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