Gauge group

A gauge group is the Lie group whose local symmetry transformations leave a gauge theory's Lagrangian invariant. The simplest case is U(1), the group of complex phase rotations e^{iqΛ(x)/ℏ} acting on a charged matter field — this is the gauge group of electromagnetism, and the unique gauge field A_μ that compensates the local U(1) transformation is the electromagnetic four-potential. Because U(1) is a one-parameter abelian group, electromagnetism has a single gauge boson (the photon), the gauge fields commute among themselves, and the field strength tensor F^{μν} is purely linear in A^μ.

For non-abelian groups the structure is richer. SU(2) — the weak isospin gauge group — has three real parameters and three gauge bosons (W^+, W^−, W^0 before electroweak mixing). SU(3) — the strong color gauge group of QCD — has eight parameters and eight gluons. SU(N) in general has N²−1 generators T^a satisfying [T^a, T^b] = i f^{abc} T^c with structure constants f^{abc}, and the field strength tensor acquires a non-linear self-interaction term: F^{a}_{μν} = ∂_μ A^a_ν − ∂_ν A^a_μ + g f^{abc} A^b_μ A^c_ν. The self-interaction means non-abelian gauge bosons carry charge under the very gauge group they mediate — gluons interact with gluons, W bosons interact with W bosons, in a way photons never interact with photons. The Standard Model gauge group is the product SU(3)_color × SU(2)_weak × U(1)_hypercharge, with twelve gauge bosons total before symmetry breaking and (after the Higgs mechanism) eight gluons, three weak bosons, and one photon.