Covariant Quantization of Neveu-Schwarz-Ramond ...

934 Progress of Theoretical Physics, Vol. 75, No.4, April 1986

Covariant Quantization of Neveu-Schwarz-Ramond Model Makoto ITO, Takuya Morozumi,* Shin'ichi NOJIRI*'*) and Shozo UEHARA**

Department of Physics, Tohoku University, Sendai 980 .. Department of Physics, Kyoto University, Kyoto 606 ** Research Institute for Theoretical Physics Hiroshima University, Takehara, Hiroshima 725 (Received December 9, 1985)

§ 1.

Introduction

A few years ago, Kato and Ogawa 1) presented manifestly covariant quantization of the bosonic string using BRS.invariance. They showed that a condition of nil potency of the BRS charge determines the values of the critical dimension and the leading zerointercept ao simultaneously. Then, they defined physical subspace by a modified sub· sidiary condition with the BRS charge and proved the no-ghost theorem. Recently SiegeF) made use of the BRS charge to derive the gauge-fixed action for the string field theories. As to superstring theories,3) however, it is difficult to covariantly quantize new superstring actions 4 ) using BRS invariance. 5 ) A modification of the covariant superstring action was proposed,6) but it seems hard to covariantly quantize it. 7l Some physicists considered cova.riant quantization of superstring theories with the Neveu-SchwarzRamond (NSR) models: 8 ) The BRS charges of the NSR models have been constructed from the super-Virasoro algebra of the NSR mode1. 9 ),10) The gauge-fixed second-quantized action of the superstring was proposed 10 ) using the BRS charge. Covariant canonical quantization of the NSR model, however, has not been performed so far. In this paper we perform covariant canonical quantization of the Neveu-Schwarz (NS) and the Ramond (R) models. In § 2 we quantize the lagrangians of the NSR spinning string. 12 ) We define the BRS transformation and fix the gauge in a BRSinvariant manner. In § 3 we present the BRS charge. We construct the Fock space and then propose the physical state condition with the BRS charge. In § 4 we prove the no-ghost theorem. The final section is devoted to conclusions. In the Appendix we give the proofs of Propositions presented in § 4. *)

Present address: Research Institute for Fundamental Physics, Kyoto University, Kyoto 606.

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We present manifestly covariant quantization of the Neveu-Schwarz and the Ramond models using BRS invariance. The critical dimension and the leading zero-intercept of each model are determined by requiring the nil potency of the BRS charge QB. We prove the no-ghost theorem defining physical subspace by a modified subsidiary condition with the BRS charge.

Covariant Quantization of Neveu-Schwarz-Ramond Model

§ 2.

935

Canonical quantization

The action for the NSR model is regarded as that for scalar multiplets [AP(c;) , XP(c;), FP(c;)](,u=O, 1, "', D-1) in the two-dimensional supergravity theories: 12 ),131,*1

+Z'f'mY 1 ,I. nYmXPo nA IJ +T6x 1 - PXIJ 'f'mY ,I. n m,I. ] Y 'f'n, 7}PIJ.

(2·1)

(2·2a) (2·2b)

~FP=cmomFP+ cym[( Om+ ~ WmYs- ~ bm)x p -+{FP+iyn(onAP+

~ ¢nXp)} Cs+ i ¢c+2i!jc ,

(2·10e)

the total lagrangian becomes

.L =.LO+.LGF+FP

) ( ean- rJan )+,1. +a m [ Z.( C* an C m+2'ze -1 c mn ek aC* k Y5C 'f'nC* n C m] ,

where

(2·11)


(2'9)

M. Ito, T. Morozumi, S. Nojiri and S. Uehara

938

Then c* Gm ngnk is symmetric and traceless, and c* Qm is ')'-traceless. The field equations are obtained from the variation of the action and the boundary conditions are determined from the variation of the action, invariance of the action under the BRS transformation (2·5) and the consistency of the boundary condition under the BRS transformation: [Field equations]

(JO-Jl)cl=O,

FfI.=Bm=I/Jm=bam=eam-r;am=O,

(2·13) where (2·14) Note that due to the equation eam- r;am=O,

(2·15) [Boundary Conditions] C1=JOC1=JIAfI.=c*GOl=0

at O'=O,

J[,

(2·16) where €= ±1 and €= + l( -1) corresponds to the R(NS) model. Then the relevant variables are expanded as follows:

C O(,r O' ) ,

C

1 '10 + r= 1 ~ = yJ[ r= "-' ( r;ne -im:+ 'In t e inr) cos nO' , yJ[n~l

l r, () O' = -

i ~ r= "-' ( r;ne -inr - 'In t e inr)' smnvrr ,

y J[

. GOO( r O' ) = c* ,

n~l

1 '101 "-' ~ (- t e inr) cosnv, rr r= r= r;ne -inr+ 'In yJ[n~l

yJ[


(Jo+Jl)dl =0,


,

(2·17) where k takes integer values 0, ±1, ±2, "'(half-integer values ±1/2, ±3/2, ... ) for the R (NS) model. Hence for the R(NS) model co, c* GOO, c and c* Q(c0, c* GOO) have zero modes 7}0, ifo, So, [:0 (7}0, ifo). 7}0, ifo and So are hermitian and [:0 is anti-hermitian. Note that X has zero modes boP for the R model and due to the anti-commutation relation, which is shown next, boP. are regarded as the gamma matrices in D-dimensional space. N ow we proceed to determine the canonical commutation relations. According to Dirac's method of quantization, we see that all constraints are of second class and the variables F P , O)

(3-6)

then the commutation relations become (3-7)

Note that the last terms in L NS and L R come out through normal ordering, which should be performed to make L NS and L R well-defined because the BRS charges formally defined in (3-1) contain divergences in L NS and L R. The parameters, ao NS and al, are undetermined so far. We defined the BRS transformation to be nilpotent, however, the BRS charges in (3 - 2) and (3-4), which are well-defined, are not necessarily nilpotent, (Q B NS)2=l~(D-10 7[ ~1 8

n

3+16aoNs-D+2 ) t 8 n 17n 17n (3-8)

(3-9)

Thus the nil potency of the BRS charges, which means the vanishing of the conformal anomaly, demand"l),10)

D=10 and ao Ns =1/2 for the NSmodel, D=10 and al=O

for the R model.

(3 -10)

These conditions are necessary to prove the no-ghost theorem in the next section. Now we proceed to construct the Fock space and define the physical subspace using the BR5 charges. The ground state 10, p> is defined as usual by

PoPlo, p>=pPlo, p>, anPIO, P>=17nIO, p>= 77nIO, P>=O, (n>O) bkPlo, p>= sklo, p>=

J

fklO, P>=O.

(k>O)

(3-11)


b/'t=.b~k,

f

n-l


944

As discussed in Ref. 1), th'e states of the bosonic string are doubly degenerate due to the zero modes of the FP ghosts which correspond to 710 and if0 here in the NSR model. This is also the case for the NS model because there exist the zero modes 710 and ifo. Then the Fock space for the NS model is spanned by the direct product of the doubly degenerate states,l) {1-),I+)(=7Jol-»)} with

ifol-)=o,

(3-12)

and those I=O for n in V F satisfying Qti ¢ >= 0 can be written as (4 ·17)

where P(O) is the projection operator into the subspace generated by Ani t and B nit,

where the arrows above symbol II indicate the ordering of product.

Proposition II. Any state i¢(,8» in VF satisfying (4·14) can be written as (4·19) where P(,B) is the projection operator into the transverse sector generated by the transverse operator Anit(,B) and Bn it (,B)21) defined by (A·7) in the Appendix, (4·20) Owing to Proposition n it is easy to complete the proof of the no-ghost theorem. We put ,8= 1 in (4·19), then we see that the norm of the physical state i¢> is positive semidefinite, (4·21)

=;;:O:o,

where the last inequality follows from the positive definiteness of the transverse state space.2l) This completes the proof. We make comments on the proof of the no-ghost theorem for the NS model: Conditions (4 ·1) and (4·2) should respectively be replaced by L NSiphys> = 0 ,

(4·22)

QB NSiphys> = 0 .

(4·23)

We define the subspace VL satisfying condition (4·22). The field redefinition for the NS model can be easily found by putting 8=0 in (4·6) for the Rmodel. The parameter ,8 is introduced just in the same way as (4·9) (except for bo-± because there is no zero mode in x(r,o)). We get the similar propositions, and then we can complete the proof. § 5.

Conclusions

We have performed the manifestly covariant quantization of the Neveu-Schwarz and Ramond models using BRS invariance. The BRS charges obtained here by canonical quantization of the lagrangian, coincides with those directly obtained according to the super-Virasoro algebra of the NSR mode1. 9 ),IO) Hence the nil potency of the BRS charges


(4 ·18)


948

leads to the critical dimension and the leading zero-intercepts. The structure of the BRS charges is peculiar. QB, rather than QB itself, corresponds to BRS charges in ordinary gauge theories, which will be clear from the proof of the no-ghost theorem. Using the BRS charges we can proceed to the covariant second-quantization of the NSR superstring. 10 ) In that case we must restrict the ground states of both the NS and the R sectors with the generalized G-parity and chirality operators 10 ) according to Ref. 22). Acknowledgements

Appendix - - Proof of Propositions 1, 11- Since the Qt is nilpotent and quadratic in mode variables (see (4·11) and (4·13)), we can follow the usual procedure developed by Kugo and Ojima. 20 ) We define (jJn=k- An-,

Xn=k-B n-,

(3n=Po+ A n+,

An=Po+Bn+ ,

A· - A-

1

Yn=

liCn,

Wn = I7f Dn ,

Yn=

liCn,

Wn = I7f Dn ,

_

1-

then we find [QI R, (jJn]=-Yn,

{QI R, Xn}=Wn,

[Qt, (3n]=O,

{Qt, An}=O,

{QI R, Yn}=O,

[QIR, Wn]=O,

{Qt, Yn}=(3n,

[QIR, Wn]=An,

(A·I)


We would like to thank T. ~ugo, M. Kato and M. Hayashi for discussions. We are also indebted to H. Terao for the collaboration at the early stage of this work. One of us (M. I.) thanks the members of the particle theory group of Physics Department· at Kyoto University for kind hospitality. This work was supported in part by the Grant-in-Aid for Scientific Research, Ministry of Education, Science and Culture, Japan (#60740138).

949

Covariant Quantization of Neveu-Schwarz-Ramond Model Ainj (j)m t 13m t Ym t Ym t Ani

B~j

Xmt Am t

(J)m t

Wm t

0

-8 nm

-8nm

0

8nm8ij

(j)n

0

8nm

/3n

8nm

0

Yn [c,b, c,b t]:;:= Yn

0 0

8nm

8nm

0 8 mn8ij

Bni Xn

0

8nm

An (J)n

8 nm

0

(A·2)

Thus each set of the modes ((j)n, /3n, Yn, Y n) and (Xn, An, (J)n, Wn) forms the so-called "quartet,,20) of unphysical modes. The projection operator p(n) into the sector of n unphysical modes can be written recursively,

(A·3)

with - + Yn - tp(n-l) (j)n _ Xn tp(n-l) (J)n - _ (J)n - tp(n-l) Xn ) . R (n)-l( - n (j)n tp(n-l) Yn .

(A·4)

Using these projection operators we can deduce the general solution for QIRI¢>=O as I¢>= ~ p(n)I¢> n=O

(A·5)

where Ic,b>= ~ R(n)I¢>. n=l

(A·6)

This completes the proof f Proposition I. Before going to the proof of Proposition II, we introduce the transverse operators.2l) They are given by


Wn

0

950


-~Hi(mk- fj) (mk- fj')(mk- P)-3/2] 2

(A-7)

where 00

pi= 2::.fii(a/z-n+ani'tzn) , n~l

00

Hi = bOi + 2:: (bn i' z-n + bni't zn) , n~l

From (A·S) we see that Ani(S=O) and En i (S=O) are Ani and En i defined by (4'6), respectively. The positive definiteness of transverse states is guaranteed by the commutation relation, [Ani(S) , Amjt(S)]=OijOmn, {Eni(S) , Emjt(S)}=OijOmn.

(A·9)

Since these operators are constructed so as to commute with the super-Virasoro operators Ln and Fn for the R model, they commute with QB, . (A ·10)

Owing to (A ·10) the projection operator P(S) defined by (4· 20) satisfies (A ·11)

By operating QB(S)P(S) on an arbitrary state 1¢(s»=2::~~osnl¢n>, we get (A ·12)

where P(s)=2::~~osnPn. Note that Po=p(O) in (4·1S). N ow we can prove Proposition II by the same method in Ref. 1). Equation (4 ·19) can be rewritten by (A' 13)

where


(A-S)


951 (A ·14)

We prove by mathematical induction with respect to n. From Proposition I the statement holds in the n = 0 case. Assume that the statement holds for n -::;, N -1. Then from Eqs. (4·15) and (A·13) we find

-{QB, QBdlfN-3)-(QBI)2IfN-4).

(A ·15)

Due to Eqs. (4·12) and (4·13) and IfN)E VF, the RHS of (A·15) becomes Qt~~~oPN-nl¢n) + QtQBlfN-l)+ QtQBlIfN-2). Hence (A ·16)

From Proposition I we get (A ·17)

where

(A ·18)

The second equality follows from (PO)2=pO(=P