renormalization in qft
TRANSCRIPT
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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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Renormalization inQFT
Abdullah Khalil
2 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
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 inQFT
Abdullah Khalil
3 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
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 inQFT
Abdullah Khalil
4 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
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 inQFT
Abdullah Khalil
5 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
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 inQFT
Abdullah Khalil
6 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
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 inQFT
Abdullah Khalil
7 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
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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Renormalization inQFT
Abdullah Khalil
Introduction
8 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
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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Renormalization inQFT
Abdullah Khalil
Introduction
9 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
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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Renormalization inQFT
Abdullah Khalil
Introduction
10 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
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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Renormalization inQFT
Abdullah Khalil
Introduction
Poles at the Physicalmass
11 Interpretation of therenormalizationparameter
RenormalizedParameters
Renormalization In afree theory
Renormalization in
interacting theory
RenormalizationConditions
References
African Institute for
Mathematical Sciences(AIMS)
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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Renormalization inQFT
Abdullah Khalil
Introduction
Poles at the Physicalmass
12 Interpretation of therenormalizationparameter
RenormalizedParameters
Renormalization In afree theory
Renormalization in
interacting theory
RenormalizationConditions
References
African Institute for
Mathematical Sciences(AIMS)
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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Renormalization inQFT
Abdullah Khalil
Introduction
Poles at the Physicalmass
13 Interpretation of therenormalizationparameter
RenormalizedParameters
Renormalization In afree theory
Renormalization in
interacting theory
RenormalizationConditions
References
African Institute for
Mathematical Sciences(AIMS)
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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Renormalization inQFT
Abdullah Khalil
Introduction
Poles at the Physicalmass
Interpretation of therenormalizationparameter
14 RenormalizedParameters
Renormalization In afree theory
Renormalization in
interacting theory
RenormalizationConditions
References
African Institute for
Mathematical Sciences(AIMS)
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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Renormalization inQFT
Abdullah Khalil
Introduction
Poles at the Physicalmass
Interpretation of therenormalizationparameter
RenormalizedParameters
15 Renormalization In afree theory
Renormalization in
interacting theory
RenormalizationConditions
References
African Institute for
Mathematical Sciences(AIMS)
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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Renormalization inQFT
Abdullah Khalil
Introduction
Poles at the Physicalmass
Interpretation of therenormalizationparameter
RenormalizedParameters
Renormalization In afree theory
16 Renormalization in
interacting theory
RenormalizationConditions
References
African Institute for
Mathematical Sciences(AIMS)
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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Renormalization inQFT
Abdullah Khalil
Introduction
Poles at the Physicalmass
Interpretation of therenormalizationparameter
RenormalizedParameters
Renormalization In afree theory
17 Renormalization in
interacting theory
RenormalizationConditions
References
African Institute for
Mathematical Sciences(AIMS)
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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Renormalization inQFT
Abdullah Khalil
Introduction
Poles at the Physicalmass
Interpretation of therenormalizationparameter
RenormalizedParameters
Renormalization In afree theory
Renormalization in
interacting theory
18 RenormalizationConditions
References
African Institute for
Mathematical Sciences(AIMS)
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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Renormalization inQFT
Abdullah Khalil
Introduction
Poles at the Physicalmass
Interpretation of therenormalizationparameter
RenormalizedParameters
Renormalization In afree theory
Renormalization in
interacting theory
RenormalizationConditions
19 References
African Institute for
Mathematical Sciences(AIMS)
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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