Dynamical quantum effects in Kaluza-Klein theories

Volume 142, number 1,2 PHYSICS LETTERS 12 July 1984

DYNAMICAL QUANTUM EFFECTS IN KALUZA-KLEIN THEORIES ~"

Gerald GILBERT, Bruce McCLAIN and Mark A. RUBIN Theory Group, Department of Physics, The University of Texas, Austin, TX 78712, USA

Received 3 January 1983 Revised manuscript received 5 April 1984

A general method is presented for calculating the effective action of a quantum field propagating in a time-dependent Kaluza-Klein background geometry. This method is valid when both the curvature of four-dimensional space and the logarithmic rate of change of the size of the '~internal" space are much less than the curvature of the internal space. The effective action is calculated explicitly to oneqoop order for a spinless Bose field in a Kaluza-Klein model for which the internal space is chosen to be the N-dimensional sphere S N for odd values of N. It is found that the quantum effects can be cru- cial in determining the sign of the kinetic term for the radius of the sphere.

1. Introducth)n. In the six decades since Kaluza-Klein theory was introduced, physicists have been tantalized by the possibility o f unifying gravity with the other fundamental interactions. According to Ka luza-KMn theory [1] there exist, in addition to the familiar four dimensions of spacetime, "extra" dimensions which form a compact spacelike submanifold of such small size that they cannot be observed directly. This yields an effective four- dimensional theory of gravity coupled to Yang-Mills fields, with a gauge group determined by the symmetries of the N-dimensional "internal" space, and with gauge coupling constants inversely proportional to appropriate root- mean-square circumferences of this space [2].

Any realistic cosmological model must be consistent with the observed expansion of the Universe. It is thus natural to generalize the standard Fr iedman-Robertson-Walker cosmology to Kaluza-Klein theories. Both the scale factor of the physical three-space and the scale factor(s) of the compact manifold will be time-dependent. To construct a realistic theory, it is of the utmost importance to understand the temporal evolution of the internal space. The size of the internal manifold must become unobservably small by the present epoch. In addition, its rate of change must be small enough to yield approximately constant gauge couplings. Hence, the study of Kaluza-Klein theories with a time:dependent background geometry will impose important constraints on candi-

dates for realistic models. Moreover, an understanding of the effects of time-dependence is essential for determining the stability of even

a static Kaluza-Klein model, such as the one proposed by Candelas and Weinberg [3]. In their model, the four "large" dimensions have the form of flat Minkowski spacetime M 4, while the internal dimensions form a sphere S N with a spacetime4ndependent radius. The stress-energy tensor which is required to curve the internal dimensions into a sphere is provided by the quantum-mechanical vacuum energy of massless scalar and spinor matter fields. For certain values of N and certain combinations o f matter fields, solutions to the one-loop effective Einstein equations were shown to exist. Both spin-zero and spin-one-half fields were needed to ensure that the effective four-dimensional gravitational coupling G and gauge couplingg 2 were both positive. Furthermore, it was shown that when the radius of the internal space has the value corresponding to a solution - i.e., an extremum of the effective action - the effective potential is at a minimum. This means that the solutions are stable against per- turbations in the radius r o f the internal space, with the proviso that the kinetic term for r enters the effective ac-

Supported in part by the Robert A. Welch Foundation and NSF PHY83-04629.

28 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Divison)


tion with the usual sign. To evaluate this term, the effective action must be computed in a time-varying background metric.

Recently, studies o f Candelas-Weinberg-type theories have appeared [4,5] in which time-variation o f r is considered. In these studies, however, the quantum contributions to the effective action are computed as if the spacetime were static, ignoring effects which depend on ~ (t), the time-derivative o f the scale factor of the internal space. In a Candelas-Weinberg model one expects these effects to be comparable in magnitude to the classical kinetic term * i . Indeed, we shall see that in some cases the quantum contributions change the sign of the kinetic term and cannot be neglected.

We present a general method for calculating the one-loop effective potential. For simplicity, we will do the calculation for a scalar field and take the compact manifold to be the N-dimensional sphere S N with N odd ' 2 . We allow the geometry of the four-space to be arbitrary and let the radius o f the sphere depend on the four-space coordinates. We obtain the effective action as an expansion in r 2 [(d/dr) In r] 2 and r2R, where R is the Ricci curvature scalar of four-space. Thus our method is useful when the curvature of four-space and the logarithmic rate of change of the size o f the internal manifold are both much less than 1Jr 2. The method is used to calculate the one-loop scalar contribution to the kinetic term for the radius of the internal space.

2. Calculation o f the effective action. We consider a (4 +N)-dimensional space with metric tensor

0 gAB(X, Y) = ( g ; v(x) r2(x)~ab(y)) , (2.1)

where upper case Latin indices mn from 1 to (4 + N ) , lower case Latin indices run from 5 to (4+N) and Greek indices mn from 1 to 4. The coordinates o f the four-space are labelled by xU and the coordinates o f the N-sphere are labelled by ya. Similarly, guu(x) is the metric tensor o f the four-space, while gab (Y) is the metric tensor o f the unit sphere S N.

Each of the b scalar fields ¢ is characterized by the action

S = -½ fdax dNy~f~(DAdpaAdp + m2~ 2 + ~j/~2) , (2.2)

where R is the curvature scalar of the metric tensor gAB (we use the conventions of ref. [3]). Any function o f the internal coordinates y can be expanded in eigenfunctions :film @) of the Lapla6e operator

on the sphere SN:

- (1/Vr~) D a x/ff gab ab QJ lm (Y) =k2 Cff lm (Y), (2.3)

Here the k 2 (l = 0, 1, ...) are the eigenvalues of the Laplace operator with m labelling the degeneracy; for each value of l, m runs from 1 to d l. We can choose the Cfflm(Y ) to be real and orthonormal:

f dwy 0') = a,,,a mm' " (2.4)

I f we now write

¢(x, y) = ~ qSlm (x) [r(x)]-N/2 (film (Y) l,m

we find

(2.5)

,1 For a sufficiently large number, b, of matter fields, the one-loop matter contribution to the effective action is of the same order in lib as the classical part, while graviton loop and multi-loop matter terms are suppressed by additional powers of lib.

,2 For pure gravity dimensional regulafization (or any scheme with a dimensionless regularizing parameter) yields no ultraviolet divergence at odd-loop order (in particular at one loop). This is also true for the gravity-scalar system considered in this paper [6].

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S = -½ ~fd4x~C-g(x)(Ou¢lm O~lm + [ m2 - - i e + r-2(k 2 + ~RN) + ~R l,m

+ ~[2N[3a +N(N+ 1)a,ua!U ] + r-N/2[~rN/2] 0 2 ) , (2.6)

where

a(x ) =- In r(x ) , (2.7)

and

k N = - N ( N - 1) , (2.8)

is the curvature of the sphere S N with unit radius. All geometric quantities in eq. (2.6) such as g, R and the d'alembertian are associated with the external four-space; e.g., R is the Ricci scalar constructed solely out o f g v(x).

It proves convenient to introduce the rescaling ,3

g#v -+ r2gtzv, (~lm -+ r-l(Olm , (2.9)

so that eq. (2.6) may be recast as

S[~)lm ] --~- 1 ~ f d4xx/~(~tm [_[] +M 2 _ ie + ~R + hlOlm • (2.10) l,m

We have defined the functions

h(x) =- ~[2Nr - 1 ff](ra) - 2Nar-lr-qr + N(N+ 1)a,ua,U] + r-l-N/2Vlr1+N/2 + 6~(Da + a,ua,U), (2.11)

and

M2(x) = m2r2(x) + k 2 + ~kN ,

such that M2(x) is independent o f x when m 2 = 0. The one-loop effective action is given by

exp(iS(l~) =fc-/) [~Im] exp{iS[Olm]} = I-I det -1/2(- l -q + M 2 - i e + ~R + } , (2.12) l,m

or

S(1) = i / ~ elf 2- d l t r [ l n ( - H + M 2 - i e + ~ R + h ) ] • (2.13)

We use the DeWitt-Schwinger representation of the determinant [8] to write

S(1)=~-~tdtfd4xfidSeff ' 2 "~- exp [ - i S ( M 2 - i e ) l F ( x , x , iS) , (2.14) 0

where F(x, x, iS) is the coincidence limit o f the heat kernel associated with the differential operator

- -D + ~ R + h . (2.15)

4:3 To O(h), the jacobian which arises from rescaling Calm in the functional integral [eq. (2.12)] is unity; this is because the logarithn of the jacobian is proportional to the trace anomaly for Calm [7 ], which vanishes at odd4oop order in an odd total number of dimensions [6].

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We now perform an expansion in the curvature o f the four-space and in the rate of change o f r (x ) . For this we write

, . i ~ ( iSybi(x) , F ( x , x , IS) = ~ (4rr71s)n/2 ] =0 (2.16)

where n is the analytically-continued dimension of spacetime in the sense of dimensional regularization. For the case m 2 = 0 , M 2 is independent o f x and the coefficients hi(x) are independent o f l. These coefficients can be found in ref. [8] if we make the replacement

~R -+ ~R + h . (2.17)

Then we obtain

bo(x ) - 1, bl(X ) = - ( ~ + ~)R - h . (2.18)

Performing the S-integration in eq. (2.14) and defining

S (1) = f d 4 x x -/Z-TZ?(I) (2.19) e f f - - d v --~- eff '

we find

o o r0 nj2,/ ~(I) = lim bj(x) dl(M2) -]+n/2 eff n-+4 "= 2 - ( 4 #

The first two terms in eq. (2.20) yield

• ~ ~- - C N - E N { ( 1 + 6~)R + [~ (N+2) 2 + 6~(N+2)(N+3)]a,ua,U},

where the C N and the E N are defined (ref. [3]) as

CAr =. -:-__.__~1 lim P ( - n / 2 ) ~i dl(M]) n/2 E N - 2(4rr)2n_+ 4

I f we undo the rescaling of (2.9) by writing

guy -+ r-2gu~ ,

eq. (2.20) becomes

(2.20)

(2.21)

1 lim P ( 1 - n / 2 ) ~ldl(M2) n/2-1 (2.22) 12(4rr) 2 n-+4

(2.23)

.~(1) eff = -r-4(x){CN+r2(X)EN[(l+O~)R+N[~(N+4)+6~(N+5)]a,t,a'u]+O(r4R2,r4(a,#a,U)2)}. (2.24)

The term in eq. (2.24) proportional to R agrees with the results o f ref. [9]. We see that eq. (2.24) is valid when both

(1) the logarithmic rate o f change of the radius of the N-sphere is much less than the energy required to excite a non-zero mode of the sphere:

a,ua'U ~ 1] r2 , (2.25)

(2) the curvature of the external four-space is much less than the curvature of the N-sphere:

R ~ lJr 2 . (2,26)

We now use eq. (2.24) to examine the sign o f the kinetic term for the radius of the sphere. The classical gravitational action in 4 + N dimensions is

31

Volume 142, number 1,2 PHYSICS LETTERS

1 S G = - --'--2_ f d 4 x dNy %/c~(R + A).

16rrG

Using the total action for gravity and b massless scalars,

s=so Candelas and Weinberg find that the Einstein equations

8S/~gA8 = o ,

have the static solution

r(x) = p - [8rrGoCN(N + 4 ) b I N ( N - 1)] 1/2

The constant G O in (2.30) is defined as

G o- ~/S2N, where

aN--PN f d~Vy,,/~.

ft is free-tuned to ensure that the four-space is flat:

A= N 2 ( N - 1)2(N+ 1)/8rrGoCN(N + 4)2b .

Since p is real, eq. (2.30) implies

Go@> o, which restricts the values of N and ~,

Using eq. (2.24), the effective action (2.28) becomes

12 July 1984

(2.27)

(2.28)

(2.29)

(2.30)

(2.31 a)

(2.31b)

(2.32)

(2.33)

1 fd4x~-~{[[r(x)/plN+ (1+ 6~)(161rGoENb/p2)[r(x) /p l -2]R(x) S = 167r G-----0

+ [[r(x) /p]NA+ [r(x) /p]N-2RN/p2 + (16rrGoCNb/p4) [r(x)/p] -4]

+ [ - N ( N - 1) [r(x)/p] N + (16~rGoENb/p2)N [~ (N+ 4) + 6~(N+ 5)] [r(x)/p]-2]a,ua,U} , (2.34)

where RN is the curvature of a unit N-sphere.

k N = - N ( N - 1). (2.35)

Inspection of eq. (2.34) reveals that, when r(x) = p, the inverse of the four-dimensional gravitational coupling is

(2.36)

(2.37 a)

(2.37b)

given by

1/16rrG = (1/16rrG0)[1 + (1 + 6~) 16~rGoENb/p2 ] ,

which implies

( l /G0) [1 + (1 + 6~)/~] > 0 ,

where

= 167rGoENb/p2.

32.


The inequality (2.37a) further restricts N and ~. To study the kinetic term for r(x), we must first express the effective action in a form in which the degrees of

freedom corresponding to spacetime-dependent changes in the size of the internal sphere are explicitly decoupled from those degrees of freedom corresponding to four-dimensional gravitational waves [the latter degrees of freedom being contained in the variable R(x)]. This may be achieved by rescaling as follows:

gu u -+ ~22guv , (2.38 a)

where

~2 = { [ 1 +/?(1 + 6~)] -1 [ [r(x)/p]N +/?(1 + 6~) [r(x)/p] -2 ] }-1/2. (2.38b)

We then obtain

= - l~rGfd4xx/~-g(R(x) + [[r(x)/p]N+ (1 + 6~)E[r(x)/p]-2]-2 S

× {[1 +E(1 + 6~)] [A[r(x)/p]N+ (RN/p2)[r(x)/p] N-2+ (16zrGoCN/pa)[r(x)/P] ~ ]

+ [½N(N+ 2)[r(x)/p]N([r(x)/p] N + E [r(x)l;] -2)

+ ff2(1 + 6~)[6 + ~N(N+ 4) + 6~ [6 + N(N+ 5)] ] [r(x)[p] -4] a,ua,U}) " (2.39)

For small oscillations around equilibrium,

r(x)/p ~ 1 + e(x) , e(x) ~ 1 , (2.40)

the kinetic term for e(x) from (2.3'9) is

-(1/167rG) [1 + (1 + 6~)/?] -2(N+ 2) {½N(I +/?+ (1 + 6~)/~ 2) + (1 + 6~)2E2(N+ 3)} e,ue,U. (2.41)

The classical limit (/~ = 0) ofeq. (2.41) has been found by Page [5]. In this limit (2.41) has the conventional sign for a kinetic term. The sphere will then be stable provided r(x) = p is a minimum of the effective potential.

However, in order for eq. (2.41) to have the "correct" sign when quantum effects are included, E must satisfy

½N[1 +/~+ (1 + 6~)/? 21 + (1 + 6~)2/~2(N+ 3) > 0 . (2.42)

For the case of minimal coupling (~ = 0) itis easy to see that (2.42) is satisfied for any value of/~, and thus places no new restrictions onN.

In contrast, no such general conclusions can be reached for, say, conformal coupling [~ = - ~ (iV + 2)/(N + 3)]. The validity ofeq. (2.42) then depends on the values of if] as a function of N~

As already noted, the inclusion of Fermi fields is essential to make G > 0 and g2 > 0. Thus we cannot reach any conclusions about the stability of the Candelas-Weinberg model until the one-loop fermion contribution to the kinetic term is calculated. Nevertheless, we have seen that the quantum effects cannot be neglected, and may in fact change the sign of the kinetic term for r(x) from its tree-level value.

3. Conclusion. We have presented a general method for calculating the effective action of a quantum field in a time-dependent Kaluza-Klein background. This method is valid when the curvature of four-space and the logarithmic rate of change of the internal space are both much less than the curvature of the internal space [eqs. (2.25) and (2.26)].

The resulting effective action was then used to study the sign of the kinetic term for the size of the internal space in the Candelas-Weinberg model, including contributions from Bose fields (but notFermi fields). For the case of minimal coupling, it was found that the kinetic term always has the conventional sign. For nonminimal coupling, we have shown that the sign may change, depending on the values of ~ and N.

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Generalizations of this work to include higher-spin fields, thermal effects * 4, the effects of non.vanishing che- mical potential, and applications to cosmology are now in progress.

We would like to thank Steven Weinberg and all the participants of the Theory Group Lunch Seminars for very helpful discussions. B.M. Would like to thank Steven Blau for many useful conversations on Kaluza-Klein theory;

M.A.R. would like to thank the Theoretical Physics Group of the Enrico Fermi Institute for their hospitality during the completion of this work.

G.G. wishes to dedicate this paper to the memory of Jeffrey Gilbert.

44 Phase transitions of a rather spectacular nature may be expected at high temperatures [ 10,3].

References

[1] Th. Kaluza, Sitz. Preuss. Akad.Wiss. K7 (1921) 966; O. Klein, Z. Phys. 37 (1926) 895; A. Zee, in: Proc. Fourth Summer Institute on Grand unified theories and related topics, eds. M. Kanuma and T. Maskawa (World Scientific, Singapore, 1981).

[2] S. Weinberg, Phys. Lett. 125B (1983) 265. [3] P. Candelas and S. Weinberg, Texas preprint UTTG-6-83 (Revised) (1983). [4] S. Randjbar-Daemi, A. Salam and J. Strathdee, Intern. Center for Theoretical Physics preprint IC/83/208 (1983). [5 ] D. Page, On the stability of spheres in extra dimensions supported by quantum fluctuations of scalar fields (1983), unpublished. [6] M.J. Duff and DJ. Toms, in: Unification of the fundamental particle interactions - II, eds. J. Ellis and F. Ferrara (Plenum,

New York, 1983). [7] K. Fujikawa, Phys. Rev. Lett. 44 (1980) 1733. [8 ] B. DeWitt, in: Relativity, groups and topology (Gordon and Breach, New York and London, 1964);

L.S. Brown, Phys. Rev. D15 (1977) 1469. [9] M. Awada and D. Toms, Univ. of Wisconsin-Milwaukee preprint (1983).

[10] M.A. Rubin and B.D.B. Roth,Nucl. Phys. B226 (1983) 444.

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Dynamical quantum effects in Kaluza-Klein theories