Eletromagneti Duality, Charges,
Monopoles, Topology, ...
Juan A. Mignao
Inst. deF
isia,Univ. Fed. doRiodeJaneiro
21945-970,Riode Janeiro, RJ,Brazil
E-mail: mignaoif.ufrj.br
Reeivedon25January,2001
Inthistalkwereviewthesubjetofeletromagnetiduality,andrelatedones,as\normal"eletri
andmagnetiharges,beingdierentfrommonopoles(magnetioreletri),thetopologialontent
ofmonopoles, surfaeterms,Dirastringsandhiralstrutureintheinlusion ofmasslessspinor
eldanditsrelationtotorsion. Welimitourdisussiontothesimplest,Heavisideeletromagneti
duality,inlude somelinkswith thenonabelian gauge theoriesand make noomment onthe use
ofthis oneptinstringtheories. Old andreentontributionsonthe subjetare presented and
disussed,andpossiblelinesofresearhwillhopefullybeindiated.
I Introdution
In this talk I intend to review the notion of
eletro-magneti duality in its simplest forms. My views on
the subjet result from a long standing ollaboration
withProf. CarlosA.P.Galv~aorstatCBPFandnow
at UnB (with several artiles in preparation, and
bit-ter disussions with referees), and numerous pleasant
onversationswith Prof. Clovis Wotzasek. Given the
hugeamountofpaperin theliteratureonthesubjet,
I an't think to have overed all the available
mate-rial,andmanywillndthatimportantrefereneswere
not given enough relevane. I apologize toofor those
hereatS~aoLourenoandelsewherewhoseown
ontri-butions in theiropinion didn'treeivetheappropriate
onsideration.
The oneptof duality has reeivedprominent
at-tention in modern gauge theories. It provides
use-ful tools to onstrut solutionsto the eld equations,
namely, those whih are self-dual or anti-self-dual, or
allowstostudyregimesofthetheorywhihpreventthe
use of perturbation expansions. These features seem
partiularlyimportantforreentdevelopmentsinstring
theoryanditsderivatives,suhasbranes,M-theory[1℄.
I shallnotdealwiththeseadvanedmatters.
Dualityinlassialeletromagneti theorywas
dis-overedbyHeaviside[2℄aenturyagofortheMaxwell
equationsin vauum. Hesawthat
r^E= B
t
r^B= 1
2 E
9
>
=
>
;
: (1)
exhangeamong themselvesunderthereplaements
E! B; B!E: (2)
Thissymmetryofthesystem,duality(weshallrefer
toitasHeavisideduality),originatedalotof
speula-tionaboutits meaning: is theeletri eld tobe
on-sideredequivalentto themagnetiindutioneld,and
thereverse?.
It seemsthat Larmor[3℄generalizedHeaviside
du-alitytoaontinuoustransformation:
E
=Eos Bsin
B
=Esin+Bos
;0
2
: (3)
This emphasizes that there is omplete ambiguity
or equivalently, ontinuous freedom, in the hoie of
eletriandindution eldsfortheradiationelds.
Withtheadventofnonabeliangaugetheoriesfor
el-ementarypartilephysis,alotofworkinPhysisand
Mathematis hasbeenperformedto larifythe
mean-ingandappliabilityofduality,asexposedabove,orin
itsmodernnonabelianversions.
Letusreallthat theoriginaleldsin theMaxwell
equationshaveeletrihargesorurrentsand/ortime
varying elds as theirsoures. Evenin vauum,
ele-tri elds are onsidered the ones aelerating
ele-tri harges parallel to their diretion, whereas
mag-neti elds provide transverse aelerationfor eletri
harges. Magnetimaterials arerelatedto elementary
magneti dipoles but single isolated magneti harges
haveneverbeenobserved.
Dira [4℄, motivated by the need to explain the
quantization of the eletri harges, introdued
forthemagnetiindution eld,i. e.,
rB=
0
gÆ(x) (4)
To preserve the relation with the magneti vetor
potential,
B=r^ A (5)
a lamentary solenoid had to extend from the
posi-tion of the magneti harge to innity. In the
quan-tumregime,theunobservabilityofthisdeterminedthe
famous Dirarelationbetweenthe eletrihargeof a
testpartileandthestrengthofthemonopoleeld 1
:
qg=2n ~
0
: (6)
The topologial haraterization of the problem
[6℄[7℄ wasestablishedalmost halfaenturylater. It is
usuallystronglyassumedthattheintrodutionof
mag-netimonopolesisrelatedtoHeavisideduality.
Thereremainunwantedaspetsofthetheoryatthe
eletromagneti level. Standing high is the fat that
thelagrangianofthetheoryhangessignunderduality,
that is, Heaviside dualityis asymmetry forthe
equa-tions of motion but not for the lagrangian providing
them. Negletingsoures,
LfE;Bg=
0
2 Z
d 3
x E 2
2
B 2
(7)
Let me list the works whih somehow were
land-marksonthissubjet.
Dira's1947artileproposingastringtheoryfor
monopoles[8℄
ThetwopotentialformalismbyNisbetin1955[9℄
andCabibboandFerrariin1962[10℄
Rohrlih'sartileontheimpossibilityofa
lassi-al loaltheorywithmonopoles[11℄
Shwingerontributionsin1966-75[12℄
Zwanziger formulations in eld theory 1968-72
[13℄
The formalismof bre bundles for themagneti
monopole set forth by Greub and Petry [6℄ and
Wu andYang[7℄
DeserandTeitelboim onstrutionofanonloal
dualitygeneratorin 1976[14℄
OliveandMontonenworkin1977aboutmagneti
monopolesingaugetheories[15℄
ShwarzandSenworkin1994withanon-Lorentz
invariantlagrangian[16℄
Seiberg and Witten use of duality in
supersym-metristringtheoriesin 1995[17℄
In thefollowing,I shalldwell onsomeof this work,a
ritiism will be addressedwhen onsidered neessary
andshalltrytoframethemin aommonrealm.
II Some preliminaries
Thetalkisaboutaspeisymmetry. Whatwe
un-derstandunderthisnameisapreisedynamialidea: a
systemexhibitsasymmetrywhenunderits ationthe
propertiesofthesystemremainunaltered. Conversely,
asymmetryisanationwhihannotdisriminate
dif-ferenesonagivenphysialsystem.
To see that it is a dynamialonept, take for
in-stanethehydrogenatom: aslongasyourspetrosope
isnotableto separatethene struture,itlooks
sym-metriunder theusualspaerotationsymmetry. One
thespinofthepartilesistakenintoaount,this
sym-metrydoesn'tholdanymore.
Thebreakingof asymmetryresultsin anordering
oftheresultingphysialharateristis: itmaygrossly
preserve it, as in the preeding ase of the hydrogen
atom, ordisturb itdeeply, astheeletroweak
symme-trybreaking.
Tox notations, as shown before vetorsin
three-spae willbedenoted by boldfae haraters, A, j ,E,
B, .... ; in what follows, four dimensional tensorial
quantities willbedenoted byitalis, A,j, F,.... ;and
nally, dierentialforms byalligraphiletters,A ,J
,F ,.... .
TheeldintensitiesEandBwillbeassoiatedto
theeldstrengthtensor,F
,andatwo-form
dieren-tialformin four-dimensionalspaetime:
F =
1
2 F
dx
^dx
(8)
= F
0k dx
0
^dx k
+F
ij dx
i
^dx j
(9)
The usual inhomogeneous Maxwell equations read
(repeatedindiesindiate sums)
F
=
0 j
(10)
with j
(;j). From the homogeneous equations,
onehasBianhiidentityforF :
F
+
F
+
F
=0 (11)
Usingdierentialforms,wehaveforMaxwell
equa-tions
ÆF =
0 J
e
dF =0
(12)
1
Itisworthpointingthatinthe expressionappearsaquantumofuxwhihishalftheoneneededforthe unobservabilityofthe
solenoidintheEhrenberg-Siday-Aharonov-Bohm\experimental"setup.
Curiouslyputtingforqthevalueoftheeletroniharge,andntakenastwo,theresultingvalueforgistheinverseoftheonstant
ThesymbolsdandÆorrespondtodierential
oper-ators, thegeneralizations forantisymmetritensorsof
theusualurlanddivergeneoperatorsinthreespae.
Givenadierentialp-form!
!=w
i
1 :::i
p dx
i1
^^dx ip
(13)
theationofdisgivenby
d!=
i
k w
i
1 i
p dx
i
k
^dx i1
^^dx ip
(14)
that is, it produes a dierential (p+1)-formfrom a
dierentialp-form . The other operator performs the
opposite way, relatingadierentialp-formwith a
dif-ferential(p 1)-form,as
Æ!= g ikil
i
k w
i
1 i
p dx
i1
^(nodx il
)^dx ip
: (15)
Another operation whih will play a r^ole in what
follows is the Hodge duality operator for dierential
forms,, whih relates dierential p-forms with
dier-ential (n p)-forms, n being the total dimension of
themanifold, in mostofourtalk thefour-dimensional
spaetime. Atingonthebase monomialsitgives:
dx
1
^^dx
p
=
1
(n p)! g
11
g pp
"
1pp+1n dx
p+1
^dx n
:
(16)
p-forms into a Hilbert spae, dening properly an
in-ternal produt. With this denition, the dierential
operatorsdandÆ appearasadjointtotheother.
Itiseasytoshowthatfordierential2-forms,Hodge
duality applied to F produes the resultof Heaviside
dualityfortheelds(E;B).
In the real world where eletrons strike on TV
sreensandeletrigeneratorslightthenightsof
mod-erntimes,thepreseneofsoures,eletrihargesand
urrent densities, abrogates the validity of Heaviside
duality. LetusrealltheompletesetofMaxwell
equa-tions:
rD =
e
rB =0
r^E = B
t
r^H =j
e +
D
t 9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
(17)
Eletrihargesandtheinvarianeoftheequations
under Lorentz transformations show that the eletri
and magnetieldsannotbetakenonthesame
foot-ing. Infat,magnetields are akindof\kinemati"
eetfromhargesinmotion,ingeneral. Onlyfor
radi-ation elds botheletri andmagneti indution have
equivalentproperties.
In terms of four-dimensional eld tensors, in
va-uum, we have eqs. (10) and (11), and we may write
them in terms of dierential forms as eqs. (12). For
whatfollows,letmewritethelatterinitsdualversion:
ÆF =0
dF =
0 J
e
(18)
In the following, having in mind the extensions of
theformerequations into ageneralizedduality
frame-work,Ishalltrytoanalyzethepossibleanswerstosome
appealingquestions:
Does thegeneralization of Heaviside duality
im-ply exlusively the introdution of Dira's
mag-neti monopoles?
Shouldthemagnetieldofamagnetihargebe
thesameasobtainedfromeletriurrents?
Should the generalized Heaviside duality to be
takenasidentialtoHodgedualityfordierential
forms?
Is the present physial world emerging from a
breakingofHeavisideduality?
III Duality without soures
ThetransformationdevisedbyHeavisidehasa
dis-omforting onsequenealready at thefree-eld level.
Theenergydensityoftheeletromagnetieldremains
invariant:
u
em =
1
2
0 (E
2
+ 2
B 2
) (19)
whereastheationdensity
s= 1
2
0 (E
2
2
B 2
) (20)
hangessign. Heaviside'stransformationisasymmetry
fortheequationsofmotion,but notforthelagrangian
generatingthem.
There are two most aepted ways of overoming
this,therstoneproposedin 1976byDeserand T
eit-elboim [14℄ , and almost twodeadeslater theseond
byShwarzandSen[16℄.
DeserandTeitelboimlookedforageneratorforthe
Heaviside duality transformations that leaves the
en-ergy invariant. This they sueeded to nd, and
pro-posed anexpressionloal intime:
G= 1
2 Z
d 3
x(E r 2
r^E+ 2
Br 2
r^B) (21)
Thefatthatonehastoreurtoagenerator
non-loalinspaeissomewhatugly.
Shwarz and Sen [16℄ went along dierent lines of
thought. They proposed an ation with two sets of
eldsandpotentials(=1;2):
S= 1
Z
d 4
x(B ()i
L
E
()
i +B
()i
B ()
i
withthematrixL being
kLk=
0 1
1 0
: (23)
Theeldsaredenedthrough
E ()
i
=
0 A
()
i
i A
()
0 ;
B ()i
= " ijk
j A
()
k
;=1;2;1ijk3:(24)
Takingintoaountthegaugefreedomofthetheory
theyset
A ()
0
=0: (25)
Oneof theequationsofmotionis
B (2)
k =E
(1)
k
(26)
and it allows to reover the usual Maxwell equations
forthe=1elds. Thedualityforthepotentialshas
aloal expression
A ()
=L
A
()
(27)
theonlyawbeingtheapparentLorentznoninvariane
oftheequations. TheywereabletoprovethatLorentz
invarianeholds.
Pakman [20℄ observed that for the free elds the
equationsofmotionread:
F
[A
℄=0; (28)
with
F
=
A
A
: (29)
The last equation implies the Bianhi identities,
whih,writtenin termsoftheHodgedualof F
read
F
[A
℄=0: (30)
He proposes to introdue a new potential for the
dualtensor,Z
,
F
[Z
℄=
Z
Z
(31)
andwritingtheationintermsofthedualtensor,the
equationofmotionisnow
F
=0 (32)
whereas theorrespondingBianhi identities (the
for-merequationsofmotion)are
F
[Z
℄=0: (33)
Weshallshowbelowotherexamplesonthis lineof
thought. TheShwarz-Senationmaynowbewritten
intermsofthetwofourpotentials,A
,Z
,anditisa
kindofinterpolationforthesituationsdesribedabove.
Besides, Pakman proposes aontinuous U(1)
general-izationusingomplexeldsandpotentials.
Girotti[21℄analyzedthequantizationofthismodel
in theCoulomb gaugeusingtheDirabraket
formal-ism. He showedthat thequantized theory propagates
masslesspartiles,andexpliitlydemonstratedthatthe
theoryisgenuinelyLorentzinvariant. Infat,Iseetwo
masslesspartilespropagating.
Dwelling deeperinto thesubjet, Girotti, Marelo
Gomes,VitorRivellesandAdilsondaSilva[22℄showed
thattheformalismofShwarzandSenatthequantum
level isequivalentto theproposalofDeserand
Teitel-boim,andofourse,bothreproduethequantum
the-oryoftheMaxwelleld. Theyobtainedthehargethat
generatestheinnitesimaldualitytransformation:
Q =
1
2 R
d 3
x ijk
(
j A
()
i )A
()
k
(34)
= 1
2 R
d 3
xB ()k
A ()
k
: (35)
Its relation with aChern-Simonsterm guarantee that
itisgaugeinvariant.
AninterestinginterpretationoftheworkbyShwarz
andSenwasshowntomebyClovisWotzasek. Takethe
osillatordeompositionofmodesatagivenfrequeny,
!=!(k). Thelagrangiandensitymaybewrittenas
L=px_ 1
2 p
2 1
2 x
2
(36)
and write px_ = 1
2
(px_ xp),_ then redenep=x
1 and
x =x
2
and onereoversthe Shwarzand Senform of
thelagrangian. Theproedureis areparametrization,
arelabellingofthepotentialelds.
Otherproposalstooverometheproblemof
ovari-anearetheintrodutionofaninnitenumberof
aux-iliaryelds[18℄oranon-polynomiallagrangianof
aux-iliaryelds[19℄.
Inpreviouseditionsof this meeting we haveheard
ontributionstothisproblemfromGirottiandour
ol-leagues of USP, and others by Clovis Wotzasek,
Ra-bin Banerjee and ollaborators. The latter study the
generalization of Heaviside duality to spaetimes of
higher(even)dimensions. Therelevantextended
ele-tromagnetitheoryistheonedenedthrough
antisym-metri tensors having 2(n 1) omponents (n being
as before the number of spaetime dimensions). For
n = 4k, k:integer, the group implementing duality is
Z
2
, whereas for dimension n = 4k+2, the group is
SO(2). Byintroduing\external"additionalvariables,
theynda\dualityofduality",inthesensethatwhen
thegroupisZ
2
(n=4k forthenormalvariables), the
\external" onestransform asSO(2), andthe opposite
in the other dimensions [23℄. The new \duality"
ex-hangesinternal andexternalvariables 2
.
Notiethatallthese worksdonottakeforgranted
thattheeletromagnetieldsintroduedthroughnew
potentialsare indeed the same asthe old ones. They
either proposetoinorporatethesymmetryin
eletro-magnetism,suitablygeneralized,oranewtheorywhere
thesymmetryisimplemented.
IV Inluding soures
ThegeneralizationofMaxwellequationswithmagneti
hargesandurrentsis[12,24, 26℄:
rD =
e
rB =
0
m
r^E = 1
0 j
m B
t
r^H =j
e +
D
t
9
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
;
(37)
There are no homogeneousequations and wehave
twoonservedurrents:
rj
e +
e
t =0
rj
m +
1
2
m
t =0
9
=
;
(38)
Notiethatthemagnetihargesarerelatedtothe
ux ofthemagneti indutioneld. CallingQ
m;V the
totalmagnetihargeinavolumeV,andSthesurfae
at theboundaryofV,then
I
S
BdS=
0 Q
m;V
(39)
Theequationsimpliitlyidentifyasthesame
quan-tity the elds generated by eletri harges and
mag-neti urrents, for the eletri eld,and generated by
magnetihargesandeletriurrentsforthemagneti
indution eld. These equationsare invariantunder a
generalizationoftheHeavisidetransformations:
E! B ; B!E
D! H ; H!D
j
e
! j
m
; j
m !j
e
e
!
m
;
m
!
e 9
>
>
=
>
>
;
(40)
The free eld equations ontinue to have two degrees
offreedom. Theadditionofsouresdonothangethe
empty spae ounting of degrees of freedom, and the
addedinvarianereduesthenumberwithsoures.
In this ase, it seems pointless to speak about
Bianhi identities, whih result in the introdution of
potentials.
Intermsofdierentialforms,wehavenow
ÆF =
0 J
e
dF =
0 J
m
(41)
where J
e
is theone-formthat orrespondsto the
ele-tri urrent density and J
m
is a three-formhavingas
omponentsthe magnetiurrentdensityvetor. The
dierentmeaningsoftheurrentsisin evidene. If,as
a useful exerise, one introdues the Heaviside
trans-formations(?)inbothmembersoftheseequations,one
nds
F =?F
J
e =?J
m
J
m
= ?J
e 9
=
;
(42)
Theimportantpoint isthat Hodgedualityis a
ge-ometrial transformation, the physial ontent of the
equations is the result of imposing Heaviside duality
for the Maxwell equations, no matter in whih form
oneexpressesthem. Whentheequationssatisfy
Heav-iside duality, Maxwell equations reeive aHodge dual
symmetriform.
Tointrodue potentials, it is enoughto swith o
theeletrihargeandurrentdensities. Therstand
fourthequationsbeomehomogeneous,andsuggestthe
introdutionofamagneti(pseudo)salarpotential, ,
andan eletri(pseudo)vetorpotential,W , with the
naturaldenitionfortheelds:
E = r'
A
t
+r^W
B = r + 1
2
W
t
+r^A 9
>
=
>
;
: (43)
There is no onit with usual three-dimensional
vetoranalysisin takingthedivergene oftheseelds,
onearrivesatordinaryPoissonequationsforthe
poten-tials'and ineahase. TheimpositionofHeaviside
dualityisahievedwith nomagnetimonopoles.
SineMaxwellequationsareinvariantunder
Heav-iside transformations, the potentialshave
orrespond-inglyto satisfythefollowingrelations:
W! A ; A!W
'! ; !'
: (44)
SineapplyingtwieHeavisidetransformationsone
getsaglobal minus signforallquantities involved, we
seethat HeavisidedualityIS NOT idential to Hodge
duality for dierential forms, it o
inides just for the
elds,notfortheurrentsandpotentials.
The relativisti formulation however proves to be
troublesome. Dene thefollowingfour-vetors:
A
f'; Ag
j
e
f
e ; j
e g
W
f ;W g
j
m
f
m ;j
m g
9
>
>
=
>
>
;
(45)
Calling
A
=
A
A
; W
=
W
W
;
(46)
onewritesthetensorfortheeldintensitiesas
F
=A
+"
W
: (47)
Therelativistiinvariantlagrangiandensityhaving
theextendedMaxwellequationswithHeavisideduality
asasymmetryis
L
A+W =
1
0 F
F
1
j
e A
1
j
m W
The rst term in the above expression merits a
loserlook. Itis
F F =A A W W + 1 3 " A W : (49)
Thelast termisa \surfaeterm", and itit should
be negletedunder the usual assumptions. Its
onse-queneswill bedisussedinwhat follows.
Theanonialmomentaare:
(A) k = 0 E k (50) (W) k = 0 B k (51)
Under Heavisideduality,theytransformproperly:
(W) k ! (A) k ; (A) k ! (W) k : (52)
Asbefore,thelagrangianhangessignunder
Heav-iside duality. The orresponding full anonial
hamil-tonianreads: H A+W = 1 2 0 ( 1 0 (A) k +" 0k ij W ij ) 2 1 2 0 ( 1 0 (W) k +" 0k ij A ij ) 2 + A 0 ( k (A) k )+W
0 ( k (W) k ) + 1 4 0 A ij 2 1 4 0 W ij 2 + 1 j e A 1 j m W (53)
Under Heaviside duality, the hamiltonian too
hanges sign. The only way out for this situation in
thequantizedversionof thetheory may bethe
rever-salofeitherspaeortime. Theintegratedhamiltonian
beingthefourthtotalmomentumomponent,itseems
thatthelast,timereversal,istheappropriateone.
Notie the important point that in this formalism
itisimpliitthattheeletrieldandmagneti
indu-tioneldgeneratedbyeletriormagnetihargesand
urrent densities are the same. When this translates
at the potentialdependene, wefae this troublesome
situation.
Forwhatregardsthegaugeinvarianepropertiesof
thetheory,itisinvariantunder [9,27℄
A !A 0 =A + + P W !W 0 =W + +" P (54) with 2P
= 0. The theory displays a kind of
U(1)U(1) gauge symmetry provided the \surfae"
term in the lagrangian is negleted and P
is taken
zero.
Theontinuousdualitytransformationholdsforthe
elds and related quantities. Indeed, as pointed by
Jakson[28℄thisshowsthatthediretionwhihistaken
as magneti or eletri in the eld spae is arbitrary.
Thelagrangian,however,transformsas:
L =L os2+L
0
sin2 (55)
whereL 0
isaninterestingintermediateexpression:
L 0 = 1 4 0 A W 1 8 0 " (A A +W W ) (56)
whih is a pseudo-salar quantity under parity
trans-formations. Thelast termsremindsoftheexpressions
obtainedfrom the alulationof theU(1)
eletromag-neti anomaly.
V The monopoles
The question of magneti monopoles appears often
linked with eletromagneti duality. In several
on-tributions, Oliveand ollaborators[15, 29℄ proposes a
generaltheoryinluding partiles(arriersofthe
ele-triharges)andmonopoles. Wehaveheardaboutthis
workhereand atthe\JorgeAndreSwiea"shool
ex-posedbyMaroAurelioKneipp[30℄.
The simplest version of monopole is a magneti
pointhargeattheorigin(4):
rB=
0
gÆ(x): (57)
By Gauss's theorem, the ux of B around that
hargeis: = I BdS= 0 g (58)
Taking,byassumption,
B=r^A (59)
onehas
rB=r(r^A)=0: (60)
Inhis 1947 artile, Dira[8℄reognizes the
inom-patibility of both requirements. To write a magneti
potential for the monopole, a Dira string is needed.
Toguaranteeunobservability of thestringby harged
partiles, theeletrihargeand themonopoleharge
should satisfyDira'sondition. WhatDiraproposes
is an enlargement that produe a dynamis to the
string. Hewrites
F =A + X (G y ) (61)
Thesumis overall theworldsheets swept byaDira
string. TheresultforG
heobtainsis
G
(x)=g
Z Z dd[ y y y y ℄Æ 4 (x y) (62)
Iunderstand thatdualitybetweeneletrihargesand
magnetimonopolesmeans,forDira,theexhangeof
thersttermintheeld-strengthtensorwiththe
se-ond.
The topologial approah to the theory of the
GreubandPetry[6℄andWuandYang[7℄. Itisframed
in berbundle language,and startsfrom apuntured
R 3
spae, whih gives essentiallya sphere. The
mag-netipotentialisnotgloballydened,andinthe
tran-sition region (z0) bothloal denitions are related
through agaugetransformation. Thisseems an
inno-enttrik, but thefat is that thegaugefuntion is a
(in Dirawords)non-integrablefuntion,i.e., anangle
(anexampleofalosedbutnon-exatone-forminR 2
).
A
= A
I
;z>0 (63)
A
= A
II
;z<0 (64)
A
II
= A
I
n;z0: (65)
The integern assumes anyvalue. One may reall
thatwasthroughasingulargaugetransformationwith
ananglethatBohieriandLo
inger[31℄eliminatedthe
eetofthesolenoidinthe
Ehrenberg-Siday-Aharonov-Bohmsetup.
Some time later, Ryder [37℄ studied the monopole
as a realization ofthe prinipal bundle Hopfmapping
S 3
!S 2
. Thisyields auniquevalue,n=2, in
agree-mentwith previousreultbyShwinger[38℄. Infat,it
is notonly the prinipal bundle whih is involved for
othervaluesofn,but rathertherelated\lensspaes".
TheseareobtainedfromtheS 3
sphereparametrizedby
twoomplexvariables
jz
1 j
2
+jz
2 j
2
=1 (66)
throughtheequivalenelasses(z
1 e
i2k l1=n
;z
2 e
i2k l2=n
),
with(k;l
1 ;l
2
)relativeprimenumbersandnaninteger.
Inthetwo-potentialformalismpresentedinthe
pre-vious setion, if one keeps the \surfae term" it
on-tributestotheminimization
ÆL
A+W
Æ(
W
)
=
0 "
A
; (67)
and yields the Bianhi identity for the potential eld
strength.
It is immediate that a monopole breaks Bianhi
identitylikewisea\normal"magnetihargedoes,but
makesitthroughamoreelaboratedontribution. Ina
ontributionbyP.C.R.CardosodeMello,S.Carneiro
e M. C. Nemes, they proposed anon-loal lagrangian
[32℄ for a dual theory with two potentials. One may
takeasimilarexpressionto theirsforthepotentials:
A
=A
(0) +
1
2 "
R
P x
A
(m)
dy
(68)
W
=W
(0) +
1
2 "
R
P x
W
(m)
dy
(69)
where the subsripts indiate no topologial struture
and magneti or eletri monopole, respetively, and
thelatterontributetotheEulerequation.
Thesamemehanismprodueseletrimonopoles,
dierentfromtheusualeletriharges. The
minimiza-has a ontribution from the surfae term in the
la-grangian. An eletrimonopole would breakthe
iden-tityr(r^W )=0.
For the magneti monopole, Greub and Petry
showed [6℄ that the ompatibility of the Shrodinger
equation for harged partiles and the non-null value
for the magneti ux (taking the wave funtion as a
setionofalinebundleonS 2
)limitsthevaluesforthe
hargesthroughtheDiraonditionandshowsthe
exis-teneofanintegral,real,deRhamohomologylassfor
themanifold M, in our ase, S 3
S 2
S 1
(loally).
In termsof dierential forms, onehas simultaneously
dF=0(Fisaloseddierentialform)and H
F6=0(F
isnotanexatform). Closedformswhiharenotexat
expandstheHodgeohomologygroup. Theanalogous
aseforeletrimonopoles, i. e.,dF =0, H
F 6=0
has been onsidered by Cabibbo and Ferrari [10℄ and
Dijkgraaf[33℄.
TheapproahtodualitybyOliveinmorethantwo
deadesis anextension of themeaningof
eletromag-neti duality [15℄. It is inspired by previous work by
Coleman [34℄and Mandelstam [35℄ on the relation
be-tweenthesine-GordonmodelandtheThirringmodelin
1+1spaetime. Inbrief,thefermionsin theThirring
model may be written as solitons of the sine-Gordon
model, via a so alled \duality" relation between the
two models, whih, besides, links the weak oupling
regime in one model with the strong oupling regime
intheother. Thisanalogyhasbeenpursuedfurtherin
non-abeliantheories byOliveaswell asother authors
[46,?℄.
Thequanta of theeld in one ase arethe
eletri-ally harged partiles, and the solitons are magneti
monopoles. Eletromagnetiduality, in this extended
sense, is seen as the remnant of a non-abelian gauge
symmetry, brokenby the
Higgs-Kibble-Brout-Englert-Andersonmehanism[36℄,withtheHKBEApartilesin
theadjointrepresentationoftheoriginalgaugegroup.
A mass formula is proposed for all the quanta of
the theory, whih, on aount of the dual symmetry
proposed should notdistinguish eletriand magneti
harges. Itis
M =a p
q 2
+g 2
(70)
The naturalframework for the realization of these
ideasin a quantum theory seemsto be N =4
super-symmetry. Further developments led to the onepts
ofS andTdualitywidelyirulatingnowadays. Inmy
opinion,thewholeshemeseemsplausible,butremains
tilnowhighlyspeulative. Forareentupdateonthese
matters,IrefertothereviewbyMaroAurelio[30℄.
VI Massless Dira elds as
dual symmetri eld theory with eletri and
mag-neti harged spinor elds has been studied [12, 13℄.
Fermionswereingeneralmassive,andthetwo-potential
theory is usedfor the eletromagneti eld. Magneti
hargesaretakengenerallyasmonopoles,disregarding
\normal"harges(anexeptionbeingNisbet[9℄).
Cu-riously, properties under spae inversion of the elds
were mostlynotaounted.
Wepresenthereaskethof resultsinvolving
mass-less spinor elds with \normal" eletriand magneti
harges. Thefreespinoreldlagrangianis
L =i~ 6 (71)
Itanbeseparatedin hiralontributions:
L =L
;L +L
;R
: (72)
Theinterationisnowintroduedwithbothpotentials
ofapretenseHeavisidedualeletromagnetitheoryvia
theextendedovariantderivative:
D
=
+i
q
~ A
+i
g
~ 2
5
W
: (73)
Theleft-righthiralseparationremains,with
D L;R
=
+i
q
~ A
i
g
~ 2
W
: (74)
Adierentombinationworksforeahhirality,and
theymaybequalied as\hiral"leftand right
poten-tials. TheyexhangeunderHeavisideduality
transfor-mations.
Theimpositionofdualityrestritsthevaluesofthe
eletriand magnetihargesto therelation:
g=q: (75)
underthehypothesisthatbothhargesareableto
pro-duethesameelds.
Theleftandrightpotentialsare
Z
L
=A
+W
(76)
Z
R
=A
W
(77)
and if one rewrites the potentials A
, W
in terms
ofthese rightandleft potentials, thetwo-potential
la-grangian (48) separates niely in left and right parts.
It appears asaU(1)
L
U(1)
R
symmetri model (but
what about themonopoles?), asugestion put forward
bySingleton[24℄.
The relation between spinor elds on a manifold
and monopole elds have been studied on the
two-dimensional sphere S 2
by Jayewardena [39℄ for the
Shwinger model, and by our group in Rio using
Connes-Lott nonommutative version for the abelian
theory[40℄. Itturnsoutthatthenumberofzeromodes
oftheDiraoperatorisrelatedtotheintegerdenedby
theChernharater. Besides, thenon-trivialtopology
modeltotheaxialanomalyandanon-zerovalueforthe
ondensatedesribedbythevauumexpetationvalue
ofthefermioneld, < >.
WithC.A. P.Galv~aowehaverealulatedthe
ax-ialanomalyinfour(Eulidean)dimensionsforthe
two-potential ase, a alulation performed previously by
Balahandranandollaborators[41℄. Theresultis
1
16 2
tr
5
a
2 =
1
16 2
1
2 "
A
A
+
1
3 W
W
+ 16
3 W
W
(
W
)
4
3 W
2
(W)
+ 2
3
2
(W)
(78)
The interesting observation is that the terms
ap-pearingin therstbraketarethe samepresentin an
= =4 intermediate step in the ontinuous duality
transformation for the lagrangian. This may suggest
thatalagrangiandensityinludingbothkindsofterms
mayimproveasadual symmetriversion.
Alastremark:BelyaevandShapiro[42℄have
stud-ied the axial urrentoupling asaremnant oftorsion
whengeneralrelativityistakentothelimitofatspae.
It may be interestingto look whether this maybe an
indiationof anaturalHeavisidedualitywhen gravity
istakenintoaount
VII A onsistent duality
sheme
AsI haveshown, thestritoneptofeletromagneti
duality,implyingthesameeldasproduedfrom
ele-tri ormagneti harges,arriesseveralproblems and
questions. Without monopoles, the theory has funny
properties,ensuingmainlyfromthefatthatHeaviside
duality isnot asymmetry of thelagrangian. The
im-positionofdualityasasymmetryledShwarzandSen
[16℄todevelopanadhoformalismapparentlywithout
relativistiinvariane,andageneratorforthe
transfor-mationfoundbyDeserandTeitelboim[14℄isnon-loal
in spae. Thelatter resultswere obtainedforthe free
eld ase. With twopotentials,someof thesefeatures
are tamed,but energyeases to be positive under the
transformation.
Reently,withC.A.P.Galv~aowehavelookedtothe
problem[43℄. Theideaofonsistenyresultsinan
het-erodoxproposal,abandoningtheidentityofeletriand
magneti elds havingdierenthargesassoures. In
thevauumsetor,thesymmetryallowsfreeexhange
between eletriand magnetields, in theHeaviside
A dual theory to the usual eletrodynamis,
mag-netodynamiswouldbe:
rE 0
= 0
rB 0
=
0
m
r^E 0
= 1
0 j
m B
0
t
r^B 0
= 1
2
E 0
t
9
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
;
: (79)
The primes indiate that the elds represented by
the symbols do notsatisfy the same equations as the
ones appearingintheordinarytheory.
Potentials are introdued from the homogeneous
equations:
E 0
= r^W 0
B 0
= 0
t +
1
2
W 0
t 9
=
;
: (80)
Inthree-dimensional notation, thelagrangian
den-sitythatproduestheeldequationsabovereads:
LfB 0
;E 0
g=
0
2 [
2
B 0
2
E 0
2
℄
m 0
j
m W
0
(81)
Writteninovariantfour-dimensionalnotation,one
has to introdue a seond rank antisymmetritensor,
G
suhthatitstime-spaeomponentsarerelatedto
B 0
andthespae-spaeonestoE 0
,andthe
orrespond-ing four-vetorsforthepotentialsand urrentdensity.
This tensor isnotto betaken asthedual of the
ele-tromagneti tensor in the Maxwellase, theelds are
assumedto bedierent.
Dualitynoworrespondstotheexhanges
E! B
0
; B!E
0
(82)
and theorrespondingequationsin theopposite
dire-tion.
A lagrangiandensity in four-dimensional notation
overing eletrodynamis and magnetodynamis with
theabovetransformationpropertiesis:
L =
0
4 F
F
0
4 G
G
1
j
A
1
j
W 0
+(gaugexing terms):
(83)
Thistheory hasthesymmetry asrequired,possess
a positive energyand the generatorfor the symmetry
is
M= Z
d 3
x[A
0
W
0
℄ (84)
and is perfetly loal and ovariantly written. Here,
A
and
are the usualpotentialand anonial
mo-mentum four-vetors of Maxwell theory, and 0
=
B
0
Æ .
Thoughthephysialideaof relatingdierentelds
through a \duality" transformation may sound
inap-propriate,letusrealltheimportantHodge
deompo-sitiontheoremfordierentialforms[44℄: foraompat,
orientedRiemannianmanifold,anyp dierentialform
maybedeomposed inthesum
!=!
0
+d+Æ (85)
where!
0
=0, andeah termin the sumbelongs to
avetor subspaein the spae of p dierentialforms
orthogonalto theother two. This raisesthe question
whetherin the twopotentialtheory itis legitimate to
identify the two ontributions to eld-strength tensor
asbeingequivalentphysial elds.
VIII Duality and monopoles in
non-abelian gauge theories
and phenomenology
Inthelastdeadesofevolutionofthetheoryof
ele-mentary partiles as expressed by non-abelian gauge
theories through the \standard model", monopoles
haveplayeddierentr^oles. Ijustwanttomentionsome
reent appliations of the monopoles in non-abelian
gaugetheorieswhihalled myattention.
Letmerstmakeadigression,andmentionthatJ.
Sim~oesshowedtomethatin1979Senjanoviwas
look-ingforamodelwithSU(2)
L
SU(2)
R
U(1)symmetry
for theeletroweak interations, whih wasonsistent
with the phenomenology data at the time [45℄. The
ideawasthat the right symmetrywasbrokenvia the
usualHKBEAmehanismand theharged vetor
me-sonfrom therightpartaquiredalargemass.
Goingbaktothematter,thegroupat Rutherford
Laboratory led by Chang Hong-Mo is studying
non-abeliangauge theories in adierentialgeometri
on-text[46℄. In1995,he, TsouSheungTsun and
Jaque-line Faridani proposed an extension of what is Hodge
duality in abelian theories fornon-abelianones. This
wasto take advantageof the notionof duality, in the
formoriginallyvisualizedbyDira. InDiraterms[4℄,
\theideaisso nethat onewould besurprisedif
Na-turehadmadenouseofit".
Usually,onestartsfromafreeation,
S 0
=S 0
F +S
0
M
(86)
where in the right hand side eah ontribution
orre-spondstoeld andmatterfreeations.
Interation is introdued adding an \ interation
term ", whose form is dedued from phenomenology
and/or invariane onsiderations. Chan and
ollabo-rators,instead,proposethat matter andelds are
de-termined to obey as a onstraint that the dual eld
the example of the abelian theory with a Dira eld,
theonstraintreads:
F
(x)=4~e ~
(x)
~
(x) (87)
andtheationinludestheonstraintwithaLagrange
parametereld,
S 0
=S 0
+ Z
d 4
x
f
F
4~e ~
(x)
~
(x)g: (88)
Notie that I would haveused theaxial urrent in
theonstraint.
Minimizing withrespettoF
(x)onegets
F
=2"
(
); (89)
andwithrespetto ~
(x),
(i6 m) ~
(x)= 4~e
(x)
~
(x): (90)
Infat,onegets,
F
= 4(
(x)
(x)): (91)
A new gauge invariane with the opposite parity is
gainedfrom
(x).
Theouplinge~isrelatedtotheoriginaloneviathe
Dirarelation(in theseunits)
e~e=n=4: (92)
Personally,Ian'tseehowtheyarrivetothisonlusion
forthespinortheory.
For the non-abelian theory, Chan, Tsou and
Fari-daniobtainedanextensionofHodgedualitysatisfying
ItreduestoHodgedualityin theabelianase
Applied twie goes bak to the original objet
(withpossiblyaphasefator)
Non-abelian harges for the original elds are
monopolesofthedualeld
Duality is introdued via loop variables; the main
featureisthat,liketheabelianase,onegetsasabonus
anadditionalgaugeinvarianewiththeoppositeparity.
The phenomenologial appliations provene from
thebreaking of thisadditionalsymmetry. Higgs elds
are taken as the frame vetorsof SU(3), they belong
to the fundamental representation; in the presene of
monopolestheyhavetobeadjusted foreahhart.
Thedualolourdistinguishgenerations. Infat,the
treeapproximationformassesproduesamassforone
generation,theothersbeingmassless(thinkinthemass
relations ::eletron). Theyarethenableto
alu-late theCabibbo-Kobayashi-Maskawamatrix,and the
resultisratherimpressive:
jV
rs j=
0
0;9755(0;9745 0;9757) 0;2199(0;219 0;224) 0;0044(0;002 0;005)
0;2195(0;218 0;014) 0;9746(0;9736 0;9750) 0;0452(0;036 0;046)
0:0143(0;004 0;014) 0;0431(0;034 0;046) 0;9990(0;9989 0;9993) 1
A
(93)
d
Theguresinparenthesisarethephenomenologial
values. Thevaluesforthemassesarenotsogood, but
Chan believes that, sine the alulations involve
ex-trapolationsinalogarithmisale,theyshouldbeless
reliable.
Anotherdevelopmentisdueto TanmayVahaspati
and ollaborators [47℄. The idea is that may be the
fermions of the standard model may be seen as the
monopoles of a dual bosoni Grand Uniation
The-ory. Startingfroma ~
SU(5)model,suessivebreakings
leadstothedualofusualeletromagnetism, ~
U(1). The
breakingshemeis
~
SU(5) ! [ ~
SU(3) ~
SU(2) ~
U(1) 0
℄=Z
6 (94)
! [ ~
SU(3) ~
U(1)℄=Z
3
(95)
~
Amonopole hargeisobtainedfrom thehomotopy
groupsforeahgroupin thehain:
Q (n)
=n
3 Q
3 +n
2 Q
2 +n
1 Q
1
; (97)
and stability requires n = 1;2;3;4;6, with
n
1
=n(mod:3),n
2
=n(mod:2)andn
3
aninteger.
Theassignmentsfortheknownpartilesresult:
(u;d)
L
! n=+1
d
R
! n= 2
( ;e)
L
! n= 3
u
R
! n=+4
e
R
! n= 6: 9
>
>
>
>
=
>
>
>
>
;
(98)
lem of spin has been attaked too, taking prot from
the\spinfromisospin"onstrution.
Themorereentworkon this approah byLepora
[47℄ onvergestothe views byOliveandollaborators
andChanandollaborators.
Eveninasethenalgoalisnotattained,onemay
ask whether these gures quoted from the works by
Chan, Tsouand ollaboratorsand Vahaspatiand
fol-lowersare merefunnynumerial aidents or pointto
the true struture of the theory taken for the
\stan-dard model" . In any ase, in my opinion they open
interestingpathwaysforresearh.
IX Conlusion
Inthis swift glane onthe subjet, I hope to have
made lear that at least two dierent views of it are
referredinthe sameterms. One,pureeletromagneti
Heavisideduality,doesnotinludeneessarilymagneti
monopoles. Theanswertotherstquestionaddressed
at theend ofsetion2isthenonthenegative.
The seond view introdues the notion of
dual-ity from two-dimensional spaetime Sine-Gordon and
Thirring model relationship. There seems to be
fruit-ful results of this ideas in nonabelian theories if one
takes for the solitons in this approah the magneti
monopoles.
Inmyview,thepureHeavisidedualitymaybe
im-plementedeither byimposing theidentityoftheelds
generatedbyeletriormagnetiharges,orextending
thephasespaetoinludedierenteldsforeah
dier-entkindofharge. Intherstase,theproblemsthat
appeararerelatedtothefatthatthetransformationis
notasymmetryoftheation,butofitsEuler-Lagrange
equations. OnemayoveromethesediÆultiesthrough
atimeinversioninthequantizedversion,butitremains
to see if the nal theoryis onsistent, and soremains
anopenquestionwhih istherightpathtofollow.
I hope to have set lear that Heaviside duality is
aphysialonditiononelds, potentialsandurrents.
Hodge duality, while being a useful tool to write the
resultsinaoniseform,isjust ageometrialsetting.
In any ase, it remains to show that duality may
produemeaningfulphysialresultsin aeld
theoreti-alframework. Comingbaktotheexampleofthe
hy-drogenatom,isnotlearthatwehaveanspetrosope
withenoughresolutiontoputintoevideneasymmetry
under thebrokentheoryleadingtotheMaxwell
equa-tions. Theresultsobtainedbythegroupsimplementing
theideainnonabeliantheoriesopenawindowforhope.
Aknowledgements
I wish to thank Clovis Wotzasek and Franisus
Vanhekeforhiskindreading oftheoriginaltext,and
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