Summary talk: Field Theory
M. Gomes
Institutode Fsia, UniversidadedeS~aoPaulo
C.P.66318,05389-970, S~aoPaulo,SP, Brazil
Reeivedon19April,2001
Wepresent someremarks ontheworksreportedatthe setion ofpartilesand eldsof theXXI
BrazilianNationalMeetingonPartilesandFields.
ThenationalmeetingsonPartilesandFieldshave
ourred sine1979. Therstmeeting wastook plae
inCambuquira,MinasGerais,and,asitwaswrittenin
the proeedings, \it appeared asan answer to the
in-reasingneedforinterationsbetweenphysiists
work-ing in suh areas". That meeting had the
partiipa-tion of one hundred physiists, one third being
grad-uatedPh. D. students. A large spetrum of interests
wasoveredinthepublishedommuniations(Solitons,
1/Nexpansions,bosonization,symmetriesdynamially
broken,Bethe-Salpeterandsoon)
Beforethatmeeting, mostprobablyin 75/76,I
re-all J. A. Swiea being surprised with the absene of
Brazilian physiists working on supersymmetry. As a
matterof fat,onlyin theIII meeting supersymmetry
appearedexpliitly,in apresentationbyElioAbdalla
andproeededmoreintenselyfromtheIVmeeting
on-wards, with Vitor Rivelles. At that time, there was
great interest in two dimensional theories, whih was
naturallydue to thegreat inuene ofthe works ofJ.
A. Swiea. Atthe seond meeting I gavea letureon
theCP n
modelandnonloalharges.
In the present meeting, there was a notorious
in-rease in the numberor partiipants(around300), as
well asin the numberof ontributions. Furthermore,
weshouldnotiethelargeamountofpresentedposters
whihwereabsolutelysarein therstmeetings.
Within these twenty years, Brazilian physis has
evolvedrapidlyintermsofquantity. Theinterestshave
alsosomehowhanged. Workson2+1dimensionaleld
theories, problems on the quantization of onstrained
systems with or without supersymmetry are natural
partoftheontributions. Modernsubjetsasthe
ADS-onformaltheoryorrespondeneandnonommutative
spaeshavebeguntoappear.
Forthepresent meetingI tried tolassifythe
on-tributionsin twobloks: Formalaspetsand
Applia-tions. Of ourse, this separation is not well dened,
involvingaertaindegreeofsubjetivity.
1. Properties: Formal Aspets. Here, I grouped
gated: theBRSTsymmetryandtheBatalin-Vilkowski
methods.
Thepresentedontributionsrangefromissuessuh
as dimensional redution in supersymmetri theories,
supersymmetri Yang-Mills that until this day oer
hallengesforaproperunderstanding,dynamialmass
generationandboundstates.
Asfar asonstrainedsystemsareonerned, there
have been disussions on the quantization of a
parti-le onstrained to move onthe surfaeof a D
dimen-sionalsphere (C.Wotzasekand C. Neves). This is an
old problem whih has been taken into onsideration
by many physiists. InBrazil, wehad works from the
UFRJgroup(J.BarelosNeto,R.Amorim,C.
Wotza-sekandC.Neves), ElioAbdallaandR.Banerjeeand
from H. O. Girotti. In other words, it is a high
in-terest topi. The basi issue here is transforming a
seond lass system into a rst lass one. A
paren-theses for the terminology; rst and seond lass
be-long to the lassiation of onstraintsintrodued by
P. A. M. Dira. The Poisson parentheses involving a
rstlassonstraintweaklyvanishes,i. e.,itis
propor-tionaltotheotheronstraints. Thoseonstraintswhih
have non-vanishing Poisson parenthesis among
them-selvesarealledseondlassones. ThereisadiÆulty
withseondlassonstraintsarisingfromthefatthat
theirPoissonparenthesesingeneralontainprodutsof
theanonialvariablesandsobeomeambiguousatthe
quantum level. In this ontext, people have proposed
shemes to onvert seond into rst lass onstraints.
One aspet that has to be onsidered with regard to
thisonstrainttransmutationiswhentheproblem
reap-pearselsewhere,as,forinstane,withintheappliation
of the Gupta{Bleuler method to onstrut a physial
sub-spae.
In1992, H. O. Girotti, A. J. da Silva, myself and
thethendotoratestudentsR.S.MendesandJ.R.S.
Nasimento,investigatedtheexisteneofbound states
onstituted from partilesofequalharges,in 2+1
irum-ingrole. IntheThirringmodelitenablesaninreasing
intheregionfreefromtahions,inquantum
eletrody-namis itgivesamasstothegaugeeld,avoidingthe
ourrene of infrared divergenes, and so on. In our
studyof2+1dimensionalquantumeletrodynamisthe
CSterm,whihisruialtoavoidthejustmentioned
di-vergenes,doesnotinitiallyexistbutitisgeneratedby
radiative orretions. The orresponding
nonrelativis-ti potential wasobtainedthroughthe analysis of the
twobody Moller sattering. Thebindingenergiesthat
weobtained,measuredinKelvins, werenear the
riti-altemperaturesfoundinthesuperondutoreramis.
In ourmeeting, O.M. del Cimapresented amodel in
whih the massfor thegaugeeld omes throughthe
Higgs mehanism. This is a omplementary
possibil-itytoourssineitneedsanauxiliarysalareld tobe
implemented. Anyway, it is interesting to know that
boundstatesanbeformedevenwithoutthepresene
of aCSeld.
Another basi subjet overed in our gathering is
related to the dynamial mass generation for gauge
elds, in fourdimensions. Various shemes areknown
to generate mass for gauge elds. In two dimensions
wehaveShwingermehanism,thegaugeeld
beom-ingmassive,thankstothehiralanomaly. Atually,the
Maxwellequationslearlyshowsthat. Asweknow,in
threedimensionwehavetheCStermprovidinga
topo-logialmassforthegaugeeld.Infourdimensions,the
situation is moreompliated. In priniple we would
only have the Higgs sheme, but as the Higgs
parti-le hasnotbeenfound there is still asearhgoing on
to ndadynamialbreakingofthesymmetrywithout
theelementaryHiggs(bytheway,IreallthatR.
Shel-lard had been working atively in this subjetduring
his dotoratewithJ.Cornwall). Thereisalsothe
pro-posal of using antisymmetrial elds in the so-alled
BF formalism. In the Abelian version of the model,
the Lagrangian onsists of the usual Maxwell term, a
kinetitermoftheantisymmetrialeldanda
topolog-ialterm,
L= 1
4 F
F
1
12 H
H
+ 1
2 m
F
B
(1)
where, as C. Almeida explained, H
=
B
+
B
+
B
. The eld equations show that both
gauge elds are massive. In this respet, a 1997
re-sult, due to M. Henneaux, S. Sorella and o., proving
thefailure oftheshemein thenonAbeliansituation,
shouldberemembered. Similaranalysiswasperformed
bythegroupofCearawhenstudyinganalogoustheories
in 2+1dimensions.
Thereis,evidently,anintrinsiinterestinthiskind
of development whih points towards a better
under-standingofthefoundationsoftheeldtheory. Within
this ontext a 1996 ontribution of Barelos must be
InourgatheringtheBFmodelwasonsideredinat
leasttwosituations: aspreviouslypointed out,it was
regardedbythegroupofCearaasameansof
furnish-ingtopologial theories in 2+1 dimensions, througha
proess of dimensional redution. In another
presen-tation, by the CBPF group(J. Helayel-Netoand o),
adimensional redutionfrom fourto threedimensions
wasmade. AsupersymmetrialmodelwithN =1
on-tainingtheBFtermwasanalyzedinorderto obtaina
MCStheorywithanadditionalmagnetiouplingand
withaN =2extendedsupersymmetry. Priortothat,
itwasveried,bydeLaPlatagroupamongothers,the
existeneofanassoiationbetweentheextended
super-symmetrywithN =2oftheHiggsAbelianmodeland
the appearane of Bogomol'nyi equations. These are
rstorderequationsanditssolitonsolutionssaturatea
lowerlimitfortheenergy. Ontheotherhand,inaMCS
gaugetheory with magneti oupling and with a
spe-iquadratipotentialitwasshowthat Bogomol'nyi
type ofself-dualequation may exist. Itwas then
on-jeturedthat, alsoin this situation,there shouldexist
a relationship between the self-duality limit and the
N =2supersymetryofthemodel. Itislearthat this
topiarises interestsin severalgroups(UFRJ,CPBR,
UERJandCeara)
Aninterestingproposalforthequantizationofa
rel-ativistipartileundertheinueneofanexternal
ele-tromagnetibakgroundwaspresentedby D. Gitman.
Thesheme,whihsurmountstheknowndiÆulties of
therstquantizationfor arelativstisystem,leads to
adesriptionofbothpartileand antipartileand, for
weakelds,orretlyreproduestheonepartilesetor
ofthequantumeld theory.
2. Appliations. The main appliations found in
this meeting were about the Casimir eet. This
ef-fet,whih wasproposed morethan ftyyearsago, is
due to vauum utuations in limited regions of the
spae. The intensity and the signal of the resulting
fore depend on the geometry and the spin being, in
thebosoni ase, attrationof parallelplates. The
ef-fethasbeenstudied foragreatnumberofgeometries
andthe niteondutivity ofthe plates hasalsobeen
perturbatively inorporated. This wasexplained in a
seminarofMostepanenko. Manysituations were
atu-ally analyzed by the groups of UFRJ (C. Farina, M.
V. Cougo-Pinto, A. C. Tort and ollaborators).
Tem-perature eets ontheCasimir energy,for thease of
a avity having perfet onduting walls and also for
parallel plates in salar eletrodynamis with diverse
boundaryonditionswereonsidered.
Aninterestingthreadisbeingdevelopedbyagroup
from Paraba(D. Bazeia, F. A. Brito, J. R. S.
Nasi-mentoandR.Freire),fousingonthesubjetofdefets
ineldtheories. Spontaneousbreakingofsymmetryin
theorieswith two degenerated vauums allowfor
tionswhereanothereldisnonvanishingin themiddle
ofthewallmayleadtotheformationofanother
topo-logial struture, the wall inside the rst wall. Those
solutionsarepresentin supersymmetritheories. BPS
solutions,in partiular,arepresentifthe
supersymme-try is extended. Also, juntions an be formed if the
modelpossessesmorethantwovauumonewallending
at another wall. These studies may haveimpliations
tobothosmologyandondensedmatter,asD.Bazeia
argued.
Renormalization group studies, adequate to the
analysis of ritial phenomena, were presented by a
CBPFgroup(A.Malbouisson,NogueiraC.Calanand
N.Svaiter). TheyonsideredtheGinzburg{Landau-CS
model (i. e. salar Maxwell{CS quantum
eletrody-namiswitha 4
selfinteration). Inaoneloop
alu-lationand foraertainrange oftheparametersofthe
model,theyshowedthatthereexistsatriritialpoint
inadditiontoanotheroneinfraredstable. Besidesthat,
gaugeinvarianewasanalyzedandveriedtobeunder
ontrol in spite of the sharp uto used. It would be
interestingto omparethesalingpropertieswiththat
obtained by a S~ao Paulo group (de Albuquerque, da
Silvaand M. Gomes) whih in atwo loop alulation
arrivedatsomewhatdierentresults.
TemperatureeetsintheorieswithaCStermwere
presented by A. Das. At nite temperature, the non
Abelian CS model presents a problem whose proper
solution till today has not been found: due to gauge
invariane,the oeÆientof thenon Abelian CSterm
must be quantized. However, as the temperature
in-reases nothingseemsto preventontinuoshanges in
thatoeÆient. Dasdesribeda0+1modelwherethe
problem an be disussed in extenso. Following the
itwaspossibleto demonstratetheompatibilityofthe
perturbation atnon zerotemperatureand the
quanti-zationoftheCSoeÆientin2+1dimensions. Healso
reportedsomemorereentinvestigationonthesubjet,
donetogetherwithJ.FrenkelandF.Brandt.
At S~ao Paulo, we have interest in the Aharonov{
Bohmeet asobtainedbyanonrelativistiredution
proess starting from a relativisti theory. That was
relatedindaSilva'stalk.WheneveraCSmodelis
on-sidered, a natural issue is about the renormalization
sheme to be employed sine, due to the Levi-Civita
symbol, dimensional regularization is not onvenient.
Toirumventsuh problem, in our disussionof 2+1
models we also employed a soft versionof the BPHZ
sheme.
Another, more reent interest at S~ao Paulo is on
nonommutative theories, as it was explained by V.
Rivelles. Nonommutative theories had some of its
motivation on a ertain limit of superstrings models.
Perhapsbeauseof thatorigin, theywere expeted to
be onsistent. However, due to a sui generis
ultravi-olet/infrared mixing eah situation should be
investi-gatedseparately. Inthis respet, supersymmetri
the-orieshaveabetterhaneofsurviving.
I would like to end this talk by quoting J. A.
Swiea'swordsinhiswork\SolitonsandConnement".
More than twenty years ago, Swiea wrote: \After
almost half a entury of existene the main question
about quantum eld theory seems still to be: what
does it really desribe? and not yet: does it provide
agooddesriptionofthenature?" Afterthesuessof
the standardmodel many basiquestions are without
