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Induced gravity for nonzero torsion
Volume 169B, number 4
PHYSICS LETTERS
3 April 1986
I N D U C E D GRAVITY FOR N O N Z E R O T O R S I O N G. G R E N S I N G Universiti~t Kiel, Fachbereich Physik, D-2300 Kiel, Fed. Rep. Germam' Received 2 January 1986
Quantum fluctuations of a spin one-halfparticle in a gravitational background field with nonvanishingtorsion are shown to induce again the Einstein-Hilbert term as a long wavelength effective action so that torsion disappears at the classical tree level.
1. introduction and summary. The microscopic origin of the Einstein-Hilbert action seems to have been revealed in recent years with the discovery that it is induced by quantum fluctuations of the matter fields [ 1,2]. In addition, the gravitational constant becomes calculable in principle if fundamental scalar fields are absent and if the spin one-half matter fields acquire a mass through dynamical symmetry breaking [3,4]. As a consequence, also the old renormalizability problem of gravity gets tractable in a consistent way because the Einstein-Hilbert term does not appear as a separate contribution in the gravitational action. Since theories of the type curvature squared are renormalizable in the perturbative sense [5] and in particular the scale invariant Weyl theory has the the additional property of being asymptotically free [6,7], the premise for dimensional transmutation is indeed valid. The violation of unitarity on the tree level is a dynamical problem, and restoration is expected if radiative corrections are taken into account (for a recent review, see ref. [8]). On the other hand, according to the gauge approach to gravity [9-12] there is no way to escape the conclusion that torsion must be taken serious as a new degree of freedom. Hence it is no esoteric question, whether or not torsion makes itself felt as a classical field strength, i.e., contributes to the low energy effective action. In order to answer this question, we generalize the model calculation of Zee [13] so as to include the case of nonvanishing torsion, but instead of the diagrammatic approach we choose the Schwinger0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
De Witt method [ 14]. Since the regularization procedure has to reflect the gross properties of dynamical symmetry breaking, dimensional and ~'-function techniques are not available. Instead one must use the Pauli-Villars method, and we regularize the Schwinger-De Witt determinant such that it yields the correct regularization of the individual graphs. Another problem consists in the determination of the so-called Minakshisundaram coefficients for Riemann-Cartan spaces, which has been dealt with by Goldthorpe [15] and Obukhov [16]. We avoid additional work and eventual computational errors by reducing the calculation to the case of vanishing torsion. Though one would expect the Einstein-Cartan theory as the ffmal result, the actual outcome will be that the induced effective action is again Einstein's one since all torsion dependence drops out.
2. Basics and notation. As remarked above, the concept of induced gravity only makes sense if fundamental scalar fields are forbidden. This is no essential restriction since Higgs fields are generally regarded as a phenomenological manifestation of a fermionic condensate. In the present context we thus avoid the conflict that only spin one-half fields feel torsion, scalar fields obviously do not. The introduction of spinors in the framework of general relativity requires the gauge approach, and in order to fix the notation we first collect the main ingredients (see, e.g., refs. [12,17] ). The covariant derivative for a spinor is written 333
Volume 169B, number 4
V~ = e ~ ( a ,
PHYSICS LETTERS
- ~i ETa e°'ra ~)'
(1)
3 April 1986
where 00
where e ~ denotes the inverse vierbein with e ~ e ~ v = 6uu and co~a u the spinor connection. Furthermore, the inf'mitesimal operators read ~ a~ = ¼i [ ~ , 7 a] in terms of the fiat space Dirac matrices, which thus are ,x-independent * i. The field strengths are obtained from the commutator
[V a, V ~] = -eV~ eV#(S'ruv V.r +~iR~auvY~,a),
(2)
where S~uu andR~Suu denote the torsion and curva. ture tensor, respectively. According to the minimal substitution prescription, the coupling to the gravitational gauge fields is uniquely determined through
fdx lel ( ~ i ( ~ T a V ~ k -
S=
(8)
in d = 26o dimensions. We shall first regularize the divergent integral for I¢, the computation of the Minakshisundaram coefficients an(X, x) with
An =
fdx
lel Sp an
(9)
will be given afterwards.
3. Induced effective action for nonvanishing torsion. The divergencies occur at the lower limit for n = 0, 1, ..., co and we thus introduce Pauli-Villars regulators m i with i = 1 ..... co, co + 1 such that
V~¢ 7a~k) - m~q~)
= ejdx lel ~D~,
i e x p ( - m 2 r ) ~ Anrn, (47rr) to n=0
Spexp(--rA)-
(3)
to+l
i=o
ci(m2)n=o , n = 0 , 1 ..... co,
(10)
where lel is the determinant ofea~ and D = i3'a(Va + ~ S ) - m,
(4)
with S a = S~at 3. Furthermore, we introduce D* = 75D75 so that A = D* D
(5)
gives the second order Dirac operator. The effective action is defined by means of the functional integral
where c O = 1 and m 0 = m. Here it is understood that the Dirac field has acquired its nonzero mass through dynamical symmetry breaking, with the large regulator masses simulating the softening effect (see ref. [13] ). The regularized action then follows to be 6o+1
oo
i 0f
d r r -1 Sp e x p ( - r A i )
1
An
WpV = ~
= ci
0
_
exp(iW[R,S]) = f ~ ~t~ exp(iS[~, ~0,e, ~ ] )
1
2 (4rr)¢o =
.)--~,
F ( c o - n + l ) i----'0 oo
= I A I 1/2,
(6)
where the Grassmann rules have been used. We calculate the determinant IA I of the second order operator by means of the original Schwinger-De Witt technique [14], and so the effective action is obtained from i
fdr
r -1 Sp e x p ( - r A ) ,
(7)
0 ,1 We adopt the convention that t~, u = 0, 1, 2, 3 denote the coordinate indices and c~,~ = 0, 1, 2, 3 the vierbein indices. The signature of the Minkowski metric tensor 8cz# is taken to be - 2 and as to the ~-matrices, we follow the choice of Itzykson and Zuber [18]. 334
Xlnm ]
1
1 2 (4zr)w
~ i/=O.)+1
A r/
r(n-co) (m2)n-to
(11)
Afterwards we choose mw+ 1 large compared to the remaining regulator masses so that cto+l = 0 and thus for d = 4 the standard result 2 2 i=0
c=0, ~
eim2 =0,
i=0
(12)
is recovered. In order to get rid o f the A0-term , representing the zero point energy, the effective action is normalized to unity for vanishing external fields , 2 , ,2 We thus circumvent the problem with a large induced cosmological constant.
Volume 169B, number 4
PHYSICS LETTERS S 5uv =
and so
WPV
_
1 1 2 (4z02
(i~oCim21nm2)Al+ -=
"'" '
(13)
which is the result sought for. There remains the determination of the first few Minakshisundaram coefficients. The approach of Goldthorpe [ 15] consists in generalizing the De Witt expansion to autoparallels, which are often believed to be the substitute of geodesics for R i e m a n n Caftan spaces. But the calculafional amount soon gets excessive and there is no hope to determine, e.g., a 2 without restriction to special cases such as totally antisymmetric torsion. To some extent, these difficulties have been overcome by Obukhov [16] in the attempt to obtain the trace and the axial anomaly, both being completely determined by the second coefficient , a But there is a simple device, which allows of the reduction to k n o w n results. To begin with, the spinor connection may be split into two terms ¢a2')'6~ = c~~8 # + Tq'su,
T~=-~(
1 S~
(15)
o 8
and co~r u denotes the standard Ricci rotation coefficients. Then the Dirac operator can be rewritten to give 0
D=iT~(V
1.
*
+~175S)-m,
(16)
o
with V~ the covarianr spinor derivative for vanishing torsion and S ~ = ~ eaO'rSS¢.rs. Since 75 commutes with the £at~ the additional contribution to the derivative is of strictly internal type, the structure group being chiral U(1). To make this manifest, we def'me o
+¼i7 5.S ,
(17)
so that the commutator reads o
['V ,'V~] = -ieU eV
(75Ssuv+~,Y~RT~uv),
(18)
with .3 To the best of our knowledge, we have nowhere found the explicit form of the coefficient a 1 or A 1 in the literature. In ref. [ 16 ] only the coefficient A 2 is given, but it differs from our result eq. (22) below.
(19)
the chiral field strengths. The crucial point is that on the right hand side of eq. (18) the derivative has disappeared (cf. eq. (2)). For the second order Dirac operator this property entails - a R - 75 o
sSafl~-, aft + m 2,
(20)
o
where R = R ~'fl eU eV_ is the curvature scalar for vanishing torsion ~ . Now we can make use of the standard results for the coefficients a n [14,19], in particular we obtain 1° Spa I = -sR, (21) and for the sake of completeness we also give , s 1
o
Spa 2 = -~Ra o
o
o
~ +~0(SRR o
o
o
8R fiR ~#
4
+ 7R #78R~fl'~ ) + 5S5flS5 ~fl.
(22)
As a decisive fact, all torsion dependence contained in a 1 drops out on taking the trace so that the effective action
W
S 8 + S~)
-
1. SuS*v - OvS*u), - "g(
(14)
where the entire torsion dependence is contained in
~a=Vc
3 April 1986
1 1 6 (47r)2
¢im~lnm _
fd lelg,
(23)
again coincides with the Einstein form. Up to a factor 1 this result is identical with the one of Zee [13], our sign conventions also yield a positive induced gravitational constant ,6 ,4 Hence there is no need to generalize thecalculation of the O o o an to operators of the form VaV,~+ YaVa +X, which is the procedure being followed in ref. [ 16]. ,s The integrated form A 2 yields the one-loop counterterm for a gravitational action with nonzero torsion, originating from the matter fields. Hence the inclusion of terms (VS) 2 should be taken serious in current model theories for non vanishing torsion in order to guarantee at least one loop renormalizability (cf. also refs. [20,21 ] ). Furthermore, ff one attempts to generalize the Weyl tensor to Riemann-Cartan spaces, the above derivative terms appear as well. *6 Of course, the objection that the induced gravitational constant depends on the details of dynamical symmetry breaking (see refs. [4,13] ), or is even cutoff dependent [22,23], also applies here. The safest way to circumvent these objections would be to redo the calculation for a generalized Kaluza-Klein theory, being favoured nowadays as promising models of grand unification, where for the case of vanishing torsion the induced gravitational constant comes out to be cutoff independent and finite [24, 251.
335
Volume 169B, number 4
PHYSICS LETTERS
In summary, this outcome provides for a reliable argument why torsion does not make itself felt in the low energy domain.
N o t e added in proof. After completion of this paper we learned that similar results have been obtained by Denardo and Spallucci [26]. These authors supplement the effective action by the torsion term originathag from the a 2 coefficient. But it is essential to note that the contributions from a n with n ~> 2 may be shown to vanish if Planck's constant tends to zero. Hence, it is indeed only the Einstein-Hilbert term which survives in the classical limit. References [ 1] A.D. Sakharov, Dok. Akad. Nauk. USSR 177 (1967) 70 [Soy. Phys. Dokl. 12 (1968) 1040]. [2] O. Klein, Phys. Scr. 96 (1974) 69. [3] S.L. Adler, Phys. Rev. Lett. 44 (1980) 1567. [4] S.L. Adler, Rev. Mod. Phys. 54 (1982) 729. [5] K.S. SteRe, Phys. Rev. D16 (1977) 953. [6] J. Julve and M. Tonin, Nuovo Cimento B46 (1978) 137. [7] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. 104B (1981) 377. [8] E.T. Tomboulis, in: Quantum theory of gravity, ed. S.M. Christensen (Adam Hilger, Bristol, 1984).
336
3 April 1986
[9] R. Utiyama, Phys. Rev. 101 (1956) 1597. [10] T.W. Kibble, J. Math. Phys. 2 (1961) 212. [ 11] D.W. Sciama, in: Recent developments in general relativity (Pergamon, Oxford, 1962). [ 12] F.W. Hehl, P. yon dcr Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys. 48 (1976) 393. [13] A. Zee, Phys. Rev. D23 (1981) 858. [ 14] B.S. De Witt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965). [15] W.H. Goldthorpe, Nucl. Phys. B170 [FS1] (1980) 307. [16] Yu.N. Obukhov, Nucl. Phys. B212 (1983) 237. [17] D. Grensing and G. Grensing, Phys. Rev. D28 (1983) 286. [18] C. Itzykson and J.B. Zuber, Quantum field theory (McGraw-Hill, New York, 1980). [19] P.B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem (Publish or Perish, Wilmington, 1984). [20] D.E. Neville, Phys. Rev. D21 (1980) 867, 2075. [21] E. Sezgin and P. van Nieuwenhuizen, Phys. Rev. D21 (1980) 3269. [22] F. David, Phys. Lett. 138B (1984) 383. [23] F. David and A. Strominger, Phys. Lett. 143B (1984) 125. [24] D.J. Toms, Phys. Lett. 129B (1983) 31. [25] P. Candelas and S. Weinberg, Nucl. Phys. B237 (1984) 397. [26] G. Denardo and E. Spalluchi, ICTP preprint 85/111.
PHYSICS LETTERS
3 April 1986
I N D U C E D GRAVITY FOR N O N Z E R O T O R S I O N G. G R E N S I N G Universiti~t Kiel, Fachbereich Physik, D-2300 Kiel, Fed. Rep. Germam' Received 2 January 1986
Quantum fluctuations of a spin one-halfparticle in a gravitational background field with nonvanishingtorsion are shown to induce again the Einstein-Hilbert term as a long wavelength effective action so that torsion disappears at the classical tree level.
1. introduction and summary. The microscopic origin of the Einstein-Hilbert action seems to have been revealed in recent years with the discovery that it is induced by quantum fluctuations of the matter fields [ 1,2]. In addition, the gravitational constant becomes calculable in principle if fundamental scalar fields are absent and if the spin one-half matter fields acquire a mass through dynamical symmetry breaking [3,4]. As a consequence, also the old renormalizability problem of gravity gets tractable in a consistent way because the Einstein-Hilbert term does not appear as a separate contribution in the gravitational action. Since theories of the type curvature squared are renormalizable in the perturbative sense [5] and in particular the scale invariant Weyl theory has the the additional property of being asymptotically free [6,7], the premise for dimensional transmutation is indeed valid. The violation of unitarity on the tree level is a dynamical problem, and restoration is expected if radiative corrections are taken into account (for a recent review, see ref. [8]). On the other hand, according to the gauge approach to gravity [9-12] there is no way to escape the conclusion that torsion must be taken serious as a new degree of freedom. Hence it is no esoteric question, whether or not torsion makes itself felt as a classical field strength, i.e., contributes to the low energy effective action. In order to answer this question, we generalize the model calculation of Zee [13] so as to include the case of nonvanishing torsion, but instead of the diagrammatic approach we choose the Schwinger0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
De Witt method [ 14]. Since the regularization procedure has to reflect the gross properties of dynamical symmetry breaking, dimensional and ~'-function techniques are not available. Instead one must use the Pauli-Villars method, and we regularize the Schwinger-De Witt determinant such that it yields the correct regularization of the individual graphs. Another problem consists in the determination of the so-called Minakshisundaram coefficients for Riemann-Cartan spaces, which has been dealt with by Goldthorpe [15] and Obukhov [16]. We avoid additional work and eventual computational errors by reducing the calculation to the case of vanishing torsion. Though one would expect the Einstein-Cartan theory as the ffmal result, the actual outcome will be that the induced effective action is again Einstein's one since all torsion dependence drops out.
2. Basics and notation. As remarked above, the concept of induced gravity only makes sense if fundamental scalar fields are forbidden. This is no essential restriction since Higgs fields are generally regarded as a phenomenological manifestation of a fermionic condensate. In the present context we thus avoid the conflict that only spin one-half fields feel torsion, scalar fields obviously do not. The introduction of spinors in the framework of general relativity requires the gauge approach, and in order to fix the notation we first collect the main ingredients (see, e.g., refs. [12,17] ). The covariant derivative for a spinor is written 333
Volume 169B, number 4
V~ = e ~ ( a ,
PHYSICS LETTERS
- ~i ETa e°'ra ~)'
(1)
3 April 1986
where 00
where e ~ denotes the inverse vierbein with e ~ e ~ v = 6uu and co~a u the spinor connection. Furthermore, the inf'mitesimal operators read ~ a~ = ¼i [ ~ , 7 a] in terms of the fiat space Dirac matrices, which thus are ,x-independent * i. The field strengths are obtained from the commutator
[V a, V ~] = -eV~ eV#(S'ruv V.r +~iR~auvY~,a),
(2)
where S~uu andR~Suu denote the torsion and curva. ture tensor, respectively. According to the minimal substitution prescription, the coupling to the gravitational gauge fields is uniquely determined through
fdx lel ( ~ i ( ~ T a V ~ k -
S=
(8)
in d = 26o dimensions. We shall first regularize the divergent integral for I¢, the computation of the Minakshisundaram coefficients an(X, x) with
An =
fdx
lel Sp an
(9)
will be given afterwards.
3. Induced effective action for nonvanishing torsion. The divergencies occur at the lower limit for n = 0, 1, ..., co and we thus introduce Pauli-Villars regulators m i with i = 1 ..... co, co + 1 such that
V~¢ 7a~k) - m~q~)
= ejdx lel ~D~,
i e x p ( - m 2 r ) ~ Anrn, (47rr) to n=0
Spexp(--rA)-
(3)
to+l
i=o
ci(m2)n=o , n = 0 , 1 ..... co,
(10)
where lel is the determinant ofea~ and D = i3'a(Va + ~ S ) - m,
(4)
with S a = S~at 3. Furthermore, we introduce D* = 75D75 so that A = D* D
(5)
gives the second order Dirac operator. The effective action is defined by means of the functional integral
where c O = 1 and m 0 = m. Here it is understood that the Dirac field has acquired its nonzero mass through dynamical symmetry breaking, with the large regulator masses simulating the softening effect (see ref. [13] ). The regularized action then follows to be 6o+1
oo
i 0f
d r r -1 Sp e x p ( - r A i )
1
An
WpV = ~
= ci
0
_
exp(iW[R,S]) = f ~ ~t~ exp(iS[~, ~0,e, ~ ] )
1
2 (4rr)¢o =
.)--~,
F ( c o - n + l ) i----'0 oo
= I A I 1/2,
(6)
where the Grassmann rules have been used. We calculate the determinant IA I of the second order operator by means of the original Schwinger-De Witt technique [14], and so the effective action is obtained from i
fdr
r -1 Sp e x p ( - r A ) ,
(7)
0 ,1 We adopt the convention that t~, u = 0, 1, 2, 3 denote the coordinate indices and c~,~ = 0, 1, 2, 3 the vierbein indices. The signature of the Minkowski metric tensor 8cz# is taken to be - 2 and as to the ~-matrices, we follow the choice of Itzykson and Zuber [18]. 334
Xlnm ]
1
1 2 (4zr)w
~ i/=O.)+1
A r/
r(n-co) (m2)n-to
(11)
Afterwards we choose mw+ 1 large compared to the remaining regulator masses so that cto+l = 0 and thus for d = 4 the standard result 2 2 i=0
c=0, ~
eim2 =0,
i=0
(12)
is recovered. In order to get rid o f the A0-term , representing the zero point energy, the effective action is normalized to unity for vanishing external fields , 2 , ,2 We thus circumvent the problem with a large induced cosmological constant.
Volume 169B, number 4
PHYSICS LETTERS S 5uv =
and so
WPV
_
1 1 2 (4z02
(i~oCim21nm2)Al+ -=
"'" '
(13)
which is the result sought for. There remains the determination of the first few Minakshisundaram coefficients. The approach of Goldthorpe [ 15] consists in generalizing the De Witt expansion to autoparallels, which are often believed to be the substitute of geodesics for R i e m a n n Caftan spaces. But the calculafional amount soon gets excessive and there is no hope to determine, e.g., a 2 without restriction to special cases such as totally antisymmetric torsion. To some extent, these difficulties have been overcome by Obukhov [16] in the attempt to obtain the trace and the axial anomaly, both being completely determined by the second coefficient , a But there is a simple device, which allows of the reduction to k n o w n results. To begin with, the spinor connection may be split into two terms ¢a2')'6~ = c~~8 # + Tq'su,
T~=-~(
1 S~
(15)
o 8
and co~r u denotes the standard Ricci rotation coefficients. Then the Dirac operator can be rewritten to give 0
D=iT~(V
1.
*
+~175S)-m,
(16)
o
with V~ the covarianr spinor derivative for vanishing torsion and S ~ = ~ eaO'rSS¢.rs. Since 75 commutes with the £at~ the additional contribution to the derivative is of strictly internal type, the structure group being chiral U(1). To make this manifest, we def'me o
+¼i7 5.S ,
(17)
so that the commutator reads o
['V ,'V~] = -ieU eV
(75Ssuv+~,Y~RT~uv),
(18)
with .3 To the best of our knowledge, we have nowhere found the explicit form of the coefficient a 1 or A 1 in the literature. In ref. [ 16 ] only the coefficient A 2 is given, but it differs from our result eq. (22) below.
(19)
the chiral field strengths. The crucial point is that on the right hand side of eq. (18) the derivative has disappeared (cf. eq. (2)). For the second order Dirac operator this property entails - a R - 75 o
sSafl~-, aft + m 2,
(20)
o
where R = R ~'fl eU eV_ is the curvature scalar for vanishing torsion ~ . Now we can make use of the standard results for the coefficients a n [14,19], in particular we obtain 1° Spa I = -sR, (21) and for the sake of completeness we also give , s 1
o
Spa 2 = -~Ra o
o
o
~ +~0(SRR o
o
o
8R fiR ~#
4
+ 7R #78R~fl'~ ) + 5S5flS5 ~fl.
(22)
As a decisive fact, all torsion dependence contained in a 1 drops out on taking the trace so that the effective action
W
S 8 + S~)
-
1. SuS*v - OvS*u), - "g(
(14)
where the entire torsion dependence is contained in
~a=Vc
3 April 1986
1 1 6 (47r)2
¢im~lnm _
fd lelg,
(23)
again coincides with the Einstein form. Up to a factor 1 this result is identical with the one of Zee [13], our sign conventions also yield a positive induced gravitational constant ,6 ,4 Hence there is no need to generalize thecalculation of the O o o an to operators of the form VaV,~+ YaVa +X, which is the procedure being followed in ref. [ 16]. ,s The integrated form A 2 yields the one-loop counterterm for a gravitational action with nonzero torsion, originating from the matter fields. Hence the inclusion of terms (VS) 2 should be taken serious in current model theories for non vanishing torsion in order to guarantee at least one loop renormalizability (cf. also refs. [20,21 ] ). Furthermore, ff one attempts to generalize the Weyl tensor to Riemann-Cartan spaces, the above derivative terms appear as well. *6 Of course, the objection that the induced gravitational constant depends on the details of dynamical symmetry breaking (see refs. [4,13] ), or is even cutoff dependent [22,23], also applies here. The safest way to circumvent these objections would be to redo the calculation for a generalized Kaluza-Klein theory, being favoured nowadays as promising models of grand unification, where for the case of vanishing torsion the induced gravitational constant comes out to be cutoff independent and finite [24, 251.
335
Volume 169B, number 4
PHYSICS LETTERS
In summary, this outcome provides for a reliable argument why torsion does not make itself felt in the low energy domain.
N o t e added in proof. After completion of this paper we learned that similar results have been obtained by Denardo and Spallucci [26]. These authors supplement the effective action by the torsion term originathag from the a 2 coefficient. But it is essential to note that the contributions from a n with n ~> 2 may be shown to vanish if Planck's constant tends to zero. Hence, it is indeed only the Einstein-Hilbert term which survives in the classical limit. References [ 1] A.D. Sakharov, Dok. Akad. Nauk. USSR 177 (1967) 70 [Soy. Phys. Dokl. 12 (1968) 1040]. [2] O. Klein, Phys. Scr. 96 (1974) 69. [3] S.L. Adler, Phys. Rev. Lett. 44 (1980) 1567. [4] S.L. Adler, Rev. Mod. Phys. 54 (1982) 729. [5] K.S. SteRe, Phys. Rev. D16 (1977) 953. [6] J. Julve and M. Tonin, Nuovo Cimento B46 (1978) 137. [7] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. 104B (1981) 377. [8] E.T. Tomboulis, in: Quantum theory of gravity, ed. S.M. Christensen (Adam Hilger, Bristol, 1984).
336
3 April 1986
[9] R. Utiyama, Phys. Rev. 101 (1956) 1597. [10] T.W. Kibble, J. Math. Phys. 2 (1961) 212. [ 11] D.W. Sciama, in: Recent developments in general relativity (Pergamon, Oxford, 1962). [ 12] F.W. Hehl, P. yon dcr Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys. 48 (1976) 393. [13] A. Zee, Phys. Rev. D23 (1981) 858. [ 14] B.S. De Witt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965). [15] W.H. Goldthorpe, Nucl. Phys. B170 [FS1] (1980) 307. [16] Yu.N. Obukhov, Nucl. Phys. B212 (1983) 237. [17] D. Grensing and G. Grensing, Phys. Rev. D28 (1983) 286. [18] C. Itzykson and J.B. Zuber, Quantum field theory (McGraw-Hill, New York, 1980). [19] P.B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem (Publish or Perish, Wilmington, 1984). [20] D.E. Neville, Phys. Rev. D21 (1980) 867, 2075. [21] E. Sezgin and P. van Nieuwenhuizen, Phys. Rev. D21 (1980) 3269. [22] F. David, Phys. Lett. 138B (1984) 383. [23] F. David and A. Strominger, Phys. Lett. 143B (1984) 125. [24] D.J. Toms, Phys. Lett. 129B (1983) 31. [25] P. Candelas and S. Weinberg, Nucl. Phys. B237 (1984) 397. [26] G. Denardo and E. Spalluchi, ICTP preprint 85/111.