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{{Electromagnetism}}
{{Quantum field theory}}
In the [[physics]] of [[gauge theory|gauge theories]], '''gauge fixing''' (also called '''choosing a gauge''') denotes a mathematical procedure for coping with redundant [[Degrees of freedom (physics and chemistry)|degrees of freedom]] in [[field (physics)|field]] variables. By definition, a gauge theory represents each physically distinct configuration of the system as an [[equivalence class]] of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a
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 [[perturbative expansion|orders]]. 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.{{citation needed|date=September 2015}}
== Gauge freedom ==
The archetypical gauge theory is the [[Oliver Heaviside|Heaviside]]–[[Josiah Willard Gibbs|Gibbs]] formulation of continuum [[electrodynamics]] in terms of an [[electromagnetic four-potential]], which is presented here in space/time asymmetric Heaviside notation. The [[electric field]] '''E''' and [[magnetic field]] '''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. These "field strength" variables can be expressed in terms of the [[electric potential|electric scalar potential]] <math>\varphi</math> and the [[magnetic vector potential]] '''A''' through the relations:
<math display="block">{\mathbf E} = -\nabla\varphi - \frac{\partial{\mathbf A}}{\partial t}\,, \quad {\mathbf B} = \nabla\times{\mathbf A}.</math>
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is made then '''E''' also remains the same. Hence, the '''E''' and '''B''' fields are unchanged if one takes any function {{math|''ψ''('''r''', ''t'')}} and simultaneously transforms '''A''' and ''φ'' via the transformations ({{EquationNote|1}}) and ({{EquationNote|2}}).
A particular choice of the scalar and vector potentials is a '''gauge''' (more precisely, '''gauge potential''') and a scalar function ''ψ'' used to change the gauge is called a '''gauge function'''.{{cn|date=May 2024}} The existence of arbitrary numbers of gauge functions {{math|''ψ''('''r''', ''t'')}} corresponds to the [[U(1)]] '''gauge freedom''' of this theory. 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. Nevertheless, gauge freedom is still true in these theories. For example, the Aharonov–Bohm effect depends on a [[line integral]] of '''A''' around a closed loop, and this integral is not changed by
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Gauge fixing in [[non-abelian gauge theory|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]].
=== An illustration ===
[[File:gauge.png|right|thumb|Gauge fixing of a ''twisted'' cylinder. (Note: the line is on the ''surface'' of the cylinder, not inside it.)]]
== Coulomb gauge ==
The '''Coulomb gauge''' (also known as the [[Helmholtz decomposition#Longitudinal and transverse fields|transverse gauge]]) is used in [[quantum chemistry]] and [[condensed matter physics]] and is defined by the gauge condition (more precisely, gauge fixing condition)
<math display="block">\nabla\cdot{\mathbf A}(\mathbf{r},t)=0\,.</math>
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The Coulomb gauge has a number of properties:
{{ordered list
|1= The potentials can be expressed in terms of instantaneous values of the fields and densities (in [[International System of Units]])<ref name=Stewart2003>{{cite journal |last=Stewart |first=A. M. |year=2003 |title=Vector potential of the Coulomb gauge |journal=[[European Journal of Physics]] |volume=24 |issue=5 |pages=519–524 |doi=10.1088/0143-0807/24/5/308 |bibcode = 2003EJPh...24..519S|s2cid=250880504 }}</ref>
<math display="block"> \varphi(\mathbf{r},t) = \frac{1}{4\pi \varepsilon_0} \int\frac{\mathbf{\rho}(\mathbf{r}',t)}{R} d^3\mathbf{r}'</math>
<math display="block"> \mathbf{A}(\mathbf{r},t) = \nabla \times\int\frac{ \mathbf{B}(\mathbf{r}',t)}{4\pi R} d^3\mathbf{r}'</math>
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Another expression for the vector potential, in terms of the time-retarded electric current density {{math|'''J'''('''r''', ''t'')}}, has been obtained to be:<ref name=Jackson2002>{{cite journal |last=Jackson |first=J. D. |year=2002 |title=From Lorenz to Coulomb and other explicit gauge transformations |journal=[[American Journal of Physics]] |volume=70 |issue=9 |pages=917–928 |doi=10.1119/1.1491265 |arxiv = physics/0204034 |bibcode = 2002AmJPh..70..917J |s2cid=119652556 }}</ref>
<math display="block"> \mathbf{A}(\mathbf{r},t) = \frac{1}{4\pi \varepsilon_0} \, \nabla\times\int \left[ \int_0^{R/c} \tau\, \frac{ { \mathbf{J}(\mathbf{r}', t- \tau)} \times { \mathbf{R } } }{R^3}\,
|2= Further gauge transformations that retain the Coulomb gauge condition might be made with gauge functions that satisfy {{math|1='''∇'''<sup>2</sup>''ψ'' = 0}}, but as the only solution to this equation that vanishes at infinity (where all fields are required to vanish) is {{math|1=''ψ''('''r''', ''t'') = 0}}, no gauge arbitrariness remains. Because of this, the Coulomb gauge is said to be a complete gauge, in contrast to gauges where some gauge arbitrariness remains, like the Lorenz gauge below.
|3= The Coulomb gauge is a minimal gauge in the sense that the integral of '''A'''<sup>2</sup> over all space is minimal for this gauge: All other gauges give a larger integral.<ref>{{cite journal |last1=Gubarev |first1=F. V. |last2=Stodolsky |first2=L. |last3=Zakharov |first3=V. I. |year=2001 |title=On the Significance of the Vector Potential Squared |journal=[[Physical Review Letters|Phys. Rev. Lett.]] |volume=86 |issue=11 |pages=2220–2222 |doi=10.1103/PhysRevLett.86.2220 |pmid=11289894 |arxiv = hep-ph/0010057 |bibcode = 2001PhRvL..86.2220G |s2cid=45172403 }}</ref> The minimum value given by the Coulomb gauge is <math display="block"> \int \mathbf{A}^2(\mathbf{r}, t) d^3\mathbf{r} = \
|4= In regions far from electric charge the scalar potential becomes zero. This is known as the '''radiation gauge'''. [[Electromagnetic radiation]] was first quantized in this gauge.
|5= The Coulomb gauge admits a natural Hamiltonian formulation of the evolution equations of the electromagnetic field interacting with a conserved current,{{cn|date=June 2023}} which is an advantage for the quantization of the theory. The Coulomb gauge is, however, not Lorentz covariant. If a [[Lorentz transformation]] to a new inertial frame is carried out, a further gauge transformation has to be made to retain the Coulomb gauge condition. Because of this, the Coulomb gauge is not used in covariant perturbation theory, which has become standard for the treatment of relativistic [[quantum field theories]] such as [[quantum electrodynamics]] (QED). Lorentz covariant gauges such as the Lorenz gauge are usually used in these theories. Amplitudes of physical processes in QED in the noncovariant Coulomb gauge coincide with those in the covariant Lorenz gauge.<ref>{{cite journal | last=Adkins | first=Gregory S. | title=Feynman rules of Coulomb-gauge QED and the electron magnetic moment | journal=Physical Review D | publisher=American Physical Society (APS) | volume=36 | issue=6 | date=1987-09-15 | issn=0556-2821 | doi=10.1103/physrevd.36.1929 | pages=1929–1932| pmid=9958379 | bibcode=1987PhRvD..36.1929A }}</ref>
|6= For a uniform and constant magnetic field '''B''' the vector potential in the Coulomb gauge can be expressed in the so-called '''symmetric gauge''' as
<math display="block">{\mathbf A}(\mathbf{r},t)=-\frac{1}{2} \mathbf{r}\times \mathbf{B}</math>
plus the gradient of any scalar field (the gauge function), which can be confirmed by calculating the div and curl of '''A'''. The divergence of '''A''' at infinity is a consequence of the unphysical assumption that the magnetic field is uniform throughout the whole of space. Although this vector potential is unrealistic in general it can provide a good approximation to the potential in a finite volume of space in which the magnetic field is uniform. Another common choice for homogeneous constant fields is the '''Landau gauge''' (not to be confused with the ''R''<sub>ξ</sub> Landau gauge of the next section), where <math>\mathbf B =B \hat{z}</math> and
<math display="block">\mathbf A =B (\mathbf r\cdot \hat{x}) \hat{y},</math> where <math>\hat x, \hat y, \hat z</math> are unitary vectors of the Cartesian coordinate system (z-axis aligned with the magnetic field).
|7= As a consequence of the considerations above, the electromagnetic potentials may be expressed in their most general forms in terms of the electromagnetic fields as
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}}
== Lorenz gauge ==
{{Main|Lorenz gauge condition}}
{{See also|Covariant formulation of classical electromagnetism}}
The
<math display="block">\nabla\cdot{\mathbf A} + \frac{1}{c^2}\frac{\partial\varphi}{\partial t}=0</math>
and in [[Gaussian units]] by:
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This may be rewritten as:
<math display="block">\partial_{\mu} A^{\mu} = 0.</math>
where <math>A^\mu = \left[\,\tfrac{1}{c}\varphi,\,\mathbf{A}\,\right]</math> is the [[electromagnetic four-potential]], {{math|∂<sub>''μ''</sub>}} the [[4-gradient]] [using the [[metric signature]] (+, −, −, −)].
It is unique among the constraint gauges in retaining manifest [[Lorentz invariance]]. 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". (Neither was the first to use it in calculations; it was introduced in 1888 by [[George Francis
The Lorenz gauge leads to the following inhomogeneous wave equations for the potentials:
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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 some sense:
<math display="block">\frac{ \partial^2 \psi }{ \partial t^2 } = c^2 \nabla^2\psi </math>
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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.
== ''R''<sub>''ξ''</sub>
The '''''R''<sub>''ξ''</sub> gauges''' are a generalization of the Lorenz gauge applicable to theories expressed in terms of an [[action principle]] with [[Lagrangian density]] <math>\mathcal{L}</math>. Instead of fixing the gauge by constraining the [[gauge field]] ''a priori'', via an auxiliary equation, one adds a gauge ''breaking'' term to the "physical" (gauge invariant) Lagrangian
<math display="block">\delta \mathcal{L} = -\frac{\left(\partial_{\mu} A^{\mu}\right)^2}{2 \xi}</math>
The choice of the parameter ''ξ'' determines the choice of gauge. The '''''R''<sub>ξ</sub> Landau gauge''' is classically equivalent to Lorenz gauge: it is obtained in the limit ''ξ'' → 0 but postpones taking that limit until after the theory has been quantized. It improves the rigor of certain existence and equivalence proofs. Most [[quantum field theory]] computations are simplest in the '''Feynman–'t Hooft gauge''', in which {{math|1=''ξ'' = 1}}; a few are more tractable in other ''R''<sub>ξ</sub> gauges, such as the '''[[Donald R. Yennie|Yennie]] gauge''' {{math|1=''ξ'' = 3}}.
An equivalent formulation of ''R''<sub>ξ</sub> gauge uses an [[auxiliary field]], a scalar field ''B'' with no independent dynamics:
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Forward and backward polarized radiation can be omitted in the [[asymptotic states]] of a quantum field theory (see [[Ward–Takahashi identity]]). 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<sub>ξ</sub>'' gauge generalizes well to [[Non-abelian gauge theory|non-abelian]] gauge groups such as the [[SU(3)]] of [[quantum chromodynamics|QCD]]. 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 matrix and determinant|Jacobian]] of the embedding of gauge freedom axes within the space of detailed configurations. This leads to the explicit appearance of forward and backward polarized gauge bosons in Feynman diagrams, along with [[Faddeev–Popov ghost]]s, which are even more "unphysical" in that they violate the [[spin–statistics theorem]]. 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.
== Maximal
In any non-[[Gauge theory|
* For [[SU(2)]] gauge theory in D dimensions, the maximal
* For [[SU(3)]] gauge theory in D dimensions, the maximal
This applies regularly in higher algebras (of groups in the algebras), for example the Clifford Algebra and as it is regularly.
== Less commonly used gauges ==
<!--The following synonyms are boldfaced as per WP:R#PLA-->
Various other gauges, which can be beneficial in specific situations have appeared in the literature.<ref name=Jackson2002 />
=== Weyl gauge ===
The '''Weyl gauge''' (also known as the '''Hamiltonian''' or '''temporal gauge''') is an ''incomplete'' gauge obtained by the choice
<math display="block">\varphi=0</math>
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It is named after [[Hermann Weyl]]. It eliminates the negative-norm [[Ghost (physics)|ghost]], lacks manifest [[Lorentz invariance]], and requires longitudinal photons and a constraint on states.<ref>{{cite book |last1=Hatfield |first1=Brian |title=Quantum field theory of point particles and strings |date=1992 |publisher=Addison-Wesley |isbn=0201360799 |pages=210–213}}</ref>
=== Multipolar gauge ===
The gauge condition of the '''multipolar gauge''' (also known as the '''line gauge''', '''point gauge''' or '''Poincaré gauge''' (named after [[Henri Poincaré]])) is:
<math display="block">\mathbf{r}\cdot\mathbf{A} = 0.</math>
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<math display="block"> \varphi(\mathbf{r},t) = -\mathbf{r} \cdot \int_0^1 \mathbf{E}(u \mathbf{r},t) du.</math>
=== Fock–Schwinger gauge ===
The gauge condition of the '''Fock–Schwinger gauge''' (named after [[Vladimir Fock]] and [[Julian Schwinger]]; sometimes also called the '''relativistic Poincaré gauge''') is:
<math display="block">x^{\mu}A_{\mu}=0</math>
where ''x''<sup>''μ''</sup> is the [[position four-vector]].
=== Dirac gauge ===
The nonlinear Dirac gauge condition (named after [[Paul Dirac]]) is: <math display="block">A_{\mu} A^{\mu} = k^2</math>
== References ==
{{reflist}}
== Further reading ==
* {{cite book |last1=Landau |first1=Lev |author-link=Lev Landau |last2=Lifshitz |first2=Evgeny |author-link2=Evgeny Lifshitz |year=2007 |title=The classical theory of fields |location=Amsterdam |publisher=Elsevier Butterworth Heinemann |isbn=978-0-7506-2768-9 }}
* {{cite book |last=Jackson |first=J. D. |title=Classical Electrodynamics |location=New York |publisher=Wiley |year=1999 |isbn=0-471-30932-X |edition=3rd }}
{{QED}}
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