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Cubic interaction terms for arbitrary spin
Nuclear Physics B227 (1983) 31-40 ,' North-Holland Publishing Company
CUBIC INTERACTION T E R M S FOR ARBITRARY SPIN* Anders K.H. BENGTSSON, Ingemar BENGTSSON and Lars BRINK Institute of Theoretical Physics, S-412 96 GiJteborg;, Sweden
Received 6 April 1983
Using thc light-cone formulation of dynamics, we construct cubic self-interaction terms for massless fields of arbitrary helicity X. The coupling constants have dimension [length]/` 1 and there is a gauge group for any odd X.
1. Introduction There are two kinds of s y m m e t r y generators in physics, kinematical and d y n a m i cal ones. The latter, which t r a n s f o r m the system f o r w a r d in time, will in general be n o n - l i n e a r l y ( a n d non-locally) realized on fields since their canonical expressions are m o d i f i e d b y interaction terms. To o b t a i n linear r e p r e s e n t a t i o n s of d y n a m i c a l symmetries it will be necessary to a d d auxiliary fields to the action, as is d o n e for e x a m p l e when a four-vector is used to describe a helicity-one field. U n f o r t u n a t e l y , there is no p r i n c i p l e g u a r a n t e e i n g the existence of these auxiliary fields, In fact, h i g h e r - d i m e n s i o n a l gravity theories are known, which seem to lack such a manifestly L o r e n t z c o v a r i a n t form [1]. This p r o b l e m is also e n c o u n t e r e d in s u p e r s y m m e t r y where the s u p e r s y m m e t r y generators are d y n a m i c a l within c o n v e n t i o n a l spacelike q u a n t i z a t i o n . F o r the p a r t i c u l a r l y interesting case of the N = 4 Y a n g - M i l l s theory q u i t e - s t r o n g no-go theorems [2] seem to p r e c l u d e the existence of a r e a s o n a b l e set of auxiliary fields for the full set of s u p e r s y m m e t r y generators. A m a n i f e s t l y s u p e r s y m metric form of this theory nevertheless exists [3], b u t it has not been possible to i n t r o d u c e interactions within this formulation. In the case of the N = 4 m o d e l this p r o b l e m was recently b y - p a s s e d [4] through the use of D i r a c ' s [5] light-cone (null-plane) f o r m u l a t i o n of dynamics. O n the light cone a p a r t of each s u p e r s y m m e t r y g e n e r a t o r b e c o m e s kinematical, so that no auxiliary fields are n e e d e d in o r d e r to o b t a i n a superfield f o r m u l a t i o n of the theory. F o r a further discussion of the a d v a n t a g e s of light-cone q u a n t i z a t i o n within this context, see ref. [6]. In this p a p e r we i n t e n d to use the fact that the light-cone f o r m u l a t i o n p e r m i t s one to w o r k with the physical fields exclusively (this is not always the case when the q u a n t i z a t i o n surface is space-like) in o r d e r to discuss a n o t h e r theoretical * Work supported in part by the Swedish Natural Science Research Council under contract no. S-Fo 8115-109. 31
32
A.K.H. Bengtsson et al. / Cubic interaction terms (I)
problem exhibiting great similarity with the situation in the N = 4 supersymmetric Yang-Mills theory. We will consider the question whether it is possible to have interactions in theories with helicity greater than 2. Again, manifestly covariant forms of such theories are known [7], but (apart from coupling to an external field) it has not been possible to introduce interactions within these formulations. For reviews of this question containing further references, see ref. [8]. In this paper we will argue, by exhibiting consistent three-point couplings for all massless fields with integer helicity, that this is purely an artefact of the insistence on manifest covariance and not a property of higher spins as such. The paper is organized as follows. In sect. 2 we introduce our notation and define free fields of arbitrary helicity. In sect. 3 we construct cubic self-interaction terms for all theories where such terms are possible. Sect. 4 summarizes our conclusions as well as further problems. The appendix contains the Poincar6 group in a suitable form. For further calculational details, consult ref. [9] (the case treated there is N = 1 supergravity).
2. The free fields in the light-cone frame
The fields that we will treat will all be described in the light-cone frame, i.e. we use for the Minkowski space X+=~22(xOq-x3),
X=~(xlq-iX2),
(2.1)
and similarly for the space-time derivatives. (Note that transverse indices are complexified.) In this frame x + is taken as the evolution parameter with p as the corresponding hamiltonian. Since any massless particle with spin has two degrees of freedom we will describe such particles of a definite spin X by a complex field ~ ( x ) . Furthermore we will choose the representation such that q0(x) has definite helicity - ) ~ and ~ the helicity + X. On a free field we can represent the orbital part of the Poincar6 algebra by (see the appendix for further notation)
a• p = - i a~-,
l = xJ-
p+= -i8 +, •
~8,
l+=i(xa+-x+8),
[+=i(~a+-x+O),
p = -i8,
[-=i
p= -i8,
aJ
.~--x-O
l+-=i x O+-x
, +0J~ -~-}.
(2.2)
A. K. H, Bengtsson et a l . , / Cubic interaction terms ( I )
33
In this representation we have made the substitution 0 - = 0cg/a +. This is the mass shell condition when it acts on a free field. However, since this condition is linear in 0-, the algebra still closes. It should, though, be noted that the representation of the generators is not hermitian (apart from the helicity operator) unless the mass shell condition is used. This is in contrast to the ordinary covariant formulation, and does correspond to a different way of going off-shell. This is not altogether new since for example the spin generators in the Dirac case is of this type. To transform a field with helicity X we also have to take the spin part of the Lorentz algebra into account [9]. The non-trivial ones are
8jcp = i~oj( l - X )ep,
To illustrate how these formulae are derived, we consider the Maxwell field A". Impose first the light-cone gauge A + = 0 and work out the compensating gauge transformation necessary to stay in the gauge when Lorentz transformations are performed. Then we can read off the resulting transformations of the physical degrees of freedom if we solve for the unphysical mode A -. In this way one can derive the transformations for any field with helicity X. However, it is simpler to start with 8j, knowing that the spin part has eigenvalue X and then close the algebra. Note in this context that eqs. (2.3) are quite close to the way massless particles are introduced as representations of the Lorentz group [10]. Since the free equations of motion are 00 O-qo = ~ - e p ,
(2.4)
the corresponding hamiltonian can be written as 8~
=
05 ~-w.
(2.5)
It is for this form that we will search for interaction terms, which are consistent with the Lorentz algebra. 3. Three-point
couplings
The lagrangian that we will attempt to construct is of the form
~= ½ ¢ ~ + ~( ~ ¢
+ ¢ ¢ ~ ) + o(~),
(3.1)
34
A . K . H . Bengtsson et al. / Cubic interaction terms ( I )
where a is a coupling constant and derivatives are still to be put into the interaction terms. The ensuing equation of motion for ~ is
03
8nCp = - ~ - ~ + a~q~ + a ~
+ O(a2).
(3.2)
(When we write an infinitesimal transformation we leave out the infinitesimal parameter). There will also be non-linear contributions to 8#+ , 8/ and 6 f , as follows 8 r ~ = so+ ~ - i ~ + 8 ~
+ o(~2),
~j ~ = ~o ~ + ixSh,~ + ~
+ o ( ~2 ) ,
87 q~ = 6° cp + iffS~op + 6~'~p + O( aZ ) .
(3.3)
With 6 ~ we mean the term to order a in the transformation. Here, 6~ and 8g are dynamical spin transformations of the form 62q~ = acpqv,
8~cp = aCpq~.
(3.4)
The last equations are assumptions at this stage. They are, however, true when X = 1, 2 and we will look for a similar solution in the general case. Our task now is to find the hamiltonian (3,2) together with the non-linear Lorentz transformations (3.3) such that they satisfy the correct algebra to first order in a. We then observe that terms of the type c~q0 and cpq0 do not mix in the transformations to this order. We can hence reduce the amount of labour involved by first considering only terms of the latter kind. We will make the ansatz that 8~ is a sum of terms of the form
8h~ = ~AO+"[ 0"0+~"0+%],
(3.5)
where t~, 0, ~, a and b are integers to be determined. This is not the most general form, but again it agrees with the k n o w n solutions for X = 1,2. We must now investigate what restrictions the Lorentz algebra imposes on this form. We begin by checking the algebra of 6 H with the Lorentz generators and start with the kinematical transformation 6j
[~,, 8.] °~ = o.
(3.6t
A. K. H. Bengtsson et al. / Cubic interaction terms (I)
35
Written out in detail, this becomes
8j~AO+'[ ~oa+%Jbo+°~]
~(l-
-
x),
O~O+°(Z-x)w Jb0+°w + 5 ° o + ~ o + ° ( t - x)w]
= ,~A0+'[
- ( t - X)~AO+"[5°0%~O÷°~]
= ~AO+"[[ 0 °, tl O+~ J~O+°w+ 0°0%[ 0 ~, ~1 a+°w - a~Aa+"[ ~oo%J~o+°,] = (a + b
-
X)aAO+"[ 0~O+~0hO+°cp] .
(3.7)
Thus we get our first restriction on the derivative structure in (3.5) namely
a+b=h.
(3.8)
We continue in this way. The commutator with 3/, yields /~+O+a=-l,
(3.9)
while the one with @ is trivial. The commutator with 3j~ on the other hand will mix terms in 3 H with different derivative structure, restricting the values of the coefficients A. In fact, with 3~q, of the form (3.5) we get
[ 6j,, 3tf ] "ep = aA[ iaO+"[ a +..... ~,a
l)q0~bo+°q0 ] + ibO+O[~aO+~l~(b 1,0+'"'t~q~] ] " (3.10)
We are also, because of eq. (3.4), in a position to do
[6[, 3tt ] ~ep = 0 = i(/z + 1 - ~ ) ~ z - a A O + " [ J~O+~JhO+oep] + a A [ i ( p + X)0+"[ j(a+l,O+~, l~bO+°qO ] + i ( o + X ) O * ' [ J ~ O + ~ J ' " * l ' O +'° ' ~ ] ] .
(3.11)
Since both of these commutators have to be zero, they give us a set of consistency conditions which in fact will suffice to determine 3~4) for us. Let us go through them helicity by helicity. For X = 0 we have a = b = 0 . A solution is clearly ~ = - 1 ,
36
A.K.H. Bengtsson et al. / Cubic interaction terms (I)
p = o = 0 and
3~rp = a 01-01~-[ epq)],
(3.12)
which, of course, is the coupling term in a q)3 theory. For )t = 1 the number of c~'s is one. In this case it is impossible to get zero out of both (3.10) and (3.11) unless we introduce an antisymmetric structure constant f , h q With/~ = 0, we have the solution
8~q)" = ctf"hc[ ~q)bq)c- @ ~ q ) c ] .
(3.13)
It is gratifying to see how the need for a gauge group emerges in this noncovariant formalism, and it is not difficult to check that this is the 3-point coupling for a non-abelian Yang-Mills field theory. For X = 2, we have a solution with/~ = 1, namely
8
8
J
7 7q)q) - 2 - U q ) - U q )
82 ] +
"
(3.14)
We recognize this as the three-point coupling in gravity [12]. N o t e that if X is half integer, no three-point coupling can exist because of (3.8). We have now exhausted all known cases. Nothing special happens, however, when we increase X. For X = 3, we have a solution with t~ = 2, again provided we introduce the structure constant fabc:
~H~
'~ "=
af ohiO+2 - 82 b 8 c 8 b 82 c ' - - 3q)hq)~"- 3 U q) fig-q) + 3fig-q) ~ q) -q)b - ~ q)c . (3.15)
We can go on, increasing X step by step. However, we already see Pascal's triangle [11] emerge and can write down the general solution. For ~ even: (3.16) For X odd:
3~Iq)" = a f " b ' Y",,=o( - 1 ) ' ( n ]
[~¢P
-O~-zq) ]"
(3.17)
A. K. H. Bengtsson et al. / Cubic interaction terms ( I )
37
It is straightforward to obtain the actions from here (see below). There is still some work left though. Referring back to (3.3) and (3.4), we see that we need a further dynamical piece 8~?¢pin order to close
(3.18)
[3, , 3 H ] % = 0 .
Demanding that (3.18) holds we find the following equation for 6ff9~:
2i~
aS_.
i x a / a + S H" C P + ~ 7 - 6 s C P - 3 H0 3 ,a q~=
3~-3
0.
(3.19)
The solution is 8s~qO= - 2 i a X
_l)8+,x ,,[~(X-l-,,) ~n ]
X 1 Y'. ( - 1 ) " ( X n
7 ;
¢gT;;-ePJ "
(3.20)
#1=0
This should be antisymmetrized for odd X, of course. There remains to check that 6j so obtained obeys the Lorentz algebra. This is straight forward and easy except for [Sj , 8j ] " ~ = O,
(3.21)
which is straightforward and tedious, requiring the use of various properties of Pascal's triangle, such as ( A n l ) + ( X - 1 ) = (1n ~ ) ' n
(3.22)
We can now write down actions which yield our equations of motion:
s=Sd4x{½UPDcP+a[UP L (-l)"(n~)O+~( '{x") c~" ] ]}+o(.'). .=o
(3.23) For odd X there is a structure constant. If we make use of the equality
~(-1) (Xn)O+,o+,,[O+,~.,m ¢P]
n=O
1
x
lx~[ a~
0+'~+'',=0 £ (-1)"tn][~
"' -a+, 2~
cp
">a" ] ¢P]'
(3.24)
38
A. K. H. Bengtsson et al. / Cubic interaction terms ( l )
we find that the full form of 8~cp including the cpcp terms that we have so far neglected is
a~,~=~ E (-1)" ,,=o
?' n
o +'~
- \ O+(~ ,,,
aN
0 1") + 2 ~ t ~ [¢O(x
+" .... O"qo).].
(3.25) Again, it can be checked that the algebra closes. This will also determine the missing piece in 6 f cO to be
xl
(~g{~ = -- 2i~oL E ( - - 1 ) n ( ~ - 1) 1 n ~ L tl = O
[ 0(X_____In) o +'~ "' ~°+'
2
.... 0"~
O(~k 1--,l) 2 n l)a ,lq0] " + 3 O+,~ CpO+~
(3.26)
4. Conclusions and comments
Let us summarize our results. We have found light-cone actions describing massless fields of arbitrary spin X, which lead to classical field theories in which the non-linearly realized Lorentz algebra closes to first order in the coupling constant c~. These actions are
S=fd4x ½¢[]w+,~,,=o ~ (-
In)[ w° ~ W T n ° ) + c ' c "
+O('~2) ' (4.1)
for X even and S = f d4x { ½~'~[]cp"
[
+~/~(~ ,,, ,,j,,
c.c.] o(.~)}
(4.2)
for 2, odd. There is a gauge group for any odd X and the coupling constants have dimension Ilength lx- 1. Several problems now present themselves, the first one being the question of higher orders in a. It would presumably be necessary to know these actions to all orders in a in order to prove that the higher-spin theories in fact are consistent. This may well be a difficult problem, but we foresee no problems of principle. The fact that the action (4.1) is known in closed form for X = 2 [12] is clearly encouraging.
39
A. K. H. Bengtsson et al. / Cubic interaction terms (I)
A n o t h e r obvious question is whether the higher-spin theories are realized in nature. It has been argued by Weinberg [13], who used S-matrix arguments, that no long-range forces can be generated by fields of spin higher than 2. They m a y still be there, however, although they have so far escaped observation. We have nothing whatsoever to say about this. Clearly, the dimensionful coupling constants would be difficult to handle in a q u a n t u m theory. Our conclusion is that the higher-spin theories are likely to exist, at least as classical field theories, although they m a y not have a manifestly covariant form. One of us (IB) wishes to thank Jan Willem van Holten for a discussion about previous approaches to higher-spin theory. We also wish to thank Olof Lindgren for proving a theorem about Pascal's triangle not given in ref. [11], and Arne Kihlberg for a discussion about the Lorentz group.
Appendix We will express our Lorentz generators in terms of the customary ones as j ~ - = ~ / ~ ( j + l +/j+2), aT+=d ( j + l _ / j + 2 ) , and similarly for J - . Moreover J = j12 and H = P - . We also use P = p l + ip2 and i f = p l _ ip2. The non-zero brackets of the Poincar6 group are [H,J + ]=ill,
[p+,J+
[ H , J + ] = i/7,
[H, J + ] = i P ,
] = - i p +,
[P+,J
]=iP,
[ P + , 3=-] = iff,
[P, J - ] = i l l ,
[ P , 3=+ ] = i p + ,
[p,J]= -e,
[ P, J - ] = i l l ,
[/7, j + ] = i p + ,
[ fi, j ] = ,ff
]=/J
,
[J ,]+]=
_M+
_j,
[J
[3=-, J + - ] = / ]
,
[aT--' J + ] =
-/J+
+J,
[J .J]=J
[J-,J+
[J+ ,J+]=iJ
+,
[J+, J] = - J + ,
[ J + , 3=+]= jar+ , oF+, J ] = . f + ,
References [1] N. Marcus and J.H. Schwarz, Phys. Lett. 115B (1982) 111 [2] W. Siegel and M. Ro~ek, Phys. Lett. 105B (1981) 279; B.E.W. Nilsson, G6teborg preprint 81-6, February 1981: V.O. Rivelles and J.G. Taylor, Phys. Lett. 121B (1983) 37
, J] = -J
,
.
40
A. K. H. Bengtsson et al. / Cubic interaction terms ( I )
[3] M. Sohnius, K.S, Stelle and P. West, Unification of the fundamental particle interactions, ed~ S. Ferrara, J. Ellis and P. van Nieuwenhuizen (Plenum Press 1980) [4] L. Brink, O. Lindgren and B.E.W. Nilsson, Nucl. Phys. B212 (1983) 401; S. Mandelstam, Nucl. Phys. B213 (1983) 149 [5] P.A.M. Dirac, Rev. Mod. Phys. 26 (1949) 392 [6] I. Bengtsson, G6teborg preprint 82-40, November 1982 [7] C.R. Hagen and L.P.S. Singh, Phys. Rev. D9 (1974) 898; C. Fronsdal, Phys. Rev. D18 (1978) 3624; J. Fang and C. Fronsdal, Phys. Rev. D18 (1978) 3630 [8] Supergravity, eds. P. van Nieuwenhuizen and D.Z. Freedman, (North-Holland, 1979) [9] A.K.H. Bengtsson, Nucl. Phys. B., to be publ. [10] E.P. Wigner, Ann. of Math. 40 (1939) 139 [11] Yang Hui, 1261 A.D.; Chu-Shih-Chieh, 1303 A.D.; B. Pascal, 1664 A.D. [12] J. Scherk and J. H. Schwarz, Gen. Rel. and Grav. 6 (1975) 537; M. Kaku, Nucl. Phys. B91 (1975) 99 [13] S. Weinberg, Phys. Rev. B135 (1964) 1049
CUBIC INTERACTION T E R M S FOR ARBITRARY SPIN* Anders K.H. BENGTSSON, Ingemar BENGTSSON and Lars BRINK Institute of Theoretical Physics, S-412 96 GiJteborg;, Sweden
Received 6 April 1983
Using thc light-cone formulation of dynamics, we construct cubic self-interaction terms for massless fields of arbitrary helicity X. The coupling constants have dimension [length]/` 1 and there is a gauge group for any odd X.
1. Introduction There are two kinds of s y m m e t r y generators in physics, kinematical and d y n a m i cal ones. The latter, which t r a n s f o r m the system f o r w a r d in time, will in general be n o n - l i n e a r l y ( a n d non-locally) realized on fields since their canonical expressions are m o d i f i e d b y interaction terms. To o b t a i n linear r e p r e s e n t a t i o n s of d y n a m i c a l symmetries it will be necessary to a d d auxiliary fields to the action, as is d o n e for e x a m p l e when a four-vector is used to describe a helicity-one field. U n f o r t u n a t e l y , there is no p r i n c i p l e g u a r a n t e e i n g the existence of these auxiliary fields, In fact, h i g h e r - d i m e n s i o n a l gravity theories are known, which seem to lack such a manifestly L o r e n t z c o v a r i a n t form [1]. This p r o b l e m is also e n c o u n t e r e d in s u p e r s y m m e t r y where the s u p e r s y m m e t r y generators are d y n a m i c a l within c o n v e n t i o n a l spacelike q u a n t i z a t i o n . F o r the p a r t i c u l a r l y interesting case of the N = 4 Y a n g - M i l l s theory q u i t e - s t r o n g no-go theorems [2] seem to p r e c l u d e the existence of a r e a s o n a b l e set of auxiliary fields for the full set of s u p e r s y m m e t r y generators. A m a n i f e s t l y s u p e r s y m metric form of this theory nevertheless exists [3], b u t it has not been possible to i n t r o d u c e interactions within this formulation. In the case of the N = 4 m o d e l this p r o b l e m was recently b y - p a s s e d [4] through the use of D i r a c ' s [5] light-cone (null-plane) f o r m u l a t i o n of dynamics. O n the light cone a p a r t of each s u p e r s y m m e t r y g e n e r a t o r b e c o m e s kinematical, so that no auxiliary fields are n e e d e d in o r d e r to o b t a i n a superfield f o r m u l a t i o n of the theory. F o r a further discussion of the a d v a n t a g e s of light-cone q u a n t i z a t i o n within this context, see ref. [6]. In this p a p e r we i n t e n d to use the fact that the light-cone f o r m u l a t i o n p e r m i t s one to w o r k with the physical fields exclusively (this is not always the case when the q u a n t i z a t i o n surface is space-like) in o r d e r to discuss a n o t h e r theoretical * Work supported in part by the Swedish Natural Science Research Council under contract no. S-Fo 8115-109. 31
32
A.K.H. Bengtsson et al. / Cubic interaction terms (I)
problem exhibiting great similarity with the situation in the N = 4 supersymmetric Yang-Mills theory. We will consider the question whether it is possible to have interactions in theories with helicity greater than 2. Again, manifestly covariant forms of such theories are known [7], but (apart from coupling to an external field) it has not been possible to introduce interactions within these formulations. For reviews of this question containing further references, see ref. [8]. In this paper we will argue, by exhibiting consistent three-point couplings for all massless fields with integer helicity, that this is purely an artefact of the insistence on manifest covariance and not a property of higher spins as such. The paper is organized as follows. In sect. 2 we introduce our notation and define free fields of arbitrary helicity. In sect. 3 we construct cubic self-interaction terms for all theories where such terms are possible. Sect. 4 summarizes our conclusions as well as further problems. The appendix contains the Poincar6 group in a suitable form. For further calculational details, consult ref. [9] (the case treated there is N = 1 supergravity).
2. The free fields in the light-cone frame
The fields that we will treat will all be described in the light-cone frame, i.e. we use for the Minkowski space X+=~22(xOq-x3),
X=~(xlq-iX2),
(2.1)
and similarly for the space-time derivatives. (Note that transverse indices are complexified.) In this frame x + is taken as the evolution parameter with p as the corresponding hamiltonian. Since any massless particle with spin has two degrees of freedom we will describe such particles of a definite spin X by a complex field ~ ( x ) . Furthermore we will choose the representation such that q0(x) has definite helicity - ) ~ and ~ the helicity + X. On a free field we can represent the orbital part of the Poincar6 algebra by (see the appendix for further notation)
a• p = - i a~-,
l = xJ-
p+= -i8 +, •
~8,
l+=i(xa+-x+8),
[+=i(~a+-x+O),
p = -i8,
[-=i
p= -i8,
aJ
.~--x-O
l+-=i x O+-x
, +0J~ -~-}.
(2.2)
A. K. H, Bengtsson et a l . , / Cubic interaction terms ( I )
33
In this representation we have made the substitution 0 - = 0cg/a +. This is the mass shell condition when it acts on a free field. However, since this condition is linear in 0-, the algebra still closes. It should, though, be noted that the representation of the generators is not hermitian (apart from the helicity operator) unless the mass shell condition is used. This is in contrast to the ordinary covariant formulation, and does correspond to a different way of going off-shell. This is not altogether new since for example the spin generators in the Dirac case is of this type. To transform a field with helicity X we also have to take the spin part of the Lorentz algebra into account [9]. The non-trivial ones are
8jcp = i~oj( l - X )ep,
To illustrate how these formulae are derived, we consider the Maxwell field A". Impose first the light-cone gauge A + = 0 and work out the compensating gauge transformation necessary to stay in the gauge when Lorentz transformations are performed. Then we can read off the resulting transformations of the physical degrees of freedom if we solve for the unphysical mode A -. In this way one can derive the transformations for any field with helicity X. However, it is simpler to start with 8j, knowing that the spin part has eigenvalue X and then close the algebra. Note in this context that eqs. (2.3) are quite close to the way massless particles are introduced as representations of the Lorentz group [10]. Since the free equations of motion are 00 O-qo = ~ - e p ,
(2.4)
the corresponding hamiltonian can be written as 8~
=
05 ~-w.
(2.5)
It is for this form that we will search for interaction terms, which are consistent with the Lorentz algebra. 3. Three-point
couplings
The lagrangian that we will attempt to construct is of the form
~= ½ ¢ ~ + ~( ~ ¢
+ ¢ ¢ ~ ) + o(~),
(3.1)
34
A . K . H . Bengtsson et al. / Cubic interaction terms ( I )
where a is a coupling constant and derivatives are still to be put into the interaction terms. The ensuing equation of motion for ~ is
03
8nCp = - ~ - ~ + a~q~ + a ~
+ O(a2).
(3.2)
(When we write an infinitesimal transformation we leave out the infinitesimal parameter). There will also be non-linear contributions to 8#+ , 8/ and 6 f , as follows 8 r ~ = so+ ~ - i ~ + 8 ~
+ o(~2),
~j ~ = ~o ~ + ixSh,~ + ~
+ o ( ~2 ) ,
87 q~ = 6° cp + iffS~op + 6~'~p + O( aZ ) .
(3.3)
With 6 ~ we mean the term to order a in the transformation. Here, 6~ and 8g are dynamical spin transformations of the form 62q~ = acpqv,
8~cp = aCpq~.
(3.4)
The last equations are assumptions at this stage. They are, however, true when X = 1, 2 and we will look for a similar solution in the general case. Our task now is to find the hamiltonian (3,2) together with the non-linear Lorentz transformations (3.3) such that they satisfy the correct algebra to first order in a. We then observe that terms of the type c~q0 and cpq0 do not mix in the transformations to this order. We can hence reduce the amount of labour involved by first considering only terms of the latter kind. We will make the ansatz that 8~ is a sum of terms of the form
8h~ = ~AO+"[ 0"0+~"0+%],
(3.5)
where t~, 0, ~, a and b are integers to be determined. This is not the most general form, but again it agrees with the k n o w n solutions for X = 1,2. We must now investigate what restrictions the Lorentz algebra imposes on this form. We begin by checking the algebra of 6 H with the Lorentz generators and start with the kinematical transformation 6j
[~,, 8.] °~ = o.
(3.6t
A. K. H. Bengtsson et al. / Cubic interaction terms (I)
35
Written out in detail, this becomes
8j~AO+'[ ~oa+%Jbo+°~]
~(l-
-
x),
O~O+°(Z-x)w Jb0+°w + 5 ° o + ~ o + ° ( t - x)w]
= ,~A0+'[
- ( t - X)~AO+"[5°0%~O÷°~]
= ~AO+"[[ 0 °, tl O+~ J~O+°w+ 0°0%[ 0 ~, ~1 a+°w - a~Aa+"[ ~oo%J~o+°,] = (a + b
-
X)aAO+"[ 0~O+~0hO+°cp] .
(3.7)
Thus we get our first restriction on the derivative structure in (3.5) namely
a+b=h.
(3.8)
We continue in this way. The commutator with 3/, yields /~+O+a=-l,
(3.9)
while the one with @ is trivial. The commutator with 3j~ on the other hand will mix terms in 3 H with different derivative structure, restricting the values of the coefficients A. In fact, with 3~q, of the form (3.5) we get
[ 6j,, 3tf ] "ep = aA[ iaO+"[ a +..... ~,a
l)q0~bo+°q0 ] + ibO+O[~aO+~l~(b 1,0+'"'t~q~] ] " (3.10)
We are also, because of eq. (3.4), in a position to do
[6[, 3tt ] ~ep = 0 = i(/z + 1 - ~ ) ~ z - a A O + " [ J~O+~JhO+oep] + a A [ i ( p + X)0+"[ j(a+l,O+~, l~bO+°qO ] + i ( o + X ) O * ' [ J ~ O + ~ J ' " * l ' O +'° ' ~ ] ] .
(3.11)
Since both of these commutators have to be zero, they give us a set of consistency conditions which in fact will suffice to determine 3~4) for us. Let us go through them helicity by helicity. For X = 0 we have a = b = 0 . A solution is clearly ~ = - 1 ,
36
A.K.H. Bengtsson et al. / Cubic interaction terms (I)
p = o = 0 and
3~rp = a 01-01~-[ epq)],
(3.12)
which, of course, is the coupling term in a q)3 theory. For )t = 1 the number of c~'s is one. In this case it is impossible to get zero out of both (3.10) and (3.11) unless we introduce an antisymmetric structure constant f , h q With/~ = 0, we have the solution
8~q)" = ctf"hc[ ~q)bq)c- @ ~ q ) c ] .
(3.13)
It is gratifying to see how the need for a gauge group emerges in this noncovariant formalism, and it is not difficult to check that this is the 3-point coupling for a non-abelian Yang-Mills field theory. For X = 2, we have a solution with/~ = 1, namely
8
8
J
7 7q)q) - 2 - U q ) - U q )
82 ] +
"
(3.14)
We recognize this as the three-point coupling in gravity [12]. N o t e that if X is half integer, no three-point coupling can exist because of (3.8). We have now exhausted all known cases. Nothing special happens, however, when we increase X. For X = 3, we have a solution with t~ = 2, again provided we introduce the structure constant fabc:
~H~
'~ "=
af ohiO+2 - 82 b 8 c 8 b 82 c ' - - 3q)hq)~"- 3 U q) fig-q) + 3fig-q) ~ q) -q)b - ~ q)c . (3.15)
We can go on, increasing X step by step. However, we already see Pascal's triangle [11] emerge and can write down the general solution. For ~ even: (3.16) For X odd:
3~Iq)" = a f " b ' Y",,=o( - 1 ) ' ( n ]
[~¢P
-O~-zq) ]"
(3.17)
A. K. H. Bengtsson et al. / Cubic interaction terms ( I )
37
It is straightforward to obtain the actions from here (see below). There is still some work left though. Referring back to (3.3) and (3.4), we see that we need a further dynamical piece 8~?¢pin order to close
(3.18)
[3, , 3 H ] % = 0 .
Demanding that (3.18) holds we find the following equation for 6ff9~:
2i~
aS_.
i x a / a + S H" C P + ~ 7 - 6 s C P - 3 H0 3 ,a q~=
3~-3
0.
(3.19)
The solution is 8s~qO= - 2 i a X
_l)8+,x ,,[~(X-l-,,) ~n ]
X 1 Y'. ( - 1 ) " ( X n
7 ;
¢gT;;-ePJ "
(3.20)
#1=0
This should be antisymmetrized for odd X, of course. There remains to check that 6j so obtained obeys the Lorentz algebra. This is straight forward and easy except for [Sj , 8j ] " ~ = O,
(3.21)
which is straightforward and tedious, requiring the use of various properties of Pascal's triangle, such as ( A n l ) + ( X - 1 ) = (1n ~ ) ' n
(3.22)
We can now write down actions which yield our equations of motion:
s=Sd4x{½UPDcP+a[UP L (-l)"(n~)O+~( '{x") c~" ] ]}+o(.'). .=o
(3.23) For odd X there is a structure constant. If we make use of the equality
~(-1) (Xn)O+,o+,,[O+,~.,m ¢P]
n=O
1
x
lx~[ a~
0+'~+'',=0 £ (-1)"tn][~
"' -a+, 2~
cp
">a" ] ¢P]'
(3.24)
38
A. K. H. Bengtsson et al. / Cubic interaction terms ( l )
we find that the full form of 8~cp including the cpcp terms that we have so far neglected is
a~,~=~ E (-1)" ,,=o
?' n
o +'~
- \ O+(~ ,,,
aN
0 1") + 2 ~ t ~ [¢O(x
+" .... O"qo).].
(3.25) Again, it can be checked that the algebra closes. This will also determine the missing piece in 6 f cO to be
xl
(~g{~ = -- 2i~oL E ( - - 1 ) n ( ~ - 1) 1 n ~ L tl = O
[ 0(X_____In) o +'~ "' ~°+'
2
.... 0"~
O(~k 1--,l) 2 n l)a ,lq0] " + 3 O+,~ CpO+~
(3.26)
4. Conclusions and comments
Let us summarize our results. We have found light-cone actions describing massless fields of arbitrary spin X, which lead to classical field theories in which the non-linearly realized Lorentz algebra closes to first order in the coupling constant c~. These actions are
S=fd4x ½¢[]w+,~,,=o ~ (-
In)[ w° ~ W T n ° ) + c ' c "
+O('~2) ' (4.1)
for X even and S = f d4x { ½~'~[]cp"
[
+~/~(~ ,,, ,,j,,
c.c.] o(.~)}
(4.2)
for 2, odd. There is a gauge group for any odd X and the coupling constants have dimension Ilength lx- 1. Several problems now present themselves, the first one being the question of higher orders in a. It would presumably be necessary to know these actions to all orders in a in order to prove that the higher-spin theories in fact are consistent. This may well be a difficult problem, but we foresee no problems of principle. The fact that the action (4.1) is known in closed form for X = 2 [12] is clearly encouraging.
39
A. K. H. Bengtsson et al. / Cubic interaction terms (I)
A n o t h e r obvious question is whether the higher-spin theories are realized in nature. It has been argued by Weinberg [13], who used S-matrix arguments, that no long-range forces can be generated by fields of spin higher than 2. They m a y still be there, however, although they have so far escaped observation. We have nothing whatsoever to say about this. Clearly, the dimensionful coupling constants would be difficult to handle in a q u a n t u m theory. Our conclusion is that the higher-spin theories are likely to exist, at least as classical field theories, although they m a y not have a manifestly covariant form. One of us (IB) wishes to thank Jan Willem van Holten for a discussion about previous approaches to higher-spin theory. We also wish to thank Olof Lindgren for proving a theorem about Pascal's triangle not given in ref. [11], and Arne Kihlberg for a discussion about the Lorentz group.
Appendix We will express our Lorentz generators in terms of the customary ones as j ~ - = ~ / ~ ( j + l +/j+2), aT+=d ( j + l _ / j + 2 ) , and similarly for J - . Moreover J = j12 and H = P - . We also use P = p l + ip2 and i f = p l _ ip2. The non-zero brackets of the Poincar6 group are [H,J + ]=ill,
[p+,J+
[ H , J + ] = i/7,
[H, J + ] = i P ,
] = - i p +,
[P+,J
]=iP,
[ P + , 3=-] = iff,
[P, J - ] = i l l ,
[ P , 3=+ ] = i p + ,
[p,J]= -e,
[ P, J - ] = i l l ,
[/7, j + ] = i p + ,
[ fi, j ] = ,ff
]=/J
,
[J ,]+]=
_M+
_j,
[J
[3=-, J + - ] = / ]
,
[aT--' J + ] =
-/J+
+J,
[J .J]=J
[J-,J+
[J+ ,J+]=iJ
+,
[J+, J] = - J + ,
[ J + , 3=+]= jar+ , oF+, J ] = . f + ,
References [1] N. Marcus and J.H. Schwarz, Phys. Lett. 115B (1982) 111 [2] W. Siegel and M. Ro~ek, Phys. Lett. 105B (1981) 279; B.E.W. Nilsson, G6teborg preprint 81-6, February 1981: V.O. Rivelles and J.G. Taylor, Phys. Lett. 121B (1983) 37
, J] = -J
,
.
40
A. K. H. Bengtsson et al. / Cubic interaction terms ( I )
[3] M. Sohnius, K.S, Stelle and P. West, Unification of the fundamental particle interactions, ed~ S. Ferrara, J. Ellis and P. van Nieuwenhuizen (Plenum Press 1980) [4] L. Brink, O. Lindgren and B.E.W. Nilsson, Nucl. Phys. B212 (1983) 401; S. Mandelstam, Nucl. Phys. B213 (1983) 149 [5] P.A.M. Dirac, Rev. Mod. Phys. 26 (1949) 392 [6] I. Bengtsson, G6teborg preprint 82-40, November 1982 [7] C.R. Hagen and L.P.S. Singh, Phys. Rev. D9 (1974) 898; C. Fronsdal, Phys. Rev. D18 (1978) 3624; J. Fang and C. Fronsdal, Phys. Rev. D18 (1978) 3630 [8] Supergravity, eds. P. van Nieuwenhuizen and D.Z. Freedman, (North-Holland, 1979) [9] A.K.H. Bengtsson, Nucl. Phys. B., to be publ. [10] E.P. Wigner, Ann. of Math. 40 (1939) 139 [11] Yang Hui, 1261 A.D.; Chu-Shih-Chieh, 1303 A.D.; B. Pascal, 1664 A.D. [12] J. Scherk and J. H. Schwarz, Gen. Rel. and Grav. 6 (1975) 537; M. Kaku, Nucl. Phys. B91 (1975) 99 [13] S. Weinberg, Phys. Rev. B135 (1964) 1049