Spontaneous Compactification of Extra Dimensions in Eleven-Dimensional Quantum Gravity

Fortschr. Phys. 34 (1986) 7, 441-455

Spontaneous Compactification of Extra Dimensions in Eleven-Dimensional Quantum Gravity')

NGUYEN VAN HIEU~)

International Centre for Theoretical Physics, Trieste, Italy

Abstract

The reduction of thc eleven-dimensional pure gravity theory to rz field theory in the four-dimensional Minkowski space-time by means of the spontaneous compactification of the extra dimensions is investigated. The contribution of the quantum fluctuations of the eleven-dimensional second rank symmetric tensor field to the curvatures oE the space-time and the compactified space of the extra dimensions are calculated in the one-loop approximations. It is shown that there exist the values of the cosmological constant such that the resulting four-dimensional theory is self-consistent.

1. Introduction

The promising success of the N = 8 extended supergravity [1--41 derived from the N = 1 supergravity in eleven dimensions by means of the dimensional reduction in explaining many observed experimental data concerning the quark-lepton spectrum and its symmetry properties has revived the old idea of KALUZA [5] and KLEIN [6] that an unified theory of the gravitation and other matter fields in four dimensions might be deduced from a gravity (or supergravity) theory in higher dimensions. I n many models of the Kaluza-Klein type the extra dimensions are spontaneously compactified [7- 161. However, in the existing eleven-dimensional supergravity theory [l-41 the spontaneous compactification takes place a t the classical leven if and only if the four-dimensional space-time is an anti-de Sitter one [17--321. To deduce a field theory in the four-dimensional Minkowski space-time from the eleven-dimensional supergravity (or gravity) by means of the spontaneous compactification of the extra dimensions it is necessary either to introduce the scalar fields (at the classical level) or to include the contribution of the quantum fluctuations [33-37].

The dynamical quantum effect in the Kaluza-Klein was studied in many works [35 to 511. In particular, CANDELAS and WEINBERG [45] have proposed a method for calcnlat- ing the contribution of the quantum fluctuations of the scalar fields to the curvatures of the four-dimensional space-time as well as of the spontaneously compactified space of the internal symmetry. CHODOS and MYERS [47] noted that the quantum fluctuations of the multidimensional gravitational symmetric second-rak tensor field themselves must

Permanent address: Institute of Physics, Academy of Sciences of Vietnam, Nghia Do-Tu Liem, Hanoi, Vietnam.

442 NGUYEN VAN HIEU, Spontaneous Compactification

also contribute to the curvatures. Generalizing the reasonings of CANDELAS and WEIN- BERG they have calculated the effective potential of this field in the one loop approximation. The one loop effective action of the gravitational field in higher dimensions was also studied by RANDJBAR-DAEMI and SARMADI [48, 491. However, the contribution of the quantum fluctuations of the multidimensional gravitational field to the curvatures was not considered. The study of this problem is carried out in the present work which would be the necessary beginning of the investigation of the spontaneous compactification of the eleven-dimensional quantum supergravity. We show that in the eleven- dimensional pure gravity theory with the spontaneous compactification to the direect product M 4 x 8, of the Minkowski space-time M , and the seven sphere S7 the cosmological constant might be chosen in such a way that the generalized Einstein equations including the contribution of the quantum fluctuations are satisfied on the one hand, and there exist no tachyons on the other hand. For simplificity, we work in the light cone gauge which has been used earlier by RANDJBAR-DAEMI, SALAM and STRATHDEE [50]. The problem of the gauge dependence of the effective action [51-571 will also be considered in the sequel. We use the unit system with k = c = 1.

2. Basic equations

We consider the second rank symmetric tensor field GAB of the eleven-dimensional pure gravity theory, where A , B . . . label the coordinates in eleven dimensions, and assume the following action

. P

The appearance of the cosmological constant 1 might be a consequence of the renormali- zation,as it was noted by RANDJBAR-DAEMI, SLLZIAM and STRATHDEE [50]. CANDELAS and WEINBERG [45] showed that it is necessary to introduce this cosmological constant in order to satisfy the Einstein equation. In our notations

1 rAcB = 9 G C D { a . @ B D + det G = det GAB.

- ~ D G A B } ,

I?or studying the quantum fluctuations we split GAB into two parts: the background classical gravitational field g A B and the fluctuating part %hAB

G A B = g d B + XhAB. (2)

The action S,,[g, h] and the Lagrangian U(y, h) of the fluctuating field h.lB in the background gAB are defined by the relations

Fortschr. Phys. 34 (1986) 7 443

CHODOS and MYERS [47] noted that in general

i.e. the background field g A B might not be the solution of the classical field equation. Therefore the Lagrangian U ( g , h ) must contain the terms of the first order in the fluctuating field hAB subjected to the quantization. The terms of the second order in hAB are known and may be found in Refs. [13, 50, 581. In the calculation of the effective potential the first order terms play no role, but in the study of the contribution of the quantum fluctuations to the curvatures they are essential and must be retained, as me shall see in the sequel.

equals In the one-loop approximation the effective potential of the background field

~ e f f b l = X[gl + a917 ( 5 )

Z[g] = -i In [dh] eiS.=[g.hl, (6)

$ [dhl where

denotes the path integral over the quantum field hAB in eleven dimensions. The generalized Einstein equation including the contribution of the quantum fluctuations is the field equation derived from the effective action X,ff[g, h]

I t may be written in the form

The last term in the right-hand side of Eq. (8) is the vacuum expectation value of the functional derivative of the quantum action

We consider the case when the eleven-dimensional space is spontaneously compactified into the direct product M4 x X, of the Minkowski space-time M4 and the seven sphere S , and denote the metric tensor in M4 and S7 by gab and Sup, resp. The indices a, b . . . label the coordinates of the Minkowski space-time M 4 , and the indices a, @ . . . label those of the seven sphere X,. From the generalked Einstein equations (8) in X, and M4, resp., we obtain two following relations


Instead of the last terms in Eqs. (11) and (12) for our purpose it is convenient to introduce their integrals over the space M , x 8,

From the definition (4) it follows that the functional derivative of the quantum action may be written in the form

Therefore we have

I-

(17)

s 7 d 7 " g l = - (~ ,u [g , hl) + dl'z 1/ ldet gl ( S ' f l K S ) >

0'4"gl = -2(Squ[g, hl) + J dllx l/m (gabJfab).

The expressions of the last terms in the right-hand side of Eqs. (16) and (17) will be given in the next section.

3. Light-Cone Gauge

For simplicity, we work in the light-conegauge, since in this gauge the unphysical modes and the Fadeev-Popov ghosts do not arise. The second order terms of the Lagrangian Y(g, h) have been calculated in this gauge by RANDIJBAR-DAEMI, SALAM and STRATHDEE 501. Remember that in the seven sphere AS, of the radius a

1 a2 Rapya = -- (quagsv - s.,sas) *

Consider the four-dimensional Lagrangian

2%) = I d7x ildet gU~l y(g, h ) . S?

I n the notations of Ref. [50] we have

Fortschr. Phys. 34 (1986) 7 445

where the fields h8) , h$) . . . H(n) in the Minkowski- space-time are the coefficients of the Fourier expansions of the corresponding fields hii, hii . . . H in the eleven-dimensional space in terms of the spherical harmonics in seven dimensions, 9, is the volume of the compact seven sphere, and

(x = h2.

From the expression (20) it follows that there will be no tachyons if (Y < 34.

first order terms in the right-hand side of Eq. (19) by setting In order to carry out the calculation of the path integrals it is necessary to avoid the

56

We have then

where €(2)(h) is obtained from the right-hand side of Eq. (20) after the substitution

hi!) 3 &j!).

In general we can write

Therefore we have

The calculation of the constant Z will be carried out in the next section. Note that if we omit the first order term in the right-hand side of Eq. (19) then in the

formula (24) we would loss the first terM in its right-hand side. This means that in our study it is essential to retain the first order term in the Lagrangian, as it has been done above.

In the second order approximation with respect to the field hAB the expression of the operator MAB in the right-hand side of Eq. (15) contains both the first and second order terms. It is straightforward to derive this expression and then to calculate the last terms in the right-hand side of Eqs. (16) and (17) in the light-cone gauge. We obtain following results :

J dllx mzii (s@M&7, 4)

4 Fortschr. Phys. 34 (1986) 7

446 NQUYEN VAN HIEU, Spontaneous Compactification

Fortschr. Phys. 34 (1986) 7 447

From the expressions (19)-(22) determining the Lagrangian L(h) it follows that

To simplify the calculation of the expectation values in right-hand side of Eqs. (34) and (35) it is essential to note that the operators R(h, 5.) and Q(h, 5.) at 8 = 1 coincide with the four-dimensional Lagrangian

R(h, 1) = &(h, 1) = L(h) . (36)

Therefore we have

To carry out the calculation of the path integrals in the right-hand side of Eqs. (37) and (38) we must avoid the first order terms in the expressions (27) and (28) of the operators R(h, 5.) and Q(h, 6) by rewritting

where a@@, 8) or gC2)(h, 5.) is obtained from the right-hand side of Eq. (29) or (31) by means of the substitution

448

(for &"(2)(h, l )) , respectively. Then instead of Eqs. (34) and (35) we have

NQUYEN VAN HIEU, Spontaneous Compactification

J dllz mGl (g"f lNt , (g , h)) 1 28(18-&) 1 112(286 - 2m)

= dllz -- { - - Vldet g1 x2u2 110 - 3& x2a2 (110 - 3&)2

4. Calculation of the Path Integrals

The Lagrangian f ; (2)(h) and the similary operators E ) , Q(2)(h, t) in the path integrals (23), (45) and (46) are the sums each term of which contains only one definite in- dependent field operator. Therefore in calculating these path integrals each field can be considered separately. As an example we investigate in details the contribution of the fields H(n). Those of the other ones can be calculated similarly. Thus we limit ourselves in the special case where in the expressions (20), (29) and (31) we retain only the last sums containing H("), and denote them by L,(H), R,(H, 5 ) and &,(H, f ) :

First we consider the path integral

s dl'z fldet gl Wo(&) = i In / [dh] e i a / d ' z L o ( H )

which determines the expectation value

2, s dllz I/ldet g1 = (I d4zL,(H)) = - Wo'( 1) s dllz fldet 91, (52)

as in Eq. (23). It has been given in many papers [45, 47, 59-61] W

d4k Dn(2) In [a(k2 + Mn2 - iO)], (53) 2(244 Q, Wo(") = -

Fortschr. Phya. 34 (1986) 7 449

where Dn@) denotes the degree of the degeneracy of the eigenvalue Mn2. To avoid the divergence of the integral in Eq. (53) we apply the [-function regularization [62,63] and dimensional regularization [64, 651 methods. For this purpose let us introduce a function of two variables s and 01

The comparison of this definition with the expression (53) gives

Wo(.) =

Therefore

2, = - 1 = lim {co(s) + sci(s)}, a2'~(s,

asaa a=l , s=O s-+o

(55)

where coo(4 = i'a(s, 1). (571

From the results of Refs. [45, 471 it follows that in Y dimensions

In the spirit of the dimensional regularization method we apply the reasonings of Refs. [45, 461 and calculate the sum in the right-hand side of Eq. (58) a t the values of v such that it converges and determines an analytical function of v. The values of the regularized functions c0(s) and c0'(s) in four dimensions are obtained by means of the analytical continuation to the point v = 4. In Refs. [45, 473 it was shown that the regularized value of CO'(O) is finite, as we shall also see in the sequel. Therefore cO(O) vanishes, and we have

2, = o . (60) Applying the same reasonings to the path integral in Eq. (23) with the Lagrangian L@)(h) of the form (20), we also obtain

z = o . (61) By the analogy with the effective potential W,(a) determined by Eq. (51) and corre-

sponding to the Lagrangian aLo(H), we define the functions V,(E) and U0(Q as the extended effectiye potentials corresponding to the extended Lagrangians R,(H, 6) and &OW, E )

I d % I/ldet g / V,(t) = i In [dH] e i J d ' r R o ( H J ) , (62)

J dllx I/jiC&jj u,(E) = i ln J HI eiJd'rQo(H,E). (63) In order to determine V,(&) and Uo(E) we can apply the same reasonings as we have

done in the evaluation of the the path integral (51). In v dimensions we obtain

450 NGUYER VAN HIEU, Spontaneous Compactification

It is easy to generalize this result and derive similary expressions for the extended effective potentials V(E) and U(E) defined in Eqs. (45) and (46):

1 T(-V/2) 1 2 (4n)+ 8, a” n = O

V(E) = -- - ( 3 ,f Dn(0)(E[n(n + 6) + 421 - 01)y/~

m

W + z D,,(O)(E[n(n + 6 ) + 181 - 01)y” n = 2

m + 2 2 D,(O)(E[n(n + 6) + 301 -

+ 2 z Dn(l)(E[n(n + 6) + 351 -

n = 1

8

n = l

W

n(n + 6) + 42 - a v / z

t 1 U(E) = --

n(n + 6) + 30p(E) - 01 d z

E 1 5 1

+ .E Dn(O) [ + f Dn(0) [

n = O

n(n + 6) + 18 - OL y/2

n=2

n(n + 6) + 30 - a E

W + 2 ,Z Dn(0) [ n = l

n(n + 6) + 35 - 01 “I2

E I + 2 5 D,(l) [ n = l

Her& Dn(0), Dn(l) and Dn(2) denote the degrees of the degeneracy of the corresponding scalar, transverse vector and second rank transverse traceless symmetric tensor spherical harmonics in 8,. They were given in Refs. [47,66,67]. CHODOS and MYERS have shown also that all the sums over n in the right-hand side of Eqs. (66) and (67) niay be suh- stituted by the corresponding sums taken from the common lowest value n = -1 without changing the whole expressions of the extended effective potentials V(6) and U(6). Applying the method of CANDELAS and WEINBERG we transform each sum in the right-hand side of the modified expressions (66) and (67) into an integral which can be calculated with the help of the residue theorem a t the unphysical values of v such that this

Fortschr. Phys. 34 (1986) 7 451

integral is covergent. After the analytical continuation to the physical value v = 4 we obtain

(68) 1 1

Vt(1) = -. A(7’(a), U‘(1) = - A(4)(&), Q7a4 07a4

where

7

(70) 280

i = l 27 A(4’(.) = -4 qJ&) - - yz(a).

The functions yi(a) -and yi(a), i = 1,2, . . . 7 are finite. They will be given in the Ap- pendix. The finiteness of the expression (58) can be etablished in the same way.

5. ExiBtence of Solutions

Prom the definitions (13), (14) and the relations (16), (17), (23), (43), (44), (61) and (68) we can derive the expressioQs of the last terms in the left-hand side of Eqs. (ll), (12) and then rewrite these relations in the form

where

(72) 28(18 - a) 112(285 - 2a) - - 7 7 168

B(7)(a) = - (a - 30) - - 2 2 110- 3n 110 - 301 (110 - 3a)2 ’

28(42 - 01) 784(30 - a) - 168 110- 301 + 110 - 3a (110 - 3 4 2 ’ B(4)(01) = 2(01 - 42) - 2

In particular, the parameter 01 determining the cosmological constant

for a given value of the radius a must satisfy following equation

B ( 7 ) ( ~ ) - A(’)(a) B(4)(a) A(4)(a)

(73)

(75)

Using the expressions in the Appendix and the relations (69), (70), (72), (73) we have proved that the algebraic equation (75) has many rootssatisfying the condition a 5 34. For each of these roots the ratio %,/a must take the corresponding value determined by Eq. (71).

6. Conclusion

We have shown that it is possible to construct a self-consistent theory of the pure gravity in the eleven-dimensional space spontaneously compactified into the direct product M4 X S, of the Minkowski space-time M , and the sevensphere S,. The product a of the

452

cosmological constant il and the square of the radius a of S, can be chosen in such a way that the generalized Einstein equations (including the contribution of the quantum fluctuations) are satisfied on the one hand, and there are no tachyons on the other hand. For each admissible value of a the radius a of S, is completely determined if the constant x is given.

In the eleven-dimensional supergravity beside of the metric tensor G A ~ (or the corresponding vielbein) there are other matter fields. The quantum fluctuations of the latter must also contribute to the curvatures and change the admissible values of the parameter LY. The spontaneous compactification of the eleven-dimensional supergravity will be studied in a subsequent work.

NGUYEN VAN HIEU, Spontaneous Compactification

Acknowledgements

This work has been initiated during the stay of the author a t the International Centre for Theo- retical Physics, Trieste. The author expresses his sincere appreciation tq Professor Abdus Salam for his interest in the work. He would also like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality a t the International Centre for Theoretical Physics, Trieste.

Appendix

Setting

where J J z ) and I,,(%) are the Bessel functions, we have following relations

Fortschr. Phys. 34 (1986) 7 453

and the similar expressions of yi(a), i = 1,2, , . ., 7 in terms of six functions F\;)(j3), i = 1,2, 3, c = &. Residue calculations give

1 6!

f 6x2(85x4 f 2 0 5 ~ ~ + 52) [J*)(x) + 3Os(73x4 f 110x2 + 12) ca(*)(z)

P&'(z) = - {8x4(x4 f 5x2 + 4) Ca'"(z) f l2x3(7x4 f 25x2 + 12) [ 4 ' * ' ( ~ )

+ 45 (155x4 f 1 3 0 ~ ~ + 4) C7(')(x) - 1 2 6 0 ~ ( f 1 3 ~ ~ + 5 ) C~ ' * ' (Z )

+ 630(f43x2 + 5 ) Cs(*)(x) + 28350xC:i)(x) + 14175C$:)(x)},

1 6!

Pi;'(%) = - {+4%3(x4 f 5x2 + 4) C2(*'(Z) & 2z2(13x4 f 459 + 20) C~(* ) (Z)

+ 12x1924 f 202' + 4) r4{*)(x) + 6(55x4 & 70x2 + 4) Cs'*)(x)

+ 150x(f5x2 + 1) Ca")(z) + 45(f27x2 + 5 ) C7'*'(x)

+ 1260&,(*~(x) + 63OC,"'(x)],

1 5 ! F ~ ) ( x ) = - { 8 ~ 4 ( ~ 4 f 10x2 + 9) ( 3 ( * ) ( x ) f 1 2 ~ ' ( 7 ~ ~ f 50x2 + 27) C ~ ' " ' ( S )

f 6x2(85x* f 410x2 + 117) C5(*)(x) + 30x(73x4 f 220x2 + 27) &(*)(x)

+ 45(155x4 & 260x2 + 9) C,(*)(x) + 1260x(f13s2 + 10) [s'*)(z)

+ 630(f43x2 + 10) C9(*)(x) + 2835Ox[$)(x) + 14175[::'(x)},

+ 12x(9x4 f 40x2 + 9) C4(*)(x) + 6(55x4 f 1 4 0 ~ ~ + 9) C5(*)(x)

+ 300x(f25x2 + 3) Ca'*)(z) + 45 (f27x2 + 10) r7(*)(x)

+ 1260x&,(*)(x) + 630(,'*)(x)},

1 36

f 6x2(85x4 f 697%' + 208) C5(*)(x) + 30x(73x4 f 374x2 + 48)

+ 45( 155x4 f 442x2 + 16) [,(*)(x) + 12604 f 132' + 17) C8(*)(x)

+ 630(f43x2 + 17) c9(')(z) + 28350x[\i'(s) + 14175[::'(x)},

1 36

F~:'(z) = - {8x4(x4 f 17x2 + 10) Ca'*'(z) f 12x3(7x4 f 85x2 + 48) C ~ ( * ) ( X )

l i ' $ ~ ) ( ~ ) = - {&4x3(2' 5 1'7%' + 16) c:*'(%) f 2z2(13x4 f 153%' + 80) Cii)(x)


References

[l] E. CREMMER, B. JULIA, Nucl. Phys. B 169 (1979), 141. [2] P. VAN NIEUWENHUIZEN, Phys. Rep. 68 c (1981) 189. 131 V. I. OGIEVETSKI, E. SORATCHEV, Proc. XV International School on High Energy Physics,

[4] B. DE WIT, H. ~ICOLAI, Nucl. Phys. B 208 (1983) 323. [5] TH. KALUZA, Sitzungsberichte der Preuss. Akad. Wiss., 1921, p. 966. [6] 0. KLEIN, Nature 118 (1926) 516, 7. Phys. 37 (1926) 895. [7] E. CREMMER, J. SCHERK, Nucl. Phys. B 108 (1976), 409; B 118 (1977) 68. [8] J. F. LUCIANI, Nncl. Phys. B 135 (1978) 111. [9] Z. HORVATH, L. PALLA, E. CREMMER, J. SCHERK, Nucl. Phys. B 127 (1977) 57.

Dubna, 1981.

[iO] Z. HORVATH, L. PALLA, Nuci. Phys. B 142 (1978) 327. [ill E. WITTEN, Nucl. Phys. B 186 (1981) 412.

[13] S. RANDIJBAR-DAEMI, A. SALAM, J. STRATHDEE, Nucl. Phys. B 214 (1983) 491. [I21 S. RANDJBAR-DAEMI, R. PERCACCI, Phys. Lett. 117 B (1982) 41.

[14] s. RANDJBAR-DAEMI, A. SALAM, J. STRATHDEE, Nud. Phys. B 242 (1984) 447; Phys. Lett.

[15] A. SALAM, Proc. XXII. International Conf. on High Energy Physics, Leipzig, 1984. [IS] P. H. FRAMPTON, P. J. MOXHAY, K. \TAIIIAMOTO, Phys. Lett. 144 B (1984) 354. [17] P. G. 0. FREUND, A. RUBIN, Phys. Lett. 97 B (1980) 233. [18] F. ENGLERT, Phys. Lett. 119 B (1982) 339. [19] N. BIRAN, F. ENGLERT, B. DE WIT, H. NICOLAI, Phys. Lett. 124 B (1983) 45. [20] F. ENOLERT, M. ROOMAN, P. SPINDEL, Phys. Lett. 127 B (1983) 47; 130 B (1983) 50. [21] M. A. AWADA, M. J. DUFF, C. N. POPE, Phys. Rev. Lett. 60 (1983) 294. [22] M. J. DUFF, B. E. W. NILSSON, C. N. POPE, Phys. Rev. Lett. 60 (1983) 2043; Nucl. Phys.

[23] B. BIRAN, A. CASHER, E. ENGLERT, &I. ROOMAN, P. SPINDEL, Phys. Lett. 134 B (1984) 179. [24] F. ENGLERT, H. NICOLAI, Proc. International Colloquium on Group Theoretical Methods in

[25] D. Z. FREEDMAN, H. NICOLAI, Nucl. Phys. B 237 (1984) 342. [26] L. CASTELLANI, N. P. WARNER, Phys. Lett. 130 B (1983) 47. [27] L. CASTELLANI, L. J. ROOMANS, N. P. WARNER, Nucl. Phys. B 241 (1984) 429. [28] A. CASHER, F. ENGLERT, H. NICOLAI, M. ROOMAN, Nucl. Phys. B 243 (1984) 173.

[30] L. CASTELLANI, R. D’AURIA, P. FRE, Nucl. Phys. B 239 (1984) 610. [31] L. CASTELLANI, R. D’AURIA, P. FRB, K. PILCH, P. VAN NIEUWENHUIZEN, Class. Quant. Gra-

[32] D. N. PAGE, Phys. Rev. D 28 (1983) 2976. [33] M. GELL-MANN, B. ZWIEBACH, Phys. Lett. 141 B (1984) 333. [34] J. M. GERARD, J. E. KIM, H. P. NILLES, Phys. Lett. 144 B (1984) 203. [35] M. J. DUFF, D. J. TOMS, Proc. Second Europhysics Study Conf. on Unification of the Fun-

[36] C. A. ORZALESI, G. VENTURI, Phys. Lett. 130 B (1984) 357. [37] M. J. DUFF, C. A. ORZALESI, Phys. Lstt. 122 B (1983) 37. [38] C. A. ORZALESI, Phys. Lett. 140 B (1984) 39. [39] T. APPELQUIST, A. CHODOS, Phys. Rev. D 28 (1983) 772.

124 B (1983) 345.

B 233 (1984) 433.

Physics, Trieste, 1983.

[29] F. A. BAIS, a. NICOLAI, P. VAN NIEUWENHUIZEN, Nucl. Phys. B 228 (1983) 333.

vity 1 (1984) 339.

damental Interactions, Erice, Silky, 1981.

Fortschr. Phys. 34 (1986) 7 455

[40] G. GILBERT, B. MCCLAIN, M. A. RUBIN, Phys. Lett. 142 B (1984) 28. 1411 S. MACDOWELL, M. RAKOWSKI, Phys. Lett. 144B (1984) 351. [421 G. GILBERT, B. MCCLAIN, Nucl. Phys. B 244 (1984) 173. [43] G. W. GIBBONS, H. NICOLAI, Phys. Lett. 143 B (1984) 108. [44] T. INAMI, K. YAMAGISHI, Phys. Lett. 143 B (1984) 115. [45] P. CANDELAS, S. WEINBERG, Nucl. Phys. B 237 (1984) 397. [46] P. CANDELAS, D. J. RAINE, Nucl. Phys. B 243 (1984) 157. [47] A. CHODOS, E. MYERS, Ann. Phys. 166 (1984) 412. [48] M. H. SARMADI, preprint ICTP/84/3, Trieste, 1984. [49] s. RANDJBAR-DAEMI, M. H. SARMADI, preprint ICTP/84/178, Trieste, 1984. 1501 S. RANDJBAR-DAEMI, A. SALAM, J. STRATIIDEE, preprint ICTP/84/21, Trieste, 1984. [51] K. KIKKAWA, T. KUBOTA, S. SAWADA, M. YAMASAKI, Phys. Lett. 144 B (1984) 365. [52] L. DOLAN, R. JACKIW, Phys. Rev. D 9 (1974) 2904. [53] R. FUKUDA, T. Kuao, Phys. Rev. D 13 (1976) 3469. [54] S. DEWITT, in Quantum Gravity 11, Eds. C. Isham, B. Penrose and D. Sciama, Oxford Uni-

[55] D. G. BOULWARE, Phys. Rev. D 23 (1981), 389. [56] I. J. R. AITCHISON, C. M. FRAZER, Ann. Phys. 156 (1984) 1. 1571 0. YASUDA, Phys. Lett. 127 B (1984) 52. 1581 M. J. G. VELTMAN, in Methods in Field Theory, Eds. B. Balian and J. Zinn-Zustin, North-

[59] S. COLEMAN, E. WEINBERG, Phys. Rev. D 7, (1973) 1888. [60] S. WEINBERG, Phys. Rev. D 7 (1973) 1068. [61] R. JACKIW, Phys. Rev. D 9 (1974) 1680. [62] D. B. RAY, I. M. SINUER, Adv. Math. 7 (1971) 145. [63] J. S. DOWKER, R. CRITCHLEY, Phys. Rev. D 13 (1976) 3224. [64] G'T HOOFT, M. J. G. VELTMAN, Nucl. Phys. 44 B (1972) 189. [65] G. LEIBERANDT, Rev. Mod. Phys. 47 (1975) 849. [66] S. OKUBO, J. Math. Phys. 18 (1977), 2382. 1671 M. A. RUBIN, C. R. ORDONEZ, preprint UTTG-10-83, Texas University, 1983, UTTG-5-84,

versity Press, N.Y., 1982.

Holland, Amsterdam, 1976.

Texas University, 1984.

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Spontaneous Compactification of Extra Dimensions in Eleven-Dimensional Quantum Gravity