  The paper examines the emergence of gauge fields during the evolution of a
particle with a spin that is described by a matrix Hamiltonian with n different
eigenvalues. It is shown that by introducing a spin gauge field a particle with
a spin can be described as a spin multiplet of scalar particles situated in a
non-Abelian pure gauge (forceless) field U(n). As the result, one can create a
theory of particle evolution that is gauge invariant with regards to the group
U^n(1). Due to this, in the adiabatic (Abelian) approximation the spin gauge
field is an analogue of n electromagnetic fields U(1) on the extended phase
space of the particle. These fields are force ones, and the forces of their
action enter the particle motion equations that are derived in the paper in the
general form. The motion equations describe the topological spin transport,
pumping and splitting. The Berry phase is represented in this theory
analogously to the Dirac phase of a particle in an electromagnetic field. Due
to the analogy with the electromagnetic field, the theory becomes natural in
the four-dimensional form. Besides the general theory the article considers a
number of important particular examples, both known and new.
