Historical Roots of Gauge Invariance

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Historical roots of gauge invariance

J. D. Jackson*

University of California and Lawrence Berkeley National Laboratory, Berkeley,

California 94720

L. B. Okun†

Institute of Theoretical and Experimental Physics, State Science Center of Russian Federation, 117218 Moscow, Russia

(Published 14 September 2001)

Gauge invariance is the basis of the modern theory of electroweak and strong interactions (the

so-called standard model). A number of authors have discussed the ideas and history of quantum

guage theories, beginning with the 1920s, but the roots of gauge invariance go back to the year 1820when electromagnetism was discovered and the first electrodynamic theory was proposed. We

describe the 19th century developments that led to the discovery that different forms of the vectorpotential (differing by the gradient of a scalar function) are physically equivalent, if accompanied by

a change in the scalar potential: A→AA , → /c t . L. V. Lorenz proposed the

condition A0 in the mid-1860s, but this constraint is generally misattributed to the better known

H. A. Lorentz. In the work in 1926 on the relativistic wave equation for a charged spinless particle in

an electromagnetic field by Schro ¨ dinger, Klein, and Fock, it was Fock who discovered the invariance

of the equation under the above changes in A and if the wave function was transformed accordingto → exp(ie /c). In 1929, H. Weyl proclaimed this invariance as a general principle and called

it Eichinvarianz in German and gauge invariance in English. The present era of non-Abelian gauge

theories started in 1954 with the paper by Yang and Mills on isospin gauge invariance.

CONTENTS

I. Introduction 663

II. Classical Era 665

A. Early history—Ampe ` re, Faraday, Neumann,

Weber 665

B. Vector potentials—Gauss, Kirchhoff, and

Helmholtz 668

C. Electrodynamics—Maxwell, Lorenz, and Hertz 669D. Charged-particle dynamics—Clausius, Heaviside,

and Lorentz (1892) 671

E. Lorentz: The acknowledged authority, general

gauge freedom 672

III. Dawning of the Quantum Era 673

A. 1926—Schro ¨ dinger, Klein, Fock 673

B. Weyl: Gauge invariance as a basic principle 675

IV. Physical Meaning of Gauge Invariance and

Examples of Gauges 676

A. On the physical meaning of gauge invariance in

QED and quantum mechanics 676

B. Examples of gauges 676

V. Summary and Concluding Remarks 677

Acknowledgments 677

References 677

I. INTRODUCTION

The principle of gauge invariance plays a key role inthe standard model, which describes electroweak andstrong interactions of elementary particles. Its originscan be traced to Fock (1926b), who extended the knownfreedom of choosing the electromagnetic potentials in

classical electrodynamics to the quantum mechanics of charged particles interacting with electromagnetic fields.Equations (5) and (9) of Fock’s paper are, in his nota-tion,

AA1 f ,

11

c

f

t

, Fock’s (5)

p p1e

cf ,

and

0e2 ip/h. Fock’s (9)

In present-day notation we write

A→AA , (1a)

→1

c

t , (1b)

→ exp ie /c . (1c)

Here A is the vector potential, is the scalar potential,and is known as the gauge function. The Maxwellequations of classical electromagnetism for the electricand magnetic fields are invariant under the transforma-tions (1a) and (1b) of the potentials. What Fock discov-ered was that, for the quantum dynamics, that is, theform of the quantum equation, to remain unchanged bythese transformations, the wave function must undergothe transformation (1c), whereby it is multiplied by alocal (space-time-dependent) phase. The concept wasdeclared a general principle and ‘‘consecrated’’ by Her-mann Weyl (1928, 1929a, 1929b). The invariance of a

*Electronic address: [email protected]†Electronic address: [email protected]

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theory under combined transformations such as Eqs.(1a), (1b), and (1c) is known as a gauge invariance or a

gauge symmetry and is a cornerstone in the creation of modern gauge theories.

The gauge symmetry of quantum electrodynamics(QED) is an Abelian one, described by the U(1) group.The first attempt to apply a non-Abelian gauge symme-try, SU(2)SU(1), to electromagnetic and weak interac-

tions was made by Oscar Klein (1938). But this pro-phetic paper was forgotten by the physics communityand was never cited by the author himself.

The proliferation of gauge theories in the second half of the 20th century began with the 1954 paper on non-Abelian gauge symmetries by Chen-Ning Yang andRobert L. Mills (1954). The creation of a non-Abelianelectroweak theory by Glashow, Salam, and Weinberg inthe 1960s was an important step forward, as were thetechnical developments by ’t Hooft and Veltman con-cerning dimensional regularization and renormalization.The discoveries at CERN of the heavy W and Z bosonsin 1983 established the essential correctness of the elec-

troweak theory. The very extensive and detailed mea-surements in high-energy electron-positron collisions atCERN and SLAC, in proton-antiproton collisions atFermilab, and in other experiments have brilliantly veri-fied the electroweak theory and determined its param-eters with precision.

In the 1970s a non-Abelian gauge theory of stronginteraction of quarks and gluons was created. One of itscreators, Murray Gell-Mann, gave it the name quantum

chromodynamics (QCD). This theory is based on theSU(3) group, as each quark of a given ‘‘flavor’’ (u, d, s, c,

b, t ) exists in three varieties or different ‘‘colors’’ (red,yellow, blue). The quark colors are analogs of electriccharge in electrodynamics. Eight colored gluons are ana-logs of the photon. Colored quarks and gluons are con-fined within numerous colorless hadrons.

Quantum chromodynamics and electroweak theoryform what is called today the standard model , which isthe basis of all of physics except for gravity. All experi-mental attempts to disprove the standard model have sofar failed. But one of the cornerstones of the standardmodel still awaits its experimental test. The search forthe so-called Higgs boson (or simply higgs) or its equiva-lent is of profound importance in particle physics today.In the standard model, this electrically neutral, spinlessparticle is intimately connected with the mechanism by

which quarks, leptons, and W and Z bosons acquire theirmasses. The mass of the higgs itself is not restricted bythe standard model, but general theoretical argumentsimply that the physics will be different from what is ex-pected if its mass is greater than 1 TeV/c 2. Indirect in-dications from LEP experimental data imply a muchlower mass, perhaps 100 GeV/c2, but so far there is nodirect evidence for the higgs. Discovery and study of thehiggs was a top priority for the aborted SuperconductingSuper Collider (SSC). Now it is a top priority for theLarge Hadron Collider (LHC) under construction atCERN.

The key role of gauge invariance in modern physicsmakes it desirable to trace its historical roots back to thebeginning of 19th century. This is the main goal of ourreview. In Sec. II, the central part of our article, we de-scribe the history of classical electrodynamics with a spe-cial emphasis on the freedom of choice of potentials Aand expressed in Eqs. (1a) and (1b).

It took almost a century to formulate this nonunique-ness of potentials that exists despite the uniqueness of

the electromagnetic fields. The electrostatic potential re-sulting from a distribution of charges was intimately as-sociated with the electrostatic potential energy of thosecharges and had only the trivial arbitrariness of the ad-dition of a constant. The invention of Leyden jars andthe development of voltaic piles led to the study of theflow of electricity, and in 1820 magnetism and electricitywere brought together by Oersted’s discovery of the in-fluence of a nearby current flow on a magnetic needle(see Jelved, Jackson, and Knudsen, 1998). Ampe ` re andothers rapidly explored the new phenomenon, general-ized it to current-current interactions, and developed amathematical description of the forces between closedcircuits carrying steady currents (Ampe ` re, 1827). In 1831

Faraday made the discovery that a time-varying mag-netic flux through a circuit induces current flow (Fara-day, 1839). Electricity and magnetism were truly united.

The lack of uniqueness of the scalar and vector poten-tials arose initially in the desire of Ampe ` re and others toreduce the description of the forces between actual cur-rent loops to differential expressions giving the elementof force between infinitesimal current elements, one ineach loop. Upon integration over the current flow ineach loop, one could obtain the total force. Ampe ` re be-lieved the element of force was central, that is, actingalong the line joining the two elements of current, butothers wrote down different expressions leading to thesame integrated result. In the 1840s the work of Neu-mann (1847, 1849) and Weber (1878, 1848) led to com-peting differential expressions for the elemental force,Faraday’s induction, and the energy between current el-ements, expressed in terms of different forms for thevector potential A. Over 20 years later Helmholtz (1870)ended the controversy by showing that Neumann’s andWeber’s forms for A were physically equivalent. Helm-holtz’s linear combination of the two forms with an ar-bitrary coefficient is the first example of what we nowcall a restricted class of different gauges for the vectorpotential.

The beginning of the last third of the 19th century sawMaxwell’s masterly creation of the correct complete set

of equations governing electromagnetism, unfortunatelyexpressed in a way that many found difficult to under-stand (Maxwell, 1865). Immediately after, the Danishphysicist Ludvig V. Lorenz, apparently independently of Maxwell, brilliantly developed the same basic equationsand conclusions about the kinship of light and the elec-tromagnetism of charges and currents (Lorenz, 1867b).With respect to gauge invariance, Lorenz’s contributionsare most significant. He introduced the so-called re-tarded scalar and vector potentials and showed that theysatisfied the relation almost universally known as ‘‘theLorentz condition,’’ though he preceded the Dutchphysicist H. A. Lorentz by more than 25 years.

664 J. D. Jackson and L. B. Okun: Historical roots of gauge invariance

Rev. Mod. Phys., Vol. 73, No. 3, July 2001



By the turn of the century, thanks to, among others,Clausius, Heaviside, Hertz, and Lorentz, who inventedwhat we now call microscopic electromagnetism with lo-calized charges in motion forming currents, the formalstructure of electromagnetic theory, the role of the po-tentials, the interaction with charged particles, and theconcept of gauge transformations (not yet known bythat name) were in place. Lorentz’s encyclopedia articles

(Lorentz, 1904a, 1940b) and his book (Lorentz, 1909)established him as an authority in classical electrody-namics, to the exclusion of earlier contributors such asLorenz.

The start of the 20th century saw the beginning of quantum theory, of special relativity, and of radioactivetransformations. In the 1910s attention turned increas-ingly to atomic phenomena, with the confrontation be-tween Bohr’s early quantum theory and experiment. Bythe 1920s the inadequacies of the Bohr theory were ap-parent. During 1925 and 1926, Heisenberg, Schro ¨ dinger,Born, and others invented quantum mechanics. Inevita-bly, when the interaction of charged particles with time-

varying electromagnetic fields came to be considered inquantum mechanics, the issue of the arbitrariness of thepotentials would arise. What was not anticipated washow the consequences of a change in the electromag-netic potentials on the quantum-mechanical wave func-tion would become transformed into a general principlethat defines what we now call quantum gauge fields.

In Sec. III we review the extension of the concept of gauge invariance in the early quantum era, the periodfrom the end of World War I to 1930, with emphasis onthe annus mirabilis, 1926. As in Sec. II, we explore howand why priorities for certain concepts were taken fromthe originators and bestowed on others. We also retellthe well-known story of the origin of the term ‘‘gaugetransformation.’’ In Sec. IV we discuss briefly the physi-cal meaning of gauge invariance and describe theplethora of different gauges in occasional use today, butleave the detailed description of subsequent develop-ments to others.

In writing this article we relied mainly on original ar-ticles and books, but in some instances we used second-ary sources (historical reviews and monographs).Among the many sources on the history of electromag-netism in the 19th century, we mention Reiff and Som-merfeld (1902), Whittaker (1951), Rosenfeld (1957),Buchwald (1985, 1989, 1994), Roche (1990), Hunt(1991), and Darrigol (2000). Volume 1 of Whittaker

(1951) surveys all of classical electricity and magnetism.Reiff and Sommerfeld (1902) provide an early review of some facets of the subject from Coulomb to Clausius.Roche’s article (Roche, 1990) treats the history of thevector potential and its changing significance fromGauss’s early work (1835, but unpublished until 1867) tothe present day. As his title implies, Hunt (1991) focuseson the British developments from Maxwell to 1900—inparticular, the works of George F. FitzGerald, OliverLodge, Oliver Heaviside, and Joseph Larmor—as Max-well’s theory evolved into the differential equations forthe fields that we know today. Darrigol (2000) covers the

development of theoretical and experimental electro-magnetism in the 19th century, with emphasis on thecontrast between Britain and the Continent in the inter-pretations of Maxwell’s theory. Buchwald (1985) de-scribes in detail the transition in the last quarter of the19th century from the macroscopic electromagnetictheory of Maxwell to the microscopic theory of Lorentzand others. Buchwald (1989) treats early theory and ex-periment in optics during the first part of the 19th cen-

tury. Buchwald (1994) focuses on the experimental andtheoretical work of Heinrich Hertz as he moved fromHelmholtz’s pupil to independent authority with a differ-ent world view. Rosenfeld’s essay (1957) focuses on themathematical and philosophical development of electro-dynamics from Weber to Hertz, with special emphasison Lorenz and Maxwell. His comments on Lorenz’s‘‘modern’’ outlook are very similar to ours. None of these works stress the development of the idea of gaugeinvariance.

The early history of quantum gauge theories as well asmore recent developments have been extensively docu-mented (Okun, 1986; Yang, 1986, 1987; O’Raifeartaigh,1997; O’Raifeartaigh and Straumann, 2000).

Ludvig Valentin Lorenz, of the classical era, andVladimir Aleksandrovich Fock emerge as physicistsgiven less than their due by history. The many accom-plishments of Lorenz in electromagnetism and optics aresummarized by Kragh (1991, 1992) and more generallyby Pihl (1939, 1972). Fock’s pioneering researches weredescribed recently, on the occasion of the 100th anniver-sary of his birth (Novozhilov and Novozhilov, 1999,2000; Prokhorov, 2000).

The word ‘‘gauge’’ was not used in English for trans-formations such as Eqs. (1a), (1b), and (1c) until 1929(Weyl, 1929a). It is convenient, nevertheless, to use themodern terminology even when discussing the works of

19th-century physicists. Similarly, we usually write equa-tions in a consistent modern notation, using Gaussianunits for electromagnetic quantities.

II. CLASSICAL ERA

A. Early history—Ampe ` re, Faraday, Neumann, Weber

On 21 July 1820 Oersted announced to the world hisamazing discovery that magnetic needles were deflectedif an electric current flowed in a circuit nearby, the firstevidence that electricity and magnetism were related(see Jelved, Jackson, and Knudsen, 1998). Within weeksof the news, experimenters everywhere were exploring,

extending, and making quantitative Oersted’s observa-tions, nowhere more than in France. In the fall of 1820,Biot and Savart studied the force of a current-carryinglong straight wire on magnetic poles and announcedtheir famous law—that, for a given current and polestrength, the force on a pole was perpendicular to thewire and to the radius vector, and fell off inversely as theperpendicular distance from the wire (Biot and Savart,1820a, 1820b). On the basis of a calculation of Laplacefor the straight wire and another experiment with aV-shaped wire, Biot abstracted the conclusion that theforce on a pole exerted by an increment of the current of

665J. D. Jackson and L. B. Okun: Historical roots of gauge invariance




length ds was (a) proportional to the product of the polestrength, the current, the length of the segment, thesquare of the inverse distance r between the segment

and the pole, and the sine of the angle between the di-rection of the segment and the line joining the segmentto the pole, and (b) directed perpendicular to the planecontaining those lines (Biot, 1824). We recognize this asthe standard expression for an increment of magneticfield dB times a pole strength—see, for example, Eq.(5.4) of Jackson (1998, p. 175).

At the same time, Ampe ` re, in a brilliant series of demonstrations before the French Academy, showed,among other things, that small solenoids carrying cur-rent behaved in the Earth’s magnetic field as did barmagnets, and began his extensive quantitative observa-tions of the forces between closed circuits carrying

steady currents. These continued over several years; thepapers were later collected in a memoir (Ampe ` re, 1827).

The different forms for the vector potential in classi-cal electromagnetism arose from the competing versionsof the elemental force between current elements ab-stracted from Ampe ` re’s extensive observations. Thesedifferent versions come from the possibility of addingperfect differentials to the elemental force, expressionsthat integrate to zero around closed circuits or circuitsextending to infinity. Consider the two closed circuits C and C carrying currents I and I , respectively, as shownin Fig. 1. Ampe ` re believed that the force increment dFbetween differential directed current segments Id s and

I

ds

was a central force, that is, directed along the linebetween the segments. He wrote his elemental force lawin compact form (Ampe ` re, 1827, p. 302),

dF 4k II

r

2r

s sdsds

k II

r 2 2r

2r

s s

r

s

r

sdsds, (2)

where the constant k1/c 2 in Gaussian units. The dis-tance r is the magnitude of rxx, where x and x arethe coordinates of dsnds and dsnds . In what fol-

lows we also use the unit vector r r/r . In vector nota-tion and Gaussian units, Ampe ` re’s force reads

dF II

c 2

r

r 23 r •n r •n2n•ndsds. (3)

It is interesting to note that, in the same work, Ampe ` rehas the equivalent of this expression at the bottom of p.253 in terms of the cosines defined by the scalar prod-

ucts. He preferred, however, to suppress the cosines andexpress his result in terms of the derivatives of r withrespect to ds and ds , as in Eq. (2).

The first observation we can make is that the ab-stracted increment of force dF has no physical meaningbecause it violates the continuity of charge and current.Currents cannot suddenly materialize, flow along the el-ements ds and ds, and then disappear again. The ex-pression is only an intermediate mathematical construct,perhaps useful, perhaps not, in finding actual forces be-tween real circuits. The second observation is that theform widely used at present [see Jackson, 1998, p. 177,Eq. (5.8), for the integrated expression],

dF II

c2r 2nn r dsds

II

c2r 2n r •n r n•ndsds, (4)

was first written down independently in 1845 by Neu-mann [1847, p. 64, Eq. (2)] and Grassmann (1845).1 Al-though not how these authors arrived at it, one way tounderstand its form is to recall that a charge q in non-relativistic motion with velocity v (think of a quasifreeelectron moving through the stationary positive ions in aconductor) generates a magnetic field (Bqvr/r 3).

Through the Lorentz force law F

q(E

v

B

/c), thisfield produces a force on another charge q moving withvelocity v in a second conductor. Now replace qv andqv with I nds and I nds. With its noncentral contri-bution, this equation does not agree with Ampe ` re’s, butthe meaningful issue is that the differences vanish forthe total force between two closed circuits. In fact, thefirst term in Eq. (4) contains a perfect differential,n• r /r 2dsds•“ (1/r ), which gives a zero contributionwhen integrated over the closed path C in Fig. 1. If weignore this part of dF, the residue appears as a centralforce (!) between elements, although not the same asAmpe ` re’s central force.

Faraday’s discovery in 1831 of electromagneticinduction—relative motion of a magnet near a closedcircuit induces a momentary flow of current—exposedthe direct link between electric and magnetic fields (Far-aday, 1839). The experimental basis of quasistatic elec-tromagnetism was now established, although the differ-

1A cogent discussion of Ampe ` re’s work, his running disputewith Biot, and Grassmann’s criticisms and alternative expres-sion for the force is given by Tricker (1965). Tricker also givestranslations of portions of the papers by Oersted, Ampe `re,Biot and Savart, and Grassmann.

FIG. 1. Two closed current-carrying circuits C and C withcurrents I and I , respectively.







AW x,t 1

c d3 x

1

r r ˆ r ˆ •Jx, t . (12)

As with Neumann, we attach a subscript W for Weber tothis form of the vector potential, even though he did notwrite Eqs. (11) or (12) explicitly.

B. Vector potentials—Gauss, Kirchhoff, and Helmholtz

In 1835, pondering the work of Ampe ` re and Faraday,Carl Friedrich Gauss discovered the vector potential andits connection to the induced electromotive force (inmodern notation, EdA/dt ), as well as its connectionto the magnetic field ( B“A) (Gauss, 1867; see alsoRoche, 1990). Unfortunately, his brief handwritten notescontaining the equivalent of these two equations re-mained unpublished until 1867 and so had little impacton the development of electrodynamics.

Gustav Kirchhoff was the first to write explicitly (incomponent form) the vector potential (12); he alsowrote the components of the induced current density asthe conductivity times the negative sum of the gradientof the scalar potential and the time derivative of thevector potential (Kirchhoff, 1857, p. 530). He attributedthe second term in the sum to Weber; the expression(12) became known as the Kirchhoff-Weber form of thevector potential. Kirchhoff applied his formalism to ananalysis of the telegraph and calculation of inductances.

We note in passing that Kirchhoff showed (contrary towhat is implied by Rosenfeld, 1957) that the Weber formof A and the associated scalar potential satisfy the

relation (in modern notation)“•

A /c t , the firstpublished relation between potentials in what we now

know as a particular gauge (Kirchhoff, 1857, pp. 532 and533).

In an impressive, if repetitive, series of papers, Her-mann von Helmholtz (1870, 1872, 1873, 1874) criticizedand clarified the earlier work of Neumann, Weber, andothers. He criticized Weber’s force equation for leadingto unphysical behavior of charged bodies in some cir-cumstances, but recognized that Weber’s form of themagnetic energy had validity. Helmholtz compared theNeumann and Weber forms of the magnetic energy be-tween current elements, dW pII dsds/c 2r , with

p(Neumann)n•n and p(Weber)n• r n• r and notedthat they differ by a multiple of the perfect differentialdsds( 2r / s s)dsds(n• r n• r n•n)/r . Thus ei-ther form leads to the same potential energy and forcefor closed circuits. Helmholtz then generalized Weber’sand Neumann’s expressions for the magnetic energy be-tween current elements by writing a linear combina-tion [Helmholtz, 1870, Eq. (1), p. 76, but in modernnotation],

dW II

2c2r 1 n•n1 n• r n• r dsds.

(13)

Obviously, this linear combination differs from both We-ber’s and Neumann’s expression by a multiple of theabove perfect differential and so is consistent with Am-pe ` re’s observations. The equivalent linear combinationof the vector potentials (7) and (12) is [ibid., Eq. (1a), p.76, in compressed notation]

A

1

21 AN

1

21 AW . (14)

Here 1 gives the Neumann form; 1 gives theWeber form. Helmholtz’s generalization exhibits a one-parameter class of potentials that is equivalent to a fam-ily of vector potentials of different gauges in Maxwell’selectrodynamics. In fact, in Eq. (1d) on p. 77, he writesthe connection between his generalization and the Neu-mann form (7) as (in modern notation)

A AN 1

2 ,

where

1

c r •Jx,t d3 x. (15)

Helmholtz goes on to show that satisfies 22 /c t , where is the instantaneous electrostaticpotential, and that (x,t ) and his vector potentialA (x,t ) are related by [ibid., Eq. (3a), p. 80, in modernnotation]

•A

c t . (16)

This relation contains the connection found in 1857 byKirchhoff (for 1) and formally the condition found

in 1867 by Lorenz (see below), but Helmholtz’s relationconnects only the quasistatic potentials, while Lorenz’srelation holds for the fully retarded potentials. Helm-holtz is close to establishing the gauge invariance of electromagnetism, but treats only a restricted class of gauges and lacks the transformation of the scalar as wellas the vector potential.

Helmholtz remarks rather imprecisely that the choiceof 0 leads to Maxwell’s theory. The resulting vectorpotential,

AM x,t 1

2c Jx,t

r

r r ˆ •Jx,t

r d3 x, (17)

can be identified with Maxwell only because, as Eq. (16)shows, it is the quasistatic vector potential found fromthe transverse current for “•A0, Maxwell’s preferredchoice for A. Maxwell never wrote down Eq. (17). It isrelevant for finding an approximate Lagrangian for theinteraction of charged particles, correct to order 1/c 2

(see Sec. II.D).We see in this early history the attempts to extend

Ampe ` re’s conclusions on the forces between current-carrying circuits to a comprehensive description of theinteraction of currents largely within the framework of potential energy, in analogy with electrostatics. Compet-





ing descriptions stemmed from the arbitrariness associ-ated with the postulated elemental interactions betweencurrent elements, an arbitrariness that vanished upon in-tegration over closed circuits. These differences led todifferent but equivalent forms for the vector potential.The focus was on steady-state current flow or quasistaticbehavior. Meanwhile, others were addressing the propa-gation of light and its possible connection with electric-ity, electric currents, and magnetism. That electricity wasdue to discrete charges and electric currents to discretecharges in motion was a minority view, with Weber anotable advocate. Gradually those ideas gained cre-dence and charged-particle dynamics came under study.Our story now turns to these developments and to howthe concept of different gauges was elaborated, and bywhom.

C. Electrodynamics—Maxwell, Lorenz, and Hertz

The vector potential played an important role in Max-well’s emerging formulation of electromagnetic theory

(Bork, 1967; see also Everitt, 1975). Maxwell developedan analytic description of Faraday’s intuitive idea that aconducting circuit in a magnetic field was in an ‘‘electro-tonic state,’’ ready to respond with current flow if themagnetic flux linking it changed in time (Maxwell, 1856).He introduced a vector, ‘‘electro-tonic intensity,’’ withvanishing divergence, whose curl is the magnetic field Bor, equivalently, whose line integral around the circuit isrelated by Stokes’s theorem to the magnetic flux throughthe loop. Including both time-varying magnetic fieldsand motion of the circuit through an inhomogeneousfield, Maxwell expressed the electromotive force (in ourlanguage) as cEdA/dt A/ t (v•“ )A.

Maxwell contrasts his mathematical treatment (whichhe does ‘‘not think contains even the shadow of a truephysical theory’’) with that of Weber, which he calls ‘‘aprofessedly physical theory of electro-dynamics, which isso elegant, so mathematical, and so entirely differentfrom anything in this paper . . . ’’ [Maxwell, 1856, Scien-tific Papers, Vol. 1, pp. 207 and 208]. Interestingly, in1865, while praising Weber and Carl Neumann, he dis-tances himself from them in avoiding charged particlesas sources, velocity-dependent interactions, and actionat a distance, preferring the mechanism of excited bod-ies and the propagation of effects through the ether.Specifically he states

We therefore have some reason to believe, from thephenomena of light and heat, that there is an aethe-real medium filling space and permeating bodies, ca-pable of being set in motion and of transmitting thatmotion from one point to another, and of transmittingthat motion to gross matter so as to heat it and affectit in various ways [Maxwell, 1865, Scientific Papers,Vol. 1, p. 528].

In this paper he again asserts his approach to the vectorpotential, now called ‘‘electromagnetic momentum,’’with its line integral around a circuit called the totalelectromagnetic momentum of the circuit. The central

role of the vector potential in Maxwell’s thinking can beseen in the table (op. cit., p. 561), where the componentsof ‘‘electromagnetic momentum’’ (F, G, H ) top the list of the 20 variable quantities in his equations. A similar listin his treatise (Maxwell, 1873, Vol. 2, Art. 618, p. 236)has the electromagnetic momentum second, after the co-ordinates of a point. In 1865 (Scientific Papers, Vol. 1, p.564), he explains that the term ‘‘electromagnetic mo-mentum’’ comes from the analogy of the mechanical F

dp/dt and the electromagnetic cEdA/dt , but cau-tions that it is ‘‘to be considered as illustrative, not asexplanatory.’’ 2 A curiosity is Maxwell’s (1873) use in histreatise of at least three different expressions for thesame quantity—‘‘vector potential’’ (Secs. 405, 590, 617),‘‘electrokinetic momentum’’ (Secs. 579, 590), and ‘‘elec-tromagnetic momentum’’ (Secs. 604, 618).

Helmholtz’s identification of Eq. (17) with Maxwellcomes from Maxwell’s preference for “•A0 when us-ing any vector potential.3 In 1873, Maxwell writes thevector potential A in the Neumann form (7), but withthe ‘‘total current,’’ conduction J plus displacement D/c t , instead of J alone. (We transcribe his notationinto present-day notation where appropriate.) He thenwrites what is now called the gauge transformationequation, AA“ [Maxwell, 1873, Eq. (7), Sec. 616,p. 235, 1st ed.; p. 256, 3rd ed.] and observes

The quantity disappears from the equations (A)B“A and it is not related to any physical phe-nomenon.

He goes on to say that he will set 0, remove theprime from A, and have it as the true value of thevector potential. The virtue to Maxwell of his A is that

it is the vector-potential of the electric current, stand-ing in the same relation to the electric current that thescalar potential stands to the matter of which it is the

potential.

Maxwell’s statement AA“ and the invarianceof the fields under this (gauge) transformation is one of the earliest explicit statements, more general thanHelmholtz’s, but he misses stating the accompanyingtransformation of the scalar potential because of his useof the ‘‘total current’’ as the source of the vector poten-tial. In the quasistatic limit, the elimination of the dis-placement current in vacuum in favor of the potentialsand their sources leads to Eq. (17), the form Helmholtzidentified with Maxwell.

The Danish physicist Ludvig Valentin Lorenz is per-haps best known for his pairing with the more famousDutch physicist Hendrik Antoon Lorentz in the Lorenz-Lorentz relation between index of refraction and den-sity. In fact, he was a pioneer in the theory of light and inelectrodynamics, contemporaneous with Maxwell. In1862 he developed a mathematical theory of light, using

2The aptness of the term electromagnetic momentum goesbeyond Maxwell’s analogy; in the Hamiltonian dynamics of acharged particle, the canonical momentum is PpeA/c .

3See Maxwell, 1865, Sec. 98, p. 581; Maxwell, 1873, 1st ed.,Secs. 616, 617, pp. 235 and 236; and 3rd ed., p. 256.





the basic known facts (transversality of vibrations,Fresnel’s laws), but avoiding the (unnecessary, to him)physical modeling of a mechanistic ether4 with bizarreproperties in favor of a purely phenomenological model(Lorenz, 1863). Indeed, in a Danish publication (Lorenz,1867a) he took a very modern-sounding position on theluminiferous ether, saying,

The assumption of an ether would be unreasonable

because it is a new non-substantial medium which hasbeen thought of only because light was conceived inthe same manner as sound and hence had to be amedium of exceedingly large elasticity and small den-sity to explain the large velocity of light. . . . It is mostunscientific to invent a new substance when its exis-tence is not revealed in a much more definite way[translation from Kragh, 1991, p. 4690].

That same year, two years after Maxwell (1865) butevidently independently, Lorenz published a paper en-titled ‘‘On the Identity of the Vibrations of Light withElectric Currents’’ (Lorenz, 1867b). Addressing the is-

sue of the disparities between the nature of electricity(two fluids), light (vibrations of the ether), and heat(motion of molecules) half a century after Oersted’s dis-coveries, he laments the absence of a unity of forces. Hecontinues,

Hence it would probably be best to admit that in thepresent state of science we can form no conception of the physical reason of forces and of their working inthe interior of bodies; and therefore (at present, at allevents) we must choose another way, free from allphysical hypotheses, in order, if possible, to develope[sic] theory step by step in such a manner that thefurther progress of a future time will not nullify the

results obtained [Lorenz, 1867b, p. 287].Avoiding the distasteful ether, Lorenz follows Kirch-

hoff in attributing a conductivity to material media, andalso a negligibly small but not zero conductivity for‘‘empty’’ space. He thus deals with current densitiesrather than electric fields, which he defines according toOhm’s law ( J E); many of his equations are the cus-tomary ones when divided by the conductivity . Afterstating Kirchhoff’s version of the static potentials, inwhich the vector potential is Weber’s form (12), he ob-serves that retardation is necessary to account for thefinite speed of propagation of light and, he supposes,

electromagnetic disturbances in general. He generalizesthe static scalar and vector potentials to the familiar ex-pressions, later often attributed to Lorentz, by introduc-

tion of [Lorenz, 1867b, p. 289 (Philos. Mag.)] as hisscalar potential and , , (ibid., p. 291) as the compo-nents of his vector potential. In modern notation theseare

x, t

x, t r /c

r d

3

x

;

Ax, t 1

c Jx,t r /c

r d3 x, (18)

the latter being the retarded form of the Neumann ver-sion (7). After showing that all known facts of electricityand magnetism (at that time all quasistatic) are consis-tent with the retarded potentials as much as with thestatic forms, Lorenz proceeds to derive equations for thefields that are the Maxwell equations we know, with anOhm’s law contribution for the assumed conducting me-dium. He points out that these equations are equivalentto those of his 1862 paper on light and proceeds to dis-

cuss light propagation and attenuation in metals, in di-electrics, and in empty space, and the absence of freecharge within conductors. He also works backward fromthe differential equations to obtain the retarded solu-tions for the potentials and the electric field in terms of the potentials, in order to establish completely theequivalence of his theories of light and electromagne-tism.

In the course of deriving his ‘‘Maxwell equations,’’ Lo-renz establishes that his retarded potentials are solutionsof the wave equation and also must satisfy the condition

d /dt 2(d /dxd /dyd /dz) (ibid., p. 294), orin modern notation and units,

•A1

c

t 0. (19)

This equation, now almost universally called the ‘‘Lor-entz condition,’’ can be seen to have originated with Lo-renz more than 25 years before Lorentz. In discussingthe quasistatic limit, Lorenz remarks (1867b, p. 292) thatthe retarded potentials (in modern, corrected terms, theLorenz gauge potentials) give the same fields as the in-stantaneous scalar potential and a vector potential thatis ‘‘a mean between Weber’s and Neumann’s theories,’’namely, Eq. (17), appropriate for Maxwell’s choice of “•A0. Without explicit reference, Lorenz was appar-

ently aware of and made use of what we call gaugetransformations.

Lorenz’s paper makes no reference to Maxwell. In-deed, he cites only himself, but by 1868 Maxwell hadread Lorenz’s paper and in his Treatise, at the end of thechapter giving his electromagnetic theory of light, hementions Lorenz’s work as covering essentially the sameground (Maxwell, 1873, 1st ed., Note after Sec. 805, p.398; 3rd ed., pp. 449 and 450). Although Lorenz made anumber of contributions to optics and electromagnetismduring his career (Kragh, 1991, 1992), his pioneering pa-pers were soon forgotten. A major contributing factor

4Notable in this regard, but somewhat peripheral to our his-tory of gauge invariance, was James MacCullagh’s early devel-opment of a phenomenological theory of light as disturbancespropagating in a novel form of the elastic ether, with the po-tential energy depending not on compression and distortionbut only on local rotation of the medium in order to make thelight vibrations purely transverse (MacCullagh, 1839; Whit-taker, 1951, pp. 141–144; Buchwald, 1985, Appendix 2). Mac-Cullagh’s equations correspond (when interpreted properly) toMaxwell’s equations for free fields in anisotropic media. Wethank John P. Ralston for making available his unpublishedmanuscript on MacCullagh’s work.





was surely Maxwell’s objection to the retarded potentialsof Riemann (1867) and Lorenz (1867b):

From the assumptions of both these papers we maydraw the conclusions, first, that action and reactionare not always equal and opposite, and second, thatapparatus may be constructed to generate any amountof work from its resources [Maxwell, 1868, ScientificPapers, Vol. 2, p. 137].

Given the sanctity of Newton’s third law and the conser-vation of energy, and Maxwell’s stature, such criticismwould have been devastating. It is ironic that the personwho almost invented electromagnetic momentum andwho showed that all electromagnetic effects propagatewith the speed of light did not recognize that the mo-mentum of the electromagnetic fields needed to betaken into account in Newton’s third law. Lorenz died in1891, inadequately recognized then or later. In fact, by1900 his name had disappeared from the mainstreamliterature on electromagnetism.

An interesting footnote on the Lorenz condition (19)is that in 1888, 20 years later, FitzGerald, trying to incor-

porate a finite speed of propagation into his mechanical‘‘wheel and band’’ model of the ether, was bothered bythe instantaneous character of Maxwell’s scalar poten-tial. FitzGerald’s model would accommodate no such in-stantaneous behavior. Realizing that it was a conse-quence of “•A0, he proposed Eq. (19) and found thestandard wave equation of propagation for both andA (Hunt, 1991, pp. 115–118).

The mistaken attribution of Eq. (19) to Lorentz waspointed out by O’Rahilly (1938), Roche (1990), VanBladel (1991), and others. That Lorenz, not Lorentz,was the father of the retarded potentials (18) was firstpointed out by Whittaker (1951, p. 268), but he mistak-

enly states (ibid., p. 394) that Levi-Civita was the first toshow (in 1897) that potentials defined by these integralssatisfy Eq. (19). Levi-Civita in fact does just what Lo-renz did in 1867. Lorentz’s own use of the Lorenz con-dition is discussed below.

Heinrich Hertz is most famous for his experiments inthe 1880s demonstrating the free propagation of electro-magnetic waves (Hertz, 1892), but he is equally impor-tant for his theoretical viewpoint. In 1884, beginningwith the quasistatic, instantaneous electric and magneticvector potentials of Helmholtz et al., he developed aniteration scheme that led to wave equations for the po-tentials and to the Maxwell equations in free space for

the fields (Hertz, 1896, Electric Waves, pp. 273–

290). Hisiterative approach showed one path from the action-at-a-distance potentials to the dynamical Maxwell equa-tions for the fields. Hertz states that both

Riemann in 1858 and Lorenz in 1867, with a view toassociating optical and electrical phenomena with oneanother, postulated the same or quite similar laws forthe propagation of the potentials. These investigatorsrecognized that these laws involve the addition of newterms to the forces which actually occur in electro-magnetics; and they justify this by pointing out thatthese new terms are too small to be experimentally

observable. But we see that the addition of theseterms is far from needing any apology. Indeed theirabsence would necessarily involve contradiction of principles which are quite generally accepted [Hertz,1896, p. 286].

It seems that Hertz did not fully appreciate that, whileLorenz’s path from potentials to field equations was dif-ferent in detail from his, Lorenz had accomplished the

same result 17 years earlier. Lorenz was not apologizing,but justifying his adoption of the retarded potentials asthe necessary generalization, still in agreement with theknown facts of electricity and magnetism. They were hisstarting point for obtaining his form of the Maxwellequations.

Six years later, Hertz (1892, pp. 193–268) addressedelectrodynamics for bodies at rest and in motion. Hediscussed various applications, with the fields always tothe fore and the scalar and vector potentials secondary.In this endeavor he made common cause with Heavi-side, to whom he gives prior credit (Hunt, 1991, pp. 122 –128). Both men believed the potentials were unneces-

sary and confusing. In calculations Hertz apparentlyavoided them at all costs; Heaviside used them sparingly(O’Hara and Pricha, 1987, pp. 58, 62, 66, and 67). Byusing only the fields, Hertz avoided the issue of differentforms of the potentials—his formalism was gauge invari-ant, by definition.5

D. Charged-particle dynamics—Clausius, Heaviside, and

Lorentz (1892)

We have already described Weber’s force Eq. (8) forthe interaction of charged particles. While it permitted

Weber to deduce the correct force between closedcurrent-carrying circuits, it does not even remotely agreeto order 1/c 2 with the force between two charges in mo-tion. It also implies inherently unphysical behavior, asshown by Helmholtz (1873). Weber’s work was impor-tant nevertheless in its focus on charged particles insteadof currents and its initiation of the Kirchhoff-Weberform of the vector potential.

A significant variation on charged-particle dynamics,closer to the truth than Weber’s, was proposed by Ru-dolf Julius Emanuel Clausius (1877, 1880). Struck byHelmholtz’s demonstration of the equivalence of We-ber’s and Neumann’s expressions for the interaction of

charges or current elements, Clausius chose to writeLagrange’s equations with an interaction of two chargedparticles e and e that amounts to an interaction La-grangian of the form6

5Hertz did not avoid potentials entirely. His name is associ-ated with the ‘‘polarization potentials’’ of radiation problems.

6Prior to the end of the 19th century in mechanics and thebeginning of the 20th century in electrodynamics, the compactnotation of L for T -V was rarely used in writing Lagrange’sequations. We use the modern notation L int as a convenientshorthand despite its absence in the papers cited.





L intee

r 1

v•v

c 2 . (20)

Generalized to one charge e interacting with many,treated as continuous charge and current densities( ,J), this Lagrangian reads

L intex,t 1

cv•AN x, t , (21)

where is the instantaneous Coulomb potential and AN

is the instantaneous Neumann potential (7) with a time-dependent current. The interaction (20), inherent inNeumann’s earlier work on currents, is a considerablestep forward in the context of charged-particle interac-tions, but its instantaneous action-at-a-distance structuremeans that it is not a true description, even to order1/c2. (The force deduced from it has the correctmagnetic-field coupling to order 1/c 2, but lacks some of the corresponding corrections to the electric-field contri-bution.)

In an impressive paper, Oliver Heaviside (1889) chose“•A0 (so that the instantaneous Coulomb field is ex-act) and constructed the appropriate vector potential,Eq. (17) for a point source, to give the velocity-dependent interaction correct to order 1/c 2 [Heaviside,1889, p. 328, Eq. (8)]. For two charges e and e withvelocities v and v, respectively, his results are equiva-lent to the interaction Lagrangian,

L intee

r 1

1

2c2 v•v r •v r •v . (22)

Heaviside also derived the magnetic part of the Lorentzforce. His contributions, like Lorenz’s, were largely ig-nored subsequently. Darwin (1920) derived Eq. (22) byanother method with no reference to Heaviside and ap-plied it to problems in the old quantum theory (see alsoFock, 1959).

A different approach was developed by H. A. Lorentz(1892) as part of his comprehensive statement of whatwe now call the microscopic Maxwell theory, withcharges at rest and in motion as the sole sources of elec-tromagnetic fields. His Chap. IV is devoted to the forcesbetween charged particles. The development is summa-rized on pp. 451 and 452 by statement of the microscopicMaxwell equations and the Lorentz force equation, FeEvB/c . In using D’Alembert’s principle to de-rive his equations, Lorentz employs the vector potential,but never states explicitly its form in terms of the

sources. It is clear, however, that he has retardation inmind, on the one hand, from his exhibition of the fullMaxwell equations to determine the fields caused by his and J v, and, on the other, by his words at the be-ginning of the chapter. He calls his reformulation (intranslation from the French) ‘‘a fundamental law com-parable to those of Weber and Clausius, while maintain-ing the consequences of Maxwell’s principles.’’ A fewsentences later, he stresses that the action of onecharged particle on another is propagated at the speedof light, a concept originated by Gauss in 1845, butlargely ignored for nearly 50 years.

Joseph Larmor (1900) used the principle of least ac-tion for the combined system of electromagnetic fieldsand charged particles to obtain both the Maxwell equa-tions and the Lorentz force equation. Karl Schwarzs-child, later renowned in astrophysics and general relativ-ity, independently used the same technique to discussthe combined system of particles and fields (Schwarzs-child, 1903). He was the first to write explicitly the fa-miliar Lagrangian L

int

describing the interaction of acharged particle e, with coordinate x and velocity v, withretarded external electromagnetic potentials,

L intex, t 1

cv•Ax, t , (23)

where and A are the potentials given by Eq. (18).It is curious that, to the best of the authors’ knowl-

edge, the issue of gauge invariance of this charged-particle Lagrangian did not receive general consider-ation in print until 1941 in the text by Landau andLifshitz (1941; see also Bergmann, 1946). The proof issimple. Under the gauge transformation (1a) and (1b),the Lagrangian (23) is augmented by

L inte 1

c

t

1

cv• e

c

d

dt , (24)

a total time derivative, and so makes no contribution tothe equations of motion. Perhaps this observation is tooobvious to warrant publication in other than textbooks.We note that, in deriving the approximate Lagrangianattributed to him, Darwin (1920) expands the retardedpotentials for a charged particle, which involve rx(t )x( t ), in powers of ( t t ) r /c , with coefficientsof the primed particle’s velocity, acceleration, etc., to ob-tain a tentative Lagrangian and then adds a total timederivative to obtain Eq. (22). Fock (1959) makes thesame expansion, but then explicitly makes a gauge trans-formation to arrive at Eq. (22). These equivalent proce-dures exploit the arbitrariness of Eq. (24).

E. Lorentz: The acknowledged authority, general gauge

freedom

Our focus here is on how H. A. Lorentz became iden-tified as the originator of both the condition (19) be-tween and A and the retarded solutions (18). In Chap.VI, Lorentz (1892) presents, without attribution, a theo-

rem that the integral

F x,t 1

4 1

r sx,t t r /c d3 x (25a)

is a solution of the inhomogeneous wave equation with s(x, t ) as a source term,

1

c 2

2F

t 22F sx, t . (25b)

He then uses such retarded solutions for time inte-grals of the vector potential in a discussion of dipoleradiation.





In fact, the theorem goes back to Riemann in 1858and Lorenz in 1861 and perhaps others. Riemann appar-ently read his paper containing the theorem to the Go ¨ t-tingen academy in 1858, but his death prevented publi-cation; this was not remedied until 1867. Lorenz (1867b)states Eqs. (25a) and (25b) and remarks that the dem-onstration is easy, giving as reference his paper on elasticwaves (Lorenz, 1861). It seems clear that in 1861 Lorenzwas unaware of Riemann’s oral presentation. The post-humous publication of Riemann’s note occurred simul-taneously with and adjacent to Lorenz’s 1867 paper inAnnalen der Physik und Chemie.7

Lorentz (1895, Sec. 32) quotes the theorems (25a) and(25b), citing his work (Lorentz, 1892) for proof, andthen in Sec. 33 writes the components of a vector field inthe form equivalent to Eq. (18) with J v. He does notcall his vector field ( x , y , z) the vector potential.Having obtained the wave equation for H with “J asthe source term, he merely notes that if H is defined asthe curl of his vector field, it is sufficient that the fieldsatisfy the wave equation with J as the source. We thussee Lorentz in 1895 explicitly exhibiting retarded solu-

tions, but without the condition of Eq. (19).In a festschrift volume in honor of the 25th anniver-

sary of Lorentz’s doctorate, Emil Wiechert (1900) sum-marizes the history of the wave equation and its re-tarded solutions. He cites Riemann in 1858, Poincare in1891, Lorentz (1892, 1895), and Levi-Civita in 1897. Nomention of Lorenz! In the same volume, des Coudres(1900) cites Lorentz (1892) for the theorem (25a) and(25b) and calls the retarded solutions (18) Lorentz’schenLo sungen. It is evident that by 1900 the physics commu-nity had attributed the retarded solutions for and A toLorentz, to the exclusion of others.

Additional reasons for Lorentz’s being the reference

point for modern classical electromagnetism are hismagisterial encyclopedia articles (Lorentz, 1904a, 1904b)and his book (1909). Here we find the first clear state-ment of the arbitrariness of the potentials under whatwe now call general gauge transformations. Lorentz(1904b, p. 157) first states that in order to have the po-tentials satisfy the ordinary wave equations they must berelated by

•A1

c

t . Lorentz’s (2)

He then discusses the arbitrariness in the potentials, not-ing that other potentials A0 and 0 may give the samefields but not satisfy his constraint. He then states that‘‘every other admissible pair A and ’’ can be related tothe first pair via the transformations

AA0 , 01

c ˙ . (26)

He then says that the scalar function can be found sothat A and do satisfy his Eq. (2) by solving the inho-mogeneous wave equation

2 1

c2 ¨ •A01

c0 . (27)

A reader might question whether Lorentz was herestating the general principle of what we term gauge in-variance. He presented his constraint before his state-ment of the arbitrariness of the potentials and then im-mediately restricted to a solution of Eq. (27). Thisdoubt is removed in his book (Lorentz, 1909). There hesays,

Understanding by A0 and 0 special values, we mayrepresent other values that may as well be chosen by[our Eq. (26)] where is some scalar function [empha-sis added]. We shall determine by subjecting A and to the condition [Lorentz’s Eq. (2)] which can al-

ways be fulfilled because it leads to the equation [ourEq. (27)], which can be satisfied by a proper choice of [Lorentz, 1909, Note 5].

He then proceeds to the wave equations and the re-tarded solutions in Sec. 13 of the main text. Lorentzobviously preferred potentials satisfying his constraint tothe exclusion of other choices, but he did recognize thegeneral principle of gauge invariance in classical electro-magnetism without putting stress on it.

The dominance of Lorentz’s publications as sourcedocuments is illustrated by their citation by G. A. Schottin his Adams Prize essay (Schott, 1912, p. 4), in which hequotes Eq. (19) [his Eq. (IX)] and the wave equations

for A and . He then cites Lorentz’s second encyclope-dia article (Lorentz, 1904b) and his book (Lorentz,1909) for the retarded solutions (18) [his Eqs. (X) and(XI)], which he later on the page calls ‘‘the Lorentz in-tegrals.’’

Lorentz’s domination aside, the last third of the 19thcentury saw the fundamentals of electromagnetism al-most completely clarified, with the ether soon to disap-pear. Scientists went about applying the subject withconfidence. They did not focus on niceties such as thearbitrariness of the potentials, content to follow Lorentzin use of the retarded potentials (18). It was only withthe advent of modern quantum field theory and the con-

struction of the electroweak theory and quantum chro-modynamics that the deep significance of gauge invari-ance emerged.

III. DAWNING OF THE QUANTUM ERA

A. 1926—Schro ¨ dinger, Klein, Fock

The year 1926 saw the floodgates open. Quantum me-chanics, or more precisely, wave mechanics, blossomedat the hands of Erwin Schro ¨ dinger and many others.Among the myriad contributions, we focus only on those

7Riemann (1867) showed that retardation led to the quasi-static instantaneous interactions of Weber and Kirchhoff,much as done by Lorenz (1867b), and remarked on the con-nection between the velocity of the propagation of light andthe ratio of electrostatic and electromagnetic units.





that relate to our story of the emergence of the principleof gauge invariance in quantum theory. The pace evenfor this restricted set is frantic enough. (To documentthe pace, we augment the references for the papers inthis era with submission and publication dates.) Thethread we pursue is the relativistic wave equation forspinless charged particles, popularly known nowadays asthe Klein-Gordon equation. The presence of both thescalar and vector potentials brought forth the discoveryof the combined transformations (1a), (1b), and (1c) byFock.

The relativistic wave equation for a spinless particlewith charge e interacting with electromagnetic fields isderived in current textbooks by first transforming theclassical constraint equation for a particle of 4-momentum p

( p0,p) and mass m, p p(mc ) 2, by

the substitution p→ p

eA /c , where A( A0

,A) is the 4-vector electromagnetic potential. Herewe use the metric g00

1, g ij ij . Then a quantum-

mechanical operator acting on a wave function is con-structed by the operator substitution, p

→i , where / x( 0,“). Explicitly, we have

i eA/c i eA /cmc2 . (28)

Alternatively, we divide through by 2 and write

ie A/c ie A /cmc /20.(29)

Separation of the space and time dimensions and choiceof a constant-energy solution, exp(iEt /), yields therelativistic version of the Schro ¨ dinger equation,

2c22iec A2iecA•e2A•A

Ee 2mc 22 . (30)

The second term on the left is absent if the Lorenzgauge condition A0 is chosen for the potentials.

The first of Schro ¨ dinger’s four papers (Schro ¨ dinger,1926a) was submitted on 27 January and published on 13March. It was devoted largely to the nonrelativistic time-independent wave equation and simple potential prob-lems, but in Sec. 3 he mentions the results of his study of the ‘‘relativistic Kepler problem.’’ (An English transla-tion of Schro ¨ dinger’s 1926 papers can be found in Schro ¨ -dinger, 1978.) According to his biographer (Moore,1989, pp. 194–197), Schro ¨ dinger derived the relativisticwave equation in November 1925, began solving theproblem of the hydrogen atom while on vacation atChristmas, and completed it in early January 1926. Dis-

appointed that he had not obtained the Sommerfeldfine-structure formula, he did not publish his work, butfocused initially on the nonrelativistic equation. Somemonths later, in Sec. 6 of his fourth paper (Schro ¨ dinger,1926b), he tentatively presented the relativistic equationin detail and discussed its application to the hydrogenatom and to the Zeeman effect.

Schro ¨ dinger was not the only person to consider therelativistic wave equation. In a private letter to Jordandated 12 April 1926, Pauli used the relativistic connec-tion between energy and momentum to derive a waveequation equivalent to Eq. (29) with a static potential,

then specialized to the nonrelativistic Schro ¨ dinger equa-tion, and went on to his main purpose—to show theequivalence of matrix mechanics and wave mechanics(van der Waerden, 1973). In the published literature,Oskar Klein (1926) treated a five-dimensional relativis-tic formalism and explicitly exhibited the four-dimensional relativistic wave equation for fixed energywith a static scalar potential. He showed that the non-

relativistic limit was the time-independent Schro ¨ dingerequation, but did not discuss any solutions. Before pub-lication of Klein’s paper, Fock (1926a) independently de-rived the relativistic wave equation from a variationalprinciple and solved the relativistic Kepler problem. Heobserved that Schro ¨ dinger had already commented onthe solution in his first paper. In his paper Fock did notinclude the general electromagnetic interaction. Schro ¨ -dinger comments in the introduction (‘‘Abstract’’) to hiscollected papers (Schro ¨ dinger, 1978),

V. Fock carried out the calculations quite indepen-dently in Leningrad, before my last paper [Schro ¨ -dinger, 1926b] was sent in, and also succeeded in de-riving the relativistic equation from a variationalprinciple [Z. Phys. 38, 242 (1926)].

The discovery of the symmetry under gauge transfor-mations (1a), (1b), and (1c) of the quantum-mechanicalsystem of a charged particle interacting with electromag-netic fields is due to Fock (1926b). His paper was sub-mitted on 30 July 1926 and published on 2 October 1926.In it he first discussed the special-relativistic wave equa-tion of his earlier paper with electromagnetic interac-tions and addressed the effect of the change in the po-tentials (1a) and (1b). He showed that the equation isinvariant under the change in the potentials provided

the wave function is transformed according to Eq. (1c).He went on to treat a five-dimensional general-relativistic formalism, similar to but independent of Klein’s. In a note added in proof, Fock notes that ‘‘Whilethis note was in proof, the beautiful work of Oskar Klein[published on 10 July] arrived in Leningrad’’ and thatthe principal results were identical.

That fall others contributed. Kudar (1926) wrote therelativistic equations down in covariant notation, citingKlein (1926) and Fock (1926a). He remarked that hisgeneral equation reduced to Fock’s for the Kepler prob-lem with the appropriate choice of potentials. WalterGordon (1926) discussed the Compton effect using the

relativistic wave equation to describe the scattering of light by a charged particle. He referred to Schro ¨ dinger’sfirst three papers, but not the fourth (Schro ¨ dinger,1926b), in which Schro ¨ dinger actually treats the relativ-istic equation. Gordon does not cite Klein or either of Fock’s papers.

The above paragraphs show the rapid pace of 1926,the occasional duplication, and the care taken by some,but not all, for proper acknowledgment of prior work byothers. If we go chronologically by publication dates, theKlein-Gordon equation should be known as the Klein-Fock-Schro ¨ dinger equation; if by notebooks and letters,





Schro ¨ dinger and Pauli could claim priority. Totally apartfrom the name attached to the relativistic wave equa-tion, the important point in our story is Fock’s paper onthe gauge invariance, published on 2 October 1926(Fock, 1926b).

The tale now proceeds to the enshrinement by Weylof symmetry under gauge transformations as a guidingprinciple for the construction of a quantum theory of matter (electromagnetism and gravity). Along the way,we retell the well-known story of how the seemingly in-appropriate word ‘‘gauge’’ came to be associated withthe transformations (1a), (1b), and (1c) and today’s gen-eralizations.

B. Weyl: Gauge invariance as a basic principle

Fritz London, in a short note in early 1927 (London,1927a) and soon after in a longer paper (1927b), pro-posed a quantum-mechanical interpretation of Weyl’sfailed attempt to unify electromagnetism and gravitation(Weyl, 1919). This attempt was undertaken long beforethe discovery of quantum mechanics. London noticedthat the principle of invariance of Weyl’s theory under ascale change of the metric tensor g

→ g exp( x),where ( x) is an arbitrary function of the space-timecoordinates, was equivalent in quantum mechanics tothe invariance of the wave equation under the transfor-mations (1) provided that ( x) was made imaginary. Inhis short note London cites Fock (1926b) but does notrepeat the citation in his longer paper, although he doesmention (without references) both Klein and Fock forthe relativistic wave equation in five dimensions.

To understand London’s point we note first thatWeyl’s incremental change of length scale dl l dx

leads to a formal solution l l 0 exp( x), where ( x) x dx ; the indefinite integral over the real 4-vector‘‘potential’’ is path dependent. If we return to therelativistic wave Eq. (29), we observe that a formal so-lution for a particle interacting with the electromagneticpotential A can be written in terms of the solutionwithout interaction as

expie

c x

A dx 0 , (31)

where 0 is the zero-field solution. [Recovery of Eq.(29) may be accomplished by ‘‘solving’’ for 0 and re-quiring 0(mc /)2 00.] With the gauge trans-formation of the 4-vector potential

A→ A

A

, (32)

the difference in phase factors is obviously the integralof a perfect differential, dx. Up to a constantphase, the wave functions and are thus related bythe phase transformation

exp ie x /c , (33)

which is precisely Fock’s Eq. (1c). London actually ex-pressed his argument in terms of ‘‘scale change’’ l l 0 expi( x), where i( x) is the quantity in the expo-nential in Eq. (31), and wrote /l 0 / l 0 .

The ‘‘gradient invariance’’ of Fock became identifiedby London and then by Weyl with an analog of Weyl’seichinvarianz (scale invariance), even though the formerconcerns a local phase change and the latter a coordi-nate scale change. In his famous book, Gruppentheorieund Quantenmechanik, Weyl (1928) discusses the cou-pling of a relativistic charged particle with the electro-magnetic field. He observes, without references, that theelectromagnetic equations and the relativistic Schro ¨ -

dinger Eq. (28) are invariant under the transformations(1a), (1b), and (1c). Weyl then states (in translation):

This principle of gauge invariance is quite analogous tothat previously set up by the author, on speculativegrounds, in order to arrive at a unified theory of gravi-tation and electricity.22 But I now believe that thisgauge invariance does not tie together electricity andgravitation, but rather electricity and matter in the man-ner described above [Weyl, 1928, p. 88].

His note 22 refers to his own work, to Schro ¨ dinger(1923), and to London (1927b). In the first (1928) edi-tion, the next sentence reads (again in translation):

‘‘How gravitation according to general relativity must beincorporated is not certain at present.’’ By the second(1931) edition, this sentence has disappeared, undoubt-edly because he believed that his own work in the mean-time (Weyl, 1929a, 1929b) had shown the connection. Infact, in the second edition a new Sec. 6 appears in Chap.IV, in which Weyl elaborates on how the gauge transfor-mation (1c) can only be fully understood in the contextof general relativity.

Weyl’s 1928 book and his papers in 1929 demonstratean evolving point of view unique to him. Presumablyprompted by London’s observation, he addressed the is-sue of gauge invariance in relativistic quantum mechan-ics, on the one hand, knowing that the principle obvi-ously applied to the electromagnetic fields and chargedmatter waves, and, on the other hand, wanting to estab-lish contact with his 1919 eichinvarianz. As we have justseen, in his 1928 book he presented the idea of gaugeinvariance in the unadorned version of Fock, withoutthe ‘‘benefit’’ of general relativity. But in the introduc-tion to the first of his 1929 papers on the electron andgravitation (Weyl, 1929a, p. 324), he states the ‘‘principleof gauge invariance’’ (the first use of the words in En-glish) very much as in his book, citing only it for author-ity. He then goes on to show that the conservation of electricity is a double consequence of gauge invariance(through the matter and the electromagnetic equations)

and that

This new principle of gauge invariance, which may goby the same name, has the character of general relativ-ity since it contains an arbitrary function , and cancertainly only be understood with reference to it [Weyl,1929a, p. 324].

He elaborated on this point in the expanded (German)version (Weyl, 1929b):

In special relativity one must regard this gauge-factoras a constant because here we have only a single point-independent tetrad. Not so in general relativity; every





point has its own tetrad and hence its own arbitrarygauge-factor: because by the removal of the rigid con-nection between tetrads at different points the gauge-factor necessarily becomes an arbitrary function of po-sition [translation taken from O’Raifeartaigh andStraumann, 2000, p. 7].

Nevertheless, Weyl stated in the first paper,

If our view is correct, then the electromagnetic field is a

necessary accompaniment of the matter wave field andnot of gravitation [Weyl, 1929a, p. 332].

The last sentence of Weyl (1929b) contains almost thesame words. His viewpoint about the need for generalrelativity can perhaps be understood in the sense that must be an arbitrary function in the curved space-time of general relativity, but not necessarily in special relativity,and that he wanted to provide continuity with his earlierwork. The close mathematical relation between non-Abelian gauge fields and general relativity as connec-tions in fiber bundles was not generally realized untilmuch later (see, for example, Yang, 1986;O’Raifeartaigh and Straumann, 2000).

Historically, of course, Weyl’s 1929 papers were a wa-tershed. They enshrined as fundamental the modernprinciple of gauge invariance, in which the existence of the 4-vector potentials (and field strengths) follow fromthe requirement that the matter equations be invariantunder gauge transformations such as Eq. (1c) of thematter fields. This principle is the touchstone of thetheory of gauge fields, so dominant in theoretical physicsin the second half of the 20th century. The importantdevelopments beyond 1929 can be found in the reviewsalready mentioned in our Introduction. The readershould be warned, however, of a curiosity regarding thecitation of Fock’s 1926 paper (Fock, 1926b) by

O’Raifeartaigh (1997), O’Raifeartaigh and Straumann(2000), and Yang (1986, 1987). While the volume andpage number are given correctly, the year is invariablygiven as 1927. One of the writers privately blames it onPauli (1933). Indeed, Pauli made that error, but he didgive to Fock the priority of introducing gauge invariancein quantum theory.

IV. PHYSICAL MEANING OF GAUGE INVARIANCE AND

EXAMPLES OF GAUGES

A. On the physical meaning of gauge invariance in QED

and quantum mechanics

While for electroweak theory and quantum chromo-dynamics gauge invariance is of paramount importance,its physical meaning in quantum electrodynamics per sedoes not seem to be extremely profound. A tiny mass of the photon would destroy the gauge invariance of QED,as a mass term m

2 A2 in the Lagrangian is not gaugeinvariant. At the same time the excellent agreement of QED with experiment and in particular its renormaliz-ability would not be impaired (see, for example, Kobza-rev and Okun, 1968; Goldhaber and Nieto, 1971). How-ever, the renormalizability would be destroyed by an

anomalous magnetic moment term in the Lagrangian,

¯ F , in spite of its manifest gauge invariance.What is really fundamental in electrodynamics is theconservation of electromagnetic current or, in otherwords, conservation of charge (see, for example, Okun,1986, Lecture 1). Conservation of charge makes the ef-fects caused by a possible nonvanishing mass of the pho-ton, m , proportional to m

2 and therefore negligibly

small for small enough values of m .It should be stressed that the existing upper limits onthe value of m lead in the case of nonconserved currentto such catastrophic bremsstrahlung that most of the ex-periments that search for monochromatic photons incharge-nonconserving processes become irrelevant(Okun and Zeldovich, 1978). Further study has shown(Voloshin and Okun, 1978) that reabsorption of virtualbremsstrahlung photons restores the conservation of charge (for reviews, see Okun, 1989, 1992).

As has been emphasized above, gauge invariance is amanifestation of nonobservability of A. However, inte-grals such as in Eq. (31) are observable when they aretaken over a closed path, as in the Aharonov-Bohm ef-fect (Aharonov and Bohm, 1959). The loop integral of the vector potential there can be converted by Stokes’stheorem into the magnetic flux through the loop, show-ing that the result is expressible in terms of the magneticfield, albeit in a nonlocal manner. It is a matter of choicewhether one wishes to stress the field or the potential,but the local vector potential is not an observable.

B. Examples of gauges

The gauge invariance of classical field theory and of electrodynamics in particular allows one to consider the

potential A with various gauge conditions, most of them being not Poincare invariant:

A0 0,1,23, Lorenz gauge, (34)

“•A j A j 0 j 1,2,3 ,

Coulomb gauge or radiation gauge, (35)

n A0 n2

0 , light-cone gauge, (36)

A00, Hamiltonian or temporal gauge, (37)

A30, axial gauge, (38)

x A0, Fock-Schwinger gauge, (39)

x j A j 0, Poincare gauge. (40)

An appropriate choice of gauge simplifies calculations.This is illustrated by many examples presented in text-books (see, for example, Jackson, 1998). For the quan-tum mechanics of nonrelativistic charged particles inter-acting with radiation, the Coulomb gauge is particularlyconvenient because the instantaneous scalar potentialdescribing the static interactions and binding is unquan-tized; only the transverse vector potential of the photonsis quantized. However, noncovariant gauges, character-ized by fixing a direction in Minkowski space, pose a





number of problems, which are discussed by Gaigg,Kummer, and Schweda (1990). The problems acquireadditional dimensions in quantum field theory, whereone has to deal with a space of states and with a set of operators.

In QED the gauge degree of freedom has to be fixedbefore the theory is quantized. Usually the gauge-fixingterm ( A) 2 is added to the gauge-invariant Lagrang-ian density with coefficient 1/2 (for further details seeBerestetskii, Lifshitz, and Pitaevskii, 1971; Ramond,1981; Gaigg, Kummer, and Schweda, 1990; Zinn-Justin,1993). In perturbation theory the propagator of a virtualphoton with 4-momentum k acquires the form

Dk 1

k2 g 1

kk

k2 . (41)

The most frequently used cases are

1 Feynman gauge, (42)

0 Landau gauge. (43)

In the Feynman gauge the propagator (41) is simpler,

while in the Landau gauge its longitudinal part vanishes,which is often more convenient. If calculations are car-ried out correctly, the final result will not contain thegauge parameter .

In the static (zero-frequency) limit the propagator(41) reduces to

D ij k,01

k2 ij 1k ik j

k2 , (44)

D00k,01

k2 , (45)

D0 j k,0D i0k,00. (46)

This is the propagator for the Helmholtz potential (14)and the static Coulomb potential.

Various gauges have been associated with names of physicists, a process begun by Heitler, who introducedthe term ‘‘Lorentz relation’’ in the first edition of hisbook (Heitler, 1936). In the third edition (Heitler, 1954)he used ‘‘Lorentz gauge’’ and ‘‘Coulomb gauge.’’Zumino (1960) introduced the terms ‘‘Feynman gauge,’’‘‘Landau gauge,’’ and ‘‘Yennie gauge’’ [ 3 in Eq.(41)].

V. SUMMARY AND CONCLUDING REMARKS

What is now generally known as a gauge transforma-tion of the electromagnetic potentials (1a) and (1b) wasdiscovered during the formulation of classical electrody-namics by its creators, Lorenz, Maxwell, Helmholtz, andLorentz, among others (1867–1909). The phase transfor-mation (1c) of the quantum-mechanical charged field ac-companying the transformation of the electromagneticpotentials was discovered by Fock (1926b). The term‘‘gauge’’ was applied to this transformation by Weyl(1928, 1929a, 1929b), who had used ‘‘eich-’’ a decade

earlier to denote a scale transformation in his unsuccess-ful attempt to unify gravity and electromagnetism.

Among textbooks on classical electrodynamics, thegauge invariance (1a) and (1b) was first discussed byLorentz in his influential book, Theory of Electrons(Lorentz, 1909). The first derivation of the invariance of the Lagrangian for the combined system of electromag-netic fields and charged particles was presented by Lan-dau and Lifshitz (1941); with reference to Fock, theyused Fock’s term ‘‘gradient invariance.’’

The first model of a non-Abelian gauge theory of weak, strong, and electromagnetic interactions was pro-posed by Klein (1938), who did not use the term‘‘gauge’’ and did not refer to Weyl. But this attempt wasfirmly forgotten. The modern era of gauge theoriesstarted with the paper by Yang and Mills (1954).

The history of gauge invariance resembles a randomwalk, with the roles of some important early playersstrangely diminished with time. There is a kind of echobetween the loss of interest by Klein and Lorenz in their‘‘god blessed children.’’ It is striking that the notion of gauge symmetry did not appear in the context of classi-

cal electrodynamics, but required the invention of quan-tum mechanics. And it is amusing how little the authorsof textbooks know about the history of physics. For fur-ther reading, we recommend the Resource Letter of Cheng and Li (1988).

ACKNOWLEDGMENTS

The authors thank Robert N. Cahn, David J. Griffiths,Helge Kragh, Engelbert L. Schucking, and Valentine L.Telegdi for their assistance and advice. The work of J.D.J. was supported in part by the Director, Office of

Science, Office of High Energy and Nuclear Physics, of the U.S. Department of Energy under Contract DE-AC03-76SF00098. The work of L.B.O. was supported inpart by grant RFBR # 00-15-96562, by an Alexander vonHumboldt award, and by the Theory Division, CERN.

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Historical Roots of Gauge Invariance