In this concluding lecture, I am going describe
the present status of the theory of quantum
electrodynamics. QED has been tremendously
successful as a perturbative quantum field
theory. To explain this success let me first
tell you how the results of perturbative calculations
are systematically organized.
In any perturbative framework, it is natural
to arrange all the contributions to a given
process in powers of the small parameter,
which in case of QED is the electromagnetic
coupling. For any Feynman diagram contributing
to a specific endpoint Green’s function
this power of the coupling is just the number
of interaction vertices. For any Green’s
function we just have to count the number
of vertices to find out to which order of
perturbation theory it belongs to...
But on the other hand it has become quite
common place to specify the accuracy of perturbation
results in quantum field theories in terms
of the number of loops. These two descriptions,
number of vertices and number of loops are
related, and let me show how. In your earlier
lecture I had given the topological relations
between the number of vertices, number of
loops, number of external lines, and number
of internal lines for any Feynman diagram,
and these relations where the number of loops
is number of internal lines minus the number
of vertices plus 1.
The number of vertices can also be related
to the number of photon lines and also to
the number of electron lines. If we eliminate the number
of internal photon and electron lines from
these relations, we obtained a result 
which relates the number of vertices to a
number of external lines and the number of
loops. For a given Green’s function the
number of external liner are fixed. So, each
extra loop increases the number of vertices
by 2. As a result the transition probabilities
and cross sections which are obtained by squaring
the amplitude can be expressed as power series
in the fine structure constant alpha which
is the square of the electromagnetic coupling.
This is one relation which says that one can
label the perturbative contributions either
by power of alpha or by number of loops. I
should point out two specific points in this
framework that this analysis is valid for
any quantum field theory with a single coupling
constant. When there are multiple coupling
constants in the theory or there are background
fields some reorganization of the series is
needed depending on the relative strengths
of different terms. So, we have to watch out
for that.
The second point to note is that there are
divergences in the calculations carried out
by perturbation framework, and these divergences
give physical contributions after they are
regularized properly. In case of QED the divergences
are logarithmic and so there appear terms
involving the logarithms of different energy
scales. These logarithmic terms can be controlled
essentially summed up to a large extent by
the techniques of renormalization group evolution,
which replaces the fixed coupling constant
by a running coupling. Once that is done remaining
logarithms are relatively small; they do not
disturb the hierarchy of terms in the perturbation
series. Sometimes these logarithms can be
converted to logarithms of alpha itself from
logarithms of ratios of energy scales.
That is the situation in case of non-relativistic
electrodynamics where the ratios of the scales
are essentially powers of the couplings. So,
sometimes the series in perturbation theory
can involve terms that are log alpha, but
one should realize that they are coming from
ratios of energy scales. Given this two caveats
we can easily rearrange the series where successive
terms decrease in magnitude and the perturbative
machinery makes reasonable sense. There is
another reason for expressing the results
in quantum field theory in terms of the number
of loops, and that is their connection to
powers of the Planck’s constant.
The fundamental scale of quantum effects is
given by the commutator of economically conjugate pair of variables. The simplest
example in quantum mechanics is the
commutator of position in momentum is equal
to i H cross. From the definition of the conjugate
momentum and the relation between the Lagrangian
and the Hamiltonian we can easily see that
both L and H have dimensions which are those
of Planck’s constant divided by time. The
theory actually indulges time integrals of
the Lagrangian in the Hamiltonian.
They are the action which is the integral
of the Lagrangian and the evaluation phase
which is the integral of the Hamiltonian.
Both these time integrals are naturally measured
to quantify quantum effects in units of the
Planck’s constant. As a matter of fact the
path integral formulation 
uses e raised to i as i H cross as the weight
of the trajectories, and the time evolution
operator is an exponential of minus i by H
cross integral H dt. When these powers of
Planck’s constants are explicitly included
in the formulism, remember that we had chosen
a notation at some stage setting H bar equal
to 1, but now we are putting back the factors
of H bar explicitly. Each interaction vertex 
is proportional to 1 by H cross and each propagator
which is given by the reciprocal of the differential
operator appearing in the Lagrangian or the
Hamiltonian is proportional to H cross. So,
the contributions to the Green’s functions
scale as h bar to the power of I minus v; I counts
the propagators and v counts the vertices
which is equal to H bar the power of L minus
1 by the topological relations which we have
already seen. So, the number of loops represents
the order of quantum correction 
to the classical result. Note that classical
result corresponds to extermination of the
action, for example, and so it would be represented
as exponential of the classical action divided
by H cross, and it corresponds to 
the tree diagrams 
with no loops.
This is the way perturbative results in quantum
electrodynamics, and in general in any quantum
field theory is organized. Once we have a
systematic description of the results we can
go and compare them with experimental observations;
that is indeed done in many important checks
in case of QED. Let me list some of them.
These are called the precision tests of 
the theory. Some of the most precise experimentally
measured and theoretically calculated quantities
are 
the anomalous magnetic moments 
of the electron and the muon. Atomic recoils in electromagnetic transitions 
energy spectra of hydrogen positronium 
and muonium which is the bound state of an
electron and a muon. And also condensed matter
phenomena like quantum hall effect and AC
Josephson effect.
The best measurements 
have an accuracy that is less than one part in a billion. The best calculations have been carried out 
to the order of four loops. These two have
similar accuracy to give a familiar perspective
to this high precision measurements and calculation.
I can say that such measurements to one part
in a billion amounts to measuring the distance
from Kashmir to Kanyakumari to accuracy that
is less than a millimeter. So, these are indeed
phenomenal results and the achievement of
the theory is that the experiment and theory
agree to this level of accuracy.
This is a spectacular success of QED; no other
theory in physics has been so stringently
tested and has proved to be so stunningly
accurate. Obviously, once we have found formulation
of quantum field theory I am showing that
it works very well. It is natural to apply
it to different phenomena, different observations
and build theories having similar structure.
That has indeed followed the developments
in quantum electrodynamics. We have many successors
to QED. QED is a prototype of quantum 
gauge field theory, and it was followed by
other quantum gauge field theories. The first
one to combine was Yang-Mills theories which
replace the abelian gauge group U 1 by a non-abelian
gauge group.The next one is the electroweak theory which
combines electromagnetism with weak interactions.The gauge group 
is SU 2 left cross U 1 hyper charge which
is spontaneously broken to the U 1 gauge group
of electromagnetism. The masses for the weak
bosons w and z are the result of this spontaneous
symmetry breaking, and they make the weak
interactions weak. The next one to follow
was the theory of quantum chromo dynamics
which uses the gauge group SU 3 and describes
the strong interactions. All these ingredients
put together make up the so called standard
model of particle physics.
This standard model has many parameters essentially
the masses of the particles, the couplings
of the gauge interactions and the mixing angles
which parameterize the mismatch between mass
Eigen states and gauge interaction Eigen states.
So far we have no understanding of where these
parameters have come from, but given the values
of these parameters from experimental observations.
The standard model can predict a large variety
of phenomena observed both in low energy systems
as well as high energy experiments. The situation
is such that we have not so far seen any significant
deviations from the predictions of the standard
model in all the experiments that have been
carried out. So, in that sense the standard
model is a highly successful effective field theory.
The merit of any effective theory is that
it is not going to go away as long as you
remain within its domain of application. You
can of course, explore beyond, and maybe there
are deviations in the region outside the boundary
of the effective theory. Maybe that will produce
a generalization of the theory where you able
to reach such regions outside the boundary.
But if you are going to work within the boundaries
the theory will always be useful. That is
essentially the status of high energy physics
these days.
That has not stopped physicist from working
hard; experimentalists have being trying to
set up new devices, new accelerators, new
situations where they can measure or hope
to measure deviations from predictions of
the standard model. And theories have been
busy trying to make a more general theory,
sort of a unified picture which can explain
the values of the parameters which appear
in the standard model. And also at some stage
include the theory of gravity which is not
part of the standard model, but right now
both these features, experimental searches
as well as theoretical developments are outside
practical regions where tests have been carried
out.
Let me recapitulate where all these subjects
started. It was a successful merger of the
theory of special relativity and quantum mechanics
which produce the formulism for quantum field
theory. The role played by quantum electrodynamics
has been a pioneering one in these developments.
Nowadays the subject of quantum field theory
has expanded. It has applications in high
energy physics, statistical mechanics, condensed
metal physics covering a variety of phenomena
involving different type of fields and also
different number of space time dimensions.
But the formulism has a common thread which
can be backtracked to all the developments
which took place in building the theory of
QED. The role of QED in this sense cannot
be overemphasized. In summary I can just repeat
the closing words of Richard Feynman in his
Nobel Prize lecture. “So, what happened
to the old theory that I fell in love as a
youth? Well, I would say that it has become
an old lady that has very little attractiveness
left in her, and the young today will not
have their hearts found anymore when they
look at her. But we can say the best we can
for any old woman that she has been a very
good mother and she has given birth to some
very good children.” So, that is the way it is.
