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

${\mathrm{\nabla }}_t\mathbf{v}:={\partial }_t\mathbf{v}+(\mathbf{v}\cdot \mathrm{\nabla })\mathbf{v}$

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

$D_{\mu }:={\partial }_{\mu }-ie{A}_{\mu }$

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

$\psi \mapsto e^{i\mathrm{\Lambda }}\psi$

and for the gauge potential

$A_{\mu }\mapsto A_{\mu }+\frac{1}{e}({\partial }_{\mu }\mathrm{\Lambda })$

then $D_{\mu }$ transforms as

$D_{\mu }\mapsto {\partial }_{\mu }-ie{A}_{\mu }-i({\partial }_{\mu }\mathrm{\Lambda })$,

and $D_{\mu }\psi$ transforms as

$D_{\mu }\psi \mapsto e^{i\mathrm{\Lambda }}{D}_{\mu }\psi$

and $\overline{\psi }:={\psi }^{†}{\gamma }^0$ transforms as

$\overline{\psi }\mapsto \overline{\psi }{e}^{-i\mathrm{\Lambda }}$

so that

$\overline{\psi }{D}_{\mu }\psi \mapsto \overline{\psi }{D}_{\mu }\psi$

and $\overline{\psi }{D}_{\mu }\psi$ 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

$\overline{\psi }{\partial }_{\mu }\psi \mapsto \overline{\psi }{\partial }_{\mu }\psi +i\overline{\psi }({\partial }_{\mu }\mathrm{\Lambda })\psi$.

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

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

where ${\mathrm{\Gamma }}^i{}_{jk}$ 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.)