  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.
