In physics, the gauge covariant derivative is a means of expressing how fields vary from place to place, in a way that respects how the coordinate systems used to describe a physical phenomenon can themselves change from place to place. The gauge covariant derivative is used in many areas of physics, including quantum field theory and fluid dynamics and, in a very special way, in general relativity.
If a physical theory is independent of the choice of local frames, the group of local frame changes, the gauge transformations, act on the fields in the theory while leaving unchanged the physical content of the theory. Ordinary differentiation of field components is not invariant under such gauge transformations, because they depend on the local frame. However, when gauge transformations act on fields and the gauge covariant derivative simultaneously, they preserve properties of theories that do not depend on frame choice and hence are valid descriptions of physics. Like the covariant derivative used in general relativity (which is special case), the gauge covariant derivative is an expression for a connection in local coordinates after choosing a frame for the fields involved, often in the form of index notation.
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.L.D. Faddeev, A.A. Slavnov, Gauge Fields: Introduction to Gauge Theory, (1980) Benjamin Cummings, Claude Itzykson, Jean-Bernard Zuber, Quantum Field Theory (1980) McGraw-Hill Warren Siegel, Fields (1999) ArXiv Another approach is to understand the gauge covariant derivative as a kind of connection, and more specifically, an affine connection.Richard S. Palais, The Geometrization of Physics (1981) Lecture Notes, Institute of Mathematics, National Tsing Hua UniversityM. E. Mayer, "Review: David D. Bleecker, Gauge theory and variational principles", Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 1, 83--92Alexandre Guay, Geometrical aspects of local gauge symmetry (2004) 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.Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation, (1973) W. H. Freeman and Company 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;David Bleecker, "Gauge Theory and Variational Principles " (1982) D. Reidel Publishing (See chapter 3) and, for the case of spinors, the associated bundle would be a spin bundle of the spin structure.David Bleecker, op. cit. (See Chapter 6.) 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.Meinhard E. Mayer, "Principal Bundles versus Lie Groupoids in Gauge Theory", (1990) in Differential Geometric Methods in Theoretical Physics, Volume 245 pp 793-802
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. 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.
Motivation of the covariant derivative through gauge covariance requirement
Consider a generic (possibly non-Abelian) gauge transformation acting on a component field . The main examples in field theory have a compact gauge group and we write the symmetry operator as where 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 , the infinitesimal generators of the gauge group), , as .
It acts on the field as
Now the partial derivative transforms, accordingly, as . Therefore, a kinetic term of the form in a Lagrangian is not invariant under gauge transformations.
Definition of the gauge covariant derivative
The root cause of the non gauge invariance is that in writing the field as a row vector or in index notation , we have implicitly made a choice of basis frame field i.e. a set of fields such that every field can be uniquely expressed as for functions (using Einstein summation), and assumed the frame fields are constant. Local (i.e. 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. The solution is to have the gauge transformations act not only on the fields but also on the derivative operator.
The gauge covariant derivative is a generalisation of the partial derivative that acts directly on the field (which is a function of space time with values in some abstract real or complex vector space) rather than its components with respect to a choice of frame (which are real or complex number valued functions of space time). A gauge covariant derivative is defined as an operator satisfying the product rule
for every smooth function (this is the defining property of a connection).
To go back to index notation we use the product rule
For a fixed index , 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) gauge potential
(the factor is conventional for compact gauge groups and is interpreted as a coupling constant). Conversely given the frame and a gauge potential , this uniquely defines the gauge covariant derivative. We then get
and with suppressed frame fields this gives in index notation
which by abuse of notation is often written as . This is the definition of the gauge covariant derivative as usually presented in physics.
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 on the fields (e.g. the Dirac conjugate inner product for spinors) reducing the gauge group to a unitary group, we can impose the further condition
making the Hermitian product "constant". Writing this out with respect to a local -orthonormal frame field gives , and using the above we see that must be Hermitian i.e. (motivating the extra factor ). The Hermitian matrices are (up to the factor ) the generators of the unitary group. More generally if the gauge covariant derivative preserves a gauge group acting with representation , the gauge covariant connection can be written as
where is representation of the Lie algebra associated to the group representation (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 "
is a physically meaningful definition of a field constant along a (smooth) curve. Hence the gauge covariant derivative defines (and is defined by) parallel transport.
Gauge field strength 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: . The linear map
is called the gauge field strength (loc. cit). In index notation, using the gauge potential . If is a G covariant derivative, one can interpret the latter term as a commutator in the Lie algebra of G and as Lie algebra valued (loc. cit).
Invariance under gauge transformations
The gauge covariant derivative transforms covariantly under Gauge transformations, i.e. for all
which in operator form takes the form
or
In particular (suppressing dependence on ) . Further, (suppressing indices and replacing them by matrix multiplication) if is of the form above, is of the form
or using ,
which is also of this form.
In the Hermitian case with a unitary gauge group and we have found a first order differential operator with as first order term such that .
Gauge theory 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.
Abelian Gauge Theory
the gauge covariant derivative on a complex scalar field (i.e. ) of charge is a connection. The gauge potential is a (1 x 1) matrix, i.e. a scalar.
The gauge field strength is