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Quantum Field Theory (PH-537)
M.Sc Physics 4th Semester
Department of Physics
NIT Jalandhar
Dr Arvind Kumar
Chapters:
1.Canonical Quantization
2. Klein Gordan Field
3. Dirac Field
4. Gauge Field
5. Interacting Theory and Elementary Processes
Books Recommended:
Lectures on Quantum Field Theory by Ashok Das
Advanced Quantum Mechanics by Schwabl
 Quantum Field Theory by Michio Kaku
 Quantum Field Theory by Mark Srednicki
 An Introduction to Quantum Field Theory by Peskin
 Field Quantization by W. Griener
Chapter I
Canonical Quantization
Lecture 1
Books Recommended:
Lectures on Quantum Field Theory by Ashok
Das
 Quantum Field Theory by Michio Kaku
Why Quantum Field Theory
 Quantum Mechanics + Special theory of relativity +
concept of fields
 Single particle relativistic quantum mechanics
cannot account for processes in which number and
type of particles changes
 Negative energy solutions were not explained in
relativistic quantum mechanics
 To overcome difficulties we make transition
from wave equation to the concept of fields
In single particle mechanics, one quantize the
single particle in external classical potential.
We deals with the operators corresponding to
physical observables and also the wave functions
which characterize the state of system
In QFT, we deals with the fields and these fields
are quantized. Particles are identified as different
modes of fields. Fields are treated as operators.
QFT find applications in particle physics,
condensed matter physics, statistical mechanics,
mathematics etc.
It is most successful when interactions are
small and can be treated perturbatively
e.g. QED in terms of fine structure constant α
QED predict anomalous magnetic moment
correct to six decimal places.
Dirac theory of electron coupled with
electromagnetic field lead to QED
Negative energy solutions: concept of holes
Higher order corrections to QED lead to problems of
infinites or divergence in integrals.
Earlier success of QED were lowest order corrections
QED contained integrals which diverge as x or k
i.e. negligence of space-time structure at small length
scale.
Classically electron’s self energy was plagued with
divergence. Breakdown of causality.
Renormalization of theory: divergent
integrals are absorbed into infinite rescaling
of coupling constant and masses.
QED describe electromagnetic forces only.
Need to modify for other fundamental forces
And we have
Electroweak theory
Quantum Chromodynmics
Notations/Conventions:
In three dimension Euclidean space
Position
Angular momentum
Some arbitrary vector
Scalar product:
Length of vector
In four dimensional space (where space and time are
considered on equal footing) any point is
represented by four coordinates and also a
vector in this space will have four components.
We have two kind of vectors in 4-dim space-time
Contavariant vector
Covariant vector
Above vectors are related through metric tensor
Metric tensors are defined as
And are related to each other
Metric tensors are symmetric
Defining two arbitrary vectors:
The scalar product is defined as
Above scalar product is invariant under Lorentz
transformation and is called Lorentz scalar. (Prove!)
Length of vector in Minkowski space
Length of a vector need not always to be positive as
was the case for 3-dim Euclidean space.
Note that
which is the invariant length of any point from
origin (Prove!).
The length between two point infinitesimal close to
each other is given by
is proper time.
Prove above statement!
Space-time region is time-like if
Space-like if
Light-like region if
(using c =1, natural unit)
All future processes takes place in future light cone
or forward light cone defined by
Contragradient and cogradient vectors are defined
as (using c = 1)
respectively.
We define Lorentz invariant quadratic operator
Known as D’ Alembertian operator as (c =1)
Energy and momentum are defined in terms of
energy-momentum four vector
Here we used
c = 1, otherwise, we
have to use E/c
Using above we define the Lorentz invariant scalar
We know the Einstein relationship
Where m is the mass of particle.
Last two equations define the mass as Lorentz
invariant scalar quantity
We know the operator forms
Co-ordinate representation of energy momentum
four-vector will be
Four dimension Levi-civita tensor which is
ant -symmetric four dimensional tensor
and
Natural System of Units
ћ = 1 and c =1
length dimensions
Energy density: Energy/length3 = mass4