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The 2D Dirac Equation 59 However, Lorentz boosts play a subordinate role in the discussion of elec- trons in graphene, as compared for example to the parity transformation, 1000 Pμ=0−1 0 0, (3.32) ν 00−10 0 0 0 −1 which transforms the space components, r → −r, but leaves the time com- ponent invariant, and the time-reversal transformation −1 0 0 0 Tμ=0 100, (3.33) ν 0 0 1 0 0001 which inverts the direction of time flow, t → −t, but leaves invariant the space components. Lorentz transformations of spinors In the last paragraph, we have briefly reviewed the covariance of classical relativistic mechanics, i.e. discussed how the laws of physics, in terms of scalars, vectors, and tensors, behave under Lorentz transformations. If we consider the quantum aspect of the laws of physics, we must also analyse how the state vectors (spinors) in the Hilbert space of quantum mechanics transform in a Lorentz transformation. The spinor in a transformed frame of reference must be related to that of the original one, ψ′(r′,t′) = S(L)ψ(r,t) ⇔ ψ′(r′,t′) = S(L)ψ(Λ−1)μνxν , (3.34) where (Λ−1)μν is the inverse of the matrix, which corresponds to the Lorentz transformation L, and S(L) is an N × N matrix that acts on the spinor. It is, furthermore, convenient to rewrite the Dirac equation (3.13) in a covariant manner, (pμγμ − mc) ψ(r, t) = 0, (3.35) where we have defined the γ matrices γ0 = β and γi = βαi. In order for this equation to be Lorentz invariant, i.e. valid in any frame of reference, we must have p ′μ γ μ − m c ψ ′ ( r ′ , t ′ ) = 0 ,PDF Image | Physical Properties of Graphene
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