Hello Frederick David Tombe ,

I am confused, [1 & 2] called ∇·A + (1/c2)∂ψ/∂t = 0 the "Lorentz condition", after, I believe, Hendrik Antoon Lorentz (1853-1928), not after Ludwig Valentin Lorenz (1829-1891). Can you cite the 1867 work (paper or book) by Lorenz, which you mentioned at the end of the body of your question?

[1] John David Jackson; Classical Electrodynamics; John Wiley & Sons, Inc.; 1962; pp. 180-181.

[2] Wolfgang K. H. Panofsky, Melba Phillips; Classical Electricity and Magnetism; Addison-Wesley Publishing Company, Inc.; 1955; p. 210.

Regards,

Tom Cuff

Thomas Cuff See reference [1] in this article for a link to Lorenz's 1867 paper, Article Maxwell's Displacement Current in the Two Gauges

At least, this is the paper where it is widely believed that Lorenz initiated the Lorenz gauge, in so much as he tried to bring the Coulomb electrostatic force into the EM wave equations.

Meanwhile, in later years, the Lorenz gauge was used in the Lorentz transformation of fields. Halim Boutayeb might be able to provide you with a link for the paper where H.A. Lorentz himself first introduced the Lorenz gauge, because he is very knowledgeable as regards Lorentz's original works. Whatever, the two similar names have got mixed up so badly, that textbooks sometimes refer to it as the Lorenz-Lorentz condition.

Hi Frederick David Tombe ,

Thank you for the citation to the Ludwig Lorenz gauge paper, I will try reading its translated version. Can you provide an example of a textbook(s) that employed the term "Lorenz-Lorentz condition"?

Regards,

Tom Cuff

Frederick David Tombe ,

I will check again, but I do not remember Lorentz using potentials in his papers. He was mainly focusing on electric and magnetic fields. Because Voigt derived "Lorentz transformation" by using the wave equation, it is understable that Lorenz gauge is "Lorentz invariant". Lorenz gauge leads to equations with D'Alembertian such as in the wave equation.

This presentation may help the discussion:

https://www.youtube.com/watch?v=LUE01tjDAb8

The potentials may be more important than we think

Thomas Cuff Here's a discussion on it,

https://physics.stackexchange.com/questions/721830/lorentz-gauge-or-lorenz-gauge

Meanwhile, both "Electromagnetism", by I.S. Grant & W.R. Phillips, and "The Classical Electromagnetic Field", by Leonard Eyges of M.I.T., use "Lorentz Condition". But it seems it really should be, "Lorenz Condition".

But it gets more complicated. At the top of page 182 in the Eyges book, he is talking about gauges as like constants of integration, of which there can be many. Further down the page, at equation (11.32), he writes the Lorentz condition, as I have in the opening question, apart from the fact that he uses Gaussian units. But it's not at all clear from the intervening text, what the relevance is of gauge-fixing to the Lorentz condition.

It seems that the two topics have got unnecessarily mixed up, resulting in bad terminology being used for the Lorentz condition when it is called the Lorentz gauge.

Meanwhile, in the same book, on page 354, Eyges talks about the Lorenz-Lorentz relation, which seems to be a totally unrelated topic. I have no understanding of that topic at all. It's something to do with time-harmonic fields in matter, which I know nothing about.

Anyway, I'll now read page 182 again, but so far I can't see the relevance of gauge fixing to what we called the Coulomb gauge and the Lorenz gauge.

Frederick David Tombe ,

In his book,

https://ia800303.us.archive.org/18/items/electronstheory00lorerich/electronstheory00lorerich.pdf

Lorentz mentions Lorenz only one time (see page 145), about the refractive index of a material (known as Lorentz-Lorenz equation).

In page 19, the Lorenz gauge is presented without mentioning Lorenz

Hi Frederick David Tombe and Halim Boutayeb ,

The Lorentz-Lorenz formula or law relating to the index of refraction (circa 1880) can be found in [3-5]. There also seems to be a great divergence about Lorenz's first name: Louis (Catalogue of Scientific Papers, Volume 8 {1864-1873}, p. 259); Ludwig Valentin (J. C. Poggendorff’s Biographisch-Literarisches Handwörterbuch für Mathematik, Astronomie, Physik, Chemie und verwandte Wissenschaftsgebiete. Volume 3, Part 1, p. 832); Ludwig Valentin (Larousse Dictionary of Scientists, 1994); and Ludvig (Helge Kragh 's paper in Physics in Perspective, 2018). A rose by any other name, eeh? Even the motivation for the word 'gauge' is suspect, [6].

I believe some housekeeping is in order before you can even hope to connect the dots, but that's just my opinion, your mileage may differ.

[3] Max Born, Emil Wolf; Principles of Optics, Sixth Edition; Cambridge University Press; 1980 [reissued 1997]; p. 87. Note, a footnote contains the two citations to Annalen der Physik und Chemie where they independently and nearly simultaneously (1880 versus 1881) published their expressions.

[4] Jurgen R. Meyer-Arendt; Introduction to Classical and Modern Optics; Prentice-Hall, Inc.; 1972; p. 21. Note, citations to the two papers can be found in a footnote with both having the same date 1880.

[5] Jennifer Bothamley; Dictionary of Theories; Visible Ink; 2002; p. 320.

[6] Leonard Susskind, George Hrabovsky; The Theoretical Minimum; Basic Books; 2013; p. 197.

Regards,

Tom Cuff

Thomas Cuff Halim Boutayeb , The more I look into this, the more I suspect that the textbook Lorentz transformations of fields, which talk about the Lorentz condition, are taking it from the work of Poincaré, and not directly from Lorentz himself. I did read somewhere that Poincaré named his work, "The Lorentz Transformations" in honour of H.A. Lorentz, because it was Lorentz who had initiated the investigation and brought it all close to its final form. It would follow, therefore, that Poincaré would also refer to the Lorenz condition as the Lorentz condition.

The connection between the Lorentz condition and the work of Lorenz in 1867 will of course be the fact that it's all about connecting the Coulomb electrostatic force to the speed of light. But I'm not sure if Ludvig Lorenz himself ever wrote out an equation in the form that is nowadays presented in many textbooks as the Lorentz condition, even though Lorenz's work contained it in substance.

Whatever, I can see no connection whatsoever to gauge-fixing, which some mathematical physicists have made an unnecessary industry out of. Gauge-fixing is a red herring. Someone compared it to match-fixing. Well, it's not even like match fixing as it doesn't actually fix anything.

Frederick David Tombe , Thomas Cuff ,

According to Wolfgang Engelhardt, solutions of Maxwell's inhomogeneous equations are different if Coulomb gauge or Lorenz gauge are used. This means that the gauge concept is not correct.

Article Potential Theory in Classical Electrodynamics

Article On the Solvability of Maxwell's Equations

Article Gauge Invariance in Classical Electrodynamics

Halim Boutayeb I would say that it is more the terminology that is wrong. Those differences that you mention are substantial, based on the actual physical context, and I would say that they have got absolutely nothing to do with gauge-fixing.

Hi Frederick David Tombe,

Regarding your question related to the Lorentz gauge ∇·A + (1/c2) ∂φ/∂t = 0: “what has it got to do with gauge theory?”

The gauge symmetry is:

A‘ = A + ∇ χ

φ’ = φ - ∂χ/∂t

As you know, the magnetic field B is defined as the curl of the Electromagnetic field A. So, the B field would not notice if a new term, being the new term a gradient of an arbitrary scalar function χ (r, t) is added to the definition of B because the curl of a gradient is null:

B’ = curl A’ = curl (A + ∇ χ) = curl A = B,

and

E’ = - ∇ φ’ - ∂A’/∂t = - ∇ (φ - ∂χ/∂t) - ∂(A + ∇ χ)/∂t = - ∇ φ - ∂A/∂t = E

Then, χ is a gauge variable that can be added to the equations without modifying the dynamics of the system: the fields Eand B remain the same and the electromagnetic Hamiltonian is invariant under this change.

But we have added a new variable, so we need one more equation, which can be derived from the Lorentz condition. To fix the gauge consists in solving that equation, obtaining the χ variable.

It can be observed that the Lorentz condition is as kind of conservation law for the electrostatic potential ∂φ/∂t plus an additional term ∇· A which can be seen as the variation respect to space and time (∂, ∇) of the tetravector (φ, A), which is the conserved quantity because it implies the same fields E and B than (φ’, A’). This fact is also related to the concept of symmetry in physics (Noether theorem).

The invariant function under the change generated by including the gauge field χ (with dimensions of a velocity potential) is the electromagnetic energy when the Hamiltonian does not depend explicitly on the time. As a consequence, the dynamics of the EM equations is also invariant.

We can demonstrate now that the Lorentz condition is more restrictive than the gauge symmetry.

Let’s assume that A and φ does not comply the Lorentz condition. Then, we have:

∇·A + (1/c2) ∂φ/∂t = g (r, t),

Where g (r, t) is not null at least in a point of space and in a lapse of time, otherwise the Lorentz condition would apply. Let’s select other potentials φ’ and A’ using the gauge symmetry:

∇· (A’ + ∇ χ) + (1/c2) ∂ (φ’ - ∂χ/∂t) /∂t = g (r, t)

We obtain:

∇·A’ + (1/c2) ∂φ’/∂t + [∇2 χ - (1/c2) ∂2χ/∂t2 - g (r, t)] = 0

If φ’ and A’ comply the Lorentz condition, then χ is the function that complies the following equation:

∇2 χ - (1/c2) ∂2χ/∂t2 = g (r, t);

which is always possible to achieve because the gauge transformations do not require any condition on χ, except to be continuous and derivable. We observe that χ can be deduced from the function g (r, t).

Then, the Lorentz condition for the potentials can be considered a more restrictive condition than the gauge transformations.

Arturo Rodriguez Palenzuela Yes, I can see that gauge fixing is all about adding arbitrary functions of integration. But as regards the Lorenz condition and the Coulomb gauge, they are self explanatory conditions which don't need to involve the concept of gauge fixing.

For example, you proposed a more general condition,

∇·A + (1/c2) ∂φ/∂t = g(r, t)

In what respect would this satisfy the definition of a gauge? Presumably the arbitrary function, g(r, t), vanishes under some differential operator. Which differential operator is that, and why?