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Relativistic Wave Equations 51 Influence of the space dimension on the matrix dimension N Notice that the dimension N of the matrix must not be confunded with the space dimension D. The latter influences, however, the possible choices for N. In the case of two space dimensions (D = 2), which is of interest in graphene, we have three mutually anti-commuting objects, β, α1, and α2. It happens that there exist, with the Pauli matrices, σx=0 1,σy=0 −i,σz=1 0 , (3.15) 1 0 i 0 0 −1 a set of three mutually 2 × 2 matrices, which satisfy the Clifford algebra (3.14). One may identify3 β = σz, α1 = σx, and α2 = σy, which yields the 2D Dirac Hamiltonian H2D = cσ · p + mc2σz, (3.16) where σ = (σx, σy), as in the previous chapter. If we identify c with the Fermi velocity vF , one immediately sees that this Hamiltonian, for m = 0, has the same form as the effective Hamiltonian (2.36) derived in the last chapter for the description of the low-energy properties of electrons in graphene, at the K′ point (ξ = 1). The effective Hamiltonian at the K point (ξ = −1) may be identified with a representation of the Dirac Hamiltonian in terms of the matrices β = −σz, α1 = −σx, and α2 = −σy, which are an equally valid choice for representing the Clifford algebra (3.14). For three space dimensions (D = 3), the representation of the matrices β and αi is slightly more involved. Whereas one may choose again the 2 × 2 Pauli matrices for zero-mass particles, i.e. when one may omit the matrix β, H3D = cα · p, (3.17) m=0 with e.g. α1 = σx, α2 = σy, and α3 = σz, this is no longer possible for massive 3D p1articles, as may be seen from the following argument. Let us assume that the Pauli matrices represent α1, α2, and β, respectively. One must, therefore, find a 2 × 2 matrix, α3, which anti-commutes with the Pauli matrices. However, the latter also form, together with the 2 × 2 unit matrix , a basis of all 2 × 2 matrices. One may, thus, decompose 3Notice that all other permutations are equally possible.PDF Image | Physical Properties of Graphene
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