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Quantum Electrodynamics (QED)

QED is the quantum field theory describing the interaction of light and matter, a fundamental component of the Standard Model.
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The statement of the theorem

Define the Lagrangian density LQED\mathcal{L}_{QED} for the interaction of the electron field ψ\psi (Dirac spinor) and the photon field AμA_{\mu} (gauge boson) under U(1)U(1) symmetry: LQED=ψˉ(iγμ∂μ−me)ψ−14FμνFμν\mathcal{L}_{QED} = \bar{\psi} (i \gamma^{\mu} \partial_{\mu} - m_e) \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} where Fμν=∂μAν−∂νAμF_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} is the electromagnetic field strength tensor. The equation of motion for the gauge field is derived from the variation δS/δAμ=0\delta S / \delta A_{\mu} = 0, yielding the inhomogeneous Maxwell's equations: ∂μFμν=jν\partial^{\mu} F_{\mu\nu} = j_{\nu} where jν=eψˉγνψj_{\nu} = e \bar{\psi} \gamma_{\nu} \psi is the conserved electromagnetic current, ensuring charge conservation.