On a Riemannian method in quantum theory about curved space-time

We start from a given Lorentz metric and a vector field of world lines along which observers and measure devices may move. We describe a procedure to associate one-particle Hilbert spaces and one-particle Hamiltonians to space-like hypersurfaces using a transition to a Riemannian metric. With the aid of suitable boundary conditions one can confine the particle within a world tube (box quantization about curved space-time manifolds).Dedicated to Professor Ivan Úlehla on the occasion of his sixtieth birthday.