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Scale dependence and the renormalization problem of quantum gravity
15 September 1975
PHYSICS LETTERS
Volume 58B, number 3
SCALE DEPENDENCE AND THE RENORMALIZATION PROBLEM OF QUANTUM GRAVITY S. DESER*, M.T. GRISARU,
P. Van NIEUWENHUIZEN*,
C.C. WU*
Department of Physics, Brandeis University, Waltham, Massachusetts 02154, USA Received 21 July 1975 Although Einstein theory is obtainable from a Weyl (scale) invariant model by a particular gauge choice, this im
poses no conditions on its counterterms. The absence of certain non scale-invariant counterterms in Einstein-YangMills (or Maxwell) theory is traced instead to invariance under vector field duality rotations.
In a recent Letter, Kallosh [l] has attempted to study possible restrictions on the on-shell counterterms in quantum gravity which might be traceable to a hidden Weyl (scale) invariance of Einstein theory when interacting with the manifestly scale invariant YangMills (or Maxwell) system. For this system a number of otherwise allowed, non-scale invariant counterterms are in fact absent in explicit one-loop calculations [2], and Kallosh argues that this is due to the hidden invariance. We claim here that this is not the case, and instead trace the absence of the “missing” counterterms of [2] to the invariance of the action under infinitesimal constant duality rotations of the Yang-Mills (or Maxwell) field strengths. That the scale invariance argument is not applicable can already be suspected from the fact that explicit calculations have exhibited the presence of scale dependent counterterms -k8(&yar5 $)4 in the massless Dirac-Einstein system [3] and -e2k2($yJ/)2 in the massless Dirac-Maxwell-Einstein system [4], although the matter parts are still manifestly scale invariant there. The scale invariance argument should be equally applicable to these systems and exclude these counterterms too*‘. The source of the Weyl invariance argument lies in the observation that the Einstein action S, = ~-~JdxfiR is equivalent [5] to the action ‘SW =+ jdxJ-g[ib2R
+ (ab)2]
,
(1)
which is invariant under the scaling 6g,,, = 2 Ag,,y, 6b = * Supported ln part by the National Science Foundation. *r One cannot appeal to anomalies to resolve this apparent conflict since they cannot affect the one loop infinities.
-hb. Since b is not a physical degree of freedom, one may recover Einstein theory by choosing the gauge * 2 b2 = 12~~~ or, alternatively by introducing the new field gccv=& K2b2g,,,. (c onversely one can obtain the scale invariant action (1) from the Einstein action by introducing [6] the field b through the change gMv -+K2b2gp,). Using the background field method, Kallosh expresses this equivalence as the equality of the loop function alS 5&(g)
= SZw(g, b
=aK-‘)
(2)
,
where (g, b) denote the background fields. (We suppress the matter fields.) Since SW + SyM iS scale invariant, the one loop counterterms will also be scale invariant functions ofg, the Yang:Mills fields and b. Now Kallosh obtains the relation aw(g, b) = Ciw(g, b + 6b) for variations about fields which satisfy the equations of motion, from which she concludes that the on-shell counterterms must be b-independent, but still manifestly scale invariant, which is a strong restriction. Unfortunately, the above relation appears to be incorrect. Consider the loop functional, QnF and aF+aF corresponding to gauge breaking terms F and F + 6F respectively, so that fi(g, b),=jdBd/zd$*d$ -S&T,
b)h-S.,(g,
exp i[S(g+h,
b +B)-SCp, b)
b)B+ F(g,h,b,B)+S~ost],(3)
*’ Variants of this idea could also lead to actions proportional to the square of the scalar curvature with or without an Einstein part. However, these are not renormalizable because R2 does not damp the propagators of the dynamical modes of the metric field, but essentially only of its trace.
355
PHYSICS LETTERS
Volume 58B, number 3
while aF+& F is defined through (F+ 6F) (g, h, b, B)
= F(g, h, b + 6b, B - 6b). If in a(g, b)F+6F one changes the B-integration variable to B t Sb, one find to first order in 6 b, Wg, b)F+tjF = a k, b)F (4) + 6b P&g,
b) (h) + S&,
b)
@)I ,
with the expectation values Uz>and (B) computed in the presence of backgorund fields (g, b). In eq. (10) of [ 1 ] the last term of (4) is missing. The existence of this term, which is not obviously zero for general background fields and may contribute to the counterterms, seems to invalidate the hidden invariance argument. The counterterms, although Weyl invariant, are allowed to be explicitly b-dependent. In the “Einstein” gauge b2 = I~K-~, they then need not look Weyl invariant, and are therefore not restricted *3 . It is however possible to understand the results of ref. [2] on the basis of another invariance argument. Consider for the Einstein-Yang-Mills (or Maxwell) system the infinitesimal duality (E * B) rotation FPv+FtiV+ A*FPV where *FlJv=~(-g)-1/2&‘V@F P’ This is a symmetry of the action S, - $J dxfiF because the variation of (F)2 is proportional
to F*F
*’ There exist gravitational models which are manifestly scale invariant, including one whose action is proportional to the square of the Weyl tensor or that recently proposed by Yang [ 71 with field equations D,Rhm@ = 0, rather than RbN = 0.
356
15 September 1975
which is a total divergence. Now the matter stress-tensor T,,(F) is clearly duality invariant [8], as is D,,Fp” (since D~F~v~ 0 is just the cyclic identity). Therefore this symmetry of the action allows the set of matter counterterms - [F2, T2 ,R Tfi’“,(D F“v)2] but excludes the non-invar/%t s$ - [RF2r(F2)2, (F*F)2, R pvapF”VFa@]. (The purely gravitational terms R2 and Rz,, are also allowed in any combination, not merely the Weyl invariant one R$fR2.) The set found in [(2)] has all the terms of the allowed type, and none of the forbidden ones, in exact agreement with the duality restriction. We acknowledge
discussions with D.Z. Freedman.
References [l] R. Kallosh, Phys. Letters 55B (1975) 321. [2] S. Deser and P. van Nieuwenhuizen, Phys. Rev. DlO (1974) 401; S. Deser, P. van Nieuwenhuizen and H.S. Tsao, Phys. Rev. (1974) 3337. [3] S. Deser and P. van Nieuwenhulzen, Phys. Rev. DlO (1974) 411. [4] M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Phys. Rev. D., to be published. [S] S. Deser, Ann. Phys. 59 (1970) 248; 3.L. Anderson, Phys. Rev. D3 (1971) 1689. 161 B. Zumino, in Brandeis University Lectures in Theoretical Physics, eds. S. Deser, M. Grisaru and H. Pendleton (MIT Press, Cambridge, Mass. 1970). [7] C.N. Yang, Phys. Rev. Letters 33 (1974) 445. [8] C.W. Misner and J.A. Wheeler, Ann. Phys. 2 (1957) 525.
PHYSICS LETTERS
Volume 58B, number 3
SCALE DEPENDENCE AND THE RENORMALIZATION PROBLEM OF QUANTUM GRAVITY S. DESER*, M.T. GRISARU,
P. Van NIEUWENHUIZEN*,
C.C. WU*
Department of Physics, Brandeis University, Waltham, Massachusetts 02154, USA Received 21 July 1975 Although Einstein theory is obtainable from a Weyl (scale) invariant model by a particular gauge choice, this im
poses no conditions on its counterterms. The absence of certain non scale-invariant counterterms in Einstein-YangMills (or Maxwell) theory is traced instead to invariance under vector field duality rotations.
In a recent Letter, Kallosh [l] has attempted to study possible restrictions on the on-shell counterterms in quantum gravity which might be traceable to a hidden Weyl (scale) invariance of Einstein theory when interacting with the manifestly scale invariant YangMills (or Maxwell) system. For this system a number of otherwise allowed, non-scale invariant counterterms are in fact absent in explicit one-loop calculations [2], and Kallosh argues that this is due to the hidden invariance. We claim here that this is not the case, and instead trace the absence of the “missing” counterterms of [2] to the invariance of the action under infinitesimal constant duality rotations of the Yang-Mills (or Maxwell) field strengths. That the scale invariance argument is not applicable can already be suspected from the fact that explicit calculations have exhibited the presence of scale dependent counterterms -k8(&yar5 $)4 in the massless Dirac-Einstein system [3] and -e2k2($yJ/)2 in the massless Dirac-Maxwell-Einstein system [4], although the matter parts are still manifestly scale invariant there. The scale invariance argument should be equally applicable to these systems and exclude these counterterms too*‘. The source of the Weyl invariance argument lies in the observation that the Einstein action S, = ~-~JdxfiR is equivalent [5] to the action ‘SW =+ jdxJ-g[ib2R
+ (ab)2]
,
(1)
which is invariant under the scaling 6g,,, = 2 Ag,,y, 6b = * Supported ln part by the National Science Foundation. *r One cannot appeal to anomalies to resolve this apparent conflict since they cannot affect the one loop infinities.
-hb. Since b is not a physical degree of freedom, one may recover Einstein theory by choosing the gauge * 2 b2 = 12~~~ or, alternatively by introducing the new field gccv=& K2b2g,,,. (c onversely one can obtain the scale invariant action (1) from the Einstein action by introducing [6] the field b through the change gMv -+K2b2gp,). Using the background field method, Kallosh expresses this equivalence as the equality of the loop function alS 5&(g)
= SZw(g, b
=aK-‘)
(2)
,
where (g, b) denote the background fields. (We suppress the matter fields.) Since SW + SyM iS scale invariant, the one loop counterterms will also be scale invariant functions ofg, the Yang:Mills fields and b. Now Kallosh obtains the relation aw(g, b) = Ciw(g, b + 6b) for variations about fields which satisfy the equations of motion, from which she concludes that the on-shell counterterms must be b-independent, but still manifestly scale invariant, which is a strong restriction. Unfortunately, the above relation appears to be incorrect. Consider the loop functional, QnF and aF+aF corresponding to gauge breaking terms F and F + 6F respectively, so that fi(g, b),=jdBd/zd$*d$ -S&T,
b)h-S.,(g,
exp i[S(g+h,
b +B)-SCp, b)
b)B+ F(g,h,b,B)+S~ost],(3)
*’ Variants of this idea could also lead to actions proportional to the square of the scalar curvature with or without an Einstein part. However, these are not renormalizable because R2 does not damp the propagators of the dynamical modes of the metric field, but essentially only of its trace.
355
PHYSICS LETTERS
Volume 58B, number 3
while aF+& F is defined through (F+ 6F) (g, h, b, B)
= F(g, h, b + 6b, B - 6b). If in a(g, b)F+6F one changes the B-integration variable to B t Sb, one find to first order in 6 b, Wg, b)F+tjF = a k, b)F (4) + 6b P&g,
b) (h) + S&,
b)
@)I ,
with the expectation values Uz>and (B) computed in the presence of backgorund fields (g, b). In eq. (10) of [ 1 ] the last term of (4) is missing. The existence of this term, which is not obviously zero for general background fields and may contribute to the counterterms, seems to invalidate the hidden invariance argument. The counterterms, although Weyl invariant, are allowed to be explicitly b-dependent. In the “Einstein” gauge b2 = I~K-~, they then need not look Weyl invariant, and are therefore not restricted *3 . It is however possible to understand the results of ref. [2] on the basis of another invariance argument. Consider for the Einstein-Yang-Mills (or Maxwell) system the infinitesimal duality (E * B) rotation FPv+FtiV+ A*FPV where *FlJv=~(-g)-1/2&‘V@F P’ This is a symmetry of the action S, - $J dxfiF because the variation of (F)2 is proportional
to F*F
*’ There exist gravitational models which are manifestly scale invariant, including one whose action is proportional to the square of the Weyl tensor or that recently proposed by Yang [ 71 with field equations D,Rhm@ = 0, rather than RbN = 0.
356
15 September 1975
which is a total divergence. Now the matter stress-tensor T,,(F) is clearly duality invariant [8], as is D,,Fp” (since D~F~v~ 0 is just the cyclic identity). Therefore this symmetry of the action allows the set of matter counterterms - [F2, T2 ,R Tfi’“,(D F“v)2] but excludes the non-invar/%t s$ - [RF2r(F2)2, (F*F)2, R pvapF”VFa@]. (The purely gravitational terms R2 and Rz,, are also allowed in any combination, not merely the Weyl invariant one R$fR2.) The set found in [(2)] has all the terms of the allowed type, and none of the forbidden ones, in exact agreement with the duality restriction. We acknowledge
discussions with D.Z. Freedman.
References [l] R. Kallosh, Phys. Letters 55B (1975) 321. [2] S. Deser and P. van Nieuwenhuizen, Phys. Rev. DlO (1974) 401; S. Deser, P. van Nieuwenhuizen and H.S. Tsao, Phys. Rev. (1974) 3337. [3] S. Deser and P. van Nieuwenhulzen, Phys. Rev. DlO (1974) 411. [4] M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Phys. Rev. D., to be published. [S] S. Deser, Ann. Phys. 59 (1970) 248; 3.L. Anderson, Phys. Rev. D3 (1971) 1689. 161 B. Zumino, in Brandeis University Lectures in Theoretical Physics, eds. S. Deser, M. Grisaru and H. Pendleton (MIT Press, Cambridge, Mass. 1970). [7] C.N. Yang, Phys. Rev. Letters 33 (1974) 445. [8] C.W. Misner and J.A. Wheeler, Ann. Phys. 2 (1957) 525.