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60 The Dirac Equation for Relativistic Fermions which yields, with the above definitions, S − 1 ( L ) Λ μν p ν γ μ S ( L ) − m c ψ ( r , t ) = 0 . ( 3 . 3 6 ) Comparison with Eq. (3.35), thus, imposes S−1(L)γμS(L) = Λμνγν. (3.37) Parity and time reversal Eq. (3.37) allows us to construct the matrices S(L) for the different Lorentz transformations. As already mentioned above, we are not interested, in the case of graphene, in those corresponding to Lorentz boosts. Instead, we con- centrate on the parity and time-reversal operations for the 2D Dirac equation. Notice that, in this case, the γ matrices read γ0 = σz and γi = σzσi, for i = x, y. For the parity operator, Eqs. (3.33) and (3.37) impose S−1(P)σzS(P) = σz and S−1(P)σzσiS(P) = −σzσi. One notices that the choice7 S(P) = σzP (3.38) (3.39) satisfies this equation, where P replaces r by −r in the argument of the spinor. Notice that the eigenstates of the parity operator are only eigenstates of the 2D Dirac Hamiltonian in the particle’s rest frame for m ̸= 0 and p = 0. In their rest frame, particles and anti-particles have, thus, opposite parity. The definition of the time-reversal operator is slightly more involved. Eqs. (3.34) and (3.37) impose S−1(PT)σzS(T) = −σz and S−1(T)σzσiS(T) = σzσi. In this case, one needs to accompany the 2 × 2 transformation matrix σy by the conjugation operation C : z → z∗, and one finds S(T) = σyCT , (3.40) 7An equally valid choice is σz multiplied by a phase factor exp(iφ).PDF Image | Physical Properties of Graphene
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