You will complain that we have lost an intuitive
feeling about what is going on and what our
next goal is to actually derive a different
quantity which connects with classical ideas.
We want to obtain something which we call
capital gamma of phi classical of x.
Now, what we want to say is something like
this, we want that. So, what do we mean by
such an action? So, recall our dominance by
classical paths. So, I need not have written
this want.
So, recall dominance by classical paths where
we found that the transition amplitude is
e raise to i times, the action on the classical
path right and time some other stuff. In the
classical limit this is what we found, if
somehow the action was said that it was dominated
by mass of a tennis ball or whatever then,
its action would just come out and the answer
would be just proportional to this.
So, we want here to define expect that when
I do all this integration 
I should come out with 
and overall just e rise. So, there is integration
here, but it gives this answer this is also
how some of the thermodynamic potentials are
defined Gibbs free energy, you have to average
over dp, you do the phase space integral of
the partition function, but then the left
hand side gives the free energy in the exponent
e to the minus Gibbs free energy is equal
to integral dx dp of e raise to, so beta of
course, e raised to minus beta H ok. Well
what remains is a its dependence on volume
something this is all that remains.
So, in a similar way, so the point is you
integrated out all the detailed degrees of
freedom and then you come out with a potential
which when you minimize you see I am sure
this is completely wrong, but I should have
brushed up before coming, but you can. So,
what do we have for what are the relations
p equal to minus d F by d V something like
this right and I think that is correct.
So, you want a potential what I mean is a
classical potential of some kind which when
you vary with respect to its argument you
get your desired quantities out of it. So,
after you have integrated out all the details
the final answer depends only on the extensive
variables, the volume is an extensive variable
in thermodynamics, pressure is a local variable,
intensive variable.
So, you can vary that potential with respectively
extensive variable and get a intensive variable
as the answer. Similarly here we expect that
after all the complicated stuff here is over
I come out with a clean and neat functional
which mimics the classical behavior ok. Except
for the problem that this is also a functional
of phi, so what is this phi and what is that
phi because this phi is all being integrated
over ok.
So, I have to somehow find a remainder extensive
variable which remains after all the integration
here is carried out. This is done by a clever
method called Legendre transform and beautifully
enough there is a relationship between the
extensive variable we expect classically and
the variables we get from this W of J.
So, the first observation is that the Green’s
function is a twice variation of yeah the
generating function. So, I am just trying
to announce to you with the result, we somehow
want an object gamma which is going to be
the quantum version or the quantum corrected
version of S, the good old action of classical
mechanics, but now it is function of some
slightly different argument.
So, if this was a naive field, this will be
dressed up quantum field in some sense, but
it will have classical values and the gamma
will be the effective action for that system
this is the idea. So, then you can see the
whole system as evolving as if it’s a classical
system in terms of some argument called phi
classical. So, the question is how do you
link the two?
So, as I said earlier it is like the potentials
one difference in thermodynamics. It is not
the naive classical field that you would have
put inside the ordinary action, it is dressed
up by quantum mechanics, it will be a classical
variable; it will be classical variable, but
not what you would have expected naively classically,
it encapsulated it the quantum effects and
the function gamma encapsulates all the quantum
effects.
So, that now you can evolve this quantum system
as if it is classical, you can study the quantum
system as if it is classical by just varying,
studying Euler-Lagrange equation.
So, Euler-Lagrange equation says that on the
classical trajectory variation of the action
is 0, so that is what will happen if you do
this. So, this is a tall claim nobody actually
achieves this, but the point is you can formally
define such a object and you can get reasonably
good estimates of it in perturbation theory
for all the leading terms.
So, finally, we will find that in principle
gamma is a functional of a field which has
encoded all the quantum effects, actually
it is only a variable, so I do not know where
we should not say that phi encodes anything
is just an argument of the functional. So,
is a functional which 
encodes all the quantum effect in terms of
a function phi classical we keep calling it
classical simply because we are going to treat
it as classical, such that the quantum system
can be studied as above.
So, we will find that gamma will be. So, to
give an example 
start with S of phi equal to integral d 4
x d mu phi d mu phi minus sum u of phi say
minus say lambda m square phi square minus
lambda phi to the 4 let us say let us suppose
we stop there. The S the gamma will come out
it will be, but begins with, it will look
similar, but then it will have more complicated
terms in it because the effects of the phi
to the 4 coupling with m squared phi squared
ok. So, the param and there could be a Z in
front which can also be function of phi classical,
there would be a wave function renormalization
of the fields and this is how it will look.
So, it will look similar, but it will have
captured all the quantum effects. Now, you
may ask why are we struggling to do all this
and the answer is in two parts, in the simpler
situations it allows us to determine the ground
state of the system the collective ground
state of the system ok; so, uses 
in the, so determining the.
Now, we tell children that if you put a negative
mass squared, then you have a you know symmetry
breaking potential and all this. So, all this
is high school algebra, that thing can be
rigorously defined obtained if you do this
if you calculate. So, if you put negative
mass squared, then if you calculate the effective
action correctly you will actually find that
the minimum of the potential is at the correct
nonzero value of phi and it will not be the
trivial square root lambda over m; m over
square root lambda or whatever you find from
here.
So, determining the correct ground state of
the system this is usually done by, so assuming
translation invariance. So, we said all derivative
terms to 0 and then we get gamma of phi classical
to go over to simply being equal to minus
integral of a V phi classical d 4 x minus
sign because gamma has the form of T minus
V after alright integral good old Lagrangian
definition.
The T part will get set to 0 all the derivative
terms there will I mean this gamma can be
a very unwieldy beast, it can have all kinds
of high level derivatives. All of those get
set to 0 and you will be left with this V
phi classical which has only polynomials,
so only powers of, so this has no derivatives.
So, this is phi classical d mu phi classical
if you like, but this is only function of
if you and this we call V effective, we call
it effective potential 
and the vacuum expectation value that we expect
from that translation invariant vacuum.
So, whatever that gives is the ground state
value of phi, but interestingly you can also
find other non trivial solutions. So, this
is use number 1, but you can also find 
ground states that are not translation invariant,
but time independent otherwise you will not
have a ground state. So, phi dot equal to
0, so all the quantum effects are encoded
in the coefficients of this effective potential.
I get a value for phi classical.
Yes.
That is.
That will be the expectation value of the
field.
Ok.
That will be the vacuum expectation value
of the field.
Right; such that phi g s will be equal to
sorry yeah where this will be the correct
vacuum also. Of course, vacuum is not the
correct word ground state is the correct word,
but people use the word vacuum. Vacuum means
there should be nothing, but what you begin
to find is that there are lot of thing is
lurking in the vacuum, but anyway that is
it will not have any particle like excitations
of the phi field that is what we mean by vacuum
right. So, phi will have a value, but there
will not be particles whizzing around.
So, in that sense it is not free of everything,
but it certainly does not have excitations
moving about and this is our current enigma
because the electroweak theory is supposed
to have a nonzero, Higgs field pervading us
like a like the good hold ether and the Higgs
field the observed in LHC was the first excitation
above this ether.
So, we are right now living in a Higgs ether
with a nonzero value for phi and nobody has
been able to get rid of this ether idea and
the ether idea looks; obviously, wrong because
let me some picture like this sorry I should
draw it through the center.
So, the thing is you are somewhere here, but
if it is so then this is the V effective of
electroweak theory though, so this is phi
Higgs and this is ground state value. The
ground state value determines the masses of
everything it enters into W and Z boson masses
the everybody’s mass. Now, the point is
that all of this is at TeV scale you calculate
it at least in you like 100s of this is this
value is some 200 GeV.
So, these values can be thought of in like
100s of GeV, 100 Gev, 200 GeV to the fourth
power because its density. You can set it
you can say that exactly when gs is 250 GeV
is where the electroweak vacuum is and the
that vacuum the V effective should be 0 should
be the ground state, no energy.
If you do this, then standard model works,
but there is a slight ticklish problem which
is that all this scale is in GeV scale, but
if you have tiny mistake in it to 15th decimal
point, it will have residual vacuum energy
left that vacuum energy can be seen by gravity
which would appear as cosmological constant.
So, cosmological constant today have the so
called Einstein’s famous lambda is today
approximately equal to 10 to the minus 11
electron volt to the forth power and so in
GeV units you have to put 36 here right, so
minus 47. So, today we are actually observe
vacuum density, vacuum energy density of the
order of 10 raise to minus 47 GeV to the 4,
this thing requires you to set it to 0 in
100s of GeV units, so you do not care if the
this much is residual.
But there is a conceptual problem, why exactly
this amount is leftover, if there was an error
in this thing being set to 0 exactly there
should be maybe 0.7 GeV to the fourth leftover
may be millionth of maybe MeV to the fourth
leftover, but what is left over is stupendously
small on the GeV scale and it is nonzero it
has a value.
So, you have to tune this to this, this is
like our government trying to check your one
rupee transaction through money pay you know.
So, it is as if the government’s budget
would get reset by whether you bought a chocolate
today or not well. So, that is extreme fine-tuning
that it should have been if it was really
determined by this physics, then it should
have been in some high scale at that scale
it is 0. So, you would say well there is no
other physics intervening between electroweak
and everything else we know it is electromagnetism
at lower scale which is exactly massless and
so on.
So, where does this energy come from? It should
have been ideally 0, if it was 0 then everybody
would be happy because then you would say
oh supersymmetry is a principle which requires
vacuum energy to be exactly 0. So, you would
have said oh it is 0 because of supersymmetry,
but it is nonzero and it is stuck at some
strange value.
So, now, you have to try to say yeah, but
supersymmetry is actually broken very very
far away in such a way that it shines a little
light here like Birbal’s Khichdi or something.
So, it is exactly, but then you are put then
tuned why it is over there, so that it produces
this. So, there are lot of I mean this is
how we make our living and making theories
for this. So, you are welcome to join lot
to be done here.
So, this is an open problem, but this is the
machinery that allows you to determine it
directly determine it systematically. But
there are more since I have got on the topic,
let me just say there are more interesting
things that you can compute from the effective
potential, where use do not set all derivatives
0, but you said that time derivative to 0.
So, you are time independent state, but not
translation invariant.
So, in this case you get gamma static which
is function of phi classical and its derivatives
and you can have something interesting which
is. So, now just grad phi squared and it since
it is gamma will need a minus sign in front
and plus sum U of phi classical. If you now
put your favorite Mexican hat potential, then
you can get what. So, if U is equal to, but
plus lambda phi to the 4, then you have U
that looks like this.
Then you can actually get solutions for phi
which start at let us call this yeah phi g
s let us say, so minus phi gs and plus phi
gs. So, you can actually as solutions which
are minus phi gs, for most of the time on
this side and plus phi gs here and in between
the actually interpolate as if it is well
interpolate as tan hyperbolic, you can actually
solve the differential equation and instead
of minus it will be plus well, so yeah plus
because it is actually not time derivative.
So, it will be plus d u by d phi you do it
in one dimension 1 + 1, so choose only 1 space
dimension and if you solve this equation this
non linear equation has a nice exact solution
has tan hyperbolic. So, you can come out with
non trivial ground states, these are called
kink solutions or solitonic solutions and
condensed matter physics and various and even
in optics you can find this solitonic modes
propagating, so these are called solitons.
So, all such things can be found from quantum
theory where you have approximately classical
description which you know actually derives
from some big daddy quantum theory, but after
everything is integrated out the effective
degrees of freedom that remain are this phi
classical, the extensive variable that remains
is the volume or just the total number, so
it is like this.
So, this kind of expressions can be constructed,
that is the meaning of gamma and we want to
see how to extract such a gamma out of our
W of J, right now we have a description W
in terms of J some external current. The answer
is that gamma need not always have the naive
symmetries that you see in the classic collection,
gamma can break symmetry that is spontaneous
symmetry breaking.
Yeah the quantum action will not respect the
classical symmetries ok. So, it can happen
in several different ways and well, so this
is actually the classic example the phi to
the 4 theory, but with a negative sign for
mass squared term the squared term the quadratic
term, but positive for quadric this is potential
not action. So, minus here, but plus here
is like this. So, this theory has let us if
I had only a real scalar field, it has phi
go to minus phi symmetry, but because it is
now like this you will have to do quantum
theory either here or here you can do it in
both places, but if you do it in either of
them then you are broken the symmetry.
Now, you do not have the freedom to flip phi
2 minus phi. So, then what will happen is
the excitations that you see over it, the
quantum excitations will be the small oscillations
here. And there then their interpretation
is specific to having chosen plus phi classical,
somebody else can make a choice of putting
minus phi classical is quanta will be different,
you will have to do some unitary transformation
to convert its to your description, but those
quanta will not enjoy the phi go to minus
phi degree of I mean symmetry.
So, the phi quantum need not have the classical
symmetries of S and it gets a little more
complicated as well which we will see later
if Vikram is a ambitious enough he will go
to what are called anomalies where it is not
even so trivial. Here at least you can even
see algebraically that this happens, but there
are also ways of quantum mechanics violating
classical symmetries which are hidden in the
loop expansion, but eventually you find that
the corresponding conserved current will not.
So, for every symmetry there this is of course,
a discrete symmetry you can if you make it
a complex field, then you will have a real
part imaginary part you know Real phi Im phi
then you at least have a continuous symmetry,
but there also symmetry will be broken you
no longer can rotate. But for this case we
know that this is just complex Klein Gordon
field, so there is a conserved current right
it is. So, it is equal to you know this phi
star d mu phi minus phi d mu phi star.
So, there is a conserved field, but a conserved
current corresponding to that symmetry, but
sometimes in quantum mechanics this may not
work, not equal to 0. So, it can happen that,
so this is just for example, you know, but
it can happen that d mu j mu may not come
out equal to 0, where you have to interpret
this as the full quantum expression for the
current.
No, it is not this is true to simple that
is not where it happens it happens in the
case of chiral fermions this is called anomaly.
So, it is a grand preview of what the whole
course is trying to do, I do not mind spending
a little bit of time because these ideas are
all so difficult that it is to repeat them.
But now, since we have lost all the time doing
this I request you to come prepared next time
having read a particular section of Ramond’s
second edition book the one on effective action.
