the fibre bundle formalism to quantum mechanics. Another excellent arguments and motives confirming this approach are given in .

Let ${\mathit{M}}$ be a differentiable manifold, representing in our context the space in which the (nonrelativistic) quantum-mechanical objects ‘live’, i.e.  the usual 3-dimensional coordinate space (isomorphic to $\mathbb{R}^3 $ with the corresponding structures).[^10] Let $\gamma\colon J\to {\mathit{M}}$, $J$ being an $\mathbb R$-interval, be the trajectory of an observer describing the behaviour of a quantum system at any moment $t\in J$ by a state vector $\Psi_\gamma(t)$ depending on $t$ and, possibly, on $\gamma$.[^11] For a fixed point $x=\gamma(t)\in {\mathit{M}}$ the variety of state vectors describing a quantum system and corresponding to different observers form a Hilbert space ${\mathit{F}}_{\gamma(t)}$ which *depends on $\gamma(t)=x$, but not on $\gamma$ and $t$ separately*.[^12]

\[Rem-base\] As we said above in footnote \[footnote-base\], the next considerations are completely valid mathematically if ${\mathit{M}}$ is an arbitrary differentiable manifold and $\gamma$ is a path in it. In this sense ${\mathit{M}}$ and $\gamma$ are free parameters in our theory and their concrete choice is subjected only to *physical reasons*, first of all, ones requiring adequate physical interpretation of the resulting theory. Typical candidates for ${\mathit{M}}$ are: the 3[-]{}dimensional Euclidean space $\mathbb{E}^3$ 