different fields are used in Lagrangians that are invariant under local gauge transformations. Kinetic terms involve derivatives of the fields which by the above arguments need to involve gauge covariant derivatives. Abelian Gauge Theory the gauge covariant derivative on a complex scalar field (i.e. ) of charge is a connection. The gauge potential is a (1 x 1) matrix, i.e. a scalar. The gauge field strength is The gauge potential can be interpreted as electromagnetic four-potential and the gauge field strength as the electromagnetic field tensor. Since this only involves the charge of the field and not higher multipoles like the magnetic moment (and in a loose and non unique way, because it replaces by ) this is called minimal coupling. Fora Dirac spinor field of charge the covariant derivative is also a connection (because it has to commute with the gamma matrices) and is defined as where again is interpreted as the electromagnetic four-potential and as the electromagnetic field tensor. (The minus sign is a convention valid for a Minkowski metric signature , which is common in general relativity and used below. For the particle physics convention , it is . The electron's charge is defined negative as , while the Dirac field is defined to transform positively as ) Quantum electrodynamics If a gauge transformation is given by and for the gauge potential then transforms as , and transforms as and transforms as so that and in the QED Lagrangian is therefore gauge invariant, and the gauge covariant derivative is thus named aptly. On the other hand, the non-covariant derivative would not preserve the Lagrangian's gauge symmetry, since . Quantum chromodynamics In quantum chromodynamics, the gauge covariant derivative is where is the coupling constant of the strong interaction, is the gluon gauge field, for eight different gluons , and where is one of the eight Gell-Mann matrices. The Gell-Mann matrices give a representation of the color symmetry group SU(3). For quarks, the representation is the fundamental representation, for gluons, the representation is the adjoint representation. Standard Model The covariant derivative in the Standard Model combines the electromagnetic, the weak and the strong interactions. It can be expressed in the following form: The gauge fields here belong to the fundamental representations of the electroweak Lie group times the color symmetry Lie group SU(3). The coupling constant provides the coupling of the hypercharge to the boson and the coupling via