Gauge covariant derivative

The gauge covariant derivative is a variation of the covariant derivative used in general relativity. If a theory has gauge transformations, it means that some physical properties of certain equations are preserved under those transformations. Likewise, the gauge covariant derivative is the ordinary derivative modified in such a way as to make it behave like a true vector operator, so that equations written using the covariant derivative preserve their physical properties under gauge transformations.

There are many ways in which to understand the gauge covariant derivative. The approach taken in this article is based on the historically traditional notation used in many physics textbooks. Another approach is to understand the gauge covariant derivative as a kind of connection, and more specifically, an affine connection. The affine connection is interesting because it does not require any concept of a metric tensor to be defined; the curvature of an affine connection can be understood as the field strength of the gauge potential. When a metric is available, then one can go in a different direction, and define a connection on a frame bundle. This path leads directly to general relativity; however, it requires a metric, which particle physics gauge theories do not have.

Rather than being generalizations of one-another, affine and metric geometry go off in different directions: the gauge group of (pseudo-)Riemannian geometry must be the indefinite orthogonal group O(s,r) in general, or the Lorentz group O(3,1) for space-time. This is because the fibers of the frame bundle must necessarily, by definition, connect the tangent and cotangent spaces of space-time. By contrast, the gauge groups employed in particle physics could be (in principle) any Lie group at all (and, in practice, being only U(1), SU(2) or SU(3) in the Standard Model). Note that Lie groups do not come equipped with a metric.

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