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Gauge Symmetry

The electroweak theory is based on the principle of gauge symmetry, specifically U(1) x SU(2), which dictates the interactions between particles.
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The statement of the theorem

The electroweak theory is based on the local gauge symmetry group G=U(1)Y×SU(2)LG = U(1)_Y \times SU(2)_L. Under a gauge transformation parameterized by α(x)=αY(x)1+τ⃗⋅α(x)⃗\alpha(x) = \alpha_Y(x) \mathbf{1} + \vec{\tau} \cdot \vec{\alpha(x)}, the matter fields ψ\psi transform as: \nψ′=eig′αY(x)1+igτ⃗⋅α(x)⃗/2ψ\psi' = e^{i g' \alpha_Y(x) \mathbf{1} + i g \vec{\tau} \cdot \vec{\alpha(x)} / 2} \psi \nThis transformation dictates the covariant derivative Dμ=∂μ−ig′Y2Bμ−igτ⃗2⋅W⃗μD_{\mu} = \partial_{\mu} - i g' \frac{Y}{2} B_{\mu} - i g \frac{\vec{\tau}}{2} \cdot \vec{W}_{\mu}, ensuring the Lagrangian remains invariant.