# Gauge covariant derivative

The gauge covariant derivative is 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

${\displaystyle \nabla _{t}{\mathbf {v} }:=\partial _{t}{\mathbf {v} }+({\mathbf {v} }\cdot \nabla ){\mathbf {v} }}$

where ${\displaystyle {\mathbf {v} }}$ 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

${\displaystyle D_{\mu }:=\partial _{\mu }-ieA_{\mu }}$

where ${\displaystyle A_{\mu }}$ is the electromagnetic vector potential.

(Note that this is valid for a Minkowski metric of signature ${\displaystyle (-,+,+,+)}$, which is used in this article. For ${\displaystyle (+,-,-,-)}$ the minus becomes a plus.)

### Construction of the covariant derivative throught Gauge covariance requirement

Consider a generic, possibly non abelian, Gauge transformation given by

${\displaystyle \phi (x)\rightarrow U(x)\phi (x)\equiv e^{i\alpha (x)}\phi (x),}$
${\displaystyle \phi ^{\dagger }(x)\rightarrow \phi ^{\dagger }(x)U(x)^{\dagger }\equiv \phi ^{\dagger }(x)e^{-i\alpha (x)},\qquad U^{\dagger }=U^{-1}.}$

where ${\displaystyle \alpha (x)}$ is an element of the Lie algebra associated with the Lie group of transformations, and can be expressed in terms of the generators as ${\displaystyle \alpha (x)=\alpha ^{a}(x)t^{a}}$.

The partial derivative ${\displaystyle \partial _{\mu }}$ transforms accordingly as

${\displaystyle \partial _{\mu }\phi (x)\rightarrow U(x)\partial _{\mu }\phi (x)+(\partial _{\mu }U)\phi (x)\equiv e^{i\alpha (x)}\partial _{\mu }\phi (x)+i(\partial _{\mu }\alpha )e^{i\alpha (x)}\phi (x)}$

and a kinetic term of the form ${\displaystyle \phi ^{\dagger }\partial _{\mu }\phi }$ is thus not invariant under this transformation.

We can introduce the covariant derivative ${\displaystyle D_{\mu }}$ in this context as a generalization of the partial derivative ${\displaystyle \partial _{\mu }}$ which transforms covariantly under the Gauge transformation, i.e. an object satisfying

${\displaystyle D_{\mu }\phi (x)\rightarrow D'_{\mu }\phi '(x)=U(x)D_{\mu }\phi (x),}$

which in operatorial form takes the form

${\displaystyle D'_{\mu }=U(x)D_{\mu }U^{\dagger }(x).}$

We thus compute (omitting the explicit ${\displaystyle x}$ dependences for brevity)

${\displaystyle D_{\mu }\phi \rightarrow D'_{\mu }U\phi =UD_{\mu }\phi +(\delta D_{\mu }U+[D_{\mu },U])\phi }$,

where

${\displaystyle D_{\mu }\rightarrow D'_{\mu }\equiv D_{\mu }+\delta D_{\mu },}$
${\displaystyle A_{\mu }\rightarrow A'_{\mu }=A_{\mu }+\delta A_{\mu }.}$

The requirement for ${\displaystyle D_{\mu }}$ to transform covariantly is now translated in the condition

${\displaystyle (\delta D_{\mu }U+[D_{\mu },U])\phi =0.}$

To obtain an explicit expression we make the Ansaz

${\displaystyle D_{\mu }=\partial _{\mu }-igA_{\mu },}$

from which it follows that

${\displaystyle \delta D_{\mu }\equiv -ig\delta A_{\mu }}$

and

${\displaystyle \delta A_{\mu }=[U,A_{\mu }]U^{\dagger }-{\frac {i}{g}}(\partial _{\mu }U)U^{\dagger }}$

which, using ${\displaystyle U(x)=1+i\alpha (x)+{\mathcal {O}}(\alpha ^{2})}$, takes the form

${\displaystyle \delta A_{\mu }={\frac {1}{g}}(\partial _{\mu }\alpha -ig[A_{\mu },\alpha ])+{\mathcal {O}}(\alpha ^{2})={\frac {1}{g}}D_{\mu }\alpha +{\mathcal {O}}(\alpha ^{2})}$

We have thus found an object ${\displaystyle D_{\mu }}$ such that

${\displaystyle \phi ^{\dagger }(x)D_{\mu }\phi (x)\rightarrow \phi '^{\dagger }(x)D'_{\mu }\phi '(x)=\phi ^{\dagger }(x)D_{\mu }\phi (x)}$

### Quantum electrodynamics

If a gauge transformation is given by

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

and for the gauge potential

${\displaystyle A_{\mu }\mapsto A_{\mu }+{1 \over e}(\partial _{\mu }\Lambda )}$

then ${\displaystyle D_{\mu }}$ transforms as

${\displaystyle D_{\mu }\mapsto \partial _{\mu }-ieA_{\mu }-i(\partial _{\mu }\Lambda )}$,
${\displaystyle D_{\mu }\psi \mapsto e^{i\Lambda }D_{\mu }\psi }$
${\displaystyle {\bar {\psi }}\mapsto {\bar {\psi }}e^{-i\Lambda }}$

so that

${\displaystyle {\bar {\psi }}D_{\mu }\psi \mapsto {\bar {\psi }}D_{\mu }\psi }$

and ${\displaystyle {\bar {\psi }}D_{\mu }\psi }$ 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 ${\displaystyle \partial _{\mu }}$ would not preserve the Lagrangian's gauge symmetry, since

${\displaystyle {\bar {\psi }}\partial _{\mu }\psi \mapsto {\bar {\psi }}\partial _{\mu }\psi +i{\bar {\psi }}(\partial _{\mu }\Lambda )\psi }$.

### Quantum chromodynamics

In quantum chromodynamics, the gauge covariant derivative is[1]

${\displaystyle D_{\mu }:=\partial _{\mu }-ig\,A_{\mu }^{\alpha }\,\lambda _{\alpha }}$

### Standard Model

The covariant derivative in the Standard Model can be expressed in the following form:[2]

${\displaystyle D_{\mu }:=\partial _{\mu }-i{\frac {g_{1}}{2}}\,Y\,B_{\mu }-i{\frac {g_{2}}{2}}\,\sigma _{j}\,W_{\mu }^{j}-i{\frac {g_{3}}{2}}\,\lambda _{\alpha }\,G_{\mu }^{\alpha }}$