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BRST quantization of a bilocal model inspired by the relativistic string
Volume 186, number 1
PHYSICS LETTERS B
26 February 1987
BRST QUANTIZATION OF A BILOCAL MODEL INSPIRED BY THE RELATIVISTIC STRING ~r R. CASALBUONI a,b and D. DOMINICI a,b, 1 a Istituto Nazionale di Fisica Nucleate, Sezione di Firenze, Florence, Italy b D~partement de Physique Thdorique, Universitd de Gen~ve, CH-1211 Geneva 4, Switzerland
Received 7 October 1986
A bilocal model derived projecting out a particular state of motion of the relativistic string is considered. In principle the model describes integer spin massless states. However, its BRST quantization is shown to be consistent only for three values for the spacetime dimensions, and the corresponding solutions describe a rank-2 symmetric tensor (/9 = 2), a vector (D = 4), and a scalar (D = 6). 1. Introduction. r h e recent advances in relativistic string theories * x have brought a lot of interest in the problem of quantizing constrained systems. The main problem here is to write down many equations for a single wave function (symbolically ~bal~ ) = 0). This problem can be solved, in principle, by enlarging the space o f the wave functions in order to be able to write down an equation o f the kind ~2i/I if/) = 0 in such a way that the original constraints are a consequence o f this equation and furthermore that the spurious components decouple. The practical way o f implementing this program is BRST quantization [2] , 2 . However obstructions can arise. In particular, it can be difficult to arrange the symmetries o f the problem in such a way that all the spurious components cancel out. However, it can be shown [2], that in the case o f commuting constraints a solution always exists. What we would like to stress in this paper is that obstructions can arise not only for systems with infinite number o f degrees of freedom, but for finite
Work supported by the Swiss National Science Foundation. 1 On leave of absence from Dipartimento di Fisica, Universit~ di Firenze, Florence, Italy. :l:t For a review see ref. [1]. ,2 For a review see ref. [3]. For the application of the BRST technique to the relativistic string theories see refs. [4,5 ]. A more complete set of references and a general survey of string field theory can be found in ref. [6]. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
systems too. The finite system we will discuss here has been extensively studied in ref. [7], and it consists in projecting out a particular state o f motion of the string, namely the state in which the string displaces itself in a rigid way (this will be explained in a more detailed way in the following). This particular motion is completely described by the coordinates o f the two ends o f the string. The rigidity implies that the center of mass, as the two ends of the string, moves at the velocity of light. As a consequence only massless states are described. Furthermore, the condition at the ends, o f no m o m e n t u m flow in the string direction, implies that the momenta of the ends are orthogonal to the string. This condition has the'further implication that the distance between the two ends must be light-like (in this sense we speak of rigidity, that is the square of the four-distance is fixed and equal to zero). Summarizing, the system is described b y six first class constraints. The naive quantization o f this model [7] shows that for given choices of the ordering in the relative modes one obtains a zero mass trans. verse state of helicity +X. We will show that in this system, as in the relativistic string, BRST quantization can be consistently performed only if a precise relation exists between the space-time dimensionality and the ordering constant. The study of a particular solution of ~21 if) = 0, where ~2 is the BRST operator, shows that the obstruc. 63
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PHYSICS LETTERS B
tion can be avoided only for three values of the spacetime dimensions, D = 2,4, 6, and that the corresponding solutions describe a rank-2 symmetric tensor (D = 2), a vector (D = 4) and a scalar (D = 6). At the end of the paper a more general analysis of the ~l ~ ) = 0 equation for the lower lying states is done, making use of the gauge transformation properties of the states. This analysis confirms the previous results.
2. Classical description o f the model We recall here the main features of the model we want to analyze. A complete treatment can be found in ref. [7]. If one considers a relativistic string in a rigid configuration one has dx(o, r)/do = x 2 ( r ) -- x 1 ( r ) , where x 1 and x 2 are the two end points. Then, taking the lagrangian proportional to the area spanned by the vector x 2 (r) - x I (r), subject to the condition that dx 1 and dx 2 lie on the same plane in spacetime, one gets r = -(1/4t~')
(2.1)
Using center of mass and relative coordinates x = (x 1 + x2)/2 , and z = (x 2 - Xl)/2 one can easily verify that the following primary constraints hold:
( p x , z ) = ( p x , p z ) = ( p z , z ) = e 2 x + p2 + z2/ot'2 = 0 , (2.2) where Px = -aL/O:~ a n d P z = -~L/a~.. Osing standard Poisson brackets
( x U , P x } = { z U , P ~ ) = _gUy,
this vertex is exactly the Yang-Mills vertex when one selects states of helicities -+1 and generalizes the theory to a non-abelian one.
3. Quantum constraint algebra. The algebra of constraints is better analyzed in a complex basis: a ' = (1/V~-r)(a'Pz~ - izU).
(2.3)
one gets from the stability of (2.2) the secondary constraints
L 0 = (a+' a),
£"0 = p 2 ,
L 1 = (p,a),
L 2 =a 2 ,
L_ 1 =L~,
L_ 2 =L~.
(2.4)
Notice that the constraints (2.2) and (2.4) form a closed algebra under Poisson brackets, which means that they are first class constraints. Various properties of this model were studied in ref. [7]. The quantization was performed and it was shown that one can have different solutions corre. sponding to zero mass states and helicities -+X. Using the path-integral formalism we were able to evaluate the three-particle vertex by joining two rigid strings to create a third one. It has also been shown that
[Lo,L+_n] = O,
[Ln' L - n ] = - L o b n , 1 - (4LO - 219) 5 n,2 , [L 1, L_2 ] = - 2 L _ l •
(3.3)
H = (Lo,L2,L
2) and R = ( L o , L 1 , L _ I ) form two subalgebras of the original algebra and [H, H] C H, [H, R] C R, [R, R] C R, Notice also that the onumber in the second equation (3.3) is not a real anomalous term begause it can be reabsorbed into the definition o f L o , in other words the algebra does not have non-trivial projective representations. The classical BRST operator is simply given by [2=ff'~r/~
-
~
½fav0S0.r~T0rTa,
(3.4)
a , ~, "/
where ~,~ are the first-class constraints,f~0 are the structure constants of the Lie algebra defined by the constraints, and r/a, ~ , are the ghost variables satisfying { ~ a , 770} = - 8 ~ . Eq. (3.4) goes into the corresponding quantum quantity modulus ordering problem. Here the only ambiguity can arise f r o m L 0. By putting (n > O) ,(In = c +, n - n = Cn ' ~ r ~ -iCn, ~ n : _i~n+, rl0 + CO, ~0 + do , 7 0 = _i~0, 5~0 : _ i ~ 0 , with c o = CO, d o = do, 50 = b~, d0 = d~, one finds = f, oCo + r o a o + a4~ o + ~ 3 o + ~ ,
where 64
(3.2)
Going from the classical Poisson brackets to the quantum commutators and defining a ground state for the oscillators by au[ 0) = 0, one gets the algebra
a
P~x =Pz2 = 0 .
(3.1)
It is convenient to redefine the constraints by taking appropriate linear combinations, and introduce the following notations:
[Lo,L+n ] =++-nL+n,
X { [(Xl,X2 - Xl)2 - ;~21(x 2 - Xl)211/2 + (x 1 ~+x2)} •
26 February 1987
(3.5)
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26 February 1987
~l phys) = (c~L 2 + c~L 1)l phys), £0 =L0 +~
-
-
M=4c~c2,
= S,
2 n(C+n-Cn + c-+ hert) , n=l,2
~t = c'~cl ,
(c*,z.,
+
n=l,2
+ 26~'clc 2 + 2C~Cle 1 .
(3,6)
Here/3 is the parameter taking into account the possible ambiguity in defining the L 9 operator at the quantum level. Notice that I2 = I2. The algebra of the quantities (3.6) can be easily worked out and one finds
[£0 ,M] = [£0 ,j~r] = [/0, fi] = N.fi.]
=
(3.11)
which tells us L 1 Iphys) = L 2 [phys) = 0. Notice that /~01 phys) = (L 0 + B)lphys) = 0, and in order to satisfy this equation/3 must be a positive integer. Then, eq. (3.11) requires (D = 2, ~ = 2), (D = 4, # = 1), (D = 6, # = 0). The physical states belonging to 50, as defined above, were studied completely in refs. [4-7], and the result is that the previous three cases correspond to massless particles of helicities +2, +1,0, respectively. The question arises to study more completely the equation ~21¢) = 0 in the space 9, and the transformation properties of the various spurious states following from the invariance 81 ¢) = ~IA) of the BRST equation. This will be made in the next section for the lower lying states.
=o.
4. Equations o f motion and gauge transformations. To discuss the general consequences of I21 ¢ ) = 0, withl¢) E 9 , we expand along the c o andd 0 modes:
Then, it is easy to evaluate ~22 : 122 = £ 0 M + ~"0.~ + ~2 =c~c2 (12 - 4/~- 2D) (3.7) and therefore
I¢)-- (c} + ~ e 0 + ~ d 0 + ~ c 0 d o ) l _ ) l = ) .
(4.1)
Applying 12 to I¢) one gets the following equations:
= 3 - D/2.
(3.8)
/~Oe# + f i * - M,:b~= O,
When this relation is satisfied we find ~2 = _i,0M _ L"0/~.
(3.9)
LO* + g~
+M~ = 0,
Then, a particular solution to the equation I21 ¢) = 0 can be obtained in analogy to the string case (see ref. [4]). First of all we define the ghost number operator
~ 0 - Mg' - .~@ = O,
Ng = n =~l , 2 (C+nCn_
In order to extract the physical informations from these equations one has to further expand the fields with respect to the other ghost modes, using the fact that one defines I¢) havingNg = - 1 . Then, one has the following expansions:
+ (e0e0
C-+ n Cn )
-eoco)+½(aoao-dodo).
(3.10)
Then, we take a particular Fock space by removing the fourfold degeneracy due to the Co, d o oscillators, as ~: { I - ) l : ) l ¢); c01-) =d01=) = 0); here 1¢) is a generic state in the Fock space of all the other oscillators. Notice that I-)1:)hasNg = - 1 . However, can be stable under the action of[2 only if£0[ ¢) = L01 ¢) = 0. Calling by 50 this subspace of ~ we see that [250= ~ a n d that ~2 = 0 on 50. The physical states in 50 must satisfy ~l phys) = 0, therefore requiring that the physical states are annihilated by the ghost annihilation operators c n Iphys) = 6n Iphys) = 0, one finds
~
+Log' - £0 ~ = O.
(4.2)
.1 + =+ 2 + -+ q B = ( ~ + w l C l C l +tPlClC 2 1 +-+ 2 + - + + .12c+ +-+ . . . . . +~P2C2Cl +~P2C2C2 ~v12 lC2ClC2)lU?, ~12 + - + - + = (¢1e~ + ¢2C2" + ~1 C l C l e 2 12 + - + - +
+ ¢2 C2CLC2)10), ~=(~1~+¢2~'+¢1
= ~12 g~-e~" .
~12 + - + - + ClClC2)IO),
(4.3) 65
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Here 10) is the vacuum state for the ghost oscillators. Plugging these expressions into (4.2) one gets the equations satisfied by the component fields in (4.3); these components are bilocal and therefore they have still the dependence on the bosonic oscillators a~, and a further expansion is needed. We will limit our. selves to expand only the first few components. It can be seen that the following terms in the expansion gives rise to a closed set of equations of motion:
~o= (~o + a+uAu + a+a+A uv + ...)10> =
+ ...)Io>,
41 = (41 + a ~ 4 1 # + ...)10) , 51 =('~1 + a ~ 5 1 , + ...)10) , 4 2 = (4 2 + ...)10>,
52 = (~2 + ...)10>,
(4.4)
with 10) the vacuum state for the bosonic oscillators, Here we have used the same notations for the fields and their vacuum components, but the difference should be clear from the context. Using (4.4) and (4.3) in (4.2), after some algebra, we get the following equations in the various sectors:
Scalar field: j~p=0,
26 February 1987
( t 3 - - 2 ) ~ 1 - - p 2 4 1 = ( ~ - 2 ) ~ 2-p2~b2 = 0 , --2)~o~ - pu4ul -- 242 = 0 , 1 1 ([3- 2)hta v +guv42 +~(p~4z, + pv41)=O,
0+~o]-242=0, ( f l - 2)0 + D 4 2 +pU41 = 0 ,
(4.7)
where we have decomposedAuv = huv + (1/D)glavO , with 0 the trace of Auv and huv is traceless. It is worth to notice the role played in these equations by the particular values of ~, ~ = 0, 1,2. In such cases it turns out impossible to solve explicitly for one of the fields. Notice that for the scalar sector we have the equations ~o= 0 for ~ ¢ 0 and p2~0 = 0 for/3 = 0. In the vector sector, for # = 1 we get the description of an abelian gauge field through an auxiliary scalar field. To have a better understanding of the situation it is convenient to study the response of 1~I'> to the gauge transformations 6 [• >= ~ IA>, where now Ngl A) = - 2 IA>. Then using the expansion IA) = ( a + Xc o + ~d0)l->l ~-)
(4.8)
one has
84 = - L O A + ~ X ,
p2~o= 0 .
(4.5)
8~ = -/~0 A + fi
,
Vector field : p2A u+pu51 =0,
8g=£0X+LoX.
pUA u + 5 1 = 0 ,
O - 1)Au +Pt~ 1 =0,
([J- 1)~" I - p 2 4 I = 0 . (4.6)
Tensor field: p20 + D ~ 2 + pU'~ 1 = 0 , 1 ~ 1 + pu41u - (2/D)g~vpo'~lo)= O, pZhuv + ~(pu4u
pUh,v + }[pv(O + ~o1) + 4v ] = O, p2~ol - pU~ 1 - 2~ 2 = 0 ,
(4.9)
From (4.9) it can be seen that to study the transformations of the fields appearing in (4.5}--(4.7) it is enough to expand A up to the first two terms in the ghost oscillators: A = (A16~" + A2e~" + ...)10).
(4.10)
The transformations in the various sectors are:
Scalar sector: a
=0.
(4.11)
Vector sector: fiat, = puA 1 ,
841=-p2A1,
84 I = ( 1 - ~ ) A
1.
(4.12) 66
Volume 186, number 1
PHYSICS LETTERS B
26 February 1987
transformation rules for the bilocal fields in eq. (4.14):
Tensor sector: 6h~v = ~1 (PuAv1 + pvA 1 - (2/D)guvpoAlo) ,
6A 1 = _ L _ 2 A ( 1 ) ,
6A 2 = L _ I A ( 1 ) .
(4.17)
It is easy to see that the lowest components o f A 1 and A 2 are not transformed (that is the local fields A 1 , Au and A2), and there are no further constraints on these parameters. We can now study eqs. (4.5)-(4.7):
60 =puAlv + D A 2 , &Pl = - P U A l u - 2 A 2 ' 6~ 2 = (2 -/3) A 2 ,
6~ "2 = - p 2 A 2 ,
6¢1 = ( 2 _ 3)A 1 '
6"~lu=-p2A1u .
(4.13)
where A t , A 2 , and Alu come from the expansion
/3 --- 0, D = 6. Using the gauge transformations (4.11)-(4.13) one can put ~1 = ~2 = ¢ l u = 0. Then, from the equations of motion one gets p29 = 0,
A 1 = (A 1 + 0+~A 1 + ...)10>,
A = A I ~ v = ~1 .- ~Ou 1 =-~ l = ~ 2 = 0 A 2 = (A 2 + ...)[0>.
(4.14)
One can easily check that the gauge invariance for the eqs. (4.5)-(4.7) would be valid for arbitrary/3, except for the equation for 0 in (4.7): 6(0 + ~oI - 2~ 2) = (D - 2 ) A 2 - 2(2 - / 3 ) A 2 , (4.15) which is invariant only if/3 = 3 - D / 2 . Notice also the crucial role played by/3 = 1,2 in the transformation laws. In fact at such values some of the fields become gauge invariant. This will be of vital importance in order to determine the solutions to the equations of motion. Armed with all this machinery we can now study the structure of the eqs. (4.5)-(4.7) for the various values of 3. The general strategy will be to use the arbitrary functions A 1, A 2 and A 1 to gauge to zero two scalars and one vector. However, the choice one can make depends on/3. Furthermore, one has to check if the possibility of the "gauge invariance for gauge invariance" mechanism arises here. In general we can perform a chain of transformations on the gauge parameter fields: IA) =- IA) -= IA(0)), 6 IA (0)) = ~[A(1)) ..... 6 [A(k)) = I2[A(k+l)). However, in the present situation the ghost number of IA (k)) must be equal to - k - 2. This means that in the expansion [A(k)) = A ( k ) l - ) l ~-), A (k) must have Ng = - k - 1. It is easy to check that we cannot have iV_ < - 2 , and therefore the chain must arrest at [A(1)~>. There is only one ghost operator withNg = - 2 : IA (1)) = A(1)~'5~" 10).
(4.16)
By applying ~ to IA(1)) one obtains the following
(4.18)
The theory describes a single massless scalar field. /3 =1, D =4. Now ~01 is invariant and we have to change the gauge fixing. Let us choose ~1 = ~k2 = ~1 u = 0. Then, from the equations of motion we get
p2A~z = O ,
pUA• = O ,
pu~ 1 =0.
(4.19)
The theory describes a gauge vector particle in the Lorentz gauge plus a constant scalar field. B "--2, D = 2. Let us fix the gauge choosing ¢I = 0, ¢1 = ~ = 0. Using eqs. (4.5)-(4.7) we get
p20=p2huv=O,
p#huv + p v O = O ,
p2@l = 0 ,
~(Pu t~l + Pv tpl ) = -guy O"
(4.20)
The last equation gives three independent conditions that in light-cone coordinates read (p+ = (1/x/2)(p 0
-+Pl)) p+ +l
=o,
p+~l_ + p_tk+l = - 2 0 ,
(4.21)
The first two equations can be solved to give ~+1 =f+ ( p _ ) 6 (p+),
t,b l = f _ (p+) 6 ( p _ ) .
(4.22)
67
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PHYSICS LETTERS B
Solving the zero mass equation for 0 we get 0 = O+(p_) 8(p+) + O_(p+) 6 ( p _ ) ,
(4.23)
c±6(p;),
(4.24)
and we see that apart from the constant terms the vector ~u1 is completely fixed by 0. Therefore we get a traceless massless tensor huv and a massless scalar 0. In our previous particular solution we were assuming that the wave function had no ghost components, i.e. f f l = 0; with this assumption we get 0 = 0 and the two solutions coincide. To conclude, we see that this analysis gives a result more general than we got from the particular solution. We have not done a complete analysis of the higher sectors, however they should not contain gauge invariant fields when one restricts/3 to its critical values. Therefore, we suspect that all the other fields decouple. As a last remark, we cannot exclude a more sophisticated modification of the classical BRST charge al-
68
lowing the quantization of this model for arbitrary values offl a n d D .
References
from which f± ( p ; ) = - ( 2 / p ; ) O± (p$) +
26 February 1987
[1] J.H. Schwarz, Phys. Rep. 89 (1982) 223; M. Green, Surveys High En. Phys. 3 (1983) 127. [2] C. Becchi, A. Rouet and R. Stora, Phys. Lett. B 52 (1974) 344; I.V. Tyutin, Gauge invariance in field theory and in statistical physics in the operator formulation, Lebedev preprint FIAN n. 39 (1975) [in Russian], unpublished; E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. B 55 (1975) 224; I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B 69 (1977) 309; E.S, Fradkin and T.E. Fradkina, Phys. Lett. B 72 (1978) 343. [3] M. Henneanx, Phys. Rep. 126 (1985) 1. [4] M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443. [5] W. Siegel, Phys. Lett. B 149 (1984) 157,162; B 151 (1985) 391,396. [6] P. West, CERN preprint TH. 4460/86 (1986). [7] R. Casalbuoni and G. Longhi, Nuovo Cimento 25A (1975) 482; R. Casalbuoni, D. Dominici and G. Longhi, Nuovo Cimento 32A (1976) 265.
PHYSICS LETTERS B
26 February 1987
BRST QUANTIZATION OF A BILOCAL MODEL INSPIRED BY THE RELATIVISTIC STRING ~r R. CASALBUONI a,b and D. DOMINICI a,b, 1 a Istituto Nazionale di Fisica Nucleate, Sezione di Firenze, Florence, Italy b D~partement de Physique Thdorique, Universitd de Gen~ve, CH-1211 Geneva 4, Switzerland
Received 7 October 1986
A bilocal model derived projecting out a particular state of motion of the relativistic string is considered. In principle the model describes integer spin massless states. However, its BRST quantization is shown to be consistent only for three values for the spacetime dimensions, and the corresponding solutions describe a rank-2 symmetric tensor (/9 = 2), a vector (D = 4), and a scalar (D = 6). 1. Introduction. r h e recent advances in relativistic string theories * x have brought a lot of interest in the problem of quantizing constrained systems. The main problem here is to write down many equations for a single wave function (symbolically ~bal~ ) = 0). This problem can be solved, in principle, by enlarging the space o f the wave functions in order to be able to write down an equation o f the kind ~2i/I if/) = 0 in such a way that the original constraints are a consequence o f this equation and furthermore that the spurious components decouple. The practical way o f implementing this program is BRST quantization [2] , 2 . However obstructions can arise. In particular, it can be difficult to arrange the symmetries o f the problem in such a way that all the spurious components cancel out. However, it can be shown [2], that in the case o f commuting constraints a solution always exists. What we would like to stress in this paper is that obstructions can arise not only for systems with infinite number o f degrees of freedom, but for finite
Work supported by the Swiss National Science Foundation. 1 On leave of absence from Dipartimento di Fisica, Universit~ di Firenze, Florence, Italy. :l:t For a review see ref. [1]. ,2 For a review see ref. [3]. For the application of the BRST technique to the relativistic string theories see refs. [4,5 ]. A more complete set of references and a general survey of string field theory can be found in ref. [6]. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
systems too. The finite system we will discuss here has been extensively studied in ref. [7], and it consists in projecting out a particular state o f motion of the string, namely the state in which the string displaces itself in a rigid way (this will be explained in a more detailed way in the following). This particular motion is completely described by the coordinates o f the two ends o f the string. The rigidity implies that the center of mass, as the two ends of the string, moves at the velocity of light. As a consequence only massless states are described. Furthermore, the condition at the ends, o f no m o m e n t u m flow in the string direction, implies that the momenta of the ends are orthogonal to the string. This condition has the'further implication that the distance between the two ends must be light-like (in this sense we speak of rigidity, that is the square of the four-distance is fixed and equal to zero). Summarizing, the system is described b y six first class constraints. The naive quantization o f this model [7] shows that for given choices of the ordering in the relative modes one obtains a zero mass trans. verse state of helicity +X. We will show that in this system, as in the relativistic string, BRST quantization can be consistently performed only if a precise relation exists between the space-time dimensionality and the ordering constant. The study of a particular solution of ~21 if) = 0, where ~2 is the BRST operator, shows that the obstruc. 63
Volume 186, number 1
PHYSICS LETTERS B
tion can be avoided only for three values of the spacetime dimensions, D = 2,4, 6, and that the corresponding solutions describe a rank-2 symmetric tensor (D = 2), a vector (D = 4) and a scalar (D = 6). At the end of the paper a more general analysis of the ~l ~ ) = 0 equation for the lower lying states is done, making use of the gauge transformation properties of the states. This analysis confirms the previous results.
2. Classical description o f the model We recall here the main features of the model we want to analyze. A complete treatment can be found in ref. [7]. If one considers a relativistic string in a rigid configuration one has dx(o, r)/do = x 2 ( r ) -- x 1 ( r ) , where x 1 and x 2 are the two end points. Then, taking the lagrangian proportional to the area spanned by the vector x 2 (r) - x I (r), subject to the condition that dx 1 and dx 2 lie on the same plane in spacetime, one gets r = -(1/4t~')
(2.1)
Using center of mass and relative coordinates x = (x 1 + x2)/2 , and z = (x 2 - Xl)/2 one can easily verify that the following primary constraints hold:
( p x , z ) = ( p x , p z ) = ( p z , z ) = e 2 x + p2 + z2/ot'2 = 0 , (2.2) where Px = -aL/O:~ a n d P z = -~L/a~.. Osing standard Poisson brackets
( x U , P x } = { z U , P ~ ) = _gUy,
this vertex is exactly the Yang-Mills vertex when one selects states of helicities -+1 and generalizes the theory to a non-abelian one.
3. Quantum constraint algebra. The algebra of constraints is better analyzed in a complex basis: a ' = (1/V~-r)(a'Pz~ - izU).
(2.3)
one gets from the stability of (2.2) the secondary constraints
L 0 = (a+' a),
£"0 = p 2 ,
L 1 = (p,a),
L 2 =a 2 ,
L_ 1 =L~,
L_ 2 =L~.
(2.4)
Notice that the constraints (2.2) and (2.4) form a closed algebra under Poisson brackets, which means that they are first class constraints. Various properties of this model were studied in ref. [7]. The quantization was performed and it was shown that one can have different solutions corre. sponding to zero mass states and helicities -+X. Using the path-integral formalism we were able to evaluate the three-particle vertex by joining two rigid strings to create a third one. It has also been shown that
[Lo,L+_n] = O,
[Ln' L - n ] = - L o b n , 1 - (4LO - 219) 5 n,2 , [L 1, L_2 ] = - 2 L _ l •
(3.3)
H = (Lo,L2,L
2) and R = ( L o , L 1 , L _ I ) form two subalgebras of the original algebra and [H, H] C H, [H, R] C R, [R, R] C R, Notice also that the onumber in the second equation (3.3) is not a real anomalous term begause it can be reabsorbed into the definition o f L o , in other words the algebra does not have non-trivial projective representations. The classical BRST operator is simply given by [2=ff'~r/~
-
~
½fav0S0.r~T0rTa,
(3.4)
a , ~, "/
where ~,~ are the first-class constraints,f~0 are the structure constants of the Lie algebra defined by the constraints, and r/a, ~ , are the ghost variables satisfying { ~ a , 770} = - 8 ~ . Eq. (3.4) goes into the corresponding quantum quantity modulus ordering problem. Here the only ambiguity can arise f r o m L 0. By putting (n > O) ,(In = c +, n - n = Cn ' ~ r ~ -iCn, ~ n : _i~n+, rl0 + CO, ~0 + do , 7 0 = _i~0, 5~0 : _ i ~ 0 , with c o = CO, d o = do, 50 = b~, d0 = d~, one finds = f, oCo + r o a o + a4~ o + ~ 3 o + ~ ,
where 64
(3.2)
Going from the classical Poisson brackets to the quantum commutators and defining a ground state for the oscillators by au[ 0) = 0, one gets the algebra
a
P~x =Pz2 = 0 .
(3.1)
It is convenient to redefine the constraints by taking appropriate linear combinations, and introduce the following notations:
[Lo,L+n ] =++-nL+n,
X { [(Xl,X2 - Xl)2 - ;~21(x 2 - Xl)211/2 + (x 1 ~+x2)} •
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(3.5)
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~l phys) = (c~L 2 + c~L 1)l phys), £0 =L0 +~
-
-
M=4c~c2,
= S,
2 n(C+n-Cn + c-+ hert) , n=l,2
~t = c'~cl ,
(c*,z.,
+
n=l,2
+ 26~'clc 2 + 2C~Cle 1 .
(3,6)
Here/3 is the parameter taking into account the possible ambiguity in defining the L 9 operator at the quantum level. Notice that I2 = I2. The algebra of the quantities (3.6) can be easily worked out and one finds
[£0 ,M] = [£0 ,j~r] = [/0, fi] = N.fi.]
=
(3.11)
which tells us L 1 Iphys) = L 2 [phys) = 0. Notice that /~01 phys) = (L 0 + B)lphys) = 0, and in order to satisfy this equation/3 must be a positive integer. Then, eq. (3.11) requires (D = 2, ~ = 2), (D = 4, # = 1), (D = 6, # = 0). The physical states belonging to 50, as defined above, were studied completely in refs. [4-7], and the result is that the previous three cases correspond to massless particles of helicities +2, +1,0, respectively. The question arises to study more completely the equation ~21¢) = 0 in the space 9, and the transformation properties of the various spurious states following from the invariance 81 ¢) = ~IA) of the BRST equation. This will be made in the next section for the lower lying states.
=o.
4. Equations o f motion and gauge transformations. To discuss the general consequences of I21 ¢ ) = 0, withl¢) E 9 , we expand along the c o andd 0 modes:
Then, it is easy to evaluate ~22 : 122 = £ 0 M + ~"0.~ + ~2 =c~c2 (12 - 4/~- 2D) (3.7) and therefore
I¢)-- (c} + ~ e 0 + ~ d 0 + ~ c 0 d o ) l _ ) l = ) .
(4.1)
Applying 12 to I¢) one gets the following equations:
= 3 - D/2.
(3.8)
/~Oe# + f i * - M,:b~= O,
When this relation is satisfied we find ~2 = _i,0M _ L"0/~.
(3.9)
LO* + g~
+M~ = 0,
Then, a particular solution to the equation I21 ¢) = 0 can be obtained in analogy to the string case (see ref. [4]). First of all we define the ghost number operator
~ 0 - Mg' - .~@ = O,
Ng = n =~l , 2 (C+nCn_
In order to extract the physical informations from these equations one has to further expand the fields with respect to the other ghost modes, using the fact that one defines I¢) havingNg = - 1 . Then, one has the following expansions:
+ (e0e0
C-+ n Cn )
-eoco)+½(aoao-dodo).
(3.10)
Then, we take a particular Fock space by removing the fourfold degeneracy due to the Co, d o oscillators, as ~: { I - ) l : ) l ¢); c01-) =d01=) = 0); here 1¢) is a generic state in the Fock space of all the other oscillators. Notice that I-)1:)hasNg = - 1 . However, can be stable under the action of[2 only if£0[ ¢) = L01 ¢) = 0. Calling by 50 this subspace of ~ we see that [250= ~ a n d that ~2 = 0 on 50. The physical states in 50 must satisfy ~l phys) = 0, therefore requiring that the physical states are annihilated by the ghost annihilation operators c n Iphys) = 6n Iphys) = 0, one finds
~
+Log' - £0 ~ = O.
(4.2)
.1 + =+ 2 + -+ q B = ( ~ + w l C l C l +tPlClC 2 1 +-+ 2 + - + + .12c+ +-+ . . . . . +~P2C2Cl +~P2C2C2 ~v12 lC2ClC2)lU?, ~12 + - + - + = (¢1e~ + ¢2C2" + ~1 C l C l e 2 12 + - + - +
+ ¢2 C2CLC2)10), ~=(~1~+¢2~'+¢1
= ~12 g~-e~" .
~12 + - + - + ClClC2)IO),
(4.3) 65
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Here 10) is the vacuum state for the ghost oscillators. Plugging these expressions into (4.2) one gets the equations satisfied by the component fields in (4.3); these components are bilocal and therefore they have still the dependence on the bosonic oscillators a~, and a further expansion is needed. We will limit our. selves to expand only the first few components. It can be seen that the following terms in the expansion gives rise to a closed set of equations of motion:
~o= (~o + a+uAu + a+a+A uv + ...)10> =
+ ...)Io>,
41 = (41 + a ~ 4 1 # + ...)10) , 51 =('~1 + a ~ 5 1 , + ...)10) , 4 2 = (4 2 + ...)10>,
52 = (~2 + ...)10>,
(4.4)
with 10) the vacuum state for the bosonic oscillators, Here we have used the same notations for the fields and their vacuum components, but the difference should be clear from the context. Using (4.4) and (4.3) in (4.2), after some algebra, we get the following equations in the various sectors:
Scalar field: j~p=0,
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( t 3 - - 2 ) ~ 1 - - p 2 4 1 = ( ~ - 2 ) ~ 2-p2~b2 = 0 , --2)~o~ - pu4ul -- 242 = 0 , 1 1 ([3- 2)hta v +guv42 +~(p~4z, + pv41)=O,
0+~o]-242=0, ( f l - 2)0 + D 4 2 +pU41 = 0 ,
(4.7)
where we have decomposedAuv = huv + (1/D)glavO , with 0 the trace of Auv and huv is traceless. It is worth to notice the role played in these equations by the particular values of ~, ~ = 0, 1,2. In such cases it turns out impossible to solve explicitly for one of the fields. Notice that for the scalar sector we have the equations ~o= 0 for ~ ¢ 0 and p2~0 = 0 for/3 = 0. In the vector sector, for # = 1 we get the description of an abelian gauge field through an auxiliary scalar field. To have a better understanding of the situation it is convenient to study the response of 1~I'> to the gauge transformations 6 [• >= ~ IA>, where now Ngl A) = - 2 IA>. Then using the expansion IA) = ( a + Xc o + ~d0)l->l ~-)
(4.8)
one has
84 = - L O A + ~ X ,
p2~o= 0 .
(4.5)
8~ = -/~0 A + fi
,
Vector field : p2A u+pu51 =0,
8g=£0X+LoX.
pUA u + 5 1 = 0 ,
O - 1)Au +Pt~ 1 =0,
([J- 1)~" I - p 2 4 I = 0 . (4.6)
Tensor field: p20 + D ~ 2 + pU'~ 1 = 0 , 1 ~ 1 + pu41u - (2/D)g~vpo'~lo)= O, pZhuv + ~(pu4u
pUh,v + }[pv(O + ~o1) + 4v ] = O, p2~ol - pU~ 1 - 2~ 2 = 0 ,
(4.9)
From (4.9) it can be seen that to study the transformations of the fields appearing in (4.5}--(4.7) it is enough to expand A up to the first two terms in the ghost oscillators: A = (A16~" + A2e~" + ...)10).
(4.10)
The transformations in the various sectors are:
Scalar sector: a
=0.
(4.11)
Vector sector: fiat, = puA 1 ,
841=-p2A1,
84 I = ( 1 - ~ ) A
1.
(4.12) 66
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26 February 1987
transformation rules for the bilocal fields in eq. (4.14):
Tensor sector: 6h~v = ~1 (PuAv1 + pvA 1 - (2/D)guvpoAlo) ,
6A 1 = _ L _ 2 A ( 1 ) ,
6A 2 = L _ I A ( 1 ) .
(4.17)
It is easy to see that the lowest components o f A 1 and A 2 are not transformed (that is the local fields A 1 , Au and A2), and there are no further constraints on these parameters. We can now study eqs. (4.5)-(4.7):
60 =puAlv + D A 2 , &Pl = - P U A l u - 2 A 2 ' 6~ 2 = (2 -/3) A 2 ,
6~ "2 = - p 2 A 2 ,
6¢1 = ( 2 _ 3)A 1 '
6"~lu=-p2A1u .
(4.13)
where A t , A 2 , and Alu come from the expansion
/3 --- 0, D = 6. Using the gauge transformations (4.11)-(4.13) one can put ~1 = ~2 = ¢ l u = 0. Then, from the equations of motion one gets p29 = 0,
A 1 = (A 1 + 0+~A 1 + ...)10>,
A = A I ~ v = ~1 .- ~Ou 1 =-~ l = ~ 2 = 0 A 2 = (A 2 + ...)[0>.
(4.14)
One can easily check that the gauge invariance for the eqs. (4.5)-(4.7) would be valid for arbitrary/3, except for the equation for 0 in (4.7): 6(0 + ~oI - 2~ 2) = (D - 2 ) A 2 - 2(2 - / 3 ) A 2 , (4.15) which is invariant only if/3 = 3 - D / 2 . Notice also the crucial role played by/3 = 1,2 in the transformation laws. In fact at such values some of the fields become gauge invariant. This will be of vital importance in order to determine the solutions to the equations of motion. Armed with all this machinery we can now study the structure of the eqs. (4.5)-(4.7) for the various values of 3. The general strategy will be to use the arbitrary functions A 1, A 2 and A 1 to gauge to zero two scalars and one vector. However, the choice one can make depends on/3. Furthermore, one has to check if the possibility of the "gauge invariance for gauge invariance" mechanism arises here. In general we can perform a chain of transformations on the gauge parameter fields: IA) =- IA) -= IA(0)), 6 IA (0)) = ~[A(1)) ..... 6 [A(k)) = I2[A(k+l)). However, in the present situation the ghost number of IA (k)) must be equal to - k - 2. This means that in the expansion [A(k)) = A ( k ) l - ) l ~-), A (k) must have Ng = - k - 1. It is easy to check that we cannot have iV_ < - 2 , and therefore the chain must arrest at [A(1)~>. There is only one ghost operator withNg = - 2 : IA (1)) = A(1)~'5~" 10).
(4.16)
By applying ~ to IA(1)) one obtains the following
(4.18)
The theory describes a single massless scalar field. /3 =1, D =4. Now ~01 is invariant and we have to change the gauge fixing. Let us choose ~1 = ~k2 = ~1 u = 0. Then, from the equations of motion we get
p2A~z = O ,
pUA• = O ,
pu~ 1 =0.
(4.19)
The theory describes a gauge vector particle in the Lorentz gauge plus a constant scalar field. B "--2, D = 2. Let us fix the gauge choosing ¢I = 0, ¢1 = ~ = 0. Using eqs. (4.5)-(4.7) we get
p20=p2huv=O,
p#huv + p v O = O ,
p2@l = 0 ,
~(Pu t~l + Pv tpl ) = -guy O"
(4.20)
The last equation gives three independent conditions that in light-cone coordinates read (p+ = (1/x/2)(p 0
-+Pl)) p+ +l
=o,
p+~l_ + p_tk+l = - 2 0 ,
(4.21)
The first two equations can be solved to give ~+1 =f+ ( p _ ) 6 (p+),
t,b l = f _ (p+) 6 ( p _ ) .
(4.22)
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Solving the zero mass equation for 0 we get 0 = O+(p_) 8(p+) + O_(p+) 6 ( p _ ) ,
(4.23)
c±6(p;),
(4.24)
and we see that apart from the constant terms the vector ~u1 is completely fixed by 0. Therefore we get a traceless massless tensor huv and a massless scalar 0. In our previous particular solution we were assuming that the wave function had no ghost components, i.e. f f l = 0; with this assumption we get 0 = 0 and the two solutions coincide. To conclude, we see that this analysis gives a result more general than we got from the particular solution. We have not done a complete analysis of the higher sectors, however they should not contain gauge invariant fields when one restricts/3 to its critical values. Therefore, we suspect that all the other fields decouple. As a last remark, we cannot exclude a more sophisticated modification of the classical BRST charge al-
68
lowing the quantization of this model for arbitrary values offl a n d D .
References
from which f± ( p ; ) = - ( 2 / p ; ) O± (p$) +
26 February 1987
[1] J.H. Schwarz, Phys. Rep. 89 (1982) 223; M. Green, Surveys High En. Phys. 3 (1983) 127. [2] C. Becchi, A. Rouet and R. Stora, Phys. Lett. B 52 (1974) 344; I.V. Tyutin, Gauge invariance in field theory and in statistical physics in the operator formulation, Lebedev preprint FIAN n. 39 (1975) [in Russian], unpublished; E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. B 55 (1975) 224; I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B 69 (1977) 309; E.S, Fradkin and T.E. Fradkina, Phys. Lett. B 72 (1978) 343. [3] M. Henneanx, Phys. Rep. 126 (1985) 1. [4] M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443. [5] W. Siegel, Phys. Lett. B 149 (1984) 157,162; B 151 (1985) 391,396. [6] P. West, CERN preprint TH. 4460/86 (1986). [7] R. Casalbuoni and G. Longhi, Nuovo Cimento 25A (1975) 482; R. Casalbuoni, D. Dominici and G. Longhi, Nuovo Cimento 32A (1976) 265.