  We analyse in details the problems which one faces trying to quantize a
scalar field on the spacelike cylinder being the simple example of a spacetime
with closed timelike curves. Our analysis brings to light the fact that the
usual set of positive and negative frequency solutions of the field equation
turns out to be incomplete. The consequence of this fact is that the usual
formulation of quantum field theory breaks down on such a spacetime. We
postulate the completeness principle and build on its basis the modified
quantization procedure. As an example, the Hadamard function and $<\phi^2>$ for
the scalar field on the spacelike cylinder are calculated. It is shown that the
``naive'' method of images gives the same results of calculation.
