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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, \( \alpha=1 \dots 8 \).
General relativity
In general relativity, the gauge covariant derivative is defined as
\( \nabla_j v^i := \partial_j v^i + \Gamma^i {}_{j k} v^k \)
where \( \Gamma^i {}_{j k} \) is the Christoffel symbol.
See also
Kinetic momentum
Connection (mathematics)
Minimal coupling
References
Tsutomu Kambe, Gauge Principle For Ideal Fluids And Variational Principle. (PDF file.)
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