Foundations of Quantum Field Theory. Klaus D Rothe

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      With Aμ real, we choose the polarization tensors to be real. The choice of the Fourier coefficients is then dictated by the reality of the electromagnetic field. In particle language it corresponds to the fact that the photon is its own antiparticle!

      The Fourier coefficients will eventually be identified with the creation (destruction) operators of 1-particle states. However, only two of these states can correspond to photons of helicity +1 and −1. In the Coulomb gauge this becomes manifest. This gauge, thus often referred to as the “physical gauge”, has however the drawback of not being manifestly Lorentz invariant.

5.2The radiation field in the Coulomb gauge

      The Coulomb gauge

      can be reached by performing the gauge transformation (5.3) with

      In this gauge, the free Maxwell equations for the vector potential read:

      Setting μ = 0, it follows from here that

      This is Laplace’s equation; it only has the trivial solution if we require that the vector potential tends to zero at infinity. In the Coulomb gauge we thus have in the absence of sources, A⁰ = 0. This shows that in this gauge the vector potential possesses only two degrees of freedom, corresponding to a radiation field, in agreement with our general considerations in Section 6 of Chapter 2, showing that a zero mass particle can exist only in two helicity states. Correspondingly we have for the general solution of (5.7) in the gauge (5.6)

      with , and λ the helicity, where we have used (5.13). We are allowing for both helicity states, since parity is conserved. We now show how to choose these polarization tensors for a real field.

      Define the two orthogonal vectors

      Introducing as in Section 6 of Chapter 2 the light-like standard 4-momentum k by

      describing the motion of a photon in z-direction with energy we see that the standard polarization tensors (5.9) are in fact eigenvectors of the helicity operator

      Indeed,

      Inspired by our considerations in Section 6 of Chapter 2 on the Little group, we now define the polarization tensors for a photon with general momentum by

      where are the elements of the Little group taking the standard vector into the final vector kμ (see 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:

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