Fortschritte der Physik / Progress of Physics: Volume 32, Number 5 [Reprint 2022 ed.] 9783112656143

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FORTSCHRITTE DER Volume 32 1984 PHYSIK Number 5 PROGRESS OF PHYSICS Board of Editors F. Kaschluhn

R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J. Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin J. Lopuszanski, Wroclaw A. Salam, Trieste D. V. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J. Zinn-Justin, Saclay

CONTENTS: W . MECKLENBURG

The Kaluza-Klein Idea. Status and Prospects

I P

207-260

AKADEMIE-VERLAG • BERLIN

ISSN 0015 - 8208

Fortschr. Phys., Berlin 32 (1984) 5, 207-260

EVP 1 0 , - M

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 pages in lenght. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from "1" onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the author's name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and muBt be referred too in the text and on the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawings should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawings and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written to small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary upright typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vectors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He p, ...), elementary mathematical functions like Be, Im, sin, cos, exp, ...): black underlined Greek letters: red underlined Boldface Greek letters: red underlined twice Upright Greek letters (symbols of elementary particles): red and black underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as e G, k K, o 0, p P, s S, u U, v V, w W, x X, y 7, z Z). I t will help the printer if the position of subscripts and superscripts is marked with pencil in the following way: at, 6», J f j i , Mij, WV* Please differentiate between following symbols: a, a ; a, a , oo; a, d; c, C, cr; e, l; e, £; k, K, x; x, x, X, x, X ; I, 1; o, 0, a, 0;p,Q; u, U, U; v, v, V; i we can choose it to be of the blockdiagonal form Gmn = & diag (Gmn{x), — 1). The five dimensional action for the real scalar field 0{z) we take to be 8= - j J d h |det GMN\^ ®{z)

4>{z)

(2.21)

where surface terms have been neglected. As before, z = (x, y) and n 5 is the five dimensional d'Alembertian. 81 is compact; therefore 0{z) is periodic in y, and we may write down a harmonic expansion 0(z) = 0(x, y) = i " 1 ' 2 £

-„{x). Noting that the five dimensional d'Alembertian is CJs = Q4 — (3/dy)2, we obtain from (2.21) after performing the «/-integration, Z

f d'x |det Gmn{x)\U2 y) = — &t(x> y + L). Note that this eliminates in particular the zero mode in the harmonic expansion. An alternative description is the following. Consider a circle with circumference 2L, and identify antipodal points on this circle. Twisted fields are those whose values differ in sign at antipodal points. For twisted fields, only odd n occur in the harmonic expansion. If in a similar way we identify antipodal points on S2, we obtain the real projective space RP2. For twisted fields, we keep in the harmonic expansion the modes with odd I only and for untwisted fields those with even I. In this section we have studied only free field theories. There is no coupling between different modes. Conversely, had we started with an action containing a self interaction of the scalar field, then there would occur couplings between different modes. J 2.

Harmonic expansion: General formalism

We generalize here the observations of the previous section to general coset spaces Bk — G/H. In this section, material from appendix D will be used. As a first example take H = 1 so that BK = G. Consider a scalar field 0(x, y). Under coordinate transformations y ->«/', 4>(x, y) -> (x). We would not want to do this in any case, as we want (x) to carry are presentation of G. Thus we have to require that under a coordinate transformation the world scalar y'. Now L(y') = gL(y) h"1, so that inPf.vv = / My) KUgMy) G/H

r1) •

The dependence on h cancels (because the we obtain

P

(2.32)

are representation matrices for H) and

InPiPy = DnPq(g) I%:qVD*p,(g-i),

t

(2.32')

which marks Ip%p'6- as the components of an invariant tensor. Accordingly, by Schur's lemma, In

p"'.pY = t> nn 'tpp-K-

(2.33)

The factor h"e. depends only on representations and can be found by setting n = %', p = p', and summing over p. Then dnK' = / d^^L-^y))

DpAe(L(y)) = VKdD8ee-.

(2.34)

Fortsehr. Phys. 32 (1984) 5

215

We thus obtain the desired inversion formula, D

= 1 ^ ] / ^ /

lMv))

V) •

(2-35)

G/H

As before, the dimensional reduction procedure consists effectively in replacing the fields of the higher dimensional theory by*expressions like (2.30) and integrating out the internal variables. In general, algebraic complications will occur, mainly due to the appearance of Clebsch-Gordan factors which not necessarily recombine in a simple way.

3.

Harmonic expansion: Examples

For the discussion of the Kaluza-Klein theory below it will be convenient to have certain specific examples of the above formalism available. Consider for example the one form = e/(y) dy* on coset spaces, where e/(y) are the vielbein coefficients. In appendix D we have given the transformation rule (cp. eqn. (D.12)) G is interesting in its own right. It will be discussed below. Here we turn to the case R cz G. In particular, one may search for the minimal gauge group R such that compactification of Mi+K into M4 XBK occurs. Clearly one feels that, the smaller R, the better. It has been shown that this minimal group is not bigger than H. Indeed, R may be smaller in rank than G, and need not act transitively on G/H. The available compactifying solutions which use groups R smaller than G are topologically nontrivial, unlike the CS solution discussed above. This is to be understood in the following way. For the gauge fields BM(z) = [Bm(x, y), B^x, y)), the compactifying solution takes the form BM(z) = (0, B^y)). Now B^y) are gauge fields on the internal space BK. The latter may be used as a base space for a nontrivial fibre bundle, and B^y) are the connections for this bundle. The nontrivial topology is then reflected in the existence of nonvanishing topological quantum numbers like the instanton charge in four dimensions (on S4) and its generalizations (the characteristic classes). A typical example of such a solution is the following [85, 86]. We take BK = S2. On S2, we introduce polar coordinates 0, etweiler (CD) solution has recently been much extended by Fretjnd [71]. He argues that scenarios of this type would be quite important for cosmological considerations, as the "effective" dimension of space-time might depend on time and at an early time the universe could indeed have been high-dimensional with all dimensions of comparable size. He also discusses the possibility that in eleven dimensional supergravity, preferential expansion of three spacedimensions occurs, much in continuation of the ideas put forward in connection with the Freund-Rubin model (see above). In addition, he discusses scenarios for the whole procedure to take place in hierarchial steps. 6.

A lattice calculation

It is difficult to foretell whether spontaneous compactification persists as a quantum phenomenon beyond semiclassical approximations. It has recently been suggested by Skagerstam [179] to approach this problem by studying lattice versions of high dimensional field theories. He assembles Monte Carlo data for the Wilson loop average for the following two models r (1) a five dimensional Yang-Mills system; (2)-a four dimensional Yang-Mills-Higgs system with Higgs field in the adjoint representation that corresponds to five dimensional Yang-Mills system with periodic fifth coordinate. He compares the two sets of data and finds very good agreement. Correspondingly, as far as Wilson loop averages are concerned, the two systems behave in the same way. This indicates that spontaneous compactification of the fifth coordinate may indeed occur in the five dimensional Yang-Mills system. B.

Stability considerations

1.

General remarks

Spontaneously compactifying solutions have been sought as minimal energy solutions (as much as energy is defined in gravity theories) of classical field equations. In order to ascertain further the nature of these solutions, stability considerations have to be taken into account.

222

W. MECKLENBURG, The Kaluza-Klein Idea

In the context of the Kaluza-Klein theory, this analysis has only been carried out for the most simple cases. As we will see, stability considerations are of non negligible impact. Energy-like (conserved) quantities play a prominent role in stability considerations. A typical example is Dirichlet's theorem which asserts that a solution of the equations of motion is stable if it provides a strong minimum of some conserved quantity. Note that stability is not guaranteed if the minimum is not a strong one. Also note that from the point of stability, energy may be replaced by another conserved quantity. In a quantum theory, barrier penetration from one local minimum to another may also •occur. For stability we would have to require that a minimum be a strong absolute one. If two minima have apparently the same energy, we have to demonstrate that no semiclassical decay from one minimum to the other occurs. Let us elaborate some of these statements for an example [81]. Consider a single scalar field (x, i) in 1 + 1 dimensions, with Lagrangian S£ = 1/2 — 1/2 f( — V(). The corresponding field equations are 4>ut - 4>x*x + F ' W = 0,

v'() = ^

V().

(3.16)

Suppose, these have a classical solution = (x, t) = (x) exp (iu>t). Inserting this into the field equation and retaining terms of first order only one has y>(x) = m2y){x).

(3.17)

For stability, all w2 should be non-negative, otherwise we would have exponentially growing modes. This requirement can actually be translated back into a criterion that the energy have a strong local minimum for 0 = (pcl for static solutions ) = / [ 1/2 + F($)]. It is minimized by the classical solutions so that {SEcJd4>)\^=Vcl = 0 and further the requirement a)2 ^ 0 is translated into (S^ciAWU-fci ^ 0. The above considerations are complicated by subtleties concerning degenerate ground states in general and zero mode oscillations in particular. If we have separated strong local minima of the energy then they will classically all correspond to stable equilibrium points. To proceed further we will have to resort to semiclassical procedures (see below.) I t can also happen that a minimal energy solution can be deformed continuously into another one without change of energy. Such degeneracies occur typically for Goldstonetype potentials. In the formalism above they will manifest themselves as zero frequencies. Physically, the corresponding zero mode fluctuations are associated with massless particles (Goldstone bosons), provided, the degeneracy can be associated with an internal symmetry of the Lagrangian. If no symmetry is associated with the zero mode there may be instabilities, in accordance with Dirichlet's theorem. Consider now the field equation for static solutions and differentiate once more with respect to x. One obtains ' = 0

(3.18)

and we see that y = 'ci is a zero mode solution of (3.17). This zero mode solution, which always, exists, can be related to the translational invariance of the system. In fact, consider small translations, x x + x0. Then (x, t) = 4>c\(x + ¡r0) = 4>ci{%) + ci{x) • xa whence by comparison with (3.17) we get m = 0 and y>(x) = ^I(ÎC).

For'tschr. Phys.-Bâ (1984) 5

223

For stability, the translational mode has to be the lowest mode. In more than one dimension, there is a translational zero mode in every spatial direction. Because of rotational invariance, the eigenvalue co2 = 0 is degenerate, and can therefore not be the lowest. In particular, for a spherically symmetric i with S1 having a very small circumference ( ~ 10~33 cm, Planck's length) indeed. The smallness of this internal space was thought to render it unobservable. In a field theoretic context a four dimensional theory may be obtained by harmonic expansion over ^ ; the massive particles then have masses of at least 1019 GeV (Planck's mass). Kaluza's idea then means in this language that at presently available energies only the zero mass sector of the theory would be relevant. It is mostly this sector that will be of concern to us in this chapter. Even though the zero mode approximation will be good for small energies, on principle grounds the massive modes cannot be neglected entirely [51, 52, 55, 169]. One reason is that the massive modes are likely to give effects comparable to those of quantum gravity. The other is that the 4 + K dimensional field equations are simply not compatible with the truncation implied by the Kaluza-Klein ansatz [109, 208]. Actually, it is interesting to note that if on the other hand all the massive modes are kept, one expects to obtain a consistent theory. For five and six dimensions, the corresponding investigations have been carried out [169, 162] with positive answer. These are the only known examples for a Consistent interacting theory of massive spin two fields. In the following, the classical Kaluza-Klein theory will be described from the point of view of a zero mode approximation around a spontaneously compactifying solution. Only briefly we will comment on the relation to the picture of fibre bundles. Finally, an account of the incorporation of spinors in the Kaluza-Klein framework is given. B.

The Kaluza-Klein ansatz

1.

The five-dimensional case

In this case no extra matter fields are needed for spontaneous compactification since both M 5 as well as M t X are solutions of the five dimensional vacuum Einstein equations. There is no reason in this case why one should prefer one of the solutions as a ground state. As we have seen, even stability considerations do not help. In fact, M 4 X 2*

226

W . MECKLENBURG,

The Kaluza-Klein Idea

is less likely to be. stable than M 5 . For the time being we therefore just assume that M t X St is the true ground state of. the theory.' The metric of the ground state is GMN = ( + , ; —) where we anticipate that internal dimensions have to be spacelike. The metric is forminvariant under P 4 X U(l), where P 4 is the four dimensional Poincaré group. Among the zero modes we therefore expect the gravitational field (spin 2) and one gauge field (spin 1). There will also be a massless scalar, whose existence probably reflects the invariance of the theory under rescaling of the internal radius. These are the zero modes whose masslessness is enforced by a symmetry and which therefore are expected to be present in a quantum theory. As we have anticipated, the massless modes just give the zero modes in a harmonic expansion of the metric tensor. In fact, GmN(x, y) = £ exp (iny) G$N(x) and in the present •

"

case the zero mode approximation c o n s i s t s simply in dropping all «/-dependence. The following form of the ansatz exhibiting the zero modes is convenient ri

I9mn

AmAn

2 /•

(41)

It turns out that gmn rather than Gmn = gmn — AmA„ plays the role of physical gravity (see also below). Now a straightforward calculation using the formulae provided in chapter I I yields the result for the five dimensional curvature scalar, =

Einstein

mn - X 4 ^mnF

~

^

= 1 we would just get the EinsteinMaxwell Lagrangian with unit gravitational constant. In this case the Einstein field equation i?55 = 0 reads FmnFmn = 0. Therefore the ansatz (4.1) is not compatible with the field equations. Five dimensional gravity is invariant under general coordinate transformations, whereas the zero mode approximation reflects the symmetry of the ground state. To "see this, remember first that the symmetry relating to the masslessness of

and if we identify (x) (4.7)

Since the ansatz (4.6) leads to i? 5 = i2 ElnsteiI1 — / 2 /4 FmnFmn, we identify / 2 as gravitational constant, / ~ 10~19/GeV. We therefore find that for e ~ 1 as befits a reasonable gauge coupling that we must have ^ = e// ~ 1019 GeV. I t is in this way that the Planck mass enters into the Kaluza-Klein theory. In fact, the bare mass of the dimensionally reduced theory is M a 2 = /¿2 + m02, so that the Planck mass really is the lower bound for the four dimensional mass. The relation fif = e is determined by the coupling of Am to a matter field. In the nonabelian case (see below) it is already provided by the selfcoupling of the gauge field. The ansatz for the scalar field implies that ju is the inverse radius of I t is fixed after we have set the values for / and e, that is only after matter fields have been introduced. In the nonabelian case matter fields are not necessary for this. 2.

The general case

In the previous section we have seen that in order to identify gauge transformations as coordinate transformations it was important that the five dimensional line element could be given the form ds2 = GMN dzM dzN = gmn dxm dxn — (dy + Am dxmf. From this, we read off suitable vielbein components. In fact, let EA — EMA dzM replace dzM. The vielbein components are determined by the requirement that the line element take the form ds2 = r)ABEAEB, with rjAB = diag ( + , ; —). We then have (4.8) Note that the appearance of the complete square {dy + Am dxm)2 in the line element is reflected in the appearance of the 0 in (4.8). The components E m ° are to be identified with the vierbein components of four dimensional gravity. Recall that this form of the ansatz clearly displays the gauge symmetry as a special case of invariance under general coordinate transformations. For the D-dimensional case we now make the ansatz (4.9) If we write down a harmonic expansion for the vielbein (4.9), then the first term for Ema will simply be Ema(x) by the general arguments of chapter I I , since these behave like

228

W . MECKLEKBURG,

The Kaluza-Klein Idea

components of a frame scalar under internal rotations. As we have noted above, the arguments of chapter I I provide for Ema(x, y) the ansatz V) = -Aj(x)

(4.10)

Df{L{y))

as the first term of a Fourier series. The AmP(x) will turn out to be four dimensional gauge fields associated with the gauge group G (recall that internal space is presumed to be of the form G/H). The Ema will again be the vierbein components for four dimensional gravity. These are the zero modes we expect for symmetry reasons. The ansatz for the vielbein is completed by setting for Ef simply the vielbein for BK = G/H, that is E/ = e/(y) (cp. appendix D). The ground state corresponds to the block diagonal vielbein Emà = b diag (òma, e,p{y)). As e/(y) is forminvariant under group-coordinate transformations, viz. e/1a(y) = e'/ly), and similarly 6ma is unaffected by four dimensional Poincaré transformations, the ground state is invariant under P 4 X G. The Kaluza-Klein ansatz preserves this symmetry for the zero mass fluctuations. Fully, it reads,

The metric equivalent is ^ > y ) = \AnKx)K^y)

gAy)

(4-12)

)•

In (4.12) the abbreviations gmn = Em"Enb and (Am, A„) = Aj\x) AJ\x) K&>"{y) K$'{y)g^ have been used; the coset space metric is (cp. Appendix D) g^(y) = efi"(y) e/(y) The components of the Killing fields had been introduced as Kif(y) = D^(L(y)) ef{y) and Kip(y) = gll,K^{y). The Killing fields have the fundamental property that they serve to express infinitesimal group transformations g = 1 + as coordinate transformations, viz. yP y -(- e*Ksiv(y). One thus expects that the coordinate transformations (4.13)

(xm, y) -> (x'm, y'") = {xm, y + e*(x) K*»(y))

can be re-expressed as local infinitesimal gauge transformations for the would-be gauge fields Ams(x). That this is indeed the case is shown in appendix C in the metric formalism for infinitesimal transformations. In the vielbein formalism for finite transformations the same result has been proven in ref. [169]. The calculation in appendix C is carried out for GMN, the matrix inverse of GMN. The inverse vielbein is / W )

AaHx) my)\

(414

where Eam and ej* are the matrix inverses of Em° and e/ = E0"A„s(x). The inverse metric tensor GMN then is /¡T» \A*(x)Kf{y)

A»H*)KP>(y) gt" + rfbA^A^KfKt'f

respectively, and

Aj(x)

\ '

The change of the metric tensor under coordinate transformations at a given point is òGMN{z) = G'MN(z) — GMN{z). For the coordinate transformation (4.13) we find in particular òGm"{z) - d[Am*{x) K^(y)] - K^{y) dAmi

(4.16)

229

Fortschr. Phys. 32 (1984) 5

with

»Am*(x) = éffcAmî{x)

+ Gmne\n{x).

(4.17)

Again we find that coordinate transformations can be reexpressed as gauge transformations. The curvature scalar corresponding to the Kaluza-Klein ansatz is evaluated using the zero spin connections which are given by (cp. the general formulae given in chapter II) Saibc)

=

j

w

= -Brm

¿

A

~

j

= 1 EamEbnF

Ba[M = Aj(Df(L(y))

E[bmEcf

8mEm +

J

Elc"E0f dmEn>

mn*D&y[L(y))

- Df(L(y)) n*(yj) fm

= 0 Ba.p{x) with [

(4-24)

GIH

dXy det

h${x)

t

(My))] 1 ' 2 M*> y) K*"{y)

•

(4.25)

I t then turns out that / d'xd'y =

[ d e t Gmn]112

Vg,h J

-

J

+

K{K±

d^x [ d e t gmn]112

Ri+K jyU*'0

One notices that the potential for the JBD scalars is not unlike in structure to those for nonlinear sigma models. The symmetry breaking patterns for a potential similar to that of equation (4.26) have been investigated in some detail by S C H E R K and S C H W A B Z [173]. However, their treatment of the invariance properties of y^pix) was different from the one sketched here.

Fortschr. Phys. 32 (1984) 5

231

Actually, the same remark applies also to an earlier paper by CHO and FRETJND [33], However, as has been argued by RANDJBAB-DAEMI and PERCACCI [157], it is the present treatment that seems to be the consistent one. A discussion of the symmetry breaking properties of the scalar sector of (4.26) is still lacking. It is not easy to read off from (4.26) whether the scalars are massive or not. What is needed is an investigation of the spectrum of the theory. For the six dimensional case the results of such an investigation have recently been presented [162]. In this case, the scalars turn out to be massive. Thus, the role of the J B D scalars in the Kaluza-Klein framework remains somewhat ambiguous. Experimentally they seem not to be really needed (cp. ref. [198]) except perhaps as a possible explanation for Dirac's large number hypothesis, cp. [35]. However, supergravity may enforce their incorporation, cp. [53, 71], even though they do not seem to appear in the zero mass sector of the theory. 4.

Kaluza-Klein ansatz for the CS model

Above, we have described in some detail the Kaluza-Klein ansatz (zero mode approximation) for the high dimensional metric tensor. However, since no spontaneously compactifying solution, with nonabelian internal symmetry and vanishing four dimensional cosmological constant have been found, this discussion lacks somewhat in consistency. It would be desirable to discuss the zero mode approximation for a model that actually does display the phenomenon of spontaneous compactification, like the CS model (the gravity-Yang-Mills-system) which we have described above. In fact, an early formal discussion of the zero mode approximation has already been given by LUCIANI [109]. A general observation is that the metric gauge field and the external gauge fields get mixed. However, the gauge coupling constant for the external gauge fields tends to infinity just as the masses for the massive modes; in the four dimensional theory, the two types of gauge fields therefore behave quite differently. A detailed analysis of tKe Einstein-Maxwell system in six dimensions has recently been presented by RANDJBAR-DAEMI, SALAM and STRATHDEE [162]. In particular, these authors discuss the classical stability of the ground state solution (with positive answer). They also analyze the spectrum of the theory after harmonic expansion. The massless states found are just the graviton, the SU(2) (internal) and 17(1) (external) gauge fields; there are no massless scalars. In addition, spinor zero modes are being constructed. Their existence follows from the general arguments we have reported above; the construction given here continues similar work done by HORVATH and PALLA [85] some time ago. 5.

Relation to fibre bundles

To some extent, Kaluza-Klein theories may be viewed as the physicists' way to understand fibre bundles. This connection was probably first observed by DE WITT [204] and later much elaborated on in particular by TRATJTMANN [193, 194] and also by many others [14, 24, 31, 44, 92, 103, 145, 157]. Here we will only very briefly describe in what sense the Kaluza-Klein ansatz (4.11) reflects the structure of the fibre bundle. For more detailed expositions we refer to the literature. Rougly speaking, a bundle is a space % which locally looks like the product of two spaces, M\ the base space, and F, the fibre, % ^ M X F. In our formalism above, the local loc.

232

W. MECKLENBURG, The Kaluza-Klein Idea

product structure is reflected in the possibility to choose an orthonormal set of frame vectors. There will be a projection n-.ft-^M with the properties that = M and JI~'(X) ^ F for all x 6 M. The existence of a set of homeomorphisms % W which among some other requirements, have to form a group 0, makes the bundle a fibre bundle. If the fibre space is the same as the group space, we have a "principal fibre bundle", otherwise we speak of an "associated fibre bundle". It is now a nontrivial result that a fibre bundle, once made into a Riemannian space (i.e. endowed with a metric), has a metric of the form (4.12). Consider firstly the case where the fibre is identical to the structural group. Such a principal fibre bundle underlies a gauge theory with G as a gauge group. Local triviality of the bundle means that GMN can locally be given block diagonal form, GMN = b diag [r}mn, g^,{y)), which incidentally may be recognized as the statement that locally gauge fields can be transformed away. The metric we identify as the metric part of a spontaneously compactifying solution, with g^(y) the metric of G. Note now that the zero mode fluctuations contain gauge fields corresponding to the gauge group G x G , since G admits left as well as right translations — in the framework of spontaneous compactification, G automatically takes the role of a coset space for GxG: we thus really do not have a principal fibre bundle. We thus realize that within the picture of spontaneous compactification, a principal fibre bundle cannot be expressed as a Kaluza-Klein ansatz [208]. Rather, a fibre bundle treatment of the Kaluza-Klein picture ought to employ associated bundles from the very beginning. Whereas a considerable body of work is available on metrisized principal fibre bundles, the couching of the Kaluza-Klein picture with coset spaces as fibres ("homogeneous fibres") into the language of fibre bundles has received much less attention. Early attempts are [26, 27] and [49], more recent formulations are given by [157] and [32]. Let us put some of the remarks made into a slightly more formal language. Introduce the vector fields dA = EAM dM. We noted in chapter II.A. that by means of the relation [D , 8 ] = —D[AB] D w e c a n obtain the connection coefficients. For simplicity choose Minkowskian space as base space (i.e. disregard physical gravity), Eam = dam. In this case we have [8a, 06] = Fab'(x) Kj?(y) 8^ = Fab$D$'(L(y)) 8a. Quite generally, we also have [8a, 8a] = 0 and [8a, dp] = —Q^f 8r, where the latter simply provides us with the connection coefficients on the coset space. This construction clarifies how the particular form of the Z)-dimensional vielbein relates to the appearance of the Yang-Mills density in the decomposition of the £>-dimensional curvature scalar (use the definition of the modified curvature scalar, R, at the end of section II.A.). The vector fields da and dx are referred to as "horizontal" and "vertical" vector fields respectively. It is characteristic that vertical vectors commute among themselves, horizontal vectors commute to give a vertical vector and that the commutator of a horizontal with a vertical vector vanishes. The construction of horizontal vector fields amounts to the same as the construction of a "connection on a fibre bundle". In fact, the gauge fields are to be identified as connections on the bundle. This construction realizes explicitly the local triviality of the bundle. The associated bundle corresponding to the Kaluza-Klein bundle may be viewed as a subspace of the "frame bundle" provided by the tangent vectors of if. The specific structure expressed by the properties of horizontal and vertical vectors singles out the associated fibre bundle with structural group G. A

C.

B

D

Spinors

For spinors, the process of dimensional reduction is rather involved. Only quite recently, general formulae have become available [169].

233

Fortschr. Phys. 32 (1984) 5

In the next section, we will report the result for the dimensional reduction of the Dirac Lagrangian. After that, we will discuss the issue of zero modes for the internal Dirac operator and the related question of parity violation in the dimensionally reduced theory. Explicit formulae for the dimensional reduction of the Rarita-Schwinger Lagrangian (relevant for supergravity) are not yet available. Therefore, only some general properties of the Rarita-Schwinger equation in 4 K dimensions will be discussed here. 1.

Dimensional reduction of the Dirac Lagrangian

The general form of the Dirac Lagrangian in D dimensions has been given in chapter I I to be £?d = i/2 det EfrA dAip + h.c. Here y> = y>(x, y) is an S0(1, 3 + üT)-spinor with 2[(4+k)/2] components. It carries a label associated with H\ the generators Qs of H are given by the imbedding into SO(K), Qs = —1/2 If the label on yj is exposed, ip - > y)j then the H generators will act on y>i as Dj;(27"'') rpj. For the spinor we use the ansatz y>(x, y) — D[L~x(y)} (x) which is symbolic for the Fourier expansion (explained in chapter II) Vi(x,

=

E I>l,p{L-\y))

l {x).

(4.27)

W e

Let us consider the covariant derivative on G/H. It is to be defined as (note that the imbedding H —> SO(K) involves a minus sign) day>i(x, y) = ea" d^i - j

B

&

,

(

4

.

2

8

)

Since the «/-dependence of the spinor is given through the Fourier expansion, we can eliminate the derivative 8^ In fact, using the definition of e(y), we obtain ea* 8„D(L-i(y)) - ~[D(Qa) + n*D(Qs)] Z>(2r%))

(4.29)

with = e ^ e / . Using the imbedding of H into the framegroup as well as a suitable connection on G/H, one obtains a mutual cancellation of terms containing nas. In fact, in ref. [169] the connection coefficients B«[ßy] = e^e^'f^y = nasfSßy are being used. In general, they correspond to nonvanishing torsion coefficients, Tpf = 1/2 fßy*. Note that these vanish on symmetric spaces, however. Now the imbedding of H into SO(K) is provided by —1/2 fSpy Dii(Z'A') = D,,(QS). Inserting this, one obtains for the covariant derivative of Da{L~1{y)): < U V ( £ - % ) ) = -D?qm

•

(4-30)

We see that the operation of covariant differentiation on coset spaces is turned into an algebraic operation. In order to apply the Kaluza-Klein ansatz to the Dirac Lagrangian, it is convenient to denote E ai j, — D(i7ajj) etc. by the generic symbol r„ß (similarly for r a ß y , the totally antisymmetric product of J'-matrices) and to drop labels on y>(x, y). The Dirac Lagrangian then becomes # d = J det (Em°) det (e/) yi(x) D(L(y)) + r « je." a„ - J 5 a [ B c ] r a c J j

^E" 8m + E0» d, w

{x) + h.c.

iso[BC]r^J (4.3i)

234

W. MECKLENBURG, The Kaluza-Klein Idea

Here we now have to insert the various expressions for BA[BC], as given in the KaluzaKlein picture (see above). One eventually obtains, using the connection on G/H as given in this section,

SeD = -i-det (Em°) det ( V ) y>{x) D{L(y)) X [ / ^ . - ( a s ) 3m + j

+ j

Ba[bc]r bc

F j ( z ) r abr°Dat(L-i(y))

-

Aa'(z)

- y*D(Qa) +

j U S ^

X £ ( £ - % ) ) f(x) + h.c.

(4.32)

After integration over y one gets j

VK det (Em°) yi(x) ^ { ^ « ( a O 8m + j

+

A% =

M

r»° - AjD(Q^)

F j ( x ) F(x) + h.c.

with

and

Ba[bc](x)

I

G/H

(

d/iD L{y)

= J ^ f dt*I>{L[y)) G/H

(4.33)

>

) [r°D(Q,)

-

1 f.frT*r]

J

D{L~\y)).

< 4 - 34 >

(4.35)

Even though the actual derivation of (4.33) is cumbersome, the emergence of a four dimensional Lagrangian with invariance under four dimensional frame rotations, general coordinate transformations and gauge transformations does not come as a surprise. In fact, recall that these are just the symmetries left from the D-dimensional ones after imposing the Kaluza-Klein ansatz. The results as presented here are taken from [169]. More detailed derivations for certain special cases can be found in [24, 49, 122]. Also, in a somewhat different setting, a treatment is given in [96, 103]. In equation (4.33), the first three terms in the curly bracket provide a Riemann- and gauge- covariant derivative. The terms linear in the gauge field can be traced back to originate in Ej* dM and B^^F^. In addition to these, we have a Fierz-Pauli type term and a mass term. The first indicates that we have a non-renormalizable dimensionally reduced theory, a not too surprising observation since we started off with a non-renormalizable theory (Z)-dimensional gravity) in the first place. The coupling of the FierzPauli term is small, and the term may or may not play a role physically. Incidentally, the term actually disappears under certain specific circumstances [122]. The mass term, on the other hand, is causing troubles, which we will describe in more detail below. In fact, if it does not vanish, spinors will have a mass not smaller than Planck's mass. Some final remarks are in order. For the imbedding problem, let us consider an example [169]. We choose G/H = SU(3)/ U(2) ( = CP 2 ). The adjoint representations of SU(3) decomposes under U(2) = SU(2) X U(l) according to 8 = 10 + 3 0 + + 2_1( where the indices refer to the U( 1) factor

Fortschr. Phys. 32 (1984) 5

235

(hypercharge). We identify the generators of SU(2) and Z7(l) with the 3„ and the 1 0 , respectively. Accordingly, the ^-content of SU(S)/U(2) is 2 r -)- 2_j. As SU(3)/U(2) is four dimensional, the frame group will be S0(4). The label i on fi(x, y) will therefore take on four values. Now, two four dimensional representations of 80(4) are available. 80(4) is locally isomorphic to SU(2) X SU{2). The 8U(2) X {x) = [fpe{x)) as it appears in the dimensionally reduced Dirac Lagrangian. We notice that only the index p is associated with a gauge field. This is to be expected, as the index on y>i{z, y) is a JEf-index rather than a (?-index, and the gauge invariance is with respect to the group G. This means that the internal degree of freedom of y>{x) originating in the use of higher dimensions is not being gauged. (Under certain specific circumstances there are exceptions, cp. ref. [122], A typical example requires the coset space to be of the form H X H/H.) Now, in grand unified theories also only some portion of the available symmetry appears to be gauged. Whether the structure of Kaluza-Klein theories is relevant for such situations is not yet clear. 2.

The zero mode problem and parity violation

We have seen in the previous section that the dimensionally reduced Dirac Lagrangian contains a mass term. In fact, if we decompose the D-dimensional Dirac operator, r AdA = r ada + r ada, then the internal piece will act as a four dimensional mass operator, providing the Dirac field with a mass of the order of Planck's mass. Therefore only the zero mode sector, that is the solutions of

r'd^x,

y) = 0

(4.36)

would be of interest to us. The solutions of equation (4.36) form a representation of G. This is so since the operator r*d a is C?-invariant by construction. The zero mode sector gives a parity invariant four dimensional Dirac Lagrangian, unlike the massive sector. In fact, let us choose a representation for the generalized Dirac

236

W . MECKLENBURG, The Kaluza-Klein Idea

matrices such that J10 = y° (x) 1 (cp. also appendix B), where y" are standard four dimensional Dirac matrices. From the definition of da one then reads off that {r"d x , P®} — 0 for all a, so that the eigenvalue equation for the internal Dirac operator can be given the form r*d„y) = Thmip with P 5 = y6 (g) 1. The mass term therefore has the form yP5m^>. Since no other sources of P-violation are present we indeed conclude that the massive sector of the dimensionally reduced Dirac Lagrangian is P-violating and the massless sector is not. This argument is not quite complete as it stands, since we have left out the possibility that the eigenvector of equation (4.36) come in left-right asymmetric pairs. This subtlety can presently be neglected as it turns out that P'd, has no zero modes at all. This is a consequence of the identity (4.37)

which can be derived by using the definition of the curvature tensor, [d^, dv\ y> = R ^ ^ X and its symmetry properties, the vanishing of the covariant derivative of the Pmatrices and the anticommutation relations for the P-matrices. Now, on coset spaces of compact groups, —d"da and R have the same sign, moreover (compare appendix E), R is a nonvanishing constant. Therefore the right hand side of (4.37) is never zero, thus (P«4) 2 and consequently r a d a have no zero modes. Now, from the physical point of view, we would rather have zero modes and parity violation. We want the latter, as the fundamental fermions in unified theories tend not to come in left-right symmetric pairs. Therefore the question arises how we could get out of the impass outlined above. For scalar fields, the following method has been suggested [195]. Imagine we introduce a negative (mass)2 in the D-dimensional theory which would just compensate the mass of the lightest massive mode. We can then prevent the appearance of tachyonic modes by using twisted fields. In fact, recall that for twisted fields we can arrange matters such that the lowest mode does not appear. In this way one would arrive at a massless four dimensional theory if the higher modes are neglected. Such a procedure does not work in the present case. In fact, we could have to introduce imaginary masses which would spoil hermiticity requirements. What we could still do however is to impose a chirality condition; the troublesome mass term may then disappear identically. According to our general philosophy, the chirality condition must be imposed on the high dimensional spinors yi{x, y) and it must be compatible with the requirements of local Lorentz invariance and general coordinate invariance in D dimensions. Denoting P fl+1 = /\ rD, such a constraint is given by the Weyl condition PD+IV = ±ip. Weyl spinors exist in even dimensions only (for odd dimensions, T D+i is a multiple of the unit matrix). Note that for spinors subject to the D-dimensional Weyl constraint, four dimensional chirality (eigenvalues of P 4 ) and internal chirality (eigenvalues of A r D ) are related. Another constraint by which we may obtain masslessness for spinors in four dimensions, is the Majorana condition. The Majorana condition, imposed in D dimension s, may actually also enforce chirality in four dimensions. Majorana spinors exist in 2, 3, 4, 8 and 9 mod 8 dimensions. A detailed analysis of the relation between high dimensional Weyl and Majorana conditions and four dimensional chirality conditions has recently been carried out [201], compare also [116, 30]. The net result is that Weyl spinors in D = 6, 8 mod 8, Majorana spinors in D = 9 mod 8 and Weyl-Majorana spinors in D = 2 mod 8 dimensions are promising candidates for the provision of the desired objective of chirality in four dimensions.

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237

Quite a different possibility to obtain masslessness for spinors and at the same time left-right asymmetry is indicated by the Atiyah-Singer index theorem, (cp. [59]). This theorem relates the socalled index of the Dirac operator, ind (rD), to be defined as the difference in the number of left-handed and right-handed zero modes to topological quantum numbers referring to the underlying manifold. In a typical example, the index may be expressed in a linear way through the characteristic classes. The theorem applies to compact manifolds only. This however is perfectly sufficient for our present needs, as we are interested in the zero modes of the internal Dirac operator only. A typical scenario where left-right asymmetric zero modes have actually been realized is the following [85, 162]. Recall that in the CS model (the Einstein-Yang-Mills system) an external Yang-Mills field appears. Further we have seen that spontaneously compactifying solutions exist with nontrivial topological properties coded into the would-be Higgs part B„(y) of the external Yang-Mills field. In this case, the internal Dirac operator is modified, r*da r*(da + B«(y)). This modified Dirac operator now does have zero modes, on account of the Atiyah index theorem, since the characteristic classes belonging to Ba(y) do not vanish. Also, the zero modes do not come in left-right symmetric pairB. In this context it is to be noted that actually the addition of Ba(y) spoils in any case our previous argument on the positivity of the. internal Dirac operator. I t is to be stressed that implicitly underlying the present discussion of the zero mode and parity violation problem for Dirac spinors is the assumption that fundamental fermions are (nearly) massless and come in left-right asymmetric combinations. This need not be so and actually grand unified theories with left-right symmetric fermion multiplets have been discussed, cp. [25]. As to the masses it has been suggested [15] that fundamental fermions retain their high masses, but neutral (colour neutral) bound states may turn out to be nearly massless; this would be a confinement mechanism. Further, in a supersymmetric context, the relevant high dimensional field may be a Rarita-Schwinger field. This also alters the picture, as the properties of the Dirac operator are quite different from that of the Rarita-Schwinger operator. Some properties of the latter will be discussed in the following section. 3.

Rarita-Schwinger fields

" T h e " supergravity theory, the N = 8 theory, contains in its eleven dimensional formulation just one fermionic field, a Rarita-Schwinger field ipA(x, y). As a spinor in eleven dimensions this has 32 components. These are subject to a Majorana condition in order to provide the correct counting of states for a supersymmetric multiplet. From the four dimensional point of view, ipA decomposes into eight Rarita-Schwinger fields y>a (gravitinos) and 56 Dirac fields yv The dimensional reduction of the Rarita-Schwinger Lagrangian for if = MI X G/H has not yet been carried out. This section will therefore contain only a few general remarks. The candidates for the elementary fermions of some grand unified theory may be taken to be the y«, for these we face again the zero mode as well as the left-right asymmetry problem. ABC Let us consider the Rarita-Schwinger equation in the form r dBtpc(x, y) = 0, with DFF — -f- BB. The connection BB carries a vector and a spinor representation. B y definition, rABc = (rArBrc — rArcrB + cycl.), which can be rewritten as 6(rArBra + Vac^b — flAB^c — Vcb^a)- Contracting the Rarita-Schwinger equation with F A and dA respectively, one finds two constraints. Employing these one finds that the Rarita-

238

W . MECKLENBURG,

The Kaluza-Klein Idea

Schwinger equation may be replaced by MABy>B - (r)ABrcdc

- dArB) xpB = 0.

(4.38)

Unfortunately, the Rarita-Schwinger operator MAB lacks a simple positivity property as possessed by the Dirac operator. On the other hand, the Rarita-Schwinger equation simplifies considerably if the constraint rAipA = 0 is imposed. (Such a constraint may actually be necessary in order to obtain the correct number of states.) Instead of (4.38) we then have r B dsfA = 0. This looks very nearly like a set of Dirac equations, except that ds carries also a vector representation. On the other hand, we may equally well study the Rarita-Schwinger equation in the holonomic basis, r M d M f N = o.

(4.39)

Now, because of the constraint, rMdMipN == rM(dMipN — dNyM)- Therefore, in this form, no connection coefficients will be attached to the vector index. Therefore (4.39) is just a set of decoupled Dirac equations, and we have a zero mode problem as before for the Dirac operator. Also it is likely that problems with left-right asymmetry will have to be overcome. In fact recall that for the particularly interesting case of eleven dimensions, Weyl spinors do not exist, which deprives us at least of this way to obtain left-right asymmetry in four dimensions.

Y.

High-dimensional Tang-Mills Theories

A.

Introductory remarks

Yang-Mills theories are, like gravity theories, of intrinsically geometric structure and are thus naturally formulated for space-times of arbitrary dimensions. We have seen that high dimensional gravity theories provide a possible framework for the unification of gravitational and gauge interactions. We will see now that high dimensional Yang-Mills theories provide a possible framework for a geometrical unification of gauge and Higgs fields. In fact, many people consider the Higgs sector in unified theories as unnatural, artifical and arbitrary, and many attempts have been made to improve on such features. The basic idea of the present attempt is to introduce gauge and Higgs fields as different components of a high dimensional Yang-Mills field. This clearly restricts the number of possible Higgs structures. Recall also that the CS model of spontaneous compactification contains high dimensional Yang-Mills fields. In addition, in a supersymmetric context, Yang-Mills theories often occur naturally in high dimensional space-time. All this provides ample motivation to study high-dimensional Yang-Mills theories. We will see below that one can use a high-dimensional Yang-Mills theory to unify gauge and Higgs fields in the Weinberg-Salam model. It then turns out that the characteristic mass scale determining the internal radius is not Planck's mass like in the Kaluza-Klein picture above, but rather of the order of the mass for the lightest Higgs boson, that is, probably around 100 GeV. This observation suggests the fascinating speculation that already at such low energies we may be able to actually see the internal dimensions directly since at this energy high dimensional spacetime rotations would transform four dimensional and internal coordinates into each other. It is easy to see that a high dimensional Yang-Mills theory contains a four dimensional Yang-Mills-Higgs theory. Let the gauge field in D = 4 + K dimensions be VM(x, y)

Fortschr. Phys. 32 (1984) 5

239

= [AM{x, y), Am(X, y)) and let the field strength tensor be WMN = dMVN — 8kVM

AYM> VN\- NOW choose all fields to be independent of the internal coordinates, D.^V^ = 0. The action density tr {WMNW mn ) then contains the usual kinetic term for the four dimensional Yang-Mills fields, tr (WMNWMN), and the AM serve as Higgs fields in the adjoint representation of the gauge group, with a minimal coupling term tr (Wm/tWm''), WMFT = 8MA^ - e[AM, AM], and a potential tr {W^W1") = e2 tr ([A^, A']2). Note that as a result of the D-dimensional'rotational invariance of the theory, the self couplings of the Higgs field and the gauge field appears to be the same. The Higgs potential obtained in this way provides minimal energy solutions of the field equations AM = const, [A^, AV] = 0. Such solutions are degenerate; AM may be replaced by XAP, A = const. As a result Df this degeneracy, the solution is unstable in the classical sense [121]. Quantum corrections tend to lift the degeneracy entirely. The effective potential (including first quantum corrections) in general does not have nonvanishing minimal energy solutions at all. Therefore the above nontrivial solution is not expected to trigger a Goldstone-Higgs-Kibble mechanism [125]. Certain care has to be taken for supersymmetric theories, where the following theorem holds: If a degeneracy like the one above occurs in zeroth order perturbation theory, and if supersymmetry is not spontaneously broken, then the degeneracy persists to all orders in perturbation theory. In such a case one may still entertain hope that the exact dynamics of the system is such that a Goldstone-Higgs-Kibble phenomenon persists. Early attempts to construct a unified picture for gauge and Higgs fields using a simple procedure as outlined above are given in references [63, 64, 50, 182, 187—189]. Eventually it turned out that in order to obtain a suitable Higgs sector, a number of ad hoc assumptions had to be made. There was some hope that these could be justified by the use of graded gauge groups. This however has caused new problems [57, 58]. Because of the anticommutation relations in the graded gauge algebra, the theory will contain •wrong statistics fields. Also, gauge field kinetic terms appear with wrong signs. Other attempts have been made to cure these problems by identifying the wrong statistics fields with ghosts (see below). In the following we will in some detail describe the most attractive scheme which has been advocated mostly by Manton and collaborators [29, 67, 115, 116, 30]. There a method of dimensional reduction is employed where the condition B^VM — 0 is required to hold only up to a gauge transformation. In this way, some promising results can be obtained. (Cp. also [83, 84, 176]). After that, some remarks on modified Yang-Mills theories and a possible geometrical interpretation of ghosts follow. +

B.

Dimensional reduction of Yang-Mills theories

The requirement that d^Vu = 0 is to hold up to a gauge transformation provides an interrelation between «/-dependence and gauge symmetry. In this way, the constraint is weak enough to let us probably escape the stability problems mentioned above and strong enough to restrict severely admissible Higgs sectors. In fact, it is not yet clear whether the restrictions are not too strong to allow for the description of physically sensible situations. In the following section, ©-symmetric gauge fields will be introduced as solutions to the constraint d^VM = 0 (up to a gauge transformation). The general form of the constraint for the four dimensional fields will be given in the following section, and its solutions are described in the subsequent section. A further section is devoted to a discussion of spinors. 3

Fortschr. Phys., Bd. 32, H. 5

240

W.

1.

Symmetric gauge fields

MECKLENBURG,

The Kaluza-Klein Idea

We consider Yang Mills theories with gauge group R over space-time of the form