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On BRST quantization of second class constraint algebras
Volume 213, number 1
PHYSICS LETTERS B
13 October 1988
O N BRST Q U A N T I Z A T I O N O F S E C O N D C L A S S C O N S T R A I N T A L G E B R A S ~" Antti J. N I E M I 1 CERN, CH-1211 Geneva 23, Switzerland
Received 2 June 1988
A BRST quantization of second-class constraint algebras that avoids Dirac brackets is constructed, and the BRST operator is shown to be related to the BRST operator of a first class algebra by a nonunitary canonical transformation. The transformation converts the second class algebra into an effective first class algebra with the help of an auxiliary second class algebra constructed from the dynamical Lagrange multipliers of the Dirac approach. The BRST invariant path integral for second class algebras is related to the path integral of the pertinent Dirac brackets, using the Parisi-Sourlas mechanism. As an application the possibility of string theories in subcritical dimensions is considered.
By now, the BRST quantization of first class constraint algebras can be viewed quite satisfactory [ 15 ]. Problems however do persist in the quantization of second class constraint algebras. The issues have been addressed in refs. [6,7] but the construction presented there is not completely satisfactory since Dirac brackets [8] are needed to eliminate degrees of freedom associated with the second class algebra. In certain applications, including the Green-Schwarz superstring [ 9 ] Dirac brackets can lead to nonlocal, singular a n d / o r noncovariant formulations. As a consequence it is desirable to develop a BRST quantization such that instead of Dirac brackets, the unphysical degrees of freedom specified by second class constraints are eliminated by Parisi-Sourlas mechanism [ 10,11 ] on a properly extended phase space, in analogy with first class constraints [2-5 ]. An approach to second class algebras that avoids the introduction of Dirac brackets has been proposed in refs. [ 12,13 ]. In this approach additional canonical degrees of freedom are introduced that convert the original second class algebra into an effective first class algebra. Quantization then follows the general prescription of refs. [ 2-5 ]. The approach is particuWork supported in part by the US Department of Energy Outstanding Junior Investigator Grant under contract No. DEAC02-76ER01545. On leave from Department of Physics, OSU, Columbus, OH 43210, USA. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
larly interesting as it generalizes the proposal of ref. [14] to quantize strings in subcritical dimensions with conformal anomaly cancelled by the Liouville field. Here we shall investigate aspects of the proposal presented in refs. [ 12,13 ], from the point of view o f ref. [ 5 ]. In particular we explain how the auxiliary fields that convert the second class algebra into a first class algebra emerge, and present a BRST formulation of the q u a n t u m path integral. We establish equivalence o f the BRST path integral with the path integral of the Dirac approach by the Parisi-Sourlas mechanism, and emphasize the phase space general covariance o f the quantization procedure. We consider an initial 2N dimensional phase space {F, ~} with g2 a closed and nondegenerate symplectic two-form on the manifold F [15]. In local (but not necessarily canonical) coordinates x i, i = 1.... ,2N; ~=~'~ijdxi
A dx j ,
Oiff~jk'~-Ojff~ki'JFOkff~tj=O ,
det Ig2ij [ S 0
(1)
and since d~2= 0 we can locally represent g2 by an exterior derivative of a one-form, g2= dO. This symplectic structure determines a natural isomorphism between one-forms and vectors: With f a n arbitrary observable i.e. a smooth function on F we define . . . . ~j=~,j,
(2) 4l
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where the vector field ~-is called the hamiltonian vector field offi A coordinate independent generalization of canonical Poisson brackets can then be introduced by identifying the bracket of two observables f a n d g with the Lie derivative o f g along the hamiltonian vector field o f f (or vice versa), {f, g} = ~ g = ~r ~0,g= g2~JOJOjg.
(3)
We are particularly interested in an irreducible second class constraint algebra consisting of 2 M ( M < N) linearly independent conditions ~ q/~=0,
a = l , . . . , 2M,
(4)
{~la,~gb}=~iJoi~,/aOj~/l'~cab
,
detlCablv~O.
(5)
(Notice that antisymmetry of C ~b implies that the number of ga's must be even.) Locally, (4) determines a 2 ( N - M ) dimensional submanifold of F which we call the physical subspace F*. The one-forms ~,~ua are normal to F* and characterize locally the complement of F* in F which defines the unphysical subspace i~. In order to enable the use of Darboux theorem we shall now argue that both F* and i~ are symplectic manifolds: For i~ the symplectic structure can be constructed by identifying the matrix inverse Cab of the central extension in (5) with components of the pertinent symplectic two-form ~ b : By assumption C "b is nondegenerate and comparing ( 1 ) with the Jacobi identity for (5), +
b, cco} +
c, c
i bc i i ab = ~r~,~0iC +~5v,~,OiCC~+~/'~,,aiC =0,
(6)
we conclude that Cab can be identified with components of a closed two-form on the submanifold i~, in the natural coordinates induced on i~ by the observables q/o. A symplectic structure on F* can be constructed as follows: We use the hamiltonian vector fields ~r~,, = g2~Oj~a to represent a given hamiltonian vector field ~-as a linear combination of ~ , the projection of ~ onto the tangent space ofF* and ~ , the projection of ~.onto the tangent space of the manifold i~:
=srr+
13 October 1988 r,
*-
'
°
~rT'f2ij ~c~-=0.
(7)
For consistency one can verify that Lie transport of t h e ~//a along the vector field 5r~ indeed vanishes, L~f~ua= ~rTi0AUa=0 ensuring that a canonical flow generated by ~r~, if initially on the physical submanifold F* also remains on F*. This determines the projection of observables in F onto observables in the physical subspace F*. In analogy with (3) it is natural to identify the Lie derivative of an observable g along the projected hamiltonian vector field 5r7 with the Poisson bracket of the projected observables on F*. This yields a coordinate independent generalization of the standard Dirac bracket o f f and g [ 8 ], and also determines the symplectic structure on F*: {f g}* = L~Tg= 5f7 ~0,g = {f,g}_ {f, ~a} Cab{{llb,g} = (ff2tJ--~QikOk!llaCab~Jlal~.,Ib)
OifOjg
- t2*i%fajg .
By construction O *0 has rank 2 ( N - M ) and from the Jacobi identity for Dirac brackets we conclude that its matrix inverse can be viewed as components of a symplectic two-form on F*, in coordinates induced on F* by the given coordinates on F. Since F can be locally viewed as a direct sum of F* and F, its symplectic structure can also be locally viewed as a direct sum of the symplectic structures on these submanifolds. Consequently the symplectic two-form on F admits locally the direct sum form t2=t2*+t~. The measure that determines the quantum theory on F* can be obtained using the 2 ( N - M ) form g2*A ... A g2* as a volume element. Using Darboux theorem to introduce local canonical coordinates p,, q* on F*, this yields the familiar Liouville measure (we discard an irrelevant normalization ) I-I g27", ^ ... A t'2}, - - , d / z ( F * ) ~ [ d p , d q * ] .
~t Here we take the constraints to have bosonic grading. Generalization to include constraints with fermionic grading is straightforward, affecting subsequent formulas only by grading factors. A generalization to reducible algebras can also be easily constructed. 42
(8)
t
(9)
By construction this measure is generally covariant. If instead of the local canonical cnordinates p,, q* on F* we evaluate it using the restriction of local canon-
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ical coordinates p~, qg of the original phase space F to 1-'* we get d p ( r * ) ~ [dp, dq*]
[dpgdq~] 6(qJ~).
(10)
From the transformation properties of a symplectic two-form under a general coordinate transformation x~--,.~i(x) on the phase space F, ~j~0=12kt
Ox~ OxJ
Oxk Oxt
(11)
we get x/det Is'2~J(x) I [dx] .
10xq
--' x/detlf2°(x(2))I clet[~s[ [02] = x / d e t l ~ J ( R ) I [cL?] ,
f [ d p , dq*] [dpo dq° ] 8(po +IT*) 8(q °-
t(r) )
x exp(if (p,0*+po0O)dz)
~ x/det I {~t-, ~,b} [ [dp, dqq 8 ( ~ ~) ~
13 October 1988
(12)
which establishes general covariance of the RHS measure in (10) and identifies the prefactor with the jacobian for a change of variables from ~//a to unphysical canonical momentum and coordinate variables on P. As a consequence the quantum theory can be represented using either of the equivalent formulations [ 1,16 ]
(14a)
= f [dpudqU]8(po+H) 8(q°-t(z))
× exp(i f (puOU)dz),
(14b)
where r now parametrizes the evolution instead of the original "time" t. The original dynamical theory can then be viewed as a second class constrained system in an extended phase space with additional canonical variables {Po,qO}= _ 1 and constraints ~,~ =P0 + H* = 0, ~2 = qO_ t = 0 that satisfy the Heisenberg algebra {q/~,g~}={g2,~'2}=0, {~u~,~u2}=-l, a second class algebra of the form (5). With no loss of generality we can then ignore the hamiltonian, by assuming that it has been included in the second class constraints (4), (5). Since the measure in (13a) is generally covariant and since integrals such as fP~O" in (13a) can be viewed as line integrals of one-forms 0 locally related to the pertinent symplectic two-forms by I2_- dO, we conclude that e.g. (13a) can be written in the manifestly generally covariant form
I[dp.dq*]exp(ifP.O*-H *) f [dp, dq*lexp(i f
(p,q*-H*))
(13a) ~f[IY2*^...^g2~*exp(iI0 ),
(15)
= f [dp~dq'] [dG]x/det[ {~,a, ~b}l
Xexp(ifp, gff-H-Gg/a).
(13b)
We observe that from the point of view of formal developments the hamiltonian in ( 13 ) is irrelevant and can be eliminated as follows: For a functional integral such as e.g. (13a) we introduce the equivalent form
where £2* denotes the symplectic two-form on the extended phase space of (14b), at fixed z. We shall now proceed to an alternative formulation of the quantum theory that avoids Dirac brackets. For this we use the Darboux theorem on the unphysical phase space i~ to introduce local canonical coordinates Ga, F a (notice that here a = 1, ..., M) such that ff2=CabdXaAdxb--.dGa^dFa. The constraint surface (4) is then locally characterized by Ga = F a= 0 and on this constraint surface the second class algebra (5) becomes the Heisenberg algebra
{Ga, Gb}={F~,Fb}=O, {Ga,Fb}=-~ b .
(16)
These variables Ga and F a can be viewed as unphys43
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ical canonical momentum and position variables of the phase space F. Following refs. [2-5] we supplement the phase space of G~, F ~by the 2MLagrange multipliers ~ ~ Z~, g~ of (13b) and by 4Mghost variables ~ , q~, ~ , ~ " . On this extended phase space we define a symplectic structure by postulating Poisson brackets (graded brackets for ghosts) {~r~,2b}={t/~, ~ } = { ~ , ~b}= -dg. In ref. [ 5] we have established that for a Heisenberg algebra such as (16) corresponding to a first class constraint algebra the following realization of Osp (1,112) ensures a Parisi-Sourlas cancellation between unphysical phase space variables, and equivalence of an extended phase space formulation with the reduced phase space formulation: R~IJ = q ~~p ~~- - ( - , ~ , 7~~. . ' *
foreacha
(17)
with (internal!) light cone basis defined by q ~ = ( F a, - n ~, ~/~, ~ ) , p,~ = (G~,2~, ~ , t]~) so that the symplectic structure can be summarized by {p~, q~b} = - - g ~ ~ d~. h In particular, the BRST operator for the functionals G~ is identified with R -O=q~G~ + fl~Tr~ .
(18)
Unfortunately, this operator cannot be directly applied to the present case since it only involves " h a l f " of the constraint functionals ~v~~ G~, F~: For a covariant BRST quantization of a second class algebra we need a BRST operator that involves all functionals ~v"~ Ga, F ~ in a symmetric fashion. The desired BRST operator can be constructed by implementing the following canonical transformation in ( 17 ): G.~
. ~1 ( G . + 2 ~ ) ,
).~
(Z.-G~),
F"-~ - ~1 rr"--* ~
1
(Fa-7~a), (19a) (Tr~+F ~)
(19b)
generated by the functional ½ ( G ~ F O + 2 ~ x ~ - 2 ~ F ~ + Gj r ~). Since canonical transformations preserve Poisson brackets the pertinent transformed generators ( 17 ) also define a representation ofOsp ( 1,1 [2 ), and in particular the canonically transformed BRST operator (18) is now symmetric in the generators of the second class constraint algebra (16):
R_O=
~
13 October 1988 1
.
1
rl ( Ga -}- ~ a ) -{- - ~
fl,~ ( F a -'l- 7[ a ) .
(20)
We observe that consistent with refs. [ 12,13], this nilpotent BRST operator differs from (18) in the sense that in addition to the generators Ga, F a of the second class algebra (16) we also have complementary generators 2~, g~ of the same algebra. In particular, the algebra of the generators G~+2a and F a + rr a in (20) is a first class algebra. Notice that in contrast to refs. [12,13] here the auxiliary variables 2a, 7ra that convert the original second class algebra into a first class algebra are not introduced ad hoc. Instead, from ref. [5] we conclude that these variables are the dynamical Lagrange multipliers ~a that enforce the original second class constraints in (13b). In order to reconstruct the BRST operator in terms of the original constraint functionals ~,~ it is necessary to invert the change of variables ~ - ~ G~, F ~. For this we recall [4, 5] that if the original constraint algebra (5) is also realized using a coordinate system consisting of canonical momentum and position variables, we may try to realize the inverse change of variables Ga, F ~ - ~ a as a canonical transformation on the extended phase space: Using nilpotency of the ghost variables we conclude that the generating functional of this canonical transformation can be expanded in powers of the ghost variables and for this we find it convenient to combine G~, F~-~ T a (notice that now a = 1, ..., 2 M a s in (4)), and 2~, 7r~-~ ~ and tl~, fl,-'q~ and ~ , , ~ a - o ~ ~ so that [ T ~ , T b ] = - [~", ~'] =co "~, with oY' the central extension (i.e. symplectic two-form) of the Heisenberg algebra ( 16 ) in the new variables. The BRST operator then becomes (modulo x ~ ) R - ° = t / ~ ( T a + ~ ~) .
(21)
Following refs. [4,5 ] we first assume that the original functions (4) differ from T ~ only infinitesimally, ~a=Ta+~_gTt'+O(~.2), and try to relate (21) to a BRST operator involving the original ~ using a unitary canonical transformation, R - °--. e - ~ R -°e~ R
{R -o, q)} + ~ { { R -o, ¢}, q~} + ....
(22) 44
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With • = ?]a1~~ ~ b and to first order in Eab this yields R-O--~ *
1,]a( Ta-.b ~.~ Z b . . l - ~ a . . I - ~
+ ½ ? l a r l b # C [ { T a , Ebc}+{eac, =
b)
T~}]+ ...
q,,(~ua+pa)+½~l,,~lbU'~bg~+
...
f [dpi...d~a]exp(i
vf=Ta+~.~T b , p ~ = ~ a + E ~ b ,
(24a)
{~a, vb}=cab,
(24b)
{~u",pb}+{p a, ~b}=C~b-D~b= U'/b~ ~ ,
(24C)
cab= [{T ",e~}+{e~, Tb}]gt ~ ..~ o.)ab.+ E~tn a cb -e~o9 b ca +
....
(24d)
Notice in particular, that on the constraint surface v a = 0 we have cab=D ab. On the constraint surface the RHS in (24b) then differ only by a sign while the RHS of (24c) vanishes. From the previous ansatz we conclude that for a general constraint algebra (5) the BRST operator admits an expansion in the powers of the ghost variables which has the functional form
1
abe d e -----~-.qaqbq~Ud~ # ~ + ... 3~2~
(25)
with pa functionals of the dynamical Lagrange multipliers and original phase space variables. These functionals, as well as the higher order structure functions ~d~ r rabcetc. can be determined order-by-order in powers of ghost variables, by requiring nilpotency of (25). For the p~ this gives
{V%pb}+{p%vb}+{p~',pb}=V~bp~--Cab,
f pW+noA a
+ qa~a-{~,R-°}),
(27)
with R - o the BRST operator (25), and we have used the same notation as e.g. in (23), (24). ~Vis an arbitrary ghost number ( - 1 ) functional of the extended phase space variables, and the measure in (27) is the Liouville measure, determined by the exterior 2N form on the extended phase space, ~ x t ^ ... ^ ~2~xt~dpl A ... A d ~ M
(26)
which determines pa up to natural arbitrariness [4]. The conditions derived in this way for pa, tTabc L.,,de e t c . are isomorphic to those presented in ref. [ 13 ]. The only difference is that now the subsidiary functionals p~ are constructed using the dynamical Lagrange multipliers ~a, not from variables introduced ad hoc
(28)
as a volume element for each z. This measure is manifestly generally covariant. Furthermore [2-5 ], since { ~v, R-o} in (27) can be identified with the jacobian corresponding to a noncanonical change of variables of the form
z'~z'~+{z'~,R-°}~,
R - ° = qa ( lp'a -b p a ) -l- l rla rh~ U ab ,~ c .~_
in addition to these Lagrange multipliers ,2 In order to construct an Osp ( 1,1 [2) invariant path integral that reproduces the reduced phase space path integral (13) by the Parisi-Souflas mechanism, we consider the path integral ,3
(23)
where, to first order in e~, we have identified
{pa,pb}=__Dab,
13 October 1988
(29)
with z" canonical coordinates on the extended phase space, and since the remaining z integrals in (27) can be viewed as a line integral of a one-form O such that ~2~xt= dO,
f o,
(30)
we conclude that (27) can be written in the manifestly generally covariant form f ]-] • g2ex' ^ .-. ^ g2~xte x p ( i f O ) .
(31)
~2 It is also possible to convert the original second class algebra into an effective first class algebra using auxiliary variables different from the dynamical Lagrange multipliers as in ref. [ 13 ]. However, care should then be taken to include proper ghost variables that ensure the cancellation of all excess variables by the Parisi-Souflas mechanism [ 10,11 ], as explained in ref. [ 5 ]. ~3 Notice that following the reasoning of ref. [ 14], we do not include any explicit hamiltonian here. As a consequence issues associated with the construction of a BRST invariant hamiltonian [4,13] do not arise in the present construction.
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In particular, since different T ' s correspond to different coordinate systems related by (29), the 5u independence of (27), ( 31 ) is manifest (but subject to Osp (1,112) invariant boundary conditions as in refs. [2-5 ] ), generalizing the familiar Tindependence of first class systems [2-5 ] to the present case. We now argue that as in refs. [2-5 ], by selecting properly (27), (31 ) reproduces the reduced phase space path integral (13): For this we use general covariance to evaluate (27), (31 ) in canonical coordinates where the BRST operator becomes (18). (Recall that this form is canonically equivalent to (25) [4,5].) If we then perform a change of variables of the form (29) with T=fl~F~+~,~a the independence of (27), ( 31 ) ensures fl independence of (27), (31), and in the fl~ov limit we get
f [dp~dqqS(G~)~(F ~) ×detl{G~,Fb}lexp(i f p~t~).
(32)
Since det] { G~, F b} I = d x / ~ coat' l, the equivalence of (27), (31 ) (with proper boundary conditions) and ( 13 ), ( 15 ) then follows from (10), (12) and the Parisi-Sourlas mechanism: In particular, the relation between the RHS integral in (15) and (31 ) can be understood as a direct consequence of the Osp (1,112) invariance of (31 ) and the Parisi-Sourlas relation
f dDxdOdOF(x2+O0)= f dO-2xF(x2).
(33)
Since the F a are now viewed as gauge fixing conditions we discard them, and concentrate only on the gauge invariant part of the BRST operator (20): R-°=tla(G~ +J-a). The expansion (25) that implements the inverse change of variables Ga~Ln then yields
R-°=rl"(L,,+A~)+½(n-rn)rlnrl'~n+,~,
46
(34)
where A,, are generators constructed from the Lagrange multipliers, and subject to the condition (26) with U~b the structure constants of the Virasoro algebra. But since the classical Virasoro algebra is a first class algebra, in the present case the central extension C a~'in (26) is absent. A nontrivial C "h arises in the quantum theory where the Virasoro algebra acquires the familiar central extension, i.e. determines a second class algebra. Demanding nilpotency of the quantum BRST operator then implies thatA, must satisfy (26) with a nontrivial Cab,
[Ln,Am] + [An, Lm] + [A~,Am] = (n-m)A,,+m+C,, .... C~m=-~(D-26)n3~(n+m)
(35a)
+ ~(D-2-24Ceo)n~(n+m),
(35b)
where we have included the ghost contribution to the conformal anomaly. Consider the following solution of (35):
A~ = ( n - k ) In ref. [ 12 ] it was suggested that anomalous first class algebras could be quantized as second class algebras. As an application of the present formalism we shall now investigate this, by considering the possibility of constructing nilpotent anticommuting BRST operators for bosonic strings in D # 26 without introducing the Liouville field as in ref. [ 14]. For this we recall that a gauge fixed first class constraint algebra determines a second class algebra. The present formalism then applies provided we declare half of the generators T ~ (for example F ~) as gauge fixing conditions. The remaining generators T" (i.e. Ga) are then the original first class gauge generators, expressed in local canonical coordinates of the unphysical phase space. In particular, for strings we relate G~ to the Virasoro generators Ln when expressed in such local canonical coordinates.
13 October 1988
ZCk2k_~+
I.t(n+ A ) ZCk).k+n ,
(36)
with /z and A c-numbers. Realizing the pertinent quantum operators in a Hilbert space based on a Fock vacuum definedby 2~/0) = 0 for n~<0 and # ' 1 0 ) = 0 for n > 0 we get
[A,,A,,] = (n-m)A~+m
+~[ ( 1 - 6 / t + 6 # 2) n 3
- (1-6lzA+6#2A2)n]6(n+m).
(37)
For 6/z2- 6B+ 1 = 1 3 - ½D, 6#2A 2 - 6p.A + 1 = 1 + 123o - ½D
(38)
the BRST operator is then nilpotent. The familiar D = 2 6 , c~o= 1 emerges for ~t= - 1/2_+ 1/,,/12, A = 1, but since the canonical transformation (19) cannot be realized as a proper unitary transformation in the
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q u a n t u m Hilbert space, these D = 26 strings are n o t unitarily equivalent to the conventional D = 26 string. This can be viewed as a manifestation of the nontrivial polarization dependence in q u a n t i z a t i o n [ 5,15 ]. The p a r a m e t e r / t can be related to the conformal weigths of the fields gn, 2n. Consequently (34) can be viewed as the BRST operator of a string theory augmented by a world sheet conformal field theory. Since canonical transformations preserve Poisson brackets, we can introduce further canonical transformations to construct new realizations of A, a n d it is reasonable to expect that this exhausts all possible conformal field theories based on variables n n, 2n. Since it is generally assumed that a u g m e n t i n g a string theory with different world sheet conformal field theories can be viewed as different string compactifications, one can then argue that different compactifications of the string are classically canonically equivalent but q u a n t u m mechanically unitarily inequivalent. However, the validity of this conjecture will not be investigated here. In conclusion, we have explained how the BRST q u a n t i z a t i o n of a second class algebra can be implemented. We have constructed the pertinent BRST invariant q u a n t u m path integral and established its equivalence with the reduced phase space path integral by the Parisi-Sourlas mechanism. Our approach avoids Dirac brackets a n d is closely related to that in refs. [12,13]. The only difference is that now the auxiliary fields that convert the second class algebra into an effective first class algebra are identified with the dynamical Lagrange multipliers of the reduced phase space theory. As an application we have considered the possibility to construct nilpotent BRST
13 October 1988
operators for anomalous theories, in particular bosonic string theories in subcritical dimensions.
References [ 1] L.D. Faddeev, Theor. Math. Phys. 1 (1969) 1. [2] E.S. Fradkin and G.A. Vilkovisky,Phys. Lett. B 55 (1975) 224; CERN Report TH-2332 (1977). [3] I.A. Batalin and G.A. Vilkovisky,Phys. Lett. B 69 (1977) 309. [4] M. Henneaux, Phys. Rep. 126 (1985) 1; M. Henneauxand C. Teitelboim,Commun. Math. Phys. 115 (1988) 213. [5] H. Aratyn, R. Ingermanson and A.J. Niemi, Phys. Lett. B 189 (1987) 427; A.J. Niemi, OSU report DOE-ER-01545-410. [6] E.S. Fradkin and T.E. Fradkina, Phys. Lett. B 72 (1978) 343. [7] I.A. Batalin and E.S. Fradkin, Phys. Lett. B 122 (1983) 157. [8] P.A.M. Dirac, Lectures on quantum mechanics (Yeshiva University), (Academic Press, New York, 1964). [9] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge U.P., Cambridge, 1986); E. Nissimov, S. Pacheva and S. Solomon,Nucl. Phys. B 297 (1988) 349. [ 10] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 (1979) 744. [ 11 ] B. McClain, A.J. Niemi, C. Taylor and L.C.R. Wijewardhana, Phys. Rev. Len. 49 (1982) 252; Nucl. Phys. B 217 (1983) 430; A. Kupiainenand A.J. Niemi, Phys. Lett. B 130 (1983) 380. [ 12] L.D. Faddeevand S.L. Shatashvili,Phys. Lett. B 167 ( 1986) 225. [13] I.A. Batalin and E.S. Fradkin, Nucl. Phys. B 279 (1987) 514. [ 14] A.M. Polyakov,Phys. Lett. B 103 ( 1981 ) 207. [ 15 ] N. Woodhouse,Geometric quantization (Clarendon Press, Oxford, 1980). [ 16 ] P. Senjanovic, Ann. Phys. 100 (1976) 227.
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13 October 1988
O N BRST Q U A N T I Z A T I O N O F S E C O N D C L A S S C O N S T R A I N T A L G E B R A S ~" Antti J. N I E M I 1 CERN, CH-1211 Geneva 23, Switzerland
Received 2 June 1988
A BRST quantization of second-class constraint algebras that avoids Dirac brackets is constructed, and the BRST operator is shown to be related to the BRST operator of a first class algebra by a nonunitary canonical transformation. The transformation converts the second class algebra into an effective first class algebra with the help of an auxiliary second class algebra constructed from the dynamical Lagrange multipliers of the Dirac approach. The BRST invariant path integral for second class algebras is related to the path integral of the pertinent Dirac brackets, using the Parisi-Sourlas mechanism. As an application the possibility of string theories in subcritical dimensions is considered.
By now, the BRST quantization of first class constraint algebras can be viewed quite satisfactory [ 15 ]. Problems however do persist in the quantization of second class constraint algebras. The issues have been addressed in refs. [6,7] but the construction presented there is not completely satisfactory since Dirac brackets [8] are needed to eliminate degrees of freedom associated with the second class algebra. In certain applications, including the Green-Schwarz superstring [ 9 ] Dirac brackets can lead to nonlocal, singular a n d / o r noncovariant formulations. As a consequence it is desirable to develop a BRST quantization such that instead of Dirac brackets, the unphysical degrees of freedom specified by second class constraints are eliminated by Parisi-Sourlas mechanism [ 10,11 ] on a properly extended phase space, in analogy with first class constraints [2-5 ]. An approach to second class algebras that avoids the introduction of Dirac brackets has been proposed in refs. [ 12,13 ]. In this approach additional canonical degrees of freedom are introduced that convert the original second class algebra into an effective first class algebra. Quantization then follows the general prescription of refs. [ 2-5 ]. The approach is particuWork supported in part by the US Department of Energy Outstanding Junior Investigator Grant under contract No. DEAC02-76ER01545. On leave from Department of Physics, OSU, Columbus, OH 43210, USA. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
larly interesting as it generalizes the proposal of ref. [14] to quantize strings in subcritical dimensions with conformal anomaly cancelled by the Liouville field. Here we shall investigate aspects of the proposal presented in refs. [ 12,13 ], from the point of view o f ref. [ 5 ]. In particular we explain how the auxiliary fields that convert the second class algebra into a first class algebra emerge, and present a BRST formulation of the q u a n t u m path integral. We establish equivalence o f the BRST path integral with the path integral of the Dirac approach by the Parisi-Sourlas mechanism, and emphasize the phase space general covariance o f the quantization procedure. We consider an initial 2N dimensional phase space {F, ~} with g2 a closed and nondegenerate symplectic two-form on the manifold F [15]. In local (but not necessarily canonical) coordinates x i, i = 1.... ,2N; ~=~'~ijdxi
A dx j ,
Oiff~jk'~-Ojff~ki'JFOkff~tj=O ,
det Ig2ij [ S 0
(1)
and since d~2= 0 we can locally represent g2 by an exterior derivative of a one-form, g2= dO. This symplectic structure determines a natural isomorphism between one-forms and vectors: With f a n arbitrary observable i.e. a smooth function on F we define . . . . ~j=~,j,
(2) 4l
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where the vector field ~-is called the hamiltonian vector field offi A coordinate independent generalization of canonical Poisson brackets can then be introduced by identifying the bracket of two observables f a n d g with the Lie derivative o f g along the hamiltonian vector field o f f (or vice versa), {f, g} = ~ g = ~r ~0,g= g2~JOJOjg.
(3)
We are particularly interested in an irreducible second class constraint algebra consisting of 2 M ( M < N) linearly independent conditions ~ q/~=0,
a = l , . . . , 2M,
(4)
{~la,~gb}=~iJoi~,/aOj~/l'~cab
,
detlCablv~O.
(5)
(Notice that antisymmetry of C ~b implies that the number of ga's must be even.) Locally, (4) determines a 2 ( N - M ) dimensional submanifold of F which we call the physical subspace F*. The one-forms ~,~ua are normal to F* and characterize locally the complement of F* in F which defines the unphysical subspace i~. In order to enable the use of Darboux theorem we shall now argue that both F* and i~ are symplectic manifolds: For i~ the symplectic structure can be constructed by identifying the matrix inverse Cab of the central extension in (5) with components of the pertinent symplectic two-form ~ b : By assumption C "b is nondegenerate and comparing ( 1 ) with the Jacobi identity for (5), +
b, cco} +
c, c
i bc i i ab = ~r~,~0iC +~5v,~,OiCC~+~/'~,,aiC =0,
(6)
we conclude that Cab can be identified with components of a closed two-form on the submanifold i~, in the natural coordinates induced on i~ by the observables q/o. A symplectic structure on F* can be constructed as follows: We use the hamiltonian vector fields ~r~,, = g2~Oj~a to represent a given hamiltonian vector field ~-as a linear combination of ~ , the projection of ~ onto the tangent space ofF* and ~ , the projection of ~.onto the tangent space of the manifold i~:
=srr+
13 October 1988 r,
*-
'
°
~rT'f2ij ~c~-=0.
(7)
For consistency one can verify that Lie transport of t h e ~//a along the vector field 5r~ indeed vanishes, L~f~ua= ~rTi0AUa=0 ensuring that a canonical flow generated by ~r~, if initially on the physical submanifold F* also remains on F*. This determines the projection of observables in F onto observables in the physical subspace F*. In analogy with (3) it is natural to identify the Lie derivative of an observable g along the projected hamiltonian vector field 5r7 with the Poisson bracket of the projected observables on F*. This yields a coordinate independent generalization of the standard Dirac bracket o f f and g [ 8 ], and also determines the symplectic structure on F*: {f g}* = L~Tg= 5f7 ~0,g = {f,g}_ {f, ~a} Cab{{llb,g} = (ff2tJ--~QikOk!llaCab~Jlal~.,Ib)
OifOjg
- t2*i%fajg .
By construction O *0 has rank 2 ( N - M ) and from the Jacobi identity for Dirac brackets we conclude that its matrix inverse can be viewed as components of a symplectic two-form on F*, in coordinates induced on F* by the given coordinates on F. Since F can be locally viewed as a direct sum of F* and F, its symplectic structure can also be locally viewed as a direct sum of the symplectic structures on these submanifolds. Consequently the symplectic two-form on F admits locally the direct sum form t2=t2*+t~. The measure that determines the quantum theory on F* can be obtained using the 2 ( N - M ) form g2*A ... A g2* as a volume element. Using Darboux theorem to introduce local canonical coordinates p,, q* on F*, this yields the familiar Liouville measure (we discard an irrelevant normalization ) I-I g27", ^ ... A t'2}, - - , d / z ( F * ) ~ [ d p , d q * ] .
~t Here we take the constraints to have bosonic grading. Generalization to include constraints with fermionic grading is straightforward, affecting subsequent formulas only by grading factors. A generalization to reducible algebras can also be easily constructed. 42
(8)
t
(9)
By construction this measure is generally covariant. If instead of the local canonical cnordinates p,, q* on F* we evaluate it using the restriction of local canon-
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ical coordinates p~, qg of the original phase space F to 1-'* we get d p ( r * ) ~ [dp, dq*]
[dpgdq~] 6(qJ~).
(10)
From the transformation properties of a symplectic two-form under a general coordinate transformation x~--,.~i(x) on the phase space F, ~j~0=12kt
Ox~ OxJ
Oxk Oxt
(11)
we get x/det Is'2~J(x) I [dx] .
10xq
--' x/detlf2°(x(2))I clet[~s[ [02] = x / d e t l ~ J ( R ) I [cL?] ,
f [ d p , dq*] [dpo dq° ] 8(po +IT*) 8(q °-
t(r) )
x exp(if (p,0*+po0O)dz)
~ x/det I {~t-, ~,b} [ [dp, dqq 8 ( ~ ~) ~
13 October 1988
(12)
which establishes general covariance of the RHS measure in (10) and identifies the prefactor with the jacobian for a change of variables from ~//a to unphysical canonical momentum and coordinate variables on P. As a consequence the quantum theory can be represented using either of the equivalent formulations [ 1,16 ]
(14a)
= f [dpudqU]8(po+H) 8(q°-t(z))
× exp(i f (puOU)dz),
(14b)
where r now parametrizes the evolution instead of the original "time" t. The original dynamical theory can then be viewed as a second class constrained system in an extended phase space with additional canonical variables {Po,qO}= _ 1 and constraints ~,~ =P0 + H* = 0, ~2 = qO_ t = 0 that satisfy the Heisenberg algebra {q/~,g~}={g2,~'2}=0, {~u~,~u2}=-l, a second class algebra of the form (5). With no loss of generality we can then ignore the hamiltonian, by assuming that it has been included in the second class constraints (4), (5). Since the measure in (13a) is generally covariant and since integrals such as fP~O" in (13a) can be viewed as line integrals of one-forms 0 locally related to the pertinent symplectic two-forms by I2_- dO, we conclude that e.g. (13a) can be written in the manifestly generally covariant form
I[dp.dq*]exp(ifP.O*-H *) f [dp, dq*lexp(i f
(p,q*-H*))
(13a) ~f[IY2*^...^g2~*exp(iI0 ),
(15)
= f [dp~dq'] [dG]x/det[ {~,a, ~b}l
Xexp(ifp, gff-H-Gg/a).
(13b)
We observe that from the point of view of formal developments the hamiltonian in ( 13 ) is irrelevant and can be eliminated as follows: For a functional integral such as e.g. (13a) we introduce the equivalent form
where £2* denotes the symplectic two-form on the extended phase space of (14b), at fixed z. We shall now proceed to an alternative formulation of the quantum theory that avoids Dirac brackets. For this we use the Darboux theorem on the unphysical phase space i~ to introduce local canonical coordinates Ga, F a (notice that here a = 1, ..., M) such that ff2=CabdXaAdxb--.dGa^dFa. The constraint surface (4) is then locally characterized by Ga = F a= 0 and on this constraint surface the second class algebra (5) becomes the Heisenberg algebra
{Ga, Gb}={F~,Fb}=O, {Ga,Fb}=-~ b .
(16)
These variables Ga and F a can be viewed as unphys43
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ical canonical momentum and position variables of the phase space F. Following refs. [2-5] we supplement the phase space of G~, F ~by the 2MLagrange multipliers ~ ~ Z~, g~ of (13b) and by 4Mghost variables ~ , q~, ~ , ~ " . On this extended phase space we define a symplectic structure by postulating Poisson brackets (graded brackets for ghosts) {~r~,2b}={t/~, ~ } = { ~ , ~b}= -dg. In ref. [ 5] we have established that for a Heisenberg algebra such as (16) corresponding to a first class constraint algebra the following realization of Osp (1,112) ensures a Parisi-Sourlas cancellation between unphysical phase space variables, and equivalence of an extended phase space formulation with the reduced phase space formulation: R~IJ = q ~~p ~~- - ( - , ~ , 7~~. . ' *
foreacha
(17)
with (internal!) light cone basis defined by q ~ = ( F a, - n ~, ~/~, ~ ) , p,~ = (G~,2~, ~ , t]~) so that the symplectic structure can be summarized by {p~, q~b} = - - g ~ ~ d~. h In particular, the BRST operator for the functionals G~ is identified with R -O=q~G~ + fl~Tr~ .
(18)
Unfortunately, this operator cannot be directly applied to the present case since it only involves " h a l f " of the constraint functionals ~v~~ G~, F~: For a covariant BRST quantization of a second class algebra we need a BRST operator that involves all functionals ~v"~ Ga, F ~ in a symmetric fashion. The desired BRST operator can be constructed by implementing the following canonical transformation in ( 17 ): G.~
. ~1 ( G . + 2 ~ ) ,
).~
(Z.-G~),
F"-~ - ~1 rr"--* ~
1
(Fa-7~a), (19a) (Tr~+F ~)
(19b)
generated by the functional ½ ( G ~ F O + 2 ~ x ~ - 2 ~ F ~ + Gj r ~). Since canonical transformations preserve Poisson brackets the pertinent transformed generators ( 17 ) also define a representation ofOsp ( 1,1 [2 ), and in particular the canonically transformed BRST operator (18) is now symmetric in the generators of the second class constraint algebra (16):
R_O=
~
13 October 1988 1
.
1
rl ( Ga -}- ~ a ) -{- - ~
fl,~ ( F a -'l- 7[ a ) .
(20)
We observe that consistent with refs. [ 12,13], this nilpotent BRST operator differs from (18) in the sense that in addition to the generators Ga, F a of the second class algebra (16) we also have complementary generators 2~, g~ of the same algebra. In particular, the algebra of the generators G~+2a and F a + rr a in (20) is a first class algebra. Notice that in contrast to refs. [12,13] here the auxiliary variables 2a, 7ra that convert the original second class algebra into a first class algebra are not introduced ad hoc. Instead, from ref. [5] we conclude that these variables are the dynamical Lagrange multipliers ~a that enforce the original second class constraints in (13b). In order to reconstruct the BRST operator in terms of the original constraint functionals ~,~ it is necessary to invert the change of variables ~ - ~ G~, F ~. For this we recall [4, 5] that if the original constraint algebra (5) is also realized using a coordinate system consisting of canonical momentum and position variables, we may try to realize the inverse change of variables Ga, F ~ - ~ a as a canonical transformation on the extended phase space: Using nilpotency of the ghost variables we conclude that the generating functional of this canonical transformation can be expanded in powers of the ghost variables and for this we find it convenient to combine G~, F~-~ T a (notice that now a = 1, ..., 2 M a s in (4)), and 2~, 7r~-~ ~ and tl~, fl,-'q~ and ~ , , ~ a - o ~ ~ so that [ T ~ , T b ] = - [~", ~'] =co "~, with oY' the central extension (i.e. symplectic two-form) of the Heisenberg algebra ( 16 ) in the new variables. The BRST operator then becomes (modulo x ~ ) R - ° = t / ~ ( T a + ~ ~) .
(21)
Following refs. [4,5 ] we first assume that the original functions (4) differ from T ~ only infinitesimally, ~a=Ta+~_gTt'+O(~.2), and try to relate (21) to a BRST operator involving the original ~ using a unitary canonical transformation, R - °--. e - ~ R -°e~ R
{R -o, q)} + ~ { { R -o, ¢}, q~} + ....
(22) 44
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With • = ?]a1~~ ~ b and to first order in Eab this yields R-O--~ *
1,]a( Ta-.b ~.~ Z b . . l - ~ a . . I - ~
+ ½ ? l a r l b # C [ { T a , Ebc}+{eac, =
b)
T~}]+ ...
q,,(~ua+pa)+½~l,,~lbU'~bg~+
...
f [dpi...d~a]exp(i
vf=Ta+~.~T b , p ~ = ~ a + E ~ b ,
(24a)
{~a, vb}=cab,
(24b)
{~u",pb}+{p a, ~b}=C~b-D~b= U'/b~ ~ ,
(24C)
cab= [{T ",e~}+{e~, Tb}]gt ~ ..~ o.)ab.+ E~tn a cb -e~o9 b ca +
....
(24d)
Notice in particular, that on the constraint surface v a = 0 we have cab=D ab. On the constraint surface the RHS in (24b) then differ only by a sign while the RHS of (24c) vanishes. From the previous ansatz we conclude that for a general constraint algebra (5) the BRST operator admits an expansion in the powers of the ghost variables which has the functional form
1
abe d e -----~-.qaqbq~Ud~ # ~ + ... 3~2~
(25)
with pa functionals of the dynamical Lagrange multipliers and original phase space variables. These functionals, as well as the higher order structure functions ~d~ r rabcetc. can be determined order-by-order in powers of ghost variables, by requiring nilpotency of (25). For the p~ this gives
{V%pb}+{p%vb}+{p~',pb}=V~bp~--Cab,
f pW+noA a
+ qa~a-{~,R-°}),
(27)
with R - o the BRST operator (25), and we have used the same notation as e.g. in (23), (24). ~Vis an arbitrary ghost number ( - 1 ) functional of the extended phase space variables, and the measure in (27) is the Liouville measure, determined by the exterior 2N form on the extended phase space, ~ x t ^ ... ^ ~2~xt~dpl A ... A d ~ M
(26)
which determines pa up to natural arbitrariness [4]. The conditions derived in this way for pa, tTabc L.,,de e t c . are isomorphic to those presented in ref. [ 13 ]. The only difference is that now the subsidiary functionals p~ are constructed using the dynamical Lagrange multipliers ~a, not from variables introduced ad hoc
(28)
as a volume element for each z. This measure is manifestly generally covariant. Furthermore [2-5 ], since { ~v, R-o} in (27) can be identified with the jacobian corresponding to a noncanonical change of variables of the form
z'~z'~+{z'~,R-°}~,
R - ° = qa ( lp'a -b p a ) -l- l rla rh~ U ab ,~ c .~_
in addition to these Lagrange multipliers ,2 In order to construct an Osp ( 1,1 [2) invariant path integral that reproduces the reduced phase space path integral (13) by the Parisi-Souflas mechanism, we consider the path integral ,3
(23)
where, to first order in e~, we have identified
{pa,pb}=__Dab,
13 October 1988
(29)
with z" canonical coordinates on the extended phase space, and since the remaining z integrals in (27) can be viewed as a line integral of a one-form O such that ~2~xt= dO,
f o,
(30)
we conclude that (27) can be written in the manifestly generally covariant form f ]-] • g2ex' ^ .-. ^ g2~xte x p ( i f O ) .
(31)
~2 It is also possible to convert the original second class algebra into an effective first class algebra using auxiliary variables different from the dynamical Lagrange multipliers as in ref. [ 13 ]. However, care should then be taken to include proper ghost variables that ensure the cancellation of all excess variables by the Parisi-Souflas mechanism [ 10,11 ], as explained in ref. [ 5 ]. ~3 Notice that following the reasoning of ref. [ 14], we do not include any explicit hamiltonian here. As a consequence issues associated with the construction of a BRST invariant hamiltonian [4,13] do not arise in the present construction.
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In particular, since different T ' s correspond to different coordinate systems related by (29), the 5u independence of (27), ( 31 ) is manifest (but subject to Osp (1,112) invariant boundary conditions as in refs. [2-5 ] ), generalizing the familiar Tindependence of first class systems [2-5 ] to the present case. We now argue that as in refs. [2-5 ], by selecting properly (27), (31 ) reproduces the reduced phase space path integral (13): For this we use general covariance to evaluate (27), (31 ) in canonical coordinates where the BRST operator becomes (18). (Recall that this form is canonically equivalent to (25) [4,5].) If we then perform a change of variables of the form (29) with T=fl~F~+~,~a the independence of (27), ( 31 ) ensures fl independence of (27), (31), and in the fl~ov limit we get
f [dp~dqqS(G~)~(F ~) ×detl{G~,Fb}lexp(i f p~t~).
(32)
Since det] { G~, F b} I = d x / ~ coat' l, the equivalence of (27), (31 ) (with proper boundary conditions) and ( 13 ), ( 15 ) then follows from (10), (12) and the Parisi-Sourlas mechanism: In particular, the relation between the RHS integral in (15) and (31 ) can be understood as a direct consequence of the Osp (1,112) invariance of (31 ) and the Parisi-Sourlas relation
f dDxdOdOF(x2+O0)= f dO-2xF(x2).
(33)
Since the F a are now viewed as gauge fixing conditions we discard them, and concentrate only on the gauge invariant part of the BRST operator (20): R-°=tla(G~ +J-a). The expansion (25) that implements the inverse change of variables Ga~Ln then yields
R-°=rl"(L,,+A~)+½(n-rn)rlnrl'~n+,~,
46
(34)
where A,, are generators constructed from the Lagrange multipliers, and subject to the condition (26) with U~b the structure constants of the Virasoro algebra. But since the classical Virasoro algebra is a first class algebra, in the present case the central extension C a~'in (26) is absent. A nontrivial C "h arises in the quantum theory where the Virasoro algebra acquires the familiar central extension, i.e. determines a second class algebra. Demanding nilpotency of the quantum BRST operator then implies thatA, must satisfy (26) with a nontrivial Cab,
[Ln,Am] + [An, Lm] + [A~,Am] = (n-m)A,,+m+C,, .... C~m=-~(D-26)n3~(n+m)
(35a)
+ ~(D-2-24Ceo)n~(n+m),
(35b)
where we have included the ghost contribution to the conformal anomaly. Consider the following solution of (35):
A~ = ( n - k ) In ref. [ 12 ] it was suggested that anomalous first class algebras could be quantized as second class algebras. As an application of the present formalism we shall now investigate this, by considering the possibility of constructing nilpotent anticommuting BRST operators for bosonic strings in D # 26 without introducing the Liouville field as in ref. [ 14]. For this we recall that a gauge fixed first class constraint algebra determines a second class algebra. The present formalism then applies provided we declare half of the generators T ~ (for example F ~) as gauge fixing conditions. The remaining generators T" (i.e. Ga) are then the original first class gauge generators, expressed in local canonical coordinates of the unphysical phase space. In particular, for strings we relate G~ to the Virasoro generators Ln when expressed in such local canonical coordinates.
13 October 1988
ZCk2k_~+
I.t(n+ A ) ZCk).k+n ,
(36)
with /z and A c-numbers. Realizing the pertinent quantum operators in a Hilbert space based on a Fock vacuum definedby 2~/0) = 0 for n~<0 and # ' 1 0 ) = 0 for n > 0 we get
[A,,A,,] = (n-m)A~+m
+~[ ( 1 - 6 / t + 6 # 2) n 3
- (1-6lzA+6#2A2)n]6(n+m).
(37)
For 6/z2- 6B+ 1 = 1 3 - ½D, 6#2A 2 - 6p.A + 1 = 1 + 123o - ½D
(38)
the BRST operator is then nilpotent. The familiar D = 2 6 , c~o= 1 emerges for ~t= - 1/2_+ 1/,,/12, A = 1, but since the canonical transformation (19) cannot be realized as a proper unitary transformation in the
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q u a n t u m Hilbert space, these D = 26 strings are n o t unitarily equivalent to the conventional D = 26 string. This can be viewed as a manifestation of the nontrivial polarization dependence in q u a n t i z a t i o n [ 5,15 ]. The p a r a m e t e r / t can be related to the conformal weigths of the fields gn, 2n. Consequently (34) can be viewed as the BRST operator of a string theory augmented by a world sheet conformal field theory. Since canonical transformations preserve Poisson brackets, we can introduce further canonical transformations to construct new realizations of A, a n d it is reasonable to expect that this exhausts all possible conformal field theories based on variables n n, 2n. Since it is generally assumed that a u g m e n t i n g a string theory with different world sheet conformal field theories can be viewed as different string compactifications, one can then argue that different compactifications of the string are classically canonically equivalent but q u a n t u m mechanically unitarily inequivalent. However, the validity of this conjecture will not be investigated here. In conclusion, we have explained how the BRST q u a n t i z a t i o n of a second class algebra can be implemented. We have constructed the pertinent BRST invariant q u a n t u m path integral and established its equivalence with the reduced phase space path integral by the Parisi-Sourlas mechanism. Our approach avoids Dirac brackets a n d is closely related to that in refs. [12,13]. The only difference is that now the auxiliary fields that convert the second class algebra into an effective first class algebra are identified with the dynamical Lagrange multipliers of the reduced phase space theory. As an application we have considered the possibility to construct nilpotent BRST
13 October 1988
operators for anomalous theories, in particular bosonic string theories in subcritical dimensions.
References [ 1] L.D. Faddeev, Theor. Math. Phys. 1 (1969) 1. [2] E.S. Fradkin and G.A. Vilkovisky,Phys. Lett. B 55 (1975) 224; CERN Report TH-2332 (1977). [3] I.A. Batalin and G.A. Vilkovisky,Phys. Lett. B 69 (1977) 309. [4] M. Henneaux, Phys. Rep. 126 (1985) 1; M. Henneauxand C. Teitelboim,Commun. Math. Phys. 115 (1988) 213. [5] H. Aratyn, R. Ingermanson and A.J. Niemi, Phys. Lett. B 189 (1987) 427; A.J. Niemi, OSU report DOE-ER-01545-410. [6] E.S. Fradkin and T.E. Fradkina, Phys. Lett. B 72 (1978) 343. [7] I.A. Batalin and E.S. Fradkin, Phys. Lett. B 122 (1983) 157. [8] P.A.M. Dirac, Lectures on quantum mechanics (Yeshiva University), (Academic Press, New York, 1964). [9] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge U.P., Cambridge, 1986); E. Nissimov, S. Pacheva and S. Solomon,Nucl. Phys. B 297 (1988) 349. [ 10] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 (1979) 744. [ 11 ] B. McClain, A.J. Niemi, C. Taylor and L.C.R. Wijewardhana, Phys. Rev. Len. 49 (1982) 252; Nucl. Phys. B 217 (1983) 430; A. Kupiainenand A.J. Niemi, Phys. Lett. B 130 (1983) 380. [ 12] L.D. Faddeevand S.L. Shatashvili,Phys. Lett. B 167 ( 1986) 225. [13] I.A. Batalin and E.S. Fradkin, Nucl. Phys. B 279 (1987) 514. [ 14] A.M. Polyakov,Phys. Lett. B 103 ( 1981 ) 207. [ 15 ] N. Woodhouse,Geometric quantization (Clarendon Press, Oxford, 1980). [ 16 ] P. Senjanovic, Ann. Phys. 100 (1976) 227.
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