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Chapter 3 The Dirac Equation for Relativistic Fermions As we have discussed in the last chapter, the low-energy properties of elec- trons in graphene are described by the Hamiltonian (2.36), Hqξ =ξh ̄vFq·σ which happens to be of the form of the Dirac equation describing relativistic massless fermions. We, therefore, devote this chapter to a brief introduction to relativistic quantum mechanics with a particular emphasis on two spatial dimensions, which are of interest in the context of graphene. This chapter is not meant to be a complete review of relativistic quantum mechanics of quantum field theory, and we refer the interested reader to more specialised books on this fascinating field of physics [24, 25, 26]. We have, furthermore, tried to be as concrete as possible and to avoid formal aspects related to the Lorentz invariance of relativistic space-time, which may be useful for high- energy physicists but which are less relevant for condensed-matter physicists. After retracing the historical path, which led to the elaboration of rel- ativistic quantum mechanics and quantum field theory in the late 20ies of the last century, we concentrate on the particular form of the Dirac equation in 2D (i.e. two space and one time dimensions) and its amazing physical consequences such as the localisation problem, the resulting jittery motion (“Zitterbewegung”), and Klein tunneling. 45PDF Image | Physical Properties of Graphene
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