So far in course we have seen how relativistic charge particles respond to a background electromagnetic
field. And we use the well known minimal coupling
prescription to incorporate that where t mu
was substituted by p mu minus e times a mu
to carry out all the manipulations as well
as detail calculations in presence of electromagnetic
field. Now, will look at the opposite system
where the electromagnetic field will be considered
as evolving in presence of background of charge
densities and currents.
And this charge densities and currents are
provided by various particles and their distributions
in space time. And we will see how the electromagnetic
field responds to this distribution of currents;
and after having done that will be in a position
to put both things together. So, that charge
particles and electromagnetic field mutually
evolved; according to their interaction both
at the same time in a dynamical system which
will be the complete theory of quantum electro dynamics.
So, the first is right now is to discuss the
systems of photons which are the units of
electromagnetic field. And how they respond
to the distribution of charges provided by
various particles. Now, one thing is very
obvious right at the outside; that this photons
are relativistic particles to begin with there
is no such thing as non relativistic photons.
So, all the consequences of Lorentz symmetry
which we discussed in case of particles such
that electrons or neutrinos or any other similar
objects; those are automatically satisfies
by when construction starts discussing with
photons.
So, these are objects which will automatically
described the Lorentz properties from its
definitions; we do not have to worry about
making the theory Lorentz covariant. And the
matter of fact that the starting point of
relativity was the structure of Maxwell’s
equations; in the sense the Lorentz transformations
where discovered or constructed which ever
you want to look at it. So, that they described
Maxwell’s equations in any arbitrary frame
in a covariant fashion.
So, if you use the neutrino or Galion transformations
they do not leave Maxwell’s equations in
variant. And so one had to invent new transformations
has a correct description of the symmetry
of Maxwell’s equations. A Lorentz indeed
wrote them down and it was later that Einstein
reformulated Lorentz transformation from a
different perspective of what structure of
base time is. So, we have the Lorentz symmetry
built in the Maxwell’s equation themselves.
And now we have to look at the consequences
of what all comes out of it; and that is a
atom to convert Maxwell’s equation to a
quantum language. And various things are rather
straight forward; so I can just mention them.
So, there is no rest frame for photons the
position uncertainty; which is delta x is
the same as the de Broglie.
So, photon cannot be localized better than
its wavelength also there is no photon wave function that can be used.So, all this features are actually common
in describing Schrodinger equations and non
relativistic atomic physics. But they do not
make any sense; in case of photons because
photons do not have a non relativistic description.
So, we have to forget those words in some
sense directly go to a language of what is
known as a field theory; where the concept
of particles is taken over by concepts of
fields; where various things are not necessarily
conserved in the same sense as they are in
the case of particles we saw some of those
features while dealing with Dirac equation
in case of uncertainties. And creation of
particle, anti particle pairs similar concepts
have to be used in describing the structure
of what electromagnetic field is and how it
is decomposed in case of photons; there are
various objects which are conserved.
So, particularly 
concepts of any conservation law 
remain as there are even though the language
changes. For example, we can talk about momentum
conservation, we can talk about energy conservation,
we can talk about charge conservation all
those things are fine. But you cannot relate
it to wave functions and probability density
in the same sense as is common in case of
Schrodinger equation. So, this is the feature
one has to keep in mind; there is no non relativistic
limit in this case. But there is a different
limit which is usually taken in describing
electromagnetic field.
And, that limit is a called a classical limit
which has been given another name as well
which is geometric optics. It refers to situation
where the photon wave length is much smaller
than the characteristics dimension of the
system. And in that case one can forget about
all the quantum uncertainties inherent in
the descriptions of photons. Because they
are not of any significant magnitude and one
gets a classical picture; it sometimes refer
to as geometric optics, sometime it is also
referred to as ray optics. And this is description
which is heavily used in all the phenomenon
one learns first in a optics such as interference,
diffraction, double slit experiments and so
on and so forth.
So, this limit does F exist; and it is basically
specified by criterion that the photon wavelength
mush be smaller than the dimensions over which
the propagation or evaluation of electromagnetic
field is taken into account. So, that is a
possible and it is useful also in many situations.
But we are not going to talk much about this
geometric optics; we are really interested
in the quantum features of the electromagnetic
field. So, now let us go back and write down
the Maxwell’s equations explicitly. And
I will use the Gaussian unit as they called
not the SI units; which are another set of
popular unit; it just different change of
how normalizations are taken in various quantities
in particular this symbols of epsilon 0 and
mu 0. And one can choose one convention or other as long as it has been done consistently. So, there are 4 Maxwell’s equations; 2 of
them are in homogeneous and in homogeneity comes from the background distribution of charges and currents.
And the other 2 are homogenous mistake this
has to be curl of p. And this equations are
already in a form which can be written in
a Lorentz covariant structure very easily;
and that requires definition of electromagnetic
field tensor. So, it is an anti symmetry to
index object and i and 0 are space and time
indices in the Lorentz structure. And we will
define F i 0 as E i and F i j is equal to
minus epsilon i j k B k by construction this
tensor is anti symmetric in its 2 Lorentz
indices. And with this definition this equations
are converted into a simpler looking form;
where the space and time derivatives and it
can combined and in terms of covariant derivative.And the first equation becomes 
d mu F mu nu is equal to 4 pi c times j mu;
where j mu as the time component as the density
rho and space component as the current vector
j.
So, that is much simpler looking form and
its automatically in a Lorentz covariant structure;
while the second equation looks little more
complicated. But it also has a simpler form
in terms of derivative acting on F. But now
the 3 indices and 3 indices are permuted in
a cyclic fashion and once that is done the
total result is 0. So, that is the covariant
form of Maxwell’s equation where we make
use of a very specific construction. And that
simplifies the structure; the homogenous equations
are also sometimes referred to as a Bianchi
identities. And structure then has a form
very similar to the in homogenous part.
And, so I will write it in almost similar
language. But now instead of F mu nu have
a F delta mu nu and F delta mu nu known to
be the dual tensor of F mu nu defined explicitly
by the relation involving the anti symmetric
epsilon symbol; which is F delta is half epsilon
F with the indices automatically contracted
in appropriate fashion. So, with this notations
Maxwell’s equations rather look very simple
in form it is just divergence of F in one
case where it is a electromagnetic tensor
is j. And when it is a dual tensor gives 0;
there are identities automatically buried
inside this structure by the property that
F is anti symmetric.
So, if I take one more derivative of this
in homogenous Maxwell’s equations. So, it
is now instead of one derivative over there.
Now, I will take derivative with respect both
the indices the equations says that this is
again let me correct this mistake it is equal
to divergence of j. But since the 2 derivates
acting are always symmetric they commute to
each other F is anti symmetric; and so this
divergences as to vanish. So, this is 
conservation law the current which couples
to electromagnetic field described by Maxwell’s
equations is automatically conserved. And
so we have various nice relations already
built in from the basic structure of the whole
electromagnetic anti symmetric tensor. And
that turns out to be very useful both in discussing
Lorentz property as well as other symmetries
property as well as conversations laws. So,
this is a structure and now let us proceed
a little further and try to see what are the
solutions that emerge from this equations.
So, there is a very simple structure which
is dictated by the Binachi identities; that
one can find an exact solutions of those identities.
It just says that the dual tensor has to be
have in a sub particular way or equivalently
the homogenous Maxwell’s equations can be
solved in a particular way. We know this result
from the vector calculus the divergence be
equal to 0 is solved by exactly the definition
is curl of A. And the similarly the curl of
E being proportional to del b by del t is
exactly solved with the first relation between
B and A; by the construction that E is minus
1 by c del A by del t minus gradient of a
scalar phi. And this again can be converted
into covariant language in terms of the field
strength. And it has a very suggestive looking
form that F mu nu is d mu A mu minus d nu
A mu. So, it is generalization of the curl,
but now will 4 dimensional space time.
And, this automatically also include the anti
symmetry properties of F mu nu and with this
definition we have solve half of the Maxwell’s
equation. And one can now go and plug this
back in into the other half of the Maxwell’s
equations the in homogenous one; and see what
comes out that particular result. So, this
is a very straight forward formulation which
now introduces 
this so called vector potential A mu; we have used this vector potential already in dealing with Dirac equation in converting
ordinary derivative to covariant derivatives.
But here one can see appearing from the definition
of the electromagnetic field. And one can
wonder a little bit whether this vector potential
is standard frame work or a the electromagnetic
field F mu is the standard frame work which
one is more fundamental and which one is not.
And, it turns out that 
A mu is a more fundamental for describing
what are known as gauge fields; electromagnetic
field is an example of gauge fields 
compared to the fields F mu nu and there is
a reason for it. But there is also extra thing
one has to tackle with this mu feature. And
that has to do with degrees of freedom involved
in the descriptions of the gauge field; in
the language of Maxwell’s equation there
were 6 components 3 for electric field and
3 for magnetic field. Now, we have rewritten
those objects in terms of these vector potential
as in object with 4 components. So, how many
degrees of freedom are really there in describing
the whole system? And one has to keep track
of which degrees of freedom are physical and
which are just auxiliary concepts needed for
writing the mathematics in a simpler language.
And, it turns out to be that photons have
only 2 physical 
degrees of freedom and they are known as the
2 directions of transfers polarizations. So,
all the extra indices which are floating around
they are all in some sense subsidiary to the
fundamental degrees of freedom. And once the
mathematics is through they must kind of disappear
in one form or the other; F mu nu as we have
seen anti symmetry tensor in 3 indices or
it had 6 degrees of freedom by imposing the
solutions of the homogenous equation we cut
down this 6 degrees of freedom into 4 which
now parameterize A mu. But even out of this
4 degree of freedom is A mu only 2 physicals
which are the transpose polarizations. And
the other 2 degree of freedom in particular
we call them temporal and longitudinal are
not physical in describing photons. But yet
what we will do is keep all the 4 degrees
of freedom around; so the all these temporal
longitudinal components will be there in the
explicit equations and calculations.
That is useful in maintaining the whole calculation
in a Lorentz covariant form. Because if you
projected it out directly the transpose degree
of freedom the Lorentz covariant automatically
gets reduced. So, we will keep the extra degrees
of freedom; but at the end of the calculation
whatever extra degrees of freedom produces
essentially cancels out; and becomes 0. And
only the contribution of corresponding to
the 2 physical degrees of freedom survives.
So, and this turns out to be very important
check of the whole frame work; that you calculates
various things you have physical degrees of
freedom and some auxiliary. And un physical
variables also floating around at the end
of it all the unphysical part basically goes
away it contributes 0; and only the physical
objects survives. But this is very deep principle
it is not just mathematical tricks.
And, this behavior is a consequence of a powerful
under lying symmetry which is called gauge
symmetry. So, this description is not just
a mathematical fluke but there is a principle
involving in the calculation. And that principle
plays a very important role in descriptions
of electromagnetic fields in particular; and
many gauge theories in general. So, one now
have another symmetry to worry about in addition
to the Lorentz symmetry in case of photons
the Lorentz symmetry part was almost automatic.
Because it started out with the fully relativistic
system and Maxwell’s equation already had
that symmetry built in. But this gauge symmetry
is a second symmetry which now shows up; and
it has very specific consequences which are
not. So, straight forward to see from equations
themselves; but by introducing a little bit
of extra frame work those features appear.
And they play a very important role in the
description of physics in particular how many
degrees of freedom are physical? And how many
degrees of freedom are kind of dummy variable
just taken around the right; and ultimately
not contributing to the final result.
So, now let us see this gauge symmetry in
a more quantitative fashion; remember we introduce
this auxiliary field the vector field A mu
has a solution of the homogenous Maxwell’s
equation; where F was just the curl of A that
was the only place where it appeared. And
it is very easy to see that all that algebra remain on altered; when the variables are shifted according to the prescription that A is shifted
by gradient of A arbitrary fashion of lambda.
And the scalar component is simultaneously
shifted by time derivative of the same function
lambda; and the Lorentz covariant form of
this object is A mu goes to A mu minus d mu
lambda.
So, all the equations which introduce A mu
remain unchanged explicitly because well we
have to take a curl of A to get F. But curl
of this gradient is automatically 0 it is
a mathematical identity. So, one can say that
under this change F just goes back to itself
and so nothing happens to the Maxwell’s
equations at all. So, this particular feature
brings out that this vector potential which
we introduced is not uniquely defined; there
is a arbitrary degree of freedom here labeled
just lambda which can be used to change it
in many different ways without any consequences.And this freedom to choose and that tells us that component
of A are not physical there are these unnecessary
things introduced by lambda. And we can choose
lambda cleverly to remove some of those extra
degrees of freedom. And that indeed turns
out to be true by choosing 
appropriate fashion lambda which sometimes
also referred to as fixing of a gauge; the
degrees of freedom of A mu can be reduced
to the physical ones only.
So, indeed it is true that by suitable choice
of lambda all the extra degrees of freedom
can be made to disappear. And then will have
only two transfers polarizations of photon
left which are physical. And the other staff
is only a convenience certainly there is no
unique way to say that I will choice one particular
lambda. And it will be better in some sense
or other and there are many different prescriptions
for this gauge fixing lambda exist.
And, one prescription may be convenient in
one particular circumference and another prescription
may be convenient in another circumference
that is delta case by case basis. But again
there are certain prescriptions which are
popular and they are popular; because of the
property of Lorentz covariant which is built
in inside the formulations we have discussed
so far. So, it says that divergence of A is
equal to 0 and that is automatically Lorentz
in variant by construction. And the potentials
than satisfy divergence of A plus 1 by c del
phi by del t is equal to 0; which is in the
space and time degrees of freedom written
independently compare to the covariant language.
And, then the so called in homogenous Maxwell’s they had 2 terms in terms of the equation. But now
I will write those things explicitly d mu of F mu nu is equal to 4 pi by c j nu.But now this d mu acting on A mu the derivatives
commute and that part is 0 by the so called
Lorentz gauge fixing conditions. So, out of
the 2 terms only the first one is left and
we have result which is the standard form of wave equation; and c
is the velocity of propagation for the particular
wave. And that is the way we are happily looking
at the electromagnetic field as wave and it
has a particular propagation speed. And it
automatically follows all the super position
principle interference, diffraction kind of
phenomena; and current basically becomes the
source which produces the wave.
So, this is the so called popular form of
gauge fixing; the Lorentz condition it produces
this wave equation. But even inside this wave
equation we have certain degrees of freedom
which are not completely fixed. And that is
because that condition 
Lorentz condition 
it does not fix the gauge completely. Because
again one can ask what happens when a mu goes
to A mu minus d mu lambda what happens to
this particular condition. And one can easily
see that this condition gives a extra degrees
of freedom still remaining; which I will call
residual which can be written as 
the wave operator acting on lambda a equal
to 0.
So, if I take a lambda which is solution of
wave equation and change A mu by gauge transformation
involving that particular lambda; nothing
will change in the wave equation as well.
So, A is still determinant up to the solution
of this homogenous wave equation. And so there
is something more which can be removed as
a matter of fact; when we impose one condition
of the Lorentz gauge fixing out of the 4 degrees
of freedom essentially removed one part which
can be called the longitudinal part if you
wish.
Because its divergence of A which is removed
but using this residual degree of freedom;
we can still remove one more component which
in this particular case can be called scalar
component. Because a wave operator is acting
on it and it gives 0. And when one removes
both these components form a mu the 4 degrees
of freedom reduced to 2. And then it gets
down to the only the 2 physical degrees of
freedom described by the transpose polarize.
So, we can now choice another solution to
fix the gauge completely; this operator is
already complete scalar. So, there is no way
we can do the complete gauge fixing in a Lorentz
covariant fashion. And one has to now break
the Lorentz symmetry explicitly to do complete
gauge fixing.
So, we will now put the 2 conditions explicitly
the sum of these 2 conditions gives the Lorentz
gauge fixing automatically. Because it is
a 4 divergence and the time and space components
mutually satisfied. But in addition to that
there is a separate condition for the time
and the space. And now if you choose this
particular conditions you will see that you
made a specific choice for a lambda there
are 2 separate equations to be imposed. And
sometimes this choice is referred to as a
radiation gauge fixing; and it removes both
the temporal and longitudinal components of
A.
And, the two transfers degrees of freedom
which have said at the beginning other physical
ones they are the one left behind. And that
is a example of complete gauge fixing to get
down all the way to the physical picture;
one does not do that very often all the way
down to the physical degree of freedom. But
the one step earlier which imposes the Lorentz
gauge fixing condition that is used very often
in calculation. Because it does not destroy
Lorentz invariance of the analysis. And in
that particular case there is automatic cancelation
which comes out of the result between the
temporal and longitudinal degrees of freedom.
And, that cancellation is nature of the uncultured
degree freedom dropping out of the final result.
One can see several other things also of in
this radiation gauge which are useful in describing
the physical degrees of freedom; how it behave?
And one can look at them of spatial cases;
if one look at the change in the object which I will right in momentum space in K mu A mu. And because A mu goes
to A mu minus del mu lambda and d mu is essentially
gives k mu after factor of I; the change in
this particular quantity is K square times lambda under gauge transformation. And, no we have the gauge fixing condition
where d mu A mu is the same object came A
mu d mu. So, not only this object is 0 but
any further change in it necessitated by a
further gauge transformation must also be
0. Because it is true in arbitrary frame or
arbitrary gauge transformations both ways.
And so K square lambda 
must vanish because d mu A mu can appear in
lots of calculations without any trouble;
and the lambda happens to be arbitrary functions.
And only way when one can satisfy and that
condition is satisfying that K square is equal
to 0; this is a indirect way of saying that
we have a wave equation which came out by
this particular gauge fixing condition. And
the wave equation was describing a Mass less
particle it had the desperation relation such
that the 4 momentum K square is equal to 0.
So, one can see that photons are Mass less
degrees of freedom when reduce down to physical
components; and they satisfy the Mass less
equations. And this is a relation between
the gauge symmetry and gauge choice or gauge
degrees of the freedom and mass lessness of
photons. And we have also seen in the Maxwell’s
equations that the same wave equation was
of structure which automatically satisfied
conservation of charge. So, there is a very
tight relationship between this various quantities;
gauge invariance, conservation of electric
charge and mass lessness of photon are all
interrelated properties. And in some sense
they are also equivalent in the sense that
one can derive one of them are given the other;
and this is a dip principles which will 3
examples of as the course progresses.
