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Gauge Theory

The Standard Model is built upon gauge theory, describing fundamental forces through the exchange of force-carrying particles (gauge bosons).
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The statement of the theorem

Let G\mathcal{G} be a compact Lie group (e.g., U(1)×SU(2)×SU(3)U(1) \times SU(2) \times SU(3)) defining the gauge symmetry. The action SS is defined by the integral of the Lagrangian density L\mathcal{L} over spacetime: S=∫d4xL(ϕ,A,ψ;g)S = \int d^4x \mathcal{L}(\phi, A, \psi; g) where ϕ\phi are scalar fields, ψ\psi are matter fields, and AA are gauge fields. The Lagrangian must be invariant under local gauge transformations: ψ′=eigαaTaψ,A′=A−1g∂α\psi' = e^{i g \alpha^a T^a} \psi, \quad A' = A - \frac{1}{g} \partial \alpha The gauge field strength tensor FμνF_{\mu\nu} is constructed from the covariant derivative Dμ=∂μ−igTaAμaD_{\mu} = \partial_{\mu} - i g T^a A^a_{\mu} such that the kinetic term for the gauge bosons is Lgauge=−14Tr(FμνFμν)\mathcal{L}_{gauge} = - \frac{1}{4} \text{Tr}(F_{\mu\nu} F^{\mu\nu}), ensuring the theory is renormalizable and consistent with the underlying symmetry G\mathcal{G}.