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56 The Dirac Equation for Relativistic Fermions 2D Dirac Hamiltonian (3.16), and one may even express it in terms of the chirality operator H2D = |p|hp, and Hpξ = ξH2D = ξ|p|hp , where the second expression takes into account the two-fold valley degener- acy, in terms of the valley isospin ξ = ±. The band index λ, which describes the valence and the conduction band, is, therefore, entirely determined by the chirality and the valley isospin, and one finds λ = ξη , (3.28) which is depicted in Fig. 3.2. Lorentz transformations of scalars and vectors Although the covariance of the Dirac equation and Lorentz transformations in general are less relevant for the condensed-matter physicist working on graphene, we intend to give a very brief overview on these slightly more formal aspects. They allow us to define a parity and a time-reversal operator, which are the special forms of Lorentz transformations useful in the context of graphene. In special relativity, all physical laws must be invariant under the Lorentz transformations (e.g. rotations and Lorentz “boosts”) of the D + 1 dimen- sional space time. This the meaning of the statement that all physical equa- tions must be covariant. As for the more common D-dimensional space, one may distinguish different mathematical objects in this space with respect to their behaviour under a space symmetry transformation4 R: a scalar remains invariant under this transformation, R : s → s′ = s , whereas a vector is transformed by the particular law R : v → v′ = Rv, where R is the rotation matrix associated with R. One may, furthermore define a tensor (of rank r), all the r components of which are transformed by the rotation matrix R : T i1 ,i2 ,...,ir → T ′i′1 ,i′2 ...,i′r = Ri′1 ,i1 Ri′2 ,i2 ...Ri′r ,ir T i1 ,i2 ,...,ir . 4We consider, for simplicity a rotation in D = 3, which is a symmetry operation for an isotropic space.PDF Image | Physical Properties of Graphene
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