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The conditions, under which the general description of the dynamical properties of quasiparticles is almost identical with those of real relativistic particles, are analyzed. Such analysis is, especially, actual today in connection with the growing interest in electronic properties of graphene and other nanostructures of carbon origin (fullerenes, nanotubes, etc.). The development of the traditional applications of quasiparticles (superfluidity, transfer of charge or energy) also requires a generalized analysis of dynamical properties of quasiparticles. The problem of the correlation of quantum and classical methods of description of the quasiparticles in the case of the excited states of crystals is considered. In order to focus attention on the discussed problem, the obtained results are demonstrated on the example of electronic excitations of crystals in the simplest case where other effects are neglected (phonons, defects, high density of excitations, which would require the account for interactions between them, the response of a lattice to excitations, and so forth). It is shown that such excitations can be described in three ways simultaneously. The first is the quantum description of the examined excitations in terms of wave functions and eigenvalues of energy. The second method is classical. It arises from the quantum method and is formulated in terms of the wave momentum. The third method, which follows from the second one, is also a description of the classical type, but is related to the other momentum – the mechanical one. The latter descriptions (the third or second one) make it possible to interpret the experimental data in terms of the usual relativistic dynamics. K e yw o r d s: quasiparticles, dispersive dependence, relativistic approximation, dynamical Dirac model, graphenes.

Some Aspects of Generalized Dynamics of Quasiparticles in Crystals with Unit Cell of Arbitrary Complexity