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Edit: Use this PO.org question instead.

Warning: Students, stay away from antiquities. The aim to learn is to survive.

Hi. Today the nomenclatures Feynman gauge and Landau gauge seem established, but could you explain the history? It's two-fold: 1. Who used first such gauges and how? 2. How did the terminology gain ground?

It also inevitably asks some history of the Lorenz gauges in QFT. In particular, when was it realized that the freedom (and need) of the gauge parameter ξ lies in quantum theory?

First I apologize that I have only limited access to literature on asking such questions. My preliminary studies revealed below.

1 Terminology in literature

(Schweber 1961) is not gauge-parameter conscious, and "Lorentz[sic] gauge" (in Feynman gauge) is opposed to coulomb. But it also refers to today's Yennie gauge in sec 15g, saying (Fried and Yennie 1958) found it is possible to take gauge where the photon propagator ∝ $g + 2pp/p^2$.

In (Nakanishi 1966), the word "Landau gauge" is seen, and it cites several near-past articles regarding Landau gauge quantization. (It is an important paper in canonical quantization in Landau gauge, together with (Lautrup 1967). Nakanishi was a strong proponent of the Landau gauge.)

In p. 74 of (Nakanashi 1972), it reads "Feynman gauge or Fermi gauge" and "Landau-Khalatonikov[sic] gauge, or simply Landau gauge". (Landau and Khalatnikov, 1955) is listed in the bibliography section, but I couldn't find which part of Nakanishi actually cites it. (Nakanishi 1972) is a review article, one of whose main topics is canonical quantization of EM field in arbitrary Lorenz gauge, i.e., for any gauge parameter.

In p 134 of (Itzykoson & Zuber, 1980), the words "Feynman gauge" and "Landau gauge" are used. Were the names settled at that time?

Hmm, in p 389 of (Siegel 1999), "Fermi-Feynman gauge" is introduced. (Srednicki 2007) uses the word "$R_\xi$" for QED, remarking "[it] has been historically used only in the context of spontaneous symmetry broken [...] but we will use it here as well."

2 Gauge parameter symbol

ξ is now usual. Is it due to (Fujikawa, Lee and Sanda 1972)?

For other symbols, I mention $\alpha$. (Nakanishi 1972) uses it, and even after (Fujikawa, Lee and Sanda 1972) it is sometimes used, for example and in (Siegel 1999).

3 Theory Timeline

1930 - Fermi: P. 240 of (Schweber 1961) says Fermi proposed to add $-\frac{1}{2} (\partial A)^2$ to the Langrangian. (Fermi was the first to introduce a subsidiary condition, but it was not perfect. See also Gupta and Bleuler below.) Although I haven't checked Fermi's papers, it may be better to call "Fermi-Feynman(-'t Hooft) gauge."

1948 - Feynman: Feynman simply justifies the use of Feynman gauge in the section 8 of (Feynman 1949). Before Feynman, it wasn't Lorentz covariant, and transverse photons were separated. Feynman says it's not necessary, and it's ok to do $\gamma^\mu$...$\gamma_\mu$.

1950 - Gupta & Bleuler: They say Gupta and Bleuler succeeds in covariant canonical quantization in Feynman gauge, by discovering the correct subsidiary condition.

1956 - Landau & Khalatnikov: See (Nakanishi 1972) above.

1958 - Yennie gauge: It is said (Fried and Yennie 1958) uses the "Yennie gauge" of today, $\xi = 3$, in bound state problems.

Early or mid 60's - Rise of interest in Landau gauge? See (Nakanishi 1966) above.

1966 - 67 Nakanishi & Lautrup: canonical quantization of EM field for any ξ.

1967 - Faddeev & Popov

1971 - 't Hooft: In 1971, 't Hooft used "Feynman-'t Hooft gauge" or simply "'t Hooft gauge" for broken gauge symmetry. (Fujikawa, Lee and Sanda 1972) generalized to any ξ. Its abstract uses the word "Feynman-'t Hooft gauge". (According to Weinberg. Haven't read both two.)

1972 - Still canonical quantization for arbitrary ξ is of interest, including massive vector field. See (Nakanishi 1972).

4 Loren't'z gauge (misspelling)

You may know that in the 20th century, the common spelling was "Lorentz gauge", with extra "t". I couldn't find any exceptions at my hand. The turning point might be the errata of Peskin & Schroeder. Srednicki and Siegel spell it correctly.

5 Bibliography

  • Fermi, E., Atti. Acad. Lincei. 9 (1929) 881, Atti. Acad. Lincei. 12 (1939) 431, Rev. Mod. Phys. 4 (1932) 87.
  • Feynman, R., Phys. Rev. 76 (1949) 769.
  • Fried H.M., Yennie, D.R., Phys. Rev. 112 (1958) 1391.
  • Fujikawa, Lee and Sanda, PRD 6 (1972) 2923
  • Itzykson & Zuber "Quantum field theory", 1980.
  • Landau, L. D., Khalatnikov, I. M., J. Exper. Theor. Phys. USSR 29 (1955), 89 [English translation: Sov. Phys. J. E. T. P. 2 (1956), 69].
  • Lautrup, B., Mat. Fys. Medel. Dan. Vid. Selsk. 35 (1967), No. 11.
  • Nakanishi, N., Prog. Theor. Phys. 35 (1966) 1111 (Downloadable gratis)
  • Nakanishi, N., Prog. Theor. Phys. Supple. 51 (1972) 1 (Downloadable gratis)
  • Schweber, "An introduction to relativistic quantum field theory", 1961.
  • Siegel, W., "Fields", arXiv:hep-th/9912205
  • Srednicki, (2007) Quantum field theory

6 Revisions of this question

26 Jun: Added the post 't Hooft era and the symbol ξ.

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The terms "Landau gauge" and "Feynman gauge" (among others) were introduced by Bruno Zumino. I accidentally learned about it an hour ago from David Derbes

http://motls.blogspot.com/2014/06/bruno-zumino-1923-2014.html?m=1

in this blog post about a sad event, Bruno Zumino's death a week ago. David Derbes wrote:

I met Bruno Zumino at the Scottish Universities Summer School in Physics at St. Andrews in 1976. A very jolly man. Supersymmetry and supergravity were just getting going.

By coincidence I was just reading a nice history of the first days of gauge invariance by J. D. Jackson and L. B. Okun that appeared in Rev. Mod. Phys. 73 (2001) 663 (arXiv:hep-ph/0012061). They cite Zumino's fine paper in J. Math. Phys. 1 (1960) 1, and in their paper say something that should be more widely known, imo:

"Various gauges have been associated with names of physicists, a process begun by Heitler, who introduced the term 'Lorentz relation' in the first edition of his book. In the third edition, [Heitler] used 'Lorentz gauge' and 'Coulomb gauge'. Zumino (1960) introduced the terms 'Feynman gauge', 'Landau gauge' and 'Yennie gauge'."

http://scitation.aip.org/content/aip/journal/jmp/1/1/10.1063/1.1703632 (Abstract)

Zumino's original model with Wess involved many more fields; but in "Zumino gauge" it was restricted to the few fields Lubos describes. Only fair that Bruno Zumino got his own gauge.

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  • $\begingroup$ Thanks to you and David Derbes for informing. Jackson & Okun (Wow.) is available at arXiv, and I can come by Zumino's JMP, too. Maybe I can report a bit more somewhere. As for the timing of this question, what a coincidence. We're all fine-tuned. May RIP. $\endgroup$
    – teika kazura
    Jul 6, 2014 at 2:31
  • $\begingroup$ What I'd like to know is, where I can find a reference where the Lorenz gauge quantisation of the electromagnetic field is realised in the fashion of Gupta and Bleuler, but without any particular choice for the Lagrange multiplier $\xi$. Many references, such as Itzykson & Zuber, affirm that "the results are independent of $\xi$". If by that they mean that the Faraday tensor $F^{\mu\nu}$ is, then we agree. I'd just like to see to what extent this modifies the Gupta-Bleuler procedure. I think I could do it myself but if this can be found somewhere that might save me the time. $\endgroup$
    – Vinsanity
    Oct 7, 2014 at 15:20
  • $\begingroup$ @Vinsanity - I find your question confusing because you seem to combine several things. First, the Lorenz gauge is a strict notion, either classical gauge or the "xi is infinity" limit of the R_xi gauges. Those are used for calculations of complicated diagrams with photon propagators and the physical results may be seen to be xi-independent but it's not "quite trivial" although the result may be justified more conceptually. However, the Gupta-Bleuler quantization is a treatment of the external photons which includes the unphysical polarizations and then says how to decouple those. $\endgroup$
    – Luboš Motl
    Oct 7, 2014 at 15:53
  • $\begingroup$ @Luboš I'm only concerned about the free EM field here (not to say interacting fields don't interest me, but it's not the topic of my question) for which all photons are external. The solution to the Maxwell equations for the vector potential depends on the value of $\xi$, and thus, as far as I can guess, so does the implementation of the Gupta-Bleuler procedure. Would that be correct? $\endgroup$
    – Vinsanity
    Oct 7, 2014 at 20:01
  • $\begingroup$ If you only consider the free field, then the equations for all polarizations are just "box A mu is zero" regardless of xi. The indefinite Hilbert space is the same for each xi, too. The natural normalizatoin of some polarizations etc. may depend on xi but there's no way to choose "natural" if you don't want to consider propagators and interactions. So I don't know what you mean by the xi-dependence of the GB procedure of anything else. The answer to the procedure is just the space of states and it's the same for each xi. $\endgroup$
    – Luboš Motl
    Oct 8, 2014 at 7:57
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Abraham Pais also gives the same hint on Fermi's proposal: A. Pais, ''Inward Bound: Of Matter and Forces in the Physical World'', Oxford University Press, (1986); pag. 354. I read Fermi's paper, but he never start with a Lagrangian. I think that he had it in mind, because he write an Hamiltonian WITH ALL CONIUGATED MOMENTA, and this is equivalent to start with a Lagrangian where the gauge fixing has been added.

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