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| The '''gauge covariant derivative''' is like a generalization of the [[covariant derivative]] used in [[general relativity]]. If a theory has [[gauge transformation]]s, 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.
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| ==Fluid dynamics==
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| In [[fluid dynamics]], the gauge covariant derivative of a fluid may be defined as
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| :<math> \nabla_t \mathbf{v}:= \partial_t \mathbf{v} + (\mathbf{v} \cdot \nabla) \mathbf{v}</math> | |
| where <math>\mathbf{v}</math> is a velocity [[vector field]] of a fluid.
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| ==Gauge theory==
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| In [[gauge theory]], which studies a particular class of [[field (physics)|fields]] which are of importance in [[quantum field theory]], the [[minimal coupling|minimally-coupled]] gauge covariant derivative is defined as
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| :<math> D_\mu := \partial_\mu - i e A_\mu </math>
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| where <math>A_\mu</math> is the electromagnetic [[vector potential]].
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| ===What happens to the covariant derivative under a gauge transformation=== | |
| If a gauge transformation is given by
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| :<math> \psi \mapsto e^{i\Lambda} \psi </math>
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| and for the gauge potential
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| :<math> A_\mu \mapsto A_\mu + {1 \over e} (\partial_\mu \Lambda) </math>
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| then <math> D_\mu </math> transforms as
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| :<math> D_\mu \mapsto \partial_\mu - i e A_\mu - i (\partial_\mu \Lambda) </math>,
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| and <math> D_\mu \psi </math> transforms as
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| :<math> D_\mu \psi \mapsto e^{i \Lambda} D_\mu \psi </math>
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| and <math> \bar \psi := \psi^\dagger \gamma^0 </math> transforms as
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| :<math> \bar \psi \mapsto \bar \psi e^{-i \Lambda} </math>
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| so that
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| :<math> \bar \psi D_\mu \psi \mapsto \bar \psi D_\mu \psi </math>
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| and <math> \bar \psi D_\mu \psi </math> in the QED [[Lagrangian]] is therefore gauge invariant, and the gauge covariant derivative is thus named aptly.
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| On the other hand, the non-covariant derivative <math> \partial_\mu </math> would not preserve the Lagrangian's gauge symmetry, since
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| :<math> \bar \psi \partial_\mu \psi \mapsto \bar \psi \partial_\mu \psi + i \bar \psi (\partial_\mu \Lambda) \psi </math>.
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| ===Quantum chromodynamics===<!-- This section is linked from [[Lagrangian]] -->
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| In [[quantum chromodynamics]], the gauge covariant derivative is<ref>http://www.fuw.edu.pl/~dobaczew/maub-42w/node9.html</ref>
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| :<math> D_\mu := \partial_\mu - i g \, A_\mu^\alpha \, \lambda_\alpha </math>
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| where <math>g</math> is the [[coupling constant]], <math>A</math> is the gluon [[gauge field]], for eight different gluons <math>\alpha=1 \dots 8</math>, <math>\psi</math> is a four-component [[Dirac spinor]], and where <math>\lambda_\alpha</math> is one of the eight [[Gell-Mann matrices]], <math>\alpha=1 \dots 8</math>.
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| ===Standard Model===
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| The covariant derivative in the [[Standard Model]] can be expressed in the following form:
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| :<math> 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 </math>
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| ==General relativity==
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| In [[general relativity]], the gauge covariant derivative is defined as
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| :<math> \nabla_j v^i := \partial_j v^i + \Gamma^i {}_{j k} v^k </math>
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| where <math>\Gamma^i {}_{j k}</math> is the [[Christoffel symbol]].
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| ==See also==
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| *[[Kinetic momentum]]
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| *[[Connection (mathematics)]]
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| *[[Minimal coupling]]
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| *[[Ricci calculus]]
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| == References ==
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| <references />
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| *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.)
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| [[Category:Differential geometry]]
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| [[Category:Connection (mathematics)]]
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| [[Category:Gauge theories]]
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