Compatiation of Gauge Theories and the
Gauge Invariane of Massive Modes
R. Amorim and J.Barelos-Neto
Instituto deFsia
UniversidadeFederaldoRiodeJaneiro
RJ 21945-970,CaixaPostal68528,Brasil
Reeivedon13Deember,2001
We study the gauge invariane of the massive modes in the ompatiation of gauge theories
fromD =5to D =4. Wedeal withAbelian gauge theoriesofrank oneandtwo,andwith
non-Abelianonesofrankone. WeshowthatStukelbergeldsnaturallyappearintheompatiation
mehanism,ontrarilytowhatusuallyoursinliterature wheretheyare introduedbyhand,as
a trik, to render gauge invariane for massive theories. We also show that inthe non-Abelian
ase theyappear inaverydierentway whenompared withtheir usualimplementationinthe
non-AbelianProamodel.
I Introdution
Nowadays, there is broad onsensus that
fundamen-tal theories might ome from spaetime dimension D
higher than four, probably D = 10 or D = 11. This
is mainly relatedtothe adventofstringtheories,that
are onsistent in the quantum word just at D = 10.
Further,thedualityamong knownstringtheories
sug-geststhattheymightemergefromamorefundamental
theoryatD=11. However,oneofthegreatdrawbak
ofthis ideaisthat thereisnoruleaboutsomespei
mehanismstoreahourwordatD=4.
Onemannerofgainingsomeinsightonthisproblem
istostudytheompatiationlosetoD=4andtry
tounderstandthefeaturesthatapossiblefundamental
theory shouldhavefrom thetheoretialonsistenyof
the results. It is important tomention that the
semi-nalidearelatedto thispointofviewdatesbakalong
timeago,intheworksofKaluzaandKlein[1,2℄,where
they havestartedfrom thegravitationalEinstein
the-ory atD =5and,after spontaneousompatiation,
reahed the Maxwell and Einstein theories at D = 4.
ThevetorgaugeeldA
wasoriginatedfroma
ompo-nentofthemetritensor. Itisopportunetoemphasize
that many of the reent attempts to implement
om-patiationmehanismhavetheKaluza-Kleinideaas
astrongsupport.
Wewould liketo addressthepresentpapertothis
line. Weonsidertheompatiationofgaugetheories
thequestion of gauge invarianein D =4, that shall
beretainedin theompatiationproedure,evenfor
themassivemodes. WeshallseethatStukelbergelds
[3℄,thatareusuallyintroduedasatrikinorderto
at-tainthegaugeinvarianeofmassivevetorelds,
nat-urally emergein this proedure. Even though we are
going to deal just with gauge elds of rank one and
two (that are the only ones whose number of
physi-al degrees of freedom are onsistent at D = 4), the
methodanbediretlyextendedforgaugeeldsofany
rank at higher spaetime dimensions. Conerning to
thenon-Abelianase,wementionthatthenon-Abelian
formulationoftheoriesofrankhigherthanoneannot
be diretly done. The gauge invariane, even for the
masslessase, is only ahieved by means of auxiliary
elds[4℄. Weshalldealwiththenon-Abelianasejust
forrankone. We mentionthat therole played by the
Stukelbergeldsinthisaseisverydierentfromtheir
usualimplementationinthenon-AbelianProamodel.
Our work is organizedasfollows: In Setion II we
treatthespontaneousompatiationofMaxwell
the-ory originally dened in D = 5, and show that in
the Fourierexpansion proedure, the zeromodes
or-respondto theusual Maxwelland realmasslesssalar
elds in D = 4. The nonzero modes orrespond to
omplexProa elds oupled to the appropriate
om-plex Stukelberg elds to keep the gauge invariane.
InSetion III weapplythisproedure inthetwo-form
two-formgaugetheoriesforthezeromodes. Thevetor
eldsplaytheroleofStukelbergeldsforthemassive
modes of the two-form elds. In setion IV, we
on-sider Yang-Mills theory. Although it has the Abelian
limitfoundinSetionII,aninterestinggaugestruture
is obtainedin thefull theory,where modesand gauge
multiplets are mixed in a non-trivial way in order to
keep the gauge invariane of the ation. We reserve
SetionV forsomeonludingremarks.
II Maxwell theory
Let us onsider the following ation for the Maxwell
theoryatD=5
S= 1
R Z
d 4
x Z
R
0 dx
4
1
4 F
MN
F
MN
(2.1)
wheretheoordinatex 4
desribesairleofradius R .
WeuseapitalRomanindiestoexpressthespaetime
dimensionD =5,i.e. M;N =0;;4,and adoptthe
metri onvention MN
= diag(+1; 1; 1; 1; 1).
TheMaxwellstresstensorF MN
is denedin termsof
thepotentialvetorA M
bytheusualrelation
F MN
= M
A N
N
A M
(2.2)
Theation(2.1)isinvariantunder thegauge
transfor-mation
ÆA M
= M
(2.3)
Letus split the vetorpotential A M
as A M
=A
for M = 0;;3 and A M
= for M = 4. We thus
havefortheation(2.1)
S= 1
R Z
d 4
x Z
R
0 dx
4
1
4 F
F
1
2
1
2
4
A
4 A
+
4
A
(2.4)
The rst term above annot be identied with the
Maxwell Lagrangian at D = 4 beause A
depends
on bothx
and x 4
. Following the usualproedure in
the (spontaneous) ompatiation proedure [2℄, we
taketheexpansionsofA
andinFourierharmonis,
namely
A
(x;x 4
) = +1
X
n= 1 A
(n) (x)exp
2in x
4
R
(x;x 4
) = +1
X
n= 1
(n) (x)exp
2in x
4
R
(2.5)
The replaement of these expansions into the ation
(2.4)leadsto
S= Z
d 4
x +1
X
n= 1
1
4 F
(n) F
( n) 1
2
(n)
( n)
2 2
n 2
R 2
A
(n) A
( n) +
2in
R A
(n)
( n)
(2.6)
Sine A
M
is a real quantity, we havefrom
expan-sions(2.5)thatA
( n) =A
(n) and
( n) =
(n) . Using
thisintotheation(2.6)werewriteitinamore
onve-nientway
S = Z
d 4
x n
1
4 F
(0) F
(0) 1
2
(0)
(0)
+ 1
X
n=1 h
1
2 F
(n) F
(n) 4n
2
2
R 2
A
(n) +
iR
2n
(n)
A
(n) iR
2n
(n) i o
(2.7)
WeobservethatthetwozeromodetermsatD=4
orrespond to Maxwell and real salar eld theories.
The other modes are related to omplex Proa elds
with masses givenby 2n=R . It is interestingto note
theroleplayedbytheorrespondingmodesofthesalar
elds. Theyare Stukelbergelds. Usually, these are
putbyhandasatriktomaketheProatheorygauge
invariantortoimplementtheHamiltonian embedding
proedure,duringtheonversionofseondtorst-lass
onstraints[5℄. Here, these elds naturally emerge in
ordertokeepthegaugesymmetryoftheinitialtheory.
III Abelian two-form
Thenaturalextensionofwhatwasdoneintheprevious
setion is toonsider gaugeelds ofrank two. Letus
S = 1 12R Z d 4 x Z R 0 dx 4 H MNP H MNP (3.1)
The ompletely antisymmetristress tensor H MNP
is
dened in termsof theantisymmetritwo-formgauge
eld B MN by H MNP = M B NP + P B MN + N B PM (3.2)
Thistheoryisinvariantforthe(reduible)[6,7℄gauge
transformation ÆB MN = M N N M (3.3)
Inordertoperformtheompatiationto D=4,
weonvenientlysplit thepotentialB MN as B MN = (B ;B 4 ) = (B ;A ) (3.4)
where wehave identied B 4
with A
. Again,this is
notthevetorpotentialoftheMaxwelltheorybeause
it depends on both x
and x 4
and its gauge
transfor-mation,aordingto (3.3),reads
ÆA = 4 4 (3.5)
whih is not the harateristi transformation of the
Maxwellonnetion.
Introduing(3.4)into(3.1),weobtain
S = 1 R Z d 4 x Z R 0 dx 4 1 12 H H 1 4 F F + 1 4 4 B 4 B + 1 2 F 4 B (3.6)
where,forthesamepreviousargument,F = A A
isnottheMaxwellstresstensor. Expandingthe
eldsB
,A
,aswellasthegaugeparameters
and
4
inFourierharmonis,wehave
B (x;x 4 ) = +1 X n= 1 B (n) (x)exp 2in x 4 R A (x;x 4 ) = +1 X n= 1 A (n) (x)exp 2in x 4 R (x;x 4 ) = +1 X n= 1 (n) (x) exp 2in x 4 R 4 (x;x 4 ) = +1 X n= 1 (n) (x) exp 2in x 4 R (3.7)
Introduing thesequantitiesinto (3.6)and onsidering
thatB ( n) =B (n) ,A ( n) =A (n)
,et.,weobtain
S = Z d 4 x h 1 12 H (0) H (0) 1 4 F (0) F (0) + 1 X n=1 1 6 H (n) H (n) 1 2 F (n) F (n) + 2n 2 2 R 2 B (n) B (n) 2in R F (n) B (n) i (3.8)
Dueto thegaugetransformationsofthezeromode
elds, ÆA (0) = (0) ÆB (0) = (0) (0) (3.9)
wehavethat thetwozeromodetermsof(3.8)arethe
tensorandvetor(Maxwell)theoriesatD=4[8℄. The
remainingmodesorrespondtomassiveomplextensor
gaugeeldsB
(n)
andmasslessvetoronesA
(n)
. Letus
rewritethen-modetermsofexpression(3.8)inamore
appropriateform S (n) = Z d 4 x h 1 6 H (n) H (n) 2n 2 2 R 2 iB (n) R 2n F (n) iB (n) R 2n F (n) i (3.10)
We notie that in the rank 2 theory, the
mass-less vetor gauge eld A
(n)
naturally appears as a
Stukelberg eld for themassiveantisymmetrigauge
eld iB
(n)
. Theirgaugetransformationsaregivenby
ÆA (n) = (n) 2in R (n) ÆB (n) = (n) (n) (3.11)
playtheroleofStukelbergeldsforthemassiverank
twotheory, wasalso found in theaseof Hamiltonian
embeddingmehanism[9℄.
Weouldgeneralizethisanalysisforgaugeelds of
anyrank. However,forD=4itdoesnotmakesenseto
onsidergaugeelds ofrankhigherthantwo,beause
IV Non-Abelian ase
Asalreadysaid, thenon-Abelianformulationofgauge
theorieswithranktwoorhigherannotbediretly
im-plemented. Its gaugeinvariane anonly be ahieved
withthehelpofauxiliaryelds[4℄. Consequently,sine
thesemodelsdonothavealeargaugeinvariane,their
disussionherewillbeavoided. Weshallonlyonsider
in this setion the vetorase. The non-Abelian
ver-sionoftheproeduredesribedinsetionIIshouldstart
fromtheation
S= 1
R Z
d 4
x Z
R
0 dx
4
tr
1
4 F
MN
F
MN
(4.1)
wherenow
F MN
= M
A N
N
A M
i[A M
;A N
℄ (4.2)
andthegaugepotentialstakevaluesin aSU(N)
alge-bra,whosehermitiangeneratorsareassumedtosatisfy
a normalized trae ondition. The ation (4.1) is
in-variantunderthegaugetransformation
ÆA M
=D M
(4.3)
oneonedenestheovariantderivativeas
D M
= M
i[;A M
℄ (4.4)
AgainwritingA 4
=,wenotethatF 4
= 4
A
D
permitstorewriteation(4.1)as
S= 1
R Z
d 4
x Z
R
0 dx
4
tr
1
4 F
F
1
2 D
D
1
2
4
A
4 A
+D
4
A
(4.5)
Expandingtheeldsabovein Fourierharmonisin
asimilarwaytoSe. II,weget
S= Z
d 4
xtr 1
X
n= 1 h
1
4 F
(n) F
( n) 1
2
2n
R A
(n) +iD
(n)
2n
R A
( n) iD
( n) i
(4.6)
where
(D
)
(n) =
(n) +i
+1
X
m= 1 [
(m) ;A
(n m) ℄
F
(n) =
A
(n)
A
(n) i
+1
X
m= 1 [A
(m) ;A
(n m)
℄ (4.7)
It is interestingtoobservethat thenon-Abelianharaterofthe ation(4.6)indues agaugestruture where
themodesmixedamongthemselves. Inotherwords,thegaugemultipletandthemodesformanontrivialstruture
that hasto beonsidered asawhole to preservethesymmetry ofthe ation. Atually, thetransformations (4.3)
havetheirmodeexpandedversiongivenby
ÆA
(n) =(D
)
(n)
Æ
(n) =
i2n
R
(n) i
+1
X
m= 1 h
(m) ;A
(n m) i
(4.8)
wheretheovariantderivativeisdenedin (4.7). Asaonsequeneoftheaboveequations,
ÆF
(n) = i
+1
X
m= 1 h
(m) ;F
(n m) i
Æ
2n
R A
(n) +i(D
)
(n)
= i +1
X
m= 1 h
(m) ;
2(n m)
R A
(n m) +i(D
)
(n m)
whihisasymmetryofation(4.6),asanbeveried.
It is interestingtoobservethat the roleplayed by the
Stukelberg elds hereis dierent from the usual one
presentedbythenon-AbelianProamodel[10℄. There,
theStukelbergeldisintroduedbyhandin orderto
just giveÆ[mA
+i(D
)℄=0.
V Conlusion
Inthispaperwehaveonsideredthespontaneous
om-patiationsof Maxwell,Abeliantwo-form, and
non-Abelian one-formgaugetheories from aD =5
spae-time with a ompat dimension to the usual D = 4
Minkowski spaetime. We havefoused ourattention
to the gaugeinvariane of the theories formulated at
D = 5, whih should be kept along the
ompatia-tionproedure. Asusual,therearisemassivemodesin
theproessofompatiation. Inpriniple,thegauge
invariane of these modes ould be lost. We observe,
however, that generalizedStukelberg elds naturally
emerge in order to keep the ontent of the original
gauge invariane. Although in the Abelian ases the
Stukelbergeldsorrespondtothosealreadyfoundin
the literature, the ompatiation of the Yang-Mills
theory reveals anew strutureof ompensating elds.
Beause of the nonlinearity of the ation, the gauge
struture displayed by the mode expansion of
ovari-ant derivatives, urvature tensors and gauge
transfor-mations play a remarkable feature in mixing Fourier
modesand gaugemultipletomponentsinanontrivial
way.
Aknowledgment: This work is supported in part
by ConselhoNaionalde Desenvolvimento Cientoe
Tenologio - CNPq (BrazilianResearh ageny) with
thesupport ofPRONEX66.2002/1998-9.
Referenes
[1℄ T.Kaluza, Akad. Wiss.Phys.Math. K1, 966 (1921);
O.Klein,Z.Phys.37,895(1926).
[2℄ Foramoregeneralrefereneonompatiation
meh-anism,seeforexample, L.Castellani, R.D'Auriaand
P. Fre, Supergravity and Superstrings { A
Geomet-riPerspetive(WorldSienti,1991),andreferenes
therein.
[3℄ E.C.G.Stukelberg,Helv.Phys.Ata30,209(1957).
[4℄ A.Larihi, Phys.Rev.D55, 5045 (1997); D.S.Hwang
and C.-Y Lee, J. Math. Phys. 38, 30 (1997); J.
Barelos-Neto, A. Cabo and M.B.D. Silva, Z. Phys.
C72,345(1996).
[5℄ I.A.Batalinand E.S.Fradkin,Phys.Lett. B180, 157
(1986); Nul. Phys. B279, 514 (1987); I.A. Batalin,
E.S. Fradkin, and T.E. Fradkina, ibid. B314, 158
(1989);B323,734(1989);I.A.BatalinandI.V.Tyutin,
Int.J.Mod.Phys.A6,3255(1991).
[6℄ M.KalbandP.Ramond,Phys.Rev.D9,2273 (1974).
[7℄ For a review see M. Henneaux, Phys. Rep. C126, 1
(1985).A more ompleteperspetive anbe foundin
M.HenneauxandC.Teitelboim,Quantizationofgauge
systems(PrinetonUniv.Press, 1992).
[8℄ J.Barelos-Neto, J.Math.Phys.41,6661 (2000).
[9℄ C. Bizdadea and S.O. Saliu, Phys. Lett. B368, 202
(1996).
[10℄ T.KunimasaandT.Goto,Prog.Theor.Phys.37,452
