So, let me write down the sequence of things
we have been doing and then you can see.
So, may so far is that we defined the quantum
mechanics through path integral 
which actually meant through transition amplitudes.
But, we basically checked that this gives
correct kernel 
for the free particle. Now, from this point
on we introduce this method of external current
or forcing function 
and this is a method, there is no real forcing
function.
So, this is method of forcing function which
will later become 
external current. This is Schwinger’s way
of thinking of it. We also made a transition
from transition amplitude to vacuum to vacuum
amplitude ok, again a concept essentially
due to Schwinger. Vacuum to vacuum amplitude
is apparently a fake think because why would
you what would you learn from going from vacuum
to vacuum it is like being back to square
one.
But, the point is that in fact, the vacuum
to vacuum amplitude is done in the presence
of the forcing function. So, you get a vacuum
to vacuum amplitude in the presence as a function
of this auxiliary variable and then by varying
this variable you can obtain all the information
back ok. So, but this is as a function of
functional of the forcing function 
and that is the key thing that you obtain
the vacuum to vacuum amplitude in terms of
the forcing function.
And then this force W of J becomes the generating
function generating functional of n point
functions or Green functions. So, that is
what we have done so far what we. So, we can
say what happens is that we do omega plus
infinity omega minus infinity to be equal
to integral over d q f d q i of omega plus
so, omega infinity q f t f times the more
physically transparent transition amplitude
right. This is obvious getting a and in the
presence of a current j with this what did
we call it F over there F.
So, vacuum to vacuum amplitude in the presence
of F basically takes this form and then this
becomes we saw in the limit that we take t
and this here we take t f t i to also infinity;
infinity and minus infinity, we basically
recover plus i the basic path integral is
with i integral minus infinity to infinity
dt of what are the action is we have been
writing in terms of q q dot t, but plus i
times integral q t j of t F of t. Now, at
this point itself we can observe that if we
define correlation functions as 
not phi I am sorry q t 2 q t 1 as equal to
1 over i d by d F t 2 1 over i d by d F of
t 1 of that transition amplitude 
and evaluated at F equal to 0.
This is all we did so far effectively except
for the calculation method of doing quadratic
integrals to recast the path integral in various
ways and of course, we will be using it again
and again. We also use the stationary phase
method to check that the path integral has
to become stationary on the classical path.
So, but other than that this is all there
is and if you have n points then you put n
of those and then you can recover the answer.
There is an illusion among people that path
integral is a good thing to do quantum mechanics,
this is completely wrong. The main use of
path integral is only a few after you make
transition to quantum field theory and then
to derive relations between Green’s functions.
So, QFT again has been used primarily as an
S-matrix theory. The only thing we do so,
we tell everybody to get them excited that
we are calculating n point function, but what
we really having mind is calculating scattering
of n particles.
So, all we do is there is a very formal procedure
which then conversion n point function into
the n point S-matrix n particle S-matrix.
So, we calculate those transition amplitudes
the S-matrix and not necessarily energy stationary
states. So, in fact, QFT fails completely,
I should not say completely, but QFT has not
proved to be very useful to compute any bound
states nor this path integral very useful.
I know that there is there are several textbooks
entire text books written on how path integral
is very useful in quantum mechanics.
Well, you can read them for their own value
whatever they have, but I have never read
them and I can vouch that no chemist will
need them to calculate the many electron bound
states. The chemist do use however the Green’s
function ideas because they want. So, Green’s
function ideas make it a little bit more formal
and peg it on a slightly different level,
but path integral is not going to be. So,
the only computation you can really do with
path integral is a Gaussian integral and later
we will see that it helps you to derive the
so called diagrams, so called wick contraction
at two-point function at a time.
But, there is a the kind of power that you
have in a partial differential equation which
allows you to solve I mean hope for getting
exact solutions for many different potentials
that does not exist in quantum field theory.
And in quantum field theory so, primary use
of this method has been to calculate S-matrix
elements. If you want to calculate. So, I
can even write it down here, how should I
say becomes most useful functional.
So, just Bethe-Salpeter functional equation
has been studied by lot of people and lot
of work exist, but I do not think I mean we
were never taught that it calculates baryon
and if it was then we would be not doing lattice
gauge theory. So, that is the status, but
we can also say the other use of the functional
formalism is in fact, to implement this theory
on the lattice. You can implement quantum
theory on the lattice in this functional formalism.
So, for lattice gauge theory also it is a
useful thing, but lattice gauge theory is
an just a completely numerical calculation.
It is Monte-Carlo calculation of that functional
integral because there is no the only approximation
schemes of the only exact calculation schemes
available is the Gaussian integral of the
so called steepest decent method in some approximation
you can, it is like the stationary phase,
we will see it. So, there are very few tricks
available at the functional level that allow
you any kind of coat answer. But, this trick
does allow you to obtain functional relationships,
the trick of partition full is trick of your
partition function thermo, in statistical
mechanics also when use a something similar.
So, that is really the all there is to it
ok, but it is extremely powerful for the purpose
for which we are going to do it. The conversion
from Green’s function to S-matrix is itself
quite a formal statement, but once you get
over it you get used to the idea. It is not
all that difficult. Hopefully, I will be able
to do it if I have the time.
There is a 2 I think. So, we have to interpret
this as product over all E where you have
to ordered the E, the energy spectrum because
path integrals are always ordered and that
was one thing I was going to comment here
I forgot where we wrote this Green’s function,
we actually end up calculating only the time
ordered product.
So, it is automatic in path integral that
you will get the functional method that you
will get the tang ordered product. So, this
is the W 0 and the W which is not of much
interest anymore and the other part in detail
is, well let me see if I can say without.
The only thing is the sign in D and it is
not square root because I guess of t 1 minus
t 2 there is an minus sign right the square
root is in the Fourier transform definition,
but here it is not there and E square minus
omega squared plus i epsilon right. So, everyone
knows all this and you know that this boils
down to theta.
So, where should we write i e will be equal
to omega. So, i omega t and plus theta of
t 2 minus t 1 times e raise to plus pi omega
t. Think there is a 1 over 2 i omega from
the poles right. So, because it is square
you will be taking E plus omega minus omega
and each one is a pole. So, from each of the
poles you get each of the pole irrelevant
depending on whether this is greater that
is greater than 0 and the value of the pole
is that the 2 pi goes in the integration in
the contour integral.
Thinking and it was a great discovery when
Feynman use this propagator that positive
frequency particles go forward in time. So,
omega is a positive number it is positive
square root of the omega squared. So, it gets
a minus sign which is the correct time evolution
according to Schrodinger convention of setting
energy operator to be equal to plus i d by
dt. So, with the minus i it gives correct
omega. So, this is going forward in time,
but this would give negative energy or would
go backward in time and that is the interpretation.
If this has not being told you before I might
as well spend a little time here telling you
about is going forward and backward because
this is at the heart of causality in quantum
field theory.
So, is the or it should be there in. So, this
is very general argument and the way Weinberg
puts it particularly is that. So, look at
the uncertainty principles. Uncertainty principle
there would be four statements, but look at
say d e by d d e dt uncertainty principle.
If you try to, but in relativity it is not
delta t and delta x that really matter because
one persons t is another persons mixture of
t and x and actually Weinberg writes so called
uncertainty principle which is written like
this.
So, it basically says that the space time
interval between two observations has to remain
greater than the Compton wavelength ok. So,
you can think of this you can put the m on
this side and treat it as if it is well not
really, but this is the form in which it is
written in his book and you can see that what
this is saying is that you can have single
particle interpretation only provided you
do not probe the object in space time intervals
that has smaller than the Compton wavelength
ok.
So, delta E delta p no. So, we I mean to say
delta t square minus delta x square here.
But, now in quantum mechanics you could always
probe the system more closely and then you
will lose the single particle interpretation
as you are been taught probably in most relativistic
quantum mechanics courses emphasize that you
will create particles. If you probe at this
land scales shorter than this then the value
of delta E and delta E will have to be larger
than the mass scale of the particle and you
will end up creating more particles and the
single particle interpretation will be lost.
But, we also have a more specific; more specific
statement. Suppose that I have creation of
a particle. So, now, we draw this space time
diagram and the light cone, normally if you
create a particle here it will be later found
here right. so, this is t 1 and t 2. So, it
will propagate from this to this, but quantum
mechanics only tells you some inequalities.
It does not say delta x cannot be less than
something or delta t all that you have to
is do is maintain this, but if this happens
you could also have a situation where t 1
is here and t 2 is here ok. So, t 1 prime
I do not want. So, I will remove the old t
1, t 2. So, suppose it is t 1 and t 2 like
this.
This may not be forbidden by this relation
because all have to do is adjust that the
delta t square is bigger than minus delta
x square remains bigger than this m square
and well, actually it will be negative in
that case. But, because I do not have control
in quantum mechanics it may very well happened
that I create a particle here, but destroy
over there. This is the also say Bell inequality
thing you do something here and it gets also
it determines something over there.
So, the point is that we recover the causality
correctly in this case because it is possible
for you to here the events are space like
separated. So, it is always possible to at
least reverse the time ok. For space like
separated events it is possible to re-orient.
This is a little let us see how does one recover
I would have to really tilt it a lot until
the projection onto that access reverses the
directions of t 1 and t 2 right. If it is
a space like separated interval, then I can
and if I choose my new axis to be like this
then now I have to do a projection parallel
to this axis. So, this will go, so, I have
to make the slope of this bigger than this,
so, sorry.
So, I do this and this. So, this is t 1 and
this is t 2 right, clear. t 1, t 2, but now
and this as a slope like this. So, I do draw
a new choice of axis which is highly relativistic.
So, it is approaching the light cone, very
thin close to light cone. If I now project
these, but drawing lines parallel to this
they will go and hit the time axis in the
reverse order intersected it already right.
So, that correct and this one from here to
draw like this until we hit this axis, already
there. So, if you will do this carefully in
your notebook, you will see that the projections
which are this is the new x prime axis, the
projections projection lines parallel to x
prime axis which go away and hit t prime axis,
the t 1 and t 2 are reversed and these well
known result and you can find the algebraic
expression for the Lorentz boost required
for this to happen.
So, now there is a question of causality that
you emit a particle here, but absorb it here
which is at a later time in this frame of
reference. In the other frame of reference
it look as if it got emitted first and got
absorbed later. This problem is solved by
quantum field theory because of this, because
in the other frame of reference it will look
like an opposite charged particle when backward
in time that is what the interpretation is.
So, for a charged system Q plus Q created
at t 1 and absorbed at t 2 is equivalent to
minus Q created at t 2 and absorbed at t 1.
So, this amounts to the t 1 location reducing
its charge in both the things. What is same
is thus both the charge of 1 is reduced and
that of 2 is increased. In the person who
is observing frame of reference as the clock
is ticking he considers time sequences going
forward in time strictly and in if the two
events due to the this uncertainty in some
this is what I meant to suggest.
In some space time region well actually it
will not be a circle, but some kind of hyperbola
in. When so long as this is of the order of
m anything can happen here in particular particle
can get produced and annihilated at space
like separated points. And if that happens
with a particular time sequence and if it
is space like separated in another one it
will look like creation event is happening
after the destruction event, well that is
not true because it will be in his frame in
the other observer’s frame of reference
it will be interpreted as minus Q sequentially
got created at t 2 prim,e which in this frame
of reference occurs before, t 1 prime.
We will find that if you compute the for a
massive field if. So, for massless fields
we will find that the support is only on the
light cone. So, if you are sitting at you
start with the origin as the reference the
Green’s function the Feynman propagator
will have support only on the light cone only
on these points out side of that it vanishes.
But, if you do it for a massive particle then
you find some slightly different function
which has of course, higher support here,
but it also has a little tail outside. It
is not strictly on the light cone. It is a
exponentially dying tell just like in a barrier
problem; barrier problem the wave function
penetrates under the barrier. So, in quantum
field theory the two-point Green’s function
will actually protrude into the classical
relativity forbidden zone, but exactly of
the order of the Compton wavelength and not
more. And if things happen in that region
so, if you see creation destruction events
in that region they will be resolved by this
explanation.
