 So, what we had was w of j equal to w of j equal to 0 and then e raise to minus i over 2 times this j 1 delta f 1 2 j 2 and then we argued that actually this is not what we like to look. So, there were two things one is that this w generates all the possible greens functions. So, w of j can be expanded as sum over 0 to infinity in principle right g n x 1 x 2 x n j x 1 j x 2 j x n and the zeroth order is 1 g 0 equal to 1 and so some kind of initial condition I think that is consistent because if I set n equal to 0 then none of these are there and I have this, but it has no argument at that point yeah and j is also equal to 0. So, it is what we have in front that is just a convention where the g n are the time ordered n point functions and technically it is 0 plus and 0 minus. Now, here we argued that this is slightly too much information is not packaged very well because this g n contains disconnected pieces as well. So, ideally we want to define connected greens functions. So, i to the power n over n factorial and yes integral over the d axis i to the n right otherwise it will not tally with the simple variational average definition we had made which comes out of that. When we define the connected greens function greens functions you can do it two ways geometrically which we had drawn last time you know drawing something things like this we had drawn it is sort of obvious what connected means in this simple picture pictorial terms and last time we had drawn three diagrams more because we wanted to get at what is one particle irreducible which is a little more advanced idea not concerned with it right now. So, geometrically it is clear what it means, but algebraically what you do is define it recursively that is you say that the connected greens function is or rather you say that nth level normal greens function is a product of sum over products of sub leading. So, now that notation gets a little complicated, but what we do is G c let us say R and product over all the partitions of n into R and there is a notation for that partitions of so, into R 1 R 2 R n and then sum over partitions and we claim that the two point function is the very first connected greens function with G 1 being trivial ok G 1 equal to 1 equal to G c 1, but G 2 equal to G c 2. So, G 2 as we know is this the two point function the Feynman propagator right and is a connected greens function. So, diagrammatically we draw it simply as a line and that is certainly connected then you can easily see that you can recursively build up because a three point function would have to be a product of two point functions times a point or it would have to be a graph like this and then so on. So, when you go to higher all you. So, you would have a set of points for n point function you would have some number of points and then you have to see what are the ways of connecting them and n point function is such that it will be product of different ways of partitioning the points and drawing connected graphs among them and what does that mean? It means the that one has to be made up of lower order connected graphs and so on. So, once you have defined the second the two point function you are fine ok. So, you can recursively define like this and now this is what I am not proved so far, but and I actually just meant to summarize what we have done so far is that then we can prove that and here the definitions differ a little bit here. So, here we use w equal w j equal to e raise to i times z of j such that z j actually generates only the connected greens functions rather it is a generating function of and the convention remains exactly the same integral of. So, now, I am I can do a little better sum over 0 to n yeah. So, we can put the summation outside 0 to infinity integral products from 1 to n of d 4 x j and then i to the n over n factorial g n c. So, after this the trick we used was to do this Legendre transform. So, we were then looking for a way of having a functional which is functional of a space time field not a auxiliary current. So, we wanted it in terms of more physical entities rather than some auxiliary entities. So, then we define. So, now the story of what is effective action starts after the this background from j to phi description by defining phi c to be variation of the z. So, intuitively we want to think of phi c simply as equal to the classical this c is not that c. So, that had to do with connected, but this is just classical. So, phi classical should basically be phi quantum averaged and phi quantum averaged would be equal to integral d q d p phi e to the i s right, but that would be same as doing 1 over i. So, variation with respect to i j of w of j, but we also want to factor out the w 0. So, sorry all this went into a corner, but so what we say together and this is at j equal to 0. So, phi c is I think minus i variation with respect to j. So, phi classical x is defined to be minus i variation with respect to j of x of log of and now I am going to get some kind of i directly from the exponent. So, I will fix this i we can fix it in a minute. So, we say it is the log of w and evaluated at j equal to 0 whatever w of j that remains when you vary. So, I could have as well put here w of j because I am going to set j equal to 0 in the end anyway. So, it becomes variation of this log and now the way we defined w well this is of course, correct right because that is the except that right. So, minus i times this is correct and then log of w is equal to i times z. So, that cancels this minus i and so it becomes equal to simply d by d j x of the z j evaluated at j equal to 0. So, we define the desired phi classical and then the trick which works when I think this is. So, I never fully understood, but I think this is a trick that works because phi is essentially like an extensive variable. So, you can do a Legendre transform. So, of course, you see through the you work through it comes out correct, but why you would think of Legendre transform I do not know. So, this trick is essentially due to I think man called Jonah Lassinio was I think Schwinger's student or collaborator. They did some clever things not very well known outside, but once you read the literature you will find them. So, one of them is this and then define gamma of phi c to be equal to right integral j sorry write it in this order phi c x j of x, but such evaluated at by inverting this relation where phi c is equal to sorry d z d z by d j equal to 0 sorry d z by d j evaluated at j equal to 0 and minus w of j z of j, but that again is at j c phi c phi equal to phi c I think that is one way that this is specified. So, you have to evaluate the j by inverting that relation and then we could check that this phi c in the free field case actually satisfies. So, I am now comparing what I have here. So, there is an overall minus sign this term was written after this term. So, minus this plus that unlike classical mechanics and then j is replaced by this functional. Then we can check for the free field case very easily that the gamma 0 turns out to be nothing, but the free field that box plus m square phi c is equal to j follows from the gamma. So, you can write free field gamma is nothing, but the usual. So, that is rather reassuring and it generates the interpretation that the quantum gamma the quantum action is going to be the classical one plus some quantum corrections. Now, you may say this looks rather too simple how is this going to work when I have more complicated situations and the answer is that in the presence of a potential we play further tricks. So, for an interacting theory at least minus technically here. So, at least so, long as the v is a functional only of phi and not the derivatives we can write that minus i. So, we simply claim that if I have to if I have to multiply what is going to happen as you remember the whole action is going to be I mean the path integral is going to be integral d etcetera e raised to minus free and then plus this part right. So, we are focusing on the inside of the functional integral and then the e raised to i times j occurs right. So, consider the piece in the path integral in the functional integral which looks like this. So, if I had a phi here what I did was to hit here by e raised to minus i d by d j. If I now have a whole thing like this all I say is that the d 4 x carries on, but it is not affected by this is just the measure of that times e raised to i j phi integral d 4 x. So, this is a big formal step wherever you see a phi. So, what we are trying to do is I have the full path integral which is integral d p d q or d pi d phi e raised to e raised to i times integral l free l free minus v and then plus i plus i is already here, but I could write it separately plus i integral j phi. Now, I know how to take care of l free along with j phi, but I do not know what to do with this v. So, it is like this right. So, what I do is I rewrite the e raised to minus i v d 4 x as this operator acting on what is going to come there, but now this is just a formal this is this does not involve phi anymore it only involves j. So, I pull this out completely of the path integral and I say that therefore, well we have been using Lagrangian version for long now. So, let us not worry about the pi part e raised to i times integral d 4 x l free minus v phi plus i integral j phi d 4 x becomes equal to since the v now as this operator does not involve phi I pull it out of everything including the functional integral phi to write e raised to minus i times integral d 4 x of. So, all this is completely formal now it has no meaning by itself in the exponent, but then let us see what remains here what remains is d phi e raised to i d 4 x l free plus i j, but that is our old w of j for free field theory. So, this is just a clever trick for writing this out to satisfy oneself that one can formally write something in practice it will work because this variation with respect to j. So, you can the v will be phi usually just phi to the 4 or something like this you know it will be a it is a monomial of some kind. So, you can always use it to extract Green's functions up to a particular order and they will involve not just the not just the not just two point functions you remember in free field theory the only thing we got out was two point functions or their products. So, here you will begin to get more things out and I did intend to do it, but somehow I did not I want to do it next time. So, you can use this to derive between g n and g 2 and gamma n. Now, this may look a little bit of what does this mean? The answer is that actually the endpoint function the g n will look like and I think the connected one g n c. So, g n c which has the structure of. So, it has some n points right that is what g n means, but this thing when it is connected boils down to lines like this with g 2 is inserted dot, dot, dot, but then an irreducible piece which will be gamma n and plus lower order terms what I do not this is whether it is probably the full g. So, what one finds is that the endpoint function as it can contain the lower pieces which are products of lower connected parts at the nth level you will find that the for the most non-trivial term there will be product parts, but the non-trivial term has the structure of a vertex some blob which you cannot further resolve a one particularly reducible diagram. So, this will be a times lines with the g 2 insertions on them. So, this is how the particle and vertex interpretation emerges the fully interacting quantum field theory. So, long the you still use perturbation theory because after all we pretended that you could treat v after you do the free field thing. So, you have to think of this as in some sense small otherwise these manipulations do not mean that much, but to the extent that you can try to isolate the effects of v those effects will can be broken up into an irreducible endpoint function times just propagators two point functions. So, that is the idea of defining the effective action. So, that these are the physical meanings of the various quantities that we have been introducing and gamma n is what will enter into the expansion of the gamma in terms of its argument phi n just like the g times j's where the gamma n are. So, I can leave this here because that is the it is very similar to that phi's and d 4. So, if you expand the gamma like that then those gamma n's have the significance of being the highest order non irreducible non reducible part in the endpoint greens function. Final comment is that this gamma there are no derivatives appearing here now of phi. So, it is functional only of phi, but we do know that in fact, the free part will contain d mu phi d mu phi. So, there is an alternative expansion which is a derivative expansion. So, this these gammas can contain non local and so, derivative expansion is so, such gamma n also contain non local pieces. So, an alternative so, you see it is just gamma n x 1 x 2 x n times a product of fields, but which are at different points. Ideally we want something so, if it was a local field theory and there was so, what I mean by this functional if it was a local functional then there should be a delta 4 of x 1 x 2 x n. So, that everything collapses to same point, but the general one is not like that obviously, because there are derivatives. So, an alternative definition is a derivative expansion or alternative description rather not there. Where we anticipate our traditional free field and simply say it is equal to d 4 x I think this is given in Ramon's book as well and minus V effective phi. Assuming no higher order derivatives of phi see however, that extension is also possible ok. Here by proposing that the only derivative terms I have are d mu phi d mu phi up to some functional which is function only of phi, but not of derivatives and whatever remains I call V phi V effective this is a local expression it is powers of phi at the same point x there is only one x integration now ok. So, this is a local theory because everything in it is local to one point x this kind of expansion if it is possible in the case this is possible this ansatz is possible V effective is called the effective potential and its minima or its minima give you the true vacuum expectation value of phi it they give you minima, but the expression about to going to write has both of the. So, what we mean is the true vacuum of the theory I should say yes it could be vacuum alright it is not necessarily only one true vacuum of the theory occur at d V by d phi equal to 0 and d square V by d phi squared greater than 0. So, this could be various that could be several such phi I and the d square V by d phi squared essentially as the interpretation of the mass. So, since L free had the form d phi squared half d phi squared minus a half m square phi square d square V by d phi squared is the effective mass squared at that I if you have a complex field then you will not have a half factor, but you have to take care of that. So, this is what we mean by our symmetry breaking which everybody has this is where we cannot explain to general public what is Higgs mechanism all about because it is the V effective as a function of phi c which has minima at and actually that is a complex scalar field. So, in a circle of radius mu which is approximately 250 g e V. So, now you really know what the Higgs vacuum means. So, far you are just varying V which was not the V effective, but luckily because the theory is renormalizable polynomial remains the same, but the coefficient get quantum correction. So, they are not just lambda over m squared the many, but including quantum corrections. So, but this is the real meaning of what symmetry breaking is if the effective potential develops this then it is there.