SYMMETRY IN KALUZA-KLEIN THEORY

A method is described for making harmonic expansions on the internal space of a Kaluza-Klein vacuum in cases where this space is a coset space. This method fully exploits the symmetry of the space and should be useful for the analysis of excitation spectra and, in particular, for constructing the correct zero-mode ansatz in cases where the multi-dimensional gravitational fields are coupled to matter fields of various kinds. 1. iThe Kaluza-Klein idea is that, underlying the 4-dimensional : world of low energy physics, there is a multi-dimensional structure which could he responsible for the internal symmetries that we oh! serve. It is postulated that the scale of this structure is small ' enough to render it unresolvable at presently available energies. For a dynamical mechanism to generate such a structure one turns to general relativity "generalized to more than four dimensions. That is, one supposes that the underlying theory involves a multi-dimensional : metric tensor and perhaps some "matter" fields coupled according to the usual rules of general relativity. One then argues that the •ground state values of the fields (including the metric tenser) aust, solve the classical field equations and must have as much symmetry as •possible. (The latter requirement reflects the familiar experience that more symmetrical states tend to have lower energy.) To find a isuitable ground state configuration is the main goal. How, to he isuitable it should have the geometry, at least locally, of a product ! • h X k |space, M x K , where M is a four-dimensional space of Minkpwskian signature (preferably Minkouskl space itself) and M is some ' K-dimensional compact space of very small size. Both M and M^ should admit groups of motions. If M is Minkovski space then its group of motions is the Poincare group, if it is de Sitter (or anti.de Sitter) space then the group is 0(1,1+) (or 0(2,3)). The "internal" space Vi should admit some compact group, G. |The ground state should be stable against small perturbations. The propagation of these perturbations or excitations, can be analyzed explicitly. By stability, it is meant that the excitation spectrum includes no tachyons or ghosts. Those modes which involve propagat tionion ft. will generally have large masses (proportional to the inverse size of ! *r) and hence would remain unexeited at present energies. The zero-modes, if any, would correspond to massless particles and these are important. In fact a principal aim cf the programme is to represent these zero-modes of M by local fields on H and to set up an effective Lagrangian for them. ' To analyze the excitation spectrum it is of course necessary to make full use of the ground state symmetry. The excitations will t be classified in irreducible representations of this symmetry. The general features of the procedure are familiar in lt~dimensional theories where a Lagrangian may possess some gauge symmetry local or global and the first question to be settled is always that of the vacuum, or ground state, symmetry. This involves examining the equations of motion (for the scalar fields in particular, if the vacuum is to "be Poincare invariant). Once the vacuum field configuration has been determined, Its symmetry can be found by inspection. If this vacuum symmetry is smaller than the Lagrangian symmetry, then