The Feynman kernel of a dimensionally reduced scalar field theory

We construct a consistent quantum field theory of a dimensionally reduced self-interacting scalar field. The Kaluza-Klein dimensional reduction on the well-known $\Phi^{4}$ scalar theory, on a certain $(4+n)$ spacetime with an arbitrary number of extra dimensions, induces a four dimensional reduced theory with scalar fields: the zeroth mode (`light' field) and an infinite number of KK-excited modes (`heavy' fields). This theory is quantized by Hamiltonian path integral methods. It is shown, from first principles, that non-trivial measure factors at the level of the functional measure are absent even if the whole set of heavy fields is taken into account. Hints on the regularization and renormalization process are briefly discussed.