Gauge covariant derivative

Last updated

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 general relativity.

Contents

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. [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.

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 fields to be constant without breaking local gauge symmetry.

We can introduce the gauge covariant derivative as a generalisation of the partial derivative that acts directly on the field rather than its components with respect to a choice of frame. A gauge covariant derivative is defined as an operator satisfying a 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 , 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

(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. [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 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

The gauge potential can be interpreted as electromagnetic four-potential and the gauge field strength as the electromagnetic field tensor. 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 by [12] ) this is called minimal coupling.

For a Dirac spinor field of charge the covariant derivative is also a connection (because it has to commute with the gamma matrices) and is defined as

where again is interpreted as the electromagnetic four-potential and as the electromagnetic field tensor. (The minus sign is a convention valid for a Minkowski metric signature (−, +, +, +), which is common in general relativity and used below. For the particle physics convention (+, −, −, −), it is . The electron's charge is defined negative as , while the Dirac field is defined to transform positively as )

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 is therefore gauge invariant, and the gauge covariant derivative is thus named aptly.[ citation needed ]

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 [13]

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 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: [14]

The gauge fields here belong to the fundamental representations of the electroweak Lie group times the color symmetry Lie group SU(3). 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 . Via the Higgs mechanism, these boson fields combine into the massless electromagnetic field and the fields for the three massive vector bosons and .

General relativity

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 instead of . In this special case, a choice of (local) coordinates not only gives partial derivatives , but they double as a frame of tangent vectors in which a vector field can be uniquely expressed as (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 is the Christoffel symbol defined by

.

It gives the covariant derivative

.

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 . Especially when the frame is orthonormal, such a frame is usually called a d-Bein. Then

where . 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 [ citation needed ]. 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.

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.[ citation needed ]

See also

Related Research Articles

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way. It has become vital in the building of the Standard Model.

<span class="mw-page-title-main">Stress–energy tensor</span> Tensor describing energy momentum density in spacetime

The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

<span class="mw-page-title-main">Noether's theorem</span> Statement relating differentiable symmetries to conserved quantities

Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems proven by mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries of physical space.

The Klein–Gordon equation is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a differential equation version of the relativistic energy–momentum relation .

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

<span class="mw-page-title-main">Nonlinear Schrödinger equation</span> Nonlinear form of the Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic, cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.

<span class="mw-page-title-main">Electromagnetic tensor</span> Mathematical object that describes the electromagnetic field in spacetime

In electromagnetism, the electromagnetic tensor or electromagnetic field tensor is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written concisely, and allows for the quantization of the electromagnetic field by the Lagrangian formulation described below.

In theoretical physics, the Wess–Zumino model has become the first known example of an interacting four-dimensional quantum field theory with linearly realised supersymmetry. In 1974, Julius Wess and Bruno Zumino studied, using modern terminology, dynamics of a single chiral superfield whose cubic superpotential leads to a renormalizable theory. It is a special case of 4D N = 1 global supersymmetry.

In theoretical physics, scalar electrodynamics is a theory of a U(1) gauge field coupled to a charged spin 0 scalar field that takes the place of the Dirac fermions in "ordinary" quantum electrodynamics. The scalar field is charged, and with an appropriate potential, it has the capacity to break the gauge symmetry via the Abelian Higgs mechanism.

The Newman–Penrose (NP) formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members often asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.

<span class="mw-page-title-main">Mathematical descriptions of the electromagnetic field</span> Formulations of electromagnetism

There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

<span class="mw-page-title-main">Gauge theory</span> Physical theory with fields invariant under the action of local "gauge" Lie groups

In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, do not change under local transformations according to certain smooth families of operations. Formally, the Lagrangian is invariant under these transformations.

<span class="mw-page-title-main">Dirac equation in curved spacetime</span> Generalization of the Dirac equation

In mathematical physics, the Dirac equation in curved spacetime is a generalization of the Dirac equation from flat spacetime to curved spacetime, a general Lorentzian manifold.

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

<span class="mw-page-title-main">Loop representation in gauge theories and quantum gravity</span> Description of gauge theories using loop operators

Attempts have been made to describe gauge theories in terms of extended objects such as Wilson loops and holonomies. The loop representation is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation in the context of Yang–Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of physical states. The idea is well known in the context of lattice Yang–Mills theory. Attempts to explore the continuous loop representation was made by Gambini and Trias for canonical Yang–Mills theory, however there were difficulties as they represented singular objects. As we shall see the loop formalism goes far beyond a simple gauge invariant description, in fact it is the natural geometrical framework to treat gauge theories and quantum gravity in terms of their fundamental physical excitations.

In theoretical physics, more specifically in quantum field theory and supersymmetry, supersymmetric Yang–Mills, also known as super Yang–Mills and abbreviated to SYM, is a supersymmetric generalization of Yang–Mills theory, which is a gauge theory that plays an important part in the mathematical formulation of forces in particle physics. It is a special case of 4D N = 1 global supersymmetry.

In supersymmetry, 4D supergravity is the theory of supergravity in four dimensions with a single supercharge. It contains exactly one supergravity multiplet, consisting of a graviton and a gravitino, but can also have an arbitrary number of chiral and vector supermultiplets, with supersymmetry imposing stringent constraints on how these can interact. The theory is primarily determined by three functions, those being the Kähler potential, the superpotential, and the gauge kinetic matrix. Many of its properties are strongly linked to the geometry associated to the scalar fields in the chiral multiplets. After the simplest form of this supergravity was first discovered, a theory involving only the supergravity multiplet, the following years saw an effort to incorporate different matter multiplets, with the general action being derived in 1982 by Eugène Cremmer, Sergio Ferrara, Luciano Girardello, and Antonie Van Proeyen.

In supersymmetry, type IIA supergravity is the unique supergravity in ten dimensions with two supercharges of opposite chirality. It was first constructed in 1984 by a dimensional reduction of eleven-dimensional supergravity on a circle. The other supergravities in ten dimensions are type IIB supergravity, which has two supercharges of the same chirality, and type I supergravity, which has a single supercharge. In 1986 a deformation of the theory was discovered which gives mass to one of the fields and is known as massive type IIA supergravity. Type IIA supergravity plays a very important role in string theory as it is the low-energy limit of type IIA string theory.

In supersymmetry, type I supergravity is the theory of supergravity in ten dimensions with a single supercharge. It consists of a single supergravity multiplet and a single Yang–Mills multiplet. The full non-abelian action was first derived in 1983 by George Chapline and Nicholas Manton. Classically the theory can admit any gauge group, but a consistent quantum theory resulting in anomaly cancellation only exists if the gauge group is either or . Both these supergravities are realised as the low-energy limits of string theories, in particular of type I string theory and of the two heterotic string theories.

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. 1 2 Alexandre Guay, Geometrical aspects of local gauge symmetry (2004)
  7. 1 2 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 Archived 2021-07-09 at the Wayback Machine " (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. 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)
  12. 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.
  13. "Quantum Chromodynamics (QCD)".
  14. See e.g. eq. 3.116 in C. Tully, Elementary Particle Physics in a Nutshell, 2011, Princeton University Press.