And there is actually another point here which
is that this quantum kinematics hypothesis
that every observable will be expressible
in terms of q and p can fail and it fails
royally with spin. Spin is a quantum number
for which you do not have any space time representation
and you do not have any classical analog.
So, when you get to spin you have to add it
by hand from your pocket. Amazingly enough
it has commutation relations which look exactly
like the angular momentum commutation relations
which angular momentum can be made out of
q and p.
And so and they in fact mix as well. So, you
can encounter when you go to quantum mechanics,
new systems where you can not break it down
like this to it may not have classical limit.
So, any observable that does not have a classical
limit like spin may not have a canonical representation
like this. The other corresponding point being
that in enumeration, the spin half you have
to quantize by anti commutators instead of
commutators which is very big jump. Because
it has again no classical analog right the
fermionic systems have to be quantize using
anti commentators to account for Pauli principle.
So, there are divergences from this visible
even in the very simpler examples of quantum
field theory. But in modern times we are living
in many many more complicated situations where
in fact, quantum field theory itself looks
in doubt ok. So, like this conformal field
theories that our class studying; you may
not have a Hamiltonian description at all.
So, you may not have so called canonical valuables,
but the previous part the first part is correct.
So, as you know this classical analogy was
a simplification.
But the modern things do not affect the fundamentals
of quantum mechanics because there all you
have to do is list all the possible commutators
between all the possible operators. And if
you have a recursive way of stating it, then
you are done you can state some ladder up
ladder algebras between Virasoro algebra and
so on. So if you can state that you may have
infinite number of independent of observables,
but if you can state their commutation relations;
then you have fix the quantum system.
So, not all systems will have this luxury,
but we try to make do with this for all most
everything that was known that is known at
present; except in the strongly correlated
condense matter systems. So, I am told if
I am not mistaken, but so super conductivity,
super fluidity; all of these were very exotic
states, they had to do with this bosonic enumeration
of states and Bose condensation.
And also the fractional quantum Hall effect
which has a wave function which goes by the
name of the person who got the Nobel Prize
for it Laughlin. Laughlin wave function, the
wave function exists, but there is no Hamiltonian
of which it is an eigenstate.
They do not know Hamiltonian system for which
it is in the list of its spectrum. So, we
are at that border line where this simply
stated set of principles may need some extension,
but there is nothing worried, I mean it will
be nice and exotic if you we find newer quantum
systems and just this simpler once, but there
is nothing that threatens our civilization
good. So, the second thing I can do this is
the issue of quantizing weakly coupled systems.
So, the next thing is 
and it is important to say that it works only
for weakly coupled systems. And if we talk
about 
weakly coupled electrodynamics in scattering
processes, QED scattering; QED as well as
electro weak; all of this is weakly coupled.
Electrons in a lattice miraculously displayed
this weakly coupled kind of phenomenon because
you can think of them as free gas of fermions
in the lattice, but that there are really
quasi particles.
So, both term should be put in separate; so
you can continue to think as if there are
electrons floating in the lattice. But actually
its much more complicated situation to which
in the situation where you are lucky you have
that description available is often called
the Landau liquid; Landau fermi liquid. Think
about this, you have the lattice which is
positively charged and you have electrons
which are all negatively charged and the coulomb
forces infinite range force because it goes
only as potential goes as 1 over r.
So, they are all actually interacting with
each other; every single electron is interacting
with every single other one. But because somehow
of this big positive charge that is given
by the background and the way fermi statistic
forces them to stack up above each other,
they just flow around as if they do not see
any interaction ok, but their interaction
is then hidden in their effective mass. So,
the write m star, I think it is written like
this which is dE by d k ok; I think it is
this with some h crosses to be included right.
So, you are define it like this and you known
that the dispersion relation energy as a function
of k has this band structure with gaps and
when you reach here; you have very large mass
because the curvature becomes 0. So, as the
curvature of d by dk curve become 0; the mass
becomes infinity. So, here are the fermions
do not move, here they cannot move things
like that, but these are actually quasi particles.
So, what is beautiful is that a free electron
has a charge and a mass and this complicated
system just looks as if it is a collection
of individual entities with charge and mass,
except that the mass has to be redefined which
is why we. So, this these entities we call
quasi particles, mass has to be redefined;
the charge cannot be redefined because of
gauge in variance you cannot have charge changing
in the system. The total charge maybe renormalized
once and for all, but it cannot be a k dependent
difference because then the gauge invariance
operation will fail ok. Gauge invariance requires
a phase in the exponent and the phases has
to be single valued; so it can only go from
0 to 2 pi.
So, all charges have to be compatible which
are integer multiplication. So, that says
the charge of the quasi particle to remain
exactly as the electron charge. But the mass
gets renormalized and then you can continue
to think as if you have some kind of fluid
of particles. But it is just an amazing miracle
and that breaks down when you get to systems
where you know condensation of various kinds
offer ok. So, getting back this procedure
of quantizing quantum system correctly as
and I said correctly means that you have to
have postulate numbers 7 with you; without
that you are not doing your correct job. So,
this thing was thought up Fock and Dirac along
the following lines.
So, the first observation is that observe
that the states many particles states 
have to be symmetric or anti symmetric. So,
I will write the fermion part in the bracket;
that means, that; so suppose I had five. So,
suppose we have a list of quantum numbers;
so suppose I have single particle single particles
states labeled as some set of observable alpha
i; then a general state. So, here you say
alpha 1; i for number 1 then alpha 2’s for
you know for number 2, but this is not correct
because you have to symmetrize it. So, you
really have to do of symmetry operation; sum
over the permutations 
of this 1, 2 3 n. So, the list alpha i specifies
one particular list, the list alpha j is another
list ok.
So, particle 1 could be carrying this and
2 carrying this that you also have to take
the case when 2 is here and 1 is here and
so on. So, you have to sum over all the permutations
what this means is that thus really this is
simply equal to n alpha; alpha i, n alpha
j. So, I messed up by not say; so think of
the i and j is defining different sets of
values right. In other words, the states are
labeled by how many particles there are in
a particular quantum numbers; set of quantum
numbers. You cannot tell number 1 is with
alpha i, but number 2 with alpha j; all you
can do is count.
So, if alpha i reccurs anywhere else, all
you have to do is say while there are 5 of
these that carries this quantum number, 7
of these carry this quantum number. So, the
states are labeled by this such that of course,
the n alpha i plus you know, this adds up
to n that is all you have ok. Now if this
is so, then we have a clever opportunity and
this is where our; this psychological problem
business is explained because we do have the
number operator.
So, when people are worried that I have two
electrons and then coming apart and then I
observe spin here, then that spin gets determined.
It is all to do with the fact that ultimately
there is a quantum number associated with
the number of fermions; it is 2 that state
has 2. And that can be a observed because
they do have a charge and the charge can be
observed at a distance even without disturbing
the system much.
So, you know for sure that there are 2; in
the case of photons we do not have conserve
number. So, there you have to be sure that
you are not reducing or not relating a photon,
but; so, I am saying this because there are
lot of the entanglement experiments are done
with photons, but I think they are able to
control that part. So, the number has to make
sense and the psychological conflict in this
string has to do with the fact that it is
1 state, but the number operator eigenvalue
in that state is 2 ok. So, the thing there
are two different entities; this cross product
where we take one particle states and then
string them to gather to make more is really
a mathematical device ok. If I did not have
this summation, this has no physical meaning
because if it is unsymmetrized then it has
no meaning, it is not one of a physical state.
So, the fact that we can fall back on single
particle states to string to gather a many
particles state has to do with this weak coupling.
And the famous thing that I have left out
of here is QCD; in Quantum Chromo Dynamics,
we have almost massless quarks and we have
exactly massless gluons and there is no way
you can count them individually.
So, they and they are never weakly coupled;
they are weakly coupled only when they are
being scattered at very high momenta or very
high centre of mass energy which means that
their weakly coupledness last only for fleeting
seconds; fleeting movements. For most of the
time they are blobs hadrons which are all
in a strongly coupled state and you cannot
pulled the quarks a part.
So, quarks are prime example of where this
construction will fail. So, the first step
is that due to postulate number 7; all the
states are labeled only by number; number
in each possible slot of eigenvalues that
are available.
But if this is true we observed that such
states are the complete list for such states
is mutually orthogonal orthonormal, we will
say because of they are equal then of course,
its gives 1 and 2 is complete right.
Because they are mutually orthogonal because
if the number in any slot if I take expectation
value of something else the number in any
slot is different, then it will give 0. So,
you have to an exactly same numbers in every
slot then of course, you will get mod square
of that state, but if any numbers differ then
you immediately get 0. So, that is one property,
the second property is that of course, every
possible state of the system can be formed
by linear combination of these. These and
this is all there is if you list all the possible
symmetrized states, then any possible state
can be written as a linear combination of
these; so, this is the complete set ok.
So, because of this fact that they are mutually
orthonormal and complete; you can think and
reverse that therefore, there must we are
tempted to introduce Hermitian operators of
which they are eigen sets right. Because it
is a complete and mutually orthonormal suggest
that there are some Hermitian operators of
which these are eigenvectors. These are simultaneous
eigenvectors; just occurs to me that how much
damage you do to the quantum state in a observation
process depends on the kind of observable.
So, there are things called weak observation;
so if you get in to that language then there
is lot of discussion there right.
So, we can expect existence of corresponding
operators which correspond to each number
operator corresponding to each of these; so
there is eigenvalue is the small n alpha.
And one thing I have to add the to for completeness
we will also need the vacuum state which is
a no particle state. For a single particle
there is nothing like a no particle state,
but once you introduce this collective states;
you also need for completeness a state that
are no particles state. So, we also introduce;
such that all N’s are 0 in it.
So, then we can think of the spectrum to be
fully complete with respect to all of these
because even in because some of the you will
find that many of the ends are 0; let us say
in some state. But then you have to consider
the possibility that all the N's are 0 as
well to make the whole spectrum of states
complete. So, that is the ground state alpha;
in modern times we know that much mischief’s
lies in fact this so called vacuum state.
Although, we think it is no particle it is
not strictly no particle because it keeps
doing something; it has quantum fluctuations
in it. But anyway we are coming to the end
of this construction and we will stop in 5
minutes; which is that now that you have understood
that there are number operators and which
also satisfy they 
are mutually commuting.
So, now we therefore, in reverse since we
are so familiar in quantum mechanics, we are
tempted if I have number operator to introduce
a and a dagger; out of which I can get the
numbers. So, you have to exactly saying this
to then you get 1 and other wise everything;
other things are 0. So, you can always introduce
this and that will reproduce this algebra.
So, this expression in terms of a, a dagger
is a great calculation device now.
So, this gives you now new calculus if some
time somebody ask me you know somebody was
otherwise where you accomplished engineer
and mathematics person. What is new mathematics
of quantum mechanics? Well the new mathematics
of quantum mechanics is that the a, a dagger
as the elementary operation then it allows
you to calculate S matrix elements.
So, you generate the scattering processes
by using these and using that commutators
with the Hamiltonian. So, thus so finally,
we observe 4 or and of course, anti commutators
for fermions right. And note that the observable
also is symmetric in the in terms of particles.
So, the observables have to automatically
be constructed out of a’s and a daggers
only. So, thus every operator; so what does
we need in live? So, every Hermitian unitary
or projection operator needed can be constructed
out of. And therefore, every calculation that
you need to during this theory can be boiled
down to computing commutators and anti commutators
of a’s and a daggers.
So far we so of course, so far the system
was free but we said weakly coupled. So, now,
we can introduce some interaction; so I will
end by writing that a generic interaction.
So, the free Hamiltonian right sum over while
a interaction; what should write V; alpha
k; l beta j m. So, I have some set alphas
and some set of betas which are l in number
and m in number and I introduce; it just comes
how many distraction and how many creation
operators. So, we write that many here ok.
So, a; so you can write out some expression
like this where l number of particles coming
and m number of particles go out. And we have
to characterize the V correctly so that we
get meaningful free theory; the V should effectively
give a localized. So, V can localized or properties
that the V has to basically ensure what that
means, takes is the content of all quantum
field theory; how to construct those is what
the quantum field theories about.
So, we will stop here; so incidentally if
you want to read up this part this last this;
Fock Dirac construction is given in Mursbakar’s
text book. Mursbakar’s books, with some
variation in notation is the same physics
the same things he talks about and very nicely
moves on to the so called many body theory.
But we will break from here onwards, we will
go on to the field notation; of course, as
you know fields can be recovered now by doing
Fourier transform from the alpha space to
a coordinate space. So, generically the alpha
space is simply moment of k; the normal way
of labeling free particles is their momentum
in spin.
So, if your Fourier transform from momentum
to coordinate space; you get the space time
fields. Mostly most books almost all the books
tried to tell you that the field is some entity
given by God and that we shall try to quantize
it. And they say look do not be afraid of
a field because you have seen coupled oscillators
and then, but this is actually the construction.
So, after they do it; they automatically get
Bose and Fermi systems because they put commutators
and anti commutators.
But why you did it has to do with this underlined
postulate 7. So, this is the real construction
of a weakly coupled field theory, but to lot
of people it seems that if you start directly
with the field, then you are doing some more
advanced and may be you have you are accessing
a non weakly coupled theory as well by making
those postulates, but I do not think so.
