# Gauge covariant derivative

The gauge covariant derivative is like 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.

### What happens to the covariant derivative under a gauge transformation

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]

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