In the last lecture, I worked out the solution
of Dirac particle reflected form a step function
barrier. And today we will try to understand
its interpretation in terms of the process
involved. And we will quickly see that we
will have to extend our language to incorporate
all the things that go on.
So, the process is characterised in terms
of the transmission and reflection amplitudes.
And last time I calculated them as the expectation
values of the currents for each of the 3 terms.
And now we would like to evaluate the values
of these ratios of currents for different
values of the potential height.
So, let us look at the first case which is
the kinetic energy E minus m C square is larger
than the height of the barrier; in this case
the transmitted momentum is real. And the
values of the ratio parameter I defined last
time lies between 0 and 1. And these 2 properties;
then allow us to calculate the range of values
available to the reflection current and the
transmitted current. And we will easily obtain
that the reflected current is less than the
incident current. And the transmitted current
is positive and as has to be true; the some
of the reflected and transmitted currents
matches with the value of the incident current.
So, this is perfectly legitimate situation;
it corresponds to partial transmission. And
it is similar to the non relativistic case;
where, the energy is larger than the height
of the barrier; the second case we can easily
look at is now increase the height of the
barrier. So that it becomes larger than the
kinetic energy and interval which I want to
first look at corresponds to the barrier height;
being between E plus m C square. And E minus
m C square which means, the kinetic energy
is less than potential energy. But the difference
is smaller than 2 times m C square.
In this particular case we can easily see;
that the transmitted momentum becomes imaginary,
this just is a simple chat from the dispersion
relation. And so r is also imaginary; and
the consequence then is that 
the reflected current is equal to the incident current; and the transmitted
current vanishes. So, this is again consistent
with the conservation of the current it corresponds
to total reflection; the wave function on
the positive side of z axis where; the barrier
is exponentially decays with z. Because the
p 2 is imaginary it is not a propagating solution.
And this situation is similar to the non relativistic
case; where the kinetic energy falls below
the potential energy.
So, these things as easily understandable;
but we are left with the third possibility
which is navel as far as the relativistic
quantum mechanics is concerned. And that is the situation where; 
the potential barrier is higher than kinetic energy. But the difference
is larger than 2m C square. And in this particular
case again; the dispersion relation will tell
you the p 2 is real; and we will have the
convention that p 2 corresponds to a wave
travelling along the positive z axis. And
in that particular situation; the value of
r becomes negative; and this as a peculiar
result that the reflected current is proportional
to 1 minus r; and r is less than 0.
So, it becomes actually larger than the incident
current which had only 1 plus r and. So, j
r is greater than j i; and the transmitted
current which is proportional to r now is
negative. So, this is something which has
no analogue in the non relativistic case,
it would seem that there is a extra contribution
somehow of a wave coming from the positive
z axis side. And going to the negative side
corresponding to this negative value of the
transmitted current; but that kind of situation
does not make sense physically. Because we
have the principle of locality and relativity;
you cannot have information travelling faster
than the speed of the light and correlations
between casually disconnected region.
So, there is no way of the transmitted part
of the space to know what is coming from the
incident part? And so one cannot have simultaneously;
something coming from the left as well as,
something coming from the right. So, the interpretation
has to go beyond just a simple minded single
particle description. And that feature is
what basically is labelled as a paradox it
is a Klein paradox; that you send a current
from the negative z axis side; you obtain
a reflected current larger than that. And
that is; because there is a current coming
from the positive z axis side and adding on
to the incident current; and the together
they give a larger value for the reflected
part. So, the way out is not to try to make
current come from positive z axis region.
But interpret at the negative value of the
current, but still propagating in the plus
z direction indicated by this value of p 2.
So, there is a wave going; but that wave somehow
corresponding to negative value of current;
and the wave which can happen is the direction
of propagation remains the same. But the charge
of the particle is opposite and that will
flip the charge of the current which gets
transmitted. So, the resolution corresponds
to 
the process known as pair creation; and the
components of the pair going off in opposite
directions. And for example, if you are talking
about electron wave function what happens
is the incident electron came from the negative
z axis tide. But it produced a pair e minus
and e plus. These 2 electrons get reflected
with making j r bigger than j i and the positron
gets transmitted. And because it has a opposite
charge; it will correspond to a negative value
of the current relative to the sign of the
electron current.
So, this is the peculiarity; but we have to
pay a price. And the price is that we can
no longer talk about the whole process in
terms of a single particle we immediately
have to go to a many particle description.
And this is a necessity required by relativistic
dynamics there is no way around it. So,the single particle description has to be given up in favour of what can be
called a many-body theory. And in some sense
these labels correspond to the quantum mechanics
changing over to quantum field theory. And
this is a unique phenomena of merging quantum
mechanics with relativity. And so the Dirac
equation thus; indicates its own limitations,
we started formulating the Dirac equation
as a single particle equation; we worked out
various consequences etc.
But then we run into these navel processes;
which we can only explain by going to a many-body
description and. So, the Dirac equation is
not self sufficient; in the sense in which
Schrodinger equation is when we are dealing
with problems involving Schrodinger equations,
everything can be solved in that specific
context. And you do how no need to give up;
the 1 particle description in favour of a
many-body theory. But Dirac equation necessitates
that. And this is a important component, that
relativity forces one to go beyond a fixed
number of particles; theory to what is generally
labelled as quantum field theory. And that
is; what we will have to address in more detail
little bit later. But one can easily see that
the process is quite legitimate of a positron.
So, the conditions which I had is this V 0
greater than e plus m C square, can also be
written as minus c minus m C square; which
corresponds to the kinetic energy of a positron,
with energy minus e. And this is less than
minus V 0 sorry, it is not less than; it is
greater than; that means, the potential height
felt by the; this is the potential for e plus.
And because the positron has operative charge
its potential felt by is also opposite, and
its kinetic energy exceeds the potential.
And, you can have a genuinely propagating
wave; that wave runs off on the positive z
axis type to infinity, that is; the interpretation
of j t. So, this is a caveat which one has
to understand in case of a relativistic quantum
mechanics. And we have a resolution only if
we invoke this peculiar process of pair creation
whenever, the barrier is large enough.
So that, it can provide the energy needed
for pair creation which is 2m C square. So,
there is change in the potential at the barrier
which has to be greater than 2m C square and
another feature which is not. So, obvious
from the previous analysis; because we took
only a step function barrier, the barrier
was concentrated in a very small space region.
But suppose if you take a smoother potential,
then the barrier certainly has to exceed in
this particular range. But also we need in
a second condition; which is this range of
variation that has to be less than the Compton
wavelength, this is a rough inequality not.
So, precise and this can be understood either
by doing a full calculation; but having a
gentle barrier or even by appealing to the
uncertainty principle; we know that we cannot
localize the particles arbitrarily without
its momentum becoming very much uncertain.
And when the particle is localized to a region
of the order of Compton wavelength its momentum
become equal to m c. And so the energy corresponding
to that particular uncertainty is uncertain
by m C square and so whenever this region
is very small, we can have quantum fluctuations
of in energy which are of the order m C square.
So, this quantum uncertainty allows delta
E of the size of m C square over regions of
Compton wavelength. And we can then have pairs
coming out because this energy is comparable
tom C square. And this pairs are often referred
to as virtual pairs in the background often
called the vacuum. So, quantum vacuum will
always have this virtual pairs hanging around
over very small distances; what the external
potential can help is that, if the potential
is changing very fast over the size of whatever,
this virtual pairs are then these virtual
pairs can be pulled apart and converted to
real pairs. And once these pairs become real,
than we will see consequences as in the case
of paradox; there will be a value of current
which becomes observable.
Hence, so one needs this particular condition,
unless that you need a large variation of
the potential over a size; which is of the
order of Compton wavelength, if that is; available
then we will have this virtual pairs coming
out of this particular vacuum. And one can
even look at this as a spontaneous process.
So, this is a this one can be understood in
the previous derivation, that we can have
no incident wave at all; you can put j i is
equal to 0.But you can still obtain non 0
value for the reflected current, as well as
the transmitted current; pairs are just popping
out of the vacuum one of them runs upon one
side; and the another one on the other side.
And, this limit of the calculation with j
i equal to 0 is a smooth limit. So that is,
indication of a spontaneous process; one can
also observe a 1 more feature, that is; the
pairs are getting produced. But somebody is
paying the cost of this 2m C square energy;
when you treat an external potential, we do
not consider any variation. And we but in
reality there is a feedback; and that is taking
out 2 m C square energy weakens; the potential
height and that ultimately stops pair creation
process. So, it is a spontaneous process;
but every time the pair comes out the potential
loses energy.
And, so it becomes smaller in height and ultimately
it will become small enough in height. So
that, no further pair creation will takes
place. So, the procedure in real world with
a feedback does have a ultimate limit in the
sense, that pair creation cannot just go on
forever; that is just a simple consequence
of energy conservation. So, this is a general
background and its interpretation.
And, now we can look at some simple examples,
how we can see this particular process; one
example is to just look at quantum electrodynamics;
which is what we have been solving using the
Dirac equation. But we want such large potentials
or equivalently; large electric fields. And
that is the ratio of this how much potential
is needed to the distance over which is needed.
And you can work it out this it can be expressed
as a gradient of potential has to be larger
than the ratio of these 2 particular quantity
2m C square and h cross by m c. So, we need
the electric fields which satisfy this particular
condition.
Now, such large fields are not that easily
available; we need special configurations
to produce the m. And that requires getting
a large charge in a small enough region; one
way to look at this condition; is to just
look at the solution of the hydrogen atom
which we have worked out earlier.
So, in that particular case we have roughly
that for nuclear charge becoming of the order of 137; we have the
total energy for the ground state of the hydrogen
atom roughly equal to 0. And the so called
Bohr radius 
becomes order of Planck constant divided by
m C which is nothing but the Compton wavelength.
So, we have the condition roughly satisfied;
though the potential is large enough. So,
the binding energy is of the order of m C
square; that is essentially what is available
to the electron.
And, that energy is available over the Compton
wavelength. Now, you may argue, that you need
a little more energy in the sense that it;
this will give m C square and not 2m C square.
So, then you can say that well, we will have
to go to nuclear charges; which are larger
than 137;that is a more detail, estimate of
what nuclear charges will be needed to see
this particular process; but roughly this
is the case.
So, in this kind of situation; if you had
a nuclear charge with large enough value,
there will be spontaneous e plus, e minus
creation coming out right from the innermost
orbit of the electron. And that will be easily
visible because the atom will suddenly start
producing; this radiation e minus will fall
on this nucleus reducing the particular field;
and e plus will be emitted as in the case
of beta decay. And we will have a spontaneous
signature. But this particular condition;
that nuclear charge is of the order of 1 over
the fine structure constant is not easy to
obtain in practice. So, more detailed estimates
indicate that you need the desired greater
than about 170 for 
and we cannot have a single nucleus of this
particular charge.
But what has been argued as this situation;
can be realised 
in collisions of 2 heavy nuclei. Then we need
the nuclear charge on each 1 of the 2 heavy
nuclear of the order of 80; which is certainly
available in practice. But this collision
will not be a static situation; it will be
only for a short duration, these two things
have to come together. And in that particular
duration you will have this instability in
the potential; and you will have particle
e plus e minus productions and that is somewhat
messy. And also we only included the electromagnetic
interaction in this particular calculation
once; you start colliding to nuclei lots of
other process also start happening because
nuclei forces themselves come into play.
So, experimental realization is a messy we
can certainly collide 2 nucleuses. But then
identifying the signature out of all the things
which go on; that is a part which is messy.
And the reason is the collision dynamics and
nuclear forces. So, certainly you can see
lots of things going on. But identifying that
this particular e plus e minus creation corresponded
to the kind of feature we saw in the Dirac
equation; that is not clear enough or rather
some ambiguities are left inside allocating
various species to various kind of interaction.
So, this is 1 particular situation not easy
to construct and not easy to interpret because
of certain caveats. So, identification of
source of pair creation is difficult.
So, now let us look at a different example,
which corresponds to the theory of Q C D in
this particular case the interaction is the
strong interaction; that of quantum chromo
dynamics or which binds the protons and neutrons
together; and also produces then nuclear process
the interactions is strong. So, it certainly
produces strong fields what is seen in processes
of Q C D; is that the potential which binds
the various components together the components
are the quarks. So, potential binding quarks
is 
confining by confining it means, that the
separation of the components become large
the potential goes to infinity. And that certainly
is the situation required for satisfying Klein
paradox the potential does go to infinity.
So, you will have regions where you can produce
2 m C square of energy. And so the particle
antiparticle pairs; that region to be small
enough. But that is possible because the coupling
involved in the Q C D is certainly large.
So, you have large gradients available and
then the process; which is q q bar, quark.
And anti-quark dynamical pair creation is
possible; when the binding system is perturbed
by certain kind of excitation; you just push
it high enough of the energy by say hitting
it with some external radiation. If that kick
is large enough, you will have this spontaneous
pair creation possible. And the bound states
in Q C D are called hadrons.
And, when they are sufficiently excited you
have possibility of creating quark, anti quarkpairs. And this is actually seen very often
in many 
collision process; which occur at different
kind of particle accelerator. And we have
easy identification also about the quarks
and anti quarks coming out. And you can actually
see them as a certain jets coming from that
collision region. So, here the identification
is actually much straight forward. What is
not clear cut is that this is the theory with
strong interactions. So, the rates of how
frequently this process occurs they are not
very easy to calculate quantitatively, that
the pair creation occurs.
But you can ask how frequently it should occur
based on the fundamental equations involved?
And because the potential is very strong one
cannot use the standard machinery of perturbation
theory to calculate these rates. And that
is the sort of limiting part of the verification.
But 
there are so can make some models and construction
qualitative expectations and they agree with
experimental results. So, this is a second
example I would go further. And give a third
example also which corresponds to theory of
gravity this has a certain difference compared
to the previous cases; for 2 reasons, 1 is
we do not have a complete quantum theory of
gravity. And another thing is the gravitational
interaction behaves differently than electromagnetic
or Q C D interactions. And that difference
comes because of the fact that in Q E D or
Q C D the particle and anti particle have
opposite charges. But in case of gravity the
particle and anti particle have same value
of mass.
So, one has to include the effect of gravity.
But as a slightly different operator in the
Dirac equation here; the masses of particle
and anti particle are equal. And so once we
include this potential it has to behave the
same way as the mass term in the Hamiltonian
behave. So, the potential appears in the Dirac equation 
by the replacement, that you have the term
beta m C square, going to beta m C square
plus V gravity. And this is the slightly different
prescription compared to electrodynamics potential
change; the Hamiltonian without this extra
factor of beta. And because of that the shift
of energy used for particle and antiparticle
who is in opposite direction in case of gravity;
the shift is in the same direction. And change
in the mathematics is this extra factor of
beta; the potential will couple in the same
way as the mass term is there in the equation.
Once, you now understand this particular caveat
you can go back to the reflection from a barrier
problem. And see what this kind of potential
will do and here; you see a slightly different
feature whenever 
the potential of this particular type is positive.
And 
the case C which I considered for pair creation;
in case of electric potential it gives the
same result as case B. 
And what this implies is there is total reflection
and no pair creation in this particular situation.
So, a positive potential barrier does not
do anything navel with the potential is a
gravitational potential. But we already know
that gravity is a different kind of a interaction;
the potential of gravity happens to be actually
negative. And then we can have the conditions
for pair creation satisfied. Because the potential
can drop by 2m C square over a certain region,
that can give sufficient energy to be released
for pair creation.
So, gravitation is attractive and the pair
creation is possible if the potential drops
sufficiently rapidly. And then the 2 m C square
can be released over a distance scale of the
order of Compton wavelength. And now we can
make an estimate about what is the condition
or how strong this field has to be? And that
can be done qualitatively looking at large
changes of the order of m C square of the
for the potential gradient; and that corresponds
to the situation which arises in case of black
holes.
So, this is possible 
in case of black holes where the field is
sufficiently strong. And the pair creation
is possible if V gravitation sufficiently,
strong is this field has to be? And that can
be done qualitatively looking at large changes
of the order of m C square of the for the
potential radiant. And that corresponds to
holes where the fully efficient strong.
And, pair creation of this particular type
has a special name it is called hawking radiation.
So, we can now easily workout the estimate;
the condition for a formation of black hole
is that the gravitational potential energy
which is G M square by R is the radius of
the black hole; and M is the mass has to be
of the order of the rest mass energy, these
are just order of magnitude estimate. And
that implies the mass and the radius of the
black hole are related by 
this simple relation mass is proportional
to the radius.
Now, we want the second condition also which
has to be related to how fast the potential
is changing. So, what is the gravitational
potential in this field produced by a black
hole? So, field produced by such a black hole
it is given by V gravity is minus G mass M
is the test object and divided by its distance.
And once you plug in this order of magnitude
estimate then it is ratio of the black hole
radius to the position of the test particle.
So, it can have 
this delta V of the order of m C square when
the change occurs over distance of the order
of the radius; and the of course value of
the radius is that of the black hole.
So, this condition is indeed satisfied and
in that particular case. So, pairs are spontaneously
produced, provided this lambda Compton is
then larger than the black hole radius. We
needed large variation over a distance of
the order of Compton wavelength we saw that
the variation is available at the size of
the black hole radius. And so the particles
which will have Compton wavelength larger
than; that will certainly get emitted by this
pair creation processes. And that now gives
a generic picture of this hawking radiation;
that the standard black holes of the so called
just produced black holes from neutron stars
have 
radius which are of the order of 10 kilometres.
And the only particles which we know of Compton
wavelength larger than this are the mass less
particle.
So, they are photons; and nothing much else
we do not anything resembling a the electron
positron pair at this particular case. But
again because of the peculiarity of gravitational
interactions you can now see what will happen
with the feedback of this radiation onto the
black hole itself? So, certainly some energy
is going away that will decrease the mass
of the black hole; it will also decrease the
radius of the black hole because of this condition
that G M is proportional to R C square, and
so as radiation comes out the black holes
start shrinking both in mass as and in radius.
And, the spontaneous pair creation condition;
then becomes possible for particles with smaller
and smaller Compton wavelength. Once, the
radius has shrunk to sufficiently small size
there may be, more than just photon and graviton
which will have wavelength larger than particular
size and they will get emitted. So, the radiation
due to pair creation actually accelerates
R shrinks. And more and more particles start
gets emitted until at the end the black hole
just disappears; all the energy which has
now is part of the radiation. And it is all
gone and you are left with nothing at the
end.
So, the feedback decreases both M and R and
the radiation loss 
speeds up with more and more particles getting
emitted even the ones with larger masses;
once the radius becomes sufficiently small
till black hole totally evaporates. So, this
is again a sensible limit for case of a feedback;
that the radiation ultimately stops. But the
radiation actually accelerates. And then stops
quite unlike what happens in the electro dynamic
case? So, this is the illustration of this
pair creation processes in various kind of
different interactions different backgrounds.
And their peculiar features certainly; we
can hope to get more details if we do have
a more quantitative analysis what I have presented
are just the qualitative estimates and order
of magnitude pictures.
But they are quite sufficient for understanding
the general principles involved; exact results
are actually not available to the detail which
we want for various reasons in case of Q C
D; we do not have ways to solve equations
exactly in case of gravity, we do not have
a complete theory of gravity in the quantum
region. So, we do not actually know literally
what happens at the end points of the process;
it has to rely on quantum gravity which we
do not know. But in the so called in between
region which is called semi classical or qualitative
understanding of the mechanism the behaviour
is expected.
And, it works in several cases when we can
design experiments to see them; of course
the hawking radiation is something which we
have not observed in any particular science.
But efforts are underway to construct systems
in condensed matter of physics; where the
same kind of situations can be created. And
then we will be able to see some of these
particles, anti particle, pair creation process.
And those are again very interesting examples
I will describe an example of recent interest
that is the property of grapheme; where Dirac
equation plays a very important role in a
later lecture; before that I will continue
with more properties of Dirac equation in
the next lecture.
