The gauge covariant derivative is a variation of the
covariant derivative used in
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
,
quantum field theory and
fluid dynamics. If a theory has
gauge transformation
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
s, 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. Another approach is to understand the gauge covariant derivative as a kind of
connection, and more specifically, an
affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
.
[Alexandre 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 In physics, field strength means the ''magnitude'' of a vector-valued field (e.g., in volts per meter, V/m, for an electric field ''E'').
For example, an electromagnetic field results in both electric field strength and magnetic 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
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
. This path leads directly to general relativity; however, it requires a metric, which
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
gauge theories
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
do not have.
Rather than being generalizations of one-another, affine and metric geometry go off in different directions: the
gauge group
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
of (
pseudo-
The prefix pseudo- (from Greek ψευδής, ''pseudes'', "false") is used to mark something that superficially appears to be (or behaves like) one thing, but is something else. Subject to context, ''pseudo'' may connote coincidence, imitation, ...
)
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
''must'' be the
indefinite orthogonal group
In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . It is also called the p ...
O(s,r) in general, or the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
O(3,1) for
space-time. This is because the fibers of the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
must necessarily, by definition, connect the
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
and
cotangent space
In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
s 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)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
,
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
and
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
. 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
In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection.
Definition
Let ''G' ...
on a
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of an
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...
for the
principal fiber bundle of the gauge theory; and, for the case of spinors, the associated bundle would be a
spin bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\col ...
of the
spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathematical ...
. 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.
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
In theoretical physics, the Coleman–Mandula theorem is a no-go theorem stating that spacetime and internal symmetries can only combine in a trivial way. This means that the charges associated with internal symmetries must always transform as Lor ...
. However, a premise of this theorem is violated by the
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ...
s (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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
and
Riemannian manifolds; for example, treating gauge covariance as
equivariance
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
,
representations of Lie algebras
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is ...
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
:
where
is a velocity
vector field of a fluid.
Gauge theory
In
gauge theory, which studies a particular class of
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
which are of importance in
quantum field theory, the
minimally-coupled gauge covariant derivative is defined as
:
where
is the
electromagnetic four-potential
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
.
(This is valid for a Minkowski
metric signature
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
, which is common in
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and used below. For the
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
convention , it is
. The
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
's charge is defined negative as
, while the Dirac field is defined to transform positively as
)
Construction of the covariant derivative through gauge covariance requirement
Consider a generic (possibly non-Abelian) Gauge transformation, defined by a symmetry operator
, acting on a field
, such that
:
:
where
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,
, as
.
The partial derivative
transforms, accordingly, as
:
and a kinetic term of the form
is thus not invariant under this transformation.
We can introduce the covariant derivative
in this context as a generalization of the partial derivative
which transforms covariantly under the Gauge transformation, i.e. an object satisfying
:
which in operatorial form takes the form
:
We thus compute (omitting the explicit
dependencies for brevity)
:
,
where
:
.
The requirement for
to transform covariantly is now translated in the condition
:
To obtain an explicit expression, we follow
QED and make the Ansatz
:
where the vector field
satisfies,
:
from which it follows that
:
and
:
which, using
, takes the form
:
We have thus found an object
such that
:
Quantum electrodynamics
If a gauge transformation is given by
:
and for the gauge potential
:
then
transforms as
:
,
and
transforms as
:
and
transforms as
:
so that
:
and
in the QED
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
is therefore gauge invariant, and the gauge covariant derivative is thus named aptly.
On the other hand, the non-covariant derivative
would not preserve the Lagrangian's gauge symmetry, since
:
.
Quantum chromodynamics
In
quantum chromodynamics, the gauge covariant derivative is
:
where
is the
coupling constant of the strong interaction,
is the gluon
gauge field, for eight different gluons
, and where
is one of the eight
Gell-Mann matrices
The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics.
They span the Lie algebra of the SU(3) group in t ...
. The Gell-Mann matrices give a
representation of the
color symmetry group
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
. For quarks, the representation is the
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group
or Lie algebra whose highest weig ...
, for gluons, the representation is the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
.
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:
[See e.g. eq. 3.116 in C. Tully, ''Elementary Particle Physics in a Nutshell'', 2011, Princeton University Press.]
:
The gauge fields here belong to the
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group
or Lie algebra whose highest weig ...
s of the
electroweak
In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...
Lie group times the
color symmetry Lie group
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
. The coupling constant
provides the coupling of the hypercharge
to the
boson and
the coupling via the three vector bosons
to the weak isospin, whose components are written here as the
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
. Via the
Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other be ...
, these boson fields combine into the massless electromagnetic field
and the fields for the three massive vector bosons
and
.
General relativity
In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the gauge covariant derivative is defined as
:
where
is the
Christoffel symbol
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing dist ...
. More formally, this derivative can be understood as the
Riemannian connection
In mathematics, a metric connection is a connection in a vector bundle ''E'' equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along a ...
on a
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
. The "gauge freedom" here is the arbitrary choice of a
coordinate frame at each point in
space-time.
See also
*
Kinetic momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
*
Connection (mathematics) In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a ''parallel'' and consistent manner. There are var ...
*
Minimal coupling In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between fields which involves only the charge distribution and not higher multipole moments of the charge distribution. This minimal coupling is in contrast to, ...
*
Ricci calculus
References
*Tsutomu Kambe,
Gauge Principle For Ideal Fluids And Variational Principle'. (PDF file.)
Differential geometry
Connection (mathematics)
Gauge theories