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On the renormalization problem of quantum gravity
Volume 55B, number 3
PHYSICS LETTERS
ON THE RENORMALIZATION
17 February 1975
PROBLEM OF QUANTUM
GRAVITY
R. KALLOSH P.N. Lebedev Physical Institute, Moscow, USSR
Received 5 November 1974 Strong restrictions on possible counterterms are obtained using the Weyl gauge symmetrical form of the Einstein gravity. These restrictions are confirmed by the available one-loop calculations. Arguments in favor of the renormalizability of the quantum gravity are given.
In the classical gravity Weyl's theory [ 1] has attracted attention since 1918. The scalar-tensor theory of gravitation avoids the great complications that would arise with Weyl's action principle, but retains Weyl's symmetry. To our knowledge it was Anderson [2] who first pointed out that the scalar-tensor model of Brans-Dicke with the parameter ca = - 3 / 2 is completely equivalent in some gauge to the Einstein tensor theory, Quite recently there has been a renewed interest in Weyl's theory [3, 4]. In the present note we report the results for the renormalization problem of the Einstein. Yang-MiUs system considered in the form possessing the spontaneously broken symmetry of the Weyl's type. We consider the action S[~/] =
(1)
where the abstract notation ~0i means (gu~,'°a' b), gravitational tensor field, o~, (a =u 1,2, 3) the Yang-Mills vector field and b some scalar field. In eq. (1) R[g] is the curvature scalar and F~v Iv] is equal to auoa - avoau+fCgbc0#0 b cv , • The total invariance group of (1) inclucles tlaree gauge groups. Action (1) is invariant under the infinitesimal transformations
guy being the
6~0i = R i [~018~a ,
c~= 0, 1..... 7
-- 2 g j ,
(2)
The first gauge group (a = 0, 1, 2, 3) corresponds to the general coordinate transformations. The second
ao , -- o ,
ab -- - b o
(3)
with 6~7 = o. Eq. (3) gives the detailed form of R~ [¢]. The functional differentiation with respect to the field variables will be denoted by a comma, followed by latin indices. Then the invariance condition of(l)
S,i [~] R /
[~] = 0,
,v = 0~ 1.... ,7
(4)
means in particular that not all the equations of motion for the classical fields are independent, e.g.
S,R7
- f d4x x/~(b2R
- 6auba~,bgUV+¼ l~uuFaUV)
group (tx = a + 3 = 4, 5, 6) is the Yang-Mills gauge group 4=. The third group (t~ = 7) is the Weyl gauge group
6S ag~,,,
6S --fib
It follows that there is no physical degree of freedom connected with the scalar field. In the quantum theory an additional gauge condition can be imposed, analogous to the tensor (or vector) field gauge conditions permitted by the coordinate (or the Yang-Mills) gauge group. The theory (1) does not contain dimensional parameters. But they can appear through the spontaneous breaking of Weyl's symmetry when the theory (1) is considered in the gauge b = IlK, where is the gravitational constant K2 = 16~rG and where the theory (1) is the usual Einstein-Yang-Mills system
s = - f d4xx/~ ( ~ R +¼ f~uF auu)
(5)
Action (5) is not invariant under the group (3), 4=Our result applies equally to the abelian case f = 0, i.e. to the Einstein-MaxweU system.
* For the detailed form of Ri[~](~ ~ 7) see e.g. [7]. 321
Volume 55B, number 3
PHYSICS LETTERS
17 February 1975
only x-independent o are permitted ¢. In general we can pass into any other gauge where the scalar field b' is not a constant. The Jacobian of the transformation is determined by the methods of Faddeev-Popov Fradkin-Tyutin [6] or De Witt [7]. We shall use De Witt's [7, 8] background functional for radiative corrections. The abstract notation for integration variables qbi m e a n s (hu~,, V~,B).
and the function yi[¢] can be calculated, or its structure can be obtained on general grounds. The analysis of diver~nces wilt be performed in some G-invariant gauge F G, where
~2[~0]~ = fd(I)idff*dff exp i ( S [ ~ + (I)]-S[¢]
/a 1 a 5(hv,t~-~ho~,v), 8(V~*# )}, t he stars denote the co-
~ [ ~ ] f , = a [ ¢ l f 2 +s,;[~l Y;[~]
~[~] = ~[~ + ~ 1 ,
(7')
(8)
and 8~0 is defined in eq. (2),/~G = {8 (B),
-S,i[¢]cbi + ½~'i/'~i(Id + ~*a ~ r o ~ ) ,
(6)
Here/~is some gauge condition and ~k*'~ ~" ~t~ is the corresponding action of fictitious particleL The main assumption of our work is that a regularization procedure exists which preserves all the formal symmetry properties (including the Weyl symmetry) of the functional (6). As was shown in [7] and in detail in [8] such functionals are G- and F-invariant when the classical equations S,i[¢ ] = 0 hold. G-invariance means the invariance of the functional under group transformations of the background field ~p. The F-invariance means the independence of the type of the gauge condition chosen inside the functional integral. The use of the G- and/7-invafiance in this approach takes the place of the generalized Ward identities [8]. Therefore both these invafiances permit the investigation of the counterterms in the perturbation theory. In general for arbitrary background fields ¢ not satisfying the equations S,i[~] = 0 the functionals in two different gauges F and F + 6/~ are related as follows [81
I2 [~01FG = I2 [¢] fG+6 f + S,i [~01XiSb
(9)
The B-integration shows that (dimensional regularisation [9] is applied, i.e. local measure of integration is equal to unity)
~2[g,v, b] PG+~f = ~2[g,v, b + 8b] p a
(10)
and thus in the G-invariant gauges
~2[g,v, b]8(B) = ~2 [g, o, b = 1/r]~(e) +S,i[g, b, b]XiSb
(11)
We note that ~2 [g, v, b = 1/r ] ~ (B) is just the usual background functional for the Einstein theory. It follows from (11) that
~2[g,b, l / r ] =~2[g,o, 1/r + 8b(x)]
a [tp] k = a [tp] k+8/~ + S , i [tp] X i [tp, 8/~]
S,i[~o]xi= s,i[~ol(R~/,~8~'~[~o,~,8~},
covariant :F differentiation with respect to the background fields g,v, of, b. Thus the first restriction on possible counterterms is G-invariance (eq. (8)) in all three groups. Now we consider the second gauge fig + 8F = F G [B-Sb, b + 8b], 8b being arbitrary function of x, i.e. only the scalar part of the gauge condition is changed and also the co-covariant derivatives in the rest of the gauge condition are now with respect to b + 5b. Eq. (7) gives
(7)
+ S,i[g,v, 1/r]XiSb.
(12)
The brackets ( ) mean the averaging with the integrand (6). Eq. (7) follows from the change in the integration variables like 8¢P i = R / [~0 + dP]6~ a, etc., with specially chosen parameters 8~ff. Taking into account eq. (4) and also the equality R~ [~0 + qb] = R / [~0] + R i f p J we are led to eq. (7). Also the finite variations of the gauge conditions can be performed with the result
Eq. (11) is our main statement. The variation of the counterterms with respect to the field b must be proportional to S i [g, o, b] (in particular the counterterms can be b-independent). The G-invariance property is not very restrictive, there are many counterterms G-invariant under the simultaneous Weyl group transformation ofguu and b. But the F-invariance the
4: The group (3) was investigated in the Einstein theory by Fradkin and Vilkovisky [5].
* The co-covaxiant derivatives of in-tensors contain the Weyl gauge invariant ChristoffeU symbol *P~v = l ~ v -- g ~ k v - gv~k~ + g u v k a ; kta -= ata logb.
322
Volume 55B, number 3
PHYSICS LETTERS
consequence of which is now expressed in eq. (11) is very restrictive. In the one-loop approximation of the Einstein-Yang-MiUs system only two counterterms are consistent with eqs. (8) and (11).
f
Vz-ic¢ cz3 [= 2 f d4x vrL-ff(RuvR"~'-½R2),
f2f d4xx/2-~F~z,FaUV
o3) h=4] (14)
where C~/6 is the Weyl tensor (conformal curvature tensor) ~ Ot ,~ c ~ - R ~ + ~ 1 (8~-88R~+R~g~-R~g~)
R
+ (.- 1)(n- 2) (8~g~-~ ~ga6), n being the dimension of the space. In other gauges not invariant in all three groups (e.g. used in calculations [ 1 0 - 1 2 ] ) as foUows from eq. (7') with guy, o~ arbitrary and b = 1/~¢, additional counterterms can appear. They contain S.i[g, o, l/K] (or in detail b-~;vu and RU~' - -~gUVR + ½ K2T~v) as multiplier and in the one-loop approximation these additional counterterms are (from general and YangMills covariance) . R 2, K2R/lb' T tw, K2 (Fa;u) 2 /.~/,, 2, K4T/zV" the semicolons denote covafiant differentiation with respect to the fields gu~,,o~. Thus the Weyl symmetry is the reason for the cancellation (which leaves 6 counterterms from 13 possible and only 2 of them survive for theS-matrix) observed in the calculations of Deser, Tsao, Nieuwenhuizen [ 12]. So in the one-loop approximation eqs. (11), (12) are confirmed [ 11, 12] and the dimensional regularization [9] appears to be consistent with large symmetry *. * The dimensional regularization [9] fails, i.e. anomalies appear, if quantities particular to 4-dimensional space, such as 3'5 or scaling behavior play an essential role in the symmetry. We can show that the Weyl gauge symmetrical theory can be constructed in an arbitrary dimensional space (in contrast tO the scaling invariance offd4x(al~alz~/2 + ~,~4) in the 4dimensional space). Therefore we hope that the dimensional regularization will give no anomalies also in the higher approximations.
17 February 1975
To consider the higher approximations we must take the action in the form
S=S " +o:f d4x
4z-dc c
(15)
where S is defined in eq. (1) and a is some dimensionless constant. The new term in (15) will be treated as a perturbation (no terms k - 4 in the propagator). All the G- and F-invariance properties remain, only eqs. ( 7 - 9 ) , ( 1 1 - 1 2 ) now contain the new contribution from (13). Preliminary investigations show that our eqs. (11), (12) forbid all the non-renormalizable counterterms and the theory seems to be renormalizable in case the appropriate regularization exists. But the situation in more than one-loop approximation is strange since eqs. (11), (12) restrict also the finite terms and therefore the radiative corrections are now powers in a dimensionless constant c~, and the increasing powers of r seems to appear only though the iterative solution of the new equations of motion. This question needs further investigation. I am grateful to V. Fainberg, E. Fradkin, V. Frolov, B. Yoffe, and I. Tyutin for valuable discussions.
References [1] H. Weyl, Raum, Zeit. Materie, 5. Auflage, J. Springer, Berlin, 1923. [2] J.L. Anderson, Phys. Rev. D3 (1971) 1689. [3] P.A.M. Dirac, Proc. R. Soc. Lond. A333 (1973) 403. [4] O.P. Freund, Annals of Phys. 84 (1974) 440. [5] E. Fradkin and G. Vilkovisky, Phys. Rev. D8 (1973) 4241. [6] L. Faddeev and B. Popov, Phys. Lett. 25B (1967) 29. E. Fradkin and I. Tyutin, Phys. Rev. D2 (1970) 2841. [7] B.S. De Witt, Phys. Rev. 162 (1967) 1195. J. Honerkamp, Nucl. Phys. B48 (1972) 269. [8] R.E. Kallosh, Nucl. Phys. B 78 (1974) 293. [9] G. 't Hooft and M. Veltman, Nuclear Phys. B44 (1972) 189. [10] G. 't Hooft and M. Veltman, Ann. Inst. H. Poincar~, in press. [ 11 ] S. Deser and P. Van Nieuwenhuizen, Phys. Rev. Letters 32 (1974) 245. [12] S. Deser, H.-S. Tsao and P. Van Nieuwenhuizen, Phys. Lett. B50 (1974) 491.
323
PHYSICS LETTERS
ON THE RENORMALIZATION
17 February 1975
PROBLEM OF QUANTUM
GRAVITY
R. KALLOSH P.N. Lebedev Physical Institute, Moscow, USSR
Received 5 November 1974 Strong restrictions on possible counterterms are obtained using the Weyl gauge symmetrical form of the Einstein gravity. These restrictions are confirmed by the available one-loop calculations. Arguments in favor of the renormalizability of the quantum gravity are given.
In the classical gravity Weyl's theory [ 1] has attracted attention since 1918. The scalar-tensor theory of gravitation avoids the great complications that would arise with Weyl's action principle, but retains Weyl's symmetry. To our knowledge it was Anderson [2] who first pointed out that the scalar-tensor model of Brans-Dicke with the parameter ca = - 3 / 2 is completely equivalent in some gauge to the Einstein tensor theory, Quite recently there has been a renewed interest in Weyl's theory [3, 4]. In the present note we report the results for the renormalization problem of the Einstein. Yang-MiUs system considered in the form possessing the spontaneously broken symmetry of the Weyl's type. We consider the action S[~/] =
(1)
where the abstract notation ~0i means (gu~,'°a' b), gravitational tensor field, o~, (a =u 1,2, 3) the Yang-Mills vector field and b some scalar field. In eq. (1) R[g] is the curvature scalar and F~v Iv] is equal to auoa - avoau+fCgbc0#0 b cv , • The total invariance group of (1) inclucles tlaree gauge groups. Action (1) is invariant under the infinitesimal transformations
guy being the
6~0i = R i [~018~a ,
c~= 0, 1..... 7
-- 2 g j ,
(2)
The first gauge group (a = 0, 1, 2, 3) corresponds to the general coordinate transformations. The second
ao , -- o ,
ab -- - b o
(3)
with 6~7 = o. Eq. (3) gives the detailed form of R~ [¢]. The functional differentiation with respect to the field variables will be denoted by a comma, followed by latin indices. Then the invariance condition of(l)
S,i [~] R /
[~] = 0,
,v = 0~ 1.... ,7
(4)
means in particular that not all the equations of motion for the classical fields are independent, e.g.
S,R7
- f d4x x/~(b2R
- 6auba~,bgUV+¼ l~uuFaUV)
group (tx = a + 3 = 4, 5, 6) is the Yang-Mills gauge group 4=. The third group (t~ = 7) is the Weyl gauge group
6S ag~,,,
6S --fib
It follows that there is no physical degree of freedom connected with the scalar field. In the quantum theory an additional gauge condition can be imposed, analogous to the tensor (or vector) field gauge conditions permitted by the coordinate (or the Yang-Mills) gauge group. The theory (1) does not contain dimensional parameters. But they can appear through the spontaneous breaking of Weyl's symmetry when the theory (1) is considered in the gauge b = IlK, where is the gravitational constant K2 = 16~rG and where the theory (1) is the usual Einstein-Yang-Mills system
s = - f d4xx/~ ( ~ R +¼ f~uF auu)
(5)
Action (5) is not invariant under the group (3), 4=Our result applies equally to the abelian case f = 0, i.e. to the Einstein-MaxweU system.
* For the detailed form of Ri[~](~ ~ 7) see e.g. [7]. 321
Volume 55B, number 3
PHYSICS LETTERS
17 February 1975
only x-independent o are permitted ¢. In general we can pass into any other gauge where the scalar field b' is not a constant. The Jacobian of the transformation is determined by the methods of Faddeev-Popov Fradkin-Tyutin [6] or De Witt [7]. We shall use De Witt's [7, 8] background functional for radiative corrections. The abstract notation for integration variables qbi m e a n s (hu~,, V~,B).
and the function yi[¢] can be calculated, or its structure can be obtained on general grounds. The analysis of diver~nces wilt be performed in some G-invariant gauge F G, where
~2[~0]~ = fd(I)idff*dff exp i ( S [ ~ + (I)]-S[¢]
/a 1 a 5(hv,t~-~ho~,v), 8(V~*# )}, t he stars denote the co-
~ [ ~ ] f , = a [ ¢ l f 2 +s,;[~l Y;[~]
~[~] = ~[~ + ~ 1 ,
(7')
(8)
and 8~0 is defined in eq. (2),/~G = {8 (B),
-S,i[¢]cbi + ½~'i/'~i(Id + ~*a ~ r o ~ ) ,
(6)
Here/~is some gauge condition and ~k*'~ ~" ~t~ is the corresponding action of fictitious particleL The main assumption of our work is that a regularization procedure exists which preserves all the formal symmetry properties (including the Weyl symmetry) of the functional (6). As was shown in [7] and in detail in [8] such functionals are G- and F-invariant when the classical equations S,i[¢ ] = 0 hold. G-invariance means the invariance of the functional under group transformations of the background field ~p. The F-invariance means the independence of the type of the gauge condition chosen inside the functional integral. The use of the G- and/7-invafiance in this approach takes the place of the generalized Ward identities [8]. Therefore both these invafiances permit the investigation of the counterterms in the perturbation theory. In general for arbitrary background fields ¢ not satisfying the equations S,i[~] = 0 the functionals in two different gauges F and F + 6/~ are related as follows [81
I2 [~01FG = I2 [¢] fG+6 f + S,i [~01XiSb
(9)
The B-integration shows that (dimensional regularisation [9] is applied, i.e. local measure of integration is equal to unity)
~2[g,v, b] PG+~f = ~2[g,v, b + 8b] p a
(10)
and thus in the G-invariant gauges
~2[g,v, b]8(B) = ~2 [g, o, b = 1/r]~(e) +S,i[g, b, b]XiSb
(11)
We note that ~2 [g, v, b = 1/r ] ~ (B) is just the usual background functional for the Einstein theory. It follows from (11) that
~2[g,b, l / r ] =~2[g,o, 1/r + 8b(x)]
a [tp] k = a [tp] k+8/~ + S , i [tp] X i [tp, 8/~]
S,i[~o]xi= s,i[~ol(R~/,~8~'~[~o,~,8~},
covariant :F differentiation with respect to the background fields g,v, of, b. Thus the first restriction on possible counterterms is G-invariance (eq. (8)) in all three groups. Now we consider the second gauge fig + 8F = F G [B-Sb, b + 8b], 8b being arbitrary function of x, i.e. only the scalar part of the gauge condition is changed and also the co-covariant derivatives in the rest of the gauge condition are now with respect to b + 5b. Eq. (7) gives
(7)
+ S,i[g,v, 1/r]XiSb.
(12)
The brackets ( ) mean the averaging with the integrand (6). Eq. (7) follows from the change in the integration variables like 8¢P i = R / [~0 + dP]6~ a, etc., with specially chosen parameters 8~ff. Taking into account eq. (4) and also the equality R~ [~0 + qb] = R / [~0] + R i f p J we are led to eq. (7). Also the finite variations of the gauge conditions can be performed with the result
Eq. (11) is our main statement. The variation of the counterterms with respect to the field b must be proportional to S i [g, o, b] (in particular the counterterms can be b-independent). The G-invariance property is not very restrictive, there are many counterterms G-invariant under the simultaneous Weyl group transformation ofguu and b. But the F-invariance the
4: The group (3) was investigated in the Einstein theory by Fradkin and Vilkovisky [5].
* The co-covaxiant derivatives of in-tensors contain the Weyl gauge invariant ChristoffeU symbol *P~v = l ~ v -- g ~ k v - gv~k~ + g u v k a ; kta -= ata logb.
322
Volume 55B, number 3
PHYSICS LETTERS
consequence of which is now expressed in eq. (11) is very restrictive. In the one-loop approximation of the Einstein-Yang-MiUs system only two counterterms are consistent with eqs. (8) and (11).
f
Vz-ic¢ cz3 [= 2 f d4x vrL-ff(RuvR"~'-½R2),
f2f d4xx/2-~F~z,FaUV
o3) h=4] (14)
where C~/6 is the Weyl tensor (conformal curvature tensor) ~ Ot ,~ c ~ - R ~ + ~ 1 (8~-88R~+R~g~-R~g~)
R
+ (.- 1)(n- 2) (8~g~-~ ~ga6), n being the dimension of the space. In other gauges not invariant in all three groups (e.g. used in calculations [ 1 0 - 1 2 ] ) as foUows from eq. (7') with guy, o~ arbitrary and b = 1/~¢, additional counterterms can appear. They contain S.i[g, o, l/K] (or in detail b-~;vu and RU~' - -~gUVR + ½ K2T~v) as multiplier and in the one-loop approximation these additional counterterms are (from general and YangMills covariance) . R 2, K2R/lb' T tw, K2 (Fa;u) 2 /.~/,, 2, K4T/zV" the semicolons denote covafiant differentiation with respect to the fields gu~,,o~. Thus the Weyl symmetry is the reason for the cancellation (which leaves 6 counterterms from 13 possible and only 2 of them survive for theS-matrix) observed in the calculations of Deser, Tsao, Nieuwenhuizen [ 12]. So in the one-loop approximation eqs. (11), (12) are confirmed [ 11, 12] and the dimensional regularization [9] appears to be consistent with large symmetry *. * The dimensional regularization [9] fails, i.e. anomalies appear, if quantities particular to 4-dimensional space, such as 3'5 or scaling behavior play an essential role in the symmetry. We can show that the Weyl gauge symmetrical theory can be constructed in an arbitrary dimensional space (in contrast tO the scaling invariance offd4x(al~alz~/2 + ~,~4) in the 4dimensional space). Therefore we hope that the dimensional regularization will give no anomalies also in the higher approximations.
17 February 1975
To consider the higher approximations we must take the action in the form
S=S " +o:f d4x
4z-dc c
(15)
where S is defined in eq. (1) and a is some dimensionless constant. The new term in (15) will be treated as a perturbation (no terms k - 4 in the propagator). All the G- and F-invariance properties remain, only eqs. ( 7 - 9 ) , ( 1 1 - 1 2 ) now contain the new contribution from (13). Preliminary investigations show that our eqs. (11), (12) forbid all the non-renormalizable counterterms and the theory seems to be renormalizable in case the appropriate regularization exists. But the situation in more than one-loop approximation is strange since eqs. (11), (12) restrict also the finite terms and therefore the radiative corrections are now powers in a dimensionless constant c~, and the increasing powers of r seems to appear only though the iterative solution of the new equations of motion. This question needs further investigation. I am grateful to V. Fainberg, E. Fradkin, V. Frolov, B. Yoffe, and I. Tyutin for valuable discussions.
References [1] H. Weyl, Raum, Zeit. Materie, 5. Auflage, J. Springer, Berlin, 1923. [2] J.L. Anderson, Phys. Rev. D3 (1971) 1689. [3] P.A.M. Dirac, Proc. R. Soc. Lond. A333 (1973) 403. [4] O.P. Freund, Annals of Phys. 84 (1974) 440. [5] E. Fradkin and G. Vilkovisky, Phys. Rev. D8 (1973) 4241. [6] L. Faddeev and B. Popov, Phys. Lett. 25B (1967) 29. E. Fradkin and I. Tyutin, Phys. Rev. D2 (1970) 2841. [7] B.S. De Witt, Phys. Rev. 162 (1967) 1195. J. Honerkamp, Nucl. Phys. B48 (1972) 269. [8] R.E. Kallosh, Nucl. Phys. B 78 (1974) 293. [9] G. 't Hooft and M. Veltman, Nuclear Phys. B44 (1972) 189. [10] G. 't Hooft and M. Veltman, Ann. Inst. H. Poincar~, in press. [ 11 ] S. Deser and P. Van Nieuwenhuizen, Phys. Rev. Letters 32 (1974) 245. [12] S. Deser, H.-S. Tsao and P. Van Nieuwenhuizen, Phys. Lett. B50 (1974) 491.
323