Foundations of Quantum Field Theory. Klaus D Rothe

Foundations of Quantum Field Theory

Foundations of Quantum Field Theory - Klaus D Rothe

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href="#uf5c05b08-0eb6-5a72-a997-3d4c1c92950e">Chapter 2),

figure

      The boost matrix figure represents a boost of figure 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

figure

      where we have used

figure

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

figure

      we easily verify the following properties of the polarization tensors:

figure

      The last property follows from the following manipulations:

figure

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

figure

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