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The gauge covariant derivative is a generalization 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.

Fluid dynamics

In fluid dynamics, the gauge covariant derivative of a fluid may be defined as

\( \nabla_t \mathbf{v}:= \partial_t \mathbf{v} + (\mathbf{v} \cdot \nabla) \mathbf{v} \)

where \mathbf{v} is a velocity vector field of a fluid.

Gauge theory

In gauge theory, which studies a particular class of fields which are of importance in quantum field theory, the minimally-coupled gauge covariant derivative is defined as

\( D_\mu := \partial_\mu - i e A_\mu \)

where \( A_\mu is the electromagnetic vector potential.

(Note that this is valid for a Minkowski metric of signature (-, +, +, +), which is used in this article. For (+, -, -, -) the minus becomes a plus.)
Construction of the covariant derivative throught the comparator
Construction of the covariant derivative throught Gauge covariance requirement

Consider a generic, possibly non abelian, Gauge transformation given by

\( \phi(x) \rightarrow U(x) \phi(x) \equiv e^{i\alpha(x)} \phi(x), \)
\( \phi^\dagger(x) \rightarrow \phi^\dagger(x) U(x)^\dagger \equiv \phi^\dagger(x) e^{-i\alpha(x)}, \qquad U^\dagger = U^{-1}. \)

where \( \alpha(x) \) is an element of the Lie algebra associated with the Lie group of transformations, and can be expressed in terms of the generators as \( \alpha(x) = \alpha^a(x) t^a. \)

The partial derivative \( \partial_\mu \)transforms accordingly as

\( \partial_\mu \phi(x) \rightarrow U(x) \partial_\mu \phi(x) + (\partial_\mu U) \phi(x) \equiv e^{i\alpha(x)} \partial_\mu \phi(x) + i (\partial_\mu \alpha) e^{i\alpha(x)} \phi(x) \)

and a kinetic term of the form \( \phi^\dagger \partial_\mu \phi \) is thus not invariant under this transformation.

We can introduce the covariant derivative \( D_\mu \) in this context as a generalization of the partial derivative \( \partial_\mu \) which transforms covariantly under the Gauge transformation, i.e. an object satisfying

\( D_\mu \phi(x) \rightarrow D'_\mu \phi'(x) = U(x) D_\mu \phi(x), \)

which in operatorial form takes the form

\( D'_\mu = U(x) D_\mu U^\dagger(x). \)

We thus compute (omitting the explicit x dependences for brevity)

\( D_\mu \phi \rightarrow D'_\mu U \phi = UD_\mu \phi + (\delta D_\mu U + [D_\mu,U])\phi, \)

where

\( D_\mu \rightarrow D'_\mu \equiv D_\mu + \delta D_\mu, \)
\( A_\mu \rightarrow A'_\mu = A_\mu + \delta A_\mu. \)

The requirement for \( D_\mu \)to transform covariantly is now translated in the condition

\( (\delta D_\mu U + [D_\mu,U])\phi = 0. \)

To obtain an explicit expression we make the Ansatz

\( D_\mu = \partial_\mu - ig A_\mu, \)

from which it follows that

\( \delta D_\mu \equiv -ig \delta A_\mu \)

and

\( \delta A_\mu = [U,A_\mu]U^\dagger -\frac{i}{g} (\partial_\mu U)U^\dagger \)

which, using \( U(x) = 1 + i \alpha(x) + \mathcal{O}(\alpha^2) \), takes the form

\( \delta A_\mu = \frac{1}{g} ( \partial_\mu \alpha - ig [A_\mu,\alpha] ) + \mathcal{O}(\alpha^2) = \frac{1}{g} D_\mu \alpha + \mathcal{O}(\alpha^2) \)

We have thus found an object \( D_\mu \)such that

\( \phi^\dagger(x) D_\mu \phi(x) \rightarrow \phi'^\dagger(x) D'_\mu \phi'(x) = \phi^\dagger(x) D_\mu \phi(x) \)

Quantum electrodynamics

If a gauge transformation is given by

\( \psi \mapsto e^{i\Lambda} \psi \)

and for the gauge potential

\( A_\mu \mapsto A_\mu + {1 \over e} (\partial_\mu \Lambda) \)

then \( D_\mu \) transforms as

\( D_\mu \mapsto \partial_\mu - i e A_\mu - i (\partial_\mu \Lambda) , \)

and \( D_\mu \psi \) transforms as

\( D_\mu \psi \mapsto e^{i \Lambda} D_\mu \psi \)

and \( \bar \psi := \psi^\dagger \gamma^0 \)transforms as

\( \bar \psi \mapsto \bar \psi e^{-i \Lambda} \)

so that

\( \bar \psi D_\mu \psi \mapsto \bar \psi D_\mu \psi \)

and \( \bar \psi D_\mu \psi \) 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 \partial_\mu would not preserve the Lagrangian's gauge symmetry, since

\( \bar \psi \partial_\mu \psi \mapsto \bar \psi \partial_\mu \psi + i \bar \psi (\partial_\mu \Lambda) \psi . \)

Quantum chromodynamics

In quantum chromodynamics, the gauge covariant derivative is[1]

\( D_\mu := \partial_\mu - i g \, A_\mu^\alpha \, \lambda_\alpha \)

where g is the coupling constant, A is the gluon gauge field, for eight different gluons \( \alpha=1 \dots 8, \psi \)is a four-component Dirac spinor, and where \(\lambda_\alpha \) is one of the eight Gell-Mann matrices, \( \alpha=1 \dots 8. \)

Standard Model

The covariant derivative in the Standard Model can be expressed in the following form:[2]

\( D_\mu := \partial_\mu - i \frac{g_1}{2} \, Y \, B_\mu - i \frac{g_2}{2} \, \sigma_j \, W_\mu^j - i \frac{g_3}{2} \, \lambda_\alpha \, G_\mu^\alpha \)

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Gauge covariant derivative