Foundations of Quantum Field Theory. Klaus D Rothe

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      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 (boost in the +z direction)

      where we have set

      Equation (2.20) just represents the first two leading terms of the expansion of the finite Lorentz boost (2.15) in the x³-direction around θ = 0. We thus identify

⁰ⁱ with the generators for pure velocity transformations along the z-axis. Notice that these matrices are anti-hermitian. On the other hand, the hermitian matrices

^ij generate rotations in the (ij)-plane:

      the generators of rotations about the z-, x- and y-axis respectively. One verifies from (2.18) that the generators of Lorentz transformations satisfy the Lie algebra

      Explicitly

      This algebra simplifies if we define the generators

      which СКАЧАТЬ