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Gauge fixing

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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.

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. 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. 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. The existence of arbitrary numbers of gauge functions \psi(\mathbf{x},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.

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.)

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. 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. 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. Solutions of this wave equation with the transverse current set to zero correspond classically to transversely polarized electromagnetic radiation in free space. 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. 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:

\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. 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 = e j^\nu, where jν 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 Klein-Gordon equation, and hence that the Lorenz gauge condition allows transversely, longitudinally, and "time-like" polarized waves in the four-potential. 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. Classically, these identities are equivalent to the continuity equation \partial_\mu j^\mu = 0.

   
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