Dynamics of the scalar field in 5-dimensional Kaluza-Klein theory

Using the language of differential forms, the Kaluza-Klein theory in 4+1 dimensions is derived. This theory unifies electromagnetic and gravitational interactions in four dimensions when the extra space dimension is compactified. Without any ad-hoc assumptions about the five-dimensional metric, the theory also contains a scalar field coupled to the other fields. By a conformal transformation the theory is transformed from the Jordan frame to the Einstein frame where the physical content is more manifest. Including a cosmological constant in the five-dimensional formulation, it is seen to result in an exponential potential for the scalar field in four dimensions. A similar potential is also found from the Casimir energy in the compact dimension. The resulting scalar field dynamics mimics realistic models recently proposed for cosmological quintessence.