Gauge theories are important as the successful field theories explaining the dynamics of elementary particles . Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1) and has one gauge field, the electromagnetic four-potential , with the photon being the gauge boson. The Standard Model is a non-abelian gauge theory with the symmetry group U(1) × SU(2) × SU(3) and has a total of twelve gauge bosons: the photon , three weak bosons and eight gluons .

This article describes the mathematics of the Standard Model of particle physics , a gauge quantum field theory containing the internal symmetries of the unitary product group SU (3) × SU(2) × U(1) . The theory is commonly viewed as containing the fundamental set of particles – the leptons , quarks , gauge bosons and the Higgs particle .

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