the definition of the gauge covariant derivative as usually presented in physics. The gauge covariant derivative is often assumed to satisfy additional conditions making additional structure "constant" in the sense that the covariant derivative vanishes. For example if we have a Hermitian product on the fields (e.g the Dirac conjugate inner product for spinors) reducing the gauge group to a unitary group, we can impose the further condition making the Hermitian product "constant". Writing this out with respect to a local -orthonormal frame field gives , and using the above we see that must be Hermitian i.e. (motivating the extra factor ). The Hermitian matrices are (up to the factor ) the generators of the unitary group. More generally if the gauge covariant derivative preserves a gauge group acting with representation , the gauge covariant connection can be written as where is representation of the Lie algebra associated to the group representation (loc. cit.). Note that including the gauge covariant derivative (or its gauge potential), as a physical field, "field with zero gauge covariant derivative along the tangent of a curve " is a physically meaningful definition of a field constant along a (smooth) curve. Hence the gauge covariant derivative defines (and is defined by) parallel transport. Gauge Field Strength Unlike the partial derivatives, the gauge covariant derivatives do not commute. However they almost do in the sense that the commutator is not an operator of order 2 but of order 0, i.e. is linear over functions: . The linear map is called the gauge field strength (loc. cit). In index notation, using the gauge potential . If is a G covariant derivative, one can interpret the latter term as a commutator in the Lie algebra of G and as Lie algebra valued (loc. cit). Invariance under gauge transformations The gauge covariant derivative transforms covariantly under Gauge transformations, i.e. for all which in operator form takes the form or In particular (suppressing dependence on ) . Further, (suppressing indices and replacing them by matrix multiplication) if is of the form above, is of the form or using , which is also of this form. In the Hermitian case with a unitary gauge group and we have found a first order differential operator with as first order term such that . Gauge theory In gauge theory, which studies a particular class of fields which are of importance in quantum field theory,