So, consider vector V, good old physicist
notation, which is in the carrier space of
SO 3. Associate a matrix and now I put this
what is this called it is called blackboard
font in tex. So, you double this line. So,
V equal to V dot tau, where tau are the sigma
matrix or pauli matrices. And let me write
over here a little note about the notation
tau i and sigma i will both be used for 
the same set of matrices the Pauli matrices.
Sigma if it has to signify spin spin and Physics,
tau in group theory. At least that is the
convention in my mind and you will find that
that is roughly true in many books. So, it
is just there the same matrices, it is just
that we call them tau 1, tau 2, tau 3. So,
we identify we define 2 by 2 this is of course,
2 by 2 complex matrices, V made up of this.
So, if you want we can write this out in detail
what it will become is you know. So, V 1 times
tau 1 will basically put the V 1 here then
V 2 times tau 2. So, minus i V 2 and plus
i V 2 and V 3 times tau 3 so, V 3 minus V
3. So, this is what this matrix looks like.
So, out of our 3d vector we create this 2
by 2 complex matrix like this. What is interesting
about this is that we can define a 
we can define a inner product 
V comma W equal to trace of V times W ok.
And, we will see that this reproduces V dot
W, because. So, how does it work?
Remember that, trace of tau i tau j was equal
to 2 times delta i j right, because the tau
matrices themselves, are traceless the sigma
matrices are the Pauli matrices are just traceless.
And, if you take a product then you remember
that they actually satisfy tau i tau j equal
to epsilon i j k times tau k, they basically
give the third Pauli matrix if you take tau
1 and tau 2 it gives tau 3 and so on. If you
take trace of that you will again get 0.
So, only if the 2 taus are same delta i j,
then it becomes identity matrix tau squared
is so, this has this depends on while trace
tau i equal to 0 and trace and tau i tau j
is tau k if i not equal to j right. So, since
if product of 2 taus gives you a third tau
and if trace of all taus is 0, then this is
always going to produce 0 unless i and j coincide
and we get identity in which case we get 2.
And therefore, if we take this trace of trace
V W equal to trace of so, now, we will take
out V i W k trace of we get tau i tau k right,
because V is just sum also we could have written
here sum over i V i tau i that is what it
is. So, V i are numbers and tracing has to
be done over these matrices ok.
So, we can pull out the numbers and do trace
over this and this trace is equal to 2 times
delta i j. So, except for a factor 2 we got
the inner product back k. So, I have to supply
a factor 2 or you can define it as half.
So, we do not have to worry. So, we checked
that this is equal to 2, trace this is of
course, twice so, you define the inner product
to be half of that. So, that it takes exactly
the same inner product right. So, what is
going on from the 3D language, 3 dimensional
rows and columns we switch to some 2 by 2
complex matrices, but we are recovering the
same geometrical or mathematical properties
of this V and W. In fact, we can now check
that these act as SU 2 representations.
So, if u belongs to SU 2, so, compare O V
equal to O acting on V 1 V 2 V 3 for SO 3
now the same where O is belonging O belonging
to this sorry O belonging to SO 3. So, the
same real numbers V 1, V 2, V 3, which appeared
as column vectors and with one helping of
O, are now transforming here by 2 helpings
of u. If you have an SU 2 element then there
is a adjoint action or Simi an action like
a similarity transformation.
So, u acts through similarity transformation,
what is called “adjoint action” whereas,
O acts through left action left multiplication.
So, on the same carrier space we can have
we can have both kinds of actions either the
left action by the real matrices O or the
adjoint action which looks like a similarity
transformation by the SU 2 matrices. And,
which act that that action SU 2 actions preserves
this trace operation.
Check that the inner product is preserved
by the SU 2 action, because we need to check
that trace of V u V W I am sorry trace V W
well is going to be invariant, it is obvious
you do not have to compute anything right,
because you insert these in the trace firstly,
this u inverse u is going to multiply in the
middle and then under trace you can always
cyclically change the order of the matrices.
So, it will become. So, it preserves the inner
product. In other words it preserves the metric
meant for the 3 dimensional vectors, you know
in the in metric language 
metric language this V dot W was actually
V transpose the delta matrix of delta metric
of Pythagorean metric.
So, it basically preserved this delta and
we see that here SU 2 is preserving the same
delta. So, actually the two groups are have
a very close correspondence, but it is not
isomorphism.
“Double valuedness” of SU 2. So, this
double is of course, with respect to SO 3.
And this has been a source of great mystery
and mysticism even for very important physicists
in quantum mechanics, because the tau matrices
or sigma matrices were first invented for
spin. And this double valuedness remained
a big mystery for everyone from Pauli to later
Wheeler and Roger Penrose ok. So, from Pauli
to Penrose they have all been amused by this
and Penrose of course, has a whole programme
of quantising gravity using the representation
similar to what I said, but for the Lorentz
group not rotation group ok.
So, what is this double valuedness? The point
is that SU 2, u n cap theta was equal to cos
theta by 2 times identity, note lot of books
do not write this identity, but it is important
to remember 
it is there ok. This is how the group works
out to be.
Now, let us look at the range over which this
theta goes ok? So, we can see that when theta
is equal to pi by 2 sorry 2 pi what am I saying.
So, you when you go to theta equal to 2 pi
you get cos pi. So, you get a minus 1 and
sin of pi is 0. So, at 2 pi you do not return
to identity, while the O elements let us say
O generated by theta along z axis. So, what
was our notation that R R z of 2 pi, which
would be exponential of 2 pi times l Z right,
but what is this exponential was just cos
theta sin theta etcetera. So, it was just
cos 2 pi cos 2 pi minus sin 2 pi and sin 2
pi. So, that is just equal to identity. 3
there is a third 0 0 1 ok.
So, it was just equal to 3 by 3 identity.
So, R z 2 pi or R x, R y anything you want,
it just basically became cosine on the diagonal
and with the whichever axis that was not being
turned this one just migrates here if you
change to x or y migrates up the diagonal,
but the other elements are just cos and sin
and exactly at 2 pi the you get back to cosines
becoming 1 and the sins becoming 0 with no
overall sign appearing, but here at 2 pi these
2 by 2 complex matrices of SU 2 do not return
to identity, but only to minus 1 ok. And so,
what we find is that there is a 2 valuedness,
there are 2 SU 2 matrices and of course, we
know that u n cap of 4 pi would be equal to
1, if you put 4 pi then you will get back
identity.
So, there are 2 SU 2 matrices for every O
matrix. And, that is because of the adjoint
action u V u inverse, which is how you transform
any V is same as minus u V minus u inverse.
So, if u carries out the required rotation.
So, does minus u and minus u is actually a
distinct matrix it is not just u itself. So,
u and minus u are both doing to vectors V
the same thing that the same thing and for
which there will be only 1 corresponding rotation
matrix ordinary rotation matrix.
So, there is a 2 valuedness and that was what
was observed here both the 2 pi value and
4 pi value will look like identity operation
as far as the V matrices are concerned, whether
you put u n cap of 2 pi which would give minus
1 and a minus 1. So, it will look it will
look like ordinary rotation of a 3d vector
and so, will 4 pi.
However, the representation which is the complex
2D vector,
The 2 dimensional complex carrier space 
does know the difference between u and minus
u, in quantum mechanics we simply call this
spinners we call these spin matrices or spin
wave functions times psi, where psi and xi
are 2 dimensional complex vectors 
and phi is space part well you want to put
t you put t space time part.
So, we split the wave function of the electron.
So, this is of course, non-relative all non-relativistic,
but you can split the wave function into a
spin part and a space part and you will find
this language in lot of the nuclear physics
and condense matter physics literature. And,
now we come to explaining how young Pauli
was maybe 23 years older something like that
when he invented the Pauli Matrices. It is
very simple. If you 
we know that there are spin up and spin down,
this was what is what has come to be called
stern gerlach I should not say what has come
to be called, but I think historically it
was the experiment that was performed by stern
gerlach stern and gerlach, but that was not
origin of Pauli’s considerations he had
other thing.
But, I am just saying what I meant by saying?
What is come to be called is what we see what
can see most clearly through stern gerlach
experiment? Is that, if you send a mixed beam
of electrons through an arrangement of magnets
North and South Pole, then this beam will
split exactly into 2 parts. You have to repair
an arbitrary, you just have you are heating
some silver or something like that.
So, you are just some vapour of silver coming
out it is all mixed up, but now if you apply
this magnetic field classically you would
have expected that, it will range over you
will see on the screen you should see a continuous
set of possibilities; this spin is here, spin
is here, spin is here, but quantum mechanics
just mix out one and the other. So, this magnetic
field is inhomogeneous.
So, that it actually can separate out the
dipole moments as you know, right homogeneous
field would not do much, but in homogeneous
field we will separate out the dipole moments,
but when that happens classically you expect
the spin to be anything. So, it will pick
out any projection that is perpendicular to
these magnetic fields. So, it will get attached
to it, but quantum mechanically you either
find this or you find this and nothing else,
it is impossible to get anything in between.
So, this is the classic experiment that tells
you that regardless of what kind of initial
state you start with, if you make a measurement
you only find the eigen values. The measurements
do not return continuous values of the observable
in any one observation. The way you recover
the sort of port average is if you do a lot
of experiments, then of course, the weighted
average of the these 2 together will equal
the weighted average here the average spin
here, but as the answer will come only as
an average, but any 1 process of 1 observation
if event will only return either spin up or
spin down.
So, we know that there are spin up and spin
down states so, what would we do logically
and if we believed now the next point.
The very very important point of quantum mechanics
linear superposition principle. So, generically
ok. So, the very crucial thing is because
there is linear superposition you write vectors
vectors are linear spaces. So, if this state
was supposed to be obeying linear superposition,
then it should be expressible as linear superposition
of a basis and that basis if it is 2 dimensional
would be just 1 0 and 0 1 ok.
I just I do not want to launch here into a
big lecture, but I just want to tell you very
briefly, that linear superposition principle
is the main positive content of quantum mechanics.
Unfortunately the uncertainty principle is
emphasised, there is nothing uncertain about
quantum mechanics, it is pretty certain, it
is commutation relations will give you the
Heisenberg uncertainty relations and it is
it is linear superposition principle, which
is very wrongly sold as uncertainty principle.
It is true that your classical expectations
are not born out you will not be able to measure
position very accurately or momentum very
accurately, but to couch this as a principle
is wrong, the principle is linear superposition,
which allows you to have Fourier transform
Fourier series representations.
And this delta p delta x relations can be
derived in any Fourier transform theory, the
width of the function in x-space will be inverse
of the width of the function in the complementary
space. So, that is just a result of Fourier
transform and Fourier transform works, whenever
there is linear superposition, whenever you
can ex obtain functions as linear superposition’s
of basis functions like sine and cosine like
done in Fourier analysis.
So, it has all to do will linear superposition
and nothing else ok. So, likewise the so,
called wave particle duality is just an oversold
point they are just particles you can, the
wave people are confused, because if you measure
a momentum eigen state then it it is characterized
by it is wave number.
So, essentially when people say wave what
they actually mean is a momentum eigen state
and a momentum eigen state I may as well write
this down. So, these are the 2 big misnomers
of quantum mechanics and sold to public and
public has brought it, you will find innumerable
philosophical discussions going on about this,
it is all garbage.
So, the “Wave particle duality” basically
is position description or momentum description
and we do know that the there is this complimentarity,
you cannot have both. When you have momentum
P what am I writing momentum P, you can associate
with it. So, you can associate with it a wave
number k, which is equal to p over h cross,
this is because you had the fundamental constant
h cross available ok. So, to momentum p you
associate a number of dimension 1 over length
and which you define in your wisdom to be
equal to 2 phi by lambda ok.
So, this lambda is a product of your fertile
imagination, there is no such lambda. The
main point is that there is a fundamental
number h cross which allows you to think of
momentum as a length scale ok. And once you
introduce a length scale you say oh my god,
but particle either here or there you know
it is a wavelength like this there is no wavelength
it is a momentum Eigen state either you have
a momentum description or. In fact, it neither
of them completely precise, because of the
earlier part the linear superposition, but
the main principle of quantum mechanics is
that there is linear superposition and in
any measurement you can observe only 1 Eigen
value, you cannot observe all pos mixture
of all Eigen value it is just that x is a
continuous space.
So, the set of Eigen value is continuous.
So, you can come out with any 1 of them with
some weightage. If and so, that settles this
issue of what is the wave there is no wave
the wave is essentially a brain wave ok. And
that is because it is possible to associate
a wavelength lambda with a momentum Eigen
state. So, that the real truths of quantum
mechanics is momentum Eigen states.
Now, coming back to this because of linear
superposition, you can write spin as a linear
superposition of up or down vectors and in
fact, those are the observed Eigen states.
And so, you what you observe of course, is
the magnetic moment physically, but that it
is up to a mu multiplying this spin vector
it is the same thing. So, now, what does 1
see, we can always go from up to down states
right, physically up can go to down by application
of a magnetic field, if you apply a magnetic
field you can flip the spin.
How will you represent this in quantum mechanics?
How will you represent it mathematically well
it is a linear space? So, there should be
a linear operator ok. So, which linear operator
will do these implements this? So, I am very
sorry to say, but this business of wave particle
duality all the books, that claim themselves
to be modern quantum mechanics they miss it
completely.
They write fat books and they are very popular,
because they reinforce what people like to
hear you know miracles are liked by people.
If you tell them the truth they do not like
it because it is a little too simple to understand,
but if you tell them something is very difficult
to understand and in fact, tell them that
it is something never understandable then
people are drawn to it like honey you know
honey bees to honey.
So, most people reinforce this idea that quantum
mechanics is not understandable. So, then
people rush to them. If you tell them that
by the way what I am telling is nothing new
it is what is written by Dirac in his 1929
book, you have to read Dirac’s book, please
do not read any other. So, there are only
2 books to read in quantum mechanics, Dirac
and Schiff of course, Landau and Lifshitz
is always there volume 3.
So, these are the books to read in quantum
mechanics and they will tell you unadorned
truths, which are Dirac’s is the best and
written in 1929 when people were still puzzling
over the interpretations of quantum mechanics.
So, essentially that is what, but nobody learns
from Dirac. And the somewhere I tried to tell
someone and they said do you really believe
Dirac, I felt like telling really believe
all the else anyway that is how it is?.
So, how do we get from down to up or up to
down we apply a lowering operator, equal to
something acting on 1 0 and; obviously, this
can be done by this row into this column.
So, I have to have 1 right. So, this into
this 0 and this into this will give 1. So,
this produces this, similarly of course, you
can do the reverse by putting a 0 1 1 0 0
on the down. So, you can convert down to up
and up to down. And the 2 states are Eigen
states of can be distinguished 
this matrix 1 0 0 minus 1, which on 1 0 will
produce plus 1 0 and will produce minus 0
1.
So, if you apply this matrix on this or on
this, you can distinguish the up spin and
down spin as Eigen states of this matrix 1
minus 1. So, we have a fundamental set of
3 matrices 0 1 sorry 0 0 1 0 0 1 0 0 and 1
minus 1 0 0. Now with good mathematical sense
you will say why do not I make some symmetric
matrices out of this. So, equivalently what
happens if I symmetrize I get 0 1 1 0 which
is sum of these 2 divided by 2, but the minus
1 if I make I will get 0 say minus 1 1 0,
but I also wanted to be a hermitian matrix.
So, I make it minus i i 0 it is simply i times
the difference and then there is 0 1 0 minus
1 ok. So, these are the equivalent hermitian
set 
and what is nice about this is that these
3 hermitian matrices exhaust the basis for
unitary matrices in 2 dimensions.
It, is quite a remarkable thing, because we
had this a a star and b and minus b star this
is how we could characterise a unitary matrix.
And, then that we wrote out as equal to a
r plus i a imaginary part b r plus i b imaginary
part minus b minus i b imaginary part sorry
with plus sign now. And a r minus i a i. So,
it just became equal to a r times identity
and plus this a i times i a i times tau 3
plus i b i times tau 2 b r times tau 2 and
plus i b i times tau 1.
So, we have 3 hermitian matrices, which help
us to understand the quantum mechanics of
spins which obeys linear superposition principle
and has this property that only Eigen values
are returned in as a result of measurements.
And amusingly the 3 hermitian matrices are
the basis for the corresponding unitary matrices
this is an accident of 2 dimensions ok.
But, a very significant 1 because helps us
to do lot of things. So, we have seen the
origins of both the types of representations.
In last 5 minutes maybe I tried to therefore,
explain to the topology of SU 2 and SU 3 which
I find the most most imaginative exercise
usually not done in physics books, but done
in Schiff’s book.
Topology of SO 3 verses SU 2 the point is
that so, we will try to draw these 2 spheres
and this will be SO 3, and although I promised
you that topology has no sense of distance
in it we will draw double the size here. So,
we want to represent something like R n cap
theta is exponent theta times theta dot L
ok. So, for simplicity just take this so,
suppose you drew axis corresponding to L x.
So, these are not real space axis L y and
L z then the n cap vector.
So, this theta is often written as n cap times
theta, n cap is a vector in this space it
has composed. So, it has dotted with l right.
So, you can dot it with the L x, L y, L z.
So, this is n cap and the rotation amount
theta you can show by drawing along it how
much you rotate. So, theta values, but they
stop at theta equal to pi because if you do
a if you do a n cap rota pi rotation and minus
n cap pi rotation you have actually covered
2 pi rotation about that n cap axis.
So, by convention so, you can take theta equal
to so, theta belongs to let us say 0 to 2
pi equivalently belongs to minus pi to pi
ok. And so, that is what we do we make a ray
a ball of radius pi. Now, to make SU 2 we
draw the sigma x, sigma y, sigma z axis for
the generators of that. So, Schiff has the
discussion in his very elegant language, it
is so compact that you will mess it if you
have flipping through the pages and has no
diagram.
So, put n cap, here we have u n cap theta
as the exponent of i times, theta times, sigma
by 2 matrices dotted with n cap. Except that
now theta goes from 0 to 4 pi. So, we let
theta belong to minus 2 pi to 2 pi and actually
it does not matter. So, we just put minus
2 pi to plus 2 pi. So, including the outer
surface of this sphere ok, the big difference
is that in SO 3, we have to exclude the minus
pi, because it will reproduce the same thing
as plus pi. I have rotated by pi around this
way if I rotate it minus pi I would have come
to the same point. So, to not to repeat topologically
have a unique set of points I have to leave
out this, but as far as SU 2 is concerned
both minus 2 pi and plus 2 pi give 1 just
gives the same element.
So, there is no ambiguity. In fact, the SU
2 is a space such that the whole of I do not
want to clutter up this image, but the whole
of the outer surface is one point. So, you
have to think of so, if you think in terms
of forget about n cap because it takes various
values, but you let theta increase you get
a sequence of 2 spheres, until you reach the
outer most 2 sphere, which is corresponding
to 2 pi, but that whole sphere is actually
just 1 point. Topologically is the same thing
the connectivity is such that all of these
points are actually just 1 point, they are
not distinct point. It is just limitation
of our visualization in 3D. In fact, drawing
it on 2D, that we think of this thing as distinct
points, but mathematically they become just
1 point in the group sense they are 1 and
the same object.
They are same element of the group it is identity
element minus 1. Thank you, identities at
the origin is minus the identity. Is topologically
the same point minus the identity on the SO
3 space the antipodal points will be corresponding
points and you have to leave out 1 of them.
So, the punch line, topologically this object
is S 3, it has the sphere of 3 dimensions,
which we cannot normally visualize within
3 dimensional space.
And the analogy is the following if you had
a disk you draw on it circles of growing size
until you reach the outer most circumference,
but you identify circumference to be 1 point,
then what will you get it is like travelling
from south pole of a 2 sphere ball to north
pole you have made the outer you folded it
up and it is like some of the sweets made
you take this and also make it into a same
point. So, topologically you have created
a 2 dimensional shell out of a 2 dimensional
disk. This thing that we did is a sequence
of 2 spheres such that the outer most 2 sphere
is joint to be 1 point. So, it is a 3 dimensional
shell in 4 dimensional real space ok.
So, it is topologically 
what is called S 3. So, S 1 is circle, S 2
is sphere spherical shell in 2 dimension and
S 3 is spherical shell of dimension 3, it’s
intrinsic dimensionality is 3 just as the
intrinsic dimensionality of a shell is 2.
So, 2 sphere although it occupies 3 dimensions
it is internsic ant walking on it only detects
2 dimensions. So, an ant walking on this will
detect 3 dimensions, but it will be a continuous
and homogeneous space, you will not find any
joint anywhere. So, topologically the S 2
is actually in S 3 whereas, topologically
SO 3 is not and it is a slightly more complicated
space, but what happens because of the double
covering is that SU 2 double covers this.
So, there are 2 copies of SO 3 inside SU 2
and after that covering it becomes a simply
connected space, SO 3 is not simply connected,
but SU 2 is.
So, we will see that next time.
