case in sect. 5. (The complete analysis [3] requires = f,,,. = 0 in addition to UJ = fop = 0 where a, /1 = 1,... k and ,
In ref. [3] it was shown that the above generalized BRST quantization always requires the auxiliary condition ~Iph)
=
0,
(7.15)
where (7.16) where in turn =
NA
=
NB
=
NB
1
-
NA,
~
(17App17A)
(7.17)
0
(7.18)
A=k+ I
Since =
due to the conditions fop
=
f,~= 0, ~ is a gauge generator. Eq.
allows, therefore, for a condition of the form I~Iph)= ñlph),
(7.15)
(7.19)
where ñ is a number. In fact, the FP ghost fixing procedure in sect. 5 will at the end determine this number. The importance of (7.15) lies in the fact that QIph)
=
~Iph) = 0
(7.20)
R. Marnelius / BRST quantization
239
may be written as
oIph)
=
dlph)
=
0,
(7.21)
where d
[NB,Q]
(7.22)
Q2.
(7.23)
=
[NA,Q],
=
d~= 0, [ö,d]+
=
are nilpotent operators: =
It is a pleasure to thank Stephen Hwang for helpful discussions. Appendix A The algebra of auxiliary conditions considered in sect. 3 In sect. 3 we first considered the algebra of Q, Q, K, K and N (~N~)given by =
[Q,K]
=
~2
[Q,~]+=
=
Q, Q,
[Q,K]_ =
[N,Q] = [N,K]=-2K,
[N,Q] = [N,K]=2K,
[K,K]=N.
(A.l)
Then we considered q and c~defined by [Q,q]~= —iK, [Q,~]÷
=
iK,
(A.2)
which require [N,q]_
=
q, [N,cfl_= —ci.
(A.3)
We first found the realization q
=
p/aVa,
~
=
(A.4)
?~aVa
which satisfy q2
=
~2
=
[q,~]~
=
0,
[Q,c~]÷ = D—i~N, [Q,q]~
(A.5) =
—D—i~N,
(A.6)
R. Marnelius / BRST quantization
240
defining the operator D. Eq. (A.4) yields D
The complete algebra of Q, (A.3), (A.5)—(A.6) and
K, K, N, q,
~,
[D,Q]
=
i~Q,
[D,Q]_
=
[D,q]_
=
—i~q, [D,~]
=
=
-iK,
=
=
—~,
[D,K] [K,q]_
(A.7)
paVa + i~(p/aPa+ p~a~)
=
[D,K]
t~
and D is then given by eqs. (A. 1)—
i~Q, -iK,
[K,~]_= [K,qJ
[K,~~]_ = —q,
=
0
(A.8)
and constitutes an OSp(l, 112) algebra. Now the realization (A.4) of (A.2) is not unique. We may also choose q
=
p/ava +
a,
q
=
p/aVa +
a,
(A.9)
where [Q,a]~= [Q,ã]~
0, [N,a]
=
=
a, [N,ã]
=
—ã
(A.10)
without affecting (A.2) and (A.3). If we in addition require that the previous algebraic properties (A.6) should be retained, then we must have
[Q,a]~
=
—[Q,a]~,
(A.1l)
which is solved by a
=
[Q,r]_,a
=
[Q,r]_, [N,r]
=
0.
(A.12)
D is then not given by (A.7). Instead we have D-+D’
=
[Q, [Q,r]],
(A.13)
i~Q, [D’,~] = i~,
(A.14)
D + T, T
=
implying [D’,Q]
=
consistent with the first line in (A.8). However, in order to retain the whole previous OSp algebra r has to satisfy some additional conditions. From (A.8) we find that these conditions fix the algebra of r, a, a and T to be [r,a]_
=
[a,ã]+
=
[T,a]_
=
[r,ã]_
= [r,T] = 0, 2 = a2 = 0, a [T,ã]_= 0,
(A.15)
R. Marnelius / BRST quantization
241
where the last two lines follow from the first line. In addition we must have [r,K]
=
[r,K]_
=
0,
[r,q]
[r,~]_
=
=
0,
(A.16)
where q,7 may either be given by (A.5) or (A.9) due to (A.15). However, there is no condition on the commutator [r,D]. For the mixed commutators between r, a, a, T and the previous OSp generators we find then (the nonzero part)
[a,K]
=
—[r,D]_,
=
a,
[q,a]÷ [ã,K]_= a,
=
[r,D]_,
[a,D] = —i~a+[Q,[r,D]], [ã,D]_= —i~a+[Q,[r,D]], [q,T]_ = ia— [Q,[r,D]], [t~,T]_ = iã— [Q,[r,D]], [D,T]_ = —[Q, [Q,ir— [r,D]]],
(A.17)
which becomes a closed algebra if [r,D] can be expressed in terms of the already defined operators. Notice that if [r,D]
=
ir,
(A.18)
then T commutes with all operators. There are several choices of r which satisfy the above conditions: r = v2, [r,D] = 2ir, r = p/a, [r,D] = —ir.
(A.19)
However, these choices cannot make q, c~and D’ conserved. The interesting choice is instead (A.20)
r=X’~va,
which satisfies (A. 18). x a are gauge-fixing conditions to çua. If we define the matrix operators Ma’~’by [Wa,Xb]_
then the consistency conditions [r,a] =
0,
=
=
(A.21)
~Ma”,
[r, a]
=
0 on r require
[Xa,M~b] + [Xb,MCa]
=
0.
(A.22)
242
R. Marnelius / BRST quantization
This inserted into the Jacobi identities for [X”,Mb’i
Wa, X’~and
=
0.
X’~requires in turn (A.23)
Above we have made the natural assumption that Mab does not depend on the Lagrange multipliers. Eqs. (A.22) and (A.23) make it natural to let Ma” only depend on X’~. References [1] S. Hwang and R. Marnelius, Nucl. Phys. B315 (1989) 638; B320 (1989) 476 [2] R. Marnelius, Nucl. Phys. B294 (1987) 685; Generalized BRST quantization, Proc. mt. meeting on geometrical and algebraic aspects of nonlinear field theory, Amalfi, Italy, 1988, ed. S. De Filippo, M. Marinaro, G. Marmo and G. Vilasi (North-Holland, Amsterdam, 1989) [3] R. Marnelius, Nucl. Phys. B370 (1992) 165 [4] R. Marnelius, Phys. Lett. B99 (1981) 467 [5] G. Curci and R. Ferrari, Phys. Lett. B63 (1976) 91; I. Ojima, Prog. Theor. Phys. Lett. 64 (1980) 625 [6] S. Hwang, Nuci. Phys. B231 (1984) 386 [7] S. Hwang, Nucl. Phys. B322 (1989) 107 [8] I. A. Batalin and G. A. Vilkovisky, Phys. Lett. B69 (1977) 309 [9] H. Aratyn, R. Ingermanson and A. J. Niemi, Phys. Lett. Bl89 (1987) 427; NucI. Phys. B307 (1988) 157 [10] I.B. Frenkel, H. Garland and G. J. Zuckerman, Proc. NatI. Acad. Sci. 83 (1986) 8442 [11] M. Spiegelglas, Nuci. Phys. B283 (1987) 205 [12] A. V. Razumov and G. N. Rybkin, NucI. Phys. B332 (1990) 209 [13] W. Kalau and J. W. van Holten, Nucl. Phys. B36 1 (1991) 233 [14] J. Bognár, Indefinite inner product spaces (Springer-Verlag, Berlin, 1974)
Nuclear Physics B 372 (1992) 218—242 North-Holland
Auxiliary conditions in BRST quantization Robert Marnelius institute of Theoretical Physics, Chalmers University of Technology, S-412 96 Goteborg, Sweden Received 31 October 1991 Accepted for publication 10 December 1991
A general structure of allowed auxiliary conditions in both ordinary and generalized BRST quantization is presented. Such conditions are shown to fall into two classes which always should imply each other. Both antiBRST invariance and coBRST invariance are shown to fit into the framework. AntiBRST invariance is also shown to imply an OSp symmetry. A previously given algorithm for solving BRST cohomologies is formulated in terms of allowed auxiliary conditions.
1. Introduction In BRST quantization one starts from a BRST-invariant model and projects out BRST-invariant states in the quantization. The BRST invariance is reflected in the conservation of a nilpotent charge Q (Q2 = 0), and the projection is effected by the BRST condition Qiph)
=
0,
(1.1)
where the BRST-invariant states ph) are called physical states. Often one combines (1.1) with additional conditions. Typical examples are antiBRST invariance or a fixing of the ghost number. In this report the general structure of such conditions is presented. It will be shown that there are only two forms of allowed additional conditions: One may either impose
[Q,C]Iph)=O
ciph)
or
=
0
(1.2) (1.3)
where (1.2) will be called a gauge invariance condition and (1.3) a gauge slicing condition. Notice that eq. (1.2) is the consistency condition of (1.1) and (1.3). It will be proposed that consistent conditions of the type (1.2) 0550-3213/92/$05.00
© 1992
—
Elsevier Science Publishers B.V. All rights reserved
R. Marnelius / BRST quantization
219
always allow for conditions of the type (1.3) without affecting the physical content. This makes the two forms have equivalent implications and it requires that all operators of the forms (1.2) and (1.3) form a closed algebra. In order to be a fundamental auxiliary condition [Q,C] and/or C must be conserved operators, i.e. they must generate symmetry transformations. When this is not the case such additional conditions may still be used to investigate the physical content of the BRST cohomology although at a particular instant. In sect. 2 the general structure of allowed auxiliary conditions is presented. In sect. 3 antiBRST invariance is shown to imply an OSp symmetry, and in sect. 4 coBRST invariance is investigated. In sect. 5 the simplifying analysis of ref. [11 is viewed in terms of auxiliary conditions, and in sect. 6 this formulation is combined with antiBRST invariance. In sect. 7, finally, some of the previous results are generalized to the extended BRST formulation proposed in refs. [2,3] which includes anomalous gauge theories and theories with second-class constraints. 2. Auxiliary conditions in ordinary BRST quantization Consider the additional condition Riph)
=
0
(2.1)
to the BRST condition (1.1). The meaning of such an additional condition depends on what type of operator R is chosen to be. One may divide R into three different types of operators which we call A-, B- and C-operators (cf. ref. [4]). They are defined by [Q,A] =0,
A
[Q,C],
[Q,B] =0,
B= [Q,C].
(2.2)
A and B are physical operators which map physical states on physical states while C is an unphysical operator in general. The only exception is the ghost number operator N~,satisfying [N~,Q].=Q.
(2.3)
One may show that any operator ~ that maps physical states on physical states may be written as =
~AnNg~h,
(2.4)
220
R. Marnelius / BRST quantization
where A~ are A- or B-operators. A-operators are genuine physical operators while B-operators are gauge generators since they generate unobservable changes of the physical states: Bjph)
=
Q~).
(2.5)
By means of the Jacobi identities one finds [4] [B,A]
=
B’,
[B,B’]
=
B”.
(2.6)
Notice that Q itself is a B-operator due to (2.3). Notice also that there is no unique separation of the operator algebra into A-, B- and C-operators without additional conditions since the definition (2.2) is unaffected by the transformations A
—~
A + B,
C—~C-4-A+B.
(2.7)
If we now consider the case when R is an A-operator in (2.1) then we have AIph)
=
0,
(2.8)
which obviously is consistent with the BRST condition (1.1) due to (2.2). However, this condition is quite ad hoc. It is like imposing AI~)= 0 in any ordinary quantum theory. It definitely changes the physical theory. Such conditions are therefore excluded. Consider next an auxiliary condition of the form BIph)
=
0,
(2.9)
which also is consistent with (1.1) due to (2.2). Since the B-operators are gauge generators this condition may be viewed as a gauge invariance condition. Since Biph) = QI ) for any choice of B, it naively seems as if (2.9) only eliminates unmeasurable states and does not kill any degrees of freedom. If this were true then one could impose (2.9) for all possible B-operators which is algebraically consistent due to (2.6). However, this is always too strong and therefore inconsistent (see sects. 5 and 6). On the other hand, we may always impose at least some conditions of the form (2.9) without affecting the physical content. The problem is only to determine exactly which constraints that are allowed. Below we shall give general algebraic conditions for possible choices of subsets of B-operators that may be used to restrict the physical states. The definite choice of such a subset is then determined by the representation of the original state space. However, before we give these conditions we have to discuss the implication of eq. (2.9).
R. Marnelius / BRST quantization
When B
221
[Q,C],condition (2.9) becomes QCIph)
=
0.
(2.10)
This seems to imply that the unphysical operator C is lifted into the class of physical operators. This is a correct interpretation provided QC~Iph)= 0
(2.11)
for any integer n, or equivalently if [B,C]Iph)
=
0,
[B,C],C]Iph)= 0,
[... [B,C],C],...,C]Iph)
=
0.
(2.12)
If C has become a physical operator then it seems as if the degrees of freedom of the physical state space has increased. This can obviously not be the case. An unphysical degree of freedom cannot be transformed into a physical degree of freedom. An additional condition can only decrease the degrees of freedom or leave them unaffected. Therefore, we should always expect that a condition like CIph)
=
0
(2.13)
is implied by (2.9) or that it at least may be imposed without affecting the physical content. The last property means that the physical states which does not satisfy (2.13) are e.g. zero-norm states. Notice though that C is only determined up to transformations of the type (2.7) and that there might exist several sectors projected out by different choices of C. For instance, if the state space allows for eigenstates to C then we may have (C—A)~ph)= 0, where
,~
(2.14)
is a constant. In such a case the value of must be chosen properly. ,~
A typical example is the fixing of the ghost number which is effected by (N~h—n)Iph)=0. (2.15) Due to eq. (2.3) the BRST condition always allows for a condition of this type. Indeed, the genuine physical states (those states that contribute to measurable quantities) have a definite ghost number n at least for topologically trivial
R. Marnelius / BRST quantization
222
gauge theories. Therefore, eq. (2.15) is not necessary to impose, and the wrong choice of n is of course disastrous. On the other hand, for topologically nontrivial models we may have genuine physical states with different ghost numbers. In such a case eq. (2.15) will exclude certain sectors of the theory. (A topological trivial model is a model for which the BRST condition (1.1) by itself projects out one canonical complete sector, while for a topological nontrivial model it yields several canonically complete sectors [1].) We believe that consistent choices of auxiliary conditions of the form (2.9) always require the conditions (2.12) and therefore always imply conditions of the form (2.13). With eq. (2.13) we have arrived at the third type of additional conditions which we will call gauge slicing conditions. If we instead of (2.9) impose (2.13) together with the BRST condition (1.1), then consistency requires both (2.9) and (2.12). Thus, we have found two ways leading to the same result: We may either add a condition like B~ph)= 0 or Ciph) = 0 to the BRST condition (1.1). The consistency conditions which have to be fulfilled make the two approaches equivalent in general. We propose now the following algebraic criteria for consistent choices of auxiliary conditions: Consider first the case when we start with several conditions like (2.9): B1Iph)
=
0,
i
=
1,... ,l.
(2.16)
The first consistency condition is that B satisfy a subalgebra of (2.6), i.e. [B1,B1] = .tjfBk + ,~Q. Secondly, if B
=
(2.17)
[Q, CJ then we also demand that [C1, B1 I
=
~z~f Ck +
iJIIkBk +
[C1,C~] = p11k Ck +
jiQ,
+ pQ,
(2.18)
where the freedom (2.7) for the choices of C1 may be used to achieve such a closed algebra. This algebra will then allow for CiIph)
=
0,
i
=
1,... ,l.
(2.19)
Equivalently, if we start with (2.19) then this will automatically require (2.16) with B [Q, C1] as well as a closed algebra of C,,B, and Q. In fact, independently from which constraints we start from, the algebraic analysis will determine which additional constraints may be added and the final algebra will always be a closed algebra of the above form. We have only to make sure of that no A-operators couple.
R. Marnelius / BRST quantization
223
One may notice that the above algebraic conditions exclude the possibility that all B-operators are used to constrain the physical states since all B-and C-operators do not form a closed algebra. In fact, if the original state space is nondegenerate then the maximal closed algebra will at most involve half of the B- and C- operators (cf sects. 5 and 6). Notice finally that a subset of B’s and C’s forming a closed algebra can only be used as constraint operators if the state space representation allows for nontrivial solutions of (2.16) and (2. 19). In the following analysis we will always start from some possible constraints and look for further constraints from the above algebraic conditions. 3. AntiBRST invariance AntiBRST invariance is the first auxiliary condition considered in BRST quantization. It was found to be useful in Yang—Mills theories [5]. AntiBRST transformations are generated by a nilpotent operator Q which anticommutes with the BRST charge Q. Q has opposite ghost number to Q. In the following we shall consider finite number of degrees of freedom and the general hamiltonian formulation of ref. [61 where the BRST charge is chosen to be of the BFVform [8] (a,b,c= 1,...,m
Q where
=
V~a
~~iU6c”Pa?l”P1’—~iUa~+PaP’~,
(3.1)
and ~ are fermionic Faddeev—Popov (FP) ghosts and antighosts Ta and Pa their conjugate momenta, respectively, and [~ap 6]~ = [~a,~b]+ = 0. (3.2) ,~°
~
and Pa have ghost number + 1 while ~ and Pa have ghost number —1.
Wa
are bosonic constraint operators satisfying [Wa,Wb]—
=
~Ua~Wc,
(3.3)
where U~gare constants. Pa are conjugate momenta to the Lagrange multipliers Va:
[Pa,Vb]_
=
iôab.
(3.4)
For simplicity Wa and Pa are restricted to be bosonic. It is straightforward to generalize the following treatment to fermionic Wa and Pa and bosonic ,j’~,Pa and ~a,p0 (see e.g. ref. [7]). According to ref. [6] the antiBRST charge Q, satisfying ~2 = 0, is given by
Q
[Q,K]_ = Wa
iU~g~c?~?~— ~iUJi)~
_PaP~~ iU~P~n~’ + Ua~VcPa?~b, —
(3.5)
224
R. Marnelius / BRST quantization
where K
=
(3.6)
Pail’~ + ~Ua~VC~a~b.
Since K has ghost number —2 and Q has + 1, Q has ghost number —1. AntiBRST invariance requires Qiph)
=
0.
(3.7)
This is a condition of the type (2.9) in sect. 2 due to eq. (3.5). Since (see ref. [6]) [Q,K]
=
0,
(3.8)
the consistency condition (2.12) is satisfied, which implies that the unphysical operator K is lifted into the class of physical operators. We may therefore impose the gauge slicing condition KIph)
=
0
(3.9)
without affecting the physical content. Indeed, K
=
(3.10)
i[~aVa,Q]+
implies KIph) = QI ) so KIph) ~ 0 is an unmeasurable state. In order to investigate whether or not K is conserved consider the generic BRST and antiBRST-invariant hamiltonian H=H0-i- [Q,[Q,R]].
(3.11)
One easily finds that the conservation of K requires [H0,K]
=
0,
[R,K]
=
0.
(3.12)
The first condition is satisfied provided H0 does not contain Lagrange multipliers or FP ghosts. This follows if [H0,Wa] = 0 for all a [6], which we assume from now on. Concerning the second relation we notice that the rather general R given in ref. [61 is written as R
=
~vav”
—
~c5,Ja~a +
s,
(3.13)
where a is a real constant and where S does not contain the FP ghosts, the Lagrange multipliers and Wa. This expression satisfies therefore the last equality in eq. (3.12) from which we conclude that K is conserved in general.
R. Marnelius / BRST quantization
225
In ref. [6] it was also shown that there is a dual relation to (3.5), namely
Q= [Q,K],
(3.14)
where K
=
(3.15)
Pap/a + ~Ta~ch/ahlb.
Since [Q,K]
=
0,
(3.16)
BRST and antiBRST invariance imply that we should be able to impose KIph)
=
0.
(3.17)
Indeed, K
=
(3.18)
j[Q,p/aV~]~
so Kiph) ~ 0 is unmeasurable. (Notice that although K is a physical B-operator it is unphysical from the point of view of Q.) By means of the Jacobi identities the hamiltonian (3.11) may be written as H
=
H0—
~ [Q,R]],
(3.19)
which implies that K is conserved if [H0,K]
=
0,
[R,K]
=
0.
(3.20)
Thus, K is conserved when K is conserved. (The first equality is satisfied by construction and the second by an R of the form (3.13).) Now we have [6] [K,K]_
= p/ap~_p/ap~~Ng~,
(3.21)
where N~is the ghost number operator. Thus, eqs. (3.6) and (3.17) imply N~Iph) = 0,
(3.22)
which may be viewed as a consistency condition for (3.6) and (3.17). This condition is always satisfied by genuine physical states when we have antiBRST invariance since [6] N~ = j[Q,,~aVa]÷+ ~[Q,p/av~]÷
(3.23)
R. Marnelius / BRST quantization
226
implies Iph,n) =QI)+QI)
(3.24)
when n ~ 0. Thus, physical states with ghost number n ~ 0 never contribute to any measurable quantities. The expressions (3.10), (3.18) and (3.23) for K,K and N~imply that the conditions (3.9), (3.17) and (3.22) may in a generalized sense be viewed as gauge invariance conditions which should allow for p/aV~~ph) = ,~aVaIph)= 0.
(3.25)
However, consistency requires DIph) where im
LI
= = =
=
0
(3.26)
rQ ,P)—a VaJ+ 1
[Q
lxr + l~1Vg)~ p/aV~]÷ — ~~Ngh
paVa + i~(~Pa + p/apa,
(3.27)
which is a generalized B-operator due to (3.23). Eq. (3.26) is automatically satisfied if (3.25) is satisfied. The constraint operators Q, Q, K, K, N~,~ ~aVa and D satisfy a closed OSp(l, 112) algebra (see appendix A). This algebra is close to the OSp(l, 112) algebra given in ref. [9]. Both contain the conserved subalgebra of Q, Q, K, K and N~.However, the generators q, t~and D differ from those in ref. [9]and they are not conserved by the hamiltonian (3.11). Their conservation requires [R,D]_= iR
(3.28)
up to terms which are both BRST and antiBRST invariant. Now the conditions (3.25) are not a unique consequence of KIph) K~ph)= 0. We may for instance make the replacement q
?laVa —~
=
,JaVa +
[Q,r]_,
=
p/a~+
[Q,r]_,
=
(3.29)
which changes D to D-~D’ = [Q4]+-i~N~ = =
—[Q,q]~ l~Ngh [Q,[Q,r]].
D+
(3.30)
R. Marnelius / BRST quantization
227
In appendix A it is shown that r may be chosen such that also Q, Q, K, K, Ngh, q, t~and D’ satisfy an OSp(l, 112) algebra. The conditions for this is given by eqs. (A.15) and (A.l6) in appendix A. The conservation of q,t~and D’ requires then (3.28) with D replaced by D’ and [H0,r] = 0. An appropriate choice of r is r where
/3 is a constant,
(3.31)
JiXaVa,
=
and x a are gauge choices to Wa [Wa,Xb]_
=
(3.32)
jMa~’.
The consistency conditions (A. 15) and (A. 16) together with the Jacobi identities require (see appendix A) X~z,Xbl_=0, [Xa,MbC]_=O, Wa,Mb(i_ [Wb,M,f] = iUa:MdC,
(3.33)
—
where the last line follows from the Jacobi identities of Wa, Wb and Xc. In topologically trivial models, Mab has also an inverse matrix operator (M’ ) ~‘ satisfying 6(M’)~ = ö~1. (3.34) =
Ma
The conditions (3.33) make it natural to assume that Mab only depends on X’~. Mab (x) and (M (x ) ) when it exists, should then be given by power series in X~zand should be possible to choose such that Mab(0) = (M1 (0))~= ó~1. q, c~and D’ are now conserved if —~
~
[I~IO,Xa]_ = 0,
(3.35)
Va
and if /3 = —~ifor R = ~v2 + S(~)under certain conditions on X’~and S, or /3 = —~ifor R = ~ There exists no OSp(1, 112) symmetry for a hamiltonian of the form (3.11) when R is given by eq. (3.13) with a ~ 0. Thus, Yang—Mills theories in the Feynman gauge are not OSp invariant. For /3 = ~i our OSp (1, 112) symmetry coincide with that of ref. [9] for special choices of x a One may now ask oneself if it is not possible to proceed by imposing —
aiph)
=
a~ph)= 0,
a~[Q,rJ_,
a~[~,r]_.
(3.36)
This would then imply (3.25) again and that the physical state space is independent of the choice of gauge slicing operators q and in K = i [~,Q] t~
+
R. Marnelius / BRST quantization
228
and K = i[Q,q]~. Since aiph) = Ql ) and alph) = Ql ) eq. (3.36) does not exclude any genuine physical states within sectors allowing for (3.36). Consistency requires Tlph)
=
0,
T~[Q,ã]÷
=
—[Q,a]~.
(3.37)
One may then go even one step further and impose
rlph)
0,
=
(3.38)
which requires no further consistency conditions due to the algebraic properties in appendix A. We have thus arrived at an extended OSp invariance of the physical state space. (In sect. 6 it is shown that all these constraints are satisfied within the gauge slicing procedure of sect. 5.) However, this is not a fundamental invariance since p/aVg, p~aVa,a, a, D, T and r are not conserved. Interestingly enough this extended OSp algebra is conserved if r = /JXaVa and H = [Q,[Q,r]] (see appendix A). 4. CoBRST invariance Like the antiBRST charge Q the coBRST charge, here denoted Q*, is nilpotent and has ghost number minus one. However, it does not commute with the BRST charge Q. We have (4.1) imposing
Q*Iph)
=
0
(4.2)
0
(4.3)
requires ~Iph)
=
for consistency, and (4.3) allows for (4.2). In the following we use the definition of coBRST given in refs. [10,11] (see also refs. [12,13]). This definition requires the original state space Q to be an inner product space on which there exists a metric operator p/ satisfying p/2 = 1 and such that (u~p/~v) is a positive definite scalar product for any choice of state vectors lu), Iv) E Q. We then have the isomorphism (u~p/Iv)= (ulv),
(4.4)
R. Marnelius / BRST quantization
229
where Iv), lu) may be considered to be state vectors in a Hilbert space 11. (Q is then a so-called Krein space [14].) Within this framework Q* is defined to be the hermitian conjugate of Q in 7-1, which implies (4.5)
=
in Q. With this definition one finds the equivalence Aiph)
0 ~ Qiph)
=
=
Q*Iph)
0.
=
(4.6)
This follows from
(uI’j A lu)
=
(~~Q*p/Q*~~) + (uIQp/QIu),
(4.7)
and since the last two brackets only vanish for QIph) = Q*Iph) = 0 due to eq. (4.4). Thus, in this case we have a strict proof that (4.3) implies (4.2). In addition, we have the Hodge decomposition for any state Iu) e Q:
lu) where AIu)0
=
=
u)~+ QIVi) + Q*1v2),
(4.8)
0. The proof (see ref. [13]) is based on the equality (uIQnIv)
=
(4.9)
(u~p/Q*~v),
which implies that KerQ (KerQ*) and ImQ* (ImQ) are orthogonal. CoBRST invariance is very useful in order to determine the physical degrees of freedom since we only need to solve AIu) = 0. However, at the fundamental level it seems not so useful since Q* and A are in general not conserved. Comparing Q* with Q in the antiBRST case there is an obvious difference in the fact that Q* ph) = 0 is a gauge slicing condition while Qiph) = 0 is a gauge invariance condition. However, this difference is slightly fictitious. The reason is the equalities A =52,
Q*p/s+Q
(4.10)
where S~ [Q,p/]. This means that Q*Iph) =
=
0 is equivalent to
(4.11) (i~)
=
0 is inconsistent with
1)
Siph)
=
0,
(4.12)
R. Marnelius / BRST quantization
230
which is a gauge invariance condition. Since the consistency conditions (2.12) are satisfied this condition implies in turn that we have conditions like (172 = 1) l7Iph)
=
±Iph).
(4.13)
The plus sign for all physical states implies that the genuine physical state space is a Hilbert space. The reason why one should not expect S and ~ to be conserved is due to the fact that ‘1 usually severely breaks the covariance of the model (particularly when we have Lorentz covariance). 5. Simplifying conditions for solving BRST cohomologies Auxiliary conditions need not be conserved. They may simply be used to investigate the physical content of a model, although at a particular instant. CoBRST is useful in this respect. Another case is the simplified analysis set up in refs. [1,3]. There it was shown that, at least for topologically trivial models, it is sufficient to consider states of the form lu)
=
IIvl)IG)
(5.1)
when analyzing the BRST condition QIph) = 0. IM) denotes matter states and IG) FP ghost states. Condition (5.1) may be viewed in terms of auxiliary conditions. In order to demonstrate this, consider a general first rank theory with finite number of degrees of freedom. The BRST charge is (a, b, c = l,...,m
Q— —
Wap/
a
a’r, bc bc raP) 17
1
—
~
1 —
~l
ba 17 ab
where Uat are constants and Wa are hermitian constraint operators satisfying the Lie algebra (3.3). p/a and Pa are Grassmann odd, hermitian FP ghosts satisfying (3.2). These properties imply that Q in eq. (5.2) is hermitian and nilpotent (Q2 = 0). (The following analysis may be generalized to the case when some constraints are odd and the corresponding FP ghosts are even (see e.g. ref. [7]).) It is easy to see that (5.1) may be implemented by means of auxiliary conditions. We may for instance choose Palph)
=
p/AIph)
=
0
(5.3)
R. Marnelius / BRST quantization
231
for a = 1,... ,k; )~= k + 1,... ,m, which obviously implies (5.1). These conditions are internally consistent since P(~and 17A anticommute. However, consistency with the BRST condition requires = 0, [Q,p/A1÷Iph)= 0.
(5.4) (5.5)
[Q,Pa]+Iph)
Using eqs. (5.2) and (5.3) the last condition becomes [Q,p/A]+Iph) = _~iU(~p/ap/flIph) = 0. Since p/a I ph)
=
(5.6)
0 is inconsistent with (5.3) we find that (5.5) requires
U~=0, a,fl=l,...,k;~=k+l,...,m. 1Iph) Thus, only if this is satisfied are we allowed to impose p/’ (5.4) we notice that [Ba,Bb]_ Bam [Q,Pa]+
=
=
~UatBc,
=
Wa+j~Ua~(Pcflb_p/bPc),
(5.7) 0. Concerning
(5.8)
and therefore eq. (5.7) implies =
~UcJBy, a,fl,y
=
1,... ,k
(5.9)
which means that eq. (5.4) is at least consistent to impose by itself. By means of eq. (5.2) and Ba = [Q,Pal we find 13P~,
B 0
where Wa + ~
=
—
(5.10)
W~+ iU0I Pap/’~ iU0~Pl —
~~UaI~
/3
=
1,... ,k;
,~
=
k + l,...m. (5.11)
Thus, eq. (5.4) implies B 0Iph)
=
W~IPh) = 0, a
=
1,... ,k
(5.12)
which definitely are additional conditions to (5.3). By means of eq. (5.3) and its consistency conditions we have arrived at exactly the conditions considered in refs. [1,3], namely (5.1) specified by eqs. (5.3), (5.7) and (5.12).
R. Marnelius / BRST quantization
232
A simple well-known example of (5.3) is the Siegel gauge in string theory: bolph)
0.
=
(5.13)
Consistency requires here the mass-shell condition [Q,bo]+Iph)
=
0.
(5.14)
These two conditions are used in the cohomology analysis in ref. [10]. The procedure in refs. [1,3] also involves the existence of dual physical states ‘(phI
=
(A1’I(G’I
(5.15)
,
satisfying
(G’IG) Here this implies (a,/i,y following)
=
=
l,
I(M’IM)I <
1,... ,k and ,u,v,t
‘(phjP~= ‘(phI 170
=
oo. =
(5.16)
k + 1,... ,m also in the
0
(5.17)
with the consistency conditions ‘(phIBA
=
0,
(5.18)
=
0.
(5.19)
The latter demands here =
0,
(5.20)
which makes eq. (5.18) consistent to impose since =
iU,~BA.
(5.21)
Furthermore, eq. (5.20) implies B~= W~+ iU~~P2p/v ~U~p/aPC, —
(5.22)
where W~is given by eq. (5.11). Eq. (5.18) yields therefore ‘(phIB2
=
‘(phIW~= 0.
(5.23)
Eqs. (5.3) and (5.17) and their consistency conditions are exactly the conditions considered in refs. [1,3].
R. Marnelius / BRST quantization
233
Another equivalent formulation of the above treatment is to start from eqs. (5.4), (5.5), (5.18) and (5.19) which are gauge invariance conditions. Eq. (5.5) implies then p/AIph) = 0 when (5.7) is valid and (5.19) implies ‘(phip/0 = 0 when (5.20) is valid. Eq. (5.4) implies (5.12), and (5.18) implies (5.23). However, P 0Iph) = 0 and ‘(phIPA = 0 are not directly implied in this formulation. On the other hand, those states which do not satisfy these conditions yield ‘(phlph) = 0 and are therefore unmeasurable. At least for topological trivial models the conditions 0 = 0 (5.24) /AIPh) = ‘(phip/ may be interpreted as gauge invariance conditions in the same way as we did for the coBRST condition in sect. 4. In order to see this notice that [Q,Xb]_
iMabp/~l,
=
(5.25)
where X’~are gauge-fixing conditions to Wa. The matrix operator Ma” is defined by (3.32), i.e. Ma’~’ i[Wa,X~!~].In addition we require the properties (3.33) which are natural to impose. This means that [Xa,Xb] = 0 and that Ma~!~ commutes with X a~For topological trivial models there is always an inverse matrix operator (M’ )~to Ma’~’. In such a case we have from (5.25) p/a
=
i(M_l)g[Q,Xb]
=
i[Q,X”] _(M’)~.
(5.26)
The conditions (5.24) should therefore be expected to imply x’~Iph) = 0 effectively for m k values of a, and ~(phIXa = 0 for k values of a. One way to achieve this in a simple manner is to choose Xa such that —
=
(M’)~ = 0,
(5.27)
which make (5.24) yield
x~Iph)= ‘(phIX0
=
0.
(5.28)
Notice that (5.27) requires [W~,X~]— =
[W~,x0]— = 0,
(5.29)
which also are natural consistency conditions between (5.28), (5.12) and (5.23). If X’~may be chosen such that Ma”(X) satisfies Mab(0) = ~ then we have ‘(phI[~~,x’~’]_Iph)= —iö~’‘(phlph).
(5.30)
R. Marnelius / BRST quantization
234
For topologically nontrivial models there exist in general no x’s such that (M’ )~exists, so eq. (5.30) then cannot be satisfied. In this case we may have ‘(phIW,~ = 0 for some a’s and/or W~Iph) = 0 for some jt’s which is inconsistent with (5.30), and which implies that there exist genuine physical states with different FP ghost dependence [1]. 6. Gauge fixing of FP ghosts and antiBRST In ref. [7] antiBRST invariance was combined with the procedure of ref. [11. Since we have shown that this procedure follows from a fixing of the FP dependence by means of auxiliary conditions, we should also be able to combine antiBRST with these constraints. In order to show this we need the BRST charge (3.1) which is the BRST charge (5.2) extended with dynamical Lagrange multipliers and corresponding antighosts. If we impose (5.3), i.e. 2Iph) = 0, (6.1) p/’ which now only partially fix the FP dependence, then the analysis of sect. 5 applies, which means that all consistency conditions derived there are still valid. (We assume that eqs. (5.7) and (5.20) are valid.) However, antiBRST invariance leads to new additional consistency conditions. We obtain P0Iph)
=
[Q,P 0]+Iph)
=
0,
[Q,p/~]+Iph)
=
0.
(6.2)
Using eq. (6.1) the first condition becomes U0~fP~,~Iph) = 0.
(6.3)
For general U this is only satisfied if ~Iph)
=
0.
(6.4)
By means of this condition the last condition in eq. (6.2) becomes P~Iph)= 0. Notice that
pa
=
(6.5) (6.6)
~
which means that (6.5) is both a gauge invariance condition allowing for (6.4) and a consistency condition for (6.4). One may easily check that [Q,~]÷Iph)= [Q,P~]_Iph)
=
0
(6.7)
K
Marnelius / BRST quantization
235
are satisfied. In order to fix the FP ghost dependence of the physical states completely we need to impose ~0Iph)= 0,
(6.8)
in addition to (6.1) and (6.4). This is a gauge invariance condition since =
(6.9)
i[Q,Va]_.
Therefore, eq. (6.8) allows in turn for v0lph)
=
0.
(6.10)
Consistency between (6.8) and antiBRST invariance requires then [~,~0]+Iph)
=
0,
(6.11)
which by means of eqs. (6.1), (6.4), (6.5) and (6.8) becomes U0P~v~Iph) = 0,
(6.12)
which is always satisfied when (6.10) is satisfied. Thus, we have arrived at the conditions P0Iph) = p/~Iph)= ~i’~Iph)= ~0Iph) W~IPh) = v0lph) = P~Iph)= 0,
=
0, (6.13)
which obviously are consistent with each other. Similarly, we find when starting from (5.17) the conditions 0 = ‘(phI~0= ‘(phI~ = 0, ‘(phIP,~= ‘(phi17 ‘(phIW~= ‘(phIv~= ‘(phiP0 = 0. (6.14) Remarkably enough these conditions imply apart from BRST and antiBRST invariance KIph)
‘(phIK
=‘
KIph) = NghIPh) = p/avalph) = ,~avaIph)= DIph) (phIK = ‘(phIN&, = ~(phIp/aVa = ~(phI~aVa = ‘(phID
=
= =
0, 0. (6.15)
where K,K,N~ and D are defined by eqs. (3.6), (3.15), (3.22) and (3.27) respectively. Thus, both the physical state space and its dual are OSp (1, 112) invariant.
236
R. Marnelius / BRST quantization
The physical state spaces allow for even further conditions as we showed at the end of sect. 5. There we found that (6.16) x~Iph)= 0, ‘(phIX0 = 0 were allowed when [WL~x’~]= [w~,x0]= 0. If X’~satisfies the consistency conditions in appendix A or equivalently (3.33), then the physical state spaces satisfy even an extended OSp symmetry given by X’~vaIph)= ~(phIXaVa = 0
and their consistency conditions as well as
(6.15)
(6.17)
(see appendix A).
7. Auxiliary conditions in generalized BRST quantization Consider again the BRST charge (5.2). If the original gauge generators Wa acquires an anomaly, i.e. if one finds [Wa,Wb]— = ~Ua~Wc+
lfab,
(7.1)
where fat, contains a factor h, then (5.2) is no longer nilpotent. Instead we have =
~lfa~,p/”p/~’.
(7.2)
In ref. [2] it was proposed that the BRST charge (5.2) may still be used to project out physical states by means of the BRST condition QIph) = 0 provided Q is conserved. (In ref. [3] it was shown that this BRST condition must be accompanied by at least one additional condition.) The resulting formulation is a proposal for BRST quantization of both anomalous gauge theories and theories with second-class constraints. The analysis of possible auxiliary conditions proceeds here as in sect. 2, although with some complications. One important complication is caused by the fact that operators of the form B~[Q,C]
(7.3)
are in general no longer physical operators when Q2 ~ 0. There are also C-operators of this form. Anyway, in the following we shall continue to call all operators of the form (7.3) B-operators although they are not physical operators in general. Now, if BIph)
=
0
(7.4)
R. Marnelius / BRST quantization
237
is added to the BRST condition this requires the nontrivial consistency condition (7.5) [Q,B]Iph) [Q2,C]Iph) = 0. For instance, on inner product spaces Biph) = QCIph) will be a zero-norm state only if (7.5) is satisfied, and B is only then a gauge generator. When (7.5) is satisfied, (7.4) will again promote C to a physical operator provided (2.11) or (2.12) is satisfied together with the consistency conditions [Q2,C],C]Iph) [[Q2,C],C],C]Iph)
[... [[Q2, C], Cl, C],..., C]iph)
=
0,
=
0,
=
0.
(7.6)
Alternatively we may impose Ciph)
=
0
(7.7)
provided eqs. (7.4), (7.5), (2.12) and (7.6) are satisfied. When we have several constraints B 1Iph) = 0,
B1
[Q,C],
(7.8)
then they are consistent to impose only if (2.17), (2.18) and [Q,B1]
=
c’B~+ cQ
(7.9)
are satisfied. In this case we may impose C,Iph)
=
0.
(7.10)
A more general possibility is that (2.17) and (7.9) are replaced by [Q,B1]
=
[B1,B~] =
c/B1
)~jfBk
+ d/C1 +
cQ,
+ K,j’Ck +~Q.
(7.11)
Thus, when we have several constraints BiIph) = 0 and C~Iph)= 0 then we require as in sect. 2 that Q, B1 and C, satisfy a closed algebra. However, its form is in general slightly different here If we consider the FP ghost fixing (5.3) in sect. 5, then we have apart from the consistency conditions given there the additional consistency conditions 2,P [Q 0]+1P11)= ifaap/’~IPh)= —if0pn~Iph)= 0, (7.12)
238
R. Marnelius / BRST quantization
where (5.3) has been used in the second equality. Obviously (7.12) can only be satisfied if f0p
=
0,
cs,/i
I,... ,k.
=
(7.13)
This condition also follows from the requirement that [Ba,Bp]Iph)
=
0,
(7.14)
where B,,, = [Q,P0 1+. Eq. (7.13) is the only additional condition required as compared to the ordinary case in sect. 5. (The complete analysis [3] requires = f,,,. = 0 in addition to UJ = fop = 0 where a, /1 = 1,... k and ,
In ref. [3] it was shown that the above generalized BRST quantization always requires the auxiliary condition ~Iph)
=
0,
(7.15)
where (7.16) where in turn =
NA
=
NB
=
NB
1
-
NA,
~
(17App17A)
(7.17)
0
(7.18)
A=k+ I
Since =
due to the conditions fop
=
f,~= 0, ~ is a gauge generator. Eq.
allows, therefore, for a condition of the form I~Iph)= ñlph),
(7.15)
(7.19)
where ñ is a number. In fact, the FP ghost fixing procedure in sect. 5 will at the end determine this number. The importance of (7.15) lies in the fact that QIph)
=
~Iph) = 0
(7.20)
R. Marnelius / BRST quantization
239
may be written as
oIph)
=
dlph)
=
0,
(7.21)
where d
[NB,Q]
(7.22)
Q2.
(7.23)
=
[NA,Q],
=
d~= 0, [ö,d]+
=
are nilpotent operators: =
It is a pleasure to thank Stephen Hwang for helpful discussions. Appendix A The algebra of auxiliary conditions considered in sect. 3 In sect. 3 we first considered the algebra of Q, Q, K, K and N (~N~)given by =
[Q,K]
=
~2
[Q,~]+=
=
Q, Q,
[Q,K]_ =
[N,Q] = [N,K]=-2K,
[N,Q] = [N,K]=2K,
[K,K]=N.
(A.l)
Then we considered q and c~defined by [Q,q]~= —iK, [Q,~]÷
=
iK,
(A.2)
which require [N,q]_
=
q, [N,cfl_= —ci.
(A.3)
We first found the realization q
=
p/aVa,
~
=
(A.4)
?~aVa
which satisfy q2
=
~2
=
[q,~]~
=
0,
[Q,c~]÷ = D—i~N, [Q,q]~
(A.5) =
—D—i~N,
(A.6)
R. Marnelius / BRST quantization
240
defining the operator D. Eq. (A.4) yields D
The complete algebra of Q, (A.3), (A.5)—(A.6) and
K, K, N, q,
~,
[D,Q]
=
i~Q,
[D,Q]_
=
[D,q]_
=
—i~q, [D,~]
=
=
-iK,
=
=
—~,
[D,K] [K,q]_
(A.7)
paVa + i~(p/aPa+ p~a~)
=
[D,K]
t~
and D is then given by eqs. (A. 1)—
i~Q, -iK,
[K,~]_= [K,qJ
[K,~~]_ = —q,
=
0
(A.8)
and constitutes an OSp(l, 112) algebra. Now the realization (A.4) of (A.2) is not unique. We may also choose q
=
p/ava +
a,
q
=
p/aVa +
a,
(A.9)
where [Q,a]~= [Q,ã]~
0, [N,a]
=
=
a, [N,ã]
=
—ã
(A.10)
without affecting (A.2) and (A.3). If we in addition require that the previous algebraic properties (A.6) should be retained, then we must have
[Q,a]~
=
—[Q,a]~,
(A.1l)
which is solved by a
=
[Q,r]_,a
=
[Q,r]_, [N,r]
=
0.
(A.12)
D is then not given by (A.7). Instead we have D-+D’
=
[Q, [Q,r]],
(A.13)
i~Q, [D’,~] = i~,
(A.14)
D + T, T
=
implying [D’,Q]
=
consistent with the first line in (A.8). However, in order to retain the whole previous OSp algebra r has to satisfy some additional conditions. From (A.8) we find that these conditions fix the algebra of r, a, a and T to be [r,a]_
=
[a,ã]+
=
[T,a]_
=
[r,ã]_
= [r,T] = 0, 2 = a2 = 0, a [T,ã]_= 0,
(A.15)
R. Marnelius / BRST quantization
241
where the last two lines follow from the first line. In addition we must have [r,K]
=
[r,K]_
=
0,
[r,q]
[r,~]_
=
=
0,
(A.16)
where q,7 may either be given by (A.5) or (A.9) due to (A.15). However, there is no condition on the commutator [r,D]. For the mixed commutators between r, a, a, T and the previous OSp generators we find then (the nonzero part)
[a,K]
=
—[r,D]_,
=
a,
[q,a]÷ [ã,K]_= a,
=
[r,D]_,
[a,D] = —i~a+[Q,[r,D]], [ã,D]_= —i~a+[Q,[r,D]], [q,T]_ = ia— [Q,[r,D]], [t~,T]_ = iã— [Q,[r,D]], [D,T]_ = —[Q, [Q,ir— [r,D]]],
(A.17)
which becomes a closed algebra if [r,D] can be expressed in terms of the already defined operators. Notice that if [r,D]
=
ir,
(A.18)
then T commutes with all operators. There are several choices of r which satisfy the above conditions: r = v2, [r,D] = 2ir, r = p/a, [r,D] = —ir.
(A.19)
However, these choices cannot make q, c~and D’ conserved. The interesting choice is instead (A.20)
r=X’~va,
which satisfies (A. 18). x a are gauge-fixing conditions to çua. If we define the matrix operators Ma’~’by [Wa,Xb]_
then the consistency conditions [r,a] =
0,
=
=
(A.21)
~Ma”,
[r, a]
=
0 on r require
[Xa,M~b] + [Xb,MCa]
=
0.
(A.22)
242
R. Marnelius / BRST quantization
This inserted into the Jacobi identities for [X”,Mb’i
Wa, X’~and
=
0.
X’~requires in turn (A.23)
Above we have made the natural assumption that Mab does not depend on the Lagrange multipliers. Eqs. (A.22) and (A.23) make it natural to let Ma” only depend on X’~. References [1] S. Hwang and R. Marnelius, Nucl. Phys. B315 (1989) 638; B320 (1989) 476 [2] R. Marnelius, Nucl. Phys. B294 (1987) 685; Generalized BRST quantization, Proc. mt. meeting on geometrical and algebraic aspects of nonlinear field theory, Amalfi, Italy, 1988, ed. S. De Filippo, M. Marinaro, G. Marmo and G. Vilasi (North-Holland, Amsterdam, 1989) [3] R. Marnelius, Nucl. Phys. B370 (1992) 165 [4] R. Marnelius, Phys. Lett. B99 (1981) 467 [5] G. Curci and R. Ferrari, Phys. Lett. B63 (1976) 91; I. Ojima, Prog. Theor. Phys. Lett. 64 (1980) 625 [6] S. Hwang, Nuci. Phys. B231 (1984) 386 [7] S. Hwang, Nucl. Phys. B322 (1989) 107 [8] I. A. Batalin and G. A. Vilkovisky, Phys. Lett. B69 (1977) 309 [9] H. Aratyn, R. Ingermanson and A. J. Niemi, Phys. Lett. Bl89 (1987) 427; NucI. Phys. B307 (1988) 157 [10] I.B. Frenkel, H. Garland and G. J. Zuckerman, Proc. NatI. Acad. Sci. 83 (1986) 8442 [11] M. Spiegelglas, Nuci. Phys. B283 (1987) 205 [12] A. V. Razumov and G. N. Rybkin, NucI. Phys. B332 (1990) 209 [13] W. Kalau and J. W. van Holten, Nucl. Phys. B36 1 (1991) 233 [14] J. Bognár, Indefinite inner product spaces (Springer-Verlag, Berlin, 1974)