Physical Properties of Graphene

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Physical Properties of Graphene ( physical-properties-graphene )

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The 2D Dirac Equation 57 We may proceed in the same manner in the case of the D + 1 dimensional space-time, where a scalar is invariant under a Lorentz transformation L, L : s → s′ = s, whereas a D + 1 vector transforms as5 L: vμ→v′μ′ =Λμ′vμ, (3.29) in terms of the (D + 1) × (D + 1) matrix Λ. The generalisation to a tensor is again straight-forward. In contrast to the above-mentioned D-dimensional space, one must clearly distinguish between vectors with an upper index and those with a lower one. This is due to the special metric of space-time 1000 gμν=0−1 0 0,  0 0 − 1 0  0 0 0 −1 and the vectors with upper and lower index are related by the metric vμ = gμνvν. Thus, if vμ = (v0,v), vμ = (v0,−v), where v regroups the D space components and v0 is the time component of the D + 1 vector. One may, furthermore, define the scalar product between two D + 1 vectors as (v,w)≡vμwμ , which must, as the name “scalar” indicates, be invariant under a Lorentz transformation. As an example to illustrate these rather formal aspects, we consider the position vector xμ = (ct,r) in space-time or the momentum vector pμ = (E/c, p), which we have already alluded to at the beginning of this chapter. The scalar products yield the mass term, pμpμ = E2/c2 − p2 = m2c2, which is indeed a scalar, and the scalar xμxμ = c2t2 − r2 . (3.30) 5We use the Einstein convention, where one sums over repeating (upper and lower) indices omitting the sum sign, vμvμ ≡Xvμvμ . μ μ

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