Feynman gauge
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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. 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 gauge transformation, equivalent to a shear along unphysical axes in configuration space. 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. 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.
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Gauge freedom
The archetypical gauge theory is the Heaviside–Gibbs formulation of continuum electrodynamics in terms of an electromagnetic fourpotential, 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 scalar potential $\backslash varphi$ and the vector potential A through the relations:
 $\{\backslash mathbf\; E\}\; =\; \backslash nabla\backslash varphi\; \; \backslash frac\{\backslash partial\{\backslash mathbf\; A\}\}\{\backslash partial\; t\}\backslash ,,\; \backslash quad\; \{\backslash mathbf\; B\}\; =\; \backslash nabla\backslash times\{\backslash mathbf\; A\}.$
If the transformation

$\backslash mathbf\{A\}\; \backslash rightarrow\; \backslash mathbf\{A\}+\backslash nabla\backslash psi$
(1)
is made, then B remains unchanged, since
 $\{\backslash mathbf\; B\}\; =\; \backslash nabla\backslash times\; (\{\backslash mathbf\; A\}+\; \backslash nabla\; \backslash psi)\; =\; \backslash nabla\backslash times\{\backslash mathbf\; A\}$.
However, this transformation changes E according to
 $\{\backslash mathbf\; E\}\; =\; \backslash nabla\backslash varphi\; \; \backslash frac\{\backslash partial\{\backslash mathbf\; A\}\}\{\backslash partial\; t\}\; \; \backslash nabla\; \backslash frac\{\backslash partial\{\backslash psi\}\}\{\backslash partial\; t\}\; =\; \backslash nabla\; \backslash left(\; \backslash varphi\; +\; \backslash frac\{\backslash partial\{\backslash psi\}\}\{\backslash partial\; t\}\backslash right)\; \; \backslash frac\{\backslash partial\{\backslash mathbf\; A\}\}\{\backslash partial\; t\}$.
If another change

$\backslash varphi\backslash rightarrow\backslash varphi\; \; \backslash frac\{\backslash partial\{\backslash psi$
(})
is made then E also remains the same.
Hence, the E and B fields are unchanged if we take any function ψ(r, t) and simultaneously transform A and φ via the transformations (1) and (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. The existence of arbitrary numbers of gauge functions 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
 $\backslash mathbf\{A\}\; \backslash rightarrow\; \backslash mathbf\{A\}\; +\; \backslash nabla\; \backslash psi\backslash ,.$
Gauge fixing in nonabelian 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
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 tell whether or not it is twisted. 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, i.e., the circular symmetry U(1) of the cross section at each point of the rod. The line is the equivalent of a gauge function; it need not be straight. Almost any line is a valid gauge fixing, i.e., 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, i.e., are gauge invariant.
Coulomb gauge
The Coulomb gauge (also known as the transverse gauge) is much used in quantum chemistry and condensed matter physics and is defined by the gauge condition (more precisely, gauge fixing condition)
 $\backslash nabla\backslash cdot\{\backslash mathbf\; A\}(\backslash mathbf\{r\},t)=0\backslash ,.$
It is particularly useful for "semiclassical" calculations in quantum mechanics, in which the vector potential is quantized but the Coulomb interaction is not.
The Coulomb gauge has a number of properties:
 The potentials can be expressed in terms of instantaneous values of the fields and densities (in SI units)
 $\backslash varphi(\backslash mathbf\{r\},t)\; =\; \backslash frac\{1\}\{4\backslash pi\; \backslash varepsilon\_0\}\; \backslash int\backslash frac\{\backslash mathbf\{\backslash rho\}(\backslash mathbf\{r\text{'}\},t)\}\{R\}d^3r\text{'}$
 $\backslash mathbf\{A\}(\backslash mathbf\{r\},t)\; =\; \backslash nabla\; \backslash times\backslash int\backslash frac\{\; \backslash mathbf\{B\}(\backslash mathbf\{r\text{'}\},t)\}\{4\backslash pi\; R\}d^3r\text{'}$
 $\backslash mathbf\{A\}(\backslash mathbf\{r\},t)\; =\; \backslash frac\{1\}\{4\backslash pi\; \backslash varepsilon\_0\}\; \backslash ,\; \backslash nabla\backslash times\backslash int\; d^3r\text{'}\; \backslash int\_0^\{R/c\}\; d\backslash tau\; \backslash tau\; \{\; \backslash mathbf\{J\}(\backslash mathbf\{r\text{'}\},\; t\backslash tau)\}\backslash times\; \{\; \backslash mathbf\{R\}\}/R^3$.
 Further gauge transformations that retain the Coulomb gauge condition might be made with gauge functions that satisfy ∇^{2 }ψ = 0, but as the only solution to this equation that vanishes at infinity (where all fields are required to vanish) is 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.
 The Coulomb gauge is a minimal gauge in the sense that the integral of A^{2} over all space is minimal for this gauge: all other gauges give a larger integral.^{[2]} The minimum value given by the Coulomb gauge is
 $\backslash int\backslash mathbf\{A\}^2(\backslash mathbf\{r\},t)d^3r\; =\; \backslash int\backslash int\backslash frac\; \{\backslash mathbf\{B\}(\backslash mathbf\{r\},t)\backslash cdot\backslash mathbf\{B\}(\backslash mathbf\{r\text{'}\},t)\}\{4\backslash pi\; R\}d^3rd^3r\text{'}$.
 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.
 The Coulomb gauge is 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. Lorentz covariant gauges such as the Lorenz gauge are used in these theories.
 For a uniform and constant magnetic field B the vector potential in the Coulomb gauge is
 $\{\backslash mathbf\; A\}(\backslash mathbf\{r\},t)=\{\backslash mathbf\; r\}\backslash times\{\backslash mathbf\; B\}/2$
 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
 $\backslash varphi(\backslash mathbf\{r\},t)\; =\; \backslash int\backslash frac\{\backslash nabla\text{'}\backslash cdot\{\backslash mathbf\; E\}(\backslash mathbf\{r\text{'}\},t)\}\{4\backslash pi\; R\}d^3r\text{'}\backslash frac\{\backslash partial\{\backslash psi(\backslash mathbf\{r\},t)\}\}\{\backslash partial\; t\}$
 $\backslash mathbf\{A\}(\backslash mathbf\{r\},t)\; =\; \backslash nabla\; \backslash times\backslash int\backslash frac\{\; \backslash mathbf\{B\}(\backslash mathbf\{r\text{'}\},t)\}\{4\backslash pi\; R\}d^3r\text{'}+\backslash nabla\; \backslash psi(\backslash mathbf\{r\},t)$
Lorenz gauge
The Lorenz gauge is given, in SI units, by:
 $\backslash nabla\backslash cdot\{\backslash mathbf\; A\}\; +\; \backslash frac\{1\}\{c^2\}\backslash frac\{\backslash partial\backslash varphi\}\{\backslash partial\; t\}=0$
and in Gaussian units by:
 $\backslash nabla\backslash cdot\{\backslash mathbf\; A\}\; +\; \backslash frac\{1\}\{c\}\backslash frac\{\backslash partial\backslash varphi\}\{\backslash partial\; t\}=0.$
This may be rewritten as:
 $\backslash partial^\{\backslash mu\}\; A\_\{\backslash mu\}\; =\; 0.$
where A_{μ} = (φ/c, −A) is the electromagnetic fourpotential, ∂^{μ} the 4gradient [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 F. FitzGerald.)
The Lorenz gauge leads to the following inhomogeneous wave equations for the potentials:
 $\backslash frac\{1\}\{c^2\}\backslash frac\{\backslash partial^2\backslash varphi\}\{\backslash partial\; t^2\}\; \; \backslash nabla^2\{\backslash varphi\}\; =\; \backslash frac\{\backslash rho\}\{\backslash varepsilon\_0\}$
 $\backslash frac\{1\}\{c^2\}\backslash frac\{\backslash partial^2\backslash mathbf\; A\}\{\backslash partial\; t^2\}\; \; \backslash nabla^2\{\backslash mathbf\; A\}\; =\; \backslash mu\_0\; \backslash mathbf\{J\}$
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
 $\{\; \backslash partial^2\; \backslash psi\; \backslash over\; \backslash partial\; t^2\; \}\; =\; c^2\; \backslash nabla^2\backslash psi$
These remaining gauge degrees of freedom propagate at the speed of light. To obtain a fully fixed gauge, one must add boundary conditions along the light cone of the experimental region.
Maxwell's equations in the Lorenz gauge simplify to
 $\backslash partial\_\backslash mu\; \backslash partial^\backslash mu\; A^\backslash nu\; =\; \backslash mu\_0\; j^\backslash nu$
where j^{ν} = (ρc, −j) is the fourcurrent.
Two solutions of these equations for the same current configuration differ by a solution of the vacuum wave equation
 $\backslash partial\_\backslash mu\; \backslash partial^\backslash mu\; A^\backslash nu\; =\; 0$.
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 "timelike" polarized waves in the fourpotential. The transverse polarizations correspond to classical radiation, i. e., transversely polarized waves in the field strength. To suppress the "unphysical" longitudinal and timelike polarization states, which are not observed in experiments at classical distance scales, one must also employ auxiliary constraints known as Ward identities. Classically, these identities are equivalent to the continuity equation
 $\backslash partial\_\backslash mu\; j^\backslash mu\; =\; 0$.
Many of the differences between classical and quantum electrodynamics can be accounted for by the role that the longitudinal and timelike polarizations play in interactions between charged particles at microscopic distances.
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 $\backslash mathcal\{L\}$. 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
 $\backslash delta\; \backslash mathcal\{L\}\; =\; \backslash frac\{(\backslash partial\_\{\backslash mu\}\; A^\{\backslash mu\})^2\}\{2\; \backslash xi\}$
The choice of the parameter ξ determines the choice of gauge. The Landau gauge, obtained as the limit ξ → 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.
An equivalent formulation of R_{ξ} gauge uses an auxiliary field, a scalar field B with no independent dynamics:
 $\backslash delta\; \backslash mathcal\{L\}\; =\; B\backslash ,\backslash partial\_\{\backslash mu\}\; A^\{\backslash mu\}\; +\; \backslash frac\{\backslash xi\}\{2\}\; B^2$
The auxiliary field can be eliminated by "completing the square" to obtain the previous form. 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.
Historically, the use of R_{ξ} gauges was a significant technical advance in extending quantum electrodynamics computations beyond oneloop order. 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. 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. 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 terms of the Feynman rules of the gaugefixed theory, this appears as a contribution to the photon propagator for internal lines from virtual photons of unphysical polarization.
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. 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. Similarly, one can combine the longitudinal and timelike gauge polarizations to obtain "forward" and "backward" polarizations; these are a form of light cone coordinates in which the metric is offdiagonal. 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. 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 SinItiro Tomonaga, with whom Feynman shared the 1965 Nobel Prize in Physics.
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 fourpotential in classical electrodynamics), they are often spoken of as "unphysical". But unlike the constraintbased gauge fixing procedures above, the R_{ξ} gauge generalizes well to nonabelian gauge groups such as the SU(3) of 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 nontrivial 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 ghosts, which are even more "unphysical" in that they violate the spinstatistics 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.
Maximum Abelian gauge
In any nonAbelian gauge theory, any maximum Abelian gauge is an incomplete gauge which fixes the gauge freedom outside of the maximum Abelian subgroup. Examples are
 For SU(2) gauge theory in D dimensions, the maximum Abelian subgroup is a U(1) subgroup. If this is chosen to be the one generated by the Pauli matrix σ_{3}, then the maximum Abelian gauge is that which maximizes the function
 $\backslash int\; d^Dx\; \backslash left[(A\_\backslash mu^1)^2+(A\_\backslash mu^2)^2\backslash right]\backslash ,,$
 where
 $\{\backslash mathbf\; A\}\_\backslash mu\; =\; A\_\backslash mu^a\; \backslash sigma\_a\backslash ,.$
 For SU(3) gauge theory in D dimensions, the maximum Abelian subgroup is a U(1)×U(1) subgroup. If this is chosen to be the one generated by the GellMann matrices λ_{3} and λ_{8}, then the maximum Abelian gauge is that which maximizes the function
 $\backslash int\; d^Dx\; \backslash left[(A\_\backslash mu^1)^2+(A\_\backslash mu^2)^2+(A\_\backslash mu^4)^2+(A\_\backslash mu^5)^2+(A\_\backslash mu^6)^2+(A\_\backslash mu^7)^2\backslash right]\backslash ,,$
 where
 $\{\backslash mathbf\; A\}\_\backslash mu\; =\; A\_\backslash mu^a\; \backslash lambda\_a$
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
Weyl gauge
The Weyl gauge (also known as the Hamiltonian or temporal gauge) is an incomplete gauge obtained by the choice
 $\backslash phi=0$
It is named after Hermann Weyl.
Multipolar gauge
The gauge condition of the Multipolar gauge (also known as the Line gauge, point gauge or Poincaré gauge) is:
 $\backslash mathbf\{r\}\backslash cdot\backslash mathbf\{A\}=0$.
This is another gauge in which the potentials can be expressed in a simple way in terms of the fields
 $\backslash mathbf\{A\}(\backslash mathbf\{r\},t)\; =\; \backslash mathbf\{r\}\; \backslash times\backslash int\backslash limits\_\{0\}^\{1\}\backslash mathbf\{B\}(u\; \backslash mathbf\{r\},t)\; u\; du$
 $\backslash varphi(\backslash mathbf\{r\},t)\; =\; \backslash mathbf\{r\}\backslash cdot\backslash int\backslash limits\_\{0\}^\{1\}\backslash mathbf\{E\}(u\; \backslash mathbf\{r\},t)\; du.$
Fock–Schwinger gauge
The gauge condition of the Fock–Schwinger gauge (sometimes called the relativistic Poincaré gauge) is:
 $x^\{\backslash mu\}A\_\{\backslash mu\}=0$
where x^{μ} is the position fourvector.