  An origin and necessity of so called conformal (or,Penrose-Chernikov-Tagirov)
coupling of scalar field to the metric of n-dimensional Riemannian space-time
is discussed in brief. The corresponding general-relativistic field equation
implies a one-particle (quantum mechanical) Schrodinger Hamiltonian which
depends on the space-time dimensionality n, contrary to the Hamiltonian
constructed by quantization of geodesic motion, which is the same for any value
of n. In general, the Hamiltonians can coincide only for n = 4, the
dimensionality of the ordinarily observed Universe. In view of the fundamental
role of a scalar field in various cosmological models, this fact may be of
interest for models of brane worlds where n > 4 .
