Gauge covariant derivative

The gauge covariant derivative (pronEng|ˌgeɪdʒ koʊˌvɛəriənt dɪˈrɪvətɪv) 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:$abla\_t\; mathbf\{v\}:=\; partial\_t\; mathbf\{v\}\; +\; (mathbf\{v\}\; cdot\; abla)\; mathbf\{v\}$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 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:$psi\; mapsto\; e^\{iLambda\}\; psi$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 transforms as:so that:and 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:.

Quantum chromodynamics

In quantum chromodynamics, the gauge covariant derivative is [http://www.fuw.edu.pl/~dobaczew/maub-42w/node9.html] :$D\_mu\; :=\; partial\_mu\; -\; i\; g\; ,\; A\_mu^alpha\; ,\; lambda\_alpha$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, $alpha=1\; dots\; 8$.

General relativity

In general relativity, the gauge covariant derivative is defined as:$abla\_j\; v^i\; :=\; partial\_j\; v^i\; +\; Gamma^i\; \{\}\_\{j\; k\}\; v^k$where $Gamma^i\; \{\}\_\{j\; k\}$ is the Christoffel symbol.

ee also

*Kinetic momentum

References

*Tsutomu Kambe, " [http://fluid.ippt.gov.pl/ictam04/text/sessions/docs/FM23/11166/FM23_11166.pdf Gauge Principle For Ideal Fluids And Variational Principle] ". (PDF file.)