Gauge covariant derivative

The gauge covariant derivative is a variation 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.

Overview

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] By contrast, the gauge groups employed in particle physics could be (in principle) any Lie group at all (and, in practice, being only U(1), SU(2) or SU(3) in the Standard Model). 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 hew most closely to the notation and language commonly employed in physics curriculum, touching only briefly on the more abstract connections.

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

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

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

(This is valid for a Minkowski metric signature (−, +, +, +), which is common in general relativity and used below. For the particle physics convention (+, −, −, −), it is ${\displaystyle D_{\mu }:=\partial _{\mu }+iqA_{\mu }}$. The electron's charge is defined negative as ${\displaystyle q_{e}=-|e|}$, while the Dirac field is defined to transform positively as ${\displaystyle \psi (x)\rightarrow e^{iq\alpha (x)}\psi (x).}$)

Construction of the covariant derivative through gauge covariance requirement

Consider a generic (possibly non-Abelian) Gauge transformation, defined by a symmetry operator ${\displaystyle U(x)=e^{i\alpha (x)}}$, acting on a field ${\displaystyle \phi (x)}$, such that

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

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

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

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

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

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 ${\displaystyle x}$ dependencies for brevity)

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

where

${\displaystyle D_{\mu }\rightarrow D'_{\mu }\equiv D_{\mu }+\delta D_{\mu }}$.

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 follow QED and make the Ansatz

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

where the vector field ${\displaystyle A_{\mu }}$ satisfies,

${\displaystyle A_{\mu }\rightarrow A'_{\mu }=A_{\mu }+\delta A_{\mu },}$

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

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

and ${\displaystyle D_{\mu }\psi }$ transforms as

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

and ${\displaystyle {\bar {\psi }}:=\psi ^{\dagger }\gamma ^{0}}$ transforms as

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

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[11]

${\displaystyle D_{\mu }:=\partial _{\mu }-ig_{s}\,G_{\mu }^{\alpha }\,\lambda _{\alpha }/2}$

where ${\displaystyle g_{s}}$ is the coupling constant of the strong interaction, ${\displaystyle G}$ is the gluon gauge field, for eight different gluons ${\displaystyle \alpha =1\dots 8}$, and where ${\displaystyle \lambda _{\alpha }}$ is one of the eight Gell-Mann matrices. The Gell-Mann matrices give a representation of the color symmetry group SU(3). For quarks, the representation is the fundamental representation, for gluons, the representation is the adjoint representation.

Standard Model

The covariant derivative in the Standard Model combines the electromagnetic, the weak and the strong interactions. It can be expressed in the following form:[12]

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

The gauge fields here belong to the fundamental representations of the electroweak Lie group ${\displaystyle U(1)\otimes SU(2)}$ times the color symmetry Lie group SU(3). The coupling constant ${\displaystyle g'}$ provides the coupling of the hypercharge ${\displaystyle Y}$ to the ${\displaystyle B}$ boson and ${\displaystyle g}$ the coupling via the three vector bosons ${\displaystyle W^{j}}$ ${\displaystyle (j=1,2,3)}$ to the weak isospin, whose components are written here as the Pauli matrices ${\displaystyle \sigma _{j}}$. Via the Higgs mechanism, these boson fields combine into the massless electromagnetic field ${\displaystyle A_{\mu }}$ and the fields for the three massive vector bosons ${\displaystyle W^{\pm }}$ and ${\displaystyle Z}$.

General relativity

In general relativity, the gauge covariant derivative is defined as

${\displaystyle \nabla _{j}v^{i}:=\partial _{j}v^{i}+\sum _{k}\Gamma ^{i}{}_{jk}v^{k}}$

where ${\displaystyle \Gamma ^{i}{}_{jk}}$ is the Christoffel symbol. More formally, this derivative can be understood as the Riemannian connection on a frame bundle. The "gauge freedom" here is the arbitrary choice of a coordinate frame at each point in space-time.

See also

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

References

1. L.D. Faddeev, A.A. Slavnov, Gauge Fields: Introduction to Gauge Theory, (1980) Benjamin Cummings, ISBN 0-8053-9016-2
2. Claude Itzykson, Jean-Bernard Zuber, Quantum Field Theory (1980) McGraw-Hill ISBN 0-07-032071-3
3. Warren Siegel, Fields (1999) ArXiv
4. Richard S. Palais, The Geometrization of Physics (1981) Lecture Notes, Institute of Mathematics, National Tsing Hua University
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
6. Alexandre Guay, Geometrical aspects of local gauge symmetry (2004)
7. Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation, (1973) W. H. Freeman and Company
8. David Bleecker, "Gauge Theory and Variational Principles" (1982) D. Reidel Publishing (See chapter 3)
9. David Bleecker, op. cit. (See Chapter 6.)
10. Meinhard E. Mayer, "Principal Bundles versus Lie Groupoids in Gauge Theory", (1990) in Differential Geometric Methods in Theoretical Physics, Volume 245 pp 793-802
11. http://www.fuw.edu.pl/~dobaczew/maub-42w/node9.html
12. See e.g. eq. 3.116 in C. Tully, Elementary Particle Physics in a Nutshell, 2011, Princeton University Press.

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