Gauge fixing

In the

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

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}

]

Gauge freedom

The archetypical gauge theory is the

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 scalar potential

\varphi

and the magnetic vector potential A through the relations:

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

If the transformation

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

(1)

is made, then B remains unchanged, since (with the identity $\nabla \times \nabla \psi =0$ )

{\mathbf {B} }=\nabla \times ({\mathbf {A} }+\nabla \psi )=\nabla \times {\mathbf {A} }.

However, this transformation changes E according to

\mathbf {E} =-\nabla \varphi -{\frac {\partial {\mathbf {A} }}{\partial t}}-\nabla {\frac {\partial {\psi }}{\partial t}}=-\nabla \left(\varphi +{\frac {\partial {\psi }}{\partial t}}\right)-{\frac {\partial {\mathbf {A} }}{\partial t}}.

If another change

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

(2)

is made then E also remains the same. Hence, the E and B fields are unchanged if one takes any function $ψ (r, t)$ and simultaneously transforms 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

\mathbf {A} \rightarrow \mathbf {A} +\nabla \psi \,.

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

.

An illustration

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

As an illustration of gauge fixing, one may look at a cylindrical rod and attempt to 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. In summary, to tell whether the rod is twisted, the gauge must be known. Physical quantities, such as the energy of the torsion, do not depend on the gauge, i.e., they are gauge invariant.

Coulomb gauge

The Coulomb gauge (also known as the transverse gauge) is used in quantum chemistry and condensed matter physics and is defined by the gauge condition (more precisely, gauge fixing condition)

\nabla \cdot {\mathbf {A} }(\mathbf {r} ,t)=0\,.

It is particularly useful for "semi-classical" 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 International System of Units)^[1]
$\varphi (\mathbf {r} ,t)={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\mathbf {\rho } (\mathbf {r} ',t)}{R}}d^{3}\mathbf {r} '$ $\mathbf {A} (\mathbf {r} ,t)=\nabla \times \int {\frac {\mathbf {B} (\mathbf {r} ',t)}{4\pi R}}d^{3}\mathbf {r} '$ where $ρ (r, t)$ is the electric charge density, $\mathbf {R} =\mathbf {r} -\mathbf {r} '$ and $R=\left|\mathbf {R} \right|$ (where r is any position vector in space and r′ is a point in the charge or current distribution), the $\nabla$ operates on r and d³r is the volume element at r.
The instantaneous nature of these potentials appears, at first sight, to violate causality, since motions of electric charge or magnetic field appear everywhere instantaneously as changes to the potentials. This is justified by noting 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 the 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 Lorentz covariant Lorenz gauge described below.
Another expression for the vector potential, in terms of the time-retarded electric current density $J (r, t)$ , has been obtained to be:^[2]
$\mathbf {A} (\mathbf {r} ,t)={\frac {1}{4\pi \varepsilon _{0}}}\,\nabla \times \int \left[\int _{0}^{R/c}\tau \,{\frac {{\mathbf {J} (\mathbf {r} ',t-\tau )}\times {\mathbf {R} }}{R^{3}}}\,d\tau \right]d^{3}\mathbf {r} '.$
Further gauge transformations that retain the Coulomb gauge condition might be made with gauge functions that satisfy $\nabla 2 ψ = 0$ , 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 A² over all space is minimal for this gauge: All other gauges give a larger integral.^[3] The minimum value given by the Coulomb gauge is $\int \mathbf {A} ^{2}(\mathbf {r} ,t)d^{3}\mathbf {r} =\iint {\frac {\mathbf {B} (\mathbf {r} ,t)\cdot \mathbf {B} (\mathbf {r} ',t)}{4\pi R}}d^{3}\mathbf {r} \,d^{3}\mathbf {r} '.$
In regions far from electric charge the scalar potential becomes zero. This is known as the radiation gauge.
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.^[4]
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 ${\mathbf {A} }(\mathbf {r} ,t)=-{\frac {1}{2}}\mathbf {r} \times \mathbf {B}$ 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_ξ Landau gauge of the next section), where $\mathbf {B} =B{\hat {z}}$ and $\mathbf {A} =B(\mathbf {r} \cdot {\hat {x}}){\hat {y}},$ where ${\hat {x}},{\hat {y}},{\hat {z}}$ are unitary vectors of the Cartesian coordinate system (z-axis aligned with the magnetic field).
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 $\varphi (\mathbf {r} ,t)=\int {\frac {\nabla '\cdot {\mathbf {E} }(\mathbf {r} ',t)}{4\pi R}}\operatorname {d} \!^{3}\mathbf {r} '-{\frac {\partial {\psi (\mathbf {r} ,t)}}{\partial t}}$ $\mathbf {A} (\mathbf {r} ,t)=\nabla \times \int {\frac {\mathbf {B} (\mathbf {r} ',t)}{4\pi R}}\operatorname {d} \!^{3}\mathbf {r} '+\nabla \psi (\mathbf {r} ,t)$ where $ψ (r, t)$ is an arbitrary scalar field called the gauge function. The fields that are the derivatives of the gauge function are known as pure gauge fields and the arbitrariness associated with the gauge function is known as gauge freedom. In a calculation that is carried out correctly the pure gauge terms have no effect on any physical observable. A quantity or expression that does not depend on the gauge function is said to be gauge invariant: All physical observables are required to be gauge invariant. A gauge transformation from the Coulomb gauge to another gauge is made by taking the gauge function to be the sum of a specific function which will give the desired gauge transformation and the arbitrary function. If the arbitrary function is then set to zero, the gauge is said to be fixed. Calculations may be carried out in a fixed gauge but must be done in a way that is gauge invariant.

Lorenz gauge

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.

This may be rewritten as:

\partial _{\mu }A^{\mu }=0.

where

A^{\mu }=\left[\,{\tfrac {1}{c}}\varphi ,\,\mathbf {A} \,\right]

is 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 FitzGerald

.)

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 some sense: there remains a subspace of gauge transformations which can also preserve the constraint. These remaining degrees of freedom correspond to gauge functions which satisfy the wave equation

{\frac {\partial ^{2}\psi }{\partial t^{2}}}=c^{2}\nabla ^{2}\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

\partial _{\mu }\partial ^{\mu }A^{\nu }=\mu _{0}j^{\nu }

where

j^{\nu }=\left[\,c\,\rho ,\,\mathbf {j} \,\right]

is the four-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

Ward identities. Classically, these identities are equivalent to the continuity equation

\partial _{\mu }j^{\mu }=0.

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_ξ gauges

The R_ξ gauges are a generalization of the Lorenz gauge applicable to theories expressed in terms of an

Lagrangian density

{\mathcal {L}}

. 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

\delta {\mathcal {L}}=-{\frac {\left(\partial _{\mu }A^{\mu }\right)^{2}}{2\xi }}

The choice of the parameter ξ determines the choice of gauge. The R_ξ 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 $ξ = 1$ ; 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:

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

The auxiliary field, sometimes called a Nakanishi–Lautrup 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

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

.

The photon propagator, which is the multiplicative factor corresponding to an internal photon in the

circularly polarized basis. 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. 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.

Sin-Itiro Tomonaga, with whom Feynman shared the 1965 Nobel Prize in Physics

.

Forward and backward polarized radiation can be omitted in the

BRST formalism

of quantization.

Maximal abelian gauge

In any non-abelian gauge theory, any maximal abelian gauge is an incomplete gauge which fixes the gauge freedom outside of the maximal abelian subgroup. Examples are

For
Pauli matrix
σ₃, then the maximal abelian gauge is that which maximizes the function $\int d^{D}x\left[\left(A_{\mu }^{1}\right)^{2}+\left(A_{\mu }^{2}\right)^{2}\right]\,,$ where ${\mathbf {A} }_{\mu }=A_{\mu }^{a}\sigma _{a}\,.$
For
SU(3) gauge theory in D dimensions, the maximal abelian subgroup is a U(1)×U(1) subgroup. If this is chosen to be the one generated by the Gell-Mann matrices
λ₃ and λ₈, then the maximal abelian gauge is that which maximizes the function $\int d^{D}x\left[\left(A_{\mu }^{1}\right)^{2}+\left(A_{\mu }^{2}\right)^{2}+\left(A_{\mu }^{4}\right)^{2}+\left(A_{\mu }^{5}\right)^{2}+\left(A_{\mu }^{6}\right)^{2}+\left(A_{\mu }^{7}\right)^{2}\right]\,,$ where ${\mathbf {A} }_{\mu }=A_{\mu }^{a}\lambda _{a}$

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

Various other gauges, which can be beneficial in specific situations have appeared in the literature.[2]

Weyl gauge

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

\varphi =0

It is named after

Lorentz invariance, and requires longitudinal photons and a constraint on states.^[5]

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:

\mathbf {r} \cdot \mathbf {A} =0.

This is another gauge in which the potentials can be expressed in a simple way in terms of the instantaneous fields

\mathbf {A} (\mathbf {r} ,t)=-\mathbf {r} \times \int _{0}^{1}\mathbf {B} (u\mathbf {r} ,t)u\,du

\varphi (\mathbf {r} ,t)=-\mathbf {r} \cdot \int _{0}^{1}\mathbf {E} (u\mathbf {r} ,t)du.

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:

x^{\mu }A_{\mu }=0

where x^μ is the

position four-vector

.

Dirac gauge

The nonlinear Dirac gauge condition (named after Paul Dirac) is:

A_{\mu }A^{\mu }=k^{2}

References

S2CID 250880504
.

^
S2CID 119652556
.

S2CID 45172403
.

PMID 9958379
.

ISBN 0201360799
.

Further reading

ISBN 978-0-7506-2768-9
.

Jackson, J. D. (1999). Classical Electrodynamics (3rd ed.). New York: Wiley.
ISBN 0-471-30932-X
.

v
t
e
Quantum electrodynamics
Formalism

Euler–Heisenberg Lagrangian

Feynman diagram

Gupta–Bleuler formalism

Path integral formulation

Particles

Dual photon

Electron

Faddeev–Popov ghost

Photon

Positron

Positronium

Virtual particles

Concepts

Anomalous magnetic dipole moment

Furry's theorem

Klein–Nishina formula

Landau pole

QED vacuum

Self-energy

Schwinger limit

Uehling potential

Vacuum polarization

Vertex function

Ward–Takahashi identity

Processes

Bhabha scattering

Breit–Wheeler process

Bremsstrahlung

Compton scattering

Delbrück scattering

Lamb shift

Møller scattering

Schwinger effect

Photon-photon scattering

See also: Template:Quantum mechanics topics

Retrieved from "https://en.wikipedia.org/w/index.php?title=Gauge_fixing&oldid=1202753476"

[Stewart2003-1] S2CID 250880504
.

[Jackson2002-2] 
S2CID 119652556
.

[3] S2CID 45172403
.

[4] PMID 9958379
.

[5] ISBN 0201360799
.

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