of Prugovečki’s and our approaches to the quantum theory will be done elsewhere.

Another geometric approach to quantum mechanics is proposed in [@Graudenz-94] and partially in [@Graudenz-96], the letter of which is, with a few exceptions, almost a review of the former. These works suggest two ideas which are quite important for us. First, the quantum evolution could be described as a (kind of) parallel transport in an infinitely dimensional (Hilbert) fibre bundle over the space-time. And second, the concrete description of a quantum system should explicitly depend on (the state of) the observer with respect to which it is depicted (or who ‘investigates’ it). These ideas are incorporated and developed in our work.

From the known to the author literature, the closest to the approach developed in this work is  which contains an excellent motivation for applying the fibre bundle technique to nonrelativistic quantum mechanics.[^4] Generally said, in this paper the evolution of a quantum system is described as a ‘generalized parallel transport’ of appropriate objects in a Hilbert fibre bundle over the 1[-]{}dimensional manifold $\mathbb{R}_+:=\{t :  t\in\mathbb{R}, t\ge0\}$, interpreted as a ‘time’ manifold (space). We shall comment on reference  in the second part of this series, after developing 