Volume 187, number 1,2 PHYSICS LETTERS B 19 March 1987

HAMILTONIAN FORMALISM

OF THE GAUGE INVARIANT FREE CLOSED BOSONIC STRING FIELD THEORY

C. BATLLE and J. GOMIS Departament d'Estructura i Constituents de la Matdria, Universitat de Barcelona, Diagonal 64 7, 08028 Barcelona, Spain

Received 28 November 1986

The hamiltonlan analyms of the gauge invariant free closed bosonic string with no restrictions on the gauge parameters is performed. All the constraints of the theory are first class. Also, we study the hamfltonian construction of the gauge transformations and give the BRST operator of the field theory.

1. Introduction. The construction o f a gauge invariant field theory of strings is a subject o f recent interest. The action for the free open bosonic string field theory can be written [ 1,2] in terms of the BRST operator o f the first quantized theory [3]. The extension to open superstrings has also been done [4,5]. For the closed bosonic string an action without restrictions on the gauge parameters has been proposed [6,7]. That action can also be written in terms of the corresponding BRST operator using an auxiliary state. The hamiltonian analysis of the gauge invariant theory is very important in order to understand the gauge structure o f the theory. Bengtsson [8] studied the hamiltonian constraints for the open string. In this paper we analyse the closed string case.

2. The action. The BRST operator o f the first quantized closed bosonic string is given by

Q = Q R + QL , (1)

whose expansion on zero modes and time derivatives is

QR,L = ~I ~R'La0~0 + 3 R ' L b R ' L ~ 0 v v -- - ~ R ' L f R ' L + l a R ' L a O + q R ' L • (2)

We can rewrite (1) in the form

a = ( 1 / V ~ ) C ~ ° a ° +ix/-2aa° + x / 2 q + ( I / x / ~ ) C ~ b + + ( 1 / x / ~ ) C o b - - ( 1 / V r 2 ) C ~ f + - ( 1 / v r 2 ) c ~ f - , (3)

where

C~ = (I/%/2)(fl~ +/3k) , C~ = (I/w/2)(~ R + ~L), a = (I/V~)(a R + aL), q = (I/%/2)(q R + qL),

b -+ =b R +b L , f-+ =fR _+re. (4)

The zero mode algebra is

= = 1 . ( 5 )

The theory has four vacua [9,6]. We can pass from a vacuum to another using the zero mode operators. In this paper we will only exhibit the I-+> vacuum, which obeys

C0-1-+>=Co' I - + > = 0 , < - + I C o ' C o - I - + > = i . (6)

The ghost number operator is given by

0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. 61 (North-Holland Physics Publishing Division)

Volume 187, number 1,2 PHYSICS LETTERS B 19 March 1987

+ - - + ---- I 1 (C d Co - + + Qgh = ~ - C 0 C~ ) - i ( C ( e ~ - C ( C 6 ) + Qgh , (7)

where only the zero mode contributions have been explicitly shown. The action for the closed free bosonic string field without restrictions on the parameters of the gauge transformation is formulated in terms of a state I¢) with zero ghost number and an auxiliary state It/) with ghost number - 1 , and it reads [7]

A = - ( i /2x /~) ((¢1 [C0-, Q][@ + (1/x/2-)(~ IQlr/) - (1/x/~(r~lQl~)} • (8)

Notice that tlus action differs from that of ref. [6] in a redefinition of the auxiliary state [r~). It is invariant under

81~)=Qle> , ~ l r l )=b- l e>+Qle '> , (9)

[e) and [e') being arbitrary infinitesimal states with ghost number - 1 and - 2 , respectively. The invariance follows from Q2 = 0 (for D = 26) and {Co-, Q) = (1 /V~ ' )b - . The condition Q2 = 0 implies

a 2 = _ ~ f + , q 2 = ~ ( b + f + + b - f - ) ,

{a ,q} = [ b + , a ] = [b-,a] = [ y + , a ] = [f-,a] = [ b + , q ] = [b-,q] = b e + , q ]

= h e - , q ] = [ b + , b - ] = [ b + , f - ] = [ b - , f + ] = [ f + , f - I = 0 . (10)

We expand the fields in the form

I~>= (¢ + sc~ + R e ( + I ¢ -- + cff + + (11)

Performing the zero mode algebra in (8) we get 1

A - ~ - a (~1~) - } (q~lb+¢) + 2i(~laS) - 2(¢1qS) + ~-<¢lb-T> + ½ ( S I b - R ) - ~ (Sir+S>

- 1- (~l/~) + } (Olb+ta) - i (¢ lav) + (¢[qv) + } ( S l f - ~ > + i(Sla/~) + (S Iq/a) +} (¢[f- /~) z

+ ~ ( S l f + v > - ~ ~ ( R l b + a ) + i ( i ~ l a { 3 ) - ~ ( R l b - v ) - ( R t q f l )

1 - i(Tla&) - (T lqa) + ~ (T[f+[3) - ~ ( T [ b - g ) . (12)

3. The constraints. In ordertto construct the hamiltonian theory we consider canonical momenta (in the fol- lowing all the states should be understood as kets, unless explicitly stated):

I 1 p ¢ = ¢ + 2 i a S - ~ l J - i a v , P s = 0 , p R = - - ~ & + i a / J , P T = 0 ,

p~,=-}k+iar, pt~ = 0 ,

We get four primary constraints

PS =PT =Po = Pv = 0 ,

and four FL-projectable * 1 velocities

= - 2 p u - 2iaS, 1~ = -2p,~ + 2iaT,

The canonical hamiltoman is

p. =-i po=O.

= --2PR + 2iaO, /J = --2p¢ -- 4pu -- 2lay.

(13)

(14)

(15)

,1 The analysis of the lagrangian and hamiltoman constraints of a singular system is given in ref. [ 10].

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Volume 187, number 1,2 PHYSICS LETTERS B 19 March 1987

He = - 2 ( p u l P u) - 2(pulPed) - 2(PR IP~) + ~ (~lb+q ~) - ~- (q~lb+/~) - ½ CRib+a)

+ (S [2iapo + 2q¢ - ~- b - R - ~ f - a - q#) + (TI - 2lap R - ~ b-q~ + qa + ~ b - u )

+ (31 - 2iapa - ~ f - ¢ + q g ) + (vl2iap u - q¢a + ½ b - R ) . (16)

Using canonical Poisson brackets we can study the stability of the primary constraints. It should be noticed that a Poisson bracket only makes sense between a "ket" and a "bra", in such a way that (I), ( I )) gives a "ket". We get four chains of constraints

PS PS - 2 i a p o - 2 q q ~ + ½ b - R + ~ f - a + q u , i ~ s = 2 i a b + ~ - 2 q p ~ - i a b + l a - b - p a - f - p R p's = b +" , = , - P s ,

PT, /~T = 2iaPR + ~ b - qb - qa - ½ b - la , i;T = iab + a + b - P u + 2qPR + b-Pc; , ~b'T = -b+ibT ,

P3' PtJ =2iapce + ~ f - ( o - q R ' P 3 = i a b + R - f - P p + 2qp~ , 7e=-b+ ,ba ,

Pv, Pv =-2uzpu + q ¢ - ½ b - R , P, = - i a b + ¢ - 2qpu + b - P a , P'v = -b+Pv • (17)

Each chain ends with a tertiary constraint. Summing up, there are twelve constraints which are alifirst class. How- ever, as in the open string case, these constraints are not independent, giving rise to the appearance of "metacon- straints". Systems with metaconstraints are known in the literature as reducible systems [11]. Indeed, we get the following eight metaconstraints:

Atl) ~q/~s + ia/~'s + ~ f - b T + ~ b - b 3 = O , A~2) -=-lab+ks +q/~s + ~ f - P T + ½ b - p 3 - 0 ,

A(21) _ 1 _ . - ~b ps - qbT -- iaPT - ~ b - b v =O, A(22)=--½b- i~s+iab+~T-q~;T--½b- iJv=0,

A~l) - -q~3 - iab"# + ½f-~)u = O , A~2) - iab*~o - q~# + ~ f - ~ v = O ,

A ( 1 ) = - - ½ b - p p + ql)v + iai;v =--0, A(2)=---½b-iJo-- iab+bv + q l J u - O . (18)

It can be seen that the metaconstraints are stable

~}1) =A}2), ~}2)=_b+A(1), i= 1 ,2 ,3 ,4 . (19)

We can express (18) in a more compact way by defining a vector

V c =COs,P'S,bT,i~T,[93,[JO, bv,i3"v),

and a matrix

q

-iab+

- ½ b -

0 M =

} I - o

q o

0 - q - i a

1 _ - ~ b iab + - q

1 _ ~b 0 0 0

0 ~b - 0 0

0 0 - ~ b - 0 1 _ 0 0 0 - ~ b

1 - q - i a ~ f - 0

1 iab + - q 0 ~ f -

- ~ b - 0 q ia

0 -½ b - - iab + q

(20)

(21)

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Volume 187, number 1,2

Then the metaconstraints are given by

A =MV c = 0 .

It turns out that (see eq. (10))

PHYSICS LETTERS B 19March 1987

(22)

M2 = 0 , (23)

and then the metaconstraints obey the same relations as the constraints ("metametaconstraints") and so on. The same features are present in the open string case [8]. When this occurs the system is said to be infinitely reducible. The appearance of metaconstraints implies the existence of ghosts for ghosts in the quantum theory. In our case an infinite chain of ghosts for ghosts.., for ghosts will be present [1 ].

4. Gauge transformations, Next we will construct the gauge transformations (9) of our system using Castellani's algorithm [12]. We look for generators having the form

M G = D ~Cnla(n~>, (24)

n=0

where (n) means the nth time derivative. The pieces G n can be obtained starting with the primary constraints and following an iterative procedure. Usu-

ally, every primary first-class constraint gives rise to a generator of the form (24) and thus to an independent gauge transformation. We will see that systems with metaconstraints are more involved.

When applied to our case, the algorithm can start, for instance, with

GM = Ps • (25)

Then G M_ 1 must obey

GM_ 1 + (GM,Hc} = C(1)ps , (26)

where C(1) is a factor to be chosen ,2 . Then

GM-1 = - {Ps, He } + C(1)ps = -/~S + C(1)ps • (27)

The next piece is such that

GM_ 2 + {GM_ 1,Hc) = C(2)p S , (28)

that is

GM-2 = P'S + C(1)/~S + C(2)ps , (29)

where C(2) = - {C (1), Hc) + if(2). Going on we have

GM-3 = -'b'S - C(1)/~S - C(2)pS - { C(1), H c ) P S + O(Ps)" (30)

The algorithm ends when G 1 = 0 on PS = 0. We see that we can implement this condition if we choose C(1) = 0 and C (2) = b +. ThenM = 2 and the generator has three pieces:

G1 = (PslA) - (/~slA) + (Ps + b+Ps IA)" (31a)

Similarly, starting with the other primary constraints we could construct

63 = (PTIP) -- (PTIb) + (/;T + b+PT Ip), (31b)

,2 In fact we should add to the right-hand side arbitrary combinations of all the primary constraints, as in the Yang-Mills case. However, in our case the chains do not mix and this is not necessary.

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Volume 187, number 1,2 PHYSICS LETTERS B 19March 1987

G5 = (Pt3IA') - (/~alX t) + (/~# + b+pt~lA'), (31c)

G7 = <PvIP") - (/~vl/~') + (/~v + b+PvlP'). (31d)

When no metaconstraints are present this gives all the gauge transformations of the theory. However, it can be checked that with the generators (31) we cannot construct the whole gauge transformation (9). The metacon- straints allow us to start a chain with a primary constraint multiplied by a nontrivial factor. For instance, try

GM = lap s • (32)

Then

GM-1 = -ia/~s + C(1)ps + C(2)pT + C(3)P# + C(4)Pu, (33)

and

GM_ 2 = iai~s - C(1)/~ s - C(2)/~T - C(3)/~# - C(4)/~ v + primary constraints. (34)

Looking at (18) we see that we can end the algorithm if we choose C(1) = - q , C(2) = - ½ f - , C (3) = - ½b- , C (4) = 0 and then we get a generator with two pieces

1 1 G2 = (iaPslX) - (ia/~s + qPS + ~ f - P T + ~ b -p# lx ) .

In the same way we can construct

G4 = (lap T I~) - (iaibT + ½ b-Ps + qPT + ½ b-pvl ~o>,

1 t G 6 = (iapt~lX') - (ia/~# + qp# - ~ f -Pv l× ) ,

• " t | t G8 = (lapvl6° ) -- (ia/)v + qPv -- ~ b-pfjI6o ) .

(35a)

(35b)

(35c)

(35d) Now it can be checked that the generators Gt, i = 1, ..., 8, give the whole gauge transformation (9) * a with the identification

l e) = [-Vr2A - (1/x/~-) xC~ + X/2pC~ - (1/x/~') t~C~C~- ] I - + ) , (36)

and similarly for l e'>.

5. The BRST generator. The knowledge o f the canonical constraints o f a theory and their relations (structure functions and metaconstraints) allows us to construct its BRST generator (see ref. [13] for a review). For each constraint and for each metaconstraint in the infinite tower produced b y M 2 = 0 we must introduce a canonical pair o f ghosts. An index (i) will distinguish the four chains generated by the four primary constraints and also the four chains of metaconstraints at each level. Then the canonical pairs are

(~ (0 7~(i) ) (c(i) ° ( ' )~ (Cn (0, 5 ~(0) (37) 0, aux 0,aux , - n ' ~ n " '

for n = 0 . . . . . oo. The bar indicates negative ghost number. A ghost in the nth level has -+ (n + 1) ghost number. Us- ing a matrix notat ion the BRST generator is given by

QB~RST = (~(i?auxIP(i)) + M _n-1

The extended BRST invariant hamiltonian is then

, a A chain starting wi th iqp S will also give a two-piece generator, bu t this is only a redefinit ion of the gauge parameter.

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Volume 187, number 1,2 PHYSICS LETTERS B 19 March 1987

4

"=Hc+ ~ l'-'0,aux'[¢~(i) 1~(i)~+0" n=0 ~ [(sD(i)l~(i))-(-1)n(c(OIb+C(nO)]) (39)

Finally, a gauge fixed hamiltonian can be constructed from QBRST and H:

H e f t H - { ~ k , FT = QBRST ) , (40)

where ~ is the fermionic gauge fixing function. Due to the structure of the primary constraints, the natural gauge fixing is

4 ¢ = ~ (C0(i)lh.(O), (41)

i--1

where x(i) = S, T, fl, v. The fmal result is

Hef f = - 2 ( p ~ IP~) - 2(p~ Ipo) - 2(PR [Pa) + [ (¢ Ib+¢) - ~- (~lb+/a) - ~ (R Ib+a)

+ ~ [(~(O[~(i))-(-1)n(C(ni)lb+ff(i)) ] . n=0

(42)

This hamiltonian can also be obtained as the canonical hamiltonian associated to the gauge fixed BRST invariant lagrangian [14] in the gauge where S = T =/~ = v = 0. In order to obtain Siegel's gauge fixed action [15] another fermionic gauge timing ~ must be taken in such a way that all the components of 17) be eliminated.

6. Comments. Our analysis shows that all the constraints of the free closed bosonic string field are fftrst class, like in the open case. The auxiliary state 177) is a necessary ingredient in order to get the infinite chain of metacon- straints, i.e. the infinite set of ghosts for ghosts.., for ghosts, which guarantees the correct count of degrees of freedom.

We think that our analysis can be of some interest to construct the gauge transformations of the interacting string in the hamiltonian formalism.

We acknowledge useful discussions with Dr. K. Wu. This work has been partially supported by CAICYT pro- ject number AE 86-0016.

References

[1 ] A. Neveu, H. Nicolai and P. West, Phys. Lett. B 167 (1986) 307. [2] E. Witten, Nucl. Phys. B 268 (1986) 253. [3] M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443;

S. Hwang, Phys. Rev. D 28 (1983) 253. [4] Y. Kazama, A. Neveu, H. Nicolai and P. West, Nucl. Phys. B 276 (1986) 366. [5] E. Witten, Nucl. Phys. B 276 (1986) 291. [6] A. Ballestrero and E. Maina, Phys. Lett. B 180 (1986) 53. [7] C. Batlle and J. Gomis, Universitat de Barcelona preprint UB-FTFP-9/86. [8] I. Bengtsson, Phys. Lett. B 172 (1986) 342. [9] A. Neveu and P. West, CERN preprint TH-4358/86.

[10] C. BatUe, J. Gomu, J. Pons and N. Rom~in-Roy, CCNY preprmt HEP-12/86, to be pubhshed in J. Math. Phys. [11] I.A. Batalin and E.S. Fradkln, Phys. Lett. B 122 (1983) 157. [12] L. Castellani, Ann. Phys. (NY) 143 (1982) 357. [13] M. Henneaux, Phys. Rep. 126 (1985) 1. [14] T. Kugo and S. Uehara, Nucl. Phys. B 197 (1982) 378. [15 ] W. Siegel, Phys. Lett. B 151 (1985) 396.

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