Gauge theory

A classical field theory is a gauge theory when its description has a local redundancy represented by gauge transformations. A gauge field defines the covariant derivative that compares matter fields at different points, and its gauge field strength measures the curvature of that comparison. Electromagnetism is the familiar Abelian example; Yang-Mills theory generalizes it to a non-Abelian internal group. Both classical equations and their quantum field theory can have this structure. Physically equivalent gauge descriptions must give the same observables, even though the gauge potential changes.

An interaction whose gauge boson couples to a matter current fixed by a gauge symmetry. In the Standard Model, the photon, gluons, W boson and Z boson have different flavour, charge and chirality selection rules.

A theory embedding the Standard Model gauge factors in a larger gauge group. Particular embeddings can relate couplings and matter multiplets or predict proton decay. The unification hypothesis alone does not specify a unique mass spectrum or an experimentally established extension.

The SU(6) representations $15={\Lambda }^{2}6$ and $\overline{6}$ branch under SU(5) as $10\oplus 5$ and $\overline{5}\oplus 1$. Their cubic gauge anomaly coefficients are $2$ and $-1$ in fundamental normalization. Thus $15\oplus 2\overline{6}$ is anomaly-free and gives one Standard Model family $10\oplus \overline{5}$, a vectorlike gauge spectrum $5\oplus \overline{5}$, and two singlets. With these representations and their conjugates, precisely one net chiral family requires $m_{15}-m_{\overline{15}}=1$ and $m_{\bar{6}}-m_6=2$, so this is the smallest such choice.

A quark-lepton unification framework with conventional minimal gauge group $SU(4{)}_C\times SU(2{)}_L\times SU(2{)}_R$. leptons supply a fourth color charge, and a right-handed neutrino completes each family. fermion masses additionally depend on symmetry breaking and Yukawa couplings.

The charge relation combines left and right weak isospin with the generator distinguishing quarks from leptons inside the extended color charge group.

For the right-handed neutrino, $T_{3R}=1/2$ and $B-L=-1$, so $Y=T_{3R}+(B-L)/2=0$. Its colour-singlet and left-weak-singlet assignments make it sterile under the Standard Model after symmetry breaking.

A gauge theory whose left- and right-chiral fermions transform differently under its gauge group. The weak charged current gives a physical example. A chiral representation assignment must satisfy the relevant gauge anomaly cancellation conditions to define a consistent quantum theory.

A formulation of a gauge theory with discretized spacetime. Matter fields are assigned to sites and gauge-group elements to links between nearby sites. Products of link variables around closed paths provide gauge-invariant quantities. This structure preserves the local gauge symmetry while making nonperturbative numerical calculations possible. Physical predictions require control of finite spacing, finite volume and statistical errors rather than treating one finite grid as the continuum theory.

Localizing a parameter of a global symmetry produces its Noether current times a derivative of that parameter. Introduce a gauge field with an inhomogeneous transformation and couple it to the Noether current to cancel this variation. Recompute the remaining variation and add the required interaction and transformation terms. Closure of the local symmetry algebra constrains the completion. For supersymmetry, its supercurrent couples to the gravitino, and the translation in the supersymmetry algebra requires a vierbein.