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52 1 The Dirac Equation for Relativistic Fermions +aσ +aσ +aσ,intermsoftherealnumbersa,a,a,and α3=a0 1x 2y 3z 012 a3. The anti-commutation requirement {α3, σi} = 2(a0 + ai) = 0 is satisfied only if we choose ai = −a0, for all values of i. However, this is at odds with the condition α32 = 1 because one would have α32 = 2a0(2 − σx − σy − σz). It is, therefore, impossible to represent the matrices β and αi by 2×2 matrices. Due to the conditi1 representing massive 3D pa1 on that N must be even, the lowest possible value for rticles is N = 4, and indeed one may satisfy the Clifford algebra (3.14) if one chooses, e.g., the so-called standard representa- tion i and αi= i , 1 00σ (3.18) ay be verified in a straight- forward manner. As a consequence, the Dirac Hamiltonian has also a 4 × 4 β= 0− where each component is a 2 × 2 matrix, as m1 σ0 matrix structure, in terms of the 2-spinors φ=φ1 and φ2 χ=χ1 . χ2 cp·σ H3D = 2 , mc m̸=0 cp · σ −mc2 (3.19) ( 3 . 2 0 ) and the quantum states must be represented by 4-spinors, Ψ = χφ In order to obtain a better understanding of this spinor form, we may inves- tigate the most simple case of a particle in its rest frame, i.e. with p = 0, in which case the Hamiltonian (3.19) becomes diagonal. The 2-spinors φ and χ are, therefore, decoupled, mc2φ = Eφ and − mc2χ = Eχ. One immediately notices that the components φ correspond to components of positive energy (particle components) and χ to those of negative energy (anti-particle components). One may furthermore show that the components φ1, φ2 and χ1, χ2 of the 2-spinors φ and χ, respectively, correspond to the different possible spin states of the particle [24, 25].PDF Image | Physical Properties of Graphene
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