So, in this course we will be studying quantum
field theory, you might ask you have
already learnt a quantum mechanics in your
under graduate in the first year course. So,
why do you need to study quantum field theory?
The motivation for studying quantum
field theory is that suppose you consider
a particle the way you do it in your quantum
mechanics is you write down Schrodinger equation.
And you solve this Schrodinger
equation you find the solution to this Schrodinger
equation, which is the probability
which should. However, there are process which
for example, if you consider the decay
process; let us assume that you consider the
mu and . mu minus
going to e minus plus mu e bar mu nu. Then
there is no way you can understand a
process like this in your a non relativistic
quantum mechanics.
..
So, the question that you can ask is suppose
you write down the you want to consider
this generalization relativistic generalization;
relativistic generalization of the
Schrodinger equations Klein and Gordon I have
already done that. And Dirac also a
relativistic generalization; Klein and Gordon
generalized it for a scalar field and Dirac
generalize it for a spinner; what you find
here is that if you want to interprete the
Klein
Gordon or the Dirac equation the way you interpreted
the Schrodinger equation then you
run into various inconsistencies.
So, if you want to interpret as a particle
mechanics; then you run into various in
consistencies. For example, the probability
amplitude a does not give the definite
positive probability and so on. So, you are
force to introduce quantum field theory. In
this lecture we will study quantum field theory
in much more detail; what I will do that is
I will give of the references that will be
used in this course. And then we will quickly
discuss scalar field theory, classical field
theory and then we will study our how to
quantize classical field theory?
..
So, the reference are 
quantum field theory by Srednicki; then quantum
field theory by
Itzykson and Zuber and an introduction to
quantum field theory by Pesktin and
Schroeder.
.
Quantum field theory by Mandal and Shaw then
quantum theory of fields by Weinberg
gauge theories in particle physics by Aitcheson
and Hey. And hey quantum field theory
in a nutshell by A Zee; then quantum field
theory by Ryder ramanand and so on. So,
almost everything that will be discussed in
this course will be borrowed from one or two
.of these books. So, what I will do now is
I will briefly review classical field theory.
And
then I will discuss how to quantize classical
field theory and what are their application
of
the quantum field theory?
.
So, let us briefly discuss classical filed
theory. So, the classical field theories that
we are
going to quantize various. So, we are going
to discuss the classical field theory feature.
For example, local in the sense that the equation
of motions content finite numbers of
that vectors; equations of motions contains
finite number; they will also discuss field
theory which are relativistic 
we will impose Laurence and variance.
..
And, we will also require that the energy
have a lower bound. So, we should have we
will consider the field theory which have
positive definite; let us first discuss the
Lagrangian formulation. So, in Lagrangian
formulation what you do; you consider the
action which is S integration L d t; the Lagrangian
itself we can write down as L as an
integration of L over the entire space d cube
x; over this L here is known as the
Lagrangian density.
.
.L is the Lagrangian density; in general L
is a function of the field which I will denote
as
pi and its derivatives. So, the action can
actually be express is S equal to integration
of L
pi del u pi d 4 x; the volume element d 4
x is d t times d cube x. 
We will find the
equation of motion from this action by using
the principle of least action.
.
So, what we will do is that we will set the
variation of the action to be 0 with the
restriction that the variation of the field
delta pi equal to 0 at boundary; this condition
will give us the equations of motion. So,
let us derive the equations of motions from
this
condition. So, what is delta S? Delta S is
integration d 4 x delta L.
..
And, this is equal to 
del L over del pi times delta pi plus del
L over del del mu pi delta of
del mu pi. Here, I am assuming that the lag
range in contains only single derivative of
the
field pi as well as itself function pi. If
there are multiple derivative for example,
if the
Lagrangian contains second derivatives of
field pi. Then you will have one more term
and so on; however we will not consider that
case at this moment.
.
So, let us rewrite the second term; here you
can see that the second term can actually
be
written as del mu of del L over del del mu
pi delta pi 
minus del mu of del L over del mu
.pi delta pi. Here, I am assuming that this
delta actually commutes with del mu; then
you
can write this term as a total derivative
minus this term. So, now what I can do is
that I
can substitute this here.
.
Then, what I get is delta S is equal to integration
d 4 x del L over del pi delta pi minus
del mu of del over del del mu pi delta pi
plus del mu del L over del del mu pi delta
pi.
Let us now focus at the last term; you can
see that this will give you a surface
integration. However, here we are imposing
the boundary condition that the variation
of
the field is 0 at the boundary. So, when we
use this condition this term will actually
vanish. So, the last term in this equation
become 0.
..
Therefore, delta S is actually equal to 
delta S is equal to d 4 x times del L of our
del pi
minus del mu of del L over del del mu pi times
delta pi; however delta pi is arbitrary.
Therefore, the integrant must be equal to
0 if delta S is equal to 0 then the integrant
must
be 0. So, this implies del L over del pi minus
del mu of del L over del mu pi is equal to
0;
this is our earlier langrage equation of motion.
So, let us now find the Hamiltonian for
the system; again you do it just the way you
find the Hamiltonian in mechanics; in
mechanics what you do?
.
.You have you introduce the conjugate momentum
P which is del L over del q dot. And
then you define the Hamiltonian which is a
functional of q and P to be P q dot minus
L.
.
Here, you do exactly the same thing what you
do is you introduce the conjugate
momentum density which I will denote as pi
of x; and the momentum density is defined
to be delta L over del del 0 pi. And the Hamiltonian
density so this is the momentum
density then the Hamiltonian density H pi
of x pi dot of x minus L of pi del pi. Let
us
consider a very simple example; the example
of a real scalar field and then let us derive
the equation of motion as well as Hamiltonian.
..
So, you will consider the example of a real
scalar field. And then we will think what
the
equation of motion is and what is that Hamiltonian
and so on? Before that let me explain
the notation that we will be use it throughout
the course; I will use the space favor
matrix. So, eta mu nu it is 1 minus 1 minus
1 minus 1 or in other words the invariant
length of a vector A mu A mu which is equal
to also eta mu nu A mu A nu mu; this is a
0
square minus A dot A; throughout the lecture
we will be using this method. And also we
will use natural units. So, we will set h
bar equal to C equal to 1throughout the course.
Now, let us consider a real scalar field.
.
.The Lagrangian density for real scalar field
is half pi dot square minus half del pi square
minus half m square pi square; you can rewrite
this is half del mu pi del mu pi minus half
m square pi square all right. So, what is
the equation of motion for the system? The
equation of motion is given by del L over
del pi minus del mu del L over del of del
mu pi
s equal to 0. So, let us derive each of the
term.
.
Del L over del pi is equal to m square pi
and del L over del del mu pi. So, how will
you
derive del L over del L del mu pi? This is
in the Lagrangian only this first term will
contribute. So, as you can see these terms
only depends on del mu pi del is independent
of del mu. So, we will consider this .. So,
this is equal to del of del
mu pi times half; this acting on half del
mu pi del mu pi. If you note this I have use
this
symbol nu here instead of mu; that is because
I have the free index mu here; I must say
that I am using Einstein summation convention.
So, whenever a symbol an index is
repeated it is summed over the value it takes.
..
So, for if I consider for A mu A mu the index
mu here runs from 0 over 1, 2, 3. So, A mu
A mu is that equal to A 0 A 0 plus A 1 A 1
plus A 2 A 2 plus A 3 A 3. And hence A mu
A mu is also equal to A nu A nu; this mu here
and this expression is a dummy index. So,
we can put any level we want for this here;
that is what I have done in this expression
instead of mu I have used nu here. So, this
is now equal to so this implies this del L
over
del del mu pi is just equal to half times
twice del mu pi times del of del mu pi divided
by
del of del mu pi. Now, what is this quantity
here? This is just delta mu you no. So, this
is
nothing but del mu pi delta mu nu. So, this
is just del mu pi.
.
.So, what we saw here is 
that
Student: ..
No. So, let me explain this symbol again eta
mu nu is equal to 1 0 0 0 0 minus 1 0 0 0
minus1 0 0 0 0 minus 1. And now delta mu is
the identity matrix instead of 1 minus 1
this is just 1 0 0 0 and so on; all the diagonal
elements are 1 and all of diagonal elements
are 0. Now, when you look at this; if mu is
not equal to nu this will just gives you a
0. If
mu is equal to nu this is 1.
So, however eta mu nu as well as it inverse
eta mu nu which is numerically equal to the
same thing eta 0 0 is 1 whereas eta 1 1 eta
2 2 and eta 3 3 are minus 1. However, here
you can see if no matter whether mu and nu
whether mu is 0 or 1 or 2 or 3 it always
gives you 1; if identity if mu is equal to
nu otherwise it gives you 0. Therefore, this
quantity here is has to be equal to delta
mu nu not eta mu nu; we are not using something
like this delta mu nu we are using and also
the tensional property here is such that it
is
calorie mixed tensor. If you look at the Laurence
transference property of this quantity
this does not transfer like contra variant
tensor of rate 2; it transfer likes a mixed
tensor
of rank 2. So, therefore the index structure
has to match. And also the notation that we
have we are using is such that this is equal
to delta mu nu right.
Student: ..
Thank you; this mu here is actually a contra
variant index not a co variant index.
Student: ..
No, I see ok thank you. So, this is actually
equal to eta mu nu. So, this is del mu pi
clear
let us so summaries what you have seen here.
..
What he say is I define del L over del pi
is equal to m square pi and
Student: ..
Minus m square pi thanks again minus m square
pi and del l of del mu pi is equal to del
mu pi. So, now we can substitute this in to
equation of motion. And what is see is minus
m square pi minus del mu del mu pi is equal
to 0 or in other words this is just del mu
del
mu pi plus m square pi s equal to 0; what
you define is the Klein Gordon equation. So,
this field here is this actually a Klein Gordon
field. Now, what we will do is we will find
the Hamiltonian density for this system and
you will see what you get?
..
So, you have already introduce the Hamiltonian
density H equals to del pi of x del L over
del of del 0 x pi minus L of pi del mu pi.
So, this is equal to pi of x and we can see
from
this expression that del L over del del 0
pi is again del 0 pi. So, you can use this
expression here; no no sorry this is simply
del 0 d I q I dot del 0 pi minus L. So, pi
of x is
del l of del del 0 pi this quantity is again
del 0 pi. So, we can use we can substitute
this
for pi of x. Then what we get is Hamiltonian
density H equal to pi of x del 0 pi del 0
pi
minus half del mu pi del mu pi minus half
m square pi square; this is equal to pi square
of x minus half del 0 pi square plus half
grade pi square plus half m square pi square;
we
can again substitute this for pi of x.
..
And, then what we get 
is that Hamiltonian density H (x) is equal
to half pi of x square
plus half grade pi square plus half. As you
can see this very simple system actually
satisfied all the criteria that we have a
specified in the beginning of this lecture;
in the
sense that the system will actually Laurence
and variant. The action is invariant in the
Laurence transformation, the equations of
motions are co variant under Laurence
transformation it is local because the equation
of motion only is a 2 derivative it contains
on the 2 derivative terms. Therefore, itself
local field theory and the energy density
as a
lower bound; all the terms here are positive
definite. So, this satisfied all the criteria.
So, what we will do this what we will do in
this subsequent lecture is that; we will start
quantize this very simple system. And then
we will study more and more complex field
theories. So, with this we will close today’s
lecture; tomorrow I will discuss some of the
symmetric and conservation loss. And then
we will actually start quantization
theory.
Thank you.
.
