 There is a interesting discovery that was made I should not say discover well interesting discovery that was made by Coleman and Sidney Coleman and Eric Weinberg not Steve Weinberg which I have requested Vikram to cover, but it basically starts with a Lagrangian whose V is at phi equal to 0 whose minima of V are at phi equal to 0, but after you do the loop calculations and add calculate the effective action they shift to non-zero ok. So, it is a very cute and clever example they constructed you start with a V like this, but it is a quantum electrodynamics. So, there is a coupling to a electromagnetic field. So, after it develops a logarithmic curvature like this and becomes like this. So, if a charged field acquires a non-zero vacuum expectation value it means the vacuum does not respect vacuum is quote charged and then electromagnetism will not be gauge invariance will be broken and the photon will acquire a small mass and etcetera, but which is what was eventually realized for standard model. We are actually living in a charged vacuum if you like if you want to feel strained and stretched yeah well there is because there is SU 2 charge filling the vacuum and that is why the W bosons are so massive because that where is at 250 g V ok. So, that is the real machinery behind and still perturbative ok, but the other general point I want to tell you is that we. So, here this F in general we propose that look what is the most general derivative expansion including only local fields I can make of gamma in which no higher than second derivative or square of the first derivative occurs then this is the most general expansion except that you have to supply some overall factor which involves only phi and not derivatives then you are still within that domain what this. So, this F would be would come from the quantum corrections and sometimes you have to put that kind of a overall multiplication even otherwise this is what I will come to next later. The other thing I want to tell you is that the real local expansion in terms of only local monomials of phi and d mu phi need not always be so civilized and in fact, what are there are things called effective field theories of pions and nucleons which is called chiral Lagrangian for hydronic physics. So, here we are actually living in a world where the strong force has shielded all the color charge and the only observed things are pions and nucleons then it turns out that those pions are goldstone bosons we did not talk about it so far, but maybe I should have done goldstone maybe next time we will do that before I sign off and Vikram takes over, but in this case the pions oh god. So, I am getting into a unnecessarily long lecture let me simply say that pions are described by written purely as in the exponent why it is like that it takes a long time to explain, but you put some coupling and then tau A pi A tau is. So, this is yeah this is SU 2 so which belongs to this is a SU 2 valued space time field SU 2 group valued often one writes algebra valued fields like A mu A mu A tau A the gauge field is algebra valued, but here we actually write group valued fields where this is phi of x tau A are the generators and maybe we have to put a half factor I think, but yeah because the generators are half. So, whereas, for headrons or nucleons we write a doublet right. So, this U is a 2 by 2 matrix because it is exponent of tau. So, it is a SU 2 valued matrix SU 2 matrix which is space time field this is a doublet representation and then the Lagrangian is to be constructed from all possible terms consistent with global SU 2 there is no gauge field and Lorentz invariance. So, if you do this then the Lagrangian begins it looks like one half d mu U dagger d mu U and you can have higher things like A times d mu d mu U dagger d mu d mu U dagger U you can start writing all kinds of terms or you can begin to write d mu U dagger d mu U whole squared. So, all kinds of terms begin to appear and then you will also have psi dagger or rather psi bar and then d mu. So, typically something like U dagger d mu U times this psi the big psi psi bar gamma mu psi. So, you begin to develop all kinds of terms which are all consistent with SU 2 global symmetry and of course, contract all Lorentz indices. So, you can have quite a while Lagrangian like this which will not fit into this kind of simple form and in that case you do not expect to derive any effective potential because all kinds of derivatives appear. This is called chiral Lagrangian. Reason for chiral is long to explain that has to do with this hypothesis. This was all figured out by well Weinberg first and then, but he did not put it in the formal way. So, and of course, S Weinberg and by Callen Coleman, Wesson, Zumino. So, Weinberg pointed out that you had to write all possible terms and then gave a prescription, but these people then figured out that what it amounted to was doing this. So, it is not always that you will have this. You can have all kinds of wild Lagrangians, but you can still define an effective potential, effective action. There is nothing against defining an effective action except that you have then have to include the higher derivatives as well with front factors containing local products of fields. So, the method will remain valid, but this expansion will be completely useless because there are all kinds of non-local pieces. You can then do a derivative expansion which is local. So, before this approach was developed, people were using what is called current algebra. So, because what you do observe is there is some kind of conserved charge, I mean barrier number is conserved and isospin was conserved. So, you can write the currents J mu A as equal to psi bar tau A gamma mu psi. So, these currents would obey the algebra of SU 2 and you just try to derive everything from, but you have to treat them as quantum fields. So, their local products are not well defined and then you can derive various relationships, but that was a very complicated method and this at least recast the theory into the theory of Lagrangian. I want to make some. So, I think next time I will try to do one way said was try to prove this connected. So, connected diagrams proper definition and y exponential of z regenerates the exponential of connected diagrams degenerates all possible diagrams or greens functions rather and maybe the Goldstone theorem. Today I just wanted to talk a little bit about what is called asymptotic theory and this is based on Isaacson and Joubert and if you can understand the writing at first go please come and collect a prize from me. And if you realize it, so it is like a kaon kaon you know it is called kaon in Zen Buddhism your guru gives you a mantra and then you contemplate it for 10 years and then one day you are enlightened oh this is what it means. So, that book is like that. So, it talks in terms of kaon's kaon. So, the main kind of puzzle that people still date face it is really you know. So, what is meant by continuum was developed by Cantor and Dedekind and others in 19th century and at first even within mathematical community there are a lot of reaction to that, but then they actually realized the value of you know nailing down what is meant by the continuum. But when quantum mechanics came and to be described in unbounded space it really caused a lot of problem to understanding everything and the problem still persist in a sense, but we kind of regulate them. So, the problem is like this let me draw time axis and I am drawing. So, well quantum mechanics somehow people draw like this. So, minus infinity is here and plus infinity is here. So, transition amplitude right. So, t equal to minus infinity to plus infinity. So, what we think of is that. So, let us try to draw the energy spectrum. Spectrum just means set of eigenvalues and actually it is a continuum. So, drawing this is only symbolic it is a genuine continuum, but there are levels like this and we expect that the theory is free. So, the whole assumption of perturbation theory is predicated on the assumption that asymptotically whatever that means, the particles are free non-interacting. And if you really force me to say what this asymptotically means well finally, ultimately in the days of cosmic rays they used to observe this bubble chamber diagrams right. Something crosses your plate there is some emulsion plate and you can see all kinds of nice curves if you see old cosmic ray plates. That is what a particle is ultimately that is what it means. It it produces pretty consistent tracks and the story has not changed till date except that today we have gigantic detectors with a beam pipe running through them and human beings are something like this and they produce huge tracks huge slew of tracks. So, what is meant by particle is that. So, it is what condense matter people tell you are electrons are not they are really quasi particles and nobody isolated an electron from it. It is just something you apply some voltage and something happens ok. You do not know what is flowing inside, but yeah there is a good description in terms of almost free quanta which are fermionic and it is a gauge invariance which enforces that they still have to carry unit charge. So, those quasi particles look just like electrons, but the real electrons are observed in these emulsions they are just tracks. So, we do know that there are these tracks and we do know that gluons and quarks do not produce such tracks. So, gluons and quarks do not exist in asymptotic states. So, the asymptotically there are particles and that they are non interacting, but then you also want interaction because they have to do something. So, in the region near t equal to 0 interaction turns on. So, in elementary quantum mechanics that tell you first you know you have done Fermi's golden rule and all that. So, they say well you introduce a kind of delta function theta function which turns on has a long plateau and then switches off, but the off part is going to be infinite it is going to go to infinity. So, this is where it turns on and then turns off of course, nature does not wait to turn things on and off. So, what do you mean by this? So, then people after lot of thinking have come up with the following prescriptions that keep us keep our sanity and allow us to use the mathematics we know. First is that the spectrum remains the same this is a very big assumption. So, I send in particles they interact and while they are interacting of course, they have some interaction energies and so on. What will happen is that of one free particle and another free particle they may become something else here and develop some interaction energy, but the spectrum is the same the list of Eigen values is the same. Although they will occupy different entries in the list and then come back because ultimately they are against till electron and mu 1 or whatever there. So, they will come back as free particles in the intermediate region they will get kicked off here and there, but within that same spectrum the spectrum of H and H 0. So, hypothesis one is that the spectra of H 0 and H 0 plus H i are identical. Now, for one thing this precludes any bound states because bound state would mean that there are some states below e equal to 0. So, it actually does not allow you to have bound states. So, you add a caveat that bound states if any should be added by hand which would be some finite spectrum can be added by hand, but it does not change the structure of the overall spectrum is just kind of a few things added at the bottom and you never really are able to calculate it. So, it does not matter yeah. So, but at the same time you want the field in this region to still be the free field fields phi and field in the. So, we call in and out fields right. So, the fields phi in phi out in the in and out regions obey the same obey free field equations and the same property holds for phi in the interaction region. This you know this we use already in quantum mechanics too because this is called the interaction picture. We evolve phi of x using only phi 0 H 0 here. So, x and t and this could be at in the in region with t 1 very large in negative. So, this is what we propose and then define a ok. So, we will not get into the you know the interaction picture. Now, because of this fact some kind of psychology some you can see beginning to see the contradiction because we want the asymptotic fields to obey canonical commutation relations which are equivalent to this evolution, but really the field which is in the interaction region has to obey this, but it has to somehow carry more information. So, the resolution of this is to propose that actually there is a normalization factor between the two. So, propose that phi of x is some normalization phi x in and we assume that it will be same as with somewhere between 0 and 1. Yes and a real number. So, this is sometimes called wave function renormalization. What this means is that thus while see what is the effect of acting with phi on vacuum phi has a creation and a destruction operator. So, the destruction operator acting on 0 will give 0 the creation operator will connected to a one particle state and will be exactly equal to 1 and if you want this is 0 minus and this is 0 plus. So, these are just 1 because they produce one particle state, but 1 phi 0 is equal to square root z times this and therefore, less than 1. This means that phi has more content than just one particle states, it contains all the pair productions and things like that which are not terribly a part of the interaction, but they are sort of kinematic redefinition of the vacuum. So, I am now actually borrowing words from Isaacson and Zuber. One particle states do not exhaust the content of phi the full phi. So, for example, you might get away with a p minus p. So, conserve all charges momenta everything that you have to conserve in the vacuum, but such states may be contributing to the actual phi. Secondly, to avoid the contradiction with this we also have to propose that this statement is not an operator statement. So, in terms of Hilbert space. So, I started numbering something right. So, perturbation theory non intrus spectrum is this, the fields this, then we say number 3. These are not logically independent ones, but kind of build on top of each other and with additional caveat that. So, the above that this relation star. So, this is called weak equivalence not an operator statement. It is true only by matrix element by matrix element, but not true for the whole operator right. It gets puffed up, it has more things in it than the out inner out. Plus infinity and vacuum at minus infinity. Technically, you have to keep them different because all you can do is that you see free particles in the in region, you see free particles in the out region. But you have passed through this region of interaction and states in quantum mechanics are defined only up to an overall phase. So, you do not know what overall phase it may have picked up in going from in to out. So, people technically keep it like that. For most part it does not matter, but there are interesting experiments where for example, vacuum is not really stable and things like that. So, then actually even the out state is not true vacuum either, but there can be phases like this. In fact, one famous example is Barry's phase. It does not have to do with asymptotic theory, but that is where you evolve a system through a series of changes, parametrically change Hamiltonian. You come back to the same value of the Hamiltonian, but the two states will differ by a phase. If there is somehow a topological obstruction to shrinking that path, if the path cannot be shrunk. So, the point is when although the in and initial and final Hamiltonians look identical, if you have either got a space or time region in between. So, that you do not have a way of directly comparing, then you should leave a relative phase between them. So, if you send the thing through LHC, do not expect that that electron is same as that one, may have got a phase in it. So, the next steps how to do are rather technical. So, we will stop today with just writing out this thing which you can take as homework, which is as a preparation for the next main statement. So, what I do want to do is cover what is called the Schellen Lehmann representation. So, this is some kind of Swedish A and people tell me that this is spoken Schellen. So, but as a preparation check that for a free field, the commutator phi x phi y can be written in the form and now I am slightly on thin ice, because I am using I do not personally use Isaacson and Zubat normalization, but for the time being I will use it, because I am taking it from there I tell you how it differs, which is defined to be equal to I times delta epsilon k 0 is sin of k 0. So, you can try to do this basically, you know that delta function is when it has a polynomial inside is product of delta functions at the various zeros divided by the norm of the derivative of the function at that point. So, you can carry out the d k 0 integration and you should recover the thing above. And Isaacson and Zubat notation is that. So, nobody can change the canonical relations, phi pi have to be 1, but in a a dagger you can insert some change in normalization. So, I am used to using along with Weinberg and Sudarshan and so on to just set this equal to delta 3, but Isaacson and Zubat put a 2 pi cube 2 omega k here. So, if you do not put any other normalization then this is just like harmonic oscillator, it will just raise and lower the number operator, but if you put this then there are some advantages to the what Isaacson and Zubat like about this is that this quantity together is Lorentz covariant. So, their creation destruction operators are also Lorentz covariant and so on, minor advantage is nothing very great, but this is the normalization they use. So, in that normalization this will come out like this. So, you have to do a free field expansion also with d 3 k over 2 pi cube 2 omega k etcetera then the a a dagger you use this then you should get that. So, we will start with this next time and yes if you would like to be prepared then you use read this section from Isaacson and Zubat in a chapter called Asymptotic Theory. Section called Asymptotic Theory which is in the chapter called external fields the effect of external fields or something.