Gauge fixing - Citizendia

In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Physics, a field is a Physical quantity associated to each point of Spacetime. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. In Mathematics, a local field is a special type of field that is a Locally compact Topological field with respect to a non-discrete topology Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.

Although the unphysical axes in the space of detailed configurations are a fundamental property of the physical model, there is no special set of directions "perpendicular" to them. Hence there is an enormous amount of freedom involved in taking a "cross section" representing each physical configuration by a particular detailed configuration (or even a weighted distribution of them). Judicious gauge fixing can simplify calculations immensely, but becomes progressively harder as the physical model becomes more realistic; its application to quantum field theory is fraught with complications related to renormalization, especially when the computation is continued to higher orders. In quantum field theory (QFT the forces between particles are mediated by other particles In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection In Quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system Historically, the search for logically consistent and computationally tractable gauge fixing procedures, and efforts to demonstrate their equivalence in the face of a bewildering variety of technical difficulties, has been a major driver of mathematical physics from the late nineteenth century to the present. Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics.

1 Gauge freedom
- 1.1 An illustration
2 Coulomb gauge
3 Lorenz gauge
4 R_ξ gauges
5 Maximum Abelian gauge
6 Less commonly used gauges
7 References and external links
8 References

Gauge freedom

The archetypical gauge theory is the Heaviside-Gibbs formulation of continuum electrodynamics in terms of an electromagnetic four-potential, which is presented here in space/time asymmetric Heaviside notation. Josiah Willard Gibbs ( February 11, 1839 &ndash April 28, 1903) was an American theoretical Physicist, Chemist The electromagnetic four-potential is a covariant Four-vector defined in volt·seconds/meter (and in maxwell/centimeter in parentheses The electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ of Maxwell's equations contain only "physical" degrees of freedom, in the sense that every mathematical degree of freedom in an electromagnetic field configuration has a separately measurable effect on the motions of test charges in the vicinity. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric These "field strength" variables can be expressed in terms of the scalar potential $\varphi$ and the vector potential $\mathbf{A}$ through the relations:

${\mathbf E} = -\nabla\varphi - \frac{\partial{\mathbf A}}{\partial t}$ and ${\mathbf B} = \nabla\times{\mathbf A}.$

However, the $\mathbf{E}$ and $\mathbf{B}$ fields are unchanged if we take any function $\psi(\mathbf{x},t)$ and transform $\mathbf{A}$ and $\varphi$ via:

$\mathbf{A} \rightarrow \mathbf{A} + \nabla \psi$

$\varphi \rightarrow \varphi - \frac{\partial\psi}{\partial t}$

A particular choice of the scalar and vector potentials is a gauge, and a scalar function $ψ$ used to change the gauge is called a gauge function. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field The magnetic potential provides a mathematical way to define a Magnetic field in Classical electromagnetism. The existence of arbitrary numbers of gauge functions $\psi(\mathbf{x},t)$ , corresponds to the U(1) gauge freedom of this theory. In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex Gauge fixing can be done in many ways, some of which we exhibit below.

Although classical electromagnetism is now often spoken of as a gauge theory, it was not originally conceived in these terms. The motion of a classical point charge is affected only by the electric and magnetic field strengths at that point, and the potentials can be treated as a mere mathematical device for simplifying some proofs and calculations. Not until the advent of quantum field theory could it be said that the potentials themselves are part of the physical configuration of a system. The earliest consequence to be accurately predicted and experimentally verified was the Aharonov-Bohm effect, which has no classical counterpart. The Aharonov-Bohm effect, sometimes called the Ehrenberg-Siday-Aharonov-Bohm effect is a quantum mechanical phenomenon by which a charged particle is affected by

Gauge fixing in non-abelian gauge theories, such as Yang-Mills theory and general relativity, is a rather more complicated topic; for details see Gribov ambiguity, Faddeev-Popov ghost, and frame bundle. Non-abelian may describe Non-abelian group, in mathematics a group that is not abelian (commutative Non-abelian gauge theory, in physics Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 In gauge theory, especially non-abelian gauge theories we often encounter Global problems when Gauge fixing. In Mathematics, a frame bundle is a Principal fiber bundle F( E) associated to any Vector bundle E.

An illustration

Gauge fixing of a twisted cylinder. (Note: the line is on the surface of the cylinder, not inside it. )

By looking at a cylindrical rod can one tell whether it is twisted? If the rod is perfectly cylindrical, then the circular symmetry of the cross section makes it impossible to give an answer. However, if there were a straight line drawn along the length of the rod, then one could easily say whether or not there is a twist by looking at the state of the line. Drawing a line is gauge fixing. Drawing the line spoils the gauge symmetry, ie, the circular symmetry U(1) of the cross section at each point of the rod. In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex The line is the equivalent of a gauge function; it need not be straight. Almost any line is a valid gauge fixing, ie, there is a large gauge freedom. To tell whether the rod is twisted, you need to first know the gauge. Physical quantities, such as the energy of the torsion do not depend on the gauge, ie, they are gauge invariant.

Coulomb gauge

The Coulomb gauge (also known as transverse or radiation gauge) is given by the constraint

$\nabla\cdot{\mathbf A}=0$

In the Coulomb gauge, it can be seen from Gauss' law that the scalar potential is determined simply by Poisson's equation based on the total charge density ρ (including bound charge):

$-\nabla^2 \varphi = \frac{\rho}{\varepsilon_0}$

The solution to this equation is the instantaneous Coulomb potential associated with the charge density, which appears at first glance to violate causality, since motions of electric charge appear everywhere instantaneously as changes to the Coulomb potential. In Mathematics, Poisson's equation is a Partial differential equation with broad utility in Electrostatics, Mechanical engineering and Theoretical In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses Causality (but not causation) denotes a necessary relationship between one event (called cause and another event (called effect) which is the direct consequence This is generally explained by pointing out that the scalar and vector potentials themselves do not affect the motions of charges, only the combinations of their derivatives that form the electromagnetic field strength. Although one can compute the field strengths explicitly in Coulomb gauge and demonstrate that changes in them propagate at the speed of light, it is much simpler to observe that the field strengths are unchanged under gauge transformations and to demonstrate causality in the manifestly covariant Lorenz gauge described below.

The advantage of the Coulomb gauge is that one can decouple the equations for the scalar and vector potentials, obtaining a wave equation for the vector potential in terms of a quantity called the transverse current which, like the Coulomb potential, drops rapidly to zero outside the immediate vicinity of electric charges. The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves Solutions of this wave equation with the transverse current set to zero correspond classically to transversely polarized electromagnetic radiation in free space. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. This is particularly useful for "semi-classical" calculations in quantum mechanics, in which the vector potential is quantized but the Coulomb interaction is not.

Lorenz gauge

See also: Covariant formulation of classical electromagnetism

The Lorenz gauge is given, in SI units, by:

$\nabla\cdot{\mathbf A} + \frac{1}{c^2}\frac{\partial\varphi}{\partial t}=0$

and in Gaussian units by:

$\nabla\cdot{\mathbf A} + \frac{1}{c}\frac{\partial\varphi}{\partial t}=0.$

It may be rewritten in terms of the electromagnetic four-potential:

$\partial^{\mu} A_{\mu} = 0.$

It is unique among the constraint gauges in retaining manifest Lorentz invariance. The covariant formulation of Classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular Maxwell's equations The centimetre-gram-second system ( CGS) is a system of physical units. The electromagnetic four-potential is a covariant Four-vector defined in volt·seconds/meter (and in maxwell/centimeter in parentheses In standard Physics, Lorentz covariance is a key property of Spacetime that follows from the Special theory of relativity, where it applies globally Note, however, that this gauge was originally named after the Danish physicist Ludvig Lorenz and not after Hendrik Lorentz; it is often misspelled "Lorentz gauge". Ludvig Valentin Lorenz ( January 18, 1829 - June 9, 1891) was a Danish mathematician and Physicist. Hendrik Antoon Lorentz ( July 18, 1853 &ndash February 4, 1928) was a Dutch Physicist who shared the 1902 Nobel (Neither was the first to use it in calculations; it was introduced in 1888 by George F. Fitzgerald. George Francis FitzGerald ( 3 August 1851 &ndash 21 February 1901) was an Irish professor of "natural and experimental philosophy" )

The Lorenz gauge leads to the following inhomogeneous wave equations for the potentials:

$\frac{1}{c^2}\frac{\partial^2\varphi}{\partial t^2} - \nabla^2{\varphi} = \frac{\rho}{\varepsilon_0}$

$\frac{1}{c^2}\frac{\partial^2\mathbf A}{\partial t^2} - \nabla^2{\mathbf A} = \mu_0 \mathbf{J}$

It can be seen from these equations that, in the absence of current and charge, the solutions are potentials which propagate at the speed of light.

The Lorenz gauge is incomplete in the sense that there remains a subspace of gauge transformations which preserve the constraint. These remaining degrees of freedom correspond to gauge functions which satisfy the wave equation

${ \partial^2 \psi \over \partial t^2 } = c^2 \nabla^2\psi$

These remaining gauge degrees of freedom propagate at the speed of light. The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves To obtain a fully fixed gauge, one must add boundary conditions along the light cone of the experimental region. In Special relativity, a light cone (or null cone) is the pattern describing the temporal evolution of a flash of Light in Minkowski spacetime

Maxwell's equations in the Lorenz gauge simplify to $\partial_\mu \partial^\mu A^\nu = e j^\nu$ , where $j ν$ is the four-current. In special and General relativity, the four-current is the Lorentz covariant Four-vector that replaces the Electromagnetic Current Two solutions of these equations for the same current configuration differ by a solution of the vacuum wave equation $\partial_\mu \partial^\mu A^\nu = 0$ . In this form it is clear that the components of the potential separately satisfy the Klein-Gordon equation, and hence that the Lorenz gauge condition allows transversely, longitudinally, and "time-like" polarized waves in the four-potential. The Klein–Gordon equation ( Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger Polarization ( ''Brit'' polarisation) is a property of Waves that describes the orientation of their oscillations The transverse polarizations correspond to classical radiation, i. e. , transversely polarized waves in the field strength. To suppress the "unphysical" longitudinal and time-like polarization states, which are not observed in experiments at classical distance scales, one must also employ auxiliary constraints known as Ward identities. In Quantum field theory, a Ward-Takahashi identity is an identity between Correlation functions that follows from the global or gauged symmetries of the Classically, these identities are equivalent to the continuity equation $\partial_\mu j^\mu = 0$ . A continuity equation is a Differential equation that describes the conservative transport of some kind of quantity

Many of the differences between classical and quantum electrodynamics can be accounted for by the role that the longitudinal and time-like polarizations play in interactions between charged particles at microscopic distances. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics.

$R ξ$ gauges

The $R ξ$ gauges are a generalization of the Lorenz gauge applicable to theories expressed in terms of an action principle with Lagrangian density $\mathcal{L}$ . The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system Instead of fixing the gauge by constraining the gauge field a priori via an auxiliary equation, one adds to the "physical" (gauge invariant) Lagrangian a gauge breaking term

$\delta \mathcal{L} = -\frac{(\partial_{\mu} A^{\mu})^2}{2 \xi}$

The choice of the parameter $ξ$ determines the choice of gauge. The Landau gauge, obtained as the limit $\xi \rightarrow 0$ , is classically equivalent to Lorenz gauge, but postponing taking the limit until after the theory is quantized improves the rigor of certain existence and equivalence proofs. Most quantum field theory computations are simplest in the Feynman-'t Hooft gauge, in which $ξ = 1$ ; a few are more tractable in other $R ξ$ gauges, such as the Yennie gauge $ξ = 3$ . In quantum field theory (QFT the forces between particles are mediated by other particles

An equivalent formulation of $R ξ$ gauge uses an auxiliary field, a scalar field $B$ with no independent dynamics:

$\delta \mathcal{L} = B\,\partial_{\mu} A^{\mu} + \frac{\xi}{2} B^2$

The auxiliary field can be eliminated by "completing the square" to obtain the previous form. In Physics, and especially Quantum field theory, an auxiliary field is one whose equations of motion admit a single solution From a mathematical perspective the auxiliary field is a variety of Goldstone boson, and its use has advantages when identifying the asymptotic states of the theory, and especially when generalizing beyond QED. In particle and Condensed matter physics, Goldstone bosons (also known as Nambu -Goldstone bosons) are Bosons that appear in models

Historically, the use of $R ξ$ gauges was a significant technical advance in extending quantum electrodynamics computations beyond one-loop order. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. In Physics, a one-loop Sander-Feynman diagram is a connected Feynman diagram with only one cycle ( Unicyclic) In addition to retaining manifest Lorentz invariance, the $R ξ$ prescription breaks the symmetry under local gauge transformations while preserving the ratio of functional measures of any two physically distinct gauge configurations. In standard Physics, Lorentz covariance is a key property of Spacetime that follows from the Special theory of relativity, where it applies globally This permits a change of variables in which infinitesimal perturbations along "physical" directions in configuration space are entirely uncoupled from those along "unphysical" directions, allowing the latter to be absorbed into the physically meaningless normalization of the functional integral. In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires You may also be looking for Functional integration (neurobiology or Functional integration (sociology. When $ξ$ is finite, each physical configuration (orbit of the group of gauge transformations) is represented not by a single solution of a constraint equation but by a Gaussian distribution centered on the extremum of the gauge breaking term. In Mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum or smallest value (minimum that In terms of the Feynman rules of the gauge-fixed theory, this appears as a contribution to the photon propagator for internal lines from virtual photons of unphysical polarization. Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described In Physics, a virtual particle is a particle that exists for a limited time and space introducing uncertainty in their energy and momentum due to the Heisenberg Uncertainty Polarization ( ''Brit'' polarisation) is a property of Waves that describes the orientation of their oscillations

The photon propagator, which is the multiplicative factor corresponding to an internal photon in the Feynman diagram expansion of a QED calculation, contains a factor $g μν$ corresponding to the Minkowski metric. Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity An expansion of this factor as a sum over photon polarizations involves terms containing all four possible polarizations. Transversely polarized radiation can be expressed mathematically as a sum over either a linearly or circularly polarized basis. In Electrodynamics, circular polarization (also circular polarisation) of Electromagnetic radiation is a Polarization such that the tip of the Similarly, one can combine the longitudinal and time-like gauge polarizations to obtain "forward" and "backward" polarizations; these are a form of light cone coordinates in which the metric is off-diagonal. In Relativity, light-cone coordinates is a special coordinate system where two of the coordinates x+ and x- are Null coordinates and all the An expansion of the $g μν$ factor in terms of circularly polarized (spin +/- 1) and light cone coordinates is called a spin sum. Spin sums can be very helpful both in simplifying expressions and in obtaining a physical understanding of the experimental effects associated with different terms in a theoretical calculation.

Richard Feynman used arguments along approximately these lines largely to justify calculation procedures that produced consistent, finite, high precision results for important observable parameters such as the anomalous magnetic moment of the electron. Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum In Quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of Quantum mechanics, expressed by Feynman diagrams Although his arguments sometimes lacked mathematical rigor even by physicists' standards and glossed over details such as the derivation of Ward-Takahashi identities of the quantum theory, his calculations worked, and Freeman Dyson soon demonstrated that his method was substantially equivalent to those of Julian Schwinger and Sin-Itiro Tomonaga, with whom Feynman shared the 1965 Nobel Prize in Physics. In Quantum field theory, a Ward-Takahashi identity is an identity between Correlation functions that follows from the global or gauged symmetries of the Freeman John Dyson FRS (born December 15, 1923) is an English-born American theoretical Physicist and Mathematician, famous for his Julian Seymour Schwinger ( February 12, 1918 &ndash July 16, 1994) was an American Theoretical physicist. Sin-Itiro Tomonaga or Shinichirō Tomonaga (朝永振一郎 Tomonaga Shin'ichirō, March 31, 1906 Year 1965 ( MCMLXV) was a Common year starting on Friday (link will display full calendar of the 1965 Gregorian calendar. The Nobel Prize (Nobelpriset (Nobelprisen is a Swedish prize established in the 1895 will of Swedish chemist Alfred Nobel; it was first awarded in Peace, Literature

Forward and backward polarized radiation can be omitted in the asymptotic states of a quantum field theory (see Ward-Takahashi identity). In Quantum field theory, a Ward-Takahashi identity is an identity between Correlation functions that follows from the global or gauged symmetries of the For this reason, and because their appearance in spin sums can be seen as a mere mathematical device in QED (much like the electromagnetic four-potential in classical electrodynamics), they are often spoken of as "unphysical". But unlike the constraint-based gauge fixing procedures above, the $R ξ$ gauge generalizes well to non-abelian gauge groups such as the SU(3) of QCD. Non-abelian may describe Non-abelian group, in mathematics a group that is not abelian (commutative Non-abelian gauge theory, in physics Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the The couplings between physical and unphysical perturbation axes do not entirely disappear under the corresponding change of variables; to obtain correct results, one must account for the non-trivial Jacobian of the embedding of gauge freedom axes within the space of detailed configurations. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. This leads to the explicit appearance of forward and backward polarized gauge bosons in Feynman diagrams, along with Faddeev-Popov ghosts, which are even more "unphysical" in that they violate the spin-statistics theorem. The spin-statistics theorem in Quantum mechanics relates the spin of a particle to the statistics obeyed by that particle The relationship between these entities, and the reasons why they do not appear as particles in the quantum mechanical sense, becomes more evident in the BRST formalism of quantization. (A draft of an alternate exposition has been added at BRST quantization.

Maximum Abelian gauge

In any non-Abelian gauge theory, any maximum Abelian gauge is an incomplete gauge which fixes the gauge freedom outside of the maximum Abelian subgroup. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Examples are

For SU(2) gauge theory in D dimensions, the maximum Abelian subgroup is a U(1) subgroup. Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n If this is chosen to be the one generated by the Pauli matrix σ₃, then the maximum Abelian gauge is that which maximizes the function

$\int d^Dx \left[(A_\mu^1)^2+(A_\mu^2)^2\right].$ where ${\mathbf A}_\mu = A_\mu^a \sigma_a$

For SU(3) gauge theory in D dimensions, the maximum Abelian subgroup is a U(1)×U(1) subgroup. The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n If this is chosen to be the one generated by the Gell-Mann matrices λ₃ and λ₈, then the maximum Abelian gauge is that which maximizes the function

$\int d^Dx \left[(A_\mu^1)^2+(A_\mu^2)^2+(A_\mu^4)^2+(A_\mu^5)^2+(A_\mu^6)^2+(A_\mu^7)^2\right].$ where ${\mathbf A}_\mu = A_\mu^a \lambda_a$

Less commonly used gauges

Weyl gauge

The Weyl gauge (also known as the Hamiltonian or temporal gauge) is an incomplete gauge obtained by the choice

φ = 0

It is named after Hermann Weyl. The Gell-Mann matrices, named for Murray Gell-Mann, are one possible representation of the infinitesimal generators of the Special unitary group called Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician.

Multipolar gauge

The gauge condition of the Multipolar gauge, Line gauge or Poincaré gauge is:

$\mathbf{x}\cdot\mathbf{A}=0$

where $\mathbf{x}$ is the position vector and $\mathbf{A}$ is the vector potential. In Vector calculus, a vector potential is a Vector field whose curl is a given vector field

Fock-Schwinger gauge

The gauge condition of the Fock-Schwinger gauge (sometimes called the relativistic Poincaré gauge) is:

x μ A μ = 0

where $x μ$ is the position four-vector and $A μ$ is the four-potential. The electromagnetic four-potential is a covariant Four-vector defined in volt·seconds/meter (and in maxwell/centimeter in parentheses

References and external links

Landau and Lifschitz, "The classical theory of fields"
Jackson, "Classical Electrodynamics"
J. D. Jackson, http://arxiv.org/abs/physics/0204034 (Alternate web link)

Quantum field theory

Feynman diagram
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References

In quantum field theory (QFT the forces between particles are mediated by other particles Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described The history of quantum field theory starts with its creation by Dirac when he attempted to quantize the Electromagnetic field in the late 1920s

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