 Electronic address: okun@heron.itep.ru
* Electronic address: jdj@lbl.gov
__________________________
27 References
26 Acknowledgements
25 V. Summary and Concluding Remarks
24 B. Examples of gauges
23 A. On the physical meaning of gauge invariance in QED and quantum mechanics
23 IV. Physical Meaning of Gauge Invariance, Examples
21 B. Weyl: gauge invariance as a basic principle
19 A. 1926: Schrodinger, Klein, Fock
19 III. Dawning of the Quantum Era
17 E. Lorentz: the acknowledged authority, general gauge freedom
15 D. Charged particle dynamics - Clausius, Heaviside, and Lorentz (1892) arXiv: v5 14 Mar 2001
11 C. Electrodynamics by Maxwell, Lorenz, and Hertz
9 B. Vector potentials - Kirchhoff and Helmholtz
5 A. Early history - Ampere, Neumann, Weber
5 II. Classical Era
2 I. Introduction S S S S T T T T N N N N E E E E T T T T N N N N O O O O C C C C
N. Yang and R. L. Mills.
in English. The present era of non-abelian gauge theories started in 1954 with the paper by C.
invariance as a general principle and called it Eichinvarianz in German and gauge invariance
. In 1929 H. Weyl proclaimed this     and to the vector potential of /ct potential of -
) with the accompanying additions to the scalar exp(ie/c wave function by a phase factor
Fock. The latter discovered that this equation is invariant with respect to multiplication of the
equation for charged spinless particles was formulated by E. Schrodinger, O. Klein, and V.
the final formulation of classical electrodynamics. In 1926 a relativistic quantum-mechanical
condition is attributed to H. A. Lorentz, who half a century later was one of the key figures in
was proposed L. V. Lorenz in the middle of 1860's . In most of the modern texts the latter

= 0 A = 0 was proposed by J. C. Maxwell;  A A A A         brought forth various restrictions on it.
 A A A A result in the same observable forces. The partial arbitrariness of the vector potential
Subsequent developments led to the discovery that different forms of the vector potential
electromagnetism was discovered and the first electrodynamic theory was proposed.
(the so called Standard Model). The roots of gauge invariance go back to the year 1820 when
Gauge invariance is the basis of the modern theory of electroweak and strong interactions
T T T T C C C C A A A A R R R R T T T T S S S S B B B B A A A A

ITEP, 117218, Moscow, Russia
L. B. Okun 

University of California and Lawrence Berkeley National Laboratory, Berkeley, CA 94720
J. D. Jackson *
Historical roots of gauge invariance
LBNL - 47066 12 March 2001


1


parameters with precision.
in other experiments have brilliantly verified the electroweak theory and determined its
positron collisions at CERN and at SLAC, and in proton-antiproton collisions at Fermilab and
electroweak theory. The very extensive and detailed measurements in high-energy electron-
CERN of the heavy W and Z bosons in 1983 established the essential correctness of the
and Veltman concerning dimensional regularization and renormalization. The discoveries at
in the 1960s was an important step forward, as were the technical developments by `t Hooft
(1954). The creation of a non-abelian electroweak theory by Glashow, Salam, and Weinberg
the 1954 paper on non-abelian gauge symmetries by Chen-Ning Yang and Robert L. Mills
The proliferation of gauge theories in the second half of the 20th century began with
paper was forgotten by the physics community and never cited by the author himself.
electromagnetic and weak interactions was made by Oscar Klein (1938). But this prophetic
by the U(1) group. The first attempt to apply a non-abelian gauge symmetry SU(2) x SU(1) to
The gauge symmetry of Quantum Electrodynamics (QED) is an abelian one, described
modern gauge theories.
known as a gauge invariance or a gauge symmetry and is a touchstone in the creation of
1929b). The invariance of a theory under combined transformations such as (1,a,b,c) is
concept was declared a general principle and "consecrated" by Hermann Weyl ( 1928, 1929a,
transformation (1c), whereby it is multiplied by a local (space-time dependent) phase. The
unchanged by these transformations, the wave function is required to undergo the
was that, for the quantum dynamics, that is, the form of the quantum equation, to remain
fields are invariant under the transformations (1a,b) of the potentials. What Fock discovered
function. The Maxwell equations of classical electromagnetism for the electric and magnetic
is the vector potential,  is the scalar potential, and  is known as the gauge A A A A Here

) c / ie ( exp  = ' TM 
.
(1c) t 
c
(1b) ,  TM ' =  - 1


(1a) , +  A A A A = ' A A A A TM A A A A

In present day notation we write

[Fock's (9) ] . e 0  = 
h / i p 2 and
c
, f - 1 p = p
e
t  c
, - 1 =
[Fock's (5) ] 1
f 
f +  1 A A A A = A A A A

his notation,
particles interacting with electromagnetic fields. Equations (5) and (9) of Fock's paper are, in
electromagnetic potentials in classical electrodynamics to the quantum mechanics of charged
traced to Vladimir Fock (1926b) who extended the known freedom of choosing the
describes electroweak and strong interactions of elementary particles. Its origins can be
The principle of gauge invariance plays a key role in the Standard Model which

N N N N O O O O I I I I T T T T C C C C U U U U D D D D O O O O R R R R T T T T N N N N I I I I . . . . I I I I

2


were physically equivalent. Helmholtz's linear A A A A Neumann's and Weber's forms for
. Over 20 years later Helmholtz (1870) ended the controversy by showing that A A A A potential
energy between current elements, expressed in terms of different forms for the vector
led to competing differential expressions for the elemental force, Faraday's induction, and the
integrated result. In the 1840's the work of Neumann (1847, 1849) and Weber (1878, 1848)
the two elements of current, but others wrote down different expressions leading to the same
result. Ampere believed the element of force was central, that is, acting along the line joining
one in each loop. Upon integration over the current flow in each loop, the total force would
differential expressions giving the element of force between infinitesimal current elements,
of Ampere and others to reduce the description of the forces between actual current loops to
The lack of uniqueness of the scalar and vector potentials arose initially in the desire
Electricity and magnetism were truly united.
that a time varying magnetic flux through a circuit induces current flow (Faraday, 1839).
closed circuits carrying steady currents (Ampere, 1827). In 1831 Faraday made the discovery
current-current interactions, and developed a mathematical description of the forces between
Knudsen, 1998). Ampere and others rapidly explored the new phenomenon, generalized it to
discovery of the influence of a nearby current flow on a magnetic needle (Jelved, Jackson, and
electricity and in 1820 magnetism and electricity were brought together by Oersted's
The invention of Leyden jars and the development of voltaic piles led to study of the flow of
energy of those charges and had only the trivial arbitrariness of the addition of a constant.
from a distribution of charges was intimately associated with the electrostatic potential
despite the uniqueness of the electromagnetic fields. The electrostatic potential resulting
It took almost a century to formulate this non-uniqueness of potentials that exists
expressed in equations (1a) and (1b).
and  A A A A electrodynamics with a special emphasis on the freedom of choice of potentials
In section II, the central part of our article, we describe the history of classical
historical roots, back to the beginning of 19th century. This is the main aim of our review.
The key role of gauge invariance in modern physics makes it desirable to trace its
Collider (LHC) under construction at CERN.
aborted Superconducting Super Collider (SSC). Now it is a top priority for the Large Hadron
direct evidence for the higgs. Discovery and study of the higgs was a top priority for the
, but so far there is no c LEP experimental data imply a much lower mass, perhaps 100 GeV/2 . Indirect indications from cwill be different from expected if its mass is greater than 1 TeV/2
restricted by the Standard Model, but general theoretical arguments imply that the physics
quarks, leptons and W-, Z-bosons acquire their masses. The mass of the higgs itself is not
electrically neutral, spinless particle is intimately connected with the mechanism by which
equivalent is of profound importance in particle physics today. In the Standard Model , this
its experimental test. The search for the so-called Higgs boson (or simply, higgs) or its
Model have failed up to now. But one of the cornerstones of the Standard Model still awaits
the basis of all of physics except for gravity. All experimental attempts to falsify the Standard
QCD and Electroweak Theory form what is called today the Standard Model, which is
of the photon. Colored quarks and gluons are confined within numerous colorless hadrons.
colors are analogues of electric charge in electrodynamics. Eight colored gluons are analogues
(u, d, s, c, b, t) exists in three varieties or different "colors" (red, yellow, blue). The quark
Chromodynamics (QCD). QCD is based on the SU(3) group, as each quark of given "flavor"
was created. One of its creators, Murray Gell-Mann, gave it the name Quantum
In the 1970's a non-abelian gauge theory of strong interaction of quarks and gluons


3


emphasis on the contrast between Britain and the Continent in the interpretations of
development of theoretical and experimental electromagnetism in the 19th century, with
differential equations for the fields that we know today. Darrigol (2000) covers the
Lodge, Oliver Heaviside, and Joseph Larmor, as Maxwell's theory evolved into the
developments from Maxwell to 1900, in particular, the works of George F. FitzGerald, Oliver
subject from Coulomb to Clausius. As his title implies, Hunt (1991) focuses on the British
magnetism. Reiff and Sommerfeld (1902) provide an early review of some facets of the
and Rosenfeld (1957). Volume 1 of Whittaker (1951) surveys all of classical electricity and
Reiff and Sommerfeld (1902), Hunt (1991), Darrigol (2000), Buchwald (1985, 1989, 1994),
sources on the history of electromagnetism in the 19th century we mention Whittaker (1951),
instances we used secondary sources (historical reviews and mongraphs). Among the many
In writing this article we mainly relied on original articles and books, but in some
others.
sometime use today, but leave the detailed description of subsequent developments to
the physical meaning of gauge invariance and describe the plethora of different gauges in
known story of the origin of the term "gauge transformation." In Section IV we discuss briefly
concepts were taken from the originators and bestowed on others. We also retell the well-
annus mirabilis, 1926. As in Section II, we explore how and why priorities for certain
quantum era, the period from the end of the first World War to 1930, with emphasis on the
In Section III we review the extension of the concept of gauge invariance in the early
defines what we now call quantum gauge fields.
the quantum mechanical wave function became transformed into a general principle that
not anticipated was how the consequences of a change in the electromagnetic potentials on
quantum mechanics, the issue of the arbitrariness of the potentials would arise. What was
of charged particles with time varying electromagnetic fields came to be considered in
Schrodinger, Born, and others invented quantum mechanics. Inevitably, when the interaction
the 1920s the inadequacies of the Bohr theory were apparent. In 1925-1926, Heisenberg,
phenomena, with the confrontation between Bohr's early quantum theory and experiment. By
and of radioactive transformations. In the 1910s attention turned increasingly to atomic
The start of the 20th century saw the beginning of the quantum, of special relativity,
to the exclusion of earlier contributors such as Lorenz.
b) and his book (Lorentz, 1909) established him as an authority in classical electrodynamics,
not yet known by that name, were in place. Lorentz's encyclopedia articles (Lorentz, 1904a,
the potentials, the interaction with charged particles, the concept of gauge transformations,
charges in motion forming currents, the formal structure of electromagnetic theory, the role of
Lorentz, who invented what we now call microscopic electromagnetism, with localized
By the turn of the century, thanks to, among others, Clausius, Heaviside, Hertz, and
years.
Lorentz condition, though he preceded the Dutch physicist H. A. Lorentz by more that 25
potentials and showed that they satisfied the relation almost universally known as the
contributions are most significant. He introduced the so-called retarded scalar and vector
charges and currents (Lorenz, 1867b). From the point of view of gauge invariance, Lorenz's
same basic equations and conclusions about the kinship of light and the electromagnetism of
physicist Ludvig V. Lorenz, apparently independently of Maxwell, brilliantly developed the
a way that many found difficult to understand (Maxwell, 1865). Immediately after, the Danish
the correct complete set of equations governing electromagnetism, unfortunately expressed in
The beginning of the last third of the 19th century saw Maxwell's masterly creation of
call a restricted class of different gauges for the vector potential.
combination of the two forms with an arbitrary coefficient is the first example of what we now


4


Earth's magnetic field as did bar magnets, and began his extensive quantitative observations
Academy, showed, among other things, that small solenoids carrying current behaved in the
At the same time Ampere, in a brilliant series of demonstrations before the French
for example Eq.(5.4), p 175 of Jackson (1998).
times a pole strength - see B B B B d as the standard expression for an increment of magnetic field
(b) directed perpendicular to the plane containing those lines. (Biot, 1824). We recognize this
angle between the direction of the segment and the line joining the segment to the pole, and
between the segment and the pole, and to the sine of the r square of the inverse distance
(a) proportional to the product of the pole strength, the current, the length of the segment, the
was ds the conclusion that the force on a pole exerted by an increment of the current of length
Laplace for the straight wire and another experiment with a V-shaped wire, Biot abstracted
perpendicular distance from the wire (Biot and Savart, 1820). On the basis of a calculation of
was perpendicular to the wire and to the radius vector, and fell off inversely as the
announced their famous law - that for a given current and pole strength, the force on a pole
and Savart studied the force of a current-carrying long straight wire on magnetic poles and
quantitative Oersted's observations, nowhere more than in France. In the fall of 1820, Biot
of the news being spread, experimenters everywhere were exploring, extending, and making
electricity and magnetism were related (Jelved, Jackson, and Knudsen, 1998). Within weeks
needles were deflected if an electric current flowed in a circuit nearby, the first evidence that
On 21 July 1820 Oersted announced to the world his amazing discovery that magnetic

r r r r e e e e b b b b e e e e W W W W , , , , n n n n n n n n a a a a m m m m u u u u e e e e N N N N , , , , e e e e r r r r e e e e p p p p m m m m A A A A - - - - y y y y r r r r o o o o t t t t s s s s i i i i h h h h y y y y l l l l r r r r a a a a E E E E . . . . A A A A

A A A A R R R R E E E E L L L L A A A A C C C C I I I I S S S S S S S S A A A A L L L L C C C C . . . . I I I I I I I I


in a consistent modern notation, using Gaussian units for electromagnetic quantities.
when discussing the works of 19th century physicists. Similarly, we usually write equations
1929 (Weyl, 1929a). It is convenient, nevertheless, to use the modern terminology even
The word "gauge" was not used in English for transformations such as (1,a,b,c) until
Prokhorov, 2000).
occasion of the 100th anniversary of his birth (Novozhilov and Novozhilov, 1999, 2000;
Pihl (1939, 1972). Fock's pioneering researches have been described recently, on the
electromagnetism and optics are summarized by Kragh (1991, 1992) and more generally by
as physicists given less than their due by history. The many accomplishments of Lorenz in
Ludvig Valentin Lorenz of the classical era and Vladimir Aleksandrovich Fock emerge
1997; O'Raifeartaigh and Straumann, 2000).
have been extensively documented (Okun, 1986; Yang, 1986; Yang, 1987; O'Raifeartaigh,
as well as more recent developments quantum gauge theories The early history of
invariance.
very similar to ours. None of these works stress the development of the idea of gauge
special emphasis on Lorenz and Maxwell. His comments on Lorenz's "modern" outlook are
mathematical and philosophical development of electrodynamics from Weber to Hertz, with
authority with a different world view. Rosenfeld's essay (Rosenfeld, 1957) focuses on the
theoretical work of Heinrich Hertz as he moved from Helmholtz's pupil to independent
the first part of the 19th century. Buchwald (1994) focuses on the experimental and
theory of Lorentz and others. Buchwald (1989) treats early theory and experiment in optics in
the 19th century from the macroscopic electromagnetic theory of Maxwell to the microscopic
Maxwell's theory. Buchwald (1985) describes in detail the transition in the last quarter of


5


(think of a quasi-free electron moving through the stationary positive ions in a conductor) ' ' ' ' v v v v in nonrelativistic motion with velocity q'understand its form is to recall that a charge

_________________________________
Biot and Savart, and Grassmann.
Tricker (1965). Tricker also gives translations of portions of the papers by Oersted, Ampere,
Grassmann's criticisms and alternative expression for the force, is given in the little book by
* A cogent discussion of Ampere's work, his running dispute with Biot, and also

_________________________________
and Grassmann (1845)*. Although not how these authors arrived at it, one way to
was first written down independently in 1845 by Neumann (Eq.(2), p. 64 of Neumann, 1847)

r c r c
2 2 2 2
(4) , ' ds ds ') n n n n  n n n n ( r r r r - ) n n n n  r r r r '( n n n n = ' ds ds ) r r r r ' n n n n ( n n n n = F F F F d
I I I I
' '

used at present (see Eq.(5.8), p. 177 of Jackson (1998) for the integrated expression) ,
finding actual forces between real circuits. The second observation is that the form widely
expression is only an intermediate mathematical construct, perhaps useful, perhaps not, in
and then disappear again. The ' s s s s d and s s s s d suddenly materialize, flow along the elements
physical meaning because it violates the continuity of charge and current. Currents cannot
has no F F F F d The first observation to make is that the abstracted increment of force
, as in (2). ds' and ds respect to
with r however, to suppress the cosines and express his result in terms of the derivatives of
253 in (Ampere, 1827) in terms of the cosines defined by the scalar products. He preferred,
It is interesting to note that Ampere has the equivalent of this expression at the bottom of p.

r c
2 2
(3) . ' ds ds ' n n n n  n n n n ' - 2 n n n n  r r r r n n n n  r r r r 3 = F F F F d
r r r r I I '

reads
. In vector notation and Gaussian units, Ampere's force /r r r r r = = = = r r r r we also use the unit vector
. In what follows ds' ' ' ' ' n n n n = ' ' ' ' s s s s d and ds n n n n = s s s s d are the coordinates of ' ' ' ' x x x x and x x x x , where ' ' ' ' x x x x - x x x x = r r r r
is the magnitude of r in Gaussian units. The distance c = 1/ k where the constant
2

' s  s  ' s  s  r2 ' s  s  r
(2) , ' ds ds - r 2 k = ' ds ds k = 4 dF
I I I I
r  r  r  ' r  '
2 2

in compact form (Ampere, 1827, p. 302),
force, that is, directed along the line between the segments. He wrote his elemental force law
was a central ' ' ' ' s s s s I'd and s s s s d I between differential directed current segments F F F F d increment
, respectively, as shown in Fig. 1. Ampere believed that the force I' and I carrying currents
C' and C closed circuits or circuits extending to infinity. Consider the two closed circuits
adding perfect differentials to the elemental force, expressions that integrate to zero around
Ampere's extensive observations. These different versions arise because of the possibility of
the competing versions of the elemental force between current elements abstracted from
The different forms for the vector potential in classical electromagnetism arose from
years; the papers were collected in a memoir in 1826 (Ampere, 1827).
of the forces between closed circuits carrying steady currents. These continued over several


6


), this form of the vector potential is , t ' ' ' ' x x x x ( J J J J density
For a general current C'. flowing in circuit I' is the vector potential of the current ' ' ' ' A A A A where

' C C
r c c
(6) , ' ds ) = x x x x '( A A A A with , ds ' A A A A  n n n n = W
' n n n n I I
'

, defined nowadays as W negative of the magnetic interaction energy
is the P , with both circuits kept fixed in orientation. Neumann's C defining the position of
with respect to a suitable coordinate P is now the negative gradient of C The force on circuit
. C' and C The double integral in (5) is the definition of the mutual inductance of the circuits

' C C
c c
r 2 r 2
(5) , ' ds ds - = P ; ' ds ds - = dP
I I I I
' n n n n  n n n n ' ' n n n n  n n n n '
circuits in Fig. 1) are
(over the P and its double integral dP From (4) we see that dP. a magnetic potential energy
of r r r r (the second term in (4)) that is the negative gradient with respect to F F F F d elemental force
amounts to (4). He then omits the perfect differential to arrive at an expression for the
As mentioned above, he expresses the elemental force between current elements in what
lurking, without explicit display. In the latter parts of his papers, he adopts a different tack.
products of quantities among which one can sense the vector potential or its time derivative
potential, but his formulas are always for the induced current or its integral and so are
(Neumann, 1847, 1849). He is credited by later writers as having invented the vector
induction in one circuit from the relative motion of nearby magnets and other circuits
Franz E. Neumann in 1845 and 1847 analyzed the process of electromagnetic
elements.
use of Ohm's law made it clear that they had in mind induced electric fields along the circuit
circuits interacting with magnets or other circuits. While workers spoke of induced currents,
still 34 years in the future. Research tended to continue on the behaviour of current-carrying
incomplete and Maxwell's completion of the description with the displacement current was
electromagnetism was now established, although the differential forms of the basic laws were
electric and magnetic fields (Faraday, 1839). The experimental basis of quasi-static
near a closed circuit induces a momentary flow of current - exposed the direct link between
Faraday's discovery in 1831 of electromagnetic induction - relative motion of a magnet

although not the same as Ampere's central force.
, the residue appears as a central force (!) between elements, F F F F d If we ignore this part of
in Fig. 1. C ), which gives a zero contribution when integrated over the closed path r (1/      s s s s - d
ds = r / r r r r     n n n n differential ,
2 closed circuits, the only meaningful thing. In fact, the first term in (4) contains a perfect
it does not agree with Ampere's, but the differences vanish for the total force between two
With its non-central contribution, 'ds'. n n n n I' and ds n n n n I with ' ' ' ' v v v v q' and v v v v q conductor. Now replace
in a second v v v v moving with velocity q ), this field produces a force on a similar charge c /
' ' ' ' B B B B v v v v + ` E E E E q( = F F F F Through the Lorentz force law ). /r r r r r ' ' ' ' v v v v , q' ' ' ' ' B B B B ( generates a magnetic field
3

7


to be C' over the inducing circuit A A A A and its integral A A A A d potential, we find from (10)
the elemental vector A A A A d with t, / ) A A A A (d ) c = - (1/ E E E E identify the induced electric field as
but if we ds, n n n n Weber wrote only the component of the induced force or emf along the element in this form. /e, n n n n   F F F FdE = d1848, p. 239) writes the induced electromotive force in the circuit,
the inducing current simultaneously, but for circuits with no relative motion Weber (Weber,
complicated than that just described because he treated relative motion and time variation of
. Weber's analysis was more ' ' ' ' a a a a arises from the presence of the acceleration dI'/dt where

dt r c2
(10) , ' ds ' n n n n  r r r r r r r r - = F F F F d
dI e
'

we find from Weber's force law, ds', ' ' ' ' n n n n I' due to the current element ds, n n n n the position of , x x x x If instead of the force between current elements, we consider the force on a charge at rest at expression (3).
we obtain Ampere's ds', ' ' ' ' n n n n I' = ' ' ' ' v v v v e' and 2 ds n n n n I = v v v v e in the other, and identify 2 ' ' ' ' v v v v velocities 
with e' in the one current and those  v v v v with velocities  e the forces between the charges 
(e'). If we add up e ) are the velocity and acceleration of the charge ' ' ' ' a a a a and ' ' ' ' v v v v ( a a a a and v v v v where

dt
r 2 dt
(9) , ')) v v v v - v v v v ( r r r r ( - ) ' v v v v - v v v v ( + ') a a a a - a a a a ( r r r r = ) ; ' v v v v - v v v v ( r r r r =
2 2 1 r d dr
2

explicitly as
The first term is just Coulomb's law. The ingredients of the second part can be expressed

dt r 2 dt c r
2 2 r 2 2
(8) . - + = F
dr 1 r d 1
2 2 ' ee ' ee

motion, is (Weber, 1878, p. 229)
Weber's central force law, admittedly ad hoc and incorrect as a force between charges in
places, causing confusion to the unwary. We write everything with modern conventions.
flow of only one sign of charge, led to the appearance of factors of two and four in peculiar
hypothesis, together with the convention that the current flow was measured in terms of the
law between charges to calculate forces between circuits. Parenthetically we note that this
that currents were caused by the flow of two electrical fluids. He thus needed a basic force
moving at the same speed, but in opposite directions, rather than the general view at the time
hypothesis that current flow in a wire consists of equal numbers of charges of both signs
Ampere's law for current-carrying circuits (Weber, 1878, 1848). Weber adopted the
in motion, consistent with e' and e introduced a central force law between two charges
varying currents as sources of the electromotive force in the secondary circuit. To this end, he
presented a theory of electromagnetic induction, considering both relative motion and time-
Independently and at roughly the same time as Neumann, in 1846 Wilhelm Weber

subsequent investigators, even though he never explicitly displayed (6) or (7).
here to associate it with Neumann's work, as did A A A A to N We have attached a subscript

r c
(7) ) t ',
. x x x x ( J J J J x' d = ) t , x x x x ( N A A A A
1 3 1


8


.), p. 76, in compressed notation) equation (1 ibid., (
a The equivalent linear combination of the vector potentials (7) and (12) is
multiple of the above perfect differential, and so is consistent with Ampere's observations.
Obviously, this linear combination differs from either Weber's or Neumann's expressions by a

r c 2
2
(13) . ' ds ds r r r r ' n n n n r r r r  n n n n ' + (1-) n n n n  n n n n (1+) = dW
' I I

equation (1.), p. 76, but in modern notation),
magnetic energy between current elements by writing a linear combination (Helmholtz, 1870,
closed circuits. Helmholtz then generalized the expressions of Weber and Neumann for the
. Thus either form leads to the same potential energy and force for )/r ' ' ' ' n n n n     n n n n - r r r r      ' ' ' ' n n n n r r r r      n n n n ds ds'(
r/s s') = ds ds' ( that they differ by a multiple of the perfect differential
2
and noted , , , , r r r r      ' ' ' ' n n n n r r r r      n n n n p(Weber) = and ' ' ' ' n n n n     n n n n p(Neumann) = with r , pII' ds ds'/c = dW elements,
2 Helmholtz compared the Neumann and Weber forms of the magnetic energy between current
circumstances, but recognized that Weber's form of the magnetic energy had validity.
criticized Weber's force equation for leading to unphysical behavior of charged bodies in some
1873, 1874) criticized and clarified the earlier work of Neumann, Weber, and others. He
In an impressive, if repetitive, series of papers, Hermann von Helmholtz (1870, 1872,

now know as a particular gauge (Kirchhoff, 1857, p.532-533).
, the first published relation between potentials in what we t  c = / A A A A         modern notation),
and the associated scalar potential  satisfy the relation (in A A A A 1957) that the Weber form of
We note in passing that Kirchhoff showed (contrary to what is implied by Rosenfeld,
inductances.
vector potential. Kirchhoff applied his formalism to analyze the telegraph and calculate
the sum to Weber; the expression (12) became known as the Kirchhoff-Weber form of the
derivative of the vector potential (Kirchhoff, 1857, p. 530). He attributed the second term in
conductivity times the negative sum of the gradient of the scalar potential and the time
potential (12); he also wrote the components of the induced current density as the
Gustav Kirchhoff was the first to write explicitly (in component form) the vector

z z z z t t t t l l l l o o o o h h h h m m m m l l l l e e e e H H H H d d d d n n n n a a a a f f f f f f f f o o o o h h h h h h h h c c c c r r r r i i i i K K K K - - - - s s s s l l l l a a a a i i i i t t t t n n n n e e e e t t t t o o o o p p p p r r r r o o o o t t t t c c c c e e e e V V V V . . . . B B B B

even though he did not write (11) or (12) explicitly.
for Weber to this form of the vector potential W As with Neumann, we attach a subscript

r c
(12) . ) t ', x x x x ( J J J J  r r r r r r r r ' x d ) = t , x x x x ( W A A A A
1 3 1

is , t) ' ' ' ' x x x x ( J J J J The generalization of this form of the vector potential for a current density

' C

r c r c
(11) . ' ds = A A A A ; ' ds ' n n n n  r r r r r r r r = A A A A d
I I
' n n n n  r r r r r r r r ' '


9


advocate. Gradually, those ideas gained credence and charged particle dynamics came under
electric currents to discrete charges in motion was a minority view, with Weber a notable
electricity, electric currents, and magnetism. That electricity was due to discrete charges and
Meanwhile, others were addressing the propagation of light and its possible connection with
vector potential. The focus was on steady-state current flow or quasi-static behavior.
integration over closed circuits. These differences led to different but equivalent forms for the
elemental interactions between current elements, an arbitrariness that vanished upon
Competing descriptions stemmed from the arbitrariness associated with the postulated
currents largely within the framework of potential energy, in analogy with electrostatics.
forces between current-carrying circuits to a comprehensive description of the interaction of
We see in the early history the attempts to extend Ampere's conclusions on the
- see Section II.D. c interaction of charged particles, correct to order 1/2
Maxwell never wrote down (17). It is relevant for finding an approximate Lagrangian for the
. A A A A = 0, Maxwell's preferred choice for A A A A         potential found from the transverse current for
can be identified with Maxwell only because, as (16) shows, it is the quasi-static vector

c 2
r r
(17) , ' x d + t) = , x x x x ( M A A A A
3 1
', t) x x x x ( J J J J  r r r r r r r r t) ', x x x x ( J J J J

theory. The resulting vector potential,
= 0 leads to Maxwell's  Helmholtz remarks rather imprecisely that the choice of
potential.
restricted class of gauges and lacks the transformation of the scalar as well as the vector
Helmholtz is close to establishing the gauge invariance of electromagnetism, but treats only a
quasi-static potentials, while Lorenz's relation holds for the fully retarded potentials.
condition found in 1867 by Lorenz -see below - but Helmholtz's relation connects only the
= -1) and formally the  This relation contains the connection found in 1857 by Kirchhoff (for

t  c
(16) .  - =  A A A A  


.), p. 80, in modern notation) equation (3a

ibid., ) are related by ( t , x x x x ( A A A A ) and his vector potential t , x x x x electrostatic potential, and that (
t , where  is the instantaneous c  = 2 / Helmholtz goes on to show that  satisfies 2

2
c
(15) . ' x d ) t ', x x x x ( J J J J  r r r r , where  = -  + N A A A A =  A A A A
3 1 (1 - )

connection between his generalization and the Neumann form (7) as (in modern notation),
gauges in Maxwell's electrodynamics. In fact, in equation (1d.) on p. 77, he writes the
one-parameter class of potentials that is equivalent to a family of vector potentials of different
= -1 gives Weber. Helmholtz's generalization exhibits a  = 1 gives the Neumann form; 

2 2
(14) . . . . W A A A A (1 - ) + N A A A A (1 + ) =  A A A A
1 1


10


plus J J J J in the Neumann form (7), but with the "total current," conduction ' ' ' ' A A A A potential
Sects. 616, 617, p. 235-236; 3rd ed., p. 256). In (Maxwell, 1873) he writes the vector
when using any vector potential (Maxwell, 1865, Sect. 98, p. 581; Maxwell, 1873, 1st ed.,
0 = = = = A A A A         Helmholtz's identification of (17) with Maxwell is because Maxwell preferred
electromagnetic momentum (Sects. 604, 618 ).
quantity - vector potential (Sects. 405, 590, 617), electrokinetic momentum (Sects. 579, 590),
______________________________________________________________________________
. /c A A A A + e p p p p = P P P P Hamiltonian dynamics of a charged particle the canonical momentum is
*The aptness of the term electomagnetic momentum goes beyond Maxwell's analogy; in the

______________________________________________________________________________
use in his treatise (Maxwell, 1873) of at least three different expressions for the same
that it is "to be considered as illustrative, not as explanatory." * A curiosity is Maxwell's
, but cautions /dt A A A A d = - E E E E c and the electromagnetic /dt p p p p d = F F F F the analogy of the mechanical
, Vol. 1, p. 564), he explains the use of the term "electomagnetic momentum" as a result of P.
Sc. electromagnetic momentum second, after the coordinates of a point. In (Maxwell, 1865,
the list. A similar list in his treatise (Maxwell, 1873, Vol. 2, Art. 618, p.236) has the
, the components of "Electromagnetic Momentum," top (F, G, H) quantities in his equations
p. 561) where in his list of the 20 variable op. cit., thinking is evidenced in the table (
electromagnetic momentum of the circuit. The central role of the vector potential in Maxwell's
"electromagnetic momentum," with its line integral around a circuit called the total
In this paper he again asserts his approach to the vector potential, now called
transmitting that motion to gross matter so as to heat it and affect it in various ways."
set in motion and of transmitting that motion from one point to another, and of
that there is an aethereal medium filling space and permeating bodies, capable of being
"We therefore have some reason to believe, from the phenomena of light and heat,
Vol. 1, p. 528):
Sc. P. and the propagation of effects through the ether. Specifically he states (Maxwell, 1865,
dependent interactions, and action-at-a-distance, preferring the mechanism of excited bodies
Neumann, he distances himself from them in avoiding charged particles as sources, velocity-
Interestingly, in the introduction to (Maxwell, 1865), while praising Weber and Carl
, Vol. 1, p.207-208). Sc. P. entirely different from anything in this paper,...." (Maxwell, 1856,
professedly physical theory of electro-dynamics, which is so elegant, so mathematical, and so
contains even the shadow of a true physical theory") with that of Weber, which he calls "a
. Maxwell contrasts his mathematical treatment (which he does "not think [that it] A A A A )         v v v v (
/t + A A A A  - /dt = A A A A d = - E E E E c Maxwell expressed the electromotive force (in our language) as
time-varying magnetic fields and motion of the circuit through an imhomogeneous field,
circuit is related by Stokes's theorem to the magnetic flux through the loop. Including both
or equivalently whose line integral around the B B B B divergence, whose curl is the magnetic field
in time (Maxwell, 1856). He introduced a vector, "electro-tonic intensity," with vanishing
"electro-tonic state," ready to respond with current flow if the magnetic flux linking it changed
description of Faraday's intuitive idea that a conducting circuit in a magnetic field was in an
electromagnetic theory (Bork, 1967; see also Everitt, 1975). He developed an analytic
The vector potential played an important role in Maxwell's emerging formulation of

z z z z t t t t r r r r e e e e H H H H d d d d n n n n a a a a , , , , z z z z n n n n e e e e r r r r o o o o L L L L , , , , l l l l l l l l e e e e w w w w x x x x a a a a M M M M y y y y b b b b s s s s c c c c i i i i m m m m a a a a n n n n y y y y d d d d o o o o r r r r t t t t c c c c e e e e l l l l E E E E . . . . C C C C

was elaborated, and by whose hands.
study. Our story now turns to these developments and how the concept of different gauges


11


electricity (two fluids), light (vibrations of the ether), and heat (motion of molecules) half a
(Lorenz, 1867b). On p. 287, addressing the issue of the disparities between the nature of
published a paper entitled "On the Identity of the Vibrations of Light with Electric Currents,"
That same year, two years after Maxwell (1865) but evidently independently, he
way." (translation taken from Kragh, 1991, p. 4690).
invent a new substance when its existence is not revealed in a much more definite
and small density to explain the large velocity of light. ..... It is most unscientific to
same manner as sound and hence had to be a medium of exceedingly large elasticity
substantial medium which has been thought of only because light was conceived in the
"The assumption of an ether would be unreasonable because it is a new non-
ether, saying,
publication (Lorenz, 1867a) he took a very modern sounding position on the luminiferous
properties in favor of a purely phenomenological model (Lorenz, 1863). Indeed, in a Danish

_____________________________________________________________________________
his unpublished manuscript on MacCullagh's work.
equations for free fields in anisotropic media. We thank John P. Ralston for making available
Appendix 2). MacCullagh's equations correspond (when interpreted properly) to Maxwell's
vibrations purely transverse (MacCullagh, 1839; Whittaker, 1951, p. 141-4; Buchwald, 1985,
of the medium in order to make the light local rotation compression and distortion but only on
propagating in a novel form of the elastic ether, with the potential energy depending not on
James MacCullagh's early development of a phenomenological theory of light as disturbances
* Notable in this regard, but somewhat peripheral to our history of gauge invariance, was

______________________________________________________________________________
avoiding the (unnecessary, to him) physical modeling of a mechanistic ether* with bizarre
theory of light, using the basic known facts (transversality of vibrations, Fresnel's laws), but
electrodynamics, contemporaneous with Maxwell. In 1862 he developed a mathematical
between index of refraction and density. In fact he was a pioneer in the theory of light and in
with the more famous Dutch physicist Hendrik Antoon Lorentz in the Lorenz-Lorentz relation
The Danish physicist Ludvig Valentin Lorenz is perhaps best known for his pairing

leads to (17), the form Helmholtz identified with Maxwell.
elimination of the displacement current in vacuum in favor of the potentials and their sources
of the "total current" as the source of the vector potential. In the quasi-static limit, the
he misses stating the accompanying transformation of the scalar potential because of his use
transformation is one of the earliest explicit statements, more general than Helmholtz's, but
and the invariance of the fields under this (gauge)      - ' ' ' ' A A A A = = = = A A A A Maxwell's statement
potential."
electric current that the scalar potential stands to the matter of which it is the
"it is the vector-potential of the electric current, standing in the same relation to the
is that A A A A value of the vector potential. The virtue to Maxwell of his
and have it as the true ' ' ' ' A A A A He goes on to say that he will set  = 0, remove the prime from
to any physical phenomenon."
] and it is not related A A A A     = = = = B B B B "The quantity  disappears from the equations (A) [
observes: (Maxwell, 1873, equation (7), Sect. 616, p. 235, 1st ed., p.256, 3rd ed.) and      - ' ' ' ' A A A A = = = =
A A A A where appropriate.] He then writes what is now called the gauge transformation equation
alone. [We transcribe his notation into present day notation J J J J , instead of /ct D D D D  displacement


12


apparently aware of and made use of what we call gauge transformations.
= 0. Without explicit reference, Lorenz was A A A A         appropriate for Maxwell's choice of
potential that is "a mean between Weber's and Neumann's theories," namely, (17),
gauge potentials) give the same fields as the instantaneous scalar potential and a vector
remarks (p. 292) that the retarded potentials (in modern, corrected terms, the "Lorenz"
with Lorenz more than 25 years before Lorentz. In discussing the quasi-static limit, Lorenz
This equation, now almost universally called the "Lorentz condition," is seen to originate

t c
(19) . = 0 + A A A A  
1

p. 294) or in modern notation and units, ibid., d,,/dt = -2(d/dx + d/dy + d/dz) (
retarded potentials are solutions of the wave equation and also must satisfy the condition,
In the course of deriving his "Maxwell equations," Lorenz establishes that his
equivalence of his theories of light and electromagnetism.
and the electric field in terms of the potentials in order to establish completely the
backward from the differential equations to obtain the retarded solutions for the potentials
dielectrics, in empty space, and the absence of free charge within conductors. He also works
his 1862 paper on light and proceeds to discuss light propagation and attenuation in metals, in
assumed conducting medium. He points out that these equations are equivalent to those of
fields that are the Maxwell equations we know, with an Ohm's law contribution for the
retarded potentials as much as with the static forms, he proceeds to derive equations for the
facts of electricity and magnetism (at that time all quasi-static) are consistent with the
the latter being the retarded form of the Neumann version (7). After showing that all known


r c r
(18) x' d
, t) = , x x x x ( A A A A x' ; d , t) = x x x x (
3 1
t - r/c) ', x x x x ( J J J J 3 ', t - r/c) x x x x (

notation these are
p. 291) as the components of his vector potential. In modern ibid., potential and , ,  (
attributed to Lorentz, by introduction of ,, ( Lorenz, 1867b, p.289, [Phil. Mag.]) as his scalar
He generalizes the static scalar and vector potentials to the familiar expressions, often
finite speed of propagation of light and, he supposes, electromagnetic disturbances in general.
potential is Weber's form (12), he observes that retardation is necessary to account for the
conductivity . After stating Kirchhoff's version of the static potentials, in which the vector
); many of his equations are the customary ones when divided by the E E E E =  J J J J Ohm's law (
thus deals with current densities rather than electric fields, which he defines according to
material media, and also a negligibly small but not zero conductivity for "empty" space. He
Avoiding the distasteful ether, Lorenz follows Kirchhoff in attributing a conductivity to
results obtained."
step in such a manner that the further progress of a future time will not nullify the
from all physical hypotheses, in order, if possible, to develope (sic) theory step by
of bodies; and therefore (at present, at all events) we must choose another way, free
form no conception of the physical reason of forces and of their working in the interior
"Hence it would probably be best to admit that in the present state of science we can
continues,
century after Oersted's discoveries, he laments the absence of a unity of forces. He


13


principles which are quite generally accepted."
needing any apology. Indeed their absence would necessarily involve contradiction of
experimentally observable. But we see that the addition of these terms is far from
they justify this by pointing out that these new terms are too small to be
the addition of new terms to the forces which actually occur in electromagnetics; and
propagation of the potentials. These investigators recognized that these laws involve
phenomena with one another, postulated the same or quite similar laws for the
"Riemann in 1858 and Lorenz in 1867, with a view to associating optical and electrical
p.286) states that both , . ibid ( action-at-a-distance potentials to the dynamical Maxwell equations for the fields. Hertz
p.273-290). His iterative approach showed one path from the Electric Waves, (Hertz, 1896,
wave equations for the potentials and to the Maxwell equations in free space for the fields
magnetic vector potentials of Helmholtz et al, he developed an iteration scheme that led to
theoretical viewpoint. In 1884, beginning with the quasi-static, instantaneous electric and
propagation of electromagnetic waves (Hertz, 1892), but he is equally important for his
Heinrich Hertz is most famous for his experiments in the 1880s demonstrating the free

Lorenz condition is discussed below.
satisfy (19). Levi-Civita in fact does just what Lorenz did in 1867. Lorentz's own use of the
that Levi-Civita was the first to show (in 1897) that potentials defined by these integrals
p.394) ibid., (18) was first pointed out by Whittaker (1951, p.268) but he mistakenly states (
Bladel (1991), and others. That Lorenz, not Lorentz, was the father of the retarded potentials
The mistaken attribution of (19) to Lorentz was pointed out by O'Rahilly (1938), Van
. (Hunt, 1991, p. 115-118). A A A A and  wave equation of propagation for both
= 0, he proposed (19) and found the standard A A A A         Realizing that it was a consequence of
Maxwell's scalar potential. His model would accommodate no such instantaneous behavior.
"wheel and band" model of the ether, was bothered by the instantaneous character of
later, FitzGerald, trying to incorporate a finite speed of propagation into his mechanical
An interesting footnote on the Lorenz condition (19) is that in 1888, twenty years
electromagnetism.
then or later. In fact, by 1900 his name had disappeared from the mainstream literature on
to be taken into account in Newton's third law. Lorenz died in 1891, inadequately recognized
the speed of light did not recognize that the momentum of the electromagnetic fields needed
electromagnetic momentum and who showed that all electromagnetic effects propagate with
stature, such criticism would be devastating. It is ironic that the person who almost invented
Given the sanctity of Newton's third law and the conservation of energy, and Maxwell's
Vol. 2, p. 137). Sc. P., 1868,
may be constructed to generate any amount of work from its resources." (Maxwell,
action and reaction are not always equal and opposite, and second, that apparatus
"From the assumptions of both these papers we may draw the conclusions, first, that
Lorenz (1867):
factor was surely Maxwell's objection to the retarded potentials of Riemann (1867) and
career (Kragh, 1991, 1992), his pioneering papers were soon forgotten. A major contributing
Although Lorenz made a number of contributions to optics and electromagnetism during his
same ground (Maxwell, 1973, 1st ed., Note after Sect. 805, p. 398; 3rd ed., p. 449-450).
his electromagnetic theory of light, he mentions Lorenz's work as covering essentially the
, at the end of the chapter giving Treatise 1868 Maxwell had read Lorenz's paper and in his
Lorenz's paper makes no reference to Maxwell. Indeed, he only cites himself, but by


14


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

c2 r
(20) , + -1 = int L
' ee
' v v v v  v v v v

______________________________________________________________________________
the papers cited.
int as a convenient shorthand despite its absence in L equations. We use the modern notation
was rarely used in writing Lagrange's T - V for L electrodynamics the compact notation of
* Prior to the end of the 19th century in mechanics and the beginning of the 20th century in

______________________________________________________________________________
form,*
that amounts to an interaction Lagrangian of the e' and e interaction of two charged particles
of charges or current elements, Clausius chose to write Lagrange's equations with an
demonstration of the equivalence of Weber's and Neumann's expressions for the interaction
was proposed by Rudolf Julius Emanuel Clausius (1877, 1880). Struck by Helmholtz's
A significant variation on charged particle dynamics, closer to the truth than Weber's,

currents and its initiation of the Kirchhoff-Weber form of the vector potential.
(1873). Weber's work was important nevertheless in its focus on charged particles instead of
charges in motion. It also implies inherently unphysical behavior, as shown by Helmholtz
with the force between two c carrying circuits, it does not even remotely agree to order 1/2
particles. While it permitted Weber to deduce the correct force between closed current-
We have already described Weber's force equation (8) for the interaction of charged

) ) ) ) 2 2 2 2 9 9 9 9 8 8 8 8 1 1 1 1 ( ( ( ( z z z z t t t t n n n n e e e e r r r r o o o o L L L L d d d d n n n n a a a a , , , , e e e e d d d d i i i i s s s s i i i i v v v v a a a a e e e e H H H H , , , , s s s s u u u u i i i i s s s s u u u u a a a a l l l l C C C C - - - - s s s s c c c c i i i i m m m m a a a a n n n n y y y y d d d d e e e e l l l l c c c c i i i i t t t t r r r r a a a a p p p p d d d d e e e e g g g g r r r r a a a a h h h h C C C C . . . . D D D D

______________________________________________________________________________
potentials" of radiation problems.
* Hertz did not avoid potentials entirely. His name is associated with the "polarization

______________________________________________________________________________
gauge invariant, by definition*.
only the fields, Hertz avoided the issue of different forms of the potentials - his formalism was
all costs; Heaviside used them sparingly (O'Hara and Pricha, 1987, p.58, 62, 66-67). By using
potentials were unnecessary and confusing. In calculations Hertz apparently avoided them at
Heaviside, to whom he gives prior credit (Hunt, 1991, p. 122-128). Both men believed the
and the scalar and vector potentials secondary. In this endeavor he made common cause with
at rest and in motion. He discussed various applications, with the fields always to the fore
Six years later, Hertz (Hertz, 1892, p.193-268) addressed electrodynamics for bodies
magnetism. They were his starting point for obtaining his form of the Maxwell equations.
the necessary generalization, still in agreement with the known facts of electricity and
earlier. Lorenz was not apologizing, but justifying his adoption of the retarded potentials as
equations was different in detail from his, Lorenz accomplished the same result 17 years
It seems that Hertz did not fully appreciate that, while Lorenz's path from potentials to field


15


particles and fields (Schwarzschild, 1903). He was the first to write explicitly the familiar
relativity, independently used the same technique to discuss the combined system of
Lorentz force equation. Karl Schwarzschild, later renowned in astrophysics and general
electromagnetic fields and charged particles to obtain both the Maxwell equations and the
Joseph Larmor (1900) used the principle of least action for the combined system of
for nearly 50 years.
propagated at the speed of light, a concept originated by Gauss in 1845, but largely ignored
A few sentences later, he stresses that the action of one charged particle on another is
consequences of Maxwell's principles."
"a fundamental law comparable to those of Weber and Clausius, while maintaining the
French)
words at the beginning of the chapter. He calls his reformulation (in translation from the
, and on the other by his v v v v =  J J J J Maxwell equations to determine the fields caused by his  and
clear, however, that he has retardation in mind, on the one hand from his exhibition of the full
employs the vector potential, but never states explicitly its form in terms of the sources. It is
]. In using D'Alembert's principle to derive his equations, Lorentz /c B B B B v v v v + E E E E [ e = F F F F statement of the microscopic Maxwell equations and the Lorentz force equation, to the forces between charged particles. The development is summarized on p.451-2 by at rest and in motion as the sole sources of electromagnetic fields. His chapter IV is devoted comprehensive statement of what we now call the microscopic Maxwell theory, with chargesA different approach was developed by H. A. Lorentz (1892) as part of his
Fock (1959).
with no reference to Heaviside and applied it to problems in the old quantum theory. See also
Lorenz's, were largely ignored subsequently. Darwin (1920) derived (22) by another method
Heaviside also derived the magnetic part of the Lorentz force. His contributions, like

c 22
(22) . r
) ' v v v v  r r r r v v v v  r r r r + ' v v v v  v v v v ( + 1 - = int L
1 ' e e

results are equivalent to the interaction Lagrangian,
, respectively, his ' ' ' ' v v v v and v v v v velocities with e' and e 1889, p.328, Eq.(8)). For two charges
(Heaviside, c for a point source, to give the velocity-dependent interaction correct to order 1/2
instantaneous Coulomb field is exact) and constructed the appropriate vector potential, (17)
(so that the = 0 A A A A         In an impressive paper, Oliver Heaviside (1889) chose

contribution.)
, but lacks some of the corresponding corrections to the electric field c coupling to order 1/2
. (The force deduced from it has the correct magnetic field c description, even to order 1/2
interactions, but its instantaneous action-at-a-distance structure means that it is not a true
earlier work on currents, is a considerable step forward in the context of charged particle
potential (7) with a time-dependent current. The interaction (20), inherent in Neumann's
Nis the instantaneous Neumann A A A A where  is the instantaneous Coulomb potential and

c
(21) , ) t , x x x x ( N A A A A  v v v v + ) t , x x x x ( - e = int L
1


16


occurred simultaneously with and adjacent to Lorenz's 1867 paper in Annalen*.
unaware of Riemann's oral presentation. The posthumous publication of Riemann's note
reference his paper on elastic waves (Lorenz, 1861). It seems clear that in 1861 Lorenz was
1867b), Lorenz states (25a,b) and remarks that the demonstration is easy, giving as
in 1858, but his death prevented publication, remedied only in Riemann (1867). In (Lorenz,
others. Riemann apparently read his paper containing the theorem to the Gottingen academy
In fact, the theorem goes back to Riemann in 1858 and Lorenz in 1861 and perhaps
of dipole radiation.
He then uses such retarded solutions for time integrals of the vector potential in a discussion


t  c
2 2
(25b) . ) t , x x x x ( s = F  -
2 1
F 2

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

4
r
(25a) ' x d ) c / r - t = ' t ', x ( s = ) t , x x x x ( F
3 1 1

(1892) presents, without attribution, a theorem that the integral
and the retarded solutions (18). In chapter VI, Lorentz A A A A condition (19) between  and
Our focus here is on how H. A. Lorentz became identified as the originator of both the
m m m m o o o o d d d d e e e e e e e e r r r r f f f f e e e e g g g g u u u u a a a a g g g g l l l l a a a a r r r r e e e e n n n n e e e e g g g g , , , , y y y y t t t t i i i i r r r r o o o o h h h h t t t t u u u u a a a a d d d d e e e e g g g g d d d d e e e e l l l l w w w w o o o o n n n n k k k k c c c c a a a a e e e e h h h h t t t t : : : : z z z z t t t t n n n n e e e e r r r r o o o o L L L L . . . . E E E E

equivalent procedures exploit the arbitrariness of (24).
same expansion, but then explicitly makes a gauge transformation to arrive at (22). These
Lagrangian and then adds a total time derivative to obtain (22). Fock (1959) makes the
coefficients of the primed particle's velocity, acceleration, etc., to obtain a tentative
, with r/c = - t'-t) ), in powers of ( t' ( ' ' ' ' x x x x ) - t ( x x x x = r r r r potentials for a charged particle, which involve
deriving the approximate Lagrangian attributed to him, Darwin (1920) expands the retarded
observation is too obvious to warrant publication in other than textbooks. We note that, in
a total time derivative and so makes no contribution to the equations of motion. Perhaps this

dt t 
c c c
(24) , =  v v v v + e = int L 
e 1 1
 d 

Under the gauge transformation (1a,b) the Lagrangian (23) is augmented by
the text by Landau and Lifshitz (1941). See also Bergmann (1946). The proof is simple.
of this charged particle Lagrangian did not receive general consideration in print until 1941 in
It is curious that, to the best of the authors' knowledge, the issue of gauge invariance
are the potentials given by (18). A A A A where  and

c
(23) , , t) x x x x ( A A A A  v v v v + t) , x x x x ( - e = int L
1

, with retarded external electromagnetic potentials, v v v v velocity
int
and x x x x , with coordinate e describing the interaction of a charged particle L Lagrangian


17


doubt is removed in his book (Lorentz, 1909). There, in Note 5, he says,
arbitrariness of the potentials and then immediately restricted  to a solution of (27). This
what we term gauge invariance. He stated his constraint before his statement of the
A reader might question whether Lorentz was here stating the general principle of

c
c 2
(27) . 0  + 0 A A A A  =   - 
1 1 2

(2)] by solving the inhomogeneous wave equation,
and  do satisfy [Lorentz's A A A A He then says that the scalar function  can be found so that

c
(26) .  + 0  ,  =  - 0 A A A A = A A A A
1

transformations, and  " can be related to the first pair via the A A A A admissible pair
may give the same fields, but not satisfy his constraint. He then states "every other
0 0 and  A A A A He then discusses the arbitrariness in the potentials, stating that other potentials

t c
[Lorentz's (2)]. . = - A A A A 
1

equations they must be related by
1904b), he first states that in order to have the potentials satisfy the ordinary wave
potentials under what we now call general gauge transformations. On p. 157 of (Lorentz,
book (Lorentz, 1909). Here we find the first clear statement of the arbitrariness of the
electromagnetism are his magisterial encyclopedia articles (Lorentz, 1904a, 1904b), and his
Additional reasons for Lorentz being the reference point for modern classical
to Lorentz, to the exclusion of others.
A A A A evident that by 1900 the physics community had attributed the retarded solutions for  and
theorem (25a,b) and calls the retarded solutions (18) "Lorentz'schen Losungen." It is
mention of Lorenz! In the same volume, des Coudres (1900) cites (Lorentz, 1892) for the
cites Riemann in1858, Poincare in 1891, Lorentz (1892, 1895), and Levi-Civita in 1897. No
Wiechert (1900) summarizes the history of the wave equation and its retarded solutions. He
In a festschrift volume in honor of the 25th anniversary of Lorentz's doctorate, Emil
solutions, but without the condition (19).
as source. We thus see Lorentz in 1895 explicitly exhibiting retarded J J J J wave equation with
is defined as the curl of his vector field, it is sufficient that the field satisfy the H H H H notes that if
as source term, he merely J J J J     with H H H H potential. Having obtained the wave equation for
z y x
) the vector ,  ,  . He does not call his vector field ( v v v v =  J J J J equivalent to (18) with
1892) for proof, and then in Sect. 33 writes the components of a vector field in the form
In (Lorentz, 1895, Sect. 32), Lorentz quotes the theorem (25a,b), citing (Lorentz,

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

______________________________________________________________________________


18



(28) .  ) mc ( =  ) c /  eA -   i )( c / eA -  i (
2  


. Explicitly, we have )     , - ( = /x = 
0 
, where i TM p acting on a wave function  is constructed by the operator substitution,
  ij . Then a quantum mechanical operator = - g = 1, g potential. Here we use the metric
ij 00
is the 4-vector electromagnetic ) A A A A = , (A = A , where c / eA - p TM p the substitution
0    

, by (mc) = p p , m ) and mass p p p p , p = ( p constraint equation for a particle of 4-momentum
2  0  electromagnetic fields is derived in current textbooks by first transforming the classical
interacting with e The relativistic wave equation for a spinless particle with charge

transformations (1 a,b,c) by Fock.
both the scalar and vector potentials brought forth the discovery of the combined
charged particles, popularly known nowadays as the Klein-Gordon equation. The presence of
and publication dates.] The thread we pursue is the relativistic wave equation for spinless
[To document the pace, we augment the references for the papers in this era with submission
of gauge invariance in quantum theory. The pace among this restricted set is frantic enough.
contributions, we focus only on those that relate to our story of the emergence of the principle
mechanics, blossomed at the hands of Erwin Schrodinger and many others. Among the myriad
The year 1926 saw the flood gates open. Quantum mechanics, or more precisely, wave

k k k k c c c c o o o o F F F F , , , , n n n n i i i i e e e e l l l l K K K K , , , , r r r r e e e e g g g g n n n n i i i i d d d d o o o o  r r r r h h h h c c c c S S S S : : : : 6 6 6 6 2 2 2 2 9 9 9 9 1 1 1 1 . . . . A A A A

A A A A R R R R E E E E M M M M U U U U T T T T N N N N A A A A U U U U Q Q Q Q E E E E H H H H T T T T F F F F O O O O G G G G N N N N I I I I N N N N W W W W A A A A D D D D . . . . I I I I I I I I I I I I

invariance emerged.
electroweak theory and quantum chromodynamics that the deep significance of gauge
(18). It was only with the advent of modern quantum field theory and the construction of the
arbitrariness of the potentials, content to follow Lorentz in use of the retarded potentials
went about applying the subject with confidence. They did not focus on niceties such as the
electromagnetism almost completely clarified, with the ether soon to disappear. Scientists
Lorentz's domination aside, the last third of the 19th century saw the fundamentals of
integrals."
solutions (18) [his equations (X) and (XI)], which he later on the page calls "the Lorentz
second Encyclopedia article (Lorentz, 1904b) and his book (Lorentz, 1909) for the retarded
and . He then cites Lorentz's A A A A quotes (19) [his equation (IX)] and the wave equations for
citation by G. A. Schott in his Adams Prize essay (Schott, 1912). On p. 4, Schott
The dominance of Lorentz's publications as source documents is illustrated by their
electromagnetism without putting stress on it.
other choices, but he did recognize the general principle of gauge invariance in classical
text. Lorentz obviously preferred potentials satisfying his constraint to the exclusion of
He then proceeds to the wave equations and the retarded solutions in Sect. 13 of the main
equation (27)] which can be satisfied by a proper choice of ."
[Lorentz's (2)] which can always be fulfilled because it leads to the equation [our
and  to the condition A A A A (emphasis added). We shall determine  by subjecting
where  is some scalar function [our equation (26)] may as well be chosen by
0 0
other values that special values, we may represent and  A A A A "Understanding by


19


interactions and addressed the effect of the change in the potentials (1 a,b). He showed that
first discussed the special-relativistic wave equation of his earlier paper with electromagnetic
(1926b). His paper was submitted on 30 July 1926 and published on 2 October 1926. In it he
mechanical system of a charged particle interacting with electromagnetic fields is due to Fock
The discovery of the symmetry under gauge transformations (1 a,b,c) of the quantum

, 242 (1926)." 8 8 8 8 3 3 3 3 Zeitschrift fur Physik relativistic equation from a variational principle.
paper [(Schrodinger, 1926b)] was sent in, and also succeeded in deriving the
"V. Fock carried out the calculations quite independently in Leningrad, before my last
("Abstract") to his collected papers (Schrodinger, 1978),
the general electromagnetic interaction. Schrodinger comments in the introduction
had already commented on the solution in his first paper. In his paper Fock did not include
variational principle and solved the relativistic Kepler problem. He observed that Schrodinger
Klein's paper, Fock (1926a) independently derived the relativistic wave equation from a
independent Schrodinger equation, but did not discuss any solutions. Before publication of
energy with a static scalar potential. He showed that the nonrelativistic limit was the time-
formalism and explicitly exhibited the four-dimensional relativistic wave equation for fixed
In the published literature, Oskar Klein (1926) treated a five-dimensional relativistic
- to show the equivalence of matrix mechanics and wave mechanics (van der Waerden, 1973).
then specialized to the nonrelativistic Schrodinger equation, and went on to his main purpose
energy and momentum to derive a wave equation equivalent to (29) with a static potential,
private letter to Jordan dated 12 April 1926, Pauli used the relativistic connection between
Schrodinger was not the only person to consider the relativistic wave equation. In a
hydrogen atom and to the Zeeman effect.
tentatively presented the relativistic equation in detail and discussed its application to the
equation. Some months later, in Sect. 6 of his fourth paper (Schrodinger, 1926b), he
fine-structure formula, he did not publish his work, but focused initially on the nonrelativistic
completed it in early January 1926. Disappointed that he had not obtained the Sommerfeld
1925, began solving the problem of the hydrogen atom while on vacation at Christmas, and
(Moore, 1989, p.194-197), Schrodinger derived the relativistic wave equation in November
Schrodinger's 1926 papers can be found in (Schrodinger, 1978)). According to his biographer
results of his study of the "relativistic Kepler problem." (An English translation of
independent wave equation and simple potential problems, but in Section 3 he mentions the
January and published on 13 March. It was devoted largely to the nonrelativistic time-
The first of Schrodinger's four papers (Schrodinger, 1926a) was submitted on 27
potentials.
 = 0 is chosen for the A  The second term on the left is absent if the Lorenz gauge condition 
(30) .  ) mc (
2 2 - ) e - E ( =  A A A A  A A A A e
2 2 +  A A A A c  ie 2 + )  A ( c  ie +   c  -
 2 2 2

yields the relativistic version of the Schrodinger equation, iEt/),  , exp(-
Separation of the space and time dimensions and choice of a constant energy solution,


(29) . 0 =  /) mc ( + ) c /  ie A +  )( c / ie A + (
2  


and write Alternatively, we divide through by -2


20


interaction as
can be written in terms of the solution without A with the electromagnetic potential

solution for a particle interacting formal relativistic wave equation (29), we observe that a
 is path-dependent. If we return to the indefinite integral over the real "potential"
 0 
; the dx  (x) = where (x), exp ... = ... leads to a formal solution dx d... = ... scale
 x  To understand London's point we note first that Weyl's incremental change of length
dimensions.
(without references) both Klein and Fock for the relativistic wave equation in five
(1926b) but does not repeat the citation in his longer paper, although he does mention
was made imaginary. In his short note London cites Fock (x) transformations (1) provided
equivalent in quantum mechanics to the invariance of the wave equation under the
is an arbitrary function of the space-time coordinates, was (x) , where exp (x) g
TM g principle of invariance of his theory under a scale change of the metric tensor 
undertaken long before the discovery of quantum mechanics. London noticed that Weyl's
attempt to unify electromagnetism and gravitation (Weyl, 1919). This attempt was
paper (London, 1927b), proposed a quantum mechanical interpretation of Weyl's failed
Fritz London, in a short note in early 1927 (London, 1927a) and soon after in a longer
e e e e l l l l p p p p i i i i c c c c n n n n i i i i r r r r p p p p c c c c i i i i s s s s a a a a b b b b a a a a s s s s a a a a e e e e c c c c n n n n a a a a i i i i r r r r a a a a v v v v n n n n i i i i e e e e g g g g u u u u a a a a g g g g : : : : l l l l y y y y e e e e W W W W . . . . B B B B

(1 a,b,c) and today's generalizations.
seemingly inappropriate word "gauge" came to be associated with the transformations
(electromagnetism and gravity). Along the way, we retell the well-known story of how the
transformations as a guiding principle for the construction of a quantum theory of matter
The tale now proceeds to the enshrinement by Weyl of symmetry under gauge
1926 (Fock, 1926b).
important point in our story is Fock's paper on the gauge invariance, published on 2 October
claim priority. Totally apart from the name attached to the relativistic wave equation, the
Klein-Fock-Schrodinger equation; if by notebooks and letters, Schrodinger and Pauli could
chronologically by publication dates, the Klein-Gordon equation should be known as the
care taken by some, but not all, for proper acknowledgment of prior work by others. If we go
The above paragraphs show the rapid pace of 1926, the occasional duplication, and the
relativistic equation. Gordon does not cite Klein or either of Fock's papers.
papers, but not the fourth (Schrodinger, 1926b) in which Schrodinger actually treats the
describe the scattering of light by a charged particle. He referred to Schrodinger's first three
Walter Gordon (1926) discussed the Compton effect using the relativistic wave equation to
equation reduced to Fock's for the Kepler problem with the appropriate choice of potentials.
covariant notation, citing Klein (1926) and Fock (1926a). He remarked that his general
That fall others contributed. Kudar (1926) wrote the relativistic equations down in
and that the principal results were identical.
arrived in Leningrad,"
"While this note was in proof, the beautiful work of Oskar Klein [published on 10 July]
formalism, similar to but independent of Klein. In a note added in proof, Fock notes that
transformed according to (1c). He went on to treat a five-dimensional general-relativistic
the equation is invariant under the change in the potentials provided the wave function is


21


seen, in his 1928 book he presented the idea of gauge invariance in the unadorned version of
the other hand wanting to establish contact with his 1919 "eichinvarianz." As we have just
principle obviously applied to the electromagnetic fields and charged matter waves, and on
gauge invariance in relativistic quantum mechanics, knowing on the one hand that the
unique to him. Presumably prompted by London's observation, he addressed the issue of
Weyl's 1928 book and his papers in 1929 demonstrate an evolving point of view
relativity.
on how the gauge transformation (1c) can only be fully understood in the context of general
In fact, in the second edition a new section 6 appears in Chapter IV, in which Weyl elaborates
believed that his own work in the meantime (Weyl, 1929a, 1929b) had shown the connection.
By the second (1931) edition, this sentence has disappeared, undoubtedly because he
present."
"How gravitation according to general relativity must be incorporated is not certain at
first (1928) edition, the next sentence reads (again in translation):
His note 22 refers to his own work, to Schrodinger (1923), and to London (1927b). In the
above."
in the manner described electricity and matter electricity and gravitation, but rather
. But I now believe that this gauge invariance does not tie together electricity22
author, on speculative grounds, in order to arrive at a unified theory of gravitation and
is quite analogous to that previously set up by the gauge invariance' principle of "This `
under the transformations (1a,b,c). Weyl then states on p. 88 (in translation):
that the electromagnetic equations and the relativistic Schrodinger equation (28) are invariant
relativistic charged particle with the electromagnetic field. He observes, without references,
"Gruppentheorie und Quantenmechanik" (Weyl, 1928), Weyl discusses the coupling of a
local phase change and the latter a coordinate scale change. In his famous book,
an analogue of Weyl's "eichinvarianz" (scale invariance), even though the former concerns a
The "gradient invariance" of Fock became identified by London and then by Weyl with
0 0
. /... =  /... 
0
is the quantity in the exponential in (31), and wrote i(x) where (i(x)), exp ... = ... change"
which is precisely Fock's (1c). London actually expressed his argument in terms of "scale

(33) ,  ) c )/ x ( ie ( exp = '

transformation
to a constant phase, the wave functions ' and  are thus related by the phase

. Up dx   the difference in phase factors is obviously the integral of a perfect differential, -


(32) ,   - A = A TM A
   ' 

potential
0 0  0
= 0.] With the gauge transformation of the 4-vector  (mc/) +    and requiring 
2  0 is the zero-field solution. [Recovery of (29) may be accomplished by "solving" for where 

c 
(31) , 0  dx  A i - exp = 
 e
x


22


experiment and in particular its renormalizability would not be impaired (see e.g., Kobzarev
Lagrangian is not gauge invariant. At the same time the excellent agreement of QED with

in the A the photon would destroy the gauge invariance of QED, as a mass term m
2 2 its physical meaning in QED per se does not seem to be extremely profound. A tiny mass of
While for Electroweak Theory and QCD gauge invariance is of paramount importance,

s s s s c c c c i i i i n n n n a a a a h h h h c c c c e e e e m m m m m m m m u u u u t t t t n n n n a a a a u u u u q q q q d d d d n n n n a a a a D D D D E E E E Q Q Q Q n n n n i i i i e e e e c c c c n n n n a a a a i i i i r r r r a a a a v v v v n n n n i i i i e e e e g g g g u u u u a a a a g g g g f f f f o o o o g g g g n n n n i i i i n n n n a a a a e e e e m m m m l l l l a a a a c c c c i i i i s s s s y y y y h h h h p p p p e e e e h h h h t t t t n n n n O O O O . . . . A A A A

S S S S E E E E L L L L P P P P M M M M A A A A X X X X E E E E , , , , E E E E C C C C N N N N A A A A I I I I R R R R A A A A V V V V N N N N I I I I E E E E G G G G U U U U A A A A G G G G F F F F O O O O G G G G N N N N I I I I N N N N A A A A E E E E M M M M L L L L A A A A C C C C I I I I S S S S Y Y Y Y H H H H P P P P . . . . V V V V I I I I


of introducing gauge invariance in quantum theory.
blames it on Pauli (1933). Indeed, Pauli made that error, but he did give to Fock the priority
number are given correctly, the year is invariably given as 1927. One of the writers privately
O'Raifeartaigh and Straumann (2000), and Yang (1986, 1987). While the volume and page
regarding the citation of Fock's 1926 paper (Fock, 1926b) by O'Raifeartaigh (1997),
already mentioned in the Introduction. The reader should be warned, however, of a curiosity
of the 20th century. The important developments beyond 1929 can be found in the reviews
touchstone of the theory of gauge fields, so dominant in theoretical physics in the second half
equations under gauge transformations such as (1c) of the matter fields. This principle is the
potentials (and field strengths) follow from the requirement of the invariance of the matter
fundamental the modern principle of gauge invariance, in which the existence of the 4-vector
Historically, of course, Weyl's 1929 papers were a watershed. They enshrined as
2000).
not generally realized until much later (See e.g., Yang, 1986; O'Raifeartaigh and Straumann,
between non-abelian gauge fields and general relativity as connections in fiber bundles was
and his desire to provide continuity with his earlier work. The close mathematical relation
function in the curved space-time of general relativity, but not necessarily in special relativity,
an arbitrary must be need for general relativity can perhaps be understood in the sense that 
The last sentence of (Weyl, 1929b) contains almost the same words. His viewpoint about the
the matter wave field and not of gravitation."
"If our view is correct, then the electromagnetic field is a necessary accompaniment of
Nevertheless, Weyl stated (Weyl, 1929a, p.332, below equation (8)),
O'Raifeartaigh and Straumann, 2000, p. 7).
an arbitrary function of position." (translation taken from necessarily becomes
removal of the rigid connection between tetrads at different points the gauge-factor
point has its own tetrad and hence its own arbitrary gauge-factor: because by the
we have only a single point-independent tetrad. Not so in general relativity; every
" In special relativity one must regard this gauge-factor as a constant because here
He elaborated on this point in (Weyl, 1929b):
certainly only be understood with reference to it."
it contains an arbitrary function , and can character of general relativity since
"This new principle of gauge invariance, which may go by the same name, has the
that
consequence of gauge invariance (through the matter and the electromagnetic equations) and
authority. He then goes on to show that the conservation of electricity is a double
invariance" (the first use of the words in English) very much as in his book, citing only it for
papers on the electron and gravitation (Weyl, 1929a), he states the "principle of gauge
Fock, without the "benefit" of general relativity. But in the introduction of the first of his 1929


23


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


(39) Fock-Schwinger gauge =0 , A x


3
(38) axial gauge = 0 , A

o
(37) Hamiltonian or temporal gauge = 0 , A


(36) light cone gauge = 0) , n = 0 ( A n
2 

j j
(35) = 1, 2, 3) , Coulomb gauge or radiation gauge j ( = 0 A  = A A A A        


(34) Lorenz gauge = 0 ( = 0,1, 2, 3) , A 


invariant:
with various gauge conditions, most of them being not Poincare A to consider the potential
 The gauge invariance of classical field theory and of electrodynamics in particular allows one

s s s s e e e e g g g g u u u u a a a a g g g g f f f f o o o o s s s s e e e e l l l l p p p p m m m m a a a a x x x x E E E E . . . . B B B B

field or the potential, but the local vector potential is not an observable.
field, albeit in a nonlocal manner. It is a matter of choice whether one wishes to stress the
magnetic flux through the loop, showing that the result is expressible in terms of the magnetic
loop integral of the vector potential there can be converted by Stokes's theorem into the
taken over a closed path, as in the Aharonov-Bohm effect (Aharonov and Bohm, 1959). The
. However integrals such as in eq (31) are observable when they are observability of A
As has been emphasized above, gauge invariance is a manifestation of non-
reviews, see Okun, 1989, 1992).
that reabsorption of virtual bremsstrahlung photons restores the conservation of charge (for
irrelevant (Okun and Zeldovich, 1978). Further study has shown (Voloshin and Okun, 1978)
which search for monochromatic photons in charge-nonconserving processes become
of non-conserved current to such catastrophic bremsstrahlung, that most of the experiments
 lead in the case m It should be stressed that the existing upper limits on the value of
 . m small enough values of
 
and therefore negligibly small for m , proportional to m nonvanishing mass of the photon,
2 Okun, 1986, Lecture 1). Conservation of charge makes the effects caused by a possible
conservation of electromagnetic current or in other words conservation of charge (see e.g.,
spite of its manifest gauge invariance. What is really fundamental in electrodynamics is the

, in F  be destroyed by an anomalous magnetic moment term in the Lagrangian, ƥ
 and Okun, 1968; Goldhaber and Nieto, 1971). On the other hand the renormalizability would


24


S S S S K K K K R R R R A A A A M M M M E E E E R R R R G G G G N N N N I I I I D D D D U U U U L L L L C C C C N N N N O O O O C C C C D D D D N N N N A A A A Y Y Y Y R R R R A A A A M M M M M M M M U U U U S S S S . . . . V V V V


gauge" ( = 3 in (41)).
Zumino (1960) introduced the terms "Feynman gauge," "Landau gauge," and "Yennie
1936). In the third edition (Heitler, 1954) he used "Lorentz gauge" and "Coulomb gauge."
Heitler who introduced the term "Lorentz relation" in the first edition of his book (Heitler,
Various gauges have been associated with names of physicists, a process begun by
This is the propagator for the Helmholtz potential (14) and the static Coulomb potential.


(46) . 0 = 0) , k k k k ( 0 i D = 0) , k k k k ( j 0 D

| k k k k |
2
(45) , = 0) , k k k k ( 0 0 D
1
| k k k k | | k k k k |
2 2
(44) , 1) - ( + i j = 0) , k k k k ( i j D
j k i 1
k

In the static (zero frequency) limit the propagator (41) reduces to
correctly, the final result will not contain the gauge parameter .
longitudinal part vanishes, which is often more convenient. If calculations are carried out
In the Feynman gauge the propagator (41) is simpler, while in the Landau gauge its

(43) (Landau gauge).  = 0

(42) ( Feynman gauge),  = 1

The most frequently used cases are

k k
2 2
(41) . 1) -  ( + g - = ) k ( D
k k  1 
 

acquires the form, k propagator of a virtual photon with 4-momentum
Lifshitz, and Pitaevskii, 1971; Ramond, 1981; Zinn-Justin, 1993). In perturbation theory the
with coefficient 1/2 (For futher details see Gaig, Kummer, and Schweda, 1990; Berestetskii,

is added to the gauge invariant Lagrangian density ) A Usually the gauge fixing term (
2  In QED the gauge degree of freedom has to be fixed before the theory is quantized.
and with a set of operators.
additional dimensions in quantum field theory where one has to deal with a space of states
of problems discussed in Gaig, Kummer, and Schweda (1990). The problems acquire
noncovariant gauges, characterized by fixing a direction in Minkowski space, pose a number
unquantized; only the transverse vector potential of the photons is quantized. However,
because the instantaneous scalar potential describing the static interactions and binding is
charged particles interacting with radiation, the Coulomb gauge is particularly convenient
presented in textbooks, e.g., (Jackson, 1998). For the quantum mechanics of nonrelativistic
An appropriate choice of gauge simplifies calculations. This is illustrated by many examples


25


Division, CERN.
grant RFBR # 00-15-96562, by an Alexander von Humboldt award, and by the Theory
of Energy under Contract DE-AC03-76SF00098. The work of LBO was supported in part by
Director, Office of Science, Office of High Energy and Nuclear Physics, of the U.S. Department
Telegdi for their assistance and advice. The work of JDJ was supported in part by the
The authors thank Robert N. Cahn, David J. Griffiths, Helge Kragh, and Valentine L.

S S S S T T T T N N N N E E E E M M M M E E E E G G G G D D D D E E E E L L L L W W W W O O O O N N N N K K K K C C C C A A A A

(Cheng and Li, 1988) is recommended.
text-books know about the history of physics. For a further reading the Resource Letter
required the invention of quantum mechanics. It is amusing how little the authors of
notion of gauge symmetry did not appear in the context of classical electrodynamics, but
loss of interest by O. Klein and L. Lorenz in their "god blessed children". It is striking that the
important early players strangely diminished with time. There is a kind of echo between the
The history of gauge invariance resembles a random walk, with the roles of some
started with the paper by Yang and Mills (1954).
refer to Weyl). But this attempt was firmly forgotten. The modern era of gauge theories
interactions was proposed by Klein (1938) (who did not use the term "gauge" and did not
The first model of a non-abelian gauge theory of weak, strong, and electromagnetic Fock, they used Fock's term "gradient invariance"). fields and charged particles was presented by Landau and Lifshitz (1941) (with reference toderivation of the invariance of the Lagrangian for the combined system of electromagneticdiscussed by Lorentz in his influential book, "Theory of Electrons" (Lorentz, 1909). The first In text books on classical electrodynamics the gauge invariance (1a,b) was firstand electromagnetism).
transformation in his unsuccessful attempt to unify gravity scale a decade before to denote a
1928, 1929a, 1929b) (who used "eich-" ( "gauge" was applied to this transformation by Weyl
transformation of the electromagnetic potentials was discovered by Fock (1926b). The term
phase transformation (1c) of the quantum mechanical charged field accompanying the
by its creators, Lorenz, Maxwell, Helmholtz, and Lorentz, among others (1867-1909). The
potentials (1a, b) was discovered in the process of formulation of classical electrodynamics
What is now generally known as a gauge transformation of the electromagnetic


26


537-551.
, 9 9 9 9 3 3 3 3 Darwin, C. G., 1920, "The Dynamical Motions of Charged particles," Phil. Mag., ser. 6,

(Oxford University Press) Electrodynamics from Ampere to Einstein Darrigol, O., 2000,

, 652-664. 5 5 5 5 Ueberlichtgeschwindigkeit Bewegen," Arch. Neerl. Scs., ser. 2,
des Coudres, Th., 1900, "Zur Theorie des Kraftfeldes elektrischer Ladungen, die sich mit

, 255-279 0 0 0 0 1 1 1 1 of the ponderomotive and electromotive forces," Phil. Mag., ser. 5,
Clausius, R., 1880, "On the employment of the electrodynamic potential for the determination

, 85-130. 2 2 2 2 8 8 8 8 Journal fur Math. (Crelle's Journal)
Clausius, R., 1877, "Ueber die Ableitung eines neuen elektrodynamischen Grundgestezes,"

, 586-600. 6 6 6 6 5 5 5 5
Cheng, T. P., and Ling-Fong Li, 1988, "Resource Letter: GI-1 Gauge invariance," Am. J. Phys.

(University of Chicago Press, Chicago)
The Creation of Scientific Effects, Heinrich Hertz and Electric Waves Buchwald, J. D., 1994,

Chicago)
(University of Chicago Press, The Rise of the Wave Theory of Light Buchwald, J. Z., 1989,

(University of Chicago Press, Chicago) From Maxwell to Microphysics Buchwald, J. Z., 1985,

, 210-222. 8 8 8 8 5 5 5 5 Bork, A. M, 1967, "Maxwell and the Vector Potential," Isis

Paris), p. 745.
3rd ed., vol 2 (Deterville, Precis Elementaire de Physique Experimentale, Biot, J. B., 1824,

222-223. , , , , 5 5 5 5 1 1 1 1 et de Phys, ser. 2,
, 151; "Note sur le Magnetisme de la pile de Volta," Ann. de Chimie 1 1 1 1 9 9 9 9 Chimie,
Biot, J. B., and F. Savart, 1820, "Experiences electro-magnetiques," Journal de Phys., de

p.115-117.
(Prentice-Hall, New York), Introduction to the Theory of Relativity Bergmann, P. G., 1946,

Pergamon, Oxford), Section 77. ( Part 1 , Theory
Relativistic Quantum Berestetskii, V. B., E. M. Lifshitz, and L. P. Pitaevskii, 1971,

, 175-388 [memoirs presented from 1820 to 1825]. 6 6 6 6 de l'Institut de France, ser. 2,
uniquement deduite de l'experience," Memoires de l'Academie Royale des Sciences
Ampere, A.-M., 1827, "Sur la theorie mathematique des phenomens electrodynamiques

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33


FIGURE 1, JACKSON & OKUN



O



x'

C' x


' I C
ds' n' = s' d



x - x' = r

ds n = s d I





, respectively I' and I with currents C' and C Figure 1. Two closed current carrying circuits

n n n n o o o o i i i i t t t t p p p p a a a a C C C C e e e e r r r r u u u u g g g g i i i i F F F F


34


35



