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(Super-) field theories from (super-) twistors
Volume 220, number 4
PHYSICS LETTERS B
13 April 1989
(SUPER-) FIELD THEORIES FROM (SUPER-) TWISTORS Y. E I S E N B E R G and S. S O L O M O N ~: Department of Physics, Weizmann Institute of Science, Rehovot 76100, Israel Received 11 November 1988
We obtain higher spin linearized field theories through the BRST second quantization of the classical D = 4 massless point particle. By applying the same procedure to the classical supersymmetric particles, one obtains linearized N-extended off-shell unconstrained superfield actions. By extending the procedure to D = 10, one is led to a new spacetime geometry which reduces in some limit to the usual D = 10 Minkowski geometry.
First we reformulate the classical massless point particle in terms of twistors [ 1-5 ]. Then, we show that, upon quantization of this system, one can obtain, in addition to the massless scalar field equations, equations of motion equivalent to the Dirac, Maxwell, Rarita-Schwinger and linearized Einstein field equations. Using the Siegel-Zwiebach-WittenN e v e u - W e s t ( S Z W N W ) [ 6] procedure we write the corresponding field theory actions based on the respective B F V - B R S T charges [ 7 ]. The analog procedure for the N-extended supersymmetric case [ 8 ] is also performed. The natural generalization of this procedure in D = 10 produces a field theory which implies an interesting generalization of the Minkowski geometry of the spacetime. The standard way to quantize the massless particle is to consider the first-class constraint p-~ = 0 and impose it on the wavefunction 1~2c19=0,
(1)
w h e r e / ~ - - iO/Ox. Because there is only one field equation ( 1 ) it is very easy to find the action ( K l e i n - G o r d o n ) from which it can be derived by variation: S=½ f d 4 x ¢/)p2@.
(2)
The relevant twistor reformulation of this system ~" Incumbent of the Charles Revson Fundation Career Development Chair.
562
is done by taking the case N = 0 of the models that appear in refs. [9,8 ]. The D = 4 twistorial particle is defined in terms of the canonical variables x ~', and their canonically conjugated m o m e n t a p~. In addition, we have two Majorana commuting spinors ( v , , zTa) with their canonically conjugated m o m e n t a (09", o9"). This model is described by the following first-class constraints: ~.c~ =-P.c~ - v~ ga = O,
3)
H = i ( og"v. - ~ a ~ a ) =0.
(4)
As proven in ref. [ 9], the original classical massless point-particle system and the twistorial one (3), (4) have the same physical content. Upon quantization one obtains the operators corresponding to the constraints ( 3 ), (4) [ 8 ]: ~ . a = - i 0 . a + v. f,~, /t=v,
~
0
-V,~ ~
0
(5) -c.
(6)
The appearance of the integer constant c in (6) is due to the q u a n t u m ambiguity in the ordering of co and v. The parameter c is the fundamental object in the entire construction: it determines which quantum field theory is obtained from the classical system ( 5 ), ( 6 ). More specifically, we will show that: c = 0 will give a massless scalar field, c = 1 a spinor field, c = 2 the Maxwell (i.e. linearized Y M ) field, c = 3 the Rarita-Schwinger field, and
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PHYSICS LETTERS B
c = 4 the linearized Einstein field. Note that the ambiguity c cannot exceed the number of pairs v, oJ (plus the number of pairs D, o3). Consequently, the above, are all the field theories obtainable from the classical massless point particle by the present procedure. This is a novel and amusing "explanation" of the absence of higher spin elementary objects in nature. Let us denote the operators (5), (6) by/%. The Dirac first quantization[ 10] is then performed by imposing f " ~ ( z ) =0,
(7)
where the wave function • depends on the variables z ~ (x, v, v). Let us analyze in detail the cases c= 1 and c = 2 and prove the equivalence of (7) to the usual Dirac (14) and respectively Maxwell (15) field equations. The condition that the helicity operator (6) annihilates the field ~,
13 April 1989
fields fulfill automatically the ordinary field equations [ 8 ] ig~ ~'*~A = 0"~Z~ = 0,
(14)
iD/~g~ ~P"~M= 0 ' ~ f ~ = 0,
( 15 )
which correspond to the Dirac and linearized YM field equations. It is straightforward to show that (14), (15) are in fact equivalent to (7) (i.e. (8), (11)). Above, we got the Maxwell field equations in terms of field strength. Actually one can reexpress them in terms of the potentials
A,~c~=- ½( v~A71a~ + Dav/3M~).
Indeed, since on-shell vf can freely be replaced by 0,a we get
fa~( vf, x) = D,~VAv,~ v , M "/~ = 0.~A~ + 0./~Aa, •
/~q~(z) =0,
(8)
yields the following solutions:
• =A=v.A"(v.g~,x)
for c = 1,
(9)
and
• =M=v.v/jM~/~(v~fc~,x)
for c=2.
(10)
One is left with the Dirac constraint equations related to (5)
~.~ q>(z) = O.
(ii)
These equations, when expressed in the momentum space simply replace all the v. Da by p.a. So, the fields which depend on v. and ~. only through the combination v.Da can be described on-shell solely in terms of spacetime variables p.~. The unique way to pass from the fields (9), (10) to new fields which depend on v. and g,~ only through the combination v. zT~ is
Z,~(vD, x) = @,A = g~vo~A",
( 12 )
f./~( vv, x ) = t'a f/~M= g,~D~ v. v aM ~/~.
( 13 )
By using ( 1 1 ) the dependence o f z a n d f o n vf can be eliminated. The field Za turns out to be the dotted part of the Majorana (Weyl) field, while f,~ together with its complex conjugate f./~ are the self-dual and anti-selfdual parts of the YM curvature tensor. The above
(16)
(17)
ot
which is the definition of the linearized field strength in terms of the field potentials. Until this stage we get only half of the field and their corresponding field equations. The other half is for the complex conjugate of q~: paT@*
(18)
(Z) =0,
where T stands for transposition. The field equations (8), ( 1 1 ) can be derived from an action constructed from the first quantized BFVBRST charge [ 7 ]:
Q=+c,C~c~+cffi_
0 0 02 "~ 0g~ '
( 19 )
where the variables are organized according to table 1. The role played by the antighost is to ensure the zero ghost number of the action. By using the SZWNW [6] method, we get the action Table 1 Lagrange multiplier
Ghost
Antighost
Operator
-
C
-
/Q
563
Volume 220, number 4
PHYSICS LETTERS B
¢
S= J dZ ~ * ( Z ) Q ~ ( Z ) ,
(20)
where Z - (z, 2 "'~, c, c "6, c.a). Due to the nilpotency of Q and QT the action (20) is invariant under the transformations
5(P(Z)=~QA(Z),
6~*(Z)=~*QTA*(Z),
(21)
and the equations of motion are QqO(Z) =0,
QTq~*(Z) =0.
(22)
Using the invariance (21 ) these equations of motion become equivalent on shell to the usual Dirac constraint equations ( ( 7 ) , ( 18 ) ) [ 7 ]. Observe that the gauge invariance (21) does not contain the usual Maxwell gauge invariance which is in fact realized trivially in the present formalism (see refs. [ 11,12 ] for similar features). Let us turn now to the N-extended D = 4 twistorial superparticle that appears in ref. [8] and show explicitly that the first quantized on-shell field equation derived there can analogously be derived from an action principle ~. In addition to the N = 0 variables appearing in (5), (6), the D = 4 twistorial Nextended superparticle depends on N pairs of canonically and complex conjugated Grassmann variables (~,, ~/'). The system contains also the Majorana anticommuting spinors 0 ''", G;~ and their conjugate momenta p .... p~. The constraints are
d.,, =p,,. +ip.~ 6,a, + i.,f2~,, v. =0,
(23)
d,~ =Pc~+ iO'"~P,c~-i,,/2q"fa =0,
(24)
T.~ = p . ~ + v.¢a =0,
(25)
--It
__
--II
H - - i (~o"v, - o ~ f ~ ) - ~,,q" = 0,
13 April 1989
(26) enables us to fix all the usual superspace variables, keeping only the supertwistorial ones. The classical system in such a gauge depends only on the twistorial variables and it is described solely in terms of the constraint (26) [3] H - - i (og~v, - o~azTa)- ~,,qn = 0.
(27)
The existence of such a pure-twistor covariant formulation is specific to D = 4. We will see later in the paper, that by requiring similar properties in D-- 10, one is forced to generalize the geometric structure of the Minkowski spacetime. One can also perform the partial gauge fixing of (23), (24) by putting 0=0
and
0=0.
(28)
Now the system is described by the constraints ( 2 5 ) (27). The original SUSY generators become in such a gauge P,, M,~ = p t , x , j
+v,~(a,~)~oga-#c~(G,~)/j~o- ~ -~,
Qi,=~/2v,~{,, O'a=~f2Oc~qi,
(29)
which fulfill the twistorial SUSY algebra {Q,,, Og}pu =2a{v, oa, {M,,, air }PB =
(a,~)~a,,.
(30) (31)
Recalling that v~ 0a replaces p , , in the twistorial formalism, we can see that (30) fulfills the usual supersymmetric algebra [ 8 ]. After the quantization, the operators correspondingto the constraints ( 2 3 ) - ( 2 7 ) are [8]
d,,,~=D,,. +x/2v,~a,,,
(32)
where
c7~-- Da-"+ Xf2ga ant,
(33)
{¢,, q'},,B = i6i,
~P.a = - i 0 . a +
n = 1..... N,
(26)
and PB denotes Poisson brackets. As proven in ref. [ 9 ], Brink-Schwarz [ 15,16 ] superparticle and the twistorial one ( 2 3 ) - ( 2 6 ) have the same physical content. In refs. [9,8] we have shown that the system ( 2 3 ) ,i Action principles not based on the NZWNW procedure were obtained for N~<3 extended supersymmetricfield theories in refs. [13,14]. 564
I:I=v.
v,~Oa,
- f a ~ a -a,,a"*-d.
(34) (35)
Since the Grassmann variables ~ and q are canonically and complex conjugated pairs, they correspond quantum mechanically to the operators a, a* forming a Clifford algebra: {a,, aJ*}=6i.
Volume 220, number 4
PHYSICS LETTERS B
The operators D,,,=O/00""-i0,a0,~ and I 3 ~ = 0/86~ + i0""0,~ are the covariant derivatives. The parameter c that multiplies the identity matrix in (35), will classify the SUSY multiplets [8]. It is a result of the quantum ordering ambiguity and would vanish in a naive quantization. Let us denote the operators ( ( 3 2 ) - ( 3 5 ) ) by/~". The Dirac first quantization is then performed by imposing /%q~ (~) =0,
(36)
where • is the wave function and depends on the variables ~ - (x, v, 9, 0, tT). As was shown in ref. [8] the equations (36) are equivalent to the well-known on-shell equations of motion for the different N-extended supermultiplets. The classification of the more important supermultiplets with respect to cwas given in ref. [8 ] where an explicit example ( c = - 1 ) is given to demonstrate the equivalence between the equations (36) and the linearized on-shell super-field equations (for N = 3 extended SYM). Let us now show how one can derive equations that are equivalent on-shell to (36) from an action principle. We divide the models into two categories which we treat differently. The first category includes the maximally extended SUSY theories: N = 2 scalar superfield, N = 4 SYM and N = 8 SUGRA. For these, (36) contains all the on-shell equations of motion. The second category includes the non-maximal models which consist of two multiplets. In this case, like in the N = 0 case, eqs. (36) describe just half of the onshell field equations. The second half is described by /~"Tq~* (Z, V, f ) = 0 .
(37)
The action will be of the same form as the N = 0 case except that the field becomes a 2 x component vector and Q is a 2x× 2 x matrix [ 8 ] : S = f d2~ ~ * ( 2 ) Qqb(2),
(38)
Q=p,d,+y~,~+c,C~u c~+cfi_p~p, 0
0
i)2" 0~.
0
0
02" Op,~
0
The variables in Q are organized in table 2. Due to the nilpotency of Q and its transposed QT, the action (38) is invariant under transformations of the form (21 ) and the equations of motion are of the form (22). Using the invariance (21) these equations of motion are of the form (22). Using the invariance (21) these equations of motion become equivalent on-shell to the usual Dirac constraint equations (36) and (37). The actions for the maximal models are given in the partial gauge (28) in terms of the BRST charge o~
0 0c ~
o~
(40)
Q=c'~,~c~ + c f l - O)~,~c ~O~,c~ by
S= ½J d Z K ~ ( Z ) Q ~ ( Z ) ,
(41)
where Z= ( x, v, 9, 2 °'~, c, c °"~, ?o,a). All the ghosts in Z are fermionic and therefore their integration is trivial. Consequently, S can be recasted into a form with no ghosts at all but with corresponding auxiliary fields instead. To get the equations of motion
Qclg(Z) =0,
(42)
one has to find an operator K such that
KQ = QTK. Then, the nilpotency of Q insures that the action (41 ) is invariant under the transformation
8c19=EQA,
(43)
and that eqs. (42) are equivalent to the Dirac constraint equations (36). To find K observe that in the maximal extended SUSY theories, the constituent fields of q) are pairs of complex conjugate (opposite helicity) fields. For example, in the N = 2 , c = - 1 scalar superfield, the wave function is [ 8 ] Table 2 Lagrange multiplier
where 2 - (L 2 "'~, P, P, P,/~, c, c "~, g,~) and
13 April 1989
Ghost
Antighost
Operator
c
-
fl
0
02 ~ 0 ? ~ "
(39) 565
Volume 220, number 4
• (v,f;p)=
A ,
PHYSICS LETTERS B
(44)
where A, X are (opposite helicity) spinor fields and E, E are scalar fields. We find a K o f t h e form K-RS, where S acts on q> simply by interchanging the fields within each conjugate pair:
ScI)(v, g ) =
,
(45)
while R is a diagonal matrix R = d i a g ( R j , Rj, R2, R2),
c"C~-c~a,
(46)
(47)
and on the bosonic ones as R2 R2: P . a - ' - P . a ,
c--,-c.
(48)
The direct generalization of the D = 4 twistorial system ( 23 ) - ( 27 ) to D = 10 was suggested in ref. [ 9 ]. The canonical variables are x ' , ~, G, 0o,, v'~, f~. Their canonical conjugate m o m e n t a are p., ~/", p~, p~, o)~, e?~,, (where the indices c~ transform as M W 10D Lorentz spinors while a are internal SO (8) spinor indices). The system is characterized by the first-class constraints d" - - ip" -p"lSOis + i , , ~ , v ~ = 0,
(49)
d " - - ip"-p"lSOn - i,,f 2q.~"" =O,
(50)
~"/J-p"/S +v""O~=O,
(51)
Hal,-= - , t ~; t, ...,, . ~ ..~/,- ~-=/, u . v -ac~)+~/"~>=0,
(52)
where {O/s,p"}pB={O~,p'~}pr~=d/j'~; {~-, ~/,}p~= iC,I,. C m' is the SO (8) charge conjugation matrix. As it was shown in ref. [9], the D = 10 constraints ~"/~ are not independent and, consequently, the above straightforward covariant BFV-BRST-SZWNW procedure ( 37 ) - (39) leading to a field theory cannot be implemented [7,171. Also the spacetime and twistor 566
variables are not anymore on the same footing because there is no gauge in which the system reduces to a purely twistorial covariant one. These problems can be traced to the fact that in D = 10 the number o f spacetime canonical variables (10) is not equal anymore to the number oftwistorial canonical variables (16X16). To restore in D = 10 the above nice properties of D = 4 one is lead to consider a D = 10 space whose tangent is parametrized by a 16X 16 matrix rn "/~ rather than a ten-component vector p " (somewhat similar generalizations of the tangent to the spacetime were considered in the context of general relativity in ref. [ 18 ] ) ~-~ The most general form o f a 16 X 16 spinor-indexed matrix m"lSin D = 10 is
m " n = p " ( G, )'~/S+ p","'-"3 ( a,.,~_,3 )"lJ
which acts on the fermionic fields as R~
Rl: p.~---,-p~a,
13 April 1989
+ p ......... (~r ......... )<".
(53)
In D = 4 only the first term would appear and m "i~ would provide just another parametrization of the usual Minkovski spacetime. The formula (53) suggests that in D = 10 we consider a spacetime generalizing the Minkovski space and parametrized by an entire host of tensors (and not just vectors):
X/t x., .2.3 antisymmetric, x ........ 5 antisymmetric and self-dual.
(54)
They can be expressed uniquely in terms of a 16 X 16 matrix with spinor indices: k<~/~=x'(°,)-,
+ x . . . . . 3( °- . . . . . . ).i~
+ x ......... (o" ........ ,)./~.
(55)
The canonical conjugate m o m e n t a ( p ' , p"'"-"', p ......... ) to the variables (54), can be represented by a matrix m "/~ (53). Our new D = 10 twistorial superparticle is constructed by replacing p./S by rn "ss in the constraints ( 4 9 ) - ( 5 2 ) . By this substitution, the new system of first-c/ass constraints becomes independent and is given by ~2 For more conservative alternatives of treating D= 10 see refs. [11,9,191 .
Volume 220, number 4
g"-
PHYSICS
- i p " - rn"/SO~ + i . , f 2~.v'~o =O.
(56)
~,~- -ip'~-rn/~"O/~-i.~q.f'~=O,
(57)
R eds = m
(58)
Hal,_
oe l~
+ v a a v- I S. = 0 ,
= - ~ (• ~ o .av
,~1,
- o g- ~I, v- a e ~ ) + t / " ~ ' = 0 ,
(59)
LETTERS
B
13 A p r i l 1 9 8 9
{M~,~, Mp.}pB = - ( qm, M~o + q . ~ M m,
(68)
-q.~,M,,,,-q.pM,,,.),
{O", Qa}~.B= 2 m "/s,
{M,,., Qa}p B =
(69)
(O',,v) B
C,Q/S.
(70)
One can perform again the partial gauge fixing (28) the Poisson brackets of these constraints are 0. =O. =0. {g", g/'},.r~ = 2iR '~/',
(60)
{ H "~', H ''a } e B = i ( C"~'H ~ -
C"aH"~').
( 61 )
The irreducibility of the system ( 56 ) - ( 59 ) allows us to proceed with the SZWNW BFV-BRST procedure based on the QBRsTcharge:
(71)
The partially fixed system is described by the constraints (58), (59). As in the case (41), the corresponding SZWNW-BFV-BRST action can be recasted in a form with no ghosts. The super-Poincar6 generators become m . a,
8
8
+ Z"I'Z"IC' t' 0Z.,~ -P,~fi/s OZ./j '
(62)
where X./,, Z-zJ,P,~ and p . are the ghost variables of the operators/~"~',/~ u/J, ~ . and g", respectively. At the quantum level, it was the operation b y / ~ t , which determined the helicities of the multiplet. Because the operators/~"~' did not change by passing to m "/s, the multiplet of the modified system is still N = 2 SUGRA but the fields depend now on the tensorial variables x~,,r,,_r,~, x,,,r,,r,~r,~, ~ in addition to xr,. Phenomenologically, one can make the coordinates xr,,r,,_m, x~,,~,,_mr,,r,s irrelevant at low energies through compactification. Then, one recovers at low energies the standard D = 10, N = 2 SUGRA. The appropriate generalization of the super-Poincar6 generators for the new system ( 56 ) - (59) is rn./~,
(63)
Mr,~ =m"lS2/~:,( a,,,.)'g -m/*"2;,/s( #r,.)~
+p"(o-r, ~),~O/~--p /~-" (drr,~). ~ /~0/s p + v .c~ ( a,,.).o~/s
Q"=-ip"+m"/SO/s,
a
/~- v . - a ( ar,.),~o9 a a,
0"= -ip'~+ m"0.,
(64) (65)
which fulfill the algebra { m "/~, m:"~}i.~ = 0 ,
(66)
( 72 ) a
[J
a
-a ~
l~-
a
+ V. (ar,.)~O p -- V. (a~,.)~O/~ , Q'~=.~v"~",
O " = . , / 2 f ~ . q ",
(73) (74)
which fulfill the algebra
{O", 0'}P. =2v°~; {Mu,,, Q~'}p~ = (¢,.)/~ "Q/~. If we eliminate now entirely the spacetime variables, by further gauge fixing wg, - f"~21s. = 0,
~ - v""~/,. = 0,
(75)
we are left with the purely twistorial system described by the constraints H "~' (59) [ 8 ]. In this gauge, the super-Poincar6 generators reduce to rnaa_ . . .,,a . . ,7a.
(76)
Mr '" = v a. ( ar,,.),:,ro is ~ ~,- v~, -~, ( a,,~)~,eo ~ 13- a ,,,
(77)
Q~'=x/2v".¢",
(78)
(~"=x/-2tT.~, ".
The field theory action of the type (41 ) for the system ( 5 6 ) - ( 5 9 ) can now be straightforwardly formulated in terms of the BRST charge (62). Its properties will be studied elsewhere. We gratefully acknowledge many illuminating discussions with Emil Nissimov and Svetlana Pacheva.
{ m"/', Mr,. }pB = - [m.'/~(a~,.)~-rn~",'(#,,.)~J],
(67) 567
Volume 220, number 4
PHYSICS LETTERS B
References [ I ] R . Penrose, J. Math. Phys. 8 (1967) 345; in: Quantum gravity, eds. C.J. Isham, R. Penrose and D.W. Sciama (Oxford U.P., Oxford, 1975) p. 268; Rep. Math. Phys. 12 (1977) 65; in: Quantum gravity 2, eds. C.J. lsham, R. Penrose and D.W. Sciama (Oxford U.P., Oxford, 1981 ) p. 587; R. Penrose and M.A.H. MacCallum, Phys. Rep. 6 (1973) 241. [ 2 ] A. Ferber, Nucl. Phys. B 132 ( 1978 ) 55. [31Shirafuji, Prog. Theor. Phys. 70 (1983) 18. [4] E. Witten, Phys. Lett. B 77 (1978) 394; Nucl. Phys. B 266 (1986) 245. [5 ] K.H. Bengtsson, 1. Bengtsson, M. Cederwall and N. Linden, Phys. Rev. D 36 (1987) 1766; I. Bengtsson and M. Cederwall, Nucl. Phys. B 302 (1988). [ 6 ] W. Siegel, Phys. Lett. B 149 ( 1984 ) 157; W. Siegel and B. Zwiebach, Nucl. Phys. B 263 (1986) 105; E. Winen, Nucl. Phys. B 268 (1986) 253; B 276 (1986) 291; A. Neveu and P.C. West, Phys. Lett. B 165 (1985) 63; B 182 ( 1986 ) 343; Nucl. Phys. B 293 ( 1987 ) 266. [7] E.S Fradkin and G.A. Vilkovisky, Phys. Lett. B 55 (1975) 244; CERN preprint TH-2332 ( 1977); I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B 69 (1977) 309; E.S. Fradkin and T.E. Fradkina, Phys. Left. B 72 (1978) 343; M. Henneaux, Phys. Rep. 126 (1985) 1. [8] Y. Eisenberg, Mod. Phys. Lett. A 4 (1989) 195.
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[9] Y. Eisenberg and S. Solomon, Nucl. Phys. B 309 (1988) 709. [10] P.A.M. Dirac, Lectures on Quantum mechanics (Belfer Graduate School of Science, Yeshiva University, New York, 1964). [ 11 ] E. Nissimov, S. Pacheva and S. Solomon, preprint WIS-88/ 24/MAY-PH (1988), to appear in Nucl. Phys. B; see also: WIS-88/23/MAY-PH, to appear in Intern. J. Mod. Phys. A; in: Perspectives in string theory (Copenhagen, October 1987) (World Scientific, Singapore, 1988); Nucl. Phys. B 299 (1988) 183; B 296 (1988) 462; B 297 (1988) 349. [ 12 ] E. Sokatchev, CERN preprint CERN-TH.5160/88 (August 1988). [ 13 ] A. Galperin, E. Ivanov, S. Kalizin, V. Ogievetsky and E. Sokatchev, Class. Quantum Grav. 1 (1984) 469; 2 (1985) 155; A. Galperin, E. lvanov, V. Ogievetsky and E. Sokatchev, Class. Quantum Gray. 2 ( 1985 ) 601, 617. [ 14] S. Kalitzin, E. Nissimov and S. Pacheva, Mod. Phys. Lett. A2 (1987) 651. [15] L. Brink and J.H. Schwarz, Phys. Lett. B 100 (1981) 310. [ 16] R. Casalbuoni, Nuovo Cimento 33A (1976) 389. [17] E. Nissimov and S. Pacheva, Phys. Lett. B 189 (1987) 57; L. Brink and M. Henneaux, Principles of string theory (Plenum, New York, 1988); R. Kall~sh, JETP Lett. 45 (1987) 365; Phys. Lett. B 195 (1987) 369. [18] S. Weinberg, Phys. Lett. B 138 (1984) 47. [ 19 ] S. Solomon, Phys. Lett. B 203 ( 1988 ) 86; H. Aratyn and R. lngermansson, preprint UICHEP-Pub-8842 (August 1988).
PHYSICS LETTERS B
13 April 1989
(SUPER-) FIELD THEORIES FROM (SUPER-) TWISTORS Y. E I S E N B E R G and S. S O L O M O N ~: Department of Physics, Weizmann Institute of Science, Rehovot 76100, Israel Received 11 November 1988
We obtain higher spin linearized field theories through the BRST second quantization of the classical D = 4 massless point particle. By applying the same procedure to the classical supersymmetric particles, one obtains linearized N-extended off-shell unconstrained superfield actions. By extending the procedure to D = 10, one is led to a new spacetime geometry which reduces in some limit to the usual D = 10 Minkowski geometry.
First we reformulate the classical massless point particle in terms of twistors [ 1-5 ]. Then, we show that, upon quantization of this system, one can obtain, in addition to the massless scalar field equations, equations of motion equivalent to the Dirac, Maxwell, Rarita-Schwinger and linearized Einstein field equations. Using the Siegel-Zwiebach-WittenN e v e u - W e s t ( S Z W N W ) [ 6] procedure we write the corresponding field theory actions based on the respective B F V - B R S T charges [ 7 ]. The analog procedure for the N-extended supersymmetric case [ 8 ] is also performed. The natural generalization of this procedure in D = 10 produces a field theory which implies an interesting generalization of the Minkowski geometry of the spacetime. The standard way to quantize the massless particle is to consider the first-class constraint p-~ = 0 and impose it on the wavefunction 1~2c19=0,
(1)
w h e r e / ~ - - iO/Ox. Because there is only one field equation ( 1 ) it is very easy to find the action ( K l e i n - G o r d o n ) from which it can be derived by variation: S=½ f d 4 x ¢/)p2@.
(2)
The relevant twistor reformulation of this system ~" Incumbent of the Charles Revson Fundation Career Development Chair.
562
is done by taking the case N = 0 of the models that appear in refs. [9,8 ]. The D = 4 twistorial particle is defined in terms of the canonical variables x ~', and their canonically conjugated m o m e n t a p~. In addition, we have two Majorana commuting spinors ( v , , zTa) with their canonically conjugated m o m e n t a (09", o9"). This model is described by the following first-class constraints: ~.c~ =-P.c~ - v~ ga = O,
3)
H = i ( og"v. - ~ a ~ a ) =0.
(4)
As proven in ref. [ 9], the original classical massless point-particle system and the twistorial one (3), (4) have the same physical content. Upon quantization one obtains the operators corresponding to the constraints ( 3 ), (4) [ 8 ]: ~ . a = - i 0 . a + v. f,~, /t=v,
~
0
-V,~ ~
0
(5) -c.
(6)
The appearance of the integer constant c in (6) is due to the q u a n t u m ambiguity in the ordering of co and v. The parameter c is the fundamental object in the entire construction: it determines which quantum field theory is obtained from the classical system ( 5 ), ( 6 ). More specifically, we will show that: c = 0 will give a massless scalar field, c = 1 a spinor field, c = 2 the Maxwell (i.e. linearized Y M ) field, c = 3 the Rarita-Schwinger field, and
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PHYSICS LETTERS B
c = 4 the linearized Einstein field. Note that the ambiguity c cannot exceed the number of pairs v, oJ (plus the number of pairs D, o3). Consequently, the above, are all the field theories obtainable from the classical massless point particle by the present procedure. This is a novel and amusing "explanation" of the absence of higher spin elementary objects in nature. Let us denote the operators (5), (6) by/%. The Dirac first quantization[ 10] is then performed by imposing f " ~ ( z ) =0,
(7)
where the wave function • depends on the variables z ~ (x, v, v). Let us analyze in detail the cases c= 1 and c = 2 and prove the equivalence of (7) to the usual Dirac (14) and respectively Maxwell (15) field equations. The condition that the helicity operator (6) annihilates the field ~,
13 April 1989
fields fulfill automatically the ordinary field equations [ 8 ] ig~ ~'*~A = 0"~Z~ = 0,
(14)
iD/~g~ ~P"~M= 0 ' ~ f ~ = 0,
( 15 )
which correspond to the Dirac and linearized YM field equations. It is straightforward to show that (14), (15) are in fact equivalent to (7) (i.e. (8), (11)). Above, we got the Maxwell field equations in terms of field strength. Actually one can reexpress them in terms of the potentials
A,~c~=- ½( v~A71a~ + Dav/3M~).
Indeed, since on-shell vf can freely be replaced by 0,a we get
fa~( vf, x) = D,~VAv,~ v , M "/~ = 0.~A~ + 0./~Aa, •
/~q~(z) =0,
(8)
yields the following solutions:
• =A=v.A"(v.g~,x)
for c = 1,
(9)
and
• =M=v.v/jM~/~(v~fc~,x)
for c=2.
(10)
One is left with the Dirac constraint equations related to (5)
~.~ q>(z) = O.
(ii)
These equations, when expressed in the momentum space simply replace all the v. Da by p.a. So, the fields which depend on v. and ~. only through the combination v.Da can be described on-shell solely in terms of spacetime variables p.~. The unique way to pass from the fields (9), (10) to new fields which depend on v. and g,~ only through the combination v. zT~ is
Z,~(vD, x) = @,A = g~vo~A",
( 12 )
f./~( vv, x ) = t'a f/~M= g,~D~ v. v aM ~/~.
( 13 )
By using ( 1 1 ) the dependence o f z a n d f o n vf can be eliminated. The field Za turns out to be the dotted part of the Majorana (Weyl) field, while f,~ together with its complex conjugate f./~ are the self-dual and anti-selfdual parts of the YM curvature tensor. The above
(16)
(17)
ot
which is the definition of the linearized field strength in terms of the field potentials. Until this stage we get only half of the field and their corresponding field equations. The other half is for the complex conjugate of q~: paT@*
(18)
(Z) =0,
where T stands for transposition. The field equations (8), ( 1 1 ) can be derived from an action constructed from the first quantized BFVBRST charge [ 7 ]:
Q=+c,C~c~+cffi_
0 0 02 "~ 0g~ '
( 19 )
where the variables are organized according to table 1. The role played by the antighost is to ensure the zero ghost number of the action. By using the SZWNW [6] method, we get the action Table 1 Lagrange multiplier
Ghost
Antighost
Operator
-
C
-
/Q
563
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PHYSICS LETTERS B
¢
S= J dZ ~ * ( Z ) Q ~ ( Z ) ,
(20)
where Z - (z, 2 "'~, c, c "6, c.a). Due to the nilpotency of Q and QT the action (20) is invariant under the transformations
5(P(Z)=~QA(Z),
6~*(Z)=~*QTA*(Z),
(21)
and the equations of motion are QqO(Z) =0,
QTq~*(Z) =0.
(22)
Using the invariance (21 ) these equations of motion become equivalent on shell to the usual Dirac constraint equations ( ( 7 ) , ( 18 ) ) [ 7 ]. Observe that the gauge invariance (21) does not contain the usual Maxwell gauge invariance which is in fact realized trivially in the present formalism (see refs. [ 11,12 ] for similar features). Let us turn now to the N-extended D = 4 twistorial superparticle that appears in ref. [8] and show explicitly that the first quantized on-shell field equation derived there can analogously be derived from an action principle ~. In addition to the N = 0 variables appearing in (5), (6), the D = 4 twistorial Nextended superparticle depends on N pairs of canonically and complex conjugated Grassmann variables (~,, ~/'). The system contains also the Majorana anticommuting spinors 0 ''", G;~ and their conjugate momenta p .... p~. The constraints are
d.,, =p,,. +ip.~ 6,a, + i.,f2~,, v. =0,
(23)
d,~ =Pc~+ iO'"~P,c~-i,,/2q"fa =0,
(24)
T.~ = p . ~ + v.¢a =0,
(25)
--It
__
--II
H - - i (~o"v, - o ~ f ~ ) - ~,,q" = 0,
13 April 1989
(26) enables us to fix all the usual superspace variables, keeping only the supertwistorial ones. The classical system in such a gauge depends only on the twistorial variables and it is described solely in terms of the constraint (26) [3] H - - i (og~v, - o~azTa)- ~,,qn = 0.
(27)
The existence of such a pure-twistor covariant formulation is specific to D = 4. We will see later in the paper, that by requiring similar properties in D-- 10, one is forced to generalize the geometric structure of the Minkowski spacetime. One can also perform the partial gauge fixing of (23), (24) by putting 0=0
and
0=0.
(28)
Now the system is described by the constraints ( 2 5 ) (27). The original SUSY generators become in such a gauge P,, M,~ = p t , x , j
+v,~(a,~)~oga-#c~(G,~)/j~o- ~ -~,
Qi,=~/2v,~{,, O'a=~f2Oc~qi,
(29)
which fulfill the twistorial SUSY algebra {Q,,, Og}pu =2a{v, oa, {M,,, air }PB =
(a,~)~a,,.
(30) (31)
Recalling that v~ 0a replaces p , , in the twistorial formalism, we can see that (30) fulfills the usual supersymmetric algebra [ 8 ]. After the quantization, the operators correspondingto the constraints ( 2 3 ) - ( 2 7 ) are [8]
d,,,~=D,,. +x/2v,~a,,,
(32)
where
c7~-- Da-"+ Xf2ga ant,
(33)
{¢,, q'},,B = i6i,
~P.a = - i 0 . a +
n = 1..... N,
(26)
and PB denotes Poisson brackets. As proven in ref. [ 9 ], Brink-Schwarz [ 15,16 ] superparticle and the twistorial one ( 2 3 ) - ( 2 6 ) have the same physical content. In refs. [9,8] we have shown that the system ( 2 3 ) ,i Action principles not based on the NZWNW procedure were obtained for N~<3 extended supersymmetricfield theories in refs. [13,14]. 564
I:I=v.
v,~Oa,
- f a ~ a -a,,a"*-d.
(34) (35)
Since the Grassmann variables ~ and q are canonically and complex conjugated pairs, they correspond quantum mechanically to the operators a, a* forming a Clifford algebra: {a,, aJ*}=6i.
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The operators D,,,=O/00""-i0,a0,~ and I 3 ~ = 0/86~ + i0""0,~ are the covariant derivatives. The parameter c that multiplies the identity matrix in (35), will classify the SUSY multiplets [8]. It is a result of the quantum ordering ambiguity and would vanish in a naive quantization. Let us denote the operators ( ( 3 2 ) - ( 3 5 ) ) by/~". The Dirac first quantization is then performed by imposing /%q~ (~) =0,
(36)
where • is the wave function and depends on the variables ~ - (x, v, 9, 0, tT). As was shown in ref. [8] the equations (36) are equivalent to the well-known on-shell equations of motion for the different N-extended supermultiplets. The classification of the more important supermultiplets with respect to cwas given in ref. [8 ] where an explicit example ( c = - 1 ) is given to demonstrate the equivalence between the equations (36) and the linearized on-shell super-field equations (for N = 3 extended SYM). Let us now show how one can derive equations that are equivalent on-shell to (36) from an action principle. We divide the models into two categories which we treat differently. The first category includes the maximally extended SUSY theories: N = 2 scalar superfield, N = 4 SYM and N = 8 SUGRA. For these, (36) contains all the on-shell equations of motion. The second category includes the non-maximal models which consist of two multiplets. In this case, like in the N = 0 case, eqs. (36) describe just half of the onshell field equations. The second half is described by /~"Tq~* (Z, V, f ) = 0 .
(37)
The action will be of the same form as the N = 0 case except that the field becomes a 2 x component vector and Q is a 2x× 2 x matrix [ 8 ] : S = f d2~ ~ * ( 2 ) Qqb(2),
(38)
Q=p,d,+y~,~+c,C~u c~+cfi_p~p, 0
0
i)2" 0~.
0
0
02" Op,~
0
The variables in Q are organized in table 2. Due to the nilpotency of Q and its transposed QT, the action (38) is invariant under transformations of the form (21 ) and the equations of motion are of the form (22). Using the invariance (21) these equations of motion are of the form (22). Using the invariance (21) these equations of motion become equivalent on-shell to the usual Dirac constraint equations (36) and (37). The actions for the maximal models are given in the partial gauge (28) in terms of the BRST charge o~
0 0c ~
o~
(40)
Q=c'~,~c~ + c f l - O)~,~c ~O~,c~ by
S= ½J d Z K ~ ( Z ) Q ~ ( Z ) ,
(41)
where Z= ( x, v, 9, 2 °'~, c, c °"~, ?o,a). All the ghosts in Z are fermionic and therefore their integration is trivial. Consequently, S can be recasted into a form with no ghosts at all but with corresponding auxiliary fields instead. To get the equations of motion
Qclg(Z) =0,
(42)
one has to find an operator K such that
KQ = QTK. Then, the nilpotency of Q insures that the action (41 ) is invariant under the transformation
8c19=EQA,
(43)
and that eqs. (42) are equivalent to the Dirac constraint equations (36). To find K observe that in the maximal extended SUSY theories, the constituent fields of q) are pairs of complex conjugate (opposite helicity) fields. For example, in the N = 2 , c = - 1 scalar superfield, the wave function is [ 8 ] Table 2 Lagrange multiplier
where 2 - (L 2 "'~, P, P, P,/~, c, c "~, g,~) and
13 April 1989
Ghost
Antighost
Operator
c
-
fl
0
02 ~ 0 ? ~ "
(39) 565
Volume 220, number 4
• (v,f;p)=
A ,
PHYSICS LETTERS B
(44)
where A, X are (opposite helicity) spinor fields and E, E are scalar fields. We find a K o f t h e form K-RS, where S acts on q> simply by interchanging the fields within each conjugate pair:
ScI)(v, g ) =
,
(45)
while R is a diagonal matrix R = d i a g ( R j , Rj, R2, R2),
c"C~-c~a,
(46)
(47)
and on the bosonic ones as R2 R2: P . a - ' - P . a ,
c--,-c.
(48)
The direct generalization of the D = 4 twistorial system ( 23 ) - ( 27 ) to D = 10 was suggested in ref. [ 9 ]. The canonical variables are x ' , ~, G, 0o,, v'~, f~. Their canonical conjugate m o m e n t a are p., ~/", p~, p~, o)~, e?~,, (where the indices c~ transform as M W 10D Lorentz spinors while a are internal SO (8) spinor indices). The system is characterized by the first-class constraints d" - - ip" -p"lSOis + i , , ~ , v ~ = 0,
(49)
d " - - ip"-p"lSOn - i,,f 2q.~"" =O,
(50)
~"/J-p"/S +v""O~=O,
(51)
Hal,-= - , t ~; t, ...,, . ~ ..~/,- ~-=/, u . v -ac~)+~/"~>=0,
(52)
where {O/s,p"}pB={O~,p'~}pr~=d/j'~; {~-, ~/,}p~= iC,I,. C m' is the SO (8) charge conjugation matrix. As it was shown in ref. [9], the D = 10 constraints ~"/~ are not independent and, consequently, the above straightforward covariant BFV-BRST-SZWNW procedure ( 37 ) - (39) leading to a field theory cannot be implemented [7,171. Also the spacetime and twistor 566
variables are not anymore on the same footing because there is no gauge in which the system reduces to a purely twistorial covariant one. These problems can be traced to the fact that in D = 10 the number o f spacetime canonical variables (10) is not equal anymore to the number oftwistorial canonical variables (16X16). To restore in D = 10 the above nice properties of D = 4 one is lead to consider a D = 10 space whose tangent is parametrized by a 16X 16 matrix rn "/~ rather than a ten-component vector p " (somewhat similar generalizations of the tangent to the spacetime were considered in the context of general relativity in ref. [ 18 ] ) ~-~ The most general form o f a 16 X 16 spinor-indexed matrix m"lSin D = 10 is
m " n = p " ( G, )'~/S+ p","'-"3 ( a,.,~_,3 )"lJ
which acts on the fermionic fields as R~
Rl: p.~---,-p~a,
13 April 1989
+ p ......... (~r ......... )<".
(53)
In D = 4 only the first term would appear and m "i~ would provide just another parametrization of the usual Minkovski spacetime. The formula (53) suggests that in D = 10 we consider a spacetime generalizing the Minkovski space and parametrized by an entire host of tensors (and not just vectors):
X/t x., .2.3 antisymmetric, x ........ 5 antisymmetric and self-dual.
(54)
They can be expressed uniquely in terms of a 16 X 16 matrix with spinor indices: k<~/~=x'(°,)-,
+ x . . . . . 3( °- . . . . . . ).i~
+ x ......... (o" ........ ,)./~.
(55)
The canonical conjugate m o m e n t a ( p ' , p"'"-"', p ......... ) to the variables (54), can be represented by a matrix m "/~ (53). Our new D = 10 twistorial superparticle is constructed by replacing p./S by rn "ss in the constraints ( 4 9 ) - ( 5 2 ) . By this substitution, the new system of first-c/ass constraints becomes independent and is given by ~2 For more conservative alternatives of treating D= 10 see refs. [11,9,191 .
Volume 220, number 4
g"-
PHYSICS
- i p " - rn"/SO~ + i . , f 2~.v'~o =O.
(56)
~,~- -ip'~-rn/~"O/~-i.~q.f'~=O,
(57)
R eds = m
(58)
Hal,_
oe l~
+ v a a v- I S. = 0 ,
= - ~ (• ~ o .av
,~1,
- o g- ~I, v- a e ~ ) + t / " ~ ' = 0 ,
(59)
LETTERS
B
13 A p r i l 1 9 8 9
{M~,~, Mp.}pB = - ( qm, M~o + q . ~ M m,
(68)
-q.~,M,,,,-q.pM,,,.),
{O", Qa}~.B= 2 m "/s,
{M,,., Qa}p B =
(69)
(O',,v) B
C,Q/S.
(70)
One can perform again the partial gauge fixing (28) the Poisson brackets of these constraints are 0. =O. =0. {g", g/'},.r~ = 2iR '~/',
(60)
{ H "~', H ''a } e B = i ( C"~'H ~ -
C"aH"~').
( 61 )
The irreducibility of the system ( 56 ) - ( 59 ) allows us to proceed with the SZWNW BFV-BRST procedure based on the QBRsTcharge:
(71)
The partially fixed system is described by the constraints (58), (59). As in the case (41), the corresponding SZWNW-BFV-BRST action can be recasted in a form with no ghosts. The super-Poincar6 generators become m . a,
8
8
+ Z"I'Z"IC' t' 0Z.,~ -P,~fi/s OZ./j '
(62)
where X./,, Z-zJ,P,~ and p . are the ghost variables of the operators/~"~',/~ u/J, ~ . and g", respectively. At the quantum level, it was the operation b y / ~ t , which determined the helicities of the multiplet. Because the operators/~"~' did not change by passing to m "/s, the multiplet of the modified system is still N = 2 SUGRA but the fields depend now on the tensorial variables x~,,r,,_r,~, x,,,r,,r,~r,~, ~ in addition to xr,. Phenomenologically, one can make the coordinates xr,,r,,_m, x~,,~,,_mr,,r,s irrelevant at low energies through compactification. Then, one recovers at low energies the standard D = 10, N = 2 SUGRA. The appropriate generalization of the super-Poincar6 generators for the new system ( 56 ) - (59) is rn./~,
(63)
Mr,~ =m"lS2/~:,( a,,,.)'g -m/*"2;,/s( #r,.)~
+p"(o-r, ~),~O/~--p /~-" (drr,~). ~ /~0/s p + v .c~ ( a,,.).o~/s
Q"=-ip"+m"/SO/s,
a
/~- v . - a ( ar,.),~o9 a a,
0"= -ip'~+ m"0.,
(64) (65)
which fulfill the algebra { m "/~, m:"~}i.~ = 0 ,
(66)
( 72 ) a
[J
a
-a ~
l~-
a
+ V. (ar,.)~O p -- V. (a~,.)~O/~ , Q'~=.~v"~",
O " = . , / 2 f ~ . q ",
(73) (74)
which fulfill the algebra
{O", 0'}P. =2v°~; {Mu,,, Q~'}p~ = (¢,.)/~ "Q/~. If we eliminate now entirely the spacetime variables, by further gauge fixing wg, - f"~21s. = 0,
~ - v""~/,. = 0,
(75)
we are left with the purely twistorial system described by the constraints H "~' (59) [ 8 ]. In this gauge, the super-Poincar6 generators reduce to rnaa_ . . .,,a . . ,7a.
(76)
Mr '" = v a. ( ar,,.),:,ro is ~ ~,- v~, -~, ( a,,~)~,eo ~ 13- a ,,,
(77)
Q~'=x/2v".¢",
(78)
(~"=x/-2tT.~, ".
The field theory action of the type (41 ) for the system ( 5 6 ) - ( 5 9 ) can now be straightforwardly formulated in terms of the BRST charge (62). Its properties will be studied elsewhere. We gratefully acknowledge many illuminating discussions with Emil Nissimov and Svetlana Pacheva.
{ m"/', Mr,. }pB = - [m.'/~(a~,.)~-rn~",'(#,,.)~J],
(67) 567
Volume 220, number 4
PHYSICS LETTERS B
References [ I ] R . Penrose, J. Math. Phys. 8 (1967) 345; in: Quantum gravity, eds. C.J. Isham, R. Penrose and D.W. Sciama (Oxford U.P., Oxford, 1975) p. 268; Rep. Math. Phys. 12 (1977) 65; in: Quantum gravity 2, eds. C.J. lsham, R. Penrose and D.W. Sciama (Oxford U.P., Oxford, 1981 ) p. 587; R. Penrose and M.A.H. MacCallum, Phys. Rep. 6 (1973) 241. [ 2 ] A. Ferber, Nucl. Phys. B 132 ( 1978 ) 55. [31Shirafuji, Prog. Theor. Phys. 70 (1983) 18. [4] E. Witten, Phys. Lett. B 77 (1978) 394; Nucl. Phys. B 266 (1986) 245. [5 ] K.H. Bengtsson, 1. Bengtsson, M. Cederwall and N. Linden, Phys. Rev. D 36 (1987) 1766; I. Bengtsson and M. Cederwall, Nucl. Phys. B 302 (1988). [ 6 ] W. Siegel, Phys. Lett. B 149 ( 1984 ) 157; W. Siegel and B. Zwiebach, Nucl. Phys. B 263 (1986) 105; E. Winen, Nucl. Phys. B 268 (1986) 253; B 276 (1986) 291; A. Neveu and P.C. West, Phys. Lett. B 165 (1985) 63; B 182 ( 1986 ) 343; Nucl. Phys. B 293 ( 1987 ) 266. [7] E.S Fradkin and G.A. Vilkovisky, Phys. Lett. B 55 (1975) 244; CERN preprint TH-2332 ( 1977); I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B 69 (1977) 309; E.S. Fradkin and T.E. Fradkina, Phys. Left. B 72 (1978) 343; M. Henneaux, Phys. Rep. 126 (1985) 1. [8] Y. Eisenberg, Mod. Phys. Lett. A 4 (1989) 195.
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[9] Y. Eisenberg and S. Solomon, Nucl. Phys. B 309 (1988) 709. [10] P.A.M. Dirac, Lectures on Quantum mechanics (Belfer Graduate School of Science, Yeshiva University, New York, 1964). [ 11 ] E. Nissimov, S. Pacheva and S. Solomon, preprint WIS-88/ 24/MAY-PH (1988), to appear in Nucl. Phys. B; see also: WIS-88/23/MAY-PH, to appear in Intern. J. Mod. Phys. A; in: Perspectives in string theory (Copenhagen, October 1987) (World Scientific, Singapore, 1988); Nucl. Phys. B 299 (1988) 183; B 296 (1988) 462; B 297 (1988) 349. [ 12 ] E. Sokatchev, CERN preprint CERN-TH.5160/88 (August 1988). [ 13 ] A. Galperin, E. Ivanov, S. Kalizin, V. Ogievetsky and E. Sokatchev, Class. Quantum Grav. 1 (1984) 469; 2 (1985) 155; A. Galperin, E. lvanov, V. Ogievetsky and E. Sokatchev, Class. Quantum Gray. 2 ( 1985 ) 601, 617. [ 14] S. Kalitzin, E. Nissimov and S. Pacheva, Mod. Phys. Lett. A2 (1987) 651. [15] L. Brink and J.H. Schwarz, Phys. Lett. B 100 (1981) 310. [ 16] R. Casalbuoni, Nuovo Cimento 33A (1976) 389. [17] E. Nissimov and S. Pacheva, Phys. Lett. B 189 (1987) 57; L. Brink and M. Henneaux, Principles of string theory (Plenum, New York, 1988); R. Kall~sh, JETP Lett. 45 (1987) 365; Phys. Lett. B 195 (1987) 369. [18] S. Weinberg, Phys. Lett. B 138 (1984) 47. [ 19 ] S. Solomon, Phys. Lett. B 203 ( 1988 ) 86; H. Aratyn and R. lngermansson, preprint UICHEP-Pub-8842 (August 1988).