A Scalar-Tensor Theory of Gravity Based on Non-Commutative Geometry

The unified description of gauge and Higgs fields based on non-commutative geometry due to Connes has a similar structure to Kaluza-Klein theory, though it is a 4-dimensional theory. Its extension to a theory in curved spacetime is, then, expected to give a Brans-Dicke type gravity in which Higgs-like scalar fields couple to the gravitational field. In this paper, we study a B-D type gravity using the matrix coordinate method originated by Coquereaux. In particular, the introduction of two kinds of scalar fields is discussed in detail from the viewpoint of cosmology.