Gauge covariant derivative

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 Gauge covariance requirement

Consider a generic (possibly non-Abelian) Gauge transformation given by

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

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]

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