Quantization of superparticle and superstring with Siegel's modification

Volume 212, number 4

PHYSICS LETTERS B

6 October 1988

Q U A N T I Z A T I O N OF SUPERPARTICLE AND S U P E R S T R I N G WITH SIEGEL'S MODIFICATION R. KALLOSH l, W. TROOST 2, A. VAN PROEYEN 3 lnstituut voor Theoretische Fysica, Katholieke Universiteit Leuven, B-3030 Leuven, Belgium Received 9 June 1988

The Green-Schwarz-Siegel superstring is shown to have infinitely reducible symmetry (even in the light-cone gauge). The zero-mode matrix Z,, ~'' '~,, is not invertible starting from the first stage n = 1. Therefore this theory may by truncated either on the zero stage, which leads to the GS theory, or not truncated at all, i.e. it has an infinite number of ghosts for ghosts.

1. The first applications of infinitely reducible lagrangian gauge symmetries [1-3] appeared for the first time in superparticle and superstring theories. Until recently the problems in covariant quantization of a Brink-Schwarz superparticle [4] and Green-Schwarz superstring [5] ~ were connected mainly with the difficulty to separate covariantly the first- and second-class constraints in the hamiltonian quantization. The lagrangian quantization of the Green-Schwarz superstring performed in refs. [ 1,10 ] is insensitive to the existence of first- and second-class constraints. Actually, the problem in lagrangian quantization is connected with the fact that the symmetry is infinitely reducible. In GSST the zero-mode matrix ~2 Z,,"I" 'A,,was shown to be nilpotent on shell: Z,, "l" '~,,Z,,+I"J"4 .... ~ 0 ,

n>l.

(1.1)

In fact, ' On leave of absence from Lebedev Physical Institute, Academy of Science USSR, Leninsky Prospect 53, 117 924 Moscow, USSR. 2 Bevoegdverklaard Navorser NFWO, Belgium. 3 Bevoegdverklaard Navorser NFWO, Belgium, Bitnet FGBDAI9 at BLEKUL11. n~ We will use the following set of abbreviations: BFV (BatalinFradkin-Vilkovisky) [6], BSSP (Brink-Schwarz superparticle) [4], SSP (Siegel superparticle) [7], MSSP (modified Siegel superparticle) [8], GSST (Green-Schwarz superstring) [ 5 ] and SMST (Siegel modified superstring) [ 9 ]. ~-~ We use Batalin-Fradkin-Vilkovisky definitions of reducible gauge theories [6].

428

Z , A ....... A,,=e = Jiffy,

( 1.2 )

a n d p 2 = 0 is an equation of motion. The rank of the zero-mode matrix is equal to eight for all n, since only an eight-dimensional square minor of the 16-dimensional matrix ~'"~ is invertible. This is the main reason why the truncation of gauge symmetries is possible on an arbitrary ghost-level in GSST and this leads to consistent quantization [ 1 ]. This means also that the gauge truncation procedure exists for arbitrary ghosts generation. The quantization of the Brink-Schwarz and GreenSchwarz actions with Siegel modification [7,9] was investigated recently [ 7-9,11-19 ] However, the results were either not correct, or not complete. The only exception is the hamiltonian quantization of Siegel superparticle [ 7 ] performed by Diaz and Zanelli. In this theory it appears that eqs. ( 1.1 ), (1.2) are also valid for n > 2 and therefore the truncation procedure which was used in ref. [1] for the GreenSchwarz string works also in Siegel superparticle theory [ 7 ]. The corresponding infinitely reducible symmetries may be truncated on an arbitrary ghost level. The purpose of this paper is to find the specific properties of the gauge symmetries of Siegel's modification of superstring theories. As a laboratory, we will first examine the corresponding symmetries for the modified superparticle of ref. [8 ]. Also, we will perform the quantization of these theories. In this way we are going to confirm the statements in refs. [ 7,13,16 ] that the number of degrees of free-

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dom in superparticles in ref. [4] and in ref. [7] is different. Also the counting of the degrees of freedom in Siegel superstring performed by Allen in ref. [ 14 ] and his statement that this theory is physically equivalent to the GSST will follow from the quantization procedure in a very specific way. One can distinguish two different types of gauge symmetry. The first type consists of symmetries generated by first-class constraints, T~, using Poisson brackets. They involve transformation of the coordinates of the theory, and a variation of the Lagrange multipliers with a term proportional to the derivative of the transformation parameter, 5¢*= {¢', T , } e " = R i , ~ e " , 60" = 3e ~ + f "p:,0 %/~- R "/~d~ .

( 1.3 )

The symmetries called A, B, C, ... in ref. [9] are of this type. When there is a linear relation among the first-class constraints T , oQ"",, = 0

(1.4)

then a different type of gauge symmetry is also present. This second type does not change the canonical variables ¢*, but only the Lagrange multipliers ¢-o, 8(h'~°= Q"°,,rl'~'--, 8( T~.o¢'~° ) =O .

(1.5)

Examples are the symmetries called E, F, G .... in ref. [8]. In the hamilton*an form of the BFV quantization procedure the treatment of these two types is different. In a first step, introducing the ghosts of the zeroth stage, only the first type of symmetry has to be considered. The second type of symmetry comes back in a second step (stage one) as a gauge symmetry of the ghost action of the zeroth stage. If one prefers to use a reducible (redundant) set of solutions Q ~ O of eq. (1.4), for example for reasons of locality or explicit covariance, then there are also zero modes of the first stage, i.e. solutions of Ql "",, Q2a~ oee] T,..... = 0 .

(1.6)

Then further stages are necessary, leading to further sets of ghosts for ghosts. In the lagrangian form of the BFV quantization procedure the gauge symmetries of both types are treated together in the first step.

6 October 1988

Let us illustrate this point for the two index antisymmetric tensor field A'V in d dimensions. In the hamilton*an approach the fields A " ( i = 1, ..., d - 1 ) are arbitrary canonical variables, with complex momenta ~zo, whereas A o,, are Lagrange multipliers corresponding to the constraints T, = ~Y~ro= 0, generating, with parameter ek, the transformations 8A o= 3*e1_ ~ d ,

5.4 oi= 0od.

(1.7)

These constraints are not independent, since

0*T,-0.

(1.8)

As a result, there is an extra symmetry of the second type transforming only the Lagrange multipliers: 8AOi= 0*~o .

(1.9)

This should not be considered as a separate symmetry in the hamilton*an approach [6]. In the zeroth stage, one introduces ghost and ant*ghost fields Co, and Coi. The zero mode eq. (1.8) leads to the introduction of further ghosts and Lagrange multiplier fields in stage one. Each stage has S O ( d - I ) covariance. In the lagrangian approach, S O ( d - 1 , 1 ) covariance leads one automatically to consider both types of symmetry eq. ( 1.7 ) and eq. ( 1.9 ) simultaneously, 8A'" = ~ ' e " - 0"e/~ , maintaining S O ( d - 1, 1 ) covariance at every stage. Considering this as a lagrangian in one dimension we recognise eq. (1.7) as symmetries of the first type (A, B, C) and eq. (1.9) as symmetries of the second type (E, F, G). The additional zero-stage ghosts of the lagrangian treatment are not lost in the hamilton*an approach, but reappear, after integrating over the additional variables in the linear gauge, as first level ghost (see table 1 ). It was verified [8] in particular for ant*symmetric tensors of the second and third rank, that the symmetries of the first type have global SO ( d - 1 ) symmetries, whereas including the second type one restores explicit S O ( d - 1, 1 ) symmetry. The equivalence was confirmed by explicit integration over canonical momenta of the hamilton*an gauge fixed action. In the case of the Siegel modified version of superparticle and superstring, symmetries of the second type (E, F, G, ...) are present (although in this case they are not connected to the first type by covariance 429

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Table 1 Ghosts for the antisymmetrie tensor.

6 October 1988

(2.6)

~]ot ~"~,J$ •

Stage

Lagrangian

Stage

Hamiltonian

0 0 0 0 l

ghosts Coj antighosts Col ghost Coo antighost Coo extra ghost C'L

0 0 1 1 1

1 1

ghost C, antighost Ct

1 1

Co~ghosts Co, antighosts 21 Lagrange multiplier C'I extra ghost 2't extra Lagrange multiplier Ci ghost C~ antighost

However, the E symmetry by itself has zero modes in the covariant lagrangian approach, because ~/constitutes 16 parameters. Indeed, choosing ~/to be of the form J/=~/2

(2.7)

reduces the E transformation to an on-shell trivial one in view of the constraints d~l ,~O~ P2 .

as in the example above). That they should be considered simultaneously in the lagrangian formulation was one of the main points of ref. [8]. It was also demonstrated there, that the inclusion of E, F, G symmetries leads to further zero modes of symmetries. This gives rise to a new infinite reducibility. The gauge fixed action for the theories ofrefs. [ 7,9 ] was not presented in ref. [ 8 ] since at that time it was not clear how to work with infinite reducible symmetries E, F, G ..... The infinite reducibility of these symmetries in ref. [9 ] was certainly different from that in GSST and in SSP [ 7 ]. 2. We start with Brink-Schwarz-Siegel action [ 7 ] or SSP, So=Pm(OXm-OFmoo)-½gp2+dOO-dt~',

(2.1)

(2.8)

For the same reason, the zero modes have themselves zero modes, and so on. The simplest way to quantize the theory eq. (2.1) is to truncate the E symmetry, i.e. to impose the constraint Y + r / = 0 ~ q = ½Y+Y-r/,

reducing the number of parameters to eight and eliminating the zero modes of the E symmetry. The gauge symmetries of the action eq. (2.1) are of the form 8q/=RZA0~Ao,

{a0={~, ~c", ~/+a},

RIoZIA°AI ~0 ,

TA = {A= ½PZ, B = d ¢ } .

ZIA°A, = {Z'a, Z'~,, Zba},

As a consequence of the reducibility of the first-class constraints

(2.10)

where we will use a for spinor indices taking 16 values, while a indicates the eight values of the index of the truncated spinor. We have one bosonic and 16 + 8 fermionic symmetries which form an on-shell closed algebra. These symmetries are reducible because of eq. (2.5):

and use the "lagrantonian" language [ 8 ]. The A and B symmetries are of the first type, generated through eq. ( 1.3 ) with (2.2)

(2.9)

(2.1 1)

where Z~= -dy+~ -,

2dA-/l¢=0,

(2.3)

Z~Xa=(½ff~+~)-),

Zba~--abaO.

(2.12)

there is a symmetry of the second type, called E: 5Eg=-2dq,

5Egt=~'q.

(2.4)

As already explained in ref. [ 8 ] the relation between A, B, E symmetries is not algebraic: aA ( p = --2d~/) + aB (x--- W~/)~ aE (O~),

(2.5)

where ~ means that we dropped on-shell trivial terms, i.e. transformations of the form 430

However, thanks to the truncation orE, ZIAA, has no longer zero modes so that we have a first-stage theory. We can quantize it according to the standard rules, present e.g. in ref. [ 6 ]: S = S c l - [ - ~ RJAo C~ 0 -[- C~oZAOAI Cl ml -*A t XIA,+CIA~ t. ~tl,4I + .... + C-gtAo o ZCOAo+CI

We choose the gauge fermion to be

(2.13)

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~--- Co~m)(g - 1 ) + Co(m7+7-q/+ Co(my-~'+0 -F c ( E ) ~ - y + [/]-FCl 7-~ )+C(B)

+ C(E)7-7 +C'1.

(2.14)

The integration over momenta n and non-propagating ghosts fixes 7+0=0 and 7+~=0, and eliminates

6 October 1988

3. As a preliminary step in investigating the gauge symmetries of SMST we are going to study the superparticle theory, which is a one-dimensional analog of SMST. This form of superparticle was suggested and analyzed in ref. [ 8 ]. In what follows we will call it MSSP (modified Siegel superparticle)

half of the zero-stage B ghosts. The remaining kinetic

S' =So + ½d~d~z, p .

terms are

Now the fields are

Sgf= - ½P2+ pm OX" +d~O0 ~ - OP+7- O0

Oi={pm, x",Oa, da},

"{-C (A) OC; A) -~-C(aB) oc(B)a'~-ClaOC ~ .

(2.15)

The resulting gauge fixed action shows that we have eight propagating fermionic degrees of freedom in Siegel's superparticle, since the contribution of the eight remaining zero-stage ghosts is compensated by the first-stage ghosts. The hamiltonian quantization of this theory performed in ref. [ 16 ] along the lines of Fradkin- Batalin [ 6 ] gives the same result. It uses the fact that the generators of the A, B symmetry eq. (2.2) have zero modes corresponding to the E symmetry in the lagrangian approach TAoZl AOAI = O, Z 2AIA2 ~ O, etc. The most important fact, which allows truncation of this theory on an arbitrary ghost level is that Zn A. . . . . An=fl=¢°tfl for n>~2,

(2.16)

i.e. Z is a quadratic 16-dimensional matrix. This means that the dimension of the gauge symmetry of the nth ghost level is the same as at the ( n - 1 )th level, and the rank of every Z, is eight, since $2 = 0 is the classical equation of motion. For the antisymmetric tensors Au,...~, ]11.../H-- n Z,,,,.,, .....

0~"-"~;,

....

~

....I]J

l~l--n

i.e. dim{G~_ i } > dim{G, } . For GS superstring and Siegel superparticle we have

dim{G,_ i } < dim{G~} , i.e. the dimension of ghost symmetries grows with every stage.

0~°= {g, ~u~,Z~P}.

(3.2)

In "lagrantonian" language this theory has A, B, C gauge symmetries, eq. ( 1.3 ), where now

TAo={A=½P2, B=dl~, C,~a=-½dt,~dal } .

(3.3)

The theory eq. (3.1) has also E, F, G gauge symmetries,

E:

8g=-2drl,

~=~'q,

F: 5~C'=u'~pda,

5Zo,a=-2I~rt,~u~,

A'~a'r=-A p'~'r, At"e'7]=0.

(3.4)

If we use the Fradkin-Batalin hamiltonian approach which is simpler for a one-dimensional theory, we may consider the E, F, G symmetries as zero modes of T~o. In particular we have

E: e2d-d~?¢=o, F'.

(de),da-d~a,sl~=O ,

G: do,d(Bdy)=O.

(3.5)

For lagrangian quantization we must take into account the A, B, C and the E, F, G symmetries and look for their zero modes as before. Our main concern now is the G symmetry, since both E and F symmetries are of the type already known. Let us take the simplest route, i.e. solve both A and B constraints, see e.g. ref. [15]:

dim{G,_ j } = dim{G, }, and now we start to investigate Siegel superstring theory where

(3.1)

P~

A = 0--,2P- = - - e+' B=0--,27-d=-

~iP i

p+ 7-7+d,

(3.6)

and fix the light-cone gauge

431

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X+(a,z)=P+r,

PHYSICS LETTERS B

7+0=0.

(3.7)

Z2

m

a2-

6 October 1988 _ _ t R a ~ Rb~ Rc~ Rm Rt,~Rc~ )dd2 \ ~ [ a 2 U b 2 ] ' ° c 2 --~[a2Ub2Uc2]

Then our action becomes S' =Pi 0 X ' - 0P +7- 0 0 - ½P2 + d,~Ooa + ½xtab]dadb ,

ZnO~n- I ~,, = (67~,,

(3.8)

where we denote by d, an SO ( 8 ) part of the SO (9, 1 ) spinor d,. Now we come to the most important new statement. MSSP has an infinite reducible gauge symmetry even in the light-cone gauge. This symmetry is an SO (8) part of the G symmetry which survives in the

light-cone gauge. The action eq. (3.8) has also the SO (8) part of the C symmetry, as presented in eq. (1.3), since the C symmetry generators (3.9)

form a closed algebra. The gauge symmetry of the action eq. (3.8) is therefore presented in eq. ( 1.3 ) with ~'={X',#,O~,da},

O"°=Z ~h,

(3.10)

which means 8cO.=4,bd b ,

8 c d ~ = 2 ( 7 - )abP+ ~bflc,

8cZob = 0{,b -- 4Xct~P+ (7-)Cd~bld.

(3.1 1 )

This C symmetry is reducible, since d~d~t,4~ = 0 .

(3.12)

The G symmetry takes the form 8Z"" = A ~'cdc,

( 3.13 )

where A has the same mixed symmetry as in eq. (3.4). This has an infinite tail with growing dimension of symmetry ~ A ab,c = A a b ' c d d d ,

(3.14) 8/1 ~t,,c,...c,, = A~t,,c~'c,,+ ' d,,,,+ , . Thus c~1 has three-spinor indices, a 2 has four-spinor indices, etc. In hamiltonian language we have T.o =dtadbl ,

ZlO~0o~I

432

uR[aRb]dc,_ a l Ub I

. . . . . .~bn 1 ~ .c,,l . . . . )6dd:: - [ ,~pP,l ' d q --(~[a,, oh,,

(3.15 cont'd) This situation has some similarity with the role played by the B symmetry in the covariant quantization of the BSSP action 80=$x,.

a Sb, < d,.,l 8[a,

(3.15)

(3.16)

Since p 2 = 0, these gauge transformations are not independent, which leads to a zero mode zIAOA~ =J~,

T,,, = d , db

I ¢n I ' ~bn t, b,, ] &,,

(3.17)

of the gauge invariance of the ghost action corresponding to eq. (3.16). The story repeats itself now since eq. (3.17) is of the same form as eq. (3.16), leading to an infinite sequence of ghost for ghosts Z,, ~.... , , , = ~ .

(3.18)

This sequence can be interrupted at an arbitrary stage, at the expense of explicit covariance. The key to eliminating the zero modes that give rise to the next stage is to put a condition on the parameters ~c2,and correspondingly on the ghost fields. If this condition is such that the number of parameters K~+I in eq. (3.18) is equal to the rank of $, then there are no zero modes left. In the case orB symmetry, this rank is equal to eight. The condition 7+xn+l_-_0

(3.19)

eliminates eight parameters from the 16 originally present in eq. (3.3). This procedure can be applied at an arbitrary level, and the result does not depend o n i t [1,16]. In the case we discuss here (MSSP), the rank of the corresponding matrix is zero for n >/1. This means, in particular, that any constraint on the parameter of transformations (or ghost fields) does not remove the reducibility. However, there remains the possibility of finding a condition that makes all transformations trivial on-shell, i.e. of the form eq. (2.6). This truncation cannot be done at an arbitrary level, but is automatically a zero-level truncation. The first possibility of making this zero-level trun-

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cation is to put the constraints on T, as well as on all the generators of the gauge symmetries of the ghost fields T,o=dtadbj=O,

Z,, ~'' ' , , ~ d = 0 .

(3.20)

The solution of these constraints is d~=0,

(3.21)

which brings us back to the Brink-Schwarz superparticle. The second possibility is to truncate the theory as follows: ½d~z~b - 2 b = 0 ,

(3.22)

and to integrate over 2 ~. This leads to the action eq. (3.8) where the last term is 2"d~ instead of ½z"bd,~db. Both C and G symmetries and the tail eq. (3.14) become on-shell trivial in the sense that So(all fields) ~ d ,

8~z~d,

(3.23)

and now d, is proportional to the equation of motion of,a., since for the new action Sgr . . . . +2~do,

(3.24)

8Sgr = d ~ , 82 ~

(3.25)

and all gauge symmetries of the action eq. (3.25) are now on-shell trivial, i.e. of the type eq. (2.6). Once more we come back to the Brink-Schwarz superparticle. The third and the most unusual situation for the first quantized theory is not to perform any truncation at all. The first example of infinite stage theory was presented by Batalin and Vilkovisky [6]. They have found that the solution of the master equation is in fact an infinite stage system,

(S,S)=

r T~=O,

The hessian of S ( Z ) is defined as follows: ~a = r ~ ,

0~0rS OZ~OZ ,'"

(3.26)

The solution of the master equation has the property

6 October 1988

OrS ~ l a = 0 0Z a

(3.27)

The generators are linearly dependent 9~~b 91~'c

Z~Zo

=0

(3.28)

and the matrix of the generators is nilpotent at the stationary point. In Witten's string field theory it was found by Bochicchio and Thorn [20] that an infinite set of ghosts for ghosts is present. However, in suitable variables [20] it simply means that the solution of the master equation for the string field theory is given in terms of qb(X(a), C(a) ) = Z, Is) 0.~. If we write the BRST operator for the theory eq. (3.8) simple-mindedly extending the rules presented by Fradkin-Vilkovisky [ 6 ] for a finite-state theory, we get for ~min ~ m i n ~---

cal'dadt~ Cnal...an+2 do,,+,Nn.......... +...,

+

(3.29)

n=l

Also the gauge fixing terms in the form ~ may be added in accordance with ref. [6]. Eq. (3.29) has a strong analogy with Witten's string field theory gauge fixed action, presented by Thorn and Bochicchio. On the bilinear level we have Srnin = ~ . Qq :~, where q~= q~+ + q~ is some infinite sum

¢'= }2 Is)O,+ }2 I~)0~(-) ~'. Now d plays the role of Q since dadb is a constraint. We do not try to develop further the infinite ghosts system quantization, since the unitarity problem remains a severe problem in string field theory quantization as well as here. Anyway, we hope that it becomes clear that the C, G, ... symmetries of MSSP have a nature which is very different from all other symmetries in first quantized theories. Therefore if we are not satisfied with the first two types of truncation (eq. (3.21) and eq. (3.22)) which bring the theory back to the BS superparticle, we must realize that the price for that is an infinite number of ghosts for ghosts. 4. The quantization of SMST follows the same pat433

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tern as already described for the MSSP theory, there are no really new qualitative differences. However, since now we have a two-dimensional theory, it is a little bit more complicated. The lagrangian we start from is . . . . + d"7 ~'~d ZZ=u~a+ dzY~Wd ~~=z ~-

( 4.1 )

The A, B, C, D symmetries of this theory are well known, see e.g. ref. [ 11 ], and their reducibility conditions, or E, F, G,... type symmetries were not found completely, so that all gauge symmetries of the lagrangian eq. (4.1) form an on-shell closed algebra. However, at least E, F, G and certainly more symmetries are present in eq. (4.1). But since our purpose now is to understand a simpler way to quantize this theory, we once more use the light-cone gauge and the hamiltonian treatment, which in fact was performed partially by Kamimura and Tatewaki [ 17 ]. They have shown that, in the light-cone gauge, after solving the A and B constraints, the theory still has a gauge symmetry which is related to the constraints ~3 T,o = {M~b = dodb, M + = dad'b7 +~b}.

(4.2)

The constraint M,b is the same constraint we met in MSSP, and M + is a new one, particular for d = 2 . It comes from the last term in the lagrangian, and in d = 1, i.e. the MSSP case, it vanishes. The Lorentz-covariant form of the A, B, C, D constraints is presented of course in the original papers [ 9,11,14 ]. Also in the covariant form the analysis of the physical content of the theory was presented by Allen [ 14 ]. He made the important observation that, despite the fact that one has first-class constraints, they are not all the true symmetries of the theory. In particular this is the case for the quadratic constraints dy~'~ad and dyUd ' , or, in the light-cone gauge, the 29 constraints Mab and M +. Allen's conclusion is that SMST has the same physical content as GSST. In ref. [17] on the contrary Mab and M + where treated as ordinary gauge symmetries. The very surprising conclusion the authors reached is the following: if the gauge invariance that remains in the lightcone gauge is imposed as a physical state condition, no physical state exists in SMST. We know from the preceding section that in MSSP ~3We further omit the z-indices. 434

6 October 1988

in the light-cone gauge the generator Mah = dadb is also present, and has the important property that it is reducible! Therefore it is certainly not allowed to impose it as a physical gauge condition, despite the fact that they form a closed algebra. This observation explains the absence of physical states in ref. [ 17 ]. Now we present the reducibility conditions for Mob and M + which appear besides those connected with Mat, only (see eq. (3.12 ) ): ( doT +°bd'b )dc = - ( dod¢)7 +~bd'b ,

(4.3)

or in terms of T,o, Z1 '~°,~,, Z2"',~2, ... we have Zl°e°Oel (n1+c[aab]t]t

da,)

:

(4.4)

The possibilities for quantizing this system are in fact already discussed in section three. Either one takes do = 0 which solves T,o, Z~ "° m ,..., and brings us back to GSST; or we have zabdb + ~y +abd,b = )a .

(4.5)

In this case all generators of the symmetries T~o, Zl, •.., Zn become trivial on-shell, in the sense ofeq. (2.6), and once more we are back at GSST. As in the MSSP, there is no possibility to perform a truncation of any finite ghost level, for the same reason: the rank of all Zn is zero at the stationary point. 5. The present investigation shows that the problem in covariant quantization of superstring theories is not necessarily connected with the impossibility to separate first- and second-class constraints. This happens in GSST theory but not e.g. in SSP theory, where only first-class constraints exist. The origin of the problem in all these theories is the impossibility to present the irreducible gauge symmetries of the theory without violating a global symmetry, i.e, the spacetime Lorentz group, or even the impossibility to have an irreducibility condition at all. The Lorentzcovariant form of gauge symmetries in these theories is always infinitely reducible. The only exception now is the GS formulation of the heterotic string with additional gauge fields, which allows to present the irreducible symmetries of the action and perform a consistent quantization of this theory without breaking Lorentz covariance of the theory [ 10 ]. The MSSP and SMST have even deeper problems connected with

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infinite reducibility. They contain a type of symmetry that does not allow truncation at an arbitrary finite level, but only at zero level, after which they are equivalent to BSSP and GSST respectively. These theories therefore present a new example of on-shell nilpotent generators, after the example of Batalin and Vilkovisky of the solutions to the master equation, and Witten's string field theory, quantized by Bochicchio and Thorn [20], which also has nilpotent generators, leading to a really infinite stage theory. For all these theories the standard criterion of unitarity, always an essential ingredient and guide in the quantization procedure, is lost. The realization that several theories of this new type exist may lead to further exploration. In this respect it might be instructive that the completeness condition 8 = 8 for GSST arising in ref. [ 1 ] as the result of the truncation procedure on some arbitrary ghost level 8= 16- ( 1 6 - ( 1 6 - . . . - ( 1 6 - 8 ) . . . ) ) ,

6 October 1988

reducible constraints are the same as one single firstclass constraint p = 0. Anyway, in GSST or SSP the summation method presented in eq. (5.2) seems to be a nice way of counting an infinite number of ghosts for ghosts if one wants to avoid the truncation procedure in GSST. Let us note that in the Lorentz-covariant form the F and G symmetries on Z"~ reproduce the off-shell gauge symmetry of d = 10 superspace lagrangian for linearized supergravity [25 ], as was already stressed before in refs. [ 1,8 ]. Indeed one can identify V,,,,p = aft Ym,,pZ,¢. The gauge symmetries (,4 and L) are then reproduced by G and Ftransformations eq. (3.4) with parameters A,~a,>._ - vail . , . ~ p _ ff;,,,,,p

,

( y , . ) ~I~A~ a,,,,,p = O ,

u'~/~= L,,,,, (7'"") "/3.

(5.4)

The latter one has to be modified with an on-shell trivial symmetry

(5.1) 87. " ~ = d yd~(y''np)'~(y,,,nq):,~Llft

may be obtained as a result of most natural methods of summation of infinite series [21 ]. A method due to Euler [22] defines the "sum" Y~_oan as " hmlt ' " exists. Then lim,~l_o~,,=oa~X n, w he n this

.

(5.5)

The off-shell gauge symmetry of the d = 10 superspace lagrangian for linearized supergravity - D . A ,,,,,f, + D . ( 7q tin,, ) BD Lp ) q

1-1+1-1+...~

lim .~'~

=

lira .\'~

(1--y+y2--...)

+ 280 t,. L,,pl,

1 -- 0

( l + x ) - j --'~ ,

(5.2)

1 -- 0

or, 1 6 - 16 + 16 . . . . . 8. Ces~ro's method of summation [23] gives the same result. To show that also increasing numbers of zero modes can lead to a sensible counting we consider a single pair of canonical variables (p, q) and the reducible symmetries generated by p2 and pq. Counting the number of zero modes at successive levels we obtain the Fibonacci sequence. The number of degrees of freedom corresponds to the series 1-2+1-1 -....

+2-3+5-8+

13-21 +34-55+89 (5.3)

The series 1 - - z + 2 z 2 - - 3 2 3 + 5 z 4 + . . . converges to ( 1 + z - z 2 ) - ~ in a neighbourhood of z = 0 . It is Borel summable (B) [ 21,24 ] for z = 1 and the value is one. Therefore the counting adds up to zero, which is the correct number of degrees of freedom because these

( 5.6 )

contains simultaneously the linearized gauge symmetry of graviton, gravitino and antisymmetric tensor field. Therefore it may be useful to investigate this theory further, preserving at least the G and F symmetries of the string theory [ 9 ]. R.E.K. is thankful to I. Batalin for most enlightening discussions and collaboration in the first stage of the work. She would also like to acknowledge most gratefully the hospitality extended to her at the Institute for Theoretical Physics of the Katholieke Universiteit at Leuven. R.E.K. and A.V.P. wish to thank the organizers and participants of the "Strings 88" Workshop in Maryland, where part of this paper was written and where they profited from most fruitful discussions. N o t e added. The equivalence between SMST and GSST was established by Gupta-Bleuler methods in recent work by Mikovi6 and Siegel [26 ].

435

Volume 212, number 4

PHYSICS LETTERS B

References [ 1 ] R.E. Kallosh, JETP Lett. 45 (1987) 365; Phys. Lett. B 195 (1987) 369. [2] E.P. Nissimov and S.J. Pacheva, Phys. Lett. B 189 (1987) 57. [ 3 ] L. Brink, M. Henneaux and C. Teitelboim, Nucl. Phys. B 293 (1987) 505. [4] L. Brink and J.H. Schwarz, Phys. Lett. B 100 ( 1981 ) 310. [5] M.B. Green and J.H. Schwarz, Phys. Lett. B 136 (1984) 367. [ 6 ] I.A. Batalin and G.A. Vilkovisky, Phys. Rev. D 28 (1983) 2567; I.A. Batalin and E.S. Fradkin, Phys. Lett. B 122 ( 1983 ) 157. [7] W. Siegel, Class. Quantum Grav. 2 (1985) L95. [ 8 ] I.A. Batalin, R.E. Kallosh and A. Van Proeyen, Leuven preprint KUL-TF-87/17, in: Quantum gravity, eds. M. Markov, V. Berezin and F. Frolov (World Scientific, Singapore), to be published. [9]W. Siegel, Nucl. Phys. B 263 (1985) 93. [ 10] R.E. Kallosh and M.A. Rahmanov, Covariant quantisation of the Green-Schwarz superstring, Lebedev preprint. [ 11 ] L.J. Romans, Nucl. Phys. B 281 (1987) 639. [12]S. Randjbar-Daemi, A. Salam and J.A. Strathdee, Mod. Phys. Lett. A 2 (1987) 145.

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6 October 1988

[ 13] Th.J. Allen, Mod. Phys. Lett. A 2 (1987) 209. [ 14] Th.J. Allen, preprint CALT-68-1401 (1987). [ 15 ] W. Siegel, Phys. Lett. B 205 (1988) 257. [ 16 ] A.H. Diaz and J. Zanelli, Phys. Lett. B 202 (1988) 347. [ 17 ] K. Kamimura and M. Tatewaki, Phys. Lett. B 203 ( 1988 ) 79. [ 18 ] P. Majumbar, On the covariant gauge fixing for the superparticle and superstring, University of Maryland preprint #88-131 (1987). [ 19] L. Brink and M. Henneaux, Principles of string theory (Plenum, New York, 1988 ). [20] M. Bochicchio, Phys. Lett. B 198 (1987) 475; C.B. Thorn, Nucl. Phys. B 287 (1987) 61. [ 21 ] E.T. Whittaker and G.N. Watson, A course of modern analysis (1902; reprinted Cambridge U.P., Cambridge, 1963) p. 155; B. Sz.-Nagy, Introduction to real functions and orthogonal expansions (Oxford U.P., Oxford, 1965) ch. 8.3; E.C. Titchmarsh, The theory of functions (Oxford U.P., Oxford, 1939). [22] L. Euler, Instit. calc. diff. (1755). [23] E. Ces~ro, Bull. Sci. Math. (2) XIV (1890) p. 114. [24] E. Borel, Leqons sur les sdries divergentes ( 1901 ) pp. 97115. [ 25 ] P. Howe, H. Nicolai and A. Van Proeyen, Phys. Lett. B 112 (1982) 446. [26] A.R. Mikovid and W. Siegel, preprint UMDEPP 88-218.