c# display pdf in window : Add pages to an existing pdf control software platform web page windows asp.net web browser QFT-Schwartz6-part461

8.4 Vacuumdiagrams
Alotofthecontractions willresult indiagrams wheretheexternallines justconnectto o eachother,and
allthenon-trivialstuffappearsinbubbles. Therearealsodiagramsrepresentingthetimeorderedproduct
ofnothingΩ|Ωwhicharejustthebubbles. Itisnothardtorelatethesetwo,thenwecanfactoroutthe
vacuumfluctuationsΩ|Ωandjustconcentrateontheinterestingphysics.
Call the e internal l points x
i
and the external points y
i
. The e bubbles are just functions s of the e x
i
, for
example,somebubbleswouldgive
f(x
i
)=D
F
(x
i
,x
j
)
D
F
(x
k
,x
l
)
(8.69)
ThesefactoroutcompletelyfromanyFeynmandiagram
M=f
n
(x
j
)g(y
i
)
(8.70)
Soifwesumoverallpossiblebubblesforafixedwayofconnectingtheexternallinesweget
M=[
n
f
n
(x
i
)]g(y
i
)
(8.71)
ButthissumoverbubblesisexactlyΩ|Ω.Sothethingwereallywanttocalculateis
Ω|φ(y
1
)
φ(y
i
)|Ω
Ω|Ω
=
0|T{φ(y
1
)
φ
I
(y
n
)}|0
nobubbles
(8.72)
Thisgetsridofallthebubbles.
Nowrememberhowwedefinedthetransfermatrixby
S=1+iT
(8.73)
withtheM-matrix,whichisthethingwe’reultimatelytryingtocalculate,isdefinedby
T =δ
4
(Σp)M
(8.74)
What contributes to o the 1? ? Well, , anything in which the finalstates s andinitialstates are identical l will
contributetothe1. Thisincludesnotonlythetreelevelgraph,at0thorderinperturbationtheory,where
the initialstates just movealongintofinalstates. . Butalsomodificationsofthesegraphswhereabubble
comesoutofoneofthesetriviallines.
The M matrix only describes the interestingphysics,wheresome scatteringactually happens. . Thus
whenwecalculateMweonlyhavetoevaluatediagramsinwhichalltheinitialstatesandfinalstatesare
connectedtogether,where wecanget fromany of the external y
i
s toanyof theothers bymoving g along
linesinthegraph. Thus
M=0|T{φ(y
1
)
φ
I
(y
n
)}|0
fullyconnected
(8.75)
Thesearetheonlygraphsthathaveinterestingphysicsinthem. Everythingelseis s factoredoutandnor-
malizedawayas“nothinghappens”.
8.5 Tree-level φscattering
Nowlet’sdoanexampleinφ
3
theory. TaketheLagrangiantobe
L=−
1
2
φφ−
1
2
m2φ2+
g
3!
φ3
(8.76)
Wewanttocalculate
dΩ
(φφ→φφ)=
1
64π2E
cm
2
|M|
2
(8.77)
8.5 Tree-level l φscattering
61
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whereremember
S=1+(2π)
4
δ
4
(Σp
i
)iM
(8.78)
Iftheincomingmomentaarep
1
andp
2
andtheoutgoingmomentaarep
3
andp
4
. Thereare3diagrams.
Thefirst
p
2
p
1
p
4
p
3
gives
iM
1
=(ig)Π(p
1
+p
2
)(ig)=(ig)
i
(p
1
+p
2
)2−m+iε
(ig)=
ig2
s−m2+iε
(8.79)
wheres=(p
1
+p
2
)
2
.
Thesecond
p
2
p
1
p
4
p
3
turnsinto
iM
2
=(ig)
i
(p
1
+p
3
)2−m2+iε
(ig)=
ig
2
t−m2+iε
(8.80)
wheret=(p
1
−p
3
)
2
.
Thefinaldiagram is
p
1
p
2
p
4
p
3
Thisevaluatesto
iM
3
=(ig)
i
(p
1
+p
3
)2−m2+iε
(ig)=
ig
2
u−m2+iε
(8.81)
whereu=(p
1
−p
4
)2.Sothesumis
dΩ
(φφ→φφ)=
g
4
64π2E
cm
2
1
s−m2
+
1
t−m2
+
1
u−m2
2
(8.82)
Wehavedroppedtheiε,whichisfineaslongass,t,uarenotequaltom2.Forthattohappen,theinter-
mediatescalar wouldhave togoon-shellinoneofthediagrams,whichisadegeneratesituation,usually
contributingonlyto1intheS-matrix. Theiε’swillbenecessaryforloops,butintree-leveldiagramsyou
canprettymuchignorethem.
8.5.1 Mandelstamvariables
These variables s,t,u arecalledMandelstam variables. . They y are agreat shorthand,usedalmost exclu-
sivelyin2→2scatteringandin1→3decays. For2→2scattering,withinitialmomenta p
1
andp
2
and
finalmomentap
3
andp
4
,theyaredefinedby
s=(p
1
+p
2
)2=(p
3
+p
4
)2
(8.83)
t=(p
1
−p
3
)
2
=(p
2
−p
4
)
2
(8.84)
u=(p
1
−p
4
)
2
=(p
2
−p
3
)
2
(8.85)
62
FeynmanRules
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Thesesatisfy
s+t+u=
m
i
2
(8.86)
wherem
i
aretheinvariantmassesoftheparticles.
Aswesayinthepreviousexample,s,tanducorrespondtoparticulardiagrams wherethepropagator
has invariant t mass s s, t t or r u. . So o we e say s-channel l for r annihilation n diagrams. . In n these e the e intermediate
statehasp
µ
2
=s>0. Thet-andu-channelstoscatteringdiagrams
p
2
p
1
p
4
p
3
s−channel
p
2
p
1
p
4
p
3
t−channel
p
1
p
2
p
4
p
3
u−channel
s,tanduaregreatbecausetheyareLorentzinvariant. SowecomputeM2(s,t,u)intheCM frame,and
thenwecaneasilyfindoutwhatitisinanyotherframe,forexampletheframeofthelabinwhichweare
doingtheexperiment. Soyousees,t,ucanbeveryusefulandwewillusethemalot!
8.6 Withderivativecouplings
Supposewehaveaninteractionwithderivativesinit,like
L=λφ
1
(∂
µ
φ
2
)(∂
µ
φ
3
)
(8.87)
where Ihaveincluded3different scalarfields for clarity. . Inmomentum m space,these ∂
µ
’s givefactors of
momenta.Butnowrememberthat
φ(x)=
d
3
p
(2π)3
1
p
a
p
e
ipx
+a
p
e
−ipx
(8.88)
Soiftheparticleisbeingcreated,thatisemergingfromavertex,it gets afactorof ip
µ
,andifit’sbeing
destroyed,thatisenteringfromavertex,itgetsafactorif −ip
µ
. Soa a − forincomingmomentumanda
+ for r outgoing g momentum. . Inthis s case, , it’s s quite e important t to o keep track of whether momentum m is
flowingintooroutofthevertex.
Forexample,takethediagram
φ
2
φ
1
φ
3
φ
2
φ
1
8.6 Withderivativecouplings
63
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Labelthe initialmomenta p
1
µ
andp
2
µ
andthefinalmomentaare p
1
µ
and p
2
µ
. Theexchangemomenta a is
k
µ
=p
1
µ
+p
2
µ
=p
1
+p
2
. Thediagramgives
iM=
(
−iλ
)
2
(−ip
2
µ
)(ikµ)
i
k2
ip
2
ν′
(−ikν)
(8.89)
=−iλ
2
[p
2
p
1
+(p
2
)
2
][p
2
p
1
+(p
2
)
2
]
(p
1
+p
2
)2
(8.90)
Asacrosscheck,supposeweintegratedthistermbyparts
L=−λφ
3
[(∂
µ
φ
1
)(∂
µ
φ
2
)+φ
1
φ
2
]
(8.91)
Nowour 1diagram becomes 4diagrams,from the 2types of vertices onthe two o sides. . Butwecanjust
addupthecontributionstotheverticesbeforemultiplying,whichgives
M=(iλ)
2
(−ip
2
µ
)(−ip
1
µ
)+(−ip
2
)
2
i
k2
ip
2
ν′

ip
1
ν
+
ip
2
2
(8.92)
=−iλ2
[p
2
p
1
+(p
2
)2][p
2
p
1
+(p
2
)2]
(p
1
+p
2
)2
(8.93)
whichisexactlywhatwehadabove!Sointegratingbypartsdoesn’teffectthematrixelements.
Toseethatingeneral,wejusthavetoshowthattotalderivativesdon’tcontributetomatrixelements.
Supposewehaveaterm
L=∂
µ
1
φ
n
)
(8.94)
where there e are any y number r of fields in this term. . This s would give e a contributionfrom m the derivative
actingoneachfield,whichgives acontributionthatfieldsmomenta. . Soifthevertex wouldhave givenV
withoutthederivative,addingthederivativemakesit
(
incoming
p
µ
i
outgoing
p
µ
j
)V
(8.95)
butthesumofincomingmomentaisequaltothesum of outgoingmomenta,becausemomentumiscon-
servedatthevertex. Therefore,totalderivativesdon’thaveaneffect.
Tobeprecise,totalderivativesdon’thaveaneffectinperturbationtheory. Itturnsoutatermlike
F
˜
F≡ε
µναβ
F
µν
F
αβ
=∂
µ
µναβ
A
α
β
A
ν
)
(8.96)
is atotalderivative. . Thus s if we adda a term θF
˜
F to o the Lagrangian, nothinghappens. . That t is,nothing
happensinperturbationtheory. Itturnsoutthatthereareeffectsofthistermthat t willnevershowupin
Feynmandiagrams,butareperfectlyreal. Theyhavephysicalconsequences. Forexample,thistermleads
totheviolationoftime-reversalinvariance(thestrongCPproblem),andatermlikeitmayberesponsible
forthe existenceof matter overantimatter intheuniverse. . Thesekinds s ofeffects arerelatedtopseudo-
particlescalledinstantonsandsphaleronsandareapurelynon-perturbativephenomenon.
8.7 Summary
Wesawthatinclassicalfieldtheory,thereiswaytosolvetheequationsofmotionusinggreensfunctions.
For example, , in the presence of a a classical source, this s led to o Feynman n diagrams s of the form for the
Coulombpotential
φ(r)∼A
0
e
2
r
∼−
e
2
k2
J
0
64
FeynmanRules
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Nowwehaveseenthesameamplitudereproducedwiththe“real”Feynmandiagrams
p
2
p
1
p
4
p
3
=
e
2
(p
3
−p
1
)2
=
e
2
k2
Iffact,ifyoureallythinkaboutit,whatweshowedwasthat
dσ∼M
2
e
4
k4
(8.97)
andthenweusedtheBornapproximationfromnon-relativisticquantummechanics
dσ∼V
˜
(k)
2
(8.98)
toextractV(k)=
e2
k2
. Butit’sthesamething–Coulomb’slawiscomingfromaLagrangian
L=−
1
2
AA+AJ
(8.99)
Fortheclassicalcase,wejusttookJ
µ
µ0
δ
(3)
(x),tosimulateapointcharge. FortheFeynmandiagram
case,wereplacedJbysomeinteractionJ=eφAφ,sothatthepointchargeisnowthoughtoflikeascalar
field,whichisthenon-relativisticversionofanelectron.Diagrammatically
=J
1
k2
J =
p
2
p
1
p
4
p
3
I think it is helpful alsoto see e more e directly why the e Feynman rules andthe e classical perturbation
theoryarethesame. StartingwithaLagrangianwithacubicinteraction
L=hh+λh
3
+hJ
(8.100)
Inclassicalfieldtheory,wefoundaperturbationexpansionthatlookedlike
h =
+
+
=
1
J+
1
λ(
1
J)(
1
J)+
where
1
≡G(x)isshorthandfortheGreen’sfunctionwhichsolvestheclassicalequation
G(x)=δ(x)
(8.101)
Itis also o sometimes s calledaKernel. . The Feynmanrulesare the samething, but theGreen’sfunctionis
replacedbytheFeynmanpropagator
D
F
(x,0)=
0|T{φ(x)φ(0)
}
|0
=
d4k
(2π)4
i
k2+iε
eikx
(8.102)
ButthisisalsoaGreen’sfunction,satisfying
D
F
(x,0)=δ(x)
(8.103)
But the time ordering g makes s it a a Green’s s functionfor r scattering processes. . BothGreen’s s functions are
essentiallyjust
1
k2
–don’tlettheiεdistractyoufromthisessentialpoint.
8.7 Summary
65
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WhataretheseGreen’sfunctions? Intheclassicalcase,solvingG(x)=δ(x) ) gives youG(x)whichis
thesizeofafieldifyouaddasourceitatx=0. Ifthesourceismovingaround,themovementpropagates
atthespeedoflight(notfaster!),sinceisLorentzinvariant. SotheGreen’sfunctiontellsyouhowthis
movementpropagates.Inthequantumcase,wesawthatthefieldφ(x)createsaparticleatpositionx
φ(x)|p=e
ipx
(8.104)
Thus
0|φ(x)φ(0)
|
0
tellsyoutheamplitudeforaparticlecreatedat0togettox. That’sthesamething
as what the Green’s function n is telling you. . So o both of f these e things s are e propagators. . Obviously y the
quantumonecaninterferewithitselfandpairproducethingsanddoallkindsoffunquantummechanical
stuffthat the classicalfield d can’t t do. . But t intheclassical limit, they are the same,and d wecansee e that
immediatelyintheequationsofmotion.
We also saw that the same amplitude comes from old-fashioned d perturbationtheory y (OFPT), where
thesumoverintermediatestatesincludesdifferenttimeorderings
T
if
=
e
2
E
i
−E
n
(1)
+
e
2
E
i
−E
n
(2)
=
2e
2
E
γ
k2
(8.105)
Now2E
γ
goes into o the normalizationofthefields,inthe
1
p
factor, sothis s transitionmatrixelement,
T
if
isthesamethingasourLorentz-invariantmatrixelementM,butcalculatedathirdway,usingtime-
independentperturbationtheoryfromquantummechanics.Sothere’slotsofwaystodothesamecalcula-
tion.
TheFeynmanruleshavetheadvantagesoverOFPTthat
• TheyaremanifestlyLorentzinvariant–usefulinrelativisticsettings
• 4-momentum m is conservedateachvertex,as opposedtojust3-momentum. Thisforces ustosacri-
fice having g intermediate e particles be e on-shell. . But t that’s ok, since e intermediate e particles are not
observable–thinkingoff-shellisnowthoughtofasagreatconceptualadvance.
• Theyaresystematicforhigher r ordercorrections,inwhichvirtualparticles arepair created. . Aswe
sawwithOppenheimer’s attempt at the Lambshift. . As s far asIknow,nobodyhasever beenable
tosortoutloopcorrections(renormalization)inOFPT
TheFeynmanruleshavetheadvantageoverclassicalfieldtheorythattheyarequantummechanical!
Finally,let’s break downthesteps thatledtotheFeynmanrules. . Westartedwithwantingtocalcu-
latecrosssectionsforscatteringprocesses. TheseweregivenbysomekinematicfactorstimesS-matrixele-
ments
dσ∼kinematics×f|S|i
2
(8.106)
Wefactoredouttheuninterestingpartof S-matrix
1−S=iδ4(Σp)M
(8.107)
dσ∼kinematics×M2
(8.108)
ThenwederivedtheLSZtheoremwhichsaidtoaddsomephasesandprojectoutthepolesoftheexternal
linesandtowritethematrixelementasatimeorderedproduct
M=p
i
2
e
ip
i
x
1
p
f
2
e
−ip
f
2
×0|T{φ
φ}|0
(8.109)
Thenthetimeorderedproductcouldbeexpressedinperturbationtheoryintermsofinteractionsandfree
particlepropagators
0|T{φ
φ}|0=
1
p
i
2
1
p
f
2
λ
n
1
k
1
2
λ
m
1
k
2
2
(8.110)
66
FeynmanRules
The propagators s from m the external legs canceled d the pfrom m LSZ and d the e phases give e delta a functions,
enforcingmomentumconservation.Intheendonlytheinterestingstuffcontributestothecrosssection
dσ∼kinematics×
λ
n
1
k
1
2
λ
m
1
k
2
2
(8.111)
Keepinmindthat allthemessyintermediatestepsingettingto o the Feynmanrulestook 30 years, from
1930-1960tobeworkedout. Soalthoughthefinalseemssimple,andyoumightevenhaveguesseditfrom
thinkingaboutclassicaltheory,historicallytherewasalotthatwentintoit. TheFeynmanrulesarefan-
tastic because they are not only simpleand d intuitive e (if youknow classical fieldtheory), but also exact
andsystematic.
8.7 Summary
67
Chapter9
GaugeInvariance
9.1 Introduction
Upuntil now, we havedealt withgeneral features s ofquantum fieldtheories. . For r example,wehaveseen
howtocalculate scatteringamplitudes startingfrom ageneralLagrangian. . Nowwe e willbegintoexplore
whattheLagrangianoftherealworldcouldpossiblybe. Then,ofcourse,wewilldiscusswhatitactually
is,oratleastwhatwehavefiguredoutaboutitsofar.
Agoodwaytostartunderstandingtheconsistencyrequirementsofthephysicaluniverseis withadis-
cussionofspin. Spinshouldbefamiliarfromquantummechanics. Spinisaquantumnumber,labelledby
J,whichreferstorepresentationsoftherotationgroupSO(3). FermionshavehalfintegerspinJ=
1
2
,
3
2
,
andbosonshaveintegerspinJ=0,1,2,3,
.
Spin 1particles s arecritical l toquantum electrodynamics, becausethe photonis s spin1. . Butthey y are
alsoessentialtounderstanding quantum fieldtheory ingeneral. . I I cannot emphasize enoughhow impor-
tantspin1fieldsare–ithashardtodoanycalculationorstudyanytheoreticalquestioninquantumfield
theory without t having g to deal with spin 1. . Inorder r to o understandspin 1, we first have tounderstand
someoftherequirementsofaconsistentquantumfieldtheory.
9.2 UnitaryrepresentationsofthePoincaregroup
Inquantum fieldtheory,we are interestedparticles, which h transform m covariantly under translations and
Lorentz transformations. . That t is, they shouldform representations s of the e Poincare e group. This means,
thereissomewaytowritethePoincaretransformationsothat
|ψ→P|ψ
(9.1)
Moreexplicitly,thereissomekindofbasiswecanchooseforourstates|ψ
,callit|ψ
i
sothat
i
→P
ij
j
(9.2)
Sowhenwesay representation wemeanawaytowritethetransformationsothatwhenit acts thebasis
closesintoitself.
Inaddition, we want unitary representations. . The e reasonfor this s is s that the things we compute in
fieldtheoryarematrixelements
M=ψ
1
2
(9.3)
whichshouldbe Poincareinvariant. If M is Poincareinvariant, and|ψ
1
and|ψ
2
transform covariantly
underaPoincaretransformationP,wefind
M=
ψ
1
|P
P|ψ
2
(9.4)
69
So we needP
P = = 1, , which is the definitionof unitarity. . There e are many more representations of the
Poincare groupthantheunitary ones, but the unitary ones are the only ones s we’ll beable e to compute
Poincareinvariantmatrixelementsfrom.
Asanaside,it is worthpointingoutherethat thereisanevenstronger requirementonphysicaltheo-
ries:theSmatrixmustbeunitary.RecallthattheS-matrixisdefinedby
|f
=S|i
(9.5)
Saywestartwithsome normalizedstatei|i=1,meaning that the probability for findinganything at
timet=−∞isP=i|i
2
=1. Thenf|f=
i|S
S|i
. SowebetterhaveS
S=1,or elseweendupat
t=+∞withlessthanwestarted! ThusunitarityoftheS-matrixisequivalenttoconservationofproba-
bility, whichseems s to o be aproperty ofour r universeas far as anyonecantell. . We’llseeconsequences s of
unitaryofS lateron,butfornowlet’ssticktothediscussionofspin.
TheunitaryrepresentationsofthePoincare groupwereclassifiedfor thefirst timebyEugeneWigner
in1939. Asyoumightimagine,beforethatpeopledidn’treallyknowwhattheruleswereforconstructing
physicaltheories,and by y trial anderror they were comingacross allkinds of problems. . Wigner r showed
thatunitaryrepresentationsareuniquelyclassifiedbymassmandspinj. mcantakeanyvalueandspin
isspinof3D rotations. I’mnotgoingtogothroughtheproofhere,butitis doneingreatdetailinWein-
berg’sbook.
Therulesarethatforageneralmassm
0,afieldofspinjhas2j+1polarizations,labeledbyσ=−
j,−j+1,
,j(youcanthinkof σ=j
z
ifyoulike). Soamassivefieldofspin0hasonestate,spin
1
2
has
two,spin1hasthree,andsoon. Formasslessstates,allspinshaveonlytwopolarizations:σ=±j(mass-
lessspin0hasonlyonepolarizationsince j=−j). It’snothardtoprovethis usinggrouptheory,butwe
willcontentourselves withunderstandingitusingfieldsandLagrangians. . Again,seeWeinberg’sbookfor
details.
ThewholetrickinconstructingalocalLorentzinvariantfieldtheoryistoembedthesespinstatesinto
objectswithspacetimeindices. Thatiswewanttosqueezestatesofspin0,
1
2
,1,
3
2
,2etcintoscalars,vec-
tors, tensors, andspinors. That way we canwritedownsimple looking Lagrangians s anddevelopgeneral
methods for doingcalculations. . Thenwe e endupwithacomplication:tensorshave 0,4,16,64,
,4
n
ele-
ments (forgettinghalfintegerspinfor now),butspinstateshave0,3,5,7,
,2j+1physicaldegrees of
freedom. Theembeddingofthe2j+1statesinthe4
n
dimensionaltensorscausesallthetrouble. But in
the end, nobody y has s figured out a better way to do things s (although h recent t excitement t about twistors
indicatesthattheremaybeanalternative).
9.2.1 asimpleexample
We don’t needall thatfancy language tosee the problem. . Say y we werenaiveandguessedthata spin1
stateshouldhaveaLagrangianlike
L=V
µ
V
µ
(9.6)
Thefreefieldsolutionsaresimplyplanewaves
V
µ
µ
e
ikx
(9.7)
whereforpolarizationε
µ
,any4-componentvectorisallowed. Wecanjustdeclarethatε
µ
transforms like
avector,then|V
µ
|
2
µ
2
isLorentzinvariant. However,ifwetrytomakeparticlesthisway,wefindtheir
normsare
V
µ
|V
µ
=ε
µ
2
0
2
−ε
1
2
−ε
2
2
−ε
3
2
(9.8)
ThenormisLorentzinvariant,butnotpositivedefinite!
Wecan’tgetanywherewithatheorywithoutapositivedefiniteinnerproduct(norm).Forexample,if
Ihave aparticle withε=ε
0
=(1,0,0,0),thetotalprobabilityisjust |ε
2
|
2
=1. IfIinsteadtookastate
withpolarizationε=ε
1
=(0,1,0,0)thetotalprobabilitywouldstillbe|ε2|=1. However r if Iconsider
thematthesametimeIwouldhaveastate|ψ
=|ε
0
⊕|ε
1
(atwoparticlestate),whosenormis
ψ|ψ
=
ε
0
0
+
ε
1
0
+
ε
0
1
+
ε
1
1
=1−1=0
(9.9)
70
Gauge Invariance
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