 you will complain that we have lost any 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 gamma 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? Recall our dominance by classical paths. So, I need not have written this one. 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 an overall just e raise. So, there is integration here, but it gives this answer ok. This is also how some of the thermodynamic potentials are defined gives free energy. You have to average over you do the phase stress 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 beta of course, e raise to minus beta h ok. Where what remains is 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 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 df by dv 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 respect to the 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 greens function is a twice variation of the generating function. So, I am just trying to announce to you 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 your 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 is 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 defines 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 encapsulate in 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. 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 why 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 some 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 will so m square phi square. 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. So, use is in the so determining the we tell children that if you put a negative mass square 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 you can 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 lamp square root lambda over 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 set 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 weak phi classical d 4 x minus sign because gamma has the form of t minus v after all right integral good old Lagrangian definition. The t part will get set to 0 all the derivative terms there will there I mean this gamma can be a very unwieldy beast it can have all kinds of phi 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 yes that will be the expectation value of the field that will be the vacuum expectation value of the field right right such that phi g s will be equal to sorry 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 things 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. This is our current enigma because the electro weak theory is supposed to have a nonzero Higgs field pervading us like a like the good old ether and the Higgs field they 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 right we 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 electro weak 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 T e V scale you calculate it at least in you like hundreds of this is this value is some 200 g e V. So, these values can be thought of in like hundreds of g e V 100 g e V 200 g e V to the force power because it is density. You can set it you can say that exactly when g is 250 g e V is where the electro weak vacuum is and at 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 g e V 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 has the so, called Einstein's famous lambda is today approximately equal to 10 to the minus 11 electron volt to the force power. And so, in g e V 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 g e V to the force. This thing requires you to set it to 0 in hundreds of g e V units. So, you do not care if the this much is residual, but there is a conceptual problem why exactly this amount is left over. If there was an error in this thing being set to 0 exactly there should be maybe 0.7 g e V to the force left over may be millionth of maybe m e V to the force left over. But what is left over is stupendously small on the g e V scale and it is non-zero it has a value. So, you have to tune this to this, this is like our government trying to check your 1 rupee transaction through money pay you know. So, it is as if the governments budget would get reset by whether you bought 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 electro we can 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 super symmetry is a principle which requires vacuum energy to be exactly 0. So, you would have said oh it is 0 because of super symmetry, but it is non-zero and it is stuck at some strange value. So, now you have to try to say yeah, but super symmetry 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 exactly, but then you have to then tune why it is over there. So, there 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 lot to be done here. So, this is an open problem, but this is the machinery that allows you to determine it directly at 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 you do not set all derivatives 0, but you set the time derivative to 0. So, your 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 some u of phi classical. If you now put your favorite Mexican head 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 minus phi let us call this phi G s let us say. So, minus phi G s and plus phi G s. So, you can actually have solutions which are minus phi G s for most of the time on this side and plus phi G s here. And in between the actually interpolate as if it is well interpolate as 10 hyperbolic you can actually solve the differential equation and instead of minus it will be plus well. So, 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 plus 1. So, choose only one space dimension and if you solve this equation this non-linear equation as a nice exact solution as time hyperbolic. So, you can come out with non-trivial ground states these are called kink solutions or solitonic solutions and condensed metaphysics 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 just 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 classical action. 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 fourth theory, but with a negative sign for mass square term the square term the quadratic term, but positive for quartic 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 cannot do it in both places, but if you do it in either of them then you have broken the symmetry. Now, you cannot do not have the freedom to flip phi to 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 his 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 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 real part imaginary part you know real phi in 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 there is a this is just complex client 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 too 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 also difficult that it is ok 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 Ramon's second edition book the one on effective action.