Renormalization in QFT

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Renormalization in QFT

April 2, 2015

Abdullah Khalil

African Institute for Mathematical Sciences(AIMS) South Africa

Supervised By

Prof. Robert de Mello
mailto:[email protected]


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Renormalization inQFT

Abdullah Khalil

Introduction

Poles at the Physicalmass

Interpretation of therenormalizationparameter

RenormalizedParameters

Renormalization In afree theory

Renormalization in

interacting theory

RenormalizationConditions

References

African Institute for

Mathematical Sciences(AIMS)

South Africa

Overview

Introduction

Poles at the Physical mass

Interpretation of the renormalization parameter

Renormalized Parameters

Renormalization In a free theory

Renormalization in interacting theory

Renormalization Conditions

References


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2 Introduction





Renormalization in

interacting theory


References



South Africa

IntroductionRenormalization

What is the renormalization?

Studying how a system changes under change of theobservation scale.

What does it do mathematically?

Removing the divergences that arises in a loop integral ofa given theory.

Renormalization in QFT

It is a matching between the observed quantities and theparameters that appear in a given theory.


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Renormalization in

interacting theory


References



South Africa

IntroductionGeneralized Heisenberg equation of motion

Lets remind ourselves of some important relations The generalized Heisenberg equation of motion:

dO(t)

dt = [O(t), H]

Which has a solution

O(t) =eHtO(0) eHt

In analogy

dO(x)dxi

= [O(x), Pi]

Which requires a solution

O(x) =ep.x O(0) ep.x


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Renormalization in

interacting theory


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South Africa

IntroductionGeneralized Heisenberg equation of motion

So we can generalize the Heisenberg equation of motion asfollows, for any field operator(x, t)

i(x)

x= [(x), p]

which requires a solution

(x, t) =eP.X (0) eP.X

We can also define that for any translation a

(x+a) =U(a)(x)U(a)

For any boost

(x) =U()(x)U()

Forx=0

(0) =U()(0)U()


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Renormalization in

interacting theory


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South Africa

IntroductionThe Completeness relation

The completeness relation can be written as a complete set ofall the states

I= |0 0| +

d3p

(2)3

1

2p |p p| 1-particle state

+

d3p1

(2)31

2p1

d3p2

(2)31

2p2|p1,p2 p1,p2|

2-particles state+3-particles state +. . . . . .


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Renormalization in

interacting theory


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South Africa

IntroductionTwo points correlation function

The two points correlation function is

0|T((x1)(x2)) |0 =(x0

1 x0

2 ) 0|(x1)(x2) |0+(x02 x01 ) 0|(x2)(x1) |0

We will assume that the vacuum expectation value to be Zero

0|

(x)|0

=0


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Renormalization in

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South Africa

IntroductionThe physical mass

what is the physical mass?

P

|p

=p

|p

Wherepp = (p

0)2 p p =m2pWith

p = p p +m2p


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Introduction

8 Poles at the Physicalmass




Renormalization in

interacting theory


References



South Africa

Poles at the Physical mass

Let us compute the two point correlation function by insertingthe identity

0|(x1)(x2) |0 = 0|(x1)I(x2) |0= 0|(x1) |0 0|(x2) |0+

d3p

(2)31

2p0|(x1) |p p|(x2) |0

+ n>1 d3p1. . . d

3pn

(2)3

. . . (2)3

2p1 . . . 2pn 0

|(x1)

|n

n

|(x2)

|0

The first term must vanish


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Introduction





Renormalization in

interacting theory


References



South Africa

Poles at the Physical mass Cont...

But

0|(x1) |p = 0| ePX1 (0)ePX1 |p = 0|(0) |p ep.x1

But

0|(0) |p = 0|U()(0)U() |p = 0|(0) |p =

Z

0|(0) |p =

Z ep.x1

Similarly, p|(0) |0 = Z ep.x2

Z =0|(0) |p2 Z 0


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Introduction





Renormalization in

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References



South Africa

Poles at the Physical mass Cont...

2nd term= Z

d3p

(2)32pep(x1x2)

=Z

d3p

(2)3ep(t1t2)

2pep(

x1

x2 )

Let us use a very simple trick

ep(t1t2)

2p=

dp0

2

ep0(t1t2)

(p0)2 2p

= dp02

ep0(t1t2)

(p)2 m2p

2ndterm= Z

d4p

(2)4i

p2

m2p

ep(x1x2) =Z0(x1 x2, m2p)

So the second term has a pole at the physical mass.


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Introduction


11 Interpretation of therenormalizationparameter



Renormalization in

interacting theory


References



South Africa

Z as a probability

Similarly for the 3rd term 0|(x1) |n n|(x2) |0 =

|0|(0) |n|2 ePn(x1x2)

Such thatP |p =Pn|p

Let us use another simple trick!

ePn(x1x2) =

d4p eip(x1x2)(4)(p Pn)

3rd term= d4p eip(x1x2)

(4)(p

Pn)

|0

|(0)

|n

|2

1

(2)3 Z(p2) (p0)

=Z

d4p

(2)3 eip(x1x2)

da2 (p2 a2) (a2) (p0)

By introducing a new parameter(a)and using the same trickabove!


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Abdullah Khalil

Introduction





Renormalization in

interacting theory


References



South Africa

Z as a probability Cont...

Lets do the integral over p0 where

(p2 a2) =(p0)2 p p a2= (p0)2 2a

3rd

term= d3p

(2)3ea(t1t2)

2a ep(

x 1

x 2) da2 (a2) Z

=

d4p

(2)3

p2 a2 ep(

x 1

x 2)

da2 (a2) Z

= da2 Z(a2)0(x1

x2, a

2)

Finally we get

0|(x1)(x2) |0 =Z0(x1x2, m2p)+

da2 Z(a2)0(x1x2, a2)


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Introduction





Renormalization in

interacting theory


References



South Africa

Z as a probability Cont...

Now let us study Z

t20|(t1,x1)(t2,x2) |0 |t2=t1

t20|(t2,x2)(t1,x1) |0 |t2=t1

=Z

1 +

da2(a2)

(3)(x1 x2) = 0| (t1,x1), (t1,x2) |0

=(3)(x1 x2 )

Z = 11 +

da2(a2)

Z 0 , 0 Z 1So Z describes the probability of creating a single particle.


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Introduction



14 RenormalizedParameters


Renormalization in

interacting theory


References



South Africa

Renormalized Parameters

If we look at our new propagator we can relate it to theFeynman propagator by this renormalization parameter

D(p2) = Z

p2

m2p

+

0

da2 (a2) Z

p2

a2

Such that the two point correlation function is

0|T((x1)(x2)) |0 =

d4p

(2)4ep(x1x2) D(p2)

The Fourier transform of the two point function:

It has a pole atp2 =m2p This pole has a residueZThat means the field in the

theory isnt the physical field strength


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Abdullah Khalil

Introduction




15 Renormalization In afree theory

Renormalization in

interacting theory


References



South Africa

Renormalization In a free theory

We want to create a single particle with probability 1 The field in the theory isnt the physical field strength?

=

Zp

The correlation function would be

0|T(p(x1)p(x2)) |0 =

d4p

(2)4ep(x1x2) Dp(p

2)

Dp(p2) =

p2

m2p

+

0

da2 (a2)

p2

a2

The Fourier transform of the two point function of the physicalfield:

It has a pole atp2 =m2p This pole has a residue


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Abdullah Khalil

Introduction





16 Renormalization in

interacting theory


References



South Africa

Renormalization in the interacting theory

An example of the interacting theory is the 4 theory withLagrangian density

L = 12

1

2m22 g4

But we have three undefined parameters (m, g, )So, weneed three conditions instead of two!

How can we get the third condition?


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Abdullah Khalil

Introduction





17 Renormalization in

interacting theory


References



South Africa

Renormalization in the interacting theory

0|(x1) . . . (x4) |0 = (g)4!

d4xG(x1, x)G(x1, x)G(x1, x)G(x1, x)

= (g)4!

d4x

4i=1

d4pi

(2)4

p2i

m2

epi(xix)

= (g)4! 4

i=1

d4pi

(2)4(p1+p2+p3+p4)

epixi p21

m2

p22

m2

p23

m2

p24

m2

g gp


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Abdullah Khalil

Introduction





Renormalization in

interacting theory

18 RenormalizationConditions

References



South Africa

renormalization condition

We can determine the physical parameters of any theory byapplying the following Conditions:

1. The Fourier transform of the two point function has a poleatp2 =m2p.

2. This pole has a residue 1.

3. The connected and amputated four point function in themomentum space that describes the scattering of twoparticles with momentap1 andp2 into two particles withmomentap3 andp4 at the point specified by:

p1+p2 p3 p4 p21 =p22 =p23 =p24 =m2p(p1+p2)

2 =4m2p (p1 p3)2 = (p1 p4)4 =0has a value(24gp)


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Introduction





Renormalization in

interacting theory


19 References



South Africa

References

Introduction to Quantum Field Theory.Robert de Mello Koch

Observables from Correlation Functions, March 2015.Markus A. Lutyhttp://www.physics.umd.edu/courses/Phys851/Luty/notes/observables.pdf .

An Introduction to Quantum Field Theory.Peskin & Scroeder
http://localhost/var/www/apps/conversion/tmp/scratch_4/Markus%20A.%20Lutyhttp://localhost/var/www/apps/conversion/tmp/scratch_4/Markus%20A.%20Lutyhttp://robert%20de%20mello%20koch.pdf/http://localhost/var/www/apps/conversion/tmp/scratch_4/Markus%20A.%20Lutyhttp://localhost/var/www/apps/conversion/tmp/scratch_4/Markus%20A.%20Lutyhttp://localhost/var/www/apps/conversion/tmp/scratch_4/Markus%20A.%20Lutyhttp://peskin%20%26%20scroeder.pdf/http://peskin%20%26%20scroeder.pdf/http://localhost/var/www/apps/conversion/tmp/scratch_4/Markus%20A.%20Lutyhttp://localhost/var/www/apps/conversion/tmp/scratch_4/Markus%20A.%20Lutyhttp://robert%20de%20mello%20koch.pdf/


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