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Quantum electrodynamics (QED) is the quantum field theory believed to describe electromagnetic interaction. It is the simplest example of a quantum gauge theory, where the gauge group is abelian, U(1).
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Spin-1 polarization vectors (massive particle)
The Lagrangian for a massive vector field (without sources) has the form
$$ L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{2}M^{2}A_{\mu}A^{\mu}$$
and is not gauge invariant due to the mass term. The …