A Discretized Version of Kaluza-klein Theory with Torsion and Massive Fields

We consider an internal space of two discrete points in the fifth dimension of the Kaluza-Klein theory by using the formalism of noncommutative geometry developed in a previous paper [1] of a spacetime supplemented by two discrete points. With the nonvanishing internal torsion 2-form there are no constraints implied on the vielbeins. The theory contains a pair of tensor, a pair of vector and a pair of scalar fields. Using the generalized Cartan structure equation we are able not only to determine uniquely the hermitian and metric compatible connection 1-forms, but also the nonvanishing internal torsion 2-form in terms of vielbeins. The resulting action has a rich and complex structure, a particular feature being the existence of massive modes. Thus the nonvanishing internal torsion generates a Kaluza-Klein type model with zero and massive modes. ∗ On leave of absence from the High Energy Division, Centre of Theoretical Physics, P.O.Box 429 Bo Ho 10000, Hanoi Vietnam. † Address after September 1, 1995: Physics Department, Rockerfeller University, New York, NY, USA