Talk:Gauge fixing

Would not a more accurate nomenclature be (divA = 0) for the Coulomb gauge and (divA = 0 together with scalar potential = 0) for the radiation gauge? The radiation gauge, which is used in perturbative calculations, is just the Coulomb gauge in the absence of charge and describes the free electromagnetic field. The Coulomb gauge, in the presence of charge and accordingly with non-zero scalar potential, is used in quantum chemistry. Xxanthippe 09:28, 19 June 2006 (UTC)

Good reference for gauge-fixing in context of Maxwell's equations.

Hi, I think this reference does a good job in explaining the concepts, at least for maxwell's equations. http://www.mathematik.tu-darmstadt.de/~bruhn/Maxwell-Theory.html

Questions for introduction paragraph

What is meant by "redundant degree of freedom"

It's mathematically convenient to describe a physical system in terms of local fields and act as though their "value" at each point in space(time) has physical significance. Unfortunately, this description usually has some excess global degrees of freedom which are mathematically trivial -- they don't really correspond to different physical situations because they don't enter into any calculation of a physical quantity or interaction.

The classical example is a description of the electromagnetic environment surrounding a configuration of stationary charges in terms of an electrostatic scalar potential. Two potentials which differ by a constant term describe the same physical situation. More complete local field theories tend to have larger global "symmetries" in this mathematical sense, which I chose to word in terms of "redundant" degrees of freedom in the local field formulation. They aren't really "symmetries" in a physical sense; a real physical symmetry is an observation about the physical world that doesn't seem to be mathematically necessary, such as the apparent constancy over time of parameters such as the electroweak coupling strength and the electroweak "angle".

What is an "equivalence class"

Sometimes you want to treat a set of configurations as equivalent, either because they aren't really physically different situations (as in the electrostatic potential case above) or because your experiment doesn't distinguish between them (as in the case of statistical thermodynamic calculations, in which detailed configurations or "microstates" are considered equivalent if they have the same distribution of energy, momentum, etc.). You toss these equivalent configurations in a mathematical bucket and call them an "equivalence class", just like a child learns to toss all sets of three objects into a mental bucket and label it "3".

When your equivalence classes reflect mathematical over-description of physical situations, you have to make sure that your physical predictions don't depend on which representative you picked out of the bucket. The principal risk is that, in the course of making a practical calculation with a particular representative of the class, you will apply approximation techniques that aren't insensitive to the choice of representative. (Imagine approximating an electrostatic situation by forcing the potential to drop rapidly to zero outside radius R; you may get different physical predictions for different starting potentials that differ by a constant term, because your formula for forcing the potential to zero isn't insensitive to this irrelevant degree of freedom in the potential field.)

What is "configuration space"

In this context, it's an abstract mathematical "space" consisting of all possible local field configurations -- loosely speaking, a distinct "point" for each way of assigning a value for each degree of freedom in each local field at each point in spacetime. In a gauge field theory, any two "points" in configuration space which are related by a gauge transformation describe the same physical situation.

What calculations do gauge fixing simplify? What calculations don't gauge fixing simplify?

Consider the analogy of a definite integral, say, $\int \limits _{0}^{1}xdx$. Technically the indefinite integral $\int xdx=x^{2}+C$ is only defined up to a constant term, and you ought to carry that $C$ around when you extend your calculation to multiple integrals, etc. Gauge fixing is similar to choosing $C=0$ and dropping terms with $C$ in them at an intermediate stage.

When you are dealing with integrals that don't converge and approximation techniques that can only be justified with some rather deep mathematics, it makes sense to specialize early to a "least pathological" gauge. The analogy is something like $\int \limits _{-\infty }^{\infty }x^{-1}X(x)dx$, for some $X(x)$ that differs appreciably from 1 only in the neighborhood of $x=\pm 1$. For the integral to make any sense, you have to have some prescription for handling the discontinuity in $x^{-1}$ at $x=0$. The natural thing is to declare that $\lim _{\epsilon \rightarrow 0}\int \limits _{-\epsilon }^{\epsilon }x^{-1}dx=0$, and to assign constant terms to the left and right ranges of the integral such that the "indefinite integral" equals 0 at $\pm \epsilon$. Then you can pretend that the pathology at 0 isn't there and go on with the calculation. The particular value of $\epsilon$ had better not appear in the final result; if it does, either you applied an approximation technique that wasn't "gauge" invariant or your theory was bogus to begin with.

In practice, the only field theory calculations (classical or quantum) that don't involve gauge fixing are those in which the gauge freedom drops out quite early in the game, as in some semi-classical calculations where one works directly with the classical "field strengths" rather than the potentials.

And if you could settle an argument for me, is gauge fixing compatible with the Lorentz Transformations?

Also, what is "manifest" Lorentz Invariance, and how does that compare to regular Lorentz Invariance?

The answers to these two questions are coupled. Consistent application of any gauge fixing prescription will give physical "predictions" consistent with Lorentz invariance, in the sense that two "input" configurations that are related by a Lorentz transformation will produce calculated results that are related by the same Lorentz transformation. (I put "predictions" and "input" in quotes because we aren't necessarily talking about past and future here. We have an event or a collection of events that are selected according to some set of "input" criteria, such the center-of-mass energy of a collision between two electrons and the relative sensitivity of our detectors to different outcomes; our "predictions" describe other things we can measure about the situation, which may include Lorentz-invariant statements like "if they interact at all, 99% of the time the electrons swap helicity".)

The sense in which some gauge fixings (such as the Coulomb prescription) are not "compatible" with the Lorentz transformation is that applying them in frame A will choose a different representative from a given equivalence class than applying them in frame B. This means that you have to either work the entire calculation in the same reference frame or take into account the violation of the gauge constraint in any other frame you switch to later in the calculation. In classical terms, the simplified differential equations obtained by applying the Coulomb gauge fixing prescription do not retain their form under a Lorentz boost.

A manifestly Lorentz invariant gauge fixing constraint (such as the Lorenz condition) is one that retains its form under Lorentz transformations, and may therefore be applied both before and after a change of frame as a justification for dropping terms in the calculation that are constrained to zero. This is the kind that generalizes properly to a gauge breaking prescription, in which you avoid a class of mathematical failures that result from applying a gauge constraint by softening it to a sort of likelihood weighting along the gauge freedom "axes" in configuration space.

Hope this helps, Michael K. Edwards 07:14, 5 December 2006 (UTC)

Clarification please!

"This was not well understood at first even by active researchers in the field[1] and remains inconspicuous in most textbook treatments, partly because a rigorous derivation of the photon propagator requires deeper mathematical tools than one needs for the rest of QED." This is cryptic, even POVish. The note leads nowhere. Clarification please! Xxanthippe 23:54, 5 January 2007 (UTC)

Lorenz Gauge (duplicate material)

The Lorenz Gauge has its own article which repeats most of the material here. It might make sense to either use one big article with all the gauges or give other gauges their own articles. Also, mixing four dimensional formulations with three dimensional formulations is confusing because almost-similar notation actally means different things. It would be much nicer to see all the three dimensional formulations in one article and then the four dimensional formulations together in a separate article so the reader always sees one "world view" at a time.

Continuity eqn & ward identities

Have to be more careful with wording here- we do not completely pull the ward identity out of our ass and 'enforce' it as the article implies- to my knowledge we can either have the current or axial current conserved and we choose the current, by our renormalization prescription, to be conserved which leads to the ward identity. —Preceding unsigned comment added by 128.230.52.201 (talk) 00:03, 10 February 2009 (UTC)

Why "Gauge"?

The Intro and section on Gauge Freedom with the illustrative diagram provides exactly the kind of gentle introduction that I was seeking. Thanks! But something puzzles me. Why is is the term "gauge" used? The term gauge implies that something is being measured carefully or that a scale is being assigned to something needing to be quantified, but it seems that this process has no effect on the physical quantities described by the field theory. Simple is better (talk) 18:54, 15 August 2009 (UTC)

See Gauge theory#History and importance. In principle, that's the article you should have been reading in the first place to find that fact...I'll add a "main article" link to the top to make it clearer that this is not the main article on gauge theory, only on a specific aspect of it. --Steve (talk) 02:35, 16 August 2009 (UTC)

Thank you for your help! Simple is better (talk) 19:07, 16 August 2009 (UTC)

Coulomb gauge

Because the Coulomb gauge is used almost universally in quantum chemistry and condensed matter physics I have rewritten this section to extend it and include recent material. Xxanthippe (talk) 09:27, 10 March 2010 (UTC).

Why U(1)?

The existence of arbitrary numbers of gauge functions \psi(\mathbf{x},t), corresponds to the U(1) gauge freedom of this theory.

Could it be made clearer why it is a U(1) freedom in particular? 151.200.120.52 (talk) 02:18, 20 May 2010 (UTC)

This does look wrong. The function $\psi$ looks like an element of the Poincare group which is certainly not the same as U(1). I think the U(1) turns up as the gauge group in Quantum electrodynamics#Mathematics. It is not obvious to me how you relate the gauge group and the Poincare group. Puzl bustr (talk) 10:48, 11 March 2012 (UTC)