Today, we will look at how to deal with the
interactions of electrically charged fermions.
So, let us summarize first what we have learnt
so far in the particle physics part of this
course.
So, we have said that in the relativistic
case when we are dealing with particles, which
are moving at; or having a high energies and
moving at relativistic speeches, then we have
to consider the evolution equation or the
equation of motion different from that of
the Schrödinger equation. And, what we found
as the Klein Gordon equation 
as one of the suitable equations to describe
the dynamics of particles including their
relativistic kinematics.
So, this in the notation that is now familiar
to us, I will write this as doh mu doh mu
phi plus m square phi equal to 0, where phi
is the wave function that represents the charged
particle. This is the equation which is quadratic,
in the derivative; both time and spatial derivative.
Let me for completeness sake write what this
means. This is partial derivative with respect
to time and the gradient operator. This we
have not included the factor of C, which is
usually there to keep both the terms at the
same dimension.
So, usually it is written as one over C. But
we are using a system of units which is in
which C is taken to be 1 and the Planck constant
h cross is equal to 1. Now, this is basically
the Klein Gordon equation describing whatever
particle that is represented by the wave function
phi.
Now, Dirac equation is again a relativistic
description of elementary particle or the
particles. This free particle; the one that
we have written down phi is also considered
as a free particle. There is no potential,
there is no interaction there is only this
mass term and the time derivative telling
us how this, for free particle will evolve
with time. And the kinetic energy part described
by, given by the gradient of that, in Dirac
equation we have a slightly different case
where we have gamma mu an object, which we
have defined clearly and described in detail
in some detail; and i doh mu, which is essentially
the momentum operator apart from the sign
and apart from the constant h cross, which
we are taking as 1. This acting on psi ok,
this minus m psi is equal to 0.
So, that is the Dirac equation. So, the differences
between these 2 which we have already gone
through, but let me just remind you.; gamma
is a 4 by 4 matrix 
and let me denote the indices, matrix indices
explicitly. These i doh mu which is not a
new matrix, which psi is a matrix, which is
single index that is it is 4 by 1 matrix.
So, you have a 4 by 4 matrix multiplying 4
by 4 matrix. That is what we have. Minus m
psi a. So, this is for each component of this;
a component of psi and a component of this
product. Now there is a conjugate equation
that I can describe, which is important in
our case. In the earlier case the conjugation
was just a charge conjugation operator and
phi would go to phi star that would do.
But in this case we have the Hermitian conjugate
equation. So, that is basically i doh mu psi
an object, which we will define in the minute
psi bar and gamma mu. This, if it is a, then
a b minus; rather plus m psi bar b is equal
to 0.
So, the psi bar is defined as psi dagger with
a gamma 0. So, psi bar b is a a b. psi itself
as we saw is psi 1 psi for component object
psi 1 psi 2 psi 3 psi 4. So, this a and b
runs from; this is the summary of things that
we have kind of discussed in the relativistic
quantum mechanics or relativistic particle
dynamics. We are dealing with free particles,
which means particles like electron, which
is freely propagating in the space. Without
an electric field; in a region without an
electric field or any other field.
That is described by this; this equation.
That is the kind of picture that we have.
Now today, as we promised, we are going to
look at how interaction can be understood.
We had already seen how that is done understood
in the case of particle, which obey the Klein
Gordon equation.
So, just to summarize or to give in a very
short way what is the difference between these
two; that without going into any details,
let me introduce another word, field and wave
function I will use interchangeably somewhat
loosely. So, in relativistic quantum mechanics
we usually call this wave function, whereas
a proper mathematical treatment is done in
the framework of quantum field theory, where
we will actually have to promote these wave
functions to what is called the fields.
So, there are some subtle differences, but
I do not think we should worry about that
at the moment. And that will actually take
us a little deeper into the aspects, which
are beyond of course’s description. This
is a good representation of any scalar particle
a complex field ok, phi which obeys the Klein
Gordon equation is a good representation of
the scalar particle. Whereas, fermionic field
like electron is represented by the Dirac
equation. It is not there are certain features
the spins of this is not going to be adequately
addressed in Klein Gordon equation ok. Therefore,
one has to actually consider the Dirac equation
to treat the electron with including its spin
part.
So, let me do not go into any more details
of that we will assume that electrons and
particles like electrons, the fermionic particle
half integer spin particles obey the Dirac
equation and usually they are represented
by this psi, with 4 components like that in
the Dirac notation. Whereas phi is just 2
degrees of freedom here. It is a complex number
one real part and one imaginary part.
So, let me consider the K G equation or the
scalar particle, when I say scalar wave function
that is a wave function, which represent,
which obeys the Klein Gordon equation. So,
let me write in bracket Klein Gordon equation
and represented by phi.
Now, where a current of this phi, particle
represented by phi moves that can be thought
about as current. By the way complex scalar
particle represents the charged scalar particle
ok. So, this current let me denote by letter
j mu and supposing there are photons or the
electric field, in this region; this notation
is also familiar to us.
So, j mu interacts with this. In a graphical
way we can represent it in this equation there
is a current of phi and then there is the
electric field electromagnetic field we represent
interaction by this. Where j mu is equal to
there is the 0th component which is rho and
represented by rho and j and rho itself is
defined as minus phi; sorry rho itself is
defined as i phi star time derivative of phi
minus phi time derivative of phi star, minus
i e.
This can be interpreted as the charge density.
Electric charge density and j is represented
by i e here it is a plus. Minus i e, and phi
star gradient of phi minus phi gradient of
phi star. Interpretation here is the current
that is the electric current itself that is
represented by this j.
So, in sense if you look at the electrodynamics;
the Maxwell’s equations and all the classical
electrodynamics this notation is clear there.
We usually represent charge density by a rho;
and we represent the current density by j
ok. So, that is what we have. So, this part
we had discussed earlier.
So, in case you have forgotten; maybe you
can check the previous lectures. Now coming
to; so, for electron it is I think I made
a sign earlier. Correct sign and this will
be for electron with minus e charge it is
minus i e. sorry for that.
Now what about the fermionic fields so, fermion
represented by psi they obey Dirac equation.
Again picture is the same, we have a current
and the current is interacting with A mu.
so, this is like an electron and on the way
here encounters electromagnetic field. Then
how does it interact and that is the way we
represent it.
So, this is again we can do in the same way.
Only thing is what is j mu. Here j mu is for
electron minus I the charge of that psi bar
gamma mu psi. So, psi bar is known, gamma
mu here is the Dirac matrix and psi is the
wave function. Now this again when we write
it in terms of the charge density and current
density we have rho equal to minus e psi bar
gamma 0 psi; ok. This is nothing but minus
e psi bar itself psi dagger gamma 0 ok, then
there is another gamma 0 psi, which is equal
to minus e psi dagger psi since gamma 0 square
is equal to 1. J is equal to minus e psi bar
gamma vector psi.
When, I say gamma vector, it is basically
gamma vector is defined in a sense as gamma
1 gamma 2 and gamma 3. We will go back and
remind ourselves about one thing; that is
the electromagnetic field itself. The electromagnetic
field in itself as the Maxwell’s equations.
Maxwell’s equations were written as doh
mu f mu nu equal to j nu. Well j nu is the
current density of the source of electric
field; current density of the source of the
electric field. And we can write it in terms
of rho and j. And that is what we have in
the classical electrodynamics case. F mu nu
are the field tensors, which we are again
familiar with we had discussed it earlier.
So, his is what it is.
This can be written in terms of; f mu nu is
essentially the mu; in terms of the potential
or the photon field it is doh mu A nu minus
doh nu A mu. And when you put this back we
have doh mu doh mu A nu ok, minus 
doh mu doh nu A mu 
equal to j nu. But this if you focus on this
then, this is essentially there is no preference
preferred order in which this differentiation
need to be done and this. This can be interchange
I can follow or do this differentiation first
and this the second. Rather, I can actually
simply change this order in which these to
act on A and nothing changes. Now I have something
like this. We can choose this A so, that doh
mu A mu is 0 always ok.
So, this is something which we will assume
now, or rather it is something which we had
discussed earlier. So, we will say that have
an A, we have chosen an A, which has this
particular condition satisfied. That choose
A so, that doh mu A mu that is basically the
dot product between these two, or the four
divergence of A vanishes. This can be done
always. You can choose A mu like that. And
that will not change any of the physics aspect.
The meaning, that it is the same electric
any given photon, electromagnetic field can
be represented by some A, which can always
be written in terms of some divergence less
A. In that will give you doh mu doh mu A nu
equal to j nu. Or this will give us a solution.
Ok, without going into the details we can
find the solution. This remember, what do
you mean by solution, this for given j source
the electric field electromagnetic field everywhere
represented by A mu satisfy this equation
that is what you have.
Supposing you have an electron or some distribution
of charge, moving distribution of charge,
something like that or a nucleus moving that
will create an electromagnetic field surrounding
that. Now what is that electric field that
is represented by A? how does it behave? That
is dictated by this equation.
So, can we say what are that, what is the
different type of A mu? Yes. So, without going
into the details let me give you the solution
that A nu equal to minus 1 over q square j
nu, where q square is essentially the momentum
transfer under this one. So, what do you mean
by q square? I will come to; let me just rewrite
this. So, what we have with us? A mu is equal
to minus 1 over q square j nu.
So, what does that mean? This means in a region
we have some electromagnetic represented by
A mu ok. So, it is the same index A mu. But
we know that is coming from say some electron
or a nucleus. Presence of sitting electron
or sitting nucleus. Or, moving electron or
moving nucleus. Or some complex charge density.
But that is represented by this thing. So,
there is j corresponding to this. How do we
actually feel the presence of electromagnetic
interactions? From the charge particles. Forget
about the Newtonian way of action at a distance.
We demand, like in special theory of relativity
and same demanded, that we have to have something
some cause for effects. The effect is that
we feel the electromagnetic field. The causes
is that there is some charged particles, moving
or not moving.
Now this changes in that or whatever the presence
of that has to be conveyed to us. That is
done by the emission or the propagation of
the photons or the presence of the photons;
photon field in that region. So, when for
that when a photon is emitted; we can think
about a emitting and reabsorbing, etcetera;
or just emitting, then it is emitted then
you will have some momentum p 1 here and p
2 here and q here.
So, this essentially is the momentum of the
A. Let me do not go into further details.
But essentially to complete description of
the electromagnetic field the surrounding
this in a space configuration, coordinate
picture, you will have to actually take all
such a possible q and integrate over a for
take the Fourier transform of this a to get
this spatial part of this thing. But this
is basically a particular photon is represented
in that way.
So, this is what you have. Whenever such a
momentum transfer happens you can actually
think about as; or such a current happens,
you can think about a photon which is represented
by minus 1 over q square. Now, that and this;
let us look at together for simplest for identification,
let me denote this as electron 1 and the current
due to that as j 1.
And this as some another charged particle
or electron 2 and a current due to that is
j 2. This 2 has nothing to do with these 2
for these components of j. This is another;
some just an index to represent this particle
and then identify that; that is caused by
this electron and this is talking about another
electron. That is all.
And, now let me look at these together. Here,
we were saying that an electron encountering
an electromagnetic field will interact with
the photon in this manner, with the current
representing the charged particle motion the
current it is electric charge itself and then
this is the A mu. Here, we have the source
of the current itself. So, we have say; we
are saying that that A mu is kind of coming
from this second another electron.
So, in a sense we can actually talk about
interaction of 2 electrons as some one current
j 1 mu interacting with another current j
mu 2 ok. So, this is a good picture, which
we have to keep in mind.
So, 2 electrons say electron 1 and electron
2 interacting with each other can be represented
by such a picture. I mean in actuality it
is much more complex than this. But main features
of this interaction can be captured and in
fact, calculated also in a probability for
such interaction etcetera can be calculated
or the scattering of an electron on an electron
to get another electron on another electron
can be captured very easily in by this way.
I will stop here then we will discuss further
in the next lecture.
