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Instability of flat spacetime in semiclassical gravity
Volume 114B, number 1
PHYSICS LETTERS
15 July 1982
INSTABILITY OF FLAT SPACETIME IN SEMICLASSICAL GRAVITY Hidenaga YAMAGISHI Department o f Physics, Princeton University, Princeton, NJ 08540, USA Received 22 February 1982 Revised manuscript received 1 April 1982
The semiclassical theory of gravity is considered in which an asymptotically flat background m e t r e is coupled to quantized matter. We show that, in general, there are modes with spacelike wave vectors for small metric fluctuations around flat spacefime. Besides the usual axioms of quantum field theory in flat spaeetime, the proof rests on the existence of a hard trace anomaly in the energy-momentum tensor due to matter self-couplings. Two possible interpretations of the result are discussed.
In view of the difficulty of obtaining a full quantum theory of gravity [1,2], many authors have considered the so-called semiclassical (or one-loop) approximation in which a classical background metric is coupled to quantized matter [ 1 - 4 ] . For the case of a background black-hole metric, the approximation has revealed a remarkable connection between black holes and thermodynamics [1]. However, modes with spacelike wave vectors have also been encountered for small metric fluctuations around flat spacetime [5]. In this letter, we shall show that such modes are a general feature of the semiclassical approximation. The quantity of interest is the vacuum persistence amplitude [6] for quantized matter in a background metric gab [7,8] eit[g] -------(0+10_) g.
(1)
The metric gab we take to be asymptotically fiat, so that the in and out vacuum states 10_+)can be def'med. In quantized matter, we may also include a gravitational part hab if it couples only to the classical metric gab and not to the rest of the matter fields ¢ or to itself. In other words, the path-integral representation [9] of (1) must be of the form e irig]
as
rig] = A f d a x v ~ +
l~-~fdaxRvrZ--g
- a f d4x(RabRab - IR2)Vr~ +b f d4xREx/'----g+ ....
(3)
which defines the cosmological constant A and the gravitational constant G in the semiclassical approximation. In view of the empirical evidence, we shall take them to be A = 0 and G > 0 (attractive). With zero cosmological constant, flat spacetime is a stationary point for the effective action, and we may also expand F [g] around the usual Monkowski metric
F ['r/+ h] = 8
f
d4y(OI
r*e a(x)%6 0,)1o:,
X he#(x)h ~6(y) + O(ha).
=f [de] [dh] exp (iS 1 [g]
+ iS 2 [¢,g] + iS 3 [h,g] ),
with S 3 quadratic (one-loop) in hab. We shall also assume that gauge fixing is done by the backgroundfield method [8] so that the effective action F[g] is invariant under general coordinate transformations. For slowly varying metrics, we may expand P[g]
(2)
0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
(4)
,1 The signature is + - - - . Greek indices are raised or lowered with n-
27
Volume 114B, number 1
PHYSICS LETTERS
In eq. (4), O ~ is the (renormalized) energy-momentum tensor in fiat spacetime; the T*-product indicates that seagulls must be added because of the strong singularity in the equal-time commutator [ 10,11 ] [O00(x, t), O00 (y; t)] = i [O0k(X , t) + O0k(Y, t)] axkS(x - - y ) .
(5)
Since all quantities in eq. (4) refer to fiat spacetime, we may conveniently introduce the Fourier transform
× ~ 6 (k) -=-- i f d 4 x eik'x(01T*O #(x)O n (0)10). (6) The Ward identities fix the tensor structure of ×a~a (k) to be
Xa376 (k) = } [rta~,v~ (k) + ~ lra3(k )rcy~ (k)] ~(k 2) + -~7ra~(k) lr r~ (k)x(k2),
(7)
15 July 1982
(8a)
rs~ (0) + -~ x'(0) = 0.
G - 1 = 47r×'(0)/3 = -8rr~'(0)/5, a = ~"(0)/10,
- ~O?a,k~ka/k2 + 3 perm.).
(85)
(12a)
b =-X"(0)/72.
(12b,c)
The crux of this paper is that we can translate the low-energy theorem (12a) into a statement about the zeroes of ~(k 2) and x(k 2) by a dispersion relation ,a We first note that X(k2)/k 2 cannot obey an unsubtracted relation [12], since if it did x(k2) = f dM2
1
p(M2),
J~-~ k2_M2
p(M2) ~> 0,
(13)
we would have
c-1 = _~~×'(0) = + (rl,~fjk.rk ~/k 2 + rl.r8 kak~/k2)
(I 1)
By considering the case in which the metric is both nearly flat and slowly varying, we may relate the lowenergy behavior of ~(k 2) and ×(k 2) to the parameters in eq. (3). The necessary calculation is conveniently performed in momentum space; the result is [12,13] ,2
where we have introduced
lra~(k ) = rla3 - k s k J k 2,
2 t
~(0) = x(0) = 0,
_ ~c
dM2 pfM2)< 0.
3 j M4
(14)
In perturbation theory, x ( k 2) behaves at high energies as (k2) 2 X logarithms because of the trace anomaly [ 15 ], and two subtractions are necessary **.
In terms of the two possible contractions
x(k 2) = - i f d4x eig'x(01T*O~(x)O~a(0)10>,
_X(k2)
(9a)
k2
X"(0)k 2 + f dJl//2
=x'(O)+~
~
(k2) 2 p(M2),
/1//6 k 2 _ M E
(15a) ~(k 2) + -~x(k 2) =
-i
fd4x eik'x(0l T*O #(x)Oa#(0)10>.
dM (9b)
Since × a ~ , (k) cannot have double poles if the inner product of the physical Hilbert space is positive definite, there must be a relation fi ~(0) + ~ X(0) = 0.
(15b)
Eqs. (15a) and (15b) are consistent with the observation [3,16-18] that counterterms proportional to the curvature squared must be included in the part
(10)
Furthermore, simple poles at k 2 = 0 are possible only if there are neutral massless scalars. (Single gravitons are decoupled by assumption.) It is not hard to see, however, that such poles would lead to non-local terms in the expansion (3). Therefore, we shall assume their absence and require that
28
p(M 2) = ~ .
*2 Strictly speaking, the T-product in the formulaeofref. [12] must be replaced by the T*-product as in eq. (9a), even though O~ is a scalar operator. ,3 For a general background on low-energy theorems and gravitation, see the articles by Weinberg in refs. [ 1,14]. ,4 The last subtraction must be performed away from k 2 = 0 if there are massless particles (with spin) in the matter spectrum, such as the neutrino or the photon. The modification does not affect the argument which follows.
Volume 114B, number 1
PHYSICS LETTERS
S 1 [g] of eq. (2); the divergent nature of the counterterm is reflected in the divergence of (15b), whereas the arbitrary nature of the finite part is reflected in the free parameter X"(0) of (15a). Given eqs. (15a) and (15b), it is easy to see that x(k 2) must have a zero at some k 2 = - X 2 (spacelike), since for k 2 large and negative x(k 2) is positive, whereas for k 2 small and negative, it is negative owing to ×'(0) > 0. A spacelike zero of ×(k2), however, means that the small disturbance equation around fiat spacetime (16)
f d 4 y ( 0 l T*O a(x)O.r8 (y)10)h ~n (y) = 0
will admit tachyonic solutions which are conformaUy flat , s . Specifically, if we substitute a trial solution h~ * (y) = r~"f* exp ( - i k . y ) into (16), we fred -~i(7?a~ -
k,., k Jk2)x(k 2) e -ik'x = O,
(17)
which may be satisfied i f k 2 = -X 2. We may also perform a similar analysis for ~(k2). Perturbation theory indicates two subtractions ,4
~(k2) = ~'(0) + ~"(O)k 2
k2
÷f 2 (k2)2 o(M2),
15 July 1982
in contradiction with (12a). In general, the nature of the zeroes will depend on the balance between ~'(0), ~"(0) and o(M2), except that they cannot be real negative (spacelike) if ~"(0) > 0. In the leading order of the 1IN expansion [19], the zeroes of~(k 2) and x(k 2) become poles in the gravition propagator. The spacelike zeroes are clearly undesirable; the complex zeroes, perhaps admissible ,6, but certainly unconventional [20]. Faced with this difficulty, we may entertain two interpretations as to the origin. One is that perturbation theory does not correctly indicate the high-energy behavior of ~(k 2) and x(k2); the other is that flat spacetime is not the true ground state of semiclassical gravity. If the first interpretation is true, the problem is essentially that of ordinary quantum field theory in flat spacetime; it is well known that perturbation theory often leads to tachyons in this case [21]. In particular, we may simply postulate that only one subtraction is necessary for x(k2)/k 2
x(k2) - x'(o) + f ~ 2 k2~2M2
(21)
k2_ M2
(20)
since with (21), the zeroes of x(k 2) are confined to the real interval 0 ~
,s Previous investigations [ 5,18,19 ] have been confined to the case of matter without self-couplings;hence the taehyonic mode could be avoided by taking x"(0) < 0. Some of the authors have expressed the opinion to the effect that this choice should forestall any further difficulties with x (k2); our result, however, indicates that such optimism is unjustified.
,6 Note that complex zeroes do not lead to solutions ofeq. (16) except by analytic continuation. ,7 This point is noted in ref. [13]. However, in such cases, the renormalization group itself tells us that G-1 ~ ~2 X exp (-2/bog 2) indicating the existence of non-perturbafive effects, so the argument cannot be regarded as conclusive.
e(M 2)/> 0,
(18)
and again, it is not hard to infer that ~(k 2) must have a zero in the complex k 2 plane. For, if there were no zeroes, z/~(z) would be analytic in the cut plane, and furthermore would satisfy an unsubtracted relation because of its rapid fall-off at infinity z
_
~(z)
O.
_
[The minus sign comes from Im (z/~(z)) X Im (~(z)/z).] But then, (19) leads to 1
~'(o) - f
dM 2
w(M2) > 0,
(19)
= -Iz/~(z)12
29
Volume 114B, number 1
PHYSICS LETTERS
Planck mass squared if not too many massive states contribute to their absorptive parts. In particular, although we may still work with the background-field m e t h o d [8] to incorporate higher-loop effects in gravity, our analysis for ~(k 2) and x ( k 2) breaks down: if the Einstein~Hilbert action is taken as the fundamental action, the theory is non-renormalizable, and the necessary number of subtractions grows without bound; i f R 2 type theories are chosen, positivity is lost [22]. In a sense, we are back at the beginning: a full q u a n t u m theory of gravity is more necessary than ever. This work is in an essential sense, the product of a seminar on induced gravity. I am heavily indebted to the lecturer Professor S.L. Adler and to the participants, in particular, Professor D.G. Boulware, Professor M.J. Perry, Professor E. Tomboulis, Dr. G.T. Horowitz, Dr. E. Mottola and Dr. L. Smolin. I would also like to thank Professor E. Witten for his (as usual) pertinent remarks.
References [ 1 ] S.W. Hawking and W. Israel, eds., General relativity, an Einstein centenary survey (Cambridge U.P., London, 1979). [2] Proc. Second Oxford Conf. on Quantum gravity (Oxford U.P., London), to be published. [3] R.P. Feynman, unpublished; R. Utiyama and B.S. DeWitt, J. Math. Phys. 3 (1962) 608. [4] Ya.B. Zel'dovich, JETP Lett. 6 (1967) 316,345; A.D. Sakharov, Sov. Phys. Dokl. 12 (1968) 1040. [5] G.T. Horowitz, in: Proc. Second Oxford Conf. on Quantum gravity (Oxford U.P., London), to be published; E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. 77B (1978) 262. [6] J. Schwinger, Phys. Rev. 82 (1951) 664;Proc. Nail. Acad. Sci. USA 37 (1951)452. [7] B.S. DeWitt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965). [8] D.G. Boulware, Phys. Rev. D23 (1981) 389; B.S. DeWitt, in: Proc. Second Oxford Conf. on Quantum gravity (Oxford U.P., London), to be published.
30
15 July 1982
[9] R.P. Feynman, Rev. Mod. Phys. 20 (1948) 36;Phys. Rev. 80 (1950) 440; P.T. Matthews and A. Salam, Nuovo ~mento 2 (1955) 120. [10] J. Sehwinger, Phys. Rev. 130 (1963)406, 800. [11] R. Jackiw, in: Current algebra and its applications, eds. J.J. Hopfield and A.S. wightman (Princeton U.P., Prince. ton, NJ, 1972). [12] S.L. Adler, Phys. Lett. 95B (1980) 241; A. Zee, Phys. Rev. D23 (1981) 858. [13] S.L. Adler, Rev. Mod. Phys., to be published; A. Zee, Phys. Lett. 109B (1982) 183. [14] S. Deser and K. Ford, eds., Lectures on Particles and field theory, 1964 Brandeis Summer Institute in Theoreilcal Physics (Prentice Hall, Englewood Cliffs, NJ, 1965); M. Grisaru and H. Pendleton, eds., Lectures on Elementary particles and quantum field theory, 1970 Brandeis Summer Institute in Theoretical Physics (MIT Press, Cambridge, MA, 1971). [15] S.L. Adler, J.C. Collins and A. Duncan, Phys. Rev. D15 (1977) 1712; N.K. Nielsen, Nucl. Phys. B120 (1977) 212; J.C. Collins, A. Duncan and S.D. Joglekar, Phys. Rev. D16 (1977) 438. [16] G. 't Hooft and M.T. Veltman, Ann. Inst. H. Poincard 20 (1974) 69. [17] M.R. Brown, Nucl. Phys. B56 (1973) 194; D.M. Capper, G. Leibbrandt and M. Ram~n Medrano, Phys. Rev. D8 (1973) 4320. [18] D.M. Capper and M.J. Duff, Nucl. Phys. B56 (1973) 194; S. Deser and P. van Nieuwenhuizen, Phys. Rev. D10 (1974) 401,411. [19] E. Tomboulis, Phys. Lett. 70B (1977) 361;97B (1980) 77; J.B. Hartle and G.T. Horowitz, Phys. Rev. D24 (1981) 257; B.S. Kay, Phys. Lett. 101B (1981) 241. [20] T.D. Lee and G.C. Wick, Nucl. Phys. B9 (1969) 209; T.D. Lee, in: Quanta, eds. P.G.O. Freund, C.J. Goebel and Y. Nambu (Chicago U.P., Chicago, 1970). [21] G. Barton, Introduction to advanced field theory (Interscience, New York, 1963). [22] A. Pais and G.E. Uhlenbeck, Phys. Rev. 79 (1950) 145.
PHYSICS LETTERS
15 July 1982
INSTABILITY OF FLAT SPACETIME IN SEMICLASSICAL GRAVITY Hidenaga YAMAGISHI Department o f Physics, Princeton University, Princeton, NJ 08540, USA Received 22 February 1982 Revised manuscript received 1 April 1982
The semiclassical theory of gravity is considered in which an asymptotically flat background m e t r e is coupled to quantized matter. We show that, in general, there are modes with spacelike wave vectors for small metric fluctuations around flat spacefime. Besides the usual axioms of quantum field theory in flat spaeetime, the proof rests on the existence of a hard trace anomaly in the energy-momentum tensor due to matter self-couplings. Two possible interpretations of the result are discussed.
In view of the difficulty of obtaining a full quantum theory of gravity [1,2], many authors have considered the so-called semiclassical (or one-loop) approximation in which a classical background metric is coupled to quantized matter [ 1 - 4 ] . For the case of a background black-hole metric, the approximation has revealed a remarkable connection between black holes and thermodynamics [1]. However, modes with spacelike wave vectors have also been encountered for small metric fluctuations around flat spacetime [5]. In this letter, we shall show that such modes are a general feature of the semiclassical approximation. The quantity of interest is the vacuum persistence amplitude [6] for quantized matter in a background metric gab [7,8] eit[g] -------(0+10_) g.
(1)
The metric gab we take to be asymptotically fiat, so that the in and out vacuum states 10_+)can be def'med. In quantized matter, we may also include a gravitational part hab if it couples only to the classical metric gab and not to the rest of the matter fields ¢ or to itself. In other words, the path-integral representation [9] of (1) must be of the form e irig]
as
rig] = A f d a x v ~ +
l~-~fdaxRvrZ--g
- a f d4x(RabRab - IR2)Vr~ +b f d4xREx/'----g+ ....
(3)
which defines the cosmological constant A and the gravitational constant G in the semiclassical approximation. In view of the empirical evidence, we shall take them to be A = 0 and G > 0 (attractive). With zero cosmological constant, flat spacetime is a stationary point for the effective action, and we may also expand F [g] around the usual Monkowski metric
F ['r/+ h] = 8
f
d4y(OI
r*e a(x)%6 0,)1o:,
X he#(x)h ~6(y) + O(ha).
=f [de] [dh] exp (iS 1 [g]
+ iS 2 [¢,g] + iS 3 [h,g] ),
with S 3 quadratic (one-loop) in hab. We shall also assume that gauge fixing is done by the backgroundfield method [8] so that the effective action F[g] is invariant under general coordinate transformations. For slowly varying metrics, we may expand P[g]
(2)
0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
(4)
,1 The signature is + - - - . Greek indices are raised or lowered with n-
27
Volume 114B, number 1
PHYSICS LETTERS
In eq. (4), O ~ is the (renormalized) energy-momentum tensor in fiat spacetime; the T*-product indicates that seagulls must be added because of the strong singularity in the equal-time commutator [ 10,11 ] [O00(x, t), O00 (y; t)] = i [O0k(X , t) + O0k(Y, t)] axkS(x - - y ) .
(5)
Since all quantities in eq. (4) refer to fiat spacetime, we may conveniently introduce the Fourier transform
× ~ 6 (k) -=-- i f d 4 x eik'x(01T*O #(x)O n (0)10). (6) The Ward identities fix the tensor structure of ×a~a (k) to be
Xa376 (k) = } [rta~,v~ (k) + ~ lra3(k )rcy~ (k)] ~(k 2) + -~7ra~(k) lr r~ (k)x(k2),
(7)
15 July 1982
(8a)
rs~ (0) + -~ x'(0) = 0.
G - 1 = 47r×'(0)/3 = -8rr~'(0)/5, a = ~"(0)/10,
- ~O?a,k~ka/k2 + 3 perm.).
(85)
(12a)
b =-X"(0)/72.
(12b,c)
The crux of this paper is that we can translate the low-energy theorem (12a) into a statement about the zeroes of ~(k 2) and x(k 2) by a dispersion relation ,a We first note that X(k2)/k 2 cannot obey an unsubtracted relation [12], since if it did x(k2) = f dM2
1
p(M2),
J~-~ k2_M2
p(M2) ~> 0,
(13)
we would have
c-1 = _~~×'(0) = + (rl,~fjk.rk ~/k 2 + rl.r8 kak~/k2)
(I 1)
By considering the case in which the metric is both nearly flat and slowly varying, we may relate the lowenergy behavior of ~(k 2) and ×(k 2) to the parameters in eq. (3). The necessary calculation is conveniently performed in momentum space; the result is [12,13] ,2
where we have introduced
lra~(k ) = rla3 - k s k J k 2,
2 t
~(0) = x(0) = 0,
_ ~c
dM2 pfM2)< 0.
3 j M4
(14)
In perturbation theory, x ( k 2) behaves at high energies as (k2) 2 X logarithms because of the trace anomaly [ 15 ], and two subtractions are necessary **.
In terms of the two possible contractions
x(k 2) = - i f d4x eig'x(01T*O~(x)O~a(0)10>,
_X(k2)
(9a)
k2
X"(0)k 2 + f dJl//2
=x'(O)+~
~
(k2) 2 p(M2),
/1//6 k 2 _ M E
(15a) ~(k 2) + -~x(k 2) =
-i
fd4x eik'x(0l T*O #(x)Oa#(0)10>.
dM (9b)
Since × a ~ , (k) cannot have double poles if the inner product of the physical Hilbert space is positive definite, there must be a relation fi ~(0) + ~ X(0) = 0.
(15b)
Eqs. (15a) and (15b) are consistent with the observation [3,16-18] that counterterms proportional to the curvature squared must be included in the part
(10)
Furthermore, simple poles at k 2 = 0 are possible only if there are neutral massless scalars. (Single gravitons are decoupled by assumption.) It is not hard to see, however, that such poles would lead to non-local terms in the expansion (3). Therefore, we shall assume their absence and require that
28
p(M 2) = ~ .
*2 Strictly speaking, the T-product in the formulaeofref. [12] must be replaced by the T*-product as in eq. (9a), even though O~ is a scalar operator. ,3 For a general background on low-energy theorems and gravitation, see the articles by Weinberg in refs. [ 1,14]. ,4 The last subtraction must be performed away from k 2 = 0 if there are massless particles (with spin) in the matter spectrum, such as the neutrino or the photon. The modification does not affect the argument which follows.
Volume 114B, number 1
PHYSICS LETTERS
S 1 [g] of eq. (2); the divergent nature of the counterterm is reflected in the divergence of (15b), whereas the arbitrary nature of the finite part is reflected in the free parameter X"(0) of (15a). Given eqs. (15a) and (15b), it is easy to see that x(k 2) must have a zero at some k 2 = - X 2 (spacelike), since for k 2 large and negative x(k 2) is positive, whereas for k 2 small and negative, it is negative owing to ×'(0) > 0. A spacelike zero of ×(k2), however, means that the small disturbance equation around fiat spacetime (16)
f d 4 y ( 0 l T*O a(x)O.r8 (y)10)h ~n (y) = 0
will admit tachyonic solutions which are conformaUy flat , s . Specifically, if we substitute a trial solution h~ * (y) = r~"f* exp ( - i k . y ) into (16), we fred -~i(7?a~ -
k,., k Jk2)x(k 2) e -ik'x = O,
(17)
which may be satisfied i f k 2 = -X 2. We may also perform a similar analysis for ~(k2). Perturbation theory indicates two subtractions ,4
~(k2) = ~'(0) + ~"(O)k 2
k2
÷f 2 (k2)2 o(M2),
15 July 1982
in contradiction with (12a). In general, the nature of the zeroes will depend on the balance between ~'(0), ~"(0) and o(M2), except that they cannot be real negative (spacelike) if ~"(0) > 0. In the leading order of the 1IN expansion [19], the zeroes of~(k 2) and x(k 2) become poles in the gravition propagator. The spacelike zeroes are clearly undesirable; the complex zeroes, perhaps admissible ,6, but certainly unconventional [20]. Faced with this difficulty, we may entertain two interpretations as to the origin. One is that perturbation theory does not correctly indicate the high-energy behavior of ~(k 2) and x(k2); the other is that flat spacetime is not the true ground state of semiclassical gravity. If the first interpretation is true, the problem is essentially that of ordinary quantum field theory in flat spacetime; it is well known that perturbation theory often leads to tachyons in this case [21]. In particular, we may simply postulate that only one subtraction is necessary for x(k2)/k 2
x(k2) - x'(o) + f ~ 2 k2~2M2
(21)
k2_ M2
(20)
since with (21), the zeroes of x(k 2) are confined to the real interval 0 ~
,s Previous investigations [ 5,18,19 ] have been confined to the case of matter without self-couplings;hence the taehyonic mode could be avoided by taking x"(0) < 0. Some of the authors have expressed the opinion to the effect that this choice should forestall any further difficulties with x (k2); our result, however, indicates that such optimism is unjustified.
,6 Note that complex zeroes do not lead to solutions ofeq. (16) except by analytic continuation. ,7 This point is noted in ref. [13]. However, in such cases, the renormalization group itself tells us that G-1 ~ ~2 X exp (-2/bog 2) indicating the existence of non-perturbafive effects, so the argument cannot be regarded as conclusive.
e(M 2)/> 0,
(18)
and again, it is not hard to infer that ~(k 2) must have a zero in the complex k 2 plane. For, if there were no zeroes, z/~(z) would be analytic in the cut plane, and furthermore would satisfy an unsubtracted relation because of its rapid fall-off at infinity z
_
~(z)
O.
_
[The minus sign comes from Im (z/~(z)) X Im (~(z)/z).] But then, (19) leads to 1
~'(o) - f
dM 2
w(M2) > 0,
(19)
= -Iz/~(z)12
29
Volume 114B, number 1
PHYSICS LETTERS
Planck mass squared if not too many massive states contribute to their absorptive parts. In particular, although we may still work with the background-field m e t h o d [8] to incorporate higher-loop effects in gravity, our analysis for ~(k 2) and x ( k 2) breaks down: if the Einstein~Hilbert action is taken as the fundamental action, the theory is non-renormalizable, and the necessary number of subtractions grows without bound; i f R 2 type theories are chosen, positivity is lost [22]. In a sense, we are back at the beginning: a full q u a n t u m theory of gravity is more necessary than ever. This work is in an essential sense, the product of a seminar on induced gravity. I am heavily indebted to the lecturer Professor S.L. Adler and to the participants, in particular, Professor D.G. Boulware, Professor M.J. Perry, Professor E. Tomboulis, Dr. G.T. Horowitz, Dr. E. Mottola and Dr. L. Smolin. I would also like to thank Professor E. Witten for his (as usual) pertinent remarks.
References [ 1 ] S.W. Hawking and W. Israel, eds., General relativity, an Einstein centenary survey (Cambridge U.P., London, 1979). [2] Proc. Second Oxford Conf. on Quantum gravity (Oxford U.P., London), to be published. [3] R.P. Feynman, unpublished; R. Utiyama and B.S. DeWitt, J. Math. Phys. 3 (1962) 608. [4] Ya.B. Zel'dovich, JETP Lett. 6 (1967) 316,345; A.D. Sakharov, Sov. Phys. Dokl. 12 (1968) 1040. [5] G.T. Horowitz, in: Proc. Second Oxford Conf. on Quantum gravity (Oxford U.P., London), to be published; E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. 77B (1978) 262. [6] J. Schwinger, Phys. Rev. 82 (1951) 664;Proc. Nail. Acad. Sci. USA 37 (1951)452. [7] B.S. DeWitt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965). [8] D.G. Boulware, Phys. Rev. D23 (1981) 389; B.S. DeWitt, in: Proc. Second Oxford Conf. on Quantum gravity (Oxford U.P., London), to be published.
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