Volume 149B, number 1,2,3 PHYSICS LETTERS 13 December 1984 ON THE CONSISTENCY OF THE KALUZA-KLEIN ANSATZ M.J. DUFF, B.E.W. NILSSON, C.N. POPE and N.P. WARNER t The Blaekett Laboratory,ImperialCollege,London SW7 2BZ, UK Received 16 August 1984 We show that the standard Kaluza-Kleon ansatz for masslessgauge fields is in general inconsistent with the higher-di- mensional field equations. Exceptions are provided by certain S7 compaetifications of d = 11 supergravity. In modern approaches to Kaluza-Klein theories, the extra (k) dimensions are treated as physical and are not to be regarded merely as a mathematical device. In the framework, therefore, it is essential that at every stage in the derivation of the effective four-dimension- al field theory one maintains consistency with the higher-dimensional field equations. To derive this ef- fective theory one selects the ground state of (space- time) X (compact manifold Mk) and performs a gener- alized Fourier expansion of all the fields in terms of harmonics on Mk. Provided one retains all the modes in this expansion (i.e. all the massive states) then no such problems of inconsistency can arise. Moreover, it is well known that if Mk has isometry group G then the theory includes massless Yang-Mills bosom with gauge group G. (Assuming, as we shall for the time being, that any other matter fields non.zero in the ground-state are singlets under G). In practice, however, one is often interested in ex- tracting an effective "low energy" theory by discard- ing all but a finite number of states including the massless graviton, the massless gauge fields and other matter fields which are usually (but not necessarily) massless. For historical reasons this procedure is often called the "Kaluza-Klein ansatz". Despite the name, this should not be an ad hoc procedure but should correspond to retaining only the appropriate Fourier modes, for example, the zero eigenvalue modes for the mass operators in the case of the massless particles. It I Permanent address: Lauritsen Laboratory, Caltech, Pasadena, CA 91125, USA. 90 is generally believed that the correct ansatz for the metric tensor ~MN(X, y) is guv(x,y)=guv(x) +Aua(x)A v#(x)Kma(y)Kn#(y)gmnQV), gun (x, y) = A u a(x)K m a(y)~mn (y) , ~mn(X,y) =~mn(Y), (1) where the coordinates xU (/a = 1, ..., 4) refer to space- time andy m (m = 1 .... , k) to the extra dimensions and gmn(Y) is the metric on Mk. The quantities Km~(y) are the Killing vectors corresponding to the isometrics of this metric and t~ runs over the dimension of the isometry group G. The claim that this is the correct ansatz is based on the observation that substituting (1) into the higher-dimensional action and integrating over y, one obtains the four-dimensional Emstein-Yang- Mills action with metric guy (x) and gauge potential A~(x). But as we have already emphasized the correct Kaluza-Klein ansatz must be consistent with the high- er-dimensional fieM equations and, as we shall now demonstrate, eq. (1) does not in general satisfy this criterion. As has already been stressed elsewhere [1,2] one obvious source of inconsistency is the neglect of scalar fields in eq. (1). For example, setting ~SS = 1 in the d = 5 pure gravity theory is inconsistent with the ^ R55 = 0 components of the Einstein equation which would force FuvFUV to vanish. In this case, the reme- dy is simple: one includes the single massless scalar 0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)