Gauge fixing: Difference between revisions

{{Electromagnetism}}

{{Quantum field theory}}

 

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}}

 

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>

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}}).

 

 

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

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]].

 

[[File:gauge.png|right|thumb|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'''.

 

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>

|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.

 

 

|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

}}

 

{{See also|Covariant formulation of classical electromagnetism}}

<math display="block">\nabla\cdot{\mathbf A} + \frac{1}{c^2}\frac{\partial\varphi}{\partial t}=0</math>

and in [[Gaussian units]] by:

This may be rewritten as:

<math display="block">\partial_{\mu} A^{\mu} = 0.</math>

 

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]].)

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.

 

<math display="block">\delta \mathcal{L} = -\frac{\left(\partial_{\mu} A^{\mu}\right)^2}{2 \xi}</math>

 

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_ξ'' 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.

 

 

This applies regularly in higher algebras (of groups in the algebras), for example the Clifford Algebra and as it is regularly.

 

<!--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 />

 

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>

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>

 

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>

<math display="block"> \varphi(\mathbf{r},t) = -\mathbf{r} \cdot \int_0^1 \mathbf{E}(u \mathbf{r},t) du.</math>

 

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>

 

The nonlinear Dirac gauge condition (named after [[Paul Dirac]]) is: <math display="block">A_{\mu} A^{\mu} = k^2</math>

 

{{reflist}}

 

 

{{QED}}