Foundations of Quantum Field Theory. Klaus D Rothe

2.2Classification of Lorentz transformations

      The Lorentz invariance of the scalar product vμvμ = v² allows us to divide the 4-vectors into three classes which cannot be transformed into each other by a Lorentz transformation:

      (a)vμ time-like (v² > 0)

      (b)vμ space-like (v² < 0)

      (c)vμ light-like (v² = 0)

      This means in particular that space-time separates, as far as Lorentz transformations are concerned, into three disconnected regions referring to the interior and exterior of the light cone x² = 0, as well as to the surface of the light cone itself. The trajectory of a point particle localized at the origin of the light cone at time t = 0 lies within the forward light cone; Moreover, if we attach a light cone to the particle at the point where it is momentarily localized, the tangent to the trajectory at that point does not intersect the surface of that light cone.

      The Lorentz invariant

      taken along the trajectory of the particle is just the proper time, measured in the rest frame of the particle.

      From (2.8) follow two important properties of Lorenz transformations:

      implying

      Note that this allows for four types of transformations which cannot be smoothly connected by varying continuously the parameter labelling the transformation. We thus have four possibilities characterizing the Lorentz invariance of the differential element (2.2):

      Only the first set of transformations is smoothly connected to the identity and hence form a Lie group. The remaining transformations do not have the group property. They are obtained by adjoining to the transformations in

space reflections, space-time reflections and time inversion, respectively, as represented by the matrices

      Only

represents an exact symmetry of nature.

       General form of a Lorentz boosts

      From (2.3) one has for a boost in the z-direction, taking the particle from rest to a momentum

,³

      where⁴

      Since

      we may parametrize γ and βγ as follows:

      or (2.13) now reads

      

      Note that this is a hermitian matrix! For a boost in an arbitrary direction one can show that the corresponding matrix elements are given by (note that

)

2.3Lie algebra of the Lorentz group

      Consider an infinitesimal Lorentz transformation in 3+1 dimensions. It is customary to parametrize its matrix elements as follows. In the case of

we are dealing with a six-parameter group parametrized by the “velocities” associated with boosts, and the Euler angles associated with the rotations. This is just the number of independent components of an antisymmetric second rank tensor. In analogy to the rotation group it is customary to write for the matrix elements of an infinitesimal Lorentz transformation

      Using the metric tensor as raising and lowering operators for the indices, we further have

      We may rewrite this transformation in matrix form as follows

      where

, with the property

      and the matrix elements⁵

      Here again it is implied that Lorentz indices can be raised and lowered with the aid of the metric tensor gμν = gμν. Expression (2.17) is just the infinitesimal expansion of

      

      Explicitly we have

      Hence for a pure Lorentz transformation in the −z-direction

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