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Minimal coupling in spin-2 field
Nuclear Physics B70 (1974) 356-364. North-Holland Publishing Company
MINIMAL COUPLING IN SPIN-2 FIELD John Francis REILLY
Department of Physics, Southampton University Received 24 October 1973
Abstract: It is shown that the spin-2 Lagrangian of Watanabe and Bhargava can be generalised by the introduction of a real parameter b associated with derivative ordering. After minimal coupling the spin-2 dynamical equations of Nath and Velo and Zwanziger can be derived as special cases. The constraint situation in these equations is then summarised and related to the first-order Lagrangian approach of Federbush, Chang, et al. Only one value of the parameter b gives a correct manifold of states after coupling. Finally it is shown that the auxiliary field approach to spin-2 proposed by Chang is dynamically inconsistent under minimal coupling.
1. Introduction
In 1939 Fierz and Pauli [1 ] constructed a second-order Lagrangian which could be minimally coupled to the electromagnetic field without leading to a spuriously enlarged or diminished manifold of states. Their dynamical spin-2 equation was, in this sense, consistent. Federbush [2] in 1961 noted that direct minimal coupling into a first-order Lagrangian theory of spin-2 led to twelve degrees of freedom in the dynamical equations. He concluded that such coupling was not consistent, and required the addition of a direct non-minimal interaction term in the Lagrangian. In 1964 Nath [3], again within the context of a second-order Lagrangian formalism, generalised the work o f Fierz and Pauli by introducing a one-parameter system of spin-2 Lagrangians. After minimal coupling the Nath equations retained their correct ten degrees of freedom without the need for Federbush terms. Thus the remark of Federbush appeared to be refuted. During 1966 Watanabe and Bhargava [4] produced another spin-2 second-order Lagrangian more general than that of Nath in that all subsidiary conditions, including tensor index symmetry, could be derived in the free case. The 1969 work of Velo and Zwanziger [5, 6] presented us with an analysis, again within a second-order Lagrangian formalism, of yet another spin-2 equation which this time did suffer from Federbush-type loss of constraint after minimal coupling. Thus a certain confusion emerged around the subject of spin-2 and its relation to minimal coupling.
J.F. Reilly, Spin-2 field
357
It will be shown that the spin-2 Lagrangian of Watanabe and Bhargava can be slightly generalised in such a way that the equations studied by Nath [3 ] and Velo and Zwanziger [6] follow from it as special cases• In this manner it is hoped to clarify the various spin-2 theories which have appeared, and to provide a modestly unified picture of the problem. The non-local spin-2 spproach of Chang [7] is also discussed and shown to be dynamically inconsistent under minimal coupling. Since the heart of the confusing situation with minimal coupling in spin-2 can also be found in the simpler case of spin-l, we begin with a brief note on the latter•
2. A free parameter in spin-1 The Lagrangians L(a) = -aut~vau~v
+ (1 - a ) O u t ~ v avt~ u + aaVt~v Ou~Ou + m 2 q J v ~ v
(1)
all lead to the same free Proca spin-1 equation (F-1 + m 2 ) ~ v -- a v a u ~ u
=
0,
(2)
where in eq. (1) a is a real parameter associated with the well-known derivative ordering ambiguity a a - * a a u a ~ +(1 - a ) a v a ~ .
(3)
Eq. (1) results by introducing the ambiguity depicted in eq. (3) into the standard spin-1 Lagrangian. Without interaction this is trivial. After minimal coupling it is not, and the dynamical equation is L v = ([I 2 - m 2 ) t ~ v --l-lVt~k u - ie(1 - - a ) FuV~v = 0 ,
(4)
1Iu = ia u + e A u
Just as in the spin-2 case exhibited by NatlL one may observe that the last expression in eq. (4) is associated with the ambiguity of eq. (3). It emerges after direct minimal coupling, and not through having added a Federbush-type non-minimal term to the Lagrangian. The expression proportional to F /a v~u in eq. (4) has been called a mag• neuc moment term [6, 8]. Now it so happens trtat in the case of spin-l, regardless of the value of a, the dynamical eq. (4) suffers neither from lost constraint nor acausally propagating solutions after minimal coupling. This fact was stated by Velo t Notation. The metric in use isg #v = (1, -1, -1, -1). Also h = c = 1. Greek indices run over the range 4, 1, 2, 3, while indices K and./run over 1, 2, 3.
J.F. Reilly, Spin-2 field
358
and Zwanziger [6], who will be referred to henceforth as ZV, and it has also been explicitly confirmed elsewhere [9]. ZV do not, however, associate the last term in eq. (4) with derivative order ambiguity, but interpret it from a Federbush viewpoint. Although in this way the spin-1 case is well-behaved, the significant point to be stressed and clarified is that in higher spin cases the choice of the parameters a is critical in determining whether or not the manifold of states will be enlarged or diminished after minimal coupling.
3. Additional parameter in local spin-2 Indicate by LOVB; 0) the second-order spin-2 Lagrangian of Watanabe and Bhargava [4]. Introducing the ambiguity of eq. (3) into L(WB;0), one finds after a little manipulation and collection of terms that it is useful to introduce one additional real parameter b into the system, which then reads L(WB; b ) = - a x ~ u v ) a X ~ ( ~ v ) + m 2 ~ u v ) ~ (tzv) - m2C~b*~ - m2Dt~uv]~ l~vl
+.4 a ~ 0 ~ ) a ~ +A*a ~*a~0 (~) ,
(5)
wherein the definitions are
~I~l
= ½(~
A :/:-½ ,
_
~.),
D =D* :/: 0 ,
B=I(3AA*
C = ½(6AA* + 3(A + A *) + 2 ) ,
+A +A* + I ) ,
~ = qJuu .
In passing one may note with interest that the functions B and C of A and A* exhibited above also appear in an approach to spin-~, see Cohen [10]. After making a direct minimal coupling into eq. (5), we get the dynamical equation (ii2 _ m 2) O(.u) _ ½A([itaiiv + i l u i i . ) ~ _glaV(A*iihiio~(~.a) + BII2~)
+ m2(CgUU~k +D~ luvl) - ½(1 - b) (IlUllx~(~) + IlVllxO(x~)) -½(1 +b)(IIxIIU~(xv) + l]xIIV~(xu)) = 0 .
(6)
Eq. (6) now provides some common ground from which to view the various spin-2
J.F. Reilly, Spin.2 field
359
equations which have appeared in the literature. For example we may proceed to identify the following two cases. 3.1. Case l
b=0;
A=A*;
(7)
~uv=t~uu.
Write L u " ( N ) = 0 to indicate the dynamical equation studied by Nath [3]. Substituting eq. (7) into eq. (6) we obtain L u v ( N ) = FI2I~uv _ ~(I/hFIU~hu + FIxFIV~X~) - ½(FI#lIx~xv + IlUFI~ h#) -AFlUIIVd/ + ½ieAF~
- m 2 ( ~ uv - CgUV~) - gUV(AIIxIIot~x° + BII2~) = 0 .
(8)
It can be seen that eq. (8) is symmetrised with respect to the dummy indices, and thus may be described as the symmetrised spin-2 equation. As shown by Nath, it implies ten subsidiary conditions: L4/(N) = 0 ;
3 first-order in time constraints,
(9a)
L44(N) + E L ( N ) = 0 ;
1 zeroth-order constraint,
(9b)
HxLXV(N) + E l i vL(N) = 0 , •
4 first-order constraints,
(9c)
IIv(eq. (9c)) - F L ( N ) = 0 ;
1 constraint, first order,
(9d)
a tenth constraint.
(9e)
The derivation of 9e involves manipulation of the other constraints [1,3, 8]. In eqs. (9) the definitions are E=-(A+I)(4A+2)
A
Z(N)=L
-1 ,
F=m2(4A+2)
-1 ,
"(N) .
Since there are ten constraints, this leaves ten degrees of freedom, which is consistent with spin-2. Note that although eq. (9b) is zeroth order in time, we do not simply differentiate it with respect to time and claim this asa new constraint. This is became L ~ ( N ) = 0 holds at all space-time points. Eq. (8) has been recently proposed again by Tait [8], without reference to Nath's paper [3]. I have shown elsewhere that the method of ZV [6] can be applied to confirm the presence of constraints in the system LUV(N) = 0 additional to those given in eqs. ( 9 a - c ) [9]. It transpires that for A = A *, the value b = 0 is the only value of this parameter
J.F. Reilly, SpO~-2fieM
360
in eq. (6) which leads to a correct manifold of states. All other values yield dynamical systems in which constraints are lost. Consider the second case II.
3.2. Case H b =1 ; A = A * = - I
; ~.v=~v..
(10)
Write Luv(ZV) = 0 to indicate the dynamical equation studied by Velo and Zwanziger. Substitute eq. (10) into eq. (6). The result is the unsymmetrised equation Luv(ZV) = (ii2 _ m 2) ~uv + m2gUV~ _ (iixllU~xv + iixllV~Xu ) + ½(n.n
+ nvn .)
+ W(nxn
- n2
) = 0.
(11)
Direct algebraic manipulation confirms the presence of only eight constraints in the system of eq. (11). This was further checked by ZV [6] who introduced the concept of an equivalent equation. The constraints implied by eq. (11) are L4J(zv) = 0 ;
3 first-order in time constraints,
(12a)
L44(ZV) = 0 ;
1 zeroth-order constraint,
(12b)
IIuLUV(ZV ) -- 0 ;
4 first-order constraints.
(12c)
Once again, although eq. (12b) is zeroth order in time, we do not claim that its time derivative constitutes an additional constraint. Eq. (I 1) thus has twelve degrees of freedom, unlike eq. (8) which has ten. Comparison of these two equations illustrates the important role played by the parameter b in determining the manifold of states. For the special value A = - 1 in eq. (8) we have the relation
L UV(N)A = - 1 = Luu(ZV) - ½ie(FUx ~ xu + FVx ~ xu ) .
(13)
It could thus also be said that Luv(ZV) = 0 is made consistent with spin-2 if we add the non-minimal Federbush term • ~ ux , F vu qj u x te
(14)
to the Lagrangian from which Luv(ZV) = 0 is derived. However, owing to the presence of a somewhat complicated tenth constraint in LUV(N) = 0 it does not appear possible to discuss the causality of this equation at the first quantised level using the ingenious method of ZV. At the second quantised level, Nath has shown that his equation is inconsistent. Finally one may remark that in a first-order Lagrangian approach to spin-2 there is of course no derivative order-
J.F. Reilly. Spin-2 field
361
ing ambiguity in the Lagrangian. We have the well-known Lagrangian [9, 11-13] L=hitP(DxPxit) , - ~-lrOp I~Xux +O u Vxp^")) +g~P(F° It^"Pxpo - P° ~ I'x ox, + ¼m2 ( ~ itv~ It)' -- egg) ,
(1 5)
where rx
= r,X ,0.12
hitp = ¢,u~ _ ½gU~ ~ . Pit '
Minimal coupling in eq. (15) leads to the unsymmetrised form LItu(ZV) = O. Federbush studied the first-order Lagrangian approach to spin-2, and hence we observe the relationship between his celebrated observation and the second-order Lagrangian approaches of Nath and Velo and Zwanziger. This relationship is clearly embodied in eq. (13).
4. Dynamical inconsistency of auxiliary field approach to spin-2 Chang [7] has given an approach to higher spin equations which utilises the concept of auxiliary fields. For integral spin his equations read: - m 2 ~ Itv'''x = F](P(s) ffitu...x) ,
(16)
where P(s) is a projection operator constructed from spin-1 projectors p(1)itv =guy _ l--l-10it0v .
(17)
The field is assumed to be initially symmetric and traceless. Thus all manipulations are directed to the task of constructing a Lagrangian formalism which yields the Klein-Gordon equation for the field and the supplementary condition Oit~ It~''~ = 0 ,
(18)
in the free case, together with an elimination of the auxiliary fields. For the case of spin-2 it will be demonstrated that after minimal coupling the auxiliary field can no longer be eliminated, and that the resulting dynamical system has twelve degrees of freedom. For spin-2 the eqs. (16) and (17) lead to the nonqocal structure _m2~bit v = O~it v _ 3v(Ot.b)it -- 3u(O~b)v 36
(0~) u - 0>,~ xu .
v o-v~ . ~
(19)
J.F. Reilly, Spin-2field
362
This converts into a "local" equation by introducing the auxiliary field a = ] [3 -1 0oax~b °x .
(20)
which Chang now takes to be an independent field. With eq. (20) substituted into eq. (19) we have _m2ffa v = [] ~b~ - Or(Off)~ -- Oa(Otb)Z' + ½gaUOoOx~b°x
+ (aua ~ - ¼gUVvq)Q .
(21)
The combination in the last bracket preserves the trace, but since Q is now treated as an independent variable we observe that the right hand side of eq. (21) is no longer manifestly divergence free. Multiplying au into eq. (21) yields
-m2Ou~uv = -½ ov(aoax~ °x) + ~ [E]aVQ 4=0, unless for example 0o0x~box = Q = 0 .
(22)
Operation on eq. (21) with 0uav leads to ([7 _ 2m 2) aoox~ox = a D2Q,
(23)
and Chang observes that if the Q-field obeys (1:3 + 2m2)Q = OoOx~b°x ,
(24)
then eqs. (23) and (24) together imply eq. (22). His task is therefore to build a Lagrangiah~hich will yield both eq. (2 I) and eq. (24). The result is L(C0 = ½~buv(D + m2)ff tw + (a~b)ta(O~bya + ½Q OoOxq/°h - 4~Q(D + 2m2)Q.
(25)
Now there will be serious problems if the auxiliary field Q cannot be eliminated after minimal coupling. For one tiring it is ill-defined for there is the ambiguity Q = 32D
- 1
0a0x~b a k
--, z3 ri-2ri llxq:x O
or
etc.
IIoll-2llX~°X
J.F. Reilly, Spin-2 field
363
The coupled version of eq. (21) is
+ (½ (11ullu + i/uiiu) _ ¼g~Vii2) a = 0 .
(26)
An ambiguity in the ordering of the II operators in the last bracket has been fixed by a requirement to preserve the symmetry of the equation under # - u interchange. Again the last bracket combination preserves the trace which, recall, does not follow by deduction from the Lagrangian. fluu = 0 is assumed. Writing eq. (26) as Luv(C) = 0 we have
LK4(c) = 0 ;
3 first order in time constraints,
(27a)
L44(C) = 0 ;
1 zeroth order in time constraint.
(27b)
To show that (27b) is a constraint the coupled version of eq. (24) Flo Fix ~°x = ~ (FI 2 - 2m 2) Q
(28)
is used to eliminate II 42Q • Note that although eq (28) can be used to perform a similar reduction on LKK(c) = 0 (K summed), the latter does not constitute an additional constraint since LKK(c) = L44(C). Therefore LKK(c) = 0 is not an independent constraint. Further use of eq. (28) in conjunction with,
IILUv(C) = 0
and
H~ HuLUv(C) = 0
allows us to find
-m2(IIutPUV - ~ W'Q) + ie(F~'lI u +~- IIxFXU) Q + ie(IlaF~ x + F x l I X ) t~uz' - ieF;Flx~b xu = O,
(29)
2 ie(11vIIxFXut]~uv + n FxurIx¢,u~ + rIF~11x¢vxu ) - 6 m 4 Q
+ (2ie11,,FulIU
- ½e2 Fu FU~)Q = 0 .
(30)
It is now clear that the auxiliary field Q can no longer be eliminated. Eq. (29) is first-order in time and so represents a further four constraints, while eq. (30) is second-order, containing a term
F4KFI2~4K ,
(31)
364
J.F. R eilly, Spin-2 field
which cannot be removed. The constraints present in the coupled system are therefore from eqs. (27), (29) making eight in all. Since we are dealing with a secondorder system and since ffuv is assumed symmetric-traceless, the ffU~Q system without constraint would possess 20 degrees of freedom. Thus the dynamical equations possess 20 - 8 = 12 degrees of freedom. After minimal coupling it appears that both the spin-2 and scaler Q-fields are propagating. It is easy to show that the Chang spin-] equations are also dynamically inconsistent under minimal coupling [9]. Since the complexity of the Chang equations rapidly increases with spin, one can predict confidently that all such equations are inconsistent under minimal coupling. Finally it may be noted that the local spin-2 equation proposed by da Silveira [14, 15] back in the late 1950 's is also a special case of the Watanabe-Bhargava system. Da Silveira's equation results from putting Fuv = 0, ~kuv = ffvz, and A = A * = -½ into eq. (6). Although, as Watanabe and Bhargava remark, the parameters A, A*, and D have no physical significance in eq. (6), the same cannot be said of parameters such as b in that equation which are associated with derivative ordering. I am grateful to Professor J.G. Taylor for many interesting and stimulating discussions.
References [1] M. Fierz and W. Pauli, Proc. Roy. Soc. A173 (1939) 211. [2] P. Federbush, Nuovo Cimento 19 (1961) 572. [3] L.M. Nath, Nucl. Phys. 7A (1965) 660. [4] H. Watanabe and S.C. Bhargava, Nucl. Phys. 87 (1966) 273. [5] G. Velo and D. Zwanziger, Phys. Rev. 186 (1969) 1337. [6] G. Velo and D. Zwanziger, Phys. Rev. 188 (1969) 2218. [7] S.J. Chang, Phys. Rev. 161 (1967) 1308. [8] W. Tait, Phys. Rev. D 5 (1972) 3272. [9] J.F. Reilly, Ph.D. thesis, University of Southampton (1973). [101 H.A. Cohen, Nuovo Cimento 52A (1967) 1242. [11] D. Adler, Can. J. Phys. 44 (1966) 289. [12] R. Arnowitt and S. Deser, Phys. Rev. 113 (1959) 2,745. [13] S.J. Chang, Phys. Rev. 148 (1966) 4, 1259. [14] A. Silveira, Phys. Rev. 97 (1955) 1144. [15] A. Silveira, Nuovo Cimento 3 (1956) 513.
MINIMAL COUPLING IN SPIN-2 FIELD John Francis REILLY
Department of Physics, Southampton University Received 24 October 1973
Abstract: It is shown that the spin-2 Lagrangian of Watanabe and Bhargava can be generalised by the introduction of a real parameter b associated with derivative ordering. After minimal coupling the spin-2 dynamical equations of Nath and Velo and Zwanziger can be derived as special cases. The constraint situation in these equations is then summarised and related to the first-order Lagrangian approach of Federbush, Chang, et al. Only one value of the parameter b gives a correct manifold of states after coupling. Finally it is shown that the auxiliary field approach to spin-2 proposed by Chang is dynamically inconsistent under minimal coupling.
1. Introduction
In 1939 Fierz and Pauli [1 ] constructed a second-order Lagrangian which could be minimally coupled to the electromagnetic field without leading to a spuriously enlarged or diminished manifold of states. Their dynamical spin-2 equation was, in this sense, consistent. Federbush [2] in 1961 noted that direct minimal coupling into a first-order Lagrangian theory of spin-2 led to twelve degrees of freedom in the dynamical equations. He concluded that such coupling was not consistent, and required the addition of a direct non-minimal interaction term in the Lagrangian. In 1964 Nath [3], again within the context of a second-order Lagrangian formalism, generalised the work o f Fierz and Pauli by introducing a one-parameter system of spin-2 Lagrangians. After minimal coupling the Nath equations retained their correct ten degrees of freedom without the need for Federbush terms. Thus the remark of Federbush appeared to be refuted. During 1966 Watanabe and Bhargava [4] produced another spin-2 second-order Lagrangian more general than that of Nath in that all subsidiary conditions, including tensor index symmetry, could be derived in the free case. The 1969 work of Velo and Zwanziger [5, 6] presented us with an analysis, again within a second-order Lagrangian formalism, of yet another spin-2 equation which this time did suffer from Federbush-type loss of constraint after minimal coupling. Thus a certain confusion emerged around the subject of spin-2 and its relation to minimal coupling.
J.F. Reilly, Spin-2 field
357
It will be shown that the spin-2 Lagrangian of Watanabe and Bhargava can be slightly generalised in such a way that the equations studied by Nath [3 ] and Velo and Zwanziger [6] follow from it as special cases• In this manner it is hoped to clarify the various spin-2 theories which have appeared, and to provide a modestly unified picture of the problem. The non-local spin-2 spproach of Chang [7] is also discussed and shown to be dynamically inconsistent under minimal coupling. Since the heart of the confusing situation with minimal coupling in spin-2 can also be found in the simpler case of spin-l, we begin with a brief note on the latter•
2. A free parameter in spin-1 The Lagrangians L(a) = -aut~vau~v
+ (1 - a ) O u t ~ v avt~ u + aaVt~v Ou~Ou + m 2 q J v ~ v
(1)
all lead to the same free Proca spin-1 equation (F-1 + m 2 ) ~ v -- a v a u ~ u
=
0,
(2)
where in eq. (1) a is a real parameter associated with the well-known derivative ordering ambiguity a a - * a a u a ~ +(1 - a ) a v a ~ .
(3)
Eq. (1) results by introducing the ambiguity depicted in eq. (3) into the standard spin-1 Lagrangian. Without interaction this is trivial. After minimal coupling it is not, and the dynamical equation is L v = ([I 2 - m 2 ) t ~ v --l-lVt~k u - ie(1 - - a ) FuV~v = 0 ,
(4)
1Iu = ia u + e A u
Just as in the spin-2 case exhibited by NatlL one may observe that the last expression in eq. (4) is associated with the ambiguity of eq. (3). It emerges after direct minimal coupling, and not through having added a Federbush-type non-minimal term to the Lagrangian. The expression proportional to F /a v~u in eq. (4) has been called a mag• neuc moment term [6, 8]. Now it so happens trtat in the case of spin-l, regardless of the value of a, the dynamical eq. (4) suffers neither from lost constraint nor acausally propagating solutions after minimal coupling. This fact was stated by Velo t Notation. The metric in use isg #v = (1, -1, -1, -1). Also h = c = 1. Greek indices run over the range 4, 1, 2, 3, while indices K and./run over 1, 2, 3.
J.F. Reilly, Spin-2 field
358
and Zwanziger [6], who will be referred to henceforth as ZV, and it has also been explicitly confirmed elsewhere [9]. ZV do not, however, associate the last term in eq. (4) with derivative order ambiguity, but interpret it from a Federbush viewpoint. Although in this way the spin-1 case is well-behaved, the significant point to be stressed and clarified is that in higher spin cases the choice of the parameters a is critical in determining whether or not the manifold of states will be enlarged or diminished after minimal coupling.
3. Additional parameter in local spin-2 Indicate by LOVB; 0) the second-order spin-2 Lagrangian of Watanabe and Bhargava [4]. Introducing the ambiguity of eq. (3) into L(WB;0), one finds after a little manipulation and collection of terms that it is useful to introduce one additional real parameter b into the system, which then reads L(WB; b ) = - a x ~ u v ) a X ~ ( ~ v ) + m 2 ~ u v ) ~ (tzv) - m2C~b*~ - m2Dt~uv]~ l~vl
+.4 a ~ 0 ~ ) a ~ +A*a ~*a~0 (~) ,
(5)
wherein the definitions are
~I~l
= ½(~
A :/:-½ ,
_
~.),
D =D* :/: 0 ,
B=I(3AA*
C = ½(6AA* + 3(A + A *) + 2 ) ,
+A +A* + I ) ,
~ = qJuu .
In passing one may note with interest that the functions B and C of A and A* exhibited above also appear in an approach to spin-~, see Cohen [10]. After making a direct minimal coupling into eq. (5), we get the dynamical equation (ii2 _ m 2) O(.u) _ ½A([itaiiv + i l u i i . ) ~ _glaV(A*iihiio~(~.a) + BII2~)
+ m2(CgUU~k +D~ luvl) - ½(1 - b) (IlUllx~(~) + IlVllxO(x~)) -½(1 +b)(IIxIIU~(xv) + l]xIIV~(xu)) = 0 .
(6)
Eq. (6) now provides some common ground from which to view the various spin-2
J.F. Reilly, Spin.2 field
359
equations which have appeared in the literature. For example we may proceed to identify the following two cases. 3.1. Case l
b=0;
A=A*;
(7)
~uv=t~uu.
Write L u " ( N ) = 0 to indicate the dynamical equation studied by Nath [3]. Substituting eq. (7) into eq. (6) we obtain L u v ( N ) = FI2I~uv _ ~(I/hFIU~hu + FIxFIV~X~) - ½(FI#lIx~xv + IlUFI~ h#) -AFlUIIVd/ + ½ieAF~
- m 2 ( ~ uv - CgUV~) - gUV(AIIxIIot~x° + BII2~) = 0 .
(8)
It can be seen that eq. (8) is symmetrised with respect to the dummy indices, and thus may be described as the symmetrised spin-2 equation. As shown by Nath, it implies ten subsidiary conditions: L4/(N) = 0 ;
3 first-order in time constraints,
(9a)
L44(N) + E L ( N ) = 0 ;
1 zeroth-order constraint,
(9b)
HxLXV(N) + E l i vL(N) = 0 , •
4 first-order constraints,
(9c)
IIv(eq. (9c)) - F L ( N ) = 0 ;
1 constraint, first order,
(9d)
a tenth constraint.
(9e)
The derivation of 9e involves manipulation of the other constraints [1,3, 8]. In eqs. (9) the definitions are E=-(A+I)(4A+2)
A
Z(N)=L
-1 ,
F=m2(4A+2)
-1 ,
"(N) .
Since there are ten constraints, this leaves ten degrees of freedom, which is consistent with spin-2. Note that although eq. (9b) is zeroth order in time, we do not simply differentiate it with respect to time and claim this asa new constraint. This is became L ~ ( N ) = 0 holds at all space-time points. Eq. (8) has been recently proposed again by Tait [8], without reference to Nath's paper [3]. I have shown elsewhere that the method of ZV [6] can be applied to confirm the presence of constraints in the system LUV(N) = 0 additional to those given in eqs. ( 9 a - c ) [9]. It transpires that for A = A *, the value b = 0 is the only value of this parameter
J.F. Reilly, SpO~-2fieM
360
in eq. (6) which leads to a correct manifold of states. All other values yield dynamical systems in which constraints are lost. Consider the second case II.
3.2. Case H b =1 ; A = A * = - I
; ~.v=~v..
(10)
Write Luv(ZV) = 0 to indicate the dynamical equation studied by Velo and Zwanziger. Substitute eq. (10) into eq. (6). The result is the unsymmetrised equation Luv(ZV) = (ii2 _ m 2) ~uv + m2gUV~ _ (iixllU~xv + iixllV~Xu ) + ½(n.n
+ nvn .)
+ W(nxn
- n2
) = 0.
(11)
Direct algebraic manipulation confirms the presence of only eight constraints in the system of eq. (11). This was further checked by ZV [6] who introduced the concept of an equivalent equation. The constraints implied by eq. (11) are L4J(zv) = 0 ;
3 first-order in time constraints,
(12a)
L44(ZV) = 0 ;
1 zeroth-order constraint,
(12b)
IIuLUV(ZV ) -- 0 ;
4 first-order constraints.
(12c)
Once again, although eq. (12b) is zeroth order in time, we do not claim that its time derivative constitutes an additional constraint. Eq. (I 1) thus has twelve degrees of freedom, unlike eq. (8) which has ten. Comparison of these two equations illustrates the important role played by the parameter b in determining the manifold of states. For the special value A = - 1 in eq. (8) we have the relation
L UV(N)A = - 1 = Luu(ZV) - ½ie(FUx ~ xu + FVx ~ xu ) .
(13)
It could thus also be said that Luv(ZV) = 0 is made consistent with spin-2 if we add the non-minimal Federbush term • ~ ux , F vu qj u x te
(14)
to the Lagrangian from which Luv(ZV) = 0 is derived. However, owing to the presence of a somewhat complicated tenth constraint in LUV(N) = 0 it does not appear possible to discuss the causality of this equation at the first quantised level using the ingenious method of ZV. At the second quantised level, Nath has shown that his equation is inconsistent. Finally one may remark that in a first-order Lagrangian approach to spin-2 there is of course no derivative order-
J.F. Reilly. Spin-2 field
361
ing ambiguity in the Lagrangian. We have the well-known Lagrangian [9, 11-13] L=hitP(DxPxit) , - ~-lrOp I~Xux +O u Vxp^")) +g~P(F° It^"Pxpo - P° ~ I'x ox, + ¼m2 ( ~ itv~ It)' -- egg) ,
(1 5)
where rx
= r,X ,0.12
hitp = ¢,u~ _ ½gU~ ~ . Pit '
Minimal coupling in eq. (15) leads to the unsymmetrised form LItu(ZV) = O. Federbush studied the first-order Lagrangian approach to spin-2, and hence we observe the relationship between his celebrated observation and the second-order Lagrangian approaches of Nath and Velo and Zwanziger. This relationship is clearly embodied in eq. (13).
4. Dynamical inconsistency of auxiliary field approach to spin-2 Chang [7] has given an approach to higher spin equations which utilises the concept of auxiliary fields. For integral spin his equations read: - m 2 ~ Itv'''x = F](P(s) ffitu...x) ,
(16)
where P(s) is a projection operator constructed from spin-1 projectors p(1)itv =guy _ l--l-10it0v .
(17)
The field is assumed to be initially symmetric and traceless. Thus all manipulations are directed to the task of constructing a Lagrangian formalism which yields the Klein-Gordon equation for the field and the supplementary condition Oit~ It~''~ = 0 ,
(18)
in the free case, together with an elimination of the auxiliary fields. For the case of spin-2 it will be demonstrated that after minimal coupling the auxiliary field can no longer be eliminated, and that the resulting dynamical system has twelve degrees of freedom. For spin-2 the eqs. (16) and (17) lead to the nonqocal structure _m2~bit v = O~it v _ 3v(Ot.b)it -- 3u(O~b)v 36
(0~) u - 0>,~ xu .
v o-v~ . ~
(19)
J.F. Reilly, Spin-2field
362
This converts into a "local" equation by introducing the auxiliary field a = ] [3 -1 0oax~b °x .
(20)
which Chang now takes to be an independent field. With eq. (20) substituted into eq. (19) we have _m2ffa v = [] ~b~ - Or(Off)~ -- Oa(Otb)Z' + ½gaUOoOx~b°x
+ (aua ~ - ¼gUVvq)Q .
(21)
The combination in the last bracket preserves the trace, but since Q is now treated as an independent variable we observe that the right hand side of eq. (21) is no longer manifestly divergence free. Multiplying au into eq. (21) yields
-m2Ou~uv = -½ ov(aoax~ °x) + ~ [E]aVQ 4=0, unless for example 0o0x~box = Q = 0 .
(22)
Operation on eq. (21) with 0uav leads to ([7 _ 2m 2) aoox~ox = a D2Q,
(23)
and Chang observes that if the Q-field obeys (1:3 + 2m2)Q = OoOx~b°x ,
(24)
then eqs. (23) and (24) together imply eq. (22). His task is therefore to build a Lagrangiah~hich will yield both eq. (2 I) and eq. (24). The result is L(C0 = ½~buv(D + m2)ff tw + (a~b)ta(O~bya + ½Q OoOxq/°h - 4~Q(D + 2m2)Q.
(25)
Now there will be serious problems if the auxiliary field Q cannot be eliminated after minimal coupling. For one tiring it is ill-defined for there is the ambiguity Q = 32D
- 1
0a0x~b a k
--, z3 ri-2ri llxq:x O
or
etc.
IIoll-2llX~°X
J.F. Reilly, Spin-2 field
363
The coupled version of eq. (21) is
+ (½ (11ullu + i/uiiu) _ ¼g~Vii2) a = 0 .
(26)
An ambiguity in the ordering of the II operators in the last bracket has been fixed by a requirement to preserve the symmetry of the equation under # - u interchange. Again the last bracket combination preserves the trace which, recall, does not follow by deduction from the Lagrangian. fluu = 0 is assumed. Writing eq. (26) as Luv(C) = 0 we have
LK4(c) = 0 ;
3 first order in time constraints,
(27a)
L44(C) = 0 ;
1 zeroth order in time constraint.
(27b)
To show that (27b) is a constraint the coupled version of eq. (24) Flo Fix ~°x = ~ (FI 2 - 2m 2) Q
(28)
is used to eliminate II 42Q • Note that although eq (28) can be used to perform a similar reduction on LKK(c) = 0 (K summed), the latter does not constitute an additional constraint since LKK(c) = L44(C). Therefore LKK(c) = 0 is not an independent constraint. Further use of eq. (28) in conjunction with,
IILUv(C) = 0
and
H~ HuLUv(C) = 0
allows us to find
-m2(IIutPUV - ~ W'Q) + ie(F~'lI u +~- IIxFXU) Q + ie(IlaF~ x + F x l I X ) t~uz' - ieF;Flx~b xu = O,
(29)
2 ie(11vIIxFXut]~uv + n FxurIx¢,u~ + rIF~11x¢vxu ) - 6 m 4 Q
+ (2ie11,,FulIU
- ½e2 Fu FU~)Q = 0 .
(30)
It is now clear that the auxiliary field Q can no longer be eliminated. Eq. (29) is first-order in time and so represents a further four constraints, while eq. (30) is second-order, containing a term
F4KFI2~4K ,
(31)
364
J.F. R eilly, Spin-2 field
which cannot be removed. The constraints present in the coupled system are therefore from eqs. (27), (29) making eight in all. Since we are dealing with a secondorder system and since ffuv is assumed symmetric-traceless, the ffU~Q system without constraint would possess 20 degrees of freedom. Thus the dynamical equations possess 20 - 8 = 12 degrees of freedom. After minimal coupling it appears that both the spin-2 and scaler Q-fields are propagating. It is easy to show that the Chang spin-] equations are also dynamically inconsistent under minimal coupling [9]. Since the complexity of the Chang equations rapidly increases with spin, one can predict confidently that all such equations are inconsistent under minimal coupling. Finally it may be noted that the local spin-2 equation proposed by da Silveira [14, 15] back in the late 1950 's is also a special case of the Watanabe-Bhargava system. Da Silveira's equation results from putting Fuv = 0, ~kuv = ffvz, and A = A * = -½ into eq. (6). Although, as Watanabe and Bhargava remark, the parameters A, A*, and D have no physical significance in eq. (6), the same cannot be said of parameters such as b in that equation which are associated with derivative ordering. I am grateful to Professor J.G. Taylor for many interesting and stimulating discussions.
References [1] M. Fierz and W. Pauli, Proc. Roy. Soc. A173 (1939) 211. [2] P. Federbush, Nuovo Cimento 19 (1961) 572. [3] L.M. Nath, Nucl. Phys. 7A (1965) 660. [4] H. Watanabe and S.C. Bhargava, Nucl. Phys. 87 (1966) 273. [5] G. Velo and D. Zwanziger, Phys. Rev. 186 (1969) 1337. [6] G. Velo and D. Zwanziger, Phys. Rev. 188 (1969) 2218. [7] S.J. Chang, Phys. Rev. 161 (1967) 1308. [8] W. Tait, Phys. Rev. D 5 (1972) 3272. [9] J.F. Reilly, Ph.D. thesis, University of Southampton (1973). [101 H.A. Cohen, Nuovo Cimento 52A (1967) 1242. [11] D. Adler, Can. J. Phys. 44 (1966) 289. [12] R. Arnowitt and S. Deser, Phys. Rev. 113 (1959) 2,745. [13] S.J. Chang, Phys. Rev. 148 (1966) 4, 1259. [14] A. Silveira, Phys. Rev. 97 (1955) 1144. [15] A. Silveira, Nuovo Cimento 3 (1956) 513.