Antisymmetry

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

$D_{\mu }:=\partial _{\mu }-ieA_{\mu }$

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

$D_{\mu }\mapsto \partial _{\mu }-ieA_{\mu }-i(\partial _{\mu }\Lambda )$,

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

$D_{\mu }:=\partial _{\mu }-ig\,A_{\mu }^{\alpha }\,\lambda _{\alpha }$

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:

$D_{\mu }:=\partial _{\mu }-i{\frac {g_{1}}{2}}\,Y\,B_{\mu }-i{\frac {g_{2}}{2}}\,\sigma _{j}\,W_{\mu }^{j}-i{\frac {g_{3}}{2}}\,\lambda _{\alpha }\,G_{\mu }^{\alpha }$

General relativity

In general relativity, the gauge covariant derivative is defined as

$\nabla _{j}v^{i}:=\partial _{j}v^{i}+\Gamma ^{i}{}_{jk}v^{k}$

where $\Gamma ^{i}{}_{jk}$ is the Christoffel symbol.

See also

• Kinetic momentum
• Connection (mathematics)
• Minimal coupling
• Ricci calculus

References

• Tsutomu Kambe, Gauge Principle For Ideal Fluids And Variational Principle. (PDF file.)