Foundations of Quantum Field Theory. Klaus D Rothe

Chapter 2),

      The boost matrix represents a boost of along the 3-direction to kμ = (|k|, 0, 0, |k|) and leaves the standard polarization tensors unchanged since it only affects the 0 and 3 elements, leaving the (1, 2) elements unchanged. Hence

      where we have used

      with R the matrix rotating the standard momentum into the direction of . Noting that

      we easily verify the following properties of the polarization tensors:

      The last property follows from the following manipulations:

      where we have used the Lorentz transformation property (2.6) of a covariant 4-vector. The polarization tensor satisfies the following completeness relation: