Introduction to perturbation theoryConsiderwith

theory ofthedomeasureSe(p)single real variable O

aactionand+ap +=4

partition function

zSz(p)(dde==Jap tapt) " pIRItIts

perturbativeZexpansion

is= Jadetappin

pert

IR

It is(not.
convergent

If it hadthe integral

Instead

X-

IKR

for

*,

it

(z(x)

convergence radius R30

ais-

but,

is

anZ(x) isnotbadly divergent,

Zpert(x)z(x)

0& truncation

asatz(x)obviously analytic at/forrealXX0order= 0:

negative X.

asymptotic expansion of Z(X)
+=.

&N:

See the additionalnote for a proof

Each term ofthefree

EpertEperttheory
Zree=

Similarly forwitha

correlation

actionfunction ofSE free(P) tap2=:) < P )free*correlation functions

Selfe

period=:,(f(a)) =(f(p)

isZfree-me (p)" (f(p)PFree*)free*< P )free

Digression

Free theory&free=

Se free(p) = tap,(dap< >free ==: G

Gaussian

integralWick contraction

< 001" freee g.

Sum=.

(pdbb)freeof all possible Wickcontractions+P

=

3
=

a

see the additional notes if you are not familiar with this.

Using thisZa /Zee+findwe,1=+(4))

7- )freefree

7 Pfee-Pox3

A(7)

free

=

Eipp 1973+
W:

B

00PX()22cLpp)

t+

4!

3D

Diagramatic presentation-apropagator

60

interaction

-

A:
C=:

line

us

Vertexo,

Bitco
Zar /Ze+=

Ciao
1+ofDif"

00 +sum,vacuum

200+

---,+

A

+...

diagrams↑noexternal line#=-

See the additionalnote for a proofexp(0+000+Sumofconnectedvacuumnu< PPXpert=...

(PPfree + <PP"))free
1+1- )free*+--

diagrams+--)=A+-som of+diagrams

2A+withtwo-++external lines

(Pb)

con

See the additionalnote for a proof< POPP) port=

Sum<PPPPCSee the additionalnote for a proof

diagrams

ofconn+with

(PP)

·conn

=

part

X

X+B

=+...

+(xX X++X++Q+...

disconnected part(( q -) (( q -)++++++((-+2+-+N p ) (x+3B

connected+<PP--&&

4 external lines

+...

+++-.

.

- -1)x)Generalizations
more interactions

Se(p) = ap +()

...

-free

Zee/Zee(Pm=< PP(petexp)Af==interactions++<PP)

conn

00+0-0+00+. . .

)( (b))=. . . .

+<PYone (Pconn

im6f- -++. . .

1969 Xpert <PPPYonnt <Pconn (PP)

conun

Cx
i++Q+....

*3+10

nvariables

Se(p =A j-+-free

Epert/Eree=interaction

exp(08+000++-=i(P)=imj&i-(...

inj+=fj ++(-+,=Elij=& (P P; ).

conn-Putut(a)Pj x4 3.

=in An As,i
D=- >Pete; 24e Aintre) es,(PiPinPisPia)pert=(PisPinPisPatconnt (PiPin) conn <Pin Pipcommm↑(iii)( : :4) + (4) Liz is)&+/"VisI
I /I iLis -"iinis

D+12"VisOri"Si& =i4=-inis

&,

D&

54

14i isO+iO-in

142is+im i4Li
O+-,PisEli x4 !

bil2bine~Pipx[j As j jO =t10.

P= As-CPartit-Pizx

Pip

Aris Aria

2 (4.

.

3)? 2.

Scalar field in 4-dimensions

SE(PT=Sai (t(0)2+

me=P-+ 4)-freeexp)CO 200
00 = x(Pipas)"

Epert/Zree+=

14

interaction

+

D+...

↑P-theory"(iD&CaspertiAtt
D
2.

-=-(2Plis Plan---&Jax &-cIO-·i.

=(a) blas Plasplas Play(a)

? Peo
pinp
Papie
Jarding[P( ) Pl / Pla)(44) += (P()P)11) Pla-,arcom

m+ (P(4 )b())

SI ."+ S "+II

4D&

23&t

4&-=P+&&X+&on&SZ

4I

324+- 43O&3V

[P(43)P(K4)&2Sx)a xp(x ba)

Q+91:)M2

diy Pitts (pup(s)--iconn((3)(24) + (14/(23)4

DoI/&,plan) &(3) (4)Scalarpropagator

SE free(p]+(dp()(=,1P(P(3)70 m==- + my b(x)Pls)

Important remark:Pitpl3,mu= Jax())&Sat=I: Sor
4

2=Thisdivergent.

---divergent

plas plas plusplas plastw.

2S diding Pipin Pittups)
M--3)

divergent:andisI

acanbeplan) &(3) (4)close to(in fact

theshortdistanceeach

integralisother:

dangerous

divergent (divergence.

Dealing
from

such

with

which

bewillideas

importantrenormalizationThis

divergence is the central problem in QFT

aas

regularization ,renormalization groupandthesuchsubject ofmain

thisemerges.
cource.

End of remark
reversex4+

Wick rotationi),yP --iy ;↳i m2

Pp+-iPoPoP4C-IPEM2Xu-⑨&

IP= M2N-V

im-

P(x)p(y) -i-ilpo(X -y)-iPo y9edPIP(idPo)(2π)4Ip2-

Po + m23xC

Po=·-p2-iP (x y).

-m2 +-P-1p2

i 0.

M + iO-&#M- iO

Gauge theory (egof·Diracfermion

0 A

gangefixing byTeYe=Le,

Le free =,-LE int =P,

-=.

I

-

CE free

representation R

aof G+0z(dA)"

+ c.

& D.
.

LE int (rescale Ap-eAr),10 A)
I(An-GAp .

iF@4 + mπ4+joc(dAn-JuAm) [Am Au].

.

ieFA4

perturbative expansion of
decanin

FE-iFa4 + mπ4&e ==QCD (An gauge potential of garge group G

variables.

-.

computed+ E.

+3(Ap An]C,

OCAnC]

partition/correlation functions
usingPropagators

S

se's <Ap

=

↑

=

CApa

C =Ca,,(Orthonormal bases

Sei] <R

,

J=

e 4i

~

=

Apalks Arb (3)=Jaber (d 15-1-

iPE(x 3)-+M>-=

4 : (F(3)-=iPE(k 3)

E(P5)DE PE==-0(.

.

.

..

=Ca(k)[b(3)

+.

.....

(4z m)+ mi-

iPE(s73)me(mic+iPE(x 3)--=Gab)(Vertices
3~ =4M↓-=-=.

-(Ape/ai==(GAn-JuAn) [Apitu)-Ac] (An Au],-,* 4,e(axe5 [d(tr,,e.

g.

[Apk)An(3s)-minm-

port+mi+=mu &[-=t+iE

3C

3M 11S>S~↓&-

3[Sz+&+↓S,S--E

3 M

E14~Dr↓Sx+

33++=

3S+...

&< 4(4)[(3) Ap(z) perttt~((4(5(3))prB&mSMinkowski limit(Note:Z~Apak) Any(3)=M+=

4 (4) (3):-&=------Calcs[b(3)

=dab

:

J-~&M↓-iJ)( P -(13)4d) P4,-=Gab) dip

ip(x 3)
-tio

4,

3

-

=-(AA) CAA,=- d (AAZCAR At=ie(di 4=ie/did,*.

,(Ap C.

.