in the family of fundamental physical theories based on the background of fibre bundles.

The idea of geometrization of quantum mechanics is an old one (see, e.g., [@Kibble-79] and the references therein). A good motivation for such approach is given in [@Anandan-90a; @Kibble-79]. Different geometrical structures in quantum mechanics were introduced , for instance such as inner products(s) [@Fock-FQM; @Messiah-QM; @Kibble-79; @Anandan-90b], (linear) connection [@Anandan-90a; @Anandan-90b; @Uhlmann-91a], symplectic structure [@Anandan-90a], complex structure [@Kibble-79], etc. The introduction of such structures admits a geometrical treatment of some problems, for instance, the dynamics in the (quantum) phase space [@Kibble-79] and the geometrical phase [@Anandan-90a]. In a very special case a gauge structure, i.e.  a parallel transport corresponding to a linear connection, in quantum mechanics is pointed out in [@Wilczek-Zee]. For us this work is remarkable with the fact that the equation (10) found in it is a very ‘ancient’ special version of the transformation law for the matrix-bundle Hamiltonian, derived in this investigation, which, together with the bundle (analog of the) Schrödinger equation, shows that (up to a constant) with respect to the quantum evolution the Hamiltonian plays the rôle of a gauge field (connection). In [@Uhlmann-91a; @Uhlmann-91b] one finds different (vector) bundles 