Gauge covariant derivative

The gauge covariant derivative ( /ˌɡeɪdʒ koʊˌvɛəriənt dɨˈrɪvətɪv/) 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.

1 Fluid dynamics
2 Gauge theory
- 2.1 What happens to the covariant derivative under a gauge transformation
- 2.2 Quantum chromodynamics
3 General relativity
4 See also
5 References

Fluid dynamics

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

$\nabla_t \mathbf{v}:= \partial_t \mathbf{v} %2B (\mathbf{v} \cdot \nabla) \mathbf{v}$

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

$A_\mu \mapsto A_\mu %2B {1 \over e} (\partial_\mu \Lambda)$

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

$\bar \psi \partial_\mu \psi \mapsto \bar \psi \partial_\mu \psi %2B i \bar \psi (\partial_\mu \Lambda) \psi$ .

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, $\alpha=1 \dots 8$ .