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58 The Dirac Equation for Relativistic Fermions Instead of a classification of all types of Lorentz transformations, which may be found in standard text books on relativity and quantum field the- ory, e.g. [24, 25], we simply discuss some explicit examples. The simplest example is a rotation in the space part, which does not mix time and space components. One may indeed use the D-dimensional rotation matrix R, and embed it into a (D+1)×(D+1) matrix, (RD+1)μν=1 0. 0R A less trivial example is the so-called Lorentz boost, a transformation into a frame of reference that moves at the speed v with respect to the original one, in a direction that we may choose to be the x-direction. In this case, we need to account for the invariance of the scalar product xμxμ in Eq. (3.30), and we need c2t2 − x2 = c2t′2 − x′2, because the y- and z-components are not affected by the transformation. The transformation, which respects this metric, may be expressed in terms of hyperbolic functions,6 ct′ = ctcoshζ −xsinhζ x′ = −ctsinhζ+xcoshζ , which respects the metric because of cosh2 ζ − sinh2 ζ = 1. These functions of the “angle” ζ, which is also called rapidity, may be related to the velocity v of the Lorentz boost, by v2 v cosh ζ = r1 − c2 and tanh ζ = c , which is a manifestation of the time dilatation and the Lorentz contraction in a Lorentz boost. In matrix form, we may thus describe the Lorentz boost in the x-direction by coshζ −sinhζ00 Λμν=−sinhζ coshζ 0 0. (3.31) 0010 0001 6Remember that for a metric which leaves x2 + y2 invariant, we would have chosen trigonometric functions, which define indeed the rotation.PDF Image | Physical Properties of Graphene
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