The world is viewed as a many-dimensional manifold. All fields originate in its metric field. The non-space-time dimensions are identified with the internal degrees of freedom of elementary particles. The projection onto the space-time submanifold leads to a number of vector fields with internal momenta as coupling constants. All of these except one, charge, can be eliminated by suitably choosing internal co-ordinates with respect to space-time. The presumably more quantumlike dynamics of the internal degrees of freedom leads to an adiabatic separation in which space-time dynamics is determined by averaging over internal motions. This is shown to lead to Klein-Gordon or Dirac-like wave equations for the space-time-dependent motion of the particle fields. The approach shows naturally how the macroscopic long-range fields, gravitation and electromagnetism, arise, differ from one another and from the remaining quantum-mechanical forces.