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The relativistic Foldy-Wouthuysen transformation is used for an advanceddescription of planar graphene electrons in external fields and free(2+1)-space. It is shown that the initial Dirac equation should by based on theusual $(4\times4)$ Dirac matrices but not on the reduction of matrix dimensionsand the use of $(2\times2)$ Pauli matrices. The latter approach does not agreewith the experiment. The spin of graphene electrons is not the one-value spinand takes the values $\pm1/2$. The exact Foldy-Wouthuysen Hamiltonian of agraphene electron in uniform and nonuniform magnetic fields is derived. Theexact energy spectrum agreeing with the experiment and exact Foldy-Wouthuysenwave eigenfunctions are obtained. These eigenfunctions describe multiwave(structured) states in (2+1)-space. It is proven that the Hermite-Gauss beamsexist even in the free space. In the multiwave Hermite-Gauss states, grapheneelectrons acquire nonzero effective masses dependent on a quantum number andmove with group velocities which are less than the Fermi velocity. Grapheneelectrons in a static electric field also can exist in the multiwaveHermite-Gauss states defining non-spreading coherent beams. These beams can beaccelerated and decelerated.

Foldy-Wouthuysen transformation and multiwave states of a graphene electron in external fields and free (2+1)-space