want to mention several works in which attempts are made for a (partial) formulation of nonrelativistic quantum mechanics in terms of fibre bundles.

It seems that for the first time the real bundle approach to quantum mechanics is developed in  where the single Hilbert space of quantum mechanics is replaced with an infinitely many copies of it forming a bundle space over the 1[-]{}dimensional ‘time’ manifold ([i.e. ]{}over $\mathbb{R}_+$). In this Hilbert fibre bundle the quantum evolution is (equivalently) described as a kind of ‘parallel’ transport of appropriate objects over the bundle’s base.

Analogous construction, a Hilbert bundle over the system’s phase space, is used in the Prugovečki’s approach to quantum theory (see, e.g. the references in [@Drechsler/Tuckey-96]).

In [@Wilczek-Zee] is first mentioned about the gauge, [i.e. ]{}linear connection, structure in quantum mechanics. That structure is pointed to be connected with the system’s Hamiltonian. This observation will find natural explanation in our work.

Some ideas concerning the interpretation of quantum evolution as a kind of a ‘parallel’ transport in a Hilbert bundle can also be found in [@Graudenz-94; @Reuter].

After this introduction, we want to present some non-exactly rigorous ideas and statements whose only purpose is the *motivation* for applying 