Gauge covariant derivative

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.[1][2][3] Another approach is to understand the gauge covariant derivative as a kind of connection, and more specifically, an affine connection.[4][5][6] The affine connection is interesting because it does not require any concept of a metric tensor to be defined; the curvature of an affine connection can be understood as the field strength of the gauge potential. When a metric is available, then one can go in a different direction, and define a connection on a frame bundle. This path leads directly to general relativity; however, it requires a metric, which particle physics gauge theories do not have.

Rather than being generalizations of one-another, affine and metric geometry go off in different directions: the gauge group of (pseudo-)Riemannian geometry must be the indefinite orthogonal group O(s,r) in general, or the Lorentz group O(3,1) for space-time. This is because the fibers of the frame bundle must necessarily, by definition, connect the tangent and cotangent spaces of space-time.[7] In contrast, the gauge groups employed in particle physics could in principle be any Lie group at all, although in practice the Standard Model only uses U(1), SU(2) and SU(3). 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 exterior covariant derivative on a section of an associated bundle for the principal fiber bundle of the gauge theory;[8] and, for the case of spinors, the associated bundle would be a spin bundle of the spin structure.[9] Although conceptually the same, this approach uses a very different set of notation, and requires a far more advanced background in multiple areas of differential geometry.

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 gauge groupoid as the closest description of the gauge connection in quantum field theory.[6][10]

For ordinary Lie algebras, the gauge covariant derivative on the space symmetries (those of the pseudo-Riemannian manifold 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 Coleman–Mandula theorem. However, a premise of this theorem is violated by the Lie superalgebras (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 supersymmetry.

The more mathematical approach uses an index-free notation, emphasizing the geometric and algebraic structure of the gauge theory and its relationship to Lie algebras and Riemannian manifolds; for example, treating gauge covariance as equivariance 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.[7] The physics approach also has a pedagogical advantage: the general structure of a gauge theory can be exposed after a minimal background in multivariate calculus, whereas the geometric approach requires a large investment of time in the general theory of differential geometry, Riemannian manifolds, Lie algebras, representations of Lie algebras and principle bundles 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.

Consider a generic (possibly non-Abelian) gauge transformation acting on a ${\displaystyle n}$ component field ${\displaystyle \phi =(\phi _{a})_{a=1..n}}$. The main examples in field theory have a compact gauge group and we write the symmetry operator as ${\displaystyle U(x)=e^{i\alpha (x)}}$ where ${\displaystyle \alpha (x)}$ is an element of the Lie algebra associated with the Lie group of symmetry transformations, and can be expressed in terms of the hermitian generators of the Lie algebra (i.e. up to a factor ${\displaystyle i}$, the infinitesimal generators of the gauge group), ${\displaystyle \{t_{K}\}_{K\in {\mathcal {K}}}}$, as ${\displaystyle \alpha (x)=\alpha ^{K}(x)t_{K}}$.

It acts on the field ${\displaystyle \phi (x)}$ as

${\displaystyle \phi (x)\rightarrow \phi '(x)=U(x)\phi (x)\equiv e^{i\alpha (x)}\phi (x),}$

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

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

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

Therefore, a kinetic term of the form ${\displaystyle \phi ^{\dagger }\partial _{\mu }\phi }$ in a Lagrangian is not invariant under gauge transformations.

The root cause of the non gauge invariance is that in writing the field ${\displaystyle \phi =(\phi _{1},\ldots \phi _{n})}$ as a row vector or in index notation ${\displaystyle \phi _{a}}$, we have implicitly made a choice of basis frame field i.e. a set of fields ${\displaystyle \varphi ^{1}(x),\ldots ,\varphi ^{n}(x)}$ such that every field can be uniquely expressed as ${\displaystyle \phi =\phi _{a}\varphi ^{a}}$ for functions ${\displaystyle \phi _{a}(x)}$ (using Einstein summation), and assumed the frame fields ${\displaystyle \varphi ^{a}}$ are constant. Local (i.e. ${\displaystyle x}$ 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 fields to be constant without breaking local gauge symmetry.

We can introduce the gauge covariant derivative ${\displaystyle D_{\mu }}$ as a generalisation of the partial derivative ${\displaystyle \partial _{\mu }}$ that acts directly on the field ${\displaystyle \phi }$ rather than its components ${\displaystyle \phi _{a}}$ with respect to a choice of frame. A gauge covariant derivative is defined as an operator satisfying a product rule

${\displaystyle D_{\mu }(f\phi )=(\partial _{\mu }f)\phi +f(D_{\mu }\phi )}$

for every smooth function ${\displaystyle f}$ (this is the defining property of a connection).

To go back to index notation we use the product rule

${\displaystyle D_{\mu }\phi =D_{\mu }(\phi _{a}\varphi ^{a})=(\partial _{\mu }\phi _{a})\varphi ^{a}+\phi _{a}(D_{\mu }\varphi ^{a}).}$.

For a fixed ${\displaystyle a}$, ${\displaystyle D_{\mu }\varphi ^{a}}$ is a field, so can be expanded w.r.t. the frame field. Hence a gauge covariant derivative and frame field defines a (possibly non Abelian) gauge potential

${\displaystyle D_{\mu }\varphi ^{a}=-igA_{\mu b}^{a}\varphi ^{b}}$

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

${\displaystyle D_{\mu }\phi =(D_{\mu }\phi )_{a}\varphi ^{a}=(\partial _{\mu }\phi _{a}-igA_{\mu a}^{b}\phi _{b})\varphi ^{a}}$.

and with suppressed frame fields this gives in index notation

${\displaystyle (D_{\mu }\phi )_{a}=\partial _{\mu }\phi _{a}-igA_{\mu a}^{b}\phi _{b},}$

which by abuse of notation is often written as

${\displaystyle D_{\mu }\phi _{a}=\partial _{\mu }\phi _{a}-igA_{\mu a}^{b}\phi _{b}}$.

This is the definition of the gauge covariant derivative as usually presented in physics.[11]

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 Hermitian product ${\displaystyle h}$ on the fields (e.g. the Dirac conjugate inner product ${\displaystyle {\bar {\phi }}\psi }$ for spinors) reducing the gauge group to a unitary group, we can impose the further condition

${\displaystyle \partial _{\mu }h(\phi ,\psi )=h(D_{\mu }\phi ,\psi )+h(\phi ,D_{\mu }\psi )}$

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

${\displaystyle \partial _{\mu }(\phi _{a}^{*}\psi _{a})=\sum _{a}(D_{\mu }\phi )_{a}^{*}\psi _{a}+\phi _{a}^{*}(D_{\mu }\psi )_{a}}$,

and using the above we see that ${\displaystyle A_{\mu }}$ must be Hermitian i.e. ${\displaystyle A_{\mu a}^{b}={A_{\mu b}^{a}}^{*}}$ (motivating the extra factor ${\displaystyle i}$). The Hermitian matrices are (up to the factor ${\displaystyle i}$) the generators of the unitary group. More generally if the gauge covariant derivative preserves a gauge group ${\displaystyle G}$ acting with representation ${\displaystyle \rho }$, the gauge covariant connection can be written as

${\displaystyle (D_{\mu }\phi )_{a}=\partial _{\mu }\phi _{a}-igA_{\mu }^{K}\rho '(t_{K})_{a}^{b}\phi _{b}}$

where ${\displaystyle \rho '}$ is representation of the Lie algebra associated to the group representation ${\displaystyle \rho }$ (loc. cit.).

Note that including the gauge covariant derivative (or its gauge potential), as a physical field, "field with zero gauge covariant derivative along the tangent of a curve ${\displaystyle \gamma }$"

${\displaystyle D_{\dot {\gamma }}\phi =({\frac {d}{dt}}\gamma ^{\mu })D_{\mu }\phi =0}$

is a physically meaningful definition of a field ${\displaystyle \phi }$ constant along a (smooth) curve. Hence the gauge covariant derivative defines (and is defined by) parallel transport.

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:

${\displaystyle [D_{\mu },D_{\nu }](f\phi )=(\partial _{\mu }\partial _{\nu }f)\phi +\partial _{\nu }fD_{\mu }\phi +\partial _{\mu }fD_{\nu }\phi +fD_{\mu }D_{\nu }\phi -(\mu \leftrightarrow \nu )=f[D_{\mu },D_{\nu }]\phi }$.

The linear map

${\displaystyle F_{\mu \nu }=-1/(ig)[D_{\mu },D_{\nu }]}$

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

${\displaystyle F_{\mu \nu \,b}^{\ a}=\partial _{\mu }A_{\nu b}^{a}-\partial _{\nu }A_{\mu b}^{a}-ig(A_{\mu c}^{a}A_{\nu b}^{c}-A_{\nu c}^{a}A_{\mu b}^{c})}$.

If ${\displaystyle D_{\mu }}$ is a G covariant derivative, one can interpret the latter term as a commutator in the Lie algebra of G and ${\displaystyle F_{\mu \nu }}$ as Lie algebra valued (loc. cit).

The gauge covariant derivative transforms covariantly under Gauge transformations, i.e. for all ${\displaystyle \phi }$

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

which in operator form takes the form

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

or

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

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

${\displaystyle -igF'_{\mu \nu }=[D'_{\mu },D'_{\nu }]=[UD_{\mu }U^{-1},UD_{\nu }U^{-1}]=U[D_{\mu },D_{\nu }]U^{-1}=-igUF_{\mu \nu }U^{-1}}$.

Further, (suppressing indices and replacing them by matrix multiplication) if ${\displaystyle D_{\mu }=\partial _{\mu }-igA_{\mu }}$ is of the form above, ${\displaystyle D'_{\mu }}$ is of the form

${\displaystyle D'_{\mu }=\partial _{\mu }+(\partial _{\mu }U^{-1})U-igUA_{\mu }U^{-1}}$

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

${\displaystyle D'_{\mu }=\partial _{\mu }-i\partial _{\mu }\alpha -igUA_{\mu }U^{-1}}$

which is also of this form.

In the Hermitian case with a unitary gauge group ${\displaystyle U^{-1}=U^{\dagger }}$ and we have found a first order differential operator ${\displaystyle D_{\mu }}$ with ${\displaystyle \partial _{\mu }}$ as first order term such that

${\displaystyle \phi ^{\dagger }D_{\mu }\phi \rightarrow \phi '^{\dagger }D'_{\mu }\phi '=\phi ^{\dagger }D_{\mu }\phi .}$.

In gauge theory, which studies a particular class of fields which are of importance in quantum field theory, 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.

The covariant derivative in general relativity is a special example of the gauge covariant derivative. It corresponds to the Levi Civita connection (a special Riemannian connection) on the tangent bundle (or the frame bundle) i.e. it acts on tangent vector fields or more generally, tensors. It is usually written as ${\displaystyle \nabla }$ instead of ${\displaystyle D}$. In this special case, a choice of (local) coordinates ${\displaystyle x^{1},\ldots ,x^{d}}$ not only gives partial derivatives ${\displaystyle \partial _{\mu }}$, but they double as a frame of tangent vectors ${\displaystyle \partial _{1},\ldots \partial _{d}}$ in which a vector field ${\displaystyle v}$ can be uniquely expressed as ${\displaystyle v=v^{\mu }\partial _{\mu }}$ (this uses the definition of a vector field as an operator on smooth functions that satisfies a product rule i.e. a derivation). Therefore, in this case "the internal indices are also space time indices". Up to slightly different normalisation (and notation) the gauge potential ${\displaystyle A_{\mu \nu }^{\lambda }}$ is the Christoffel symbol defined by

${\displaystyle \nabla _{\mu }\partial _{\nu }=\Gamma _{\mu \nu }^{\lambda }\partial _{\lambda }}$.

It gives the covariant derivative

${\displaystyle (\nabla _{\mu }v)^{\nu }=(\nabla _{\mu }(v^{\lambda }\partial _{\lambda }))^{\nu }=((\partial _{\mu }v^{\lambda })\partial _{\lambda }+v^{\lambda }(\nabla _{\mu }\partial _{\lambda }))^{\nu }=\partial _{\mu }v^{\nu }+\Gamma _{\mu \lambda }^{\nu }v^{\lambda }}$.

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 ${\displaystyle e_{1}=e_{1}^{\mu }\partial _{\mu },\ldots ,e_{d}=e_{d}^{\mu }\partial _{\mu }}$. Especially when the frame is orthonormal, such a frame is usually called a d-Bein. Then

${\displaystyle (\nabla _{\mu }v)^{n}=(\nabla _{\mu }(v^{\ell }e_{\ell }))^{n}=((\partial _{\mu }v^{\ell })e_{\ell }+v^{\ell }(\nabla _{\mu }e_{\ell }))^{n}=\partial _{\mu }v^{n}+\Gamma _{\mu \ell }^{n}v^{\ell }}$

where ${\displaystyle \nabla _{\mu }e_{m}=\Gamma _{\mu m}^{\ell }e_{\ell }}$. 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 space-time: 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.

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.

• Kinetic momentum
• Connection (mathematics)
• Minimal coupling
• Ricci calculus

• Tsutomu Kambe, Gauge Principle For Ideal Fluids And Variational Principle. (PDF file.)