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Consistency problems of hypergravity
Volume 86B, number 2
PHYSICS LETTERS
24 September 1979
CONSISTENCY PROBLEMS OF HYPERGRAVITY C. ARAGONE 1 and S. DESER 2
PhysicsDepartment, Brandeis University, Waltham,MA 02154, USA Received 13 April 1979 Revised manuscript received 13 July 1979
The constraints arising upon coupling a massless spin 5/2 field to gravity are analyzed. In contrast to supergravity, they depend not only on the Einstein tensor, but also on the off-shell (Weyl) components of the curvature. The latter contributions do vanish, however, for "self-dual" systems, i.e., half-flat gravitational and pure (left/right) helicity spin 5/2 fields.
There are two a priori candidates for a supersymmetric completion of general relativity, namely spin 3/2 or 5/2 fields, the former corresponding to the consistent, locally supersymmetric model of supergravity [1 ]. According to general lore on higher spin fields and to S-matrix arguments on absence of low frequency spin 5 / 2 - 2 interactions [2], spin 5/2 is not likely to produce a similarly successful theory ("hypergravity") despite the possible benefits such a model might bring. In this note, we analyze the consistency problems facing the coupled massless 5 [ 2 - 2 system, contrasting them with the 3 / 2 - 2 case, quite apart from supersymmetry considerations. We shall see that, owing to the additional index structure for spin ~>5/2, the identities which tile free field possesses because of its pure helicity +s content become more stringent local constraints on the dynamical variables of the coupled fields. This basic difficulty is encountered in coupling all gauge (massless spin/> I) fields to gravity, and is due to noncommutation of covariant derivatives. However, spin 1 escapes this problem because the identity ~(~z,F uz') - 0 can still be written in terms of ordinary (rather than covariant) derivatives, using the contravariant tensor density X/Z-ffFU v. For spin 3/2, the divergence DzRU of the Rarita-Schwinger equation 1 John Simon Guggenheim Memorial Fellow. On leave of absence from Universidad Simon Bolivar. Work supported in part by CONICIT, Venezuela, grant S1-0972. 2 Supported in part by NSF grant PHY-78-09644.
R u ==-eUVC'a%Dc,~k~ = %,.fur = 0 ,
(1)
does involve covariant derivatives but its value depends only on the Ricci rather than the Weyl tensor; this is what makes a consistent coupling at all possible, because the Ricci (unlike the Weyl) tensor is specified by the Einstein equations. Absence of the Weyl tensor is a simple consequence of index counting: the scalar D~RU has the generic formRabcdF A ~ , but there are too few indices in the I~ matrices 7~, 3'57~, o~v to involve the four-index curvature. For spin 2, described by a symmetric tensor Cur, there is also dependence only on the Ricci tensor in the "Bianchi identity" D~G~'(~p) where G~ v is the ~ field equation; it is well known [3] that a nontrivial way out of these constraints is for the ~o-field to be a small deviation from the basic background gt~" In that case, the DuG~ v constraint simply becomes an expansion of the full gravitational Bianchi identity to first order in 5guy = ~ . Consider now the spin 5/2 field, whose flat space action and field equations have been given by Schwinger [4]. Its gauge invariances and associated identities are also implicit in his eq. (3-3.114). The field equations may be cast into very simple form [5] by taking appropriate linear combinations of the original ones together with their traces and 3,-traces. We write them here in a "geometrical" notation: S~,~ = y ~ ( - a~,~uv+ a u ~ +
avCJ~,) - 3,~S2~,~,~ =0.
(2) The "Christoffel symbol" I2uu,a is 7 a orthogonal. Eqs. 161
Volume 86B, number 2
PHYSICS LETTERS
(2) are invariant under the following abelian transformations of the symmetric tensor-spinor ~kuv: =
(3)
with respect to the three independent 7-traceless v e c t o r spinors ~u(x). The detailed hamiltonian analysis which establishes the pure helicity 5/2 content of the above system will be given elsewhere [6], but the count is immediate: to each local gauge freedom in a fermionic system are associated one Lagrange multiplier, one constraint equation (and therefore one constraint variable) and one gauge function. In our case, this removes 3 X 3 components out of the 10ffuv, leaving a single spinor to describe the +5/2 helicity states. (For spin 3/2 the four eu are correspondingly reduced to one by the invariance under the one gauge 6 flu = ~ u a ( x ) . ) There is one identity (which is unfortunately trivial) that does persist in the flat to curved transition, and that is the cyclic identity on the "curvature" D o ~2uu,a - D v ~ p p , ~ . The relevant ones, however, are those associated with the invariances (3). These are given by the divergence of a linear combination of the field equation according to IU =- auSUV - ~1 OuS ~ot - ~1 ~ 7 . S U ~ - 0 ,
(4)
which are a set of three equations, since 7uIU - O. In curved space, however, with ~ -+D (we neglect torsion since it is irrelevant to t ~ basicu problem), we find
Iu=[D~',~](~u=,-½guv~) + [Dp,Du ] ~k=,+ (~2 _ D)~u = 0 ,
(5)
where ~ u - - 3 ~ a u , ~ - - - 3 ~ , P ~ # - ~ , and neglect of torsion means that [Du, 3'~] = 0. Using the Ricci identities, [Du, D~,] t~ =- -- 2 - 1 R pvab oab ~ , [D u, D , ] Vo, = Ru,:j,,, V# , Rpuab ~ ~ p ~ , a b -- ~v~pab , 0 ab -- 2 - 2 [7 a, 7 b ] ,
(6a)
we obtain
- ½ GaaTo, ffup - G u a ' ~ ¢ + 2 Gaexup,~T57 g/ - ~ R T u ~ O = O,
162
where *CtjV..~ is the dual of the Weyl tensor C. Leaving aside the possible vanishing (with appropriate torsion corrections) of the Ricci and curvature scalar terms on the Einstein mass shell, we are still left with an apparently unacceptable constraint involving the Weyl tensor * 1. However, using the spinor identities 7 ~ v ~ - 7 ~ v ~ = e,~xo3'sTx~ °+ 2 a ~ T X ~ v x (6b)
elav°tl3Oa[3 = 3"5°Pv ,
the Weyl part of (7a) reduces to -(CUU~t3 + *CUVa#75)Ta~#
+ (CU=,a(3- *CUU~#75)oa(3¢'v.
(7b)
Here we have decomposed ~uv with respect to 7-traces, "7 x . b l
"r" ¢,"=
= o,
= (':' ¢,.3 -
.,,, -",
-
Next we express the fields in terms of their irreducible self- and anti-self-dual parts C -+= ½(C+ iC) and helicity components ~b-+ = ½(1 -+ i 7 5 ) ¢ , the latter both for t~,v and ~ , . Then (7b) depends only on the combinations (C+~k+ + C - ~ k - ) , and not on if". It vanishes, therefore, for single parity solutions which are half-fiat and have fixed helicity .2 (e.g., C ÷ = 0 = ~ - ) . One is then left with the Ricci-tensor dependent parts of (7a) familiar from supergravity; their behavior is under investigation. Similar conclusions are obtained in terms of other representations [9] of spin 5/2 in terms of a nonsymmetric "vierbein" field ~ua, invariant under 6 ~bua = ~ ' a " The consistency conditions obtained here for hypergravity seem clearly more stringent than in supergravity, although it is possible that the single parity condition is itself a consequence of the field equations and regularity requirements, a conjecture which has already been raised for pure gravity in euclidean signature [10]. Even apart from these questions, consistency of the remaining terms in (7a) would require grading with a spinor-vector valued local gauge "~a(X), to generalize ,1 Non-minimal coupling cannot help here since a term ~ f R ~ ¢
- 2 I u - (- 3"C u='°~(~75+ CuW"#)'7o~~ vfs- ½G aO'Tu~k~#
7Ulu=O,
24 September 1979
(7a)
in the action would lead to D (R~) inI w %2 This result agrees with the consistency conditions on general (,4, B) representations of the Lorentz group in a background gravitational field in harmonic gauge [8], which is ~p = 0 in our notation. We note that higher invariants also vanish for single parity fields in supergravity [7].
Volume 86B, number 2
PHYSICS LETTERS
[11,12] the spinor gauge invariance of supergravity (and perhaps larger multiplets than 5/2). From the Ricci-tensor parts of (7a) one can read off the leading terms of such a transformation, namely 5 f f ~ = (D~gv+ D u ~ ) and ~e~ a is proportional to the coefficient of GUa there. Whether this invariance can be implemented without requiring the spin 5/2 stress tensor to vanish remains to be seen.
References [1 ] D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13 (1976) 3214; S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335. [2] M.T. Grisaru and H. Pendleton, Phys. Lett. 67B (1977) 323;
[3] [4] [5]
[6] [7] [8] [9] [10] [11] [12]
24 September 1979
M.T. Grisaru, H. Pendleton and P. van Nieuwenhuizen, Phys. Rev. D15 (1977) 996. C. Aragone and S. Deser, Nuovo Cimento 3A (1971) 709. J. Schwinger, Particles, sources and fields (Addison-Wesley, Reading, MA, 1970). J. Fang and C. Fronsdal, Phys. Rev. D18 (1978) 3630; see also F.A. Berends, J.W. van Holten, P. van Nieuwenhuizen and B. de Wit, Phys. Lett. 83B (1979) 188; T. Curtwright, EF179•20 preprint. C. Aragone and S. Deser, Brandeis preprint (1979). S. Christensen, S. Deser, M. Duff and M.T. Grisaru, Phys. Lett. 84B (1979) 411. S. Christensen and M. Duff, Brandeis preprint (1979). C. Aragone and S. Deser, in preparation. S.W. Hawking, 1978 Cargese Lectures (Plenum, New York, 1979). B.G. Konopel'chenko, Soy. J. Part. Nucl. 8 (1977) 57. J. Hietarinta, Phys. Rev. D13 (1976) 838.
163
PHYSICS LETTERS
24 September 1979
CONSISTENCY PROBLEMS OF HYPERGRAVITY C. ARAGONE 1 and S. DESER 2
PhysicsDepartment, Brandeis University, Waltham,MA 02154, USA Received 13 April 1979 Revised manuscript received 13 July 1979
The constraints arising upon coupling a massless spin 5/2 field to gravity are analyzed. In contrast to supergravity, they depend not only on the Einstein tensor, but also on the off-shell (Weyl) components of the curvature. The latter contributions do vanish, however, for "self-dual" systems, i.e., half-flat gravitational and pure (left/right) helicity spin 5/2 fields.
There are two a priori candidates for a supersymmetric completion of general relativity, namely spin 3/2 or 5/2 fields, the former corresponding to the consistent, locally supersymmetric model of supergravity [1 ]. According to general lore on higher spin fields and to S-matrix arguments on absence of low frequency spin 5 / 2 - 2 interactions [2], spin 5/2 is not likely to produce a similarly successful theory ("hypergravity") despite the possible benefits such a model might bring. In this note, we analyze the consistency problems facing the coupled massless 5 [ 2 - 2 system, contrasting them with the 3 / 2 - 2 case, quite apart from supersymmetry considerations. We shall see that, owing to the additional index structure for spin ~>5/2, the identities which tile free field possesses because of its pure helicity +s content become more stringent local constraints on the dynamical variables of the coupled fields. This basic difficulty is encountered in coupling all gauge (massless spin/> I) fields to gravity, and is due to noncommutation of covariant derivatives. However, spin 1 escapes this problem because the identity ~(~z,F uz') - 0 can still be written in terms of ordinary (rather than covariant) derivatives, using the contravariant tensor density X/Z-ffFU v. For spin 3/2, the divergence DzRU of the Rarita-Schwinger equation 1 John Simon Guggenheim Memorial Fellow. On leave of absence from Universidad Simon Bolivar. Work supported in part by CONICIT, Venezuela, grant S1-0972. 2 Supported in part by NSF grant PHY-78-09644.
R u ==-eUVC'a%Dc,~k~ = %,.fur = 0 ,
(1)
does involve covariant derivatives but its value depends only on the Ricci rather than the Weyl tensor; this is what makes a consistent coupling at all possible, because the Ricci (unlike the Weyl) tensor is specified by the Einstein equations. Absence of the Weyl tensor is a simple consequence of index counting: the scalar D~RU has the generic formRabcdF A ~ , but there are too few indices in the I~ matrices 7~, 3'57~, o~v to involve the four-index curvature. For spin 2, described by a symmetric tensor Cur, there is also dependence only on the Ricci tensor in the "Bianchi identity" D~G~'(~p) where G~ v is the ~ field equation; it is well known [3] that a nontrivial way out of these constraints is for the ~o-field to be a small deviation from the basic background gt~" In that case, the DuG~ v constraint simply becomes an expansion of the full gravitational Bianchi identity to first order in 5guy = ~ . Consider now the spin 5/2 field, whose flat space action and field equations have been given by Schwinger [4]. Its gauge invariances and associated identities are also implicit in his eq. (3-3.114). The field equations may be cast into very simple form [5] by taking appropriate linear combinations of the original ones together with their traces and 3,-traces. We write them here in a "geometrical" notation: S~,~ = y ~ ( - a~,~uv+ a u ~ +
avCJ~,) - 3,~S2~,~,~ =0.
(2) The "Christoffel symbol" I2uu,a is 7 a orthogonal. Eqs. 161
Volume 86B, number 2
PHYSICS LETTERS
(2) are invariant under the following abelian transformations of the symmetric tensor-spinor ~kuv: =
(3)
with respect to the three independent 7-traceless v e c t o r spinors ~u(x). The detailed hamiltonian analysis which establishes the pure helicity 5/2 content of the above system will be given elsewhere [6], but the count is immediate: to each local gauge freedom in a fermionic system are associated one Lagrange multiplier, one constraint equation (and therefore one constraint variable) and one gauge function. In our case, this removes 3 X 3 components out of the 10ffuv, leaving a single spinor to describe the +5/2 helicity states. (For spin 3/2 the four eu are correspondingly reduced to one by the invariance under the one gauge 6 flu = ~ u a ( x ) . ) There is one identity (which is unfortunately trivial) that does persist in the flat to curved transition, and that is the cyclic identity on the "curvature" D o ~2uu,a - D v ~ p p , ~ . The relevant ones, however, are those associated with the invariances (3). These are given by the divergence of a linear combination of the field equation according to IU =- auSUV - ~1 OuS ~ot - ~1 ~ 7 . S U ~ - 0 ,
(4)
which are a set of three equations, since 7uIU - O. In curved space, however, with ~ -+D (we neglect torsion since it is irrelevant to t ~ basicu problem), we find
Iu=[D~',~](~u=,-½guv~) + [Dp,Du ] ~k=,+ (~2 _ D)~u = 0 ,
(5)
where ~ u - - 3 ~ a u , ~ - - - 3 ~ , P ~ # - ~ , and neglect of torsion means that [Du, 3'~] = 0. Using the Ricci identities, [Du, D~,] t~ =- -- 2 - 1 R pvab oab ~ , [D u, D , ] Vo, = Ru,:j,,, V# , Rpuab ~ ~ p ~ , a b -- ~v~pab , 0 ab -- 2 - 2 [7 a, 7 b ] ,
(6a)
we obtain
- ½ GaaTo, ffup - G u a ' ~ ¢ + 2 Gaexup,~T57 g/ - ~ R T u ~ O = O,
162
where *CtjV..~ is the dual of the Weyl tensor C. Leaving aside the possible vanishing (with appropriate torsion corrections) of the Ricci and curvature scalar terms on the Einstein mass shell, we are still left with an apparently unacceptable constraint involving the Weyl tensor * 1. However, using the spinor identities 7 ~ v ~ - 7 ~ v ~ = e,~xo3'sTx~ °+ 2 a ~ T X ~ v x (6b)
elav°tl3Oa[3 = 3"5°Pv ,
the Weyl part of (7a) reduces to -(CUU~t3 + *CUVa#75)Ta~#
+ (CU=,a(3- *CUU~#75)oa(3¢'v.
(7b)
Here we have decomposed ~uv with respect to 7-traces, "7 x . b l
"r" ¢,"=
= o,
= (':' ¢,.3 -
.,,, -",
-
Next we express the fields in terms of their irreducible self- and anti-self-dual parts C -+= ½(C+ iC) and helicity components ~b-+ = ½(1 -+ i 7 5 ) ¢ , the latter both for t~,v and ~ , . Then (7b) depends only on the combinations (C+~k+ + C - ~ k - ) , and not on if". It vanishes, therefore, for single parity solutions which are half-fiat and have fixed helicity .2 (e.g., C ÷ = 0 = ~ - ) . One is then left with the Ricci-tensor dependent parts of (7a) familiar from supergravity; their behavior is under investigation. Similar conclusions are obtained in terms of other representations [9] of spin 5/2 in terms of a nonsymmetric "vierbein" field ~ua, invariant under 6 ~bua = ~ ' a " The consistency conditions obtained here for hypergravity seem clearly more stringent than in supergravity, although it is possible that the single parity condition is itself a consequence of the field equations and regularity requirements, a conjecture which has already been raised for pure gravity in euclidean signature [10]. Even apart from these questions, consistency of the remaining terms in (7a) would require grading with a spinor-vector valued local gauge "~a(X), to generalize ,1 Non-minimal coupling cannot help here since a term ~ f R ~ ¢
- 2 I u - (- 3"C u='°~(~75+ CuW"#)'7o~~ vfs- ½G aO'Tu~k~#
7Ulu=O,
24 September 1979
(7a)
in the action would lead to D (R~) inI w %2 This result agrees with the consistency conditions on general (,4, B) representations of the Lorentz group in a background gravitational field in harmonic gauge [8], which is ~p = 0 in our notation. We note that higher invariants also vanish for single parity fields in supergravity [7].
Volume 86B, number 2
PHYSICS LETTERS
[11,12] the spinor gauge invariance of supergravity (and perhaps larger multiplets than 5/2). From the Ricci-tensor parts of (7a) one can read off the leading terms of such a transformation, namely 5 f f ~ = (D~gv+ D u ~ ) and ~e~ a is proportional to the coefficient of GUa there. Whether this invariance can be implemented without requiring the spin 5/2 stress tensor to vanish remains to be seen.
References [1 ] D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13 (1976) 3214; S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335. [2] M.T. Grisaru and H. Pendleton, Phys. Lett. 67B (1977) 323;
[3] [4] [5]
[6] [7] [8] [9] [10] [11] [12]
24 September 1979
M.T. Grisaru, H. Pendleton and P. van Nieuwenhuizen, Phys. Rev. D15 (1977) 996. C. Aragone and S. Deser, Nuovo Cimento 3A (1971) 709. J. Schwinger, Particles, sources and fields (Addison-Wesley, Reading, MA, 1970). J. Fang and C. Fronsdal, Phys. Rev. D18 (1978) 3630; see also F.A. Berends, J.W. van Holten, P. van Nieuwenhuizen and B. de Wit, Phys. Lett. 83B (1979) 188; T. Curtwright, EF179•20 preprint. C. Aragone and S. Deser, Brandeis preprint (1979). S. Christensen, S. Deser, M. Duff and M.T. Grisaru, Phys. Lett. 84B (1979) 411. S. Christensen and M. Duff, Brandeis preprint (1979). C. Aragone and S. Deser, in preparation. S.W. Hawking, 1978 Cargese Lectures (Plenum, New York, 1979). B.G. Konopel'chenko, Soy. J. Part. Nucl. 8 (1977) 57. J. Hietarinta, Phys. Rev. D13 (1976) 838.
163