  We start to develop the quantization formalism in a hyperbolic Hilbert space.
Generalizing Born's probability interpretation, we found that unitary
transformations in such a Hilbert space represent a new class of
transformations of probabilities which describe a kind of {\it hyperbolic
interference}. The most interesting problem which was generated by our
investigation is to find experimental evidence of hyperbolic interference. The
hyperbolic quantum formalism can also be interesting as a new theory of
probabilistic waves that can be developed parallely to the standard quantum
theory. Comparative analysis of these two wave theories could be useful for
understanding of the role of various structures of the standard quantum
formalism. In particular, one of distinguishing feature of the hyperbolic
quantum formalism is the restricted validity of the superposition principle.
