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On the gravitational lagrangian with R2-terms
Volume 151B, number 5,6
PHYSICS LETTERS
21 February 1985
ON THE GRAVITATIONAL LAGRANGIAN WITH R2-TERMS
P.F. GONZ,~LEZ-DIAZ Instituto de Optica "'Daza de Valdds'" C S.I. C, Serrano 121, Madrid 28006, Spain Received 7 November 1984
A new gravitational quantum lagrangian with logarithmic dependence on R is suggested. The resulting propagator seems to have no tachyon or unpbysical state poles, nor complex conjugate poles on the physical sheet.
General relativity with higher derivative terms (GRHDT) has been claimed [1,2] as a suitable starting point to quantize gravity. However, although GRHDT preserves many of the good properties of general relativity [3], it appears now clear [4,5] that in this kind of theory one ultimately has either unitarity or renormalizability, but never both. Renormalization group techniques have been used by Julve and Tonin [6], and Salam and Strathdee [7] to move the ghost mass off to infinity, where it should become innocuous. Resummation based on the 1/N approximation has been carried out by Tomboulis [7] who has showed that the real poles vanish in a 1IN expansion when gravity is coupled to N massless fermions. The trouble now is that a complex pair of poles would appear in the physical sheet so violating the analyticity of the S-matrix. The aim of this short letter is to advance one more alternative action integral with additional R2-terms which, through a logarithmic dependence on R, could avoid the above problems. Starting from the potential for a scalar field q~, Vm = _1/,/2¢2 + ~kO~b4, propagating in a background spacetime with curvature R in the case that the scalar fiekt is conformally coupled, we can define [9] an effective gravitational potential for a Higgs field with spontaneous symmetry breaking
o = (3/4rrG) 1/2 = M * , and suitable normalization conditions:
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
l~gff(¢) = _~/d0~, 2 2 ( 1 - 3e4M*2/16~r2#20) + ¼(P0/M*)2¢4(1 - 9e4M*2/327rEp 2) + (3e4[647r 2) ¢4 In [(¢//14") 2] ,
(1)
with g = -8rrGp2¢ 2 .
(2)
In order to obtain a renormalizable quantum lagrangian, we can now generalize the relation between ¢4 and the curvature-squared so that ¢4 be connected to R 2 and RuvRUV, instead o f R 2 alone. The term RuvTeRUVTe need not be considered here due to the use of the generalized Gauss-Bonnet theorem [10]. Thus, to preserve conformal invariance [11 ], we take 64rr2G2p4¢ 4 = co(RuvRUV- 1R2) + 13~'R 2 ,
(3)
where 6o and ~"are coefficients whose values need not be specified now. Consider then the simple case where the coupling constant e is given by 3e4/321r 2 = p-to/in 2 ,..,2 .
(4)
Inserting (2), (3) and (4) into (1) we obtain the quantically-corrected action integral for the conformally coupled gravity field: Sgff=_
1 fd4x(_g)l/2 16riG
(5)
X {n - R o 1 [co(R.vR # v - ~R 2) + }~'R 2] [1 - l n ( R / R o ) ] } , 405
Volume 151B, number 5,6
PHYSICS LETTERS
in which R 0 = - 6 p ~ is obtained by including the vacuum expectation value a = +~/* in (2). Eq. (5) conrains the logarithmic dependence on R that was already introduced by DeWitt [12], Ginzburg et al. [13] and others [14]. In the region R 2 >R02 this ensures a bounce which can avoid the general relativity singularities [15] and for R 2 < R 2 it gives a power-law asymptotic behaviour of the scale factor. Such an asymptotic behaviour may be obtained in the limit t -+ oo only in the case that production of matter is allowed [15]. It is worth noting that the effective potential corresponding to (5) acquires a dynamical minimum at the disordered phase ¢ = 0 (zero curvature) with exactly the same stability as the minimum at the ordered phase $ = M * (R ---R0). Thus, even at zero temperature, quantum corrections can lead to a dynamical symmetry restoration in our Higgs model. Since in this model all observable values o f R 2 ought to be less than R 2, we can expand l n ( R / R o ) in an infinite series: l n ( R / R o ) = ( R / R 0 - 1) - ½(R/R 0 - 1) 2 + ~ ( R / R 0 - 1) 3 - ....
(6)
Inserting (6) into (5) and expressing the result in terms of tinearized theory guy = rluu + huu, we can obtain, after taking the quadratic part, the gauge-independent graviton propagator in momentum space by using the usual procedure [4] : (2) 32zrGP~w;oo Auv;ap(k) = k 2 [ k 2 R ~ l ~ ( 1 + Xn=l n - l ) - 1]
+ k 2 [ k 2 R ~ l ~ ( 1 + Z n = l n _ l ) _ ½],
406
where P ~ ! o p and P~t°~ap are, respectively, the spin-two and spin-zero orthogonal Stelle projectors [4]. It is evident that for any w, ~"¢ 0, the only zero o f the denominators is for k 2 = 0, even if we rotate to Minkowski space. Therefore, propagator (7) has neither tachyon nor ghost poles, while the theory appears to be renormalizable and analytical. Thus, i f R ~
[1] C.J. Isham, in: Quantum gravity 2: a second Oxford
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
(0) 8 ~rGP~,; op (7)
21 February 1985
[14] [15]
symposium, eds. C.J. Isham, R. Penrose and D.W. Sciama (Clarendon, Oxford, 1981); see also P. Van Nieuwenhuizen in: Quantum gravity 2: a second Oxford symposium, eds. C.J. Isham, R. Penrose and D.W. Sciama (Clarendon, Oxford, 1981). S. Weinberg, in: General relativity: an Einstein centenary survey, eds. S.W. Hawking and W. Israel (Cambridge U:P., London, England, 1979). M.V. Fishetti et al., Phys. Rev. D20 (1979) 1757. K.S. Stelle, Phys. Rev. D16 (1977) 953. E. Sezgin and P. van Nieuwenhuizen, Phys. Rev. D21 (1980) 3269. J. Julve and M. Tonin, Nuovo Cimento 46B (1978) 137. A. Salam and J. Strathdee, Phys. Rev. D18 (1978) 4480. E. Tomboulis, Phy~ Lett. 70B (1977) 361; 97B (1980) 77. P.F. Gonzglez-D~az,Phys. Lett. 141B (1984) 314. S.S. Chern, Hamburg Abh. 20 (1955) 177. L.S. Brown and J.C. Collins, Ann. Phys. (NY) 130 (1980) 215. B.S. DeWitt, Phys. Rev. 160 (1967) 1113; 162 (1967) 1239. V.L. Ginzburg, D.A. Kirzhnits and A.A. Lyubushin, Zh. Eksp. Teor. Fiz. 60 (1971) 451 [Soy. Phys. JETP 33 (1971) 242]. T. Hill, unpublished. V.T. Gurovich and A.A. Starobinskii, Zh. Eksp. Teor. Fiz. 77 (1979) 1683 [Soy. Phys. JETP 50 (1979) 844].
PHYSICS LETTERS
21 February 1985
ON THE GRAVITATIONAL LAGRANGIAN WITH R2-TERMS
P.F. GONZ,~LEZ-DIAZ Instituto de Optica "'Daza de Valdds'" C S.I. C, Serrano 121, Madrid 28006, Spain Received 7 November 1984
A new gravitational quantum lagrangian with logarithmic dependence on R is suggested. The resulting propagator seems to have no tachyon or unpbysical state poles, nor complex conjugate poles on the physical sheet.
General relativity with higher derivative terms (GRHDT) has been claimed [1,2] as a suitable starting point to quantize gravity. However, although GRHDT preserves many of the good properties of general relativity [3], it appears now clear [4,5] that in this kind of theory one ultimately has either unitarity or renormalizability, but never both. Renormalization group techniques have been used by Julve and Tonin [6], and Salam and Strathdee [7] to move the ghost mass off to infinity, where it should become innocuous. Resummation based on the 1/N approximation has been carried out by Tomboulis [7] who has showed that the real poles vanish in a 1IN expansion when gravity is coupled to N massless fermions. The trouble now is that a complex pair of poles would appear in the physical sheet so violating the analyticity of the S-matrix. The aim of this short letter is to advance one more alternative action integral with additional R2-terms which, through a logarithmic dependence on R, could avoid the above problems. Starting from the potential for a scalar field q~, Vm = _1/,/2¢2 + ~kO~b4, propagating in a background spacetime with curvature R in the case that the scalar fiekt is conformally coupled, we can define [9] an effective gravitational potential for a Higgs field with spontaneous symmetry breaking
o = (3/4rrG) 1/2 = M * , and suitable normalization conditions:
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
l~gff(¢) = _~/d0~, 2 2 ( 1 - 3e4M*2/16~r2#20) + ¼(P0/M*)2¢4(1 - 9e4M*2/327rEp 2) + (3e4[647r 2) ¢4 In [(¢//14") 2] ,
(1)
with g = -8rrGp2¢ 2 .
(2)
In order to obtain a renormalizable quantum lagrangian, we can now generalize the relation between ¢4 and the curvature-squared so that ¢4 be connected to R 2 and RuvRUV, instead o f R 2 alone. The term RuvTeRUVTe need not be considered here due to the use of the generalized Gauss-Bonnet theorem [10]. Thus, to preserve conformal invariance [11 ], we take 64rr2G2p4¢ 4 = co(RuvRUV- 1R2) + 13~'R 2 ,
(3)
where 6o and ~"are coefficients whose values need not be specified now. Consider then the simple case where the coupling constant e is given by 3e4/321r 2 = p-to/in 2 ,..,2 .
(4)
Inserting (2), (3) and (4) into (1) we obtain the quantically-corrected action integral for the conformally coupled gravity field: Sgff=_
1 fd4x(_g)l/2 16riG
(5)
X {n - R o 1 [co(R.vR # v - ~R 2) + }~'R 2] [1 - l n ( R / R o ) ] } , 405
Volume 151B, number 5,6
PHYSICS LETTERS
in which R 0 = - 6 p ~ is obtained by including the vacuum expectation value a = +~/* in (2). Eq. (5) conrains the logarithmic dependence on R that was already introduced by DeWitt [12], Ginzburg et al. [13] and others [14]. In the region R 2 >R02 this ensures a bounce which can avoid the general relativity singularities [15] and for R 2 < R 2 it gives a power-law asymptotic behaviour of the scale factor. Such an asymptotic behaviour may be obtained in the limit t -+ oo only in the case that production of matter is allowed [15]. It is worth noting that the effective potential corresponding to (5) acquires a dynamical minimum at the disordered phase ¢ = 0 (zero curvature) with exactly the same stability as the minimum at the ordered phase $ = M * (R ---R0). Thus, even at zero temperature, quantum corrections can lead to a dynamical symmetry restoration in our Higgs model. Since in this model all observable values o f R 2 ought to be less than R 2, we can expand l n ( R / R o ) in an infinite series: l n ( R / R o ) = ( R / R 0 - 1) - ½(R/R 0 - 1) 2 + ~ ( R / R 0 - 1) 3 - ....
(6)
Inserting (6) into (5) and expressing the result in terms of tinearized theory guy = rluu + huu, we can obtain, after taking the quadratic part, the gauge-independent graviton propagator in momentum space by using the usual procedure [4] : (2) 32zrGP~w;oo Auv;ap(k) = k 2 [ k 2 R ~ l ~ ( 1 + Xn=l n - l ) - 1]
+ k 2 [ k 2 R ~ l ~ ( 1 + Z n = l n _ l ) _ ½],
406
where P ~ ! o p and P~t°~ap are, respectively, the spin-two and spin-zero orthogonal Stelle projectors [4]. It is evident that for any w, ~"¢ 0, the only zero o f the denominators is for k 2 = 0, even if we rotate to Minkowski space. Therefore, propagator (7) has neither tachyon nor ghost poles, while the theory appears to be renormalizable and analytical. Thus, i f R ~
[1] C.J. Isham, in: Quantum gravity 2: a second Oxford
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
(0) 8 ~rGP~,; op (7)
21 February 1985
[14] [15]
symposium, eds. C.J. Isham, R. Penrose and D.W. Sciama (Clarendon, Oxford, 1981); see also P. Van Nieuwenhuizen in: Quantum gravity 2: a second Oxford symposium, eds. C.J. Isham, R. Penrose and D.W. Sciama (Clarendon, Oxford, 1981). S. Weinberg, in: General relativity: an Einstein centenary survey, eds. S.W. Hawking and W. Israel (Cambridge U:P., London, England, 1979). M.V. Fishetti et al., Phys. Rev. D20 (1979) 1757. K.S. Stelle, Phys. Rev. D16 (1977) 953. E. Sezgin and P. van Nieuwenhuizen, Phys. Rev. D21 (1980) 3269. J. Julve and M. Tonin, Nuovo Cimento 46B (1978) 137. A. Salam and J. Strathdee, Phys. Rev. D18 (1978) 4480. E. Tomboulis, Phy~ Lett. 70B (1977) 361; 97B (1980) 77. P.F. Gonzglez-D~az,Phys. Lett. 141B (1984) 314. S.S. Chern, Hamburg Abh. 20 (1955) 177. L.S. Brown and J.C. Collins, Ann. Phys. (NY) 130 (1980) 215. B.S. DeWitt, Phys. Rev. 160 (1967) 1113; 162 (1967) 1239. V.L. Ginzburg, D.A. Kirzhnits and A.A. Lyubushin, Zh. Eksp. Teor. Fiz. 60 (1971) 451 [Soy. Phys. JETP 33 (1971) 242]. T. Hill, unpublished. V.T. Gurovich and A.A. Starobinskii, Zh. Eksp. Teor. Fiz. 77 (1979) 1683 [Soy. Phys. JETP 50 (1979) 844].