Ludwig Valentin Lorenz is the discoverer of the "Lorenz gauge"!

The “ L or enz g auge” is named in honour of Ludwig V alen tin Lorenz! Bozhidar Z. Iliev ∗ † ‡ Short title: The “ L or enz gauge” is named in honour of L.V . Lorenz! Pro duced : → Octob er 22, 2018 http://www. a rX iv.o rg e-Print a r chive No. : a rXiv:0803.0047 [physics.hist-ph] BO / • • HO R  TM Subje c t Classes: Ele ctr o dyna mics, Gauge t he ories 2001 MSC numb ers: 01-01 , 78 -03 78A25, 81S99, 81T13 2003 P A CS numb ers: 1.65.+g, 03.50.De 03.70. +k, 11.15.-q Key-Wor ds: L or enz g auge, L or enz c ondition Ele ctr omagnetic p otentials, Gauge p otentials Gauge c onditions, Gauges, Ele ctr o dynamics, Gauge the ories, Y ang-M il ls the ories ∗ Lab orator y of Mathematical Mo deling in Physics, Institute for Nuclear Research and Nuclear Energy , Bulgarian Academy of Sciences, Boul. Tza rigra dsko chauss ´ ee 72 , 1784 Sofia, Bulgaria † E-mail address: b ozho@inr ne.bas.bg ‡ URL: h ttp://theo.inrne.bas.bg/ ∼ b ozho/ Abstract The letter reminds th e historical fact that the kno wn “Lorenz gauge” (or “Lorenz con- dition/relation”) is first m entioned in a written f orm by an d named after Ludwig V alen tin Lorenz and n ot by/afte r Hendrik Ant o on Loren tz. Bozhidar Z. Iliev: The “Lorenz gauge” is named in honour of L.V. Lorenz ! 1 1. In tro duction Let ϕ and A b e resp ectiv ely the scalar and vect or p oten tials of the classical electromagnetic field [1]. Since th ey con tain some arbitrarin ess, they can b e connected via different relations, called gauges or gauge cond itions/relatio ns. A p articular example of them b eing the “ L or enz gauge”, viz. ∇ · A + 1 c ∂ ϕ ∂ t = 0 , where c is the v elo cit y of light in v acua, ∇ :=  ∂ ∂ x , ∂ ∂ y , ∂ ∂ z  and x , y , z and t are the sp ace and time co ord inates, resp ectiv ely . I n inv arian t form, this gauge reads ∂ µ A µ = 0 with A µ , µ, ν = 0 , 1 , 2 , 3 b eing the comp on ents of the 4-v ector p oten tial and ∂ µ = η µν ∂ ∂ x ν , where η µν are the compon ents of the the s tand ard metric tensor of the Min ko wski spacetime and ( x 0 , x 1 , x 2 , x 3 ) = ( ct, x, y , z ). More generally , if A a µ , a = 1 , . . . , n ∈ N , are the 4-p oten tial of an arbitrary classical gauge field, they can b e sub jected to the Lorenz relation in the form ∂ µ A a µ = 0 . The imp ortance of the Lorenz gauge comes from its relativistic in v ariance, from a sim- plification of man y calculations in it, etc. A partial an alysis of the Lorenz gauge in qu an tum ele ctro d ynamics ca n b e f ou n d in [2, c h. I I, § 12] or [3, ch. 9, § 2] (see also [4, c h. I, § 5]). 2. The historical truth The Lorenz condition/relation and gauge are n amed in honour of the Danish theoretical physic ist Lu dwig V alen tin Lorenz (1829–189 1), who has fi rst p ublished it in 1867 [5, 6] (see also [7, pp. 268-2 69, 291] and [8, 9]). Ho wev er this condition wa s first in tro duced in lectures b y Bernh ard G. W. Riemann in 1861 as p oin ted in [7, p. 291]. It should b e noted that the L or enz cond ition/gauge is quite often erroneously referred as the Loren t z condition/gauge after the name of the Dutc h theoretical p hysicist Hendrik An to on Loren tz (1853– 1928 ) as, e.g., in [10, p. 18], [11, p. 45] and [12, p p. 421-42 2, 426, 542]. The table b elo w repr esen ts some results of searc h ing o v er the Internet for “ L or enz gauge” and “Lorentz gauge.” W e see that the situation is quite sad in fa vour of the wrong term, bu t there is a slight impr ov emen t durin g the last 3 y ears. It is remark able that in the b o ok [13, p. 58, fo otnote 1] is mentioned the Lorenz (gauge) condition, but the author contin ues to call it Loren tz (gauge) condition. The reasons for the error in r eferring to the “ L or enz gauge” as “ Loren t z gauge ” are explained and analyzed, for in stance, in [14, pp . 670–671]. 3. On the geomet ry of the gauge conditions It is known that the gauge p oten tials of a gauge field, in particular of the ele ctromagnetic one, are co efficien ts of a linear connection on a v ector bundle f rom geometrical p oin t of view [11, 15]. If a gauge field is giv en, its p oten tials are fi xed in any frame of reference. F or that reason, the imp osition of some r elations b et wee n them, in particular of the Lorenz gauge, leads to restrictions on the reference frames one can in vok e. T o b e more precise, the class of f rames in the total b undle sp ace that can b e us ed is narro w ed so that th e corresp onding gauge relations to b e satisfied. Bozhidar Z. Iliev: The “Lorenz gauge” is named in honour of L.V. Lorenz ! 2 T able 2.1: Numb er a of found searc h resu lts for “Loren t z gauge” and “Lorenz gauge” W eb d atabase b Date “Loren t z gauge” “Lorenz gauge” Ratio arXiv full record (Physics) Jul 2005 89 17 5.25 arXiv full record (Physics) F eb 2008 85 25 3.40 Go ogle Jul 2005 13700 558 24.55 Go ogle F eb 2008 22600 5500 4.11 Go ogle Sc h olar Jul 2005 2640 216 12.22 Go ogle Sc h olar F eb 2008 5030 499 10.08 arXiv exp. full text (Physics) F eb 2008 2273 345 6.59 arXiv exp. full text (Math.) F eb 2008 133 28 4.75 Y aho o F eb 2008 21700 4460 4.87 A OL F eb 2008 6020 944 6.38 Ask F eb 2008 1600 658 2.43 a The num ber of found results dep ends on may factors and may b e vari able even during one day . b The URL of arXiv is h ttp ://arxiv.org, of Go ogle is http://ww w.google.com , of Google Scholar is http://sc holar.google.com , of Y aho o is http://searc h.yahoo.com , of AOL is http://se arc h.aol.com, and of Ask is http://www .ask.com. 4. App eal instea d of a conclusion The Lorenz gauge is in current u sage and seems to b e in use in future. F or that r eason, let us r ecognize the con tribution of Ludwig V alentin Loren z in this field of physic s and restore the historical tr u th b y terming this gauge/relatio n after him! Ac kno wledgmen ts This w ork w as partially su pp orted b y the National S cience F und of Bulgaria under Gran t No. F 1515/2005 . References [1] L. D. Landau and E. M. Lifshitz. Classic al the ory of fields , volume I I of Course of the or etic al physics . P ergamon Press, Oxford, 5 edition, 1967. T r an s lation fr om Ru ssian, Nauk a, Mosco w, 1973. [2] N. N. Bogolyub o v and D. V. Sh irk o v. Intr o duction to the the ory of quantize d field s . Nauk a, Mosco w, third edition, 1976. In Russian. English translation: Wiley , New Y ork, 1980. [3] Silv an S. Sc hw eb er . An intr o duction to r elativistic quantum field the ory . Ro w, P eter- son and Co., Ev anston, Ill., Elmsford, N.Y ., 1 961. Russian translation: IL (F oreign Literature Pub.), Mosco w, 1963. [4] A. I. Akhiezer and V. B. Berestet skii. Quantum ele ctr o dynamics . Nauk a, Mosco w, 1969. In Ru ssian. English translation: Auth orized En glish ed ., rev. and enl. b y the Bozhidar Z. Iliev: The “Lorenz gauge” is named in honour of L.V. Lorenz ! 3 author, T ranslated f rom the 2d Ru ssian ed. by G.M. V olk off, New Y ork, Interscience Publishers, 1965. Other English translat ions: New Y ork, Consu ltan ts Bureau, 1957; London, Oldb our ne Press, 1964, 1962. [5] Ludw ig V alen tin Lorenz. ¨ Ub er d ie In tensit¨ at der Sch wingungen des Lic h ts mit den elektrisc hen Str¨ omen. Annalen der Physik und Chemie , 131:243–2 63, 1867. [6] Ludw ig V alen tin Lorenz. O n the identit y of the vib rations of ligh t with electrical cur - ren ts. Philosophic al Magazine , 34:287–3 01, 1867. [7] Edmund Whittak er. A history of the the ories of aeth er and ele ctricity , v olume 1. The classical theories. of Harp er tor chb o oks / The sc i enc e lib r ary . Ha rp er & brothers, New Y ork, 1960. Originally publish ed by Thomas Nelson & Son Ltd, Lond on, 1910; revised and enlarged 1951 . See also the 1989 edition: New Y ork: Do ve r, 1989. [8] J. v an Bladel. Lorenz or Lorent z? Antennas and Pr op agation Magazine, IEE E , 33:69, 1991. [9] R. Nevel s and Chang-Seok Sh in. Lorenz, Loren tz, and th e gauge. A ntennas and Pr op- agation M agazine, IEEE , 43(3):70– 71, 2001. [10] P aul Roman. Intr o duction to quantum field the ory . J ohn Wiley & Sons, Inc., New Y ork-Lond on-Sydney-T oron to, 1969. [11] M. G¨ oc k eler and T. Sc h ¨ uc ker. Differ ential ge ometry, gauge the ories, and gr avity . C am- bridge Univ. Pr ess, Cam b ridge, 1987. [12] Da vid J. Griffiths . Intr o duction to ele ctr o dynamics . Englew oo d Cliffs, NJ: Prentic e-Hall, 3 edition, 1998. [13] Bo Thid ´ e. Ele ctr omagnetic field the ory . Upsilo n Books, Up p sala, S w eden, 2001. [14] J. D. Jac kson and L. B. Oku n. Historical ro ots of gauge in v ariance. R eviews of mo d- ern physics , 73:663–680 , 2001. S ee also : http:// arXiv.org e-Prin t arc hive , E-print No. hep-ph/0012061, Decem b er 2000. [15] N. P . Konoplev a and V. N. Popov. Gauge field s . Hardw o o d Academic Pu blishers, Ch ur-London-New Y ork, second edition, 1981. T ranslation from Ru ssian: A tomizdat, Mosco w, 1972 (1 ed.), 1980 (2 ed.).