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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 v is a velocity Vector Field of a fluid.


QUANTUM FIELD THEORY

In Quantum Field Theory , the gauge covariant derivative is defined as
: D_\mu := \partial_\mu - i e A_\mu
where A 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
: A_\mu \mapsto A_\mu + {1 \over e} (\partial_\mu \Lambda)
where Λ is a Lorentz Transformation , then D_\mu transforms as
: D_\mu \mapsto \partial_\mu - i e A_\mu - i (\partial_\mu \Lambda) ,
also 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 is therefore Lorentz Covariant , so that the QED Lagrangian is 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 {Link without Title}
: 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 α=1...8, ψ is a four-component Dirac Spinor , and where \lambda_\alpha is one of the eight Gell-Mann Matrices , α=1...8.


GENERAL RELATIVITY

In General Relativity , the gauge covariant derivative is defined as
:
abla_j \mathbf{v} := \partial_j v^i + v^k \Gamma^i {}_{k j}
where Γ is the Christoffel Symbol .


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