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)

The Gauge theory article currently explains this more clearly, as does “Quantum field theory” by Mark Sredniki, a reference to the Quantum field theory article. It is not in fact a U(1) freedom, but the symmetry group is an infinite-dimensional one, which I take to be smooth maps from spacetime to U(1). One can replace U(1) by the reals (perhaps with a factor of i, the square-root of -1) by means of the exponential map. The term “U(1) gauge freedom” implicitly means invariance under this larger Lie group. Mathematical physicists deal directly with the larger group, physicists write down explicit gauge transformations (as in the Srednicki reference, p 336) which then implies the action of the larger group. Puzl bustr (talk) 11:24, 23 April 2012 (UTC)

Arbitrariness of the Lorenz Gauge and Ward Identities

>To suppress the "unphysical" longitudinal and time-like polarization states, which are not observed in experiments at classical distance scales,

Has anybody tried to determine what these longitudinal and time-like waves would look like, or done any experiments investigating their existence?

98.154.22.134 (talk) 10:18, 21 March 2013 (UTC)

Differential vs integral definitions of Coulomb gauge

I noticed a little ambiguity in defining the Coulomb gauge.

• The first definition is that the vector potential is divergence-free, which is an incomplete definition (for example we can still add any constant value to the scalar and vector potentials).
• The second definition is in complete integrals. However I notice that these integrals do not converge in all cases.
• A third definition is the vector potential in terms of integrated retarded current. However, the Coulomb gauge is not intrinsically retarded, so an advanced integral should also be possible! Are these integrals equivalent to the preceding definition, or all somewhat distinct?

My guess is that the real definition is the first, incomplete definition, since it is universal and gets us the properties we want. The integral definition is nice but it is more strict and does not necessarily equal all electromagnetic potentials that can we call "Coulomb gauge". Am I right in this thinking? It is after all what we present on Mathematical_descriptions_of_the_electromagnetic_field#Coulomb_gauge. If so I'd like to bring over some of the more general equations from that article in order to bring this section into analogous parity with Lorentz gauge, and separate the special Coulomb-integral gauge into a subsection.

Likewise a small discussion of what are the remaining freedoms in the Coulomb gauge, is worth including. If I got it right, this freedom is that any gauge transformation $\psi (r,t)$ satisfying $\nabla ^{2}\psi =0$ is valid. (Which is a bit weird, since there is no restriction in the time-dependence!) --Nanite (talk) 18:48, 7 December 2016 (UTC)

All fields are required to vanish at infinity, so that rules out constants. Xxanthippe (talk) 21:38, 7 December 2016 (UTC).

That's an interesting constraint to place on EM fields since it lets us play integration by parts... but AFAIK the vanishing of fields is not generally required with Maxwell equations (not even in integral forms). And even if you do assert vanishing on physical grounds, it is sometimes desirable to abandon the vanishing, e.g., when discussing perfect plane waves, or perfectly uniform E or B fields. --Nanite (talk) 22:16, 7 December 2016 (UTC)

You raise an interesting point, but this is the wrong forum to discuss it. The talk page is for discussion of improvements to the article, not of the subject itself. Xxanthippe (talk) 23:29, 7 December 2016 (UTC).

@Xxanthippe: I had completely forgotten about our discussion above, until just this moment. :D Sorry if I seem to be blundering in again! What do you think about moving the Coulomb gauge stuff into its own article, to put it on a level footing with the Lorenz gauge? It seems there is more to say on the topic, for example the explanation (e.g. in Jackson's Electrodynamics) for why it is called 'transverse gauge', 'radiation gauge', etc. And, we can leave a summary here as a residue. I am inclined to do this but I'm curious your opinion. --Nanite (talk) 00:02, 6 January 2021 (UTC)

It could be useful to have a page devoted to the Coulomb gauge alone, as it is so popular, but the page should be based on reliable sources and eschew independent interpretations. The current page should be retained as it compares and contrasts the various gauges used. Xxanthippe (talk) 03:16, 7 January 2021 (UTC).

Incompleteness of Weyl gauge?

I'm not sure about the "incompleteness" attribute stated on the temporal/Hamiltonian/Weyl gauge. Given that one can always perform a gauge transformation that takes $V'=V-\partial _{t}\lambda$ where $\lambda$ is the gauge transformation function, one can always find a $\lambda$ to ensure that $V'=0$ by solving the differential equation, $V-\partial _{t}\lambda =0$, as long as $V$ is continuous in time (as it is). Therefore, it is complete. If there is a formal reason for it being incomplete, a citation would be useful, because it is not obvious. — Preceding unsigned comment added by Pflammer (talk • contribs) 15:55, 7 December 2018 (UTC)

The Lorenz gauge was also mentioned to be "incomplete" in the sense that taking only potentials satisfying the gauge fixing condition, there are still multiple field configurations that are gauge-equivalent. I think this is saying the same thing? I believe the Weyl Gauge and the $r\cdot A$ gauge are also known as the "axial gauge", if that helps. Qwyxivi (talk) 02:27, 28 September 2019 (UTC)

Are there two Landau gauges?

I think the article should make the difference between Landau gauge for low energy calculations like in the Landau levels, and the Landau gauge presented here. But I do not know if they are related somehow.--ReyHahn (talk) 12:06, 10 September 2020 (UTC)

Yes, it seems there are two totally different "Landau gauges", and a brief google search tells me they are of about equal popularity. I guess it depends whether you're doing condensed matter and talking about 1930s Landau, or you're doing particle physics and talking about 1950s Landau. :-) Do we need a disambiguation page for this? --Nanite (talk) 00:50, 6 January 2021 (UTC)

I guess the Landau 1930 gauge can be added to the Coulomb gauge section, as the "Landau gauge in the Coulomb gauge (not to be confused with the Landau gauge in the sections below)".--ReyHahn (talk) 12:10, 1 July 2021 (UTC)

Hamiltonian formulation and Coulomb gauge

Point 5 in the Coulomb Gauge paragraph is a great result: "The Coulomb gauge admits a natural Hamiltonian formulation of the evolution equations of the electromagnetic field interacting with a conserved current". I think this is very little known, even in the specialist community. I could find no reference on this point, although I arrived at the same conclusion in a particular case (https://doi.org/10.1209/0295-5075/103/28004). A reference in the present wikipedia article is definitely needed. 192.54.145.139 (talk) 09:13, 10 June 2023 (UTC)

Quibble on the (potential) historiography of gauge theory

In the text one can read « 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. » I argue that this is false. Just after this the Aharonov-Bohm effect is mentioned, which can be described by a nonrelativistic Schrödinger equation (where the hamiltonian is the quantum Lorentz force hamiltonian) for a particle coupled with classical electromagnetic scalar and vector potentials. Historically it is true that this was discovered after the advent of quantum field theory (and quantum electrodynamics) but we cannot affirm that it could not have been said before that advent that the potentials are physical (in the sense used here of "measurable"). It is conceivable that the Schrödinger equation for the Lorentz force would have been studied and that physicists would have found the Aharonov-Bohm effect before QED and QFT. Of course there is only one reality, one history, so if when we say "could be" we mean "was", then the text is right, but this is not what is usually implied by the formulation used in the text: usually when we hypothesize what could have been, we have some simple model of the world where we imaginarily tweak parameters (events) and imagine the consequences; and i think that imagining possible alternative histories of the discovery of the physicality of electromagnetic potentials we can reasonably argue that it could have been said that they are physical before QFT. Thus i suggest changing this sentence to, for instance, « It was only after the advent of quantum field theory that physicists discovered that the potentials themselves are part of the physical configuration of a system. » We can date the concept of QFT and QED to 1927 by Dirac, or perhaps to Born-Heisenberg-Jordan in 1926 for free QFT, so that would have left very little time, in 1925 and 1926, to write the Lorentz-force Schrödinger equation and derive a physical effect of the potentials before the advent of QFT, but still i think my point is valid. Plm203 (talk) 22:53, 14 August 2023 (UTC)

Please don't use the Talk page to discuss your views of the topic. There are many forums elsewhere where this can be done. Xxanthippe (talk) 22:40, 15 August 2023 (UTC).

Hi, i am giving my view of a specific assertion (which i think is misleading, as i explain) made in this wiki page, isn't it where this should be done ? It is never nice to read criticism but i think my criticism is justified, and i proposed an improvement -which i don't implement myself out of respect for the more senior contributors. Thank you for your consideration. Plm203 (talk) 10:44, 16 August 2023 (UTC)

To emphasise the point of PLM203, the AB effect is equally true for a purely classical field theory coupling to a classical EM field. Eg, take the complex Klein Gordon field, minimally coupled to the EM field. It will manifest the AB effect just as well as a quantum theory will. Also, a particle theory of charges makes no sense, since the energies are all infinite, and the quantum theories are field theories not particle theories. Ie, the statement made in the paper " The earliest consequence to be accurately predicted and experimentally verified was the Aharonov–Bohm effect, which has no classical counterpart." has a final contention that is just wrong. 75.155.165.57 (talk) 04:54, 29 January 2024 (UTC)

Coulomb gauge is not the same as radiation gauge.

yes it is interesting that the two gauges might be identical in special cases, but does that observation really belong in a section devoted to essential properties of Coulomb gauge? I apologize for provocative title, but I can see where a beginning student could get that misconception from a casual reading of item four of Coulomb gauge section. I suggest that it be carefully reworded to make the invention more clear. EternalStudent2000 (talk) 15:18, 21 November 2023 (UTC)

It seems clear enough to me. If not to you then suggest an alternative. Xxanthippe (talk) 22:03, 21 November 2023 (UTC).