fibre bundles [@bp-normalF-LTP; @bp-TP-general; @bp-LTP-general], generalizing the theory of parallel transport, which is partially generalized here to the infinitely dimensional case.

The main object in quantum mechanics is the Hamiltonian (operator) which, through the Schrödinger equation, governs the evolution of a quantum system . In our novel approach its rôle is played by a suitable linear transport along paths in an appropriate (Hilbert) fibre bundle. It turns out that up to a constant the matrix-bundle Hamiltonian, which is uniquely determined by the Hamiltonian in a given field of bases, coincides with the matrix of the coefficients of this transport (cf. an analogous result in [@bp-BQM-preliminary sect. 5]). This fact, together with the replacement of the usual Hilbert space with a Hilbert fibre bundle, is the corner-stone for the possibility for the new formulation of the nonrelativistic quantum mechanics.

The present first part of our investigation is organized as follows.

In Sect. \[II\] are reviewed some facts from the quantum mechanics and partially is fixed our notation. Here, as well as throughout this work, we follow the established in the physical literature degree of rigor. But, if required, the present work can reformulate to meet the present-day mathematical standards. For this 