Gauge covariant derivative

<h2 id="overview">Overview</h2>
There are many ways to understand the gauge covariant derivative. The approach taken in this article is based on the historically traditional notation used in many physics textbooks.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> Another approach is to understand the gauge covariant derivative as a kind of <a href="/facts/Connection_(mathematics)/KURsCmIu">connection</a>, and more specifically, an <a href="/facts/Affine_connection/WRjjxSlM">affine connection</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> The affine connection is interesting because it does not require any concept of a <a href="/facts/Metric_tensor/tbqek1gJ">metric tensor</a> to be defined; the <a href="/facts/Curvature/RFCAEsj8">curvature</a> of an affine connection can be understood as the <a href="/facts/Field_strength/cIH9DzKy">field strength</a> of the gauge potential. When a metric is available, then one can go in a different direction, and define a connection on a <a href="/facts/Frame_bundle/Gugt3qyL">frame bundle</a>. This path leads directly to general relativity; however, it requires a metric, which <a href="/facts/Particle_physics/YgIuyj2t">particle physics</a> <a href="/facts/Gauge_theory/fSW3gggv">gauge theories</a> do not have.
Rather than being generalizations of one-another, affine and metric geometry go off in different directions: the <a href="/facts/Gauge_group/fSW3gggv">gauge group</a> of (<a href="/facts/Pseudo-Riemannian_geometry/VFXsVjqb">pseudo-</a>)<a href="/facts/Riemannian_geometry/pPEzjyQC">Riemannian geometry</a> must be the <a href="/facts/Indefinite_orthogonal_group/snKhwDB2">indefinite orthogonal group</a> O(s,r) in general, or the <a href="/facts/Lorentz_group/t1APc2Ye">Lorentz group</a> O(3,1) for <a href="/facts/Space-time/uRIN1x4b">space-time</a>. This is because the fibers of the <a href="/facts/Frame_bundle/Gugt3qyL">frame bundle</a> must necessarily, by definition, connect the <a href="/facts/Tangent_space/crvpbSeG">tangent</a> and <a href="/facts/Cotangent_space/INNIyG4S">cotangent spaces</a> of space-time.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> In contrast, the gauge groups employed in particle physics could in principle be any <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> at all, although in practice the <a href="/facts/Standard_Model/LhVGf9ow">Standard Model</a> only uses <a href="/facts/U(1)/kuzdWtHo">U(1)</a>, <a href="/facts/SU(2)/Qgy2orcu">SU(2)</a> and <a href="/facts/SU(3)/Qgy2orcu">SU(3)</a>. Note that Lie groups do not come equipped with a metric.
A yet more complicated, yet more accurate and geometrically enlightening, approach is to understand that the gauge covariant derivative is (exactly) the same thing as the <a href="/facts/Exterior_covariant_derivative/eKHqDq6Q">exterior covariant derivative</a> on a <a href="/facts/Section_(fiber_bundle)/DLnpXS5a">section</a> of an <a href="/facts/Associated_bundle/7JfKDcBJ">associated bundle</a> for the <a href="/facts/Principal_bundle/bbB2D94B">principal fiber bundle</a> of the gauge theory;<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> and, for the case of spinors, the associated bundle would be a <a href="/facts/Spin_bundle/xjXIskf5">spin bundle</a> of the <a href="/facts/Spin_structure/BF78kFz8">spin structure</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> Although conceptually the same, this approach uses a very different set of notation, and requires a far more advanced background in multiple areas of <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>.
The final step in the geometrization of gauge invariance is to recognize that, in quantum theory, one needs only to compare neighboring fibers of the principal fiber bundle, and that the fibers themselves provide a superfluous extra description. This leads to the idea of modding out the gauge group to obtain the <a href="/facts/Atiyah_algebroid/FWII9u27">gauge groupoid</a> as the closest description of the gauge connection in quantum field theory.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
For ordinary Lie algebras, the gauge covariant derivative on the space symmetries (those of the <a href="/facts/Pseudo-Riemannian_manifold/VFXsVjqb">pseudo-Riemannian manifold</a> and general relativity) cannot be intertwined with the internal gauge symmetries; that is, metric geometry and affine geometry are necessarily distinct mathematical subjects: this is the content of the <a href="/facts/Coleman%E2%80%93Mandula_theorem/SdJY2VHJ">Coleman–Mandula theorem</a>. However, a premise of this theorem is violated by the <a href="/facts/Lie_superalgebra/c1oxUoEl">Lie superalgebras</a> (which are not Lie algebras!) thus offering hope that a single unified symmetry can describe both spatial and internal symmetries: this is the foundation of <a href="/facts/Supersymmetry/hkWGrMet">supersymmetry</a>.
The more mathematical approach uses an index-free notation, emphasizing the geometric and algebraic structure of the gauge theory and its relationship to <a href="/facts/Lie_algebras/n2gAUkNq">Lie algebras</a> and <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifolds</a>; for example, treating gauge covariance as <a href="/facts/Equivariance/KkIkDwIH">equivariance</a> on fibers of a fiber bundle. The index notation used in physics makes it far more convenient for practical calculations, although it makes the overall geometric structure of the theory more opaque.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> The physics approach also has a pedagogical advantage: the general structure of a gauge theory can be exposed after a minimal background in <a href="/facts/Multivariate_calculus/PEs5SgUF">multivariate calculus</a>, whereas the geometric approach requires a large investment of time in the general theory of <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifolds</a>, <a href="/facts/Lie_algebras/n2gAUkNq">Lie algebras</a>, <a href="/facts/Representation_of_a_Lie_group/29R96ar2">representations of Lie algebras</a> and <a href="/facts/Principle_bundle/bbB2D94B">principle bundles</a> before a general understanding can be developed. In more advanced discussions, both notations are commonly intermixed.
This article attempts to follow more closely to the notation and language commonly employed in physics curriculum, touching only briefly on the more abstract connections.

<h2 id="motivation-of-the-covariant-derivative-through-gauge-covariance-requirement">Motivation of the covariant derivative through gauge covariance requirement</h2>
Consider a generic (possibly non-Abelian) gauge transformation acting on a 
 
 
 
 n
 
 
 {\displaystyle n}
 
 component field 
 
 
 
 ϕ
 =
 (
 
 ϕ
 
 a
 
 
 
 )
 
 a
 =
 1..
 n
 
 
 
 
 {\displaystyle \phi =(\phi _{a})_{a=1..n}}
 
. The main examples in field theory have a compact gauge group and we write the symmetry operator as 
 
 
 
 U
 (
 x
 )
 =
 
 e
 
 i
 α
 (
 x
 )
 
 
 
 
 {\displaystyle U(x)=e^{i\alpha (x)}}
 
 where 
 
 
 
 α
 (
 x
 )
 
 
 {\displaystyle \alpha (x)}
 
 is an element of the <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> associated with the <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> of symmetry transformations, and can be expressed in terms of the hermitian generators of the Lie algebra (i.e. up to a factor 
 
 
 
 i
 
 
 {\displaystyle i}
 
, the infinitesimal generators of the gauge group), 
 
 
 
 {
 
 t
 
 K
 
 
 
 }
 
 K
 ∈
 
 
 K
 
 
 
 
 
 
 {\displaystyle \{t_{K}\}_{K\in {\mathcal {K}}}}
 
, as 
 
 
 
 α
 (
 x
 )
 =
 
 α
 
 K
 
 
 (
 x
 )
 
 t
 
 K
 
 
 
 
 {\displaystyle \alpha (x)=\alpha ^{K}(x)t_{K}}
 
.
It acts on the field 
 
 
 
 ϕ
 (
 x
 )
 
 
 {\displaystyle \phi (x)}
 
 as

ϕ
        (
        x
        )
        →
        
          ϕ
          ′
        
        (
        x
        )
        =
        U
        (
        x
        )
        ϕ
        (
        x
        )
        ≡
        
          e
          
            i
            α
            (
            x
            )
          
        
        ϕ
        (
        x
        )
        ,
      
    
    {\displaystyle \phi (x)\rightarrow \phi '(x)=U(x)\phi (x)\equiv e^{i\alpha (x)}\phi (x),}

ϕ
          
            †
          
        
        (
        x
        )
        →
        ϕ
        
          
            
              
              ′
            
          
          
            †
          
        
        ≡
        
          ϕ
          
            †
          
        
        (
        x
        )
        
          U
          
            †
          
        
        (
        x
        )
        =
        
          ϕ
          
            †
          
        
        (
        x
        )
        
          e
          
            −
            i
            α
            (
            x
            )
          
        
        ,
        
        
          U
          
            †
          
        
        =
        
          U
          
            −
            1
          
        
        .
      
    
    {\displaystyle \phi ^{\dagger }(x)\rightarrow \phi {'}^{\dagger }\equiv \phi ^{\dagger }(x)U^{\dagger }(x)=\phi ^{\dagger }(x)e^{-i\alpha (x)},\qquad U^{\dagger }=U^{-1}.}

Now the partial derivative 
 
 
 
 
 ∂
 
 μ
 
 
 
 
 {\displaystyle \partial _{\mu }}
 
 transforms, accordingly, as

∂
 
 μ
 
 
 ϕ
 (
 x
 )
 →
 
 ∂
 
 μ
 
 
 
 ϕ
 ′
 
 (
 x
 )
 =
 U
 (
 x
 )
 
 ∂
 
 μ
 
 
 ϕ
 (
 x
 )
 +
 (
 
 ∂
 
 μ
 
 
 U
 )
 ϕ
 (
 x
 )
 ≡
 
 e
 
 i
 α
 (
 x
 )
 
 
 
 ∂
 
 μ
 
 
 ϕ
 (
 x
 )
 +
 i
 (
 
 ∂
 
 μ
 
 
 α
 )
 
 e
 
 i
 α
 (
 x
 )
 
 
 ϕ
 (
 x
 )
 
 
 {\displaystyle \partial _{\mu }\phi (x)\rightarrow \partial _{\mu }\phi '(x)=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)}
 
.
Therefore, a kinetic term of the form 
 
 
 
 
 ϕ
 
 †
 
 
 
 ∂
 
 μ
 
 
 ϕ
 
 
 {\displaystyle \phi ^{\dagger }\partial _{\mu }\phi }
 
 in a Lagrangian is not invariant under gauge transformations.

<h2 id="definition-of-the-gauge-covariant-derivative">Definition of the gauge covariant derivative</h2>
The root cause of the non gauge invariance is that in writing the field 
 
 
 
 ϕ
 =
 (
 
 ϕ
 
 1
 
 
 ,
 …
 
 ϕ
 
 n
 
 
 )
 
 
 {\displaystyle \phi =(\phi _{1},\ldots \phi _{n})}
 
 as a row vector or in <a href="/facts/Index_notation/yaTwqxDY">index notation</a> 
 
 
 
 
 ϕ
 
 a
 
 
 
 
 {\displaystyle \phi _{a}}
 
, we have implicitly made a choice of basis frame field i.e. a set of fields 
 
 
 
 
 φ
 
 1
 
 
 (
 x
 )
 ,
 …
 ,
 
 φ
 
 n
 
 
 (
 x
 )
 
 
 {\displaystyle \varphi ^{1}(x),\ldots ,\varphi ^{n}(x)}
 
 
such that every field can be uniquely expressed as 
 
 
 
 ϕ
 =
 
 ϕ
 
 a
 
 
 
 φ
 
 a
 
 
 
 
 {\displaystyle \phi =\phi _{a}\varphi ^{a}}
 
 
for functions 
 
 
 
 
 ϕ
 
 a
 
 
 (
 x
 )
 
 
 {\displaystyle \phi _{a}(x)}
 
 (using <a href="/facts/Einstein_summation/UXwS43wU">Einstein summation</a>), and assumed the frame fields 
 
 
 
 
 φ
 
 a
 
 
 
 
 {\displaystyle \varphi ^{a}}
 
 are constant. 
Local (i.e. 
 
 
 
 x
 
 
 {\displaystyle x}
 
 dependent) gauge invariance can be considered as invariance under the choice of frame. 
However, if one basis frame is as good as any gauge equivalent other one, we can not assume a frame field to be constant without 
breaking local gauge symmetry.
We can introduce the gauge covariant derivative 
 
 
 
 
 D
 
 μ
 
 
 
 
 {\displaystyle D_{\mu }}
 
 as 
a generalisation of the partial derivative 
 
 
 
 
 ∂
 
 μ
 
 
 
 
 {\displaystyle \partial _{\mu }}
 
 that acts directly on the field 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 rather than its components 
 
 
 
 
 ϕ
 
 a
 
 
 
 
 {\displaystyle \phi _{a}}
 
 with respect to a choice of frame. 
A gauge covariant derivative is defined as an operator satisfying a <a href="/facts/Product_rule/MyKxsP5t">product rule</a>

D
          
            μ
          
        
        (
        f
        ϕ
        )
        =
        (
        
          ∂
          
            μ
          
        
        f
        )
        ϕ
        +
        f
        (
        
          D
          
            μ
          
        
        ϕ
        )
      
    
    {\displaystyle D_{\mu }(f\phi )=(\partial _{\mu }f)\phi +f(D_{\mu }\phi )}

for every smooth function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 (this is the defining property of a connection).
To go back to index notation we use the product rule

D
 
 μ
 
 
 ϕ
 =
 
 D
 
 μ
 
 
 (
 
 ϕ
 
 a
 
 
 
 φ
 
 a
 
 
 )
 =
 (
 
 ∂
 
 μ
 
 
 
 ϕ
 
 a
 
 
 )
 
 φ
 
 a
 
 
 +
 
 ϕ
 
 a
 
 
 (
 
 D
 
 μ
 
 
 
 φ
 
 a
 
 
 )
 .
 
 
 {\displaystyle D_{\mu }\phi =D_{\mu }(\phi _{a}\varphi ^{a})=(\partial _{\mu }\phi _{a})\varphi ^{a}+\phi _{a}(D_{\mu }\varphi ^{a}).}
 
.
For a fixed 
 
 
 
 a
 
 
 {\displaystyle a}
 
, 
 
 
 
 
 D
 
 μ
 
 
 
 φ
 
 a
 
 
 
 
 {\displaystyle D_{\mu }\varphi ^{a}}
 
 is a field, so can be expanded with respect to the frame field.
Hence a gauge covariant derivative and frame field defines a (possibly non Abelian) <a href="/facts/Gauge_potential/fSW3gggv">gauge potential</a>

D
          
            μ
          
        
        
          φ
          
            a
          
        
        =
        −
        i
        g
        
          A
          
            μ
            b
          
          
            a
          
        
        
          φ
          
            b
          
        
      
    
    {\displaystyle D_{\mu }\varphi ^{a}=-igA_{\mu b}^{a}\varphi ^{b}}

(the factor 
 
 
 
 −
 i
 g
 
 
 {\displaystyle -ig}
 
 is conventional for compact gauge groups and is interpreted as a coupling constant). 
Conversely given the frame 
 
 
 
 
 φ
 
 1
 
 
 ,
 …
 
 φ
 
 n
 
 
 
 
 {\displaystyle \varphi ^{1},\ldots \varphi ^{n}}
 
 and a gauge potential 
 
 
 
 
 A
 
 μ
 b
 
 
 a
 
 
 
 
 {\displaystyle A_{\mu b}^{a}}
 
,
this uniquely defines the gauge covariant derivative. 
We then get

D
 
 μ
 
 
 ϕ
 =
 (
 
 D
 
 μ
 
 
 ϕ
 
 )
 
 a
 
 
 
 φ
 
 a
 
 
 =
 (
 
 ∂
 
 μ
 
 
 
 ϕ
 
 a
 
 
 −
 i
 g
 
 A
 
 μ
 a
 
 
 b
 
 
 
 ϕ
 
 b
 
 
 )
 
 φ
 
 a
 
 
 
 
 {\displaystyle D_{\mu }\phi =(D_{\mu }\phi )_{a}\varphi ^{a}=(\partial _{\mu }\phi _{a}-igA_{\mu a}^{b}\phi _{b})\varphi ^{a}}
 
.
and with suppressed frame fields this gives in index notation

(
        
          D
          
            μ
          
        
        ϕ
        
          )
          
            a
          
        
        =
        
          ∂
          
            μ
          
        
        
          ϕ
          
            a
          
        
        −
        i
        g
        
          A
          
            μ
            a
          
          
            b
          
        
        
          ϕ
          
            b
          
        
        ,
      
    
    {\displaystyle (D_{\mu }\phi )_{a}=\partial _{\mu }\phi _{a}-igA_{\mu a}^{b}\phi _{b},}

which by abuse of notation is often written as

D
 
 μ
 
 
 
 ϕ
 
 a
 
 
 =
 
 ∂
 
 μ
 
 
 
 ϕ
 
 a
 
 
 −
 i
 g
 
 A
 
 μ
 a
 
 
 b
 
 
 
 ϕ
 
 b
 
 
 
 
 {\displaystyle D_{\mu }\phi _{a}=\partial _{\mu }\phi _{a}-igA_{\mu a}^{b}\phi _{b}}
 
.
This is the definition of the gauge covariant derivative as usually presented in physics.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>
The gauge covariant derivative is often assumed to satisfy additional conditions making additional structure "constant" in the sense that the covariant derivative vanishes.
For example, if we have a <a href="/facts/Hermitian_product/Kp6qUA8v">Hermitian product</a> 
 
 
 
 h
 
 
 {\displaystyle h}
 
 on the fields (e.g. the Dirac conjugate inner product 
 
 
 
 
 
 
 ϕ
 ¯
 
 
 
 ψ
 
 
 {\displaystyle {\bar {\phi }}\psi }
 
 for spinors) reducing the gauge group to a unitary group, we can impose the <a href="/facts/Hermitian_connection/uQEQRbtx">further condition</a>

∂
          
            μ
          
        
        h
        (
        ϕ
        ,
        ψ
        )
        =
        h
        (
        
          D
          
            μ
          
        
        ϕ
        ,
        ψ
        )
        +
        h
        (
        ϕ
        ,
        
          D
          
            μ
          
        
        ψ
        )
      
    
    {\displaystyle \partial _{\mu }h(\phi ,\psi )=h(D_{\mu }\phi ,\psi )+h(\phi ,D_{\mu }\psi )}

making the Hermitian product "constant". Writing this out with respect to a local 
 
 
 
 h
 
 
 {\displaystyle h}
 
-orthonormal frame field gives

∂
 
 μ
 
 
 (
 
 ϕ
 
 a
 
 
 ∗
 
 
 
 ψ
 
 a
 
 
 )
 =
 
 ∑
 
 a
 
 
 (
 
 D
 
 μ
 
 
 ϕ
 
 )
 
 a
 
 
 ∗
 
 
 
 ψ
 
 a
 
 
 +
 
 ϕ
 
 a
 
 
 ∗
 
 
 (
 
 D
 
 μ
 
 
 ψ
 
 )
 
 a
 
 
 
 
 {\displaystyle \partial _{\mu }(\phi _{a}^{*}\psi _{a})=\sum _{a}(D_{\mu }\phi )_{a}^{*}\psi _{a}+\phi _{a}^{*}(D_{\mu }\psi )_{a}}
 
,
and using the above we see that 
 
 
 
 
 A
 
 μ
 
 
 
 
 {\displaystyle A_{\mu }}
 
 must be Hermitian i.e. 
 
 
 
 
 A
 
 μ
 a
 
 
 b
 
 
 =
 
 
 
 A
 
 μ
 b
 
 
 a
 
 
 
 
 ∗
 
 
 
 
 {\displaystyle A_{\mu a}^{b}={A_{\mu b}^{a}}^{*}}
 
 (motivating the extra factor 
 
 
 
 i
 
 
 {\displaystyle i}
 
). The Hermitian matrices are (up to the factor 
 
 
 
 i
 
 
 {\displaystyle i}
 
) the generators of the unitary group. More generally if the gauge covariant derivative preserves a gauge group 
 
 
 
 G
 
 
 {\displaystyle G}
 
 acting with representation 
 
 
 
 ρ
 
 
 {\displaystyle \rho }
 
, 
the gauge covariant connection can be written as

(
        
          D
          
            μ
          
        
        ϕ
        
          )
          
            a
          
        
        =
        
          ∂
          
            μ
          
        
        
          ϕ
          
            a
          
        
        −
        i
        g
        
          A
          
            μ
          
          
            K
          
        
        
          ρ
          ′
        
        (
        
          t
          
            K
          
        
        
          )
          
            a
          
          
            b
          
        
        
          ϕ
          
            b
          
        
      
    
    {\displaystyle (D_{\mu }\phi )_{a}=\partial _{\mu }\phi _{a}-igA_{\mu }^{K}\rho '(t_{K})_{a}^{b}\phi _{b}}

where 
 
 
 
 
 ρ
 ′
 
 
 
 {\displaystyle \rho '}
 
 is representation of the Lie algebra associated to the group representation 
 
 
 
 ρ
 
 
 {\displaystyle \rho }
 
 (loc. cit.).
Note that including the gauge covariant derivative (or its <a href="/facts/Gauge_potential/fSW3gggv">gauge potential</a>), as a physical field, "field with zero gauge covariant derivative along the tangent of a curve 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
"

D
          
            
              
                γ
                ˙
              
            
          
        
        ϕ
        =
        (
        
          
            d
            
              d
              t
            
          
        
        
          γ
          
            μ
          
        
        )
        
          D
          
            μ
          
        
        ϕ
        =
        0
      
    
    {\displaystyle D_{\dot {\gamma }}\phi =({\frac {d}{dt}}\gamma ^{\mu })D_{\mu }\phi =0}

is a physically meaningful definition of a field 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 constant along a (smooth) curve.
Hence the gauge covariant derivative defines (and is defined by) <a href="/facts/Parallel_transport/QX0Wfcte">parallel transport</a>.

<h2 id="gauge-field-strength">Gauge Field Strength</h2>
Unlike the partial derivatives, the gauge covariant derivatives do not commute. However they almost do in the sense that the commutator is not an operator of order 2 but of order 0, i.e. is linear over functions:

[
 
 D
 
 μ
 
 
 ,
 
 D
 
 ν
 
 
 ]
 (
 f
 ϕ
 )
 =
 (
 
 ∂
 
 μ
 
 
 
 ∂
 
 ν
 
 
 f
 )
 ϕ
 +
 
 ∂
 
 ν
 
 
 f
 
 D
 
 μ
 
 
 ϕ
 +
 
 ∂
 
 μ
 
 
 f
 
 D
 
 ν
 
 
 ϕ
 +
 f
 
 D
 
 μ
 
 
 
 D
 
 ν
 
 
 ϕ
 −
 (
 μ
 ↔
 ν
 )
 =
 f
 [
 
 D
 
 μ
 
 
 ,
 
 D
 
 ν
 
 
 ]
 ϕ
 
 
 {\displaystyle [D_{\mu },D_{\nu }](f\phi )=(\partial _{\mu }\partial _{\nu }f)\phi +\partial _{\nu }fD_{\mu }\phi +\partial _{\mu }fD_{\nu }\phi +fD_{\mu }D_{\nu }\phi -(\mu \leftrightarrow \nu )=f[D_{\mu },D_{\nu }]\phi }
 
.
The linear map

F
          
            μ
            ν
          
        
        =
        −
        1
        
          /
        
        (
        i
        g
        )
        [
        
          D
          
            μ
          
        
        ,
        
          D
          
            ν
          
        
        ]
      
    
    {\displaystyle F_{\mu \nu }=-1/(ig)[D_{\mu },D_{\nu }]}

is called the gauge field strength (loc. cit).
In index notation, using the gauge potential

F
 
 μ
 ν
 
 b
 
 
  
 a
 
 
 =
 
 ∂
 
 μ
 
 
 
 A
 
 ν
 b
 
 
 a
 
 
 −
 
 ∂
 
 ν
 
 
 
 A
 
 μ
 b
 
 
 a
 
 
 −
 i
 g
 (
 
 A
 
 μ
 c
 
 
 a
 
 
 
 A
 
 ν
 b
 
 
 c
 
 
 −
 
 A
 
 ν
 c
 
 
 a
 
 
 
 A
 
 μ
 b
 
 
 c
 
 
 )
 
 
 {\displaystyle F_{\mu \nu \,b}^{\ a}=\partial _{\mu }A_{\nu b}^{a}-\partial _{\nu }A_{\mu b}^{a}-ig(A_{\mu c}^{a}A_{\nu b}^{c}-A_{\nu c}^{a}A_{\mu b}^{c})}
 
.
If 
 
 
 
 
 D
 
 μ
 
 
 
 
 {\displaystyle D_{\mu }}
 
 is a G covariant derivative, one can interpret the latter term as a commutator in the Lie algebra of G and 
 
 
 
 
 F
 
 μ
 ν
 
 
 
 
 {\displaystyle F_{\mu \nu }}
 
 as Lie algebra valued (loc. cit).

<h2 id="invariance-under-gauge-transformations">Invariance under gauge transformations</h2>
The gauge covariant derivative transforms covariantly under Gauge transformations, i.e. for all 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }

D
          
            μ
          
        
        ϕ
        (
        x
        )
        →
        
          D
          
            μ
          
          ′
        
        
          ϕ
          ′
        
        (
        x
        )
        =
        
          D
          
            μ
          
          ′
        
        U
        (
        x
        )
        ϕ
        (
        x
        )
        =
        U
        (
        x
        )
        
          D
          
            μ
          
        
        ϕ
        (
        x
        )
        ,
      
    
    {\displaystyle D_{\mu }\phi (x)\rightarrow D'_{\mu }\phi '(x)=D'_{\mu }U(x)\phi (x)=U(x)D_{\mu }\phi (x),}

which in operator form takes the form

D
          
            μ
          
          ′
        
        U
        (
        x
        )
        =
        U
        (
        x
        )
        
          D
          
            μ
          
        
      
    
    {\displaystyle D'_{\mu }U(x)=U(x)D_{\mu }}

or

D
          
            μ
          
          ′
        
        =
        U
        (
        x
        )
        
          D
          
            μ
          
        
        
          U
          
            −
            1
          
        
        (
        x
        )
        .
      
    
    {\displaystyle D'_{\mu }=U(x)D_{\mu }U^{-1}(x).}

In particular (suppressing dependence on 
 
 
 
 x
 
 
 {\displaystyle x}
 
)

−
 i
 g
 
 F
 
 μ
 ν
 
 ′
 
 =
 [
 
 D
 
 μ
 
 ′
 
 ,
 
 D
 
 ν
 
 ′
 
 ]
 =
 [
 U
 
 D
 
 μ
 
 
 
 U
 
 −
 1
 
 
 ,
 U
 
 D
 
 ν
 
 
 
 U
 
 −
 1
 
 
 ]
 =
 U
 [
 
 D
 
 μ
 
 
 ,
 
 D
 
 ν
 
 
 ]
 
 U
 
 −
 1
 
 
 =
 −
 i
 g
 U
 
 F
 
 μ
 ν
 
 
 
 U
 
 −
 1
 
 
 
 
 {\displaystyle -igF'_{\mu \nu }=[D'_{\mu },D'_{\nu }]=[UD_{\mu }U^{-1},UD_{\nu }U^{-1}]=U[D_{\mu },D_{\nu }]U^{-1}=-igUF_{\mu \nu }U^{-1}}
 
.
Further, (suppressing indices and replacing them by matrix multiplication) if 
 
 
 
 
 D
 
 μ
 
 
 =
 
 ∂
 
 μ
 
 
 −
 i
 g
 
 A
 
 μ
 
 
 
 
 {\displaystyle D_{\mu }=\partial _{\mu }-igA_{\mu }}
 
 is of the form above,

D
 
 μ
 
 ′
 
 
 
 {\displaystyle D'_{\mu }}
 
 is of the form

D
          
            μ
          
          ′
        
        =
        
          ∂
          
            μ
          
        
        +
        (
        
          ∂
          
            μ
          
        
        
          U
          
            −
            1
          
        
        )
        U
        −
        i
        g
        U
        
          A
          
            μ
          
        
        
          U
          
            −
            1
          
        
      
    
    {\displaystyle D'_{\mu }=\partial _{\mu }+(\partial _{\mu }U^{-1})U-igUA_{\mu }U^{-1}}

or using 
 
 
 
 U
 (
 x
 )
 =
 
 e
 
 i
 α
 (
 x
 )
 
 
 
 
 {\displaystyle U(x)=e^{i\alpha (x)}}
 
,

D
          
            μ
          
          ′
        
        =
        
          ∂
          
            μ
          
        
        −
        i
        
          ∂
          
            μ
          
        
        α
        −
        i
        g
        U
        
          A
          
            μ
          
        
        
          U
          
            −
            1
          
        
      
    
    {\displaystyle D'_{\mu }=\partial _{\mu }-i\partial _{\mu }\alpha -igUA_{\mu }U^{-1}}

which is also of this form.
In the Hermitian case with a unitary gauge group 
 
 
 
 
 U
 
 −
 1
 
 
 =
 
 U
 
 †
 
 
 
 
 {\displaystyle U^{-1}=U^{\dagger }}
 
 and we have found a first order differential operator 
 
 
 
 
 D
 
 μ
 
 
 
 
 {\displaystyle D_{\mu }}
 
 with 
 
 
 
 
 ∂
 
 μ
 
 
 
 
 {\displaystyle \partial _{\mu }}
 
 as first order term such that

ϕ
 
 †
 
 
 
 D
 
 μ
 
 
 ϕ
 →
 
 ϕ
 
 ′
 
 †
 
 
 
 
 D
 
 μ
 
 ′
 
 
 ϕ
 ′
 
 =
 
 ϕ
 
 †
 
 
 
 D
 
 μ
 
 
 ϕ
 .
 
 
 {\displaystyle \phi ^{\dagger }D_{\mu }\phi \rightarrow \phi '^{\dagger }D'_{\mu }\phi '=\phi ^{\dagger }D_{\mu }\phi .}
 
.
<h2 id="gauge-theory">Gauge theory</h2>
In <a href="/facts/Gauge_theory/fSW3gggv">gauge theory</a>, which studies a particular class of <a href="/facts/Field_(physics)/FJdAKeRA">fields</a> which are of importance in <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a>, different fields are used in Lagrangians that are invariant under local gauge transformations. Kinetic terms involve derivatives of the fields which by the above arguments need to involve gauge covariant derivatives.

<h3>Abelian Gauge Theory</h3>
the gauge covariant derivative 
 
 
 
 
 D
 
 μ
 
 
 
 
 {\displaystyle D_{\mu }}
 
 on a complex scalar field 
 
 
 
 ϕ
 =
 
 ϕ
 
 1
 
 
 
 φ
 
 1
 
 
 
 
 {\displaystyle \phi =\phi _{1}\varphi ^{1}}
 
 (i.e. 
 
 
 
 n
 =
 1
 
 
 {\displaystyle n=1}
 
) of charge 
 
 
 
 q
 
 
 {\displaystyle q}
 
 is a 
 
 
 
 U
 (
 1
 )
 
 
 {\displaystyle U(1)}
 
 connection. The gauge potential 
 
 
 
 
 A
 
 μ
 
 
 
 
 {\displaystyle A_{\mu }}
 
 is a (1 x 1) matrix, i.e. a scalar.

(
        
          D
          
            μ
          
        
        ϕ
        
          )
          
            1
          
        
        =
        (
        
          ∂
          
            μ
          
        
        
          ϕ
          
            1
          
        
        −
        i
        q
        
          A
          
            μ
          
        
        
          ϕ
          
            1
          
        
        )
      
    
    {\displaystyle (D_{\mu }\phi )_{1}=(\partial _{\mu }\phi _{1}-iqA_{\mu }\phi _{1})}

The gauge field strength is

F
          
            μ
            ν
          
        
        =
        
          ∂
          
            μ
          
        
        
          A
          
            ν
          
        
        −
        
          ∂
          
            ν
          
        
        
          A
          
            μ
          
        
      
    
    {\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}

The gauge potential can be interpreted as <a href="/facts/Electromagnetic_four-potential/S8gTFNAt">electromagnetic four-potential</a> and the gauge field strength as the <a href="/facts/Faraday_tensor/MWYuKLPy">electromagnetic field tensor</a>. Since this only involves the charge of the field and not higher multipoles like the magnetic moment (and in a loose and non unique way, because it replaces 
 
 
 
 
 ∂
 
 μ
 
 
 
 
 {\displaystyle \partial _{\mu }}
 
 by 
 
 
 
 
 D
 
 μ
 
 
 
 
 {\displaystyle D_{\mu }}
 
 <a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>) this is called <a href="/facts/Minimal_coupling/wFh0TmCk">minimal coupling</a>.
For a Dirac spinor field 
 
 
 
 ψ
 
 
 {\displaystyle \psi }
 
 of charge 
 
 
 
 q
 
 
 {\displaystyle q}
 
 the covariant derivative is also a 
 
 
 
 U
 (
 1
 )
 
 
 {\displaystyle U(1)}
 
 connection (because it has to commute with the gamma matrices) and is defined as

(
        
          D
          
            μ
          
        
        ψ
        
          )
          
            α
          
        
        :=
        (
        
          ∂
          
            μ
          
        
        −
        i
        q
        
          A
          
            μ
          
        
        )
        
          ψ
          
            α
          
        
      
    
    {\displaystyle (D_{\mu }\psi )_{\alpha }:=(\partial _{\mu }-iqA_{\mu })\psi _{\alpha }}

where again 
 
 
 
 
 A
 
 μ
 
 
 
 
 {\displaystyle A_{\mu }}
 
 is interpreted as the <a href="/facts/Electromagnetic_four-potential/S8gTFNAt">electromagnetic four-potential</a> and 
 
 
 
 
 F
 
 μ
 ν
 
 
 
 
 {\displaystyle F_{\mu \nu }}
 
 as the electromagnetic field tensor. 
(The minus sign is a convention valid for a Minkowski <a href="/facts/Metric_signature/w8d57bQG">metric signature</a> (−, +, +, +), which is common in <a href="/facts/General_relativity/jVsoIg8X">general relativity</a> and used below. For the <a href="/facts/Particle_physics/YgIuyj2t">particle physics</a> convention (+, −, −, −), it is 
 
 
 
 
 D
 
 μ
 
 
 :=
 
 ∂
 
 μ
 
 
 +
 i
 q
 
 A
 
 μ
 
 
 
 
 {\displaystyle D_{\mu }:=\partial _{\mu }+iqA_{\mu }}
 
. The <a href="/facts/Electron/L0KBo5cW">electron</a>'s charge is defined negative as 
 
 
 
 
 q
 
 e
 
 
 =
 −
 
 |
 
 e
 
 |
 
 
 
 {\displaystyle q_{e}=-|e|}
 
, while the Dirac field is defined to transform positively as 
 
 
 
 ψ
 (
 x
 )
 →
 
 e
 
 i
 q
 α
 (
 x
 )
 
 
 ψ
 (
 x
 )
 .
 
 
 {\displaystyle \psi (x)\rightarrow e^{iq\alpha (x)}\psi (x).}
 
)

<h3>Quantum electrodynamics</h3>
If a gauge transformation is given by

ψ
        ↦
        
          e
          
            i
            Λ
          
        
        ψ
      
    
    {\displaystyle \psi \mapsto e^{i\Lambda }\psi }

and for the gauge potential

A
          
            μ
          
        
        ↦
        
          A
          
            μ
          
        
        +
        
          
            1
            e
          
        
        (
        
          ∂
          
            μ
          
        
        Λ
        )
      
    
    {\displaystyle A_{\mu }\mapsto A_{\mu }+{1 \over e}(\partial _{\mu }\Lambda )}

then 
 
 
 
 
 D
 
 μ
 
 
 
 
 {\displaystyle D_{\mu }}
 
 transforms as

D
 
 μ
 
 
 ↦
 
 ∂
 
 μ
 
 
 −
 i
 e
 
 A
 
 μ
 
 
 −
 i
 (
 
 ∂
 
 μ
 
 
 Λ
 )
 
 
 {\displaystyle D_{\mu }\mapsto \partial _{\mu }-ieA_{\mu }-i(\partial _{\mu }\Lambda )}
 
,
and 
 
 
 
 
 D
 
 μ
 
 
 ψ
 
 
 {\displaystyle D_{\mu }\psi }
 
 transforms as

D
          
            μ
          
        
        ψ
        ↦
        
          e
          
            i
            Λ
          
        
        
          D
          
            μ
          
        
        ψ
      
    
    {\displaystyle D_{\mu }\psi \mapsto e^{i\Lambda }D_{\mu }\psi }

and 
 
 
 
 
 
 
 ψ
 ¯
 
 
 
 :=
 
 ψ
 
 †
 
 
 
 γ
 
 0
 
 
 
 
 {\displaystyle {\bar {\psi }}:=\psi ^{\dagger }\gamma ^{0}}
 
 transforms as

ψ
              ¯
            
          
        
        ↦
        
          
            
              ψ
              ¯
            
          
        
        
          e
          
            −
            i
            Λ
          
        
      
    
    {\displaystyle {\bar {\psi }}\mapsto {\bar {\psi }}e^{-i\Lambda }}

so that

ψ
              ¯
            
          
        
        
          D
          
            μ
          
        
        ψ
        ↦
        
          
            
              ψ
              ¯
            
          
        
        
          D
          
            μ
          
        
        ψ
      
    
    {\displaystyle {\bar {\psi }}D_{\mu }\psi \mapsto {\bar {\psi }}D_{\mu }\psi }

and 
 
 
 
 
 
 
 ψ
 ¯
 
 
 
 
 D
 
 μ
 
 
 ψ
 
 
 {\displaystyle {\bar {\psi }}D_{\mu }\psi }
 
 in the QED <a href="/facts/Lagrangian_(field_theory)/p9BCUz57">Lagrangian</a> is therefore gauge invariant, and the gauge covariant derivative is thus named aptly.
On the other hand, the non-covariant derivative 
 
 
 
 
 ∂
 
 μ
 
 
 
 
 {\displaystyle \partial _{\mu }}
 
 would not preserve the Lagrangian's gauge symmetry, since

ψ
 ¯
 
 
 
 
 ∂
 
 μ
 
 
 ψ
 ↦
 
 
 
 ψ
 ¯
 
 
 
 
 ∂
 
 μ
 
 
 ψ
 +
 i
 
 
 
 ψ
 ¯
 
 
 
 (
 
 ∂
 
 μ
 
 
 Λ
 )
 ψ
 
 
 {\displaystyle {\bar {\psi }}\partial _{\mu }\psi \mapsto {\bar {\psi }}\partial _{\mu }\psi +i{\bar {\psi }}(\partial _{\mu }\Lambda )\psi }
 
.
<h3>Quantum chromodynamics</h3>
In <a href="/facts/Quantum_chromodynamics/3cA2ctyb">quantum chromodynamics</a>, the gauge covariant derivative is<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

D
          
            μ
          
        
        :=
        
          ∂
          
            μ
          
        
        −
        i
        
          g
          
            s
          
        
        
        
          G
          
            μ
          
          
            α
          
        
        
        
          λ
          
            α
          
        
        
          /
        
        2
      
    
    {\displaystyle D_{\mu }:=\partial _{\mu }-ig_{s}\,G_{\mu }^{\alpha }\,\lambda _{\alpha }/2}

where 
 
 
 
 
 g
 
 s
 
 
 
 
 {\displaystyle g_{s}}
 
 is the <a href="/facts/Coupling_constant/Queop1TR">coupling constant</a> of the strong interaction, 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is the gluon <a href="/facts/Gauge_field/fSW3gggv">gauge field</a>, for eight different gluons 
 
 
 
 α
 =
 1
 …
 8
 
 
 {\displaystyle \alpha =1\dots 8}
 
, and where 
 
 
 
 
 λ
 
 α
 
 
 
 
 {\displaystyle \lambda _{\alpha }}
 
 is one of the eight <a href="/facts/Gell-Mann_matrices/8WleYNIf">Gell-Mann matrices</a>. The Gell-Mann matrices give a <a href="/facts/Representation_of_a_Lie_group/29R96ar2">representation</a> of the <a href="/facts/Color_charge/aWS1xsCR">color symmetry</a> group <a href="/facts/SU(3)/Qgy2orcu">SU(3)</a>. For quarks, the representation is the <a href="/facts/Fundamental_representation/vgM8UkNT">fundamental representation</a>, for gluons, the representation is the <a href="/facts/Adjoint_representation/bWG4c2su">adjoint representation</a>.

<h3>Standard Model</h3>
The covariant derivative in the <a href="/facts/Standard_Model/LhVGf9ow">Standard Model</a> combines the electromagnetic, the weak and the strong interactions. It can be expressed in the following form:<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

D
          
            μ
          
        
        :=
        
          ∂
          
            μ
          
        
        −
        i
        
          
            
              g
              ′
            
            2
          
        
        Y
        
        
          B
          
            μ
          
        
        −
        i
        
          
            g
            2
          
        
        
          σ
          
            j
          
        
        
        
          W
          
            μ
          
          
            j
          
        
        −
        i
        
          
            
              g
              
                s
              
            
            2
          
        
        
          λ
          
            α
          
        
        
        
          G
          
            μ
          
          
            α
          
        
      
    
    {\displaystyle D_{\mu }:=\partial _{\mu }-i{\frac {g'}{2}}Y\,B_{\mu }-i{\frac {g}{2}}\sigma _{j}\,W_{\mu }^{j}-i{\frac {g_{s}}{2}}\lambda _{\alpha }\,G_{\mu }^{\alpha }}

The gauge fields here belong to the <a href="/facts/Fundamental_representation/vgM8UkNT">fundamental representations</a> of the <a href="/facts/Electroweak/QATz4dYF">electroweak</a> <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> 
 
 
 
 U
 (
 1
 )
 ×
 S
 U
 (
 2
 )
 
 
 {\displaystyle U(1)\times SU(2)}
 
 times the <a href="/facts/Color_charge/aWS1xsCR">color symmetry</a> Lie group <a href="/facts/SU(3)/Qgy2orcu">SU(3)</a>. The coupling constant 
 
 
 
 
 g
 ′
 
 
 
 {\displaystyle g'}
 
 provides the coupling of the hypercharge 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 to the 
 
 
 
 B
 
 
 {\displaystyle B}
 
 boson and 
 
 
 
 g
 
 
 {\displaystyle g}
 
 the coupling via the three vector bosons 
 
 
 
 
 W
 
 j
 
 
 
 
 {\displaystyle W^{j}}
 
 
 
 
 
 (
 j
 =
 1
 ,
 2
 ,
 3
 )
 
 
 {\displaystyle (j=1,2,3)}
 
 to the weak isospin, whose components are written here as the <a href="/facts/Pauli_matrices/BdeC2wos">Pauli matrices</a> 
 
 
 
 
 σ
 
 j
 
 
 
 
 {\displaystyle \sigma _{j}}
 
. Via the <a href="/facts/Electroweak_interaction/QATz4dYF">Higgs mechanism</a>, these boson fields combine into the massless electromagnetic field 
 
 
 
 
 A
 
 μ
 
 
 
 
 {\displaystyle A_{\mu }}
 
 and the fields for the three massive vector bosons 
 
 
 
 
 W
 
 ±
 
 
 
 
 {\displaystyle W^{\pm }}
 
 and 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
.

<h2 id="general-relativity">General relativity</h2>
The <a href="/facts/Covariant_derivative/sRGN0Xh6">covariant derivative</a> in <a href="/facts/General_relativity/jVsoIg8X">general relativity</a> is a special example of the gauge covariant derivative. It corresponds to the <a href="/facts/Levi_Civita_connection/16KGSQUY">Levi Civita connection</a> (a special <a href="/facts/Riemannian_connection/YyGhEDKr">Riemannian connection</a>) on the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> (or the <a href="/facts/Frame_bundle/Gugt3qyL">frame bundle</a>) i.e. it acts on tangent vector fields or more generally, tensors. It is usually written as 
 
 
 
 ∇
 
 
 {\displaystyle \nabla }
 
 instead of 
 
 
 
 D
 
 
 {\displaystyle D}
 
. In this special case, a choice of (local) coordinates 
 
 
 
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 d
 
 
 
 
 {\displaystyle x^{1},\ldots ,x^{d}}
 
 not only gives partial derivatives 
 
 
 
 
 ∂
 
 μ
 
 
 
 
 {\displaystyle \partial _{\mu }}
 
, but they double as a frame of tangent vectors 
 
 
 
 
 ∂
 
 1
 
 
 ,
 …
 
 ∂
 
 d
 
 
 
 
 {\displaystyle \partial _{1},\ldots \partial _{d}}
 
 in which a <a href="/facts/Vector_field/ccFNuPPp">vector field</a> 
 
 
 
 v
 
 
 {\displaystyle v}
 
 can be uniquely expressed as 
 
 
 
 v
 =
 
 v
 
 μ
 
 
 
 ∂
 
 μ
 
 
 
 
 {\displaystyle v=v^{\mu }\partial _{\mu }}
 
 (this uses the definition of a vector field as an operator on smooth functions that satisfies a product rule i.e. a <a href="/facts/Derivation_(differential_algebra)/vcSqpxzu">derivation</a>). Therefore, in this case "the internal indices are also space time indices". 
Up to slightly different normalisation (and notation) the gauge potential 
 
 
 
 
 A
 
 μ
 ν
 
 
 λ
 
 
 
 
 {\displaystyle A_{\mu \nu }^{\lambda }}
 
 is the <a href="/facts/Christoffel_symbol/uCyd7RWX">Christoffel symbol</a> defined by

∇
 
 μ
 
 
 
 ∂
 
 ν
 
 
 =
 
 Γ
 
 μ
 ν
 
 
 λ
 
 
 
 ∂
 
 λ
 
 
 
 
 {\displaystyle \nabla _{\mu }\partial _{\nu }=\Gamma _{\mu \nu }^{\lambda }\partial _{\lambda }}
 
.
It gives the covariant derivative

(
 
 ∇
 
 μ
 
 
 v
 
 )
 
 ν
 
 
 =
 (
 
 ∇
 
 μ
 
 
 (
 
 v
 
 λ
 
 
 
 ∂
 
 λ
 
 
 )
 
 )
 
 ν
 
 
 =
 (
 (
 
 ∂
 
 μ
 
 
 
 v
 
 λ
 
 
 )
 
 ∂
 
 λ
 
 
 +
 
 v
 
 λ
 
 
 (
 
 ∇
 
 μ
 
 
 
 ∂
 
 λ
 
 
 )
 
 )
 
 ν
 
 
 =
 
 ∂
 
 μ
 
 
 
 v
 
 ν
 
 
 +
 
 Γ
 
 μ
 λ
 
 
 ν
 
 
 
 v
 
 λ
 
 
 
 
 {\displaystyle (\nabla _{\mu }v)^{\nu }=(\nabla _{\mu }(v^{\lambda }\partial _{\lambda }))^{\nu }=((\partial _{\mu }v^{\lambda })\partial _{\lambda }+v^{\lambda }(\nabla _{\mu }\partial _{\lambda }))^{\nu }=\partial _{\mu }v^{\nu }+\Gamma _{\mu \lambda }^{\nu }v^{\lambda }}
 
.
The formal similarity with the gauge covariant derivative is more clear when the choice of coordinates is decoupled from the choice of frame of vector fields 
 
 
 
 
 e
 
 1
 
 
 =
 
 e
 
 1
 
 
 μ
 
 
 
 ∂
 
 μ
 
 
 ,
 …
 ,
 
 e
 
 d
 
 
 =
 
 e
 
 d
 
 
 μ
 
 
 
 ∂
 
 μ
 
 
 
 
 {\displaystyle e_{1}=e_{1}^{\mu }\partial _{\mu },\ldots ,e_{d}=e_{d}^{\mu }\partial _{\mu }}
 
. Especially when the frame is orthonormal, such a frame is usually called a <a href="/facts/Frame_fields_in_general_relativity/Lw90nUa6">d-Bein</a>. Then

(
        
          ∇
          
            μ
          
        
        v
        
          )
          
            n
          
        
        =
        (
        
          ∇
          
            μ
          
        
        (
        
          v
          
            ℓ
          
        
        
          e
          
            ℓ
          
        
        )
        
          )
          
            n
          
        
        =
        (
        (
        
          ∂
          
            μ
          
        
        
          v
          
            ℓ
          
        
        )
        
          e
          
            ℓ
          
        
        +
        
          v
          
            ℓ
          
        
        (
        
          ∇
          
            μ
          
        
        
          e
          
            ℓ
          
        
        )
        
          )
          
            n
          
        
        =
        
          ∂
          
            μ
          
        
        
          v
          
            n
          
        
        +
        
          Γ
          
            μ
            ℓ
          
          
            n
          
        
        
          v
          
            ℓ
          
        
      
    
    {\displaystyle (\nabla _{\mu }v)^{n}=(\nabla _{\mu }(v^{\ell }e_{\ell }))^{n}=((\partial _{\mu }v^{\ell })e_{\ell }+v^{\ell }(\nabla _{\mu }e_{\ell }))^{n}=\partial _{\mu }v^{n}+\Gamma _{\mu \ell }^{n}v^{\ell }}

where 
 
 
 
 
 ∇
 
 μ
 
 
 
 e
 
 m
 
 
 =
 
 Γ
 
 μ
 m
 
 
 ℓ
 
 
 
 e
 
 ℓ
 
 
 
 
 {\displaystyle \nabla _{\mu }e_{m}=\Gamma _{\mu m}^{\ell }e_{\ell }}
 
.
The direct analogue of the "gauge freedom" of the gauge covariant derivative is the arbitrariness of the choice of an orthonormal d-Bein at each point in <a href="/facts/Space-time/uRIN1x4b">space-time</a>: local Lorentz invariance . However, in this case the more general independence of the choice of coordinates for the definition of the Levi Civita connection gives diffeomorphism or general coordinate invariance.

<h2 id="fluid-dynamics">Fluid dynamics</h2>

In <a href="/facts/Fluid_dynamics/y7uzSCfg">fluid dynamics</a>, the gauge covariant derivative of a fluid may be defined as

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

where 
 
 
 
 
 v
 
 
 
 {\displaystyle \mathbf {v} }
 
 is a velocity <a href="/facts/Vector_field/ccFNuPPp">vector field</a> of a fluid.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Kinetic_momentum/TTkGv5eY">Kinetic momentum</a></li>
<li><a href="/facts/Connection_(mathematics)/KURsCmIu">Connection (mathematics)</a></li>
<li><a href="/facts/Minimal_coupling/wFh0TmCk">Minimal coupling</a></li>
<li><a href="/facts/Ricci_calculus/We4yudeq">Ricci calculus</a></li></ul>

<ul><li>Tsutomu Kambe, <a href="http://fluid.ippt.gov.pl/ictam04/text/sessions/docs/FM23/11166/FM23_11166.pdf">Gauge Principle For Ideal Fluids And Variational Principle</a>. (PDF file.)</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">L.D. Faddeev, A.A. Slavnov, Gauge Fields: Introduction to Gauge Theory, (1980) Benjamin Cummings, ISBN 0-8053-9016-2 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Claude Itzykson, Jean-Bernard Zuber, Quantum Field Theory (1980) McGraw-Hill ISBN 0-07-032071-3 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Warren Siegel, Fields (1999) ArXiv <a href="https://arxiv.org/abs/hep-th/9912205" target="_blank">https://arxiv.org/abs/hep-th/9912205</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Richard S. Palais, The Geometrization of Physics (1981) Lecture Notes, Institute of Mathematics, National Tsing Hua University <a href="http://vmm.math.uci.edu/GeometrizationOfPhysics.pdf" target="_blank">http://vmm.math.uci.edu/GeometrizationOfPhysics.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">M. E. Mayer, "Review: David D. Bleecker, Gauge theory and variational principles", Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 1, 83--92 <a href="http://projecteuclid.org/download/pdf_1/euclid.bams/1183550979" target="_blank">http://projecteuclid.org/download/pdf_1/euclid.bams/1183550979</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Alexandre Guay, Geometrical aspects of local gauge symmetry (2004) <a href="http://philsci-archive.pitt.edu/2133/1/geometrie.pdf" target="_blank">http://philsci-archive.pitt.edu/2133/1/geometrie.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation, (1973) W. H. Freeman and Company <a href="/wiki/Gravitation_(book)" target="_blank">/wiki/Gravitation_(book)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">David Bleecker, "Gauge Theory and Variational Principles Archived 2021-07-09 at the Wayback Machine" (1982) D. Reidel Publishing (See chapter 3) <a href="https://zulfahmed.files.wordpress.com/2014/05/88623149-bleecker-d-gauge-theory-and-variational-principles-aw-1981-ka-t-201s-pqgf.pdf" target="_blank">https://zulfahmed.files.wordpress.com/2014/05/88623149-bleecker-d-gauge-theory-and-variational-principles-aw-1981-ka-t-201s-pqgf.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">David Bleecker, op. cit. (See Chapter 6.) <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Alexandre Guay, Geometrical aspects of local gauge symmetry (2004) <a href="http://philsci-archive.pitt.edu/2133/1/geometrie.pdf" target="_blank">http://philsci-archive.pitt.edu/2133/1/geometrie.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Meinhard E. Mayer, "Principal Bundles versus Lie Groupoids in Gauge Theory", (1990) in Differential Geometric Methods in Theoretical Physics, Volume 245 pp 793-802 <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation, (1973) W. H. Freeman and Company <a href="/wiki/Gravitation_(book)" target="_blank">/wiki/Gravitation_(book)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Peskin, Michael, E.; Schroeder, Daniel, V. (1995). An introduction to Quantum Field Theory. Addison Wesley. pp. 78, 490.{{cite book}}: CS1 maint: multiple names: authors list (link) <a href="/wiki/Template:Cite_book" target="_blank">/wiki/Template:Cite_book</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Jenkins, Elisabeth E.; Manohar, Aneesh V.; Trott, Michael (2013). "On Gauge Invariance and Minimal Coupling" (PDF). Journal of High Energy Physics. 2013 (9). Springer. doi:10.1007/JHEP09(2013)063. S2CID 256013401. <a href="https://link.springer.com/content/pdf/10.1007/JHEP09(2013)063.pdf" target="_blank">https://link.springer.com/content/pdf/10.1007/JHEP09(2013)063.pdf</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">"Quantum Chromodynamics (QCD)". <a href="http://www.fuw.edu.pl/~dobaczew/maub-42w/node9.html" target="_blank">http://www.fuw.edu.pl/~dobaczew/maub-42w/node9.html</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">See e.g. eq. 3.116 in C. Tully, Elementary Particle Physics in a Nutshell, 2011, Princeton University Press. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
</ol>

Gauge covariant derivative open-in-new

Gauge covariant derivative