Nonommutative Field Theory
M. Gomes
InstitutodeFsiadaUniversidadede S~aoPaulo
CaixaPostal 66318,05315-970, S~aoPaulo,SP,Brazil
Reeivedon9Marh,2002
Nonommutativeeldtheories presentmany surprisingproperties. Asaonsequene ofthe
non-ommutativity,highmomentummodesdonotdeouplefromthephysisatlargedistanesleading
to theappearaneofinfraredsingularities evenintheorieswithoutmasslesspartiles. Being
non-integrable,theseinfraredsingularities destroy theusualperturbativeexpansions. We disusstalk
alternativeproeduresto,weexamineontrol thissituation. Inthelow momentumregimeof the
Wess-Zuminomodeltheeetoftheunderlyingnonloalityonthenonrelativistipotentialwillbe
examinedandsomephysialimpliations.
Itisarelativelyoldquestionifthereexistsornota
minimumlengthbeyondwhih nostrit loalizationis
possible. Inthe aseofapositiveanswer,interations
would have to be smeared over the minimum length,
possibly eliminating the ultraviolet divergenes of the
perturbativeseries. Suh an idea was put forward in
the forties of the last entury [1℄ but never ahieved
popularity,mainlybeauseofthesuessofthe
renor-malizationprogramforQED.Morereently,ithasbeen
arguedthatforverysmalldistanes,distanesofthe
or-derofthePlank'slength(10 33
m),themeasurement
ofoordinatesismeaninglessdueto theappearaneof
stronggravitationaleldswhihwillpreventeverything
to transmit information [2℄. Aording to the
Heisen-bergunertaintyrelations,apreisemeasurementof a
oordinate leads to a very high indeterminay in the
momentum. Inthisproess, therefore,agreat amount
of energy is transmitted to the observed system and
the orrespondingenergy-momentum tensorgenerates
a strong gravitational eld, as predited by the
Ein-stein'sequation. Themorepreisethemeasurementof
theoordinatesthebiggerwillbethegravitational
om-ingfromthereferredmeasurement. Wheneverthiseld
is sostrongasto prevent light orother signsto ome
out from the region under observation no operational
meaninganbegiventotheideaofstritloalization.
From a quantum mehanial point of view, the
aboveanalysissuggestthevalidityofthefollowing
po-sitionunertaintyrelations
q
q
j
j (1)
where
isanantisymmetrimatrixwhoseelements
areoftheorderofthesquareoftheminimumlength.In
fat,bypostulating(1)itfollowsthat,ifoneoordinate
theremainingones.
The possibility unveiled by (1) is a diret
onse-quene of
[q
;q
℄=i
; (2)
so that lassialeldsbeomenonommutative
opera-torsafterthequantizationpresribedin(2).
Nonommutative quantum eld theories have also
arised in the ontext of string theory as it has been
shownthatthedynamis ofaD-braneinthepresene
of an antisymmetrield an bedesribed, in ertain
limits,intermsofagaugetheorydeformedbyaMoyal
produt[3℄;thisprodutisaharateristiofthe
non-ommutativity. Inthis meeting,the stringsaspetsof
nonommutativespaehavebeendisussedintheProf.
Matsuotalk. Here,Ionlywouldliketoremarkthatthe
disoveryof nonommutativeeld theory asaertain
limit of stringtheory raisedthe hopes that this limit
should leadto aonsistenttheory(see[4,5℄forreent
reviews). I will disussthisaspetandwill stressthat
thisisI highlynontrivialpoint.
LeavingasidequestionsposedbythelossofLorentz
invariane, a rst observation is that a violation of
ausalityand unitarityours when the
nonommuta-tivityinvolvesthetimeoordinate[6,7℄.
Thenextobservationonernstheentanglementof
sales: smalldistanesinonediretionimplylarge
dis-tanes in the other, asfollows from (1). Usually, the
ultravioletbehaviorofaeldtheoryisunrelatedtothe
infrared one. Thus, in massive 4
there are quadrati
andlogarithmiultravioletdivergenesoexistingwith
infrared niteness. In the nonommutative ase
mat-ters aremoresubtle. Thesimplest examplewhere the
o-nonommutativeamplitudeisgivenby
Z
d 4
k os(k
p
)
k 2
m 2
: (3)
The appearane of trigonometri fators is a
hall-markof thenonommutativesituation. Itprovidesan
osillatingfator whih eetivelydamps the
ultravio-letbehavior. Theintegralturnsouttobenitebut,in
awaydierentfrom theusual situation,itdepends on
the externalmomentum. If this momentum were zero
(or ifwerezero)theintegralwould bequadratially
divergent. We arethus led to the onlusion that for
smallptheintegralshould behaveas
1
p 2
2
: (4)
When multiple insertionsof this graphare made into
a largediagram theygenerate non integrableinfrared
singularities. This feature produesthe breakdownof
theperturbativeshemein manyoftherenormalizable
theories. Weshallreturnto thispointlateron.
We now proeed to desribe some of the steps to
befollowedto build somesimpleeldtheories in
non-ommutative spae. First of all, eld operators may
beonstrutedviathesoalledWeyl-Moyal
orrespon-dene. Tothisendweintroduetheoperator
T(k)=e ik
q
; (5)
whihobeystherules[8℄:
1. T y
(k)=T( k)ifq
y
=q
.
2. T(k)T(k 0
) = T(k +k 0
)e i
2 k
k
0
whih
fol-lows immediately after the appliation of the Baker{
Hausdorfformula
e A
e B
=e A+B
e 1
2 [A;B℄
; (6)
for[A; B℄=number.
3. TrT(k)=(2) n
Q
Æ(k
)
The operatorsT(k)will be assumed to form a
ba-sisinthespaeoftheeld operatorsinanalogousway
as e ik x
form a basis for squared integrable funtions
onthe ordinary spae. More preisely, to every
suÆ-ient smooth lassial funtion (x) we assoiate the
eld operator (fornotationalsimpliity dx d n
x and
dkd n
k)
= 1
(2) n
Z
dxdkT(k)e ikx
(x); (7)
or
= Z
dk
(2) n
T(k) ~
(k); (8)
intermsoftheFouriertransform ~
(k)of(x),
~
(k)= Z
dxe ik x
(x): (9)
Wemayverifythat
(x)= Z
dk
(2) n
e ik x
Tr[T y
(k)℄: (10)
Thisformula relatesin adenitewayalassial
fun-tion(x) to agivenoperator (q). It anbe used to
onstrutthe so alled Moyalprodut of the lassial
funtions
1 and
2
, orresponding to the produt of
theoperators
1 and
2
whihareassoiatedtothem:
1 (x)
2 (x)=
Z
dk
(2) n
e ik x
Tr[
1
2 T
y
(k)℄: (11)
TheMoyalprodutisahighlynonloalfuntion
involv-inganarbitrarynumberofderivatives. Infat,fromthe
denition(11)it ispossibletoshowthat
1 (x)
2
(x)= lim
y!x e
i
2
y
x
1 (y)
2 (x)=
1 (x)
2 (x)+
i
2
(
1 (x))(
2
(x))+:::; (12)
whih, due to the antisymmetry of
, diers from the ordinary pointwise produt just by a total derivative.
The same does not happen for the Moyal produt of three or more elds. Employing the simplied notation
k
i ^k
j
1
2 k
i k
j
,onends
Z
dx
1 (x)
2 (x)
3 (x)=
Z
dk
1
(2) n
dk
2
(2) n
dk
3
(2) n
(2) n
Æ(k
1 +k
2 +k
3 )e
ik
1 ^k
2
~
1 (k
1 )
~
2 (k
2 )
~
3 (k
3
) (13)
and, moregenerally,[8℄
Z
dx
1 (x)?
2
(x)?:::?
N (x)=
Z
Y
dk
i
(2) n
(2) n
Æ(k
1 +k
2
+:::+k
N )
~
1 (k
1 )
~
2 (k
2 ):::
~
N (k
N
)exp( i X
i<j k
i ^k
j );
As seen from the above expressions, the Feynman
rules foranonommutative eld theoryare similar to
theonesintheommutativease,unlessbyosillating
trigonometrifatorsattheverties. Atually,aspeial
onsiderationisneededin theaseofgaugesymmetry:
duetothenonommutativity,evenintheU(1)aseone
should startwith anonabelianexpressionfor theeld
strengthandthenreplaetheordinaryprodutbythe
Moyalone[9,10,11℄.
The presene of the osillating fators at the
ver-tiesregularizesomeoftheultravioletdivergenesbut
produe infrared divergenes. In suh situation, we
may envisage two proedures to go ahead:
resumma-tionandextension ofthemodel. Inthe rstapproah
thefree propagatorismodiedbyinorporatingsome
part of the interation to ameliorate the infrared
be-havior [12, 13℄. This modiationleads to dispersion
relations dierent from the usual ones and has some
interestingphysialimpliationsingaugetheories[14℄.
However,thereisatroublesomeaspetofthismethod,
namely, theabsene of alearly identiable
perturba-tive parameter. A seond possibility whih does not
havethis drawbakonsistsin the additionof new
in-terationsinsuhwayimprovetheultraviolet/infrared
behaviorofthe amplitudes. Inthis respet
supersym-metri theories appear as natural andidates.
Atu-ally,ithasbeenprovedthatthenonommutative
Wess-Zuminomodelandthe2+1dimensionalnonlinear
non-ommutativesigmamodel arefreefrom thedangerous
IR/UVmixing [15,16℄.
Besides the renormalization problems ommented
before one is naturally also interested in unovering
the physial ontent of a given nonommutative eld
theory. Here, one nds unusual and perhaps
sur-prising aspets. For example, ontrarily to its
om-mutative ounterpart, nonommutative QED exhibits
asymptoti freedom. Also, the realization of
sponta-neoussymmetrybreakdownaneessaryingredientofa
nonommutative\standard"modelseemstodependon
thenatureoftheunderlyinginteration. Forthelinear
sigma model, it has been demonstrated that, at least
uptooneloop,spontaneousbreakdownmayourfor
the U(N) but not for the O(N) symmetry if N > 2
[17℄.
Followingthethreaddelineatedabove,Iwouldlike
topresentsomereentresultsonthelowenergylimitof
the nonommutative Wess-Zumino model [18℄.
Start-ing from fermion-fermion and boson-boson sattering
amplitudes, one may determine the equivalent
poten-tials whih xes the non-relativisti dynamis. They
arenonloal potentialwithpeuliarproperties.
TheLagrangiandesribingthemodelis[19℄
L =
1
2
A(
2
)A+ 1
2
B(
2
)B+ 1
2
(i6 m) + 1
2 F
2
+ 1
2 G
2
+mFA+mGB
+g(F?A?A F?B?B+G?A?B+G?B?A ? ?A ?i
5
?B): (15)
d
where A isasalareld, B is apseudo salareld,
is a Majorana spinor eld and F and G are,
respe-tively, salarand pseudosalarauxiliaryelds. Notie
that omposites bilinearin thebasields are dened
usingordinary,i. e.,notMoyalorderedproduts.
Consider thesattering of twoMajorana fermions.
Letusdesignatebyp
1 ;p
2 (p
0
1 ;p
0
2
)andby
1 ;
2 (
0
1 ;
0
2 )
thefourmomentaandz-spinomponentsofthe
inom-ing(outgoing) partiles,respetively. Wework in
en-terofmassframewherethekinematisbeomessimpler
andonehasthatp
1
=(!;~p),p
2
=(!; ~p),p 0
1
=(!;~p 0
),
p 0
2
=(!; p~ 0
),j ~p 0
j=j ~pj,and!=!( ~p)= p
~ p 2
+m 2
.
Inthelowest order ofperturbation there arethree
ontributing amplitudes assoiated to the diret,
ex-hangeand\annihilation"hannels(beauseweare
us-ingMajoranaelds). Thetotalamplitudeis
R==(2) 4
Æ (4)
(p 0
1 +p
0
2 p
1 p
2 )(T
a +T
b +T
) (16)
wherein thenonrelativistilimitonends
T L
a =
1
(2) 4
g
m
2
Æ
0
1
1 Æ
0
2
2
1
2
os m
0j k
j
+ 1
2 os p
i
ij k
j
; (17)
T L
b =
1
4
g
2
Æ
0
1 2
Æ
0
2 1
1
os m
0j k
0j
+ 1
os p i
ij k
0j
T L = 1 3(2) 4 g m 2 Æ 0 1 1 Æ 0 2 2 os m 0j p j os m 0j p j k j Æ 0 1 2 Æ 0 2 1 os m 0j p j os m 0j p j k 0j ; (19) where k j p j p 0j (k 0j p j +p 0j
)denotes themomentumtransferred inthediret (exhange) satteringwhile
thesupersriptLsignalizesthattheaboveexpressionsonlyholdtrueforthelowenergyregime.
Conerningtheboson-bosonsattering, astraightforwardalulationshowsthattheamplitudes orresponding
to thediret andexhangeproessesbeomenegligiblein thelowenergyregime,thetotalamplitudebeingjust
T L = T L = 1 6(2) 4 g m 2 os m 0j p j os m 0j p j k j +os m 0j p j k 0j : (20) d
Notiethatthemagneti partofthe
nonommuta-tivityparameter(
ij
)doesnotontributeinthe
non-relativistilimit.
Fromtheaboveexpressions,equivalent
nonrelativis-ti potentials may be determined as follows. Usually
these potentials are dened as the Fourier transform
(FT)withrespetto thetransferredmomentum ( ~
k)of
the sattering amplitudes. If the amplitudes depend
onlyon ~
ktheorresponding(FT)willbeloal
depend-ing only on therelativeoordinate~r. In ourase the
amplitudes depend notonly on ~
k but also onthe
ini-tialmomentum ~pofthe satteredpartile. Beauseof
the nonommutativityof momentum and position
op-erators, the onstrution of the potentials from these
FTmaybeaitedbyorderingambiguities. Wesolve
theseambiguitiesbyrequiringhermitiityofthe
result-ing expression. After that we hek that the resultis
an eetive potential in the sense that its momentum
spae matrixelementsorretlyreproduesthe
ampli-tudesthatwehadabinitio. Proeedinginthisway,we
areledto introdue
Æ 0 1 1 Æ 0 2 2 M F ( ~
k;~p) T L
a (
~
k;~p)+ T L
;dir (
~
k;~p) (21)
and
M B
( ~
k;~p) T L ;dir ( ~
k;p)~ ; (22)
for the fermioni and bosoni sattering amplitudes.
TheFTtransformoftheamplitudesarethengivenby
V F;B
(~r;~p) = (2) 3 Z d 3 kM F;B ( ~
k;~p)e i
~
k~r
: (23)
where the subsript dir is used to indiate that only
the diret part of the amplitudes enter in the
al-ulation. The orresponding quantum operators are
obtained through the replaements ~r ! ~
R;~p ! ~ P, where ~ Rand ~
P aretheCartesianpositionand
momen-tum operators obeying, by assumption, the anonial
ommutation relations R j ;R l = P j ;P l
= 0 and
R j ;P l
=iÆ jl
. Wehave[18℄
^ V F ( ~ R ; ~
P) = g m 2 Z d 3 k (2) 3 e ik l R l e ik l lj P j +e ik l R l e ik l lj P j 2 3 g m 2 h Æ (3) ~
R+m ~ +Æ (3) ~ R m ~ i + 1 3 g m 2 h Æ (3) ~ R m ~ e 2im ~ ~ P + e 2im ~ ~ P Æ (3) ~ R m ~ i ; (24) and ^ V B ( ~ R ; ~ P) = 1 6 g m 2 h Æ (3) ~
where ~
f
0j
;j=1;2;3g. Notiethat themagneti
omponents
ij
onlyontributetotherstlineof(24).
Thisontributionisfreefromorderingambiguities
be-auseof
k l
R l
;k m
mj P
j
= ik l
k m
mj Æ
lj
= 0: (26)
Aslaimedbefore,ithasbeenhekedthat(24)and
(25)orretlyreproduesthesatteringamplitudeswe
hadbefore. Fromthoseformsof thepotentialswean
drawthefollowingonlusions:
1. In the ase of spae/spae nonommutativity
fermions may be pitured as rods oriented
perpendi-ular to the diretion of the inoming momentum. By
ontrast, in thisase, thebosonipartiles arenot
af-fetedbythenonommutativity.
2.Intheaseoftime/spaenonommutativityboth
fermionsandbosonsareaeted. Thenonloality
man-ifest through the fat that the sattering ours only
when the partiles are at a distane of the order of
mjj.
Novestigeofbreakingofunitaritywasfoundinspite
ofthepreseneofsatteredadvanedwaves. However,
this piturewillmostertainly breakdownwhenloop
ontributionsaretakenintoonsideration.
Aknowledgements
Thiswork waspartiallysupported byFunda~aode
AmparoaPesquisadoEstadodeS~aoPaulo(FAPESP)
and Conselho Naional de Desenvolvimento Ciento
eTenologio(CNPq).
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