The gauge covariant derivative is used in many areas of physics, including quantum field theory and fluid dynamics and in a very special way general relativity.
Ordinary differentiation of field components is not invariant under such gauge transformations, because they depend on the local frame.
However, when gauge transformations act on fields and the gauge covariant derivative simultaneously, they preserve properties of theories that do not depend on frame choice and hence are valid descriptions of physics.
Like the covariant derivative used in general relativity (which is special case), the gauge covariant derivative is an expression for a connection in local coordinates after choosing a frame for the fields involved, often in the form of index notation.
The approach taken in this article is based on the historically traditional notation used in many physics textbooks.
[4][5][6] 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.
This path leads directly to general relativity; however, it requires a metric, which particle physics gauge theories do not have.
This is because the fibers of the frame bundle must necessarily, by definition, connect the tangent and cotangent spaces of space-time.
[9] Although conceptually the same, this approach uses a very different set of notation, and requires a far more advanced background in multiple areas of differential geometry.
[6][10] For ordinary Lie algebras, the gauge covariant derivative on the space symmetries (those of the pseudo-Riemannian manifold and general relativity) cannot be intertwined with the internal gauge symmetries; that is, metric geometry and affine geometry are necessarily distinct mathematical subjects: this is the content of the Coleman–Mandula theorem.
The more mathematical approach uses an index-free notation, emphasizing the geometric and algebraic structure of the gauge theory and its relationship to Lie algebras and Riemannian manifolds; for example, treating gauge covariance as equivariance on fibers of a fiber bundle.
[7] The physics approach also has a pedagogical advantage: the general structure of a gauge theory can be exposed after a minimal background in multivariate calculus, whereas the geometric approach requires a large investment of time in the general theory of differential geometry, Riemannian manifolds, Lie algebras, representations of Lie algebras and principle bundles before a general understanding can be developed.
This article attempts to follow more closely to the notation and language commonly employed in physics curriculum, touching only briefly on the more abstract connections.
Consider a generic (possibly non-Abelian) gauge transformation acting on a
The main examples in field theory have a compact gauge group and we write the symmetry operator as
A gauge covariant derivative is defined as an operator satisfying a product rule for every smooth function
To go back to index notation we use the product rule For a fixed
is conventional for compact gauge groups and is interpreted as a coupling constant).
We then get and with suppressed frame fields this gives in index notation which by abuse of notation is often written as This is the definition of the gauge covariant derivative as usually presented in physics.
[11] The gauge covariant derivative is often assumed to satisfy additional conditions making additional structure "constant" in the sense that the covariant derivative vanishes.
is a G covariant derivative, one can interpret the latter term as a commutator in the Lie algebra of G and
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
(The minus sign is a convention valid for a Minkowski metric signature (−, +, +, +), which is common in general relativity and used below.
The Gell-Mann matrices give a representation of the color symmetry group SU(3).
The covariant derivative in the Standard Model combines the electromagnetic, the weak and the strong interactions.
It can be expressed in the following form:[14] The gauge fields here belong to the fundamental representations of the electroweak Lie group
It corresponds to the Levi Civita connection (a special Riemannian connection) on the tangent bundle (or the frame bundle) i.e. it acts on tangent vector fields or more generally, tensors.
(this uses the definition of a vector field as an operator on smooth functions that satisfies a product rule i.e. a derivation).
is the Christoffel symbol defined by It gives the covariant derivative The formal similarity with the gauge covariant derivative is more clear when the choice of coordinates is decoupled from the choice of frame of vector fields
The direct analogue of the "gauge freedom" of the gauge covariant derivative is the arbitrariness of the choice of an orthonormal d-Bein at each point in space-time: local Lorentz invariance [citation needed].