Eric Myers

A theory of gravitation in more than four dimensions (Kaluza-Klein theory) is considered to see if it can explain in a natural, unified way the observed gauge and gravitational fields in four dimensions. The extra dimensions are taken to be closed and small enough that they cannot be observed directly. The possibility that quantum effects might cause the contraction of the extra dimensions is considered. The quantum effective potential (Casimir energy) of the gravitational field is calculated on the space-time manifold (Minkowski-space) x (N-sphere) to one-loop order in the loop expansion. (N must be odd for technical reasons.) A cosmological constant is included in the higher dimensional theory, although the observed cosmological constant in four dimensions is required to be zero. For positive values of the cosmological constant the effective potential is attractive, which means that the extra dimensions would tend to contract at least until they are of a size on the order of the Planck length. The minima of the effective potential, which determine the solutions to the quantum-corrected equations of motion, are located. The first case in which a solution meeting certain minimal requirements is found is N = 13, in which case the associated gauge group is SO(14). The gauge coupling constant in this theory is a predicted number. Similar solutions are found for N = 15, 17, 19, and 21. Unfortunately, in all cases the effective potential has an imaginary part, which is interpreted as an instability of the solution against quantum decay.