 Right. So, we were exploring the consequences of the fact that the generalized susceptibility chi is analytic in the upper half plane in the frequency. So, if I draw the frequency plane, this being the frequency plane with the real omega here and imaginary omega, let us call it prime. So, this is the omega prime plane out here. Then the point was that if you start with any fixed real frequency omega, we discovered that the integral over the closed contour C of the omega prime chi of omega prime over omega prime minus omega over this contour. So, all the way from minus infinity coming in and then a little indentation a semicircular indentation in the upper half plane through an radius epsilon say and then back on the real axis and all the way down and closed in this fashion. This is equal to 0 and the condition we needed for this was that this integral the integral had to vanish as omega prime went to infinity anywhere in the upper half plane. So, if chi of omega prime goes to 0, if chi of omega prime goes to 0 as mod omega prime tends to infinity in the upper half plane, then this contour C could be blown out all the way to infinity. This contribution would then vanish because this is going to give you a capital R e to the i theta. This is going to give you a capital R, the two cancel each other and then if this goes to 0, the answer goes to 0. So, a sufficient condition for this integral to converge and for this contribution to go to 0 is that this be true that chi vanish in the upper half plane. As you can see it suffices if chi vanishes along the real axis because on the imaginary axis you actually have extra convergence factors. So, if this is true then this 0 implies that the integral from minus infinity up to this point which is omega minus epsilon and this is omega plus epsilon up to infinity plus the semicircle is 0. And since the integral from this in the contribution from the semicircle is 0, anyway you end up with a statement that integral from minus infinity to omega minus epsilon the omega prime chi of omega prime over omega prime minus omega plus an integral from omega plus epsilon to infinity the same thing. These two plus this contribution, this little contribution is as follows plus an integral from on this contour, on this little contour here omega prime equal to omega plus epsilon e to the i theta. So it is a circle about the point omega. So omega prime is omega itself plus this little complex number epsilon e to the i theta. And the integration variable here is theta running from pi to 0, goes the other way here. So this is equal to epsilon e to the i theta, i d theta that is what d omega prime is, we just differentiate this quantity and then the integral runs from pi up to 0 divided by omega prime minus omega that is equal to epsilon e to the i theta and that is it. So 0 is equal to this whole thing and this cancels, this term cancels here and the integral is minus i pi times, oh sorry, the sky of omega plus epsilon e to the i theta that is of course sitting inside there. So in the limit in which epsilon goes to 0, this becomes the epsilon cancels, the theta cancels and you are left with i pi times chi of omega, right, minus i pi because this is from pi. So this finally says that i pi times chi of omega is equal to an integral from minus infinity to infinity d omega prime, chi of omega prime over omega prime minus omega leaving out a small portion which is symmetric about the point omega and this is called the principle value integral, this is called the Cauchy principle value. So this is equal to, but let me just write it as p and this stands for Cauchy principle. So this was an invention of Cauchy's, he discovered that in many cases when you have an integral with singularity on the real axis, on the axis of integration, on the contour of integration, in general the real axis, then if you leave out a small symmetrical portion about this little segment which is symmetrically situated about the singularity, in this case at omega and take the limit as epsilon goes to 0, then that can be finite and it is called the Cauchy principle value. In this case, it is finite and it is equal to the susceptibility at this point apart from this factor here. So if I bring the i pi to the right hand side over, so look at what has happened. We did an excursion, we made an excursion from real values of omega to complex values, but we are back to the real axis because this contribution went away, we are back to the real axis and what we have done is to, we have succeeded in expressing the susceptibility at any real frequency as an integral over all other frequencies except for an infinitesimal interval about that point, that frequency. This sort of thing is called a dispersion relation, we will write a simpler form of this in a minute, but notice that you are back on the real axis, so there are no complex frequencies here, it is completely physical, right. All we have to do now is to take real and imaginary parts of this. So it follows that the real part of chi of omega is equal to, it comes from the imaginary part, this is real plus i times imaginary and the i cancels and you get p over pi integral minus infinity to infinity. By the way, this, I have written this as p integral minus infinity to infinity, sometimes it is written like this, minus infinity to infinity with a slash over the integral. That notation is also used just to show you that leave out a symmetrical portion about infinitesimal symmetrical portion segment about the singularity and take the limit. That limit is not always guaranteed to exist, but in this case it does, okay. So this is equal to d omega prime, imaginary part chi of omega prime over omega prime minus omega. The singularity of the integral, at the point omega prime equal to omega is avoided by the principal value, supposed to be left out omitted. And similarly, imaginary part of chi of omega equal to, that comes from the real part but there is a minus i here, when you take this up, so it is equal to minus p over pi integral minus infinity to infinity d omega prime real part of it. So this is the reason why I asserted in the beginning that the susceptibility cannot be purely real or purely imaginary because if it were so, then being an analytic function it would just vanish identically. You cannot have this to be identically 0 or this to be identically 0 because then the whole thing is 0, okay. These relations are called dispersion relations. They are also called occasionally for historical reason. These are the people who introduced these relations first into physics, Kramers-Kronig relations. Even because it was introduced by Kramers-Kronig first in the context of refractive index, an optical susceptibility if you like. Refractive index is, the complex refractive index is just the optical susceptibility and they introduced it in that connection, okay. Now any two real functions, any two functions, real valued functions of a real variable such that you have suppose say two functions f and g, then the fact if you have f of x prime d x prime over x prime minus x principal value equal to g and g is the inverse relation, they called a pair of Hilbert transforms. So the real and imaginary parts of a causal linear retarded susceptibility form a Hilbert transform pair. So the real part is a Hilbert transform of the imaginary part and the imaginary part is a Hilbert transform of the real part. Notice there is a minus sign. So in principle if I substitute for imaginary chi of omega from here on this, we will have one more integration to do. In principle you should get an identity. In other words that intermediate integral should turn out to be a delta function, okay, which it will, I am not going to show it here but it will. So these two guys real chi and imaginary chi form a pair of Hilbert transforms. Now what was the reason for this whole thing happening? Well it happened because this is an analytic function in the upper half plane and where did the analyticity come from? What is the origin of this business? Remember this got represented as an integral 0 to infinity d tau e to the i omega tau phi of tau. So it got represented in the terms of this one sided Fourier transform and then the argument was if this is true for real omega then for omega with positive imaginary part it is certainly true and analytic. So it arose from this and where did this come from? Why is this 0 and not minus infinity or anything else? Causality. Causality. So in all cases dispersion relations are a consequence of causality. Finally that is what is doing it. The physical reason why the generalized susceptibility satisfies dispersion relations is because it is a causal response. This has profound implications in other parts of physics especially in particle physics, quantum field theory and so on anywhere where dispersion relations appear and they appear in a large number of places but finally it stays down to causality. Now you might say look this is an integral over negative frequencies also whereas physical frequencies are positive but there the fact that this is an anti-symmetric function comes to a rate. So we can convert this integral into something that runs over from 0 to infinity. So let us do that. The real part therefore equal to p over pi, there is an integral from 0 to infinity and then the portion from minus infinity to 0, I am going to write in this form. This is minus infinity to 0 plus integral 0 to infinity times d omega prime blah blah blah. In this term, so I am going to write this as p over pi, I am going to change variables to minus omega prime. Then I pull out a minus sign due to the Jacobian, this becomes infinity to 0 and those 2 minus signs cancel. So both integrals look like 0 to infinity d omega prime and then in the first integral I have chi of omega prime over omega prime minus omega, that is the integral from 0 to infinity in this formula, in this formula and then minus infinity to 0, remember I change variables to minus omega prime. So let us write that as sorry this is imaginary plus imaginary chi of minus omega prime divided by minus omega prime minus omega because I change variables but this fellow is an odd function. So it is equal to minus imaginary chi of omega prime. That minus sign cancels with this and you are left with the following, you are left with imaginary chi of omega prime times 1 over omega prime minus omega plus 1 over omega prime plus omega but this omega cancels and just gives you 2 omega prime. So therefore I have a nice representation which says that in general we have a useful representation which says real chi of omega equal to 2 over pi principal value 0 to infinity d omega prime, omega prime imaginary part of chi of omega prime because when I add these 2 I get 2 omega prime divided by omega prime squared minus omega square. Now everything is physical, it says this denominator has a simple pole at omega prime equal to omega, the one at minus omega is outside the region of integration and you have to take the principal value, the pole at omega prime equal to omega and integrate this quantity, okay. It is guaranteed to converge. At infinity this is going to produce a capital R, that is going to produce a capital R, this is going to produce R squared, they cancel each other but this fellow goes to 0 and the integral exists, okay. So this is the physical form of the dispersion relation. Similarly you do the same thing for imaginary part chi of omega equal to, there is still to be a 2 over pi and there was a minus sign which is going to persist, so minus 2 over pi principal times principal value from 0 to infinity d omega prime. Now you are going to subtract 1 over omega prime minus omega minus 1 over the other because this fellow is symmetric so it does not produce an extra minus sign. When you subtract that you get 2 omega so this becomes an omega which outside the integral and then the real part chi of omega prime, it is a very useful form of the dispersion relation. So this is the one that you use, you would use in practice. In practice what happens is the physical use this is put to is suppose you know the susceptibility or you measured the susceptibility both the real and imaginary parts because they have physical meanings, you measured it in some frequency range and you could approximate it by 0 or something like that outside this range. Then to find the susceptibility at any point outside the range you could use this formula. To first order it will be correct even if you cut it off at some upper limit here, that is a consequence of the fact that they are real and imaginary parts that is the reason, sure that is certainly true but it is not really better that these functions behave differently as in tautical, okay. Now you could ask the other question, we slurred over some point, we said chi of omega must go to 0 at infinity. Suppose it does not, you still have to deal with this situation because that can happen in many cases. Suppose it does not go to 0 but it goes let us say on the upper half plane suppose mod chi of omega goes to some number, some chi infinity as mod omega goes to infinity in the upper half plane. Suppose this were true then the contribution from the semicircle is not 0 that is all but you can still work out what the contribution is because now you notice that this infinite semicircle is here. On this semicircle chi goes to chi omega, chi sub infinity and then you have on this circle omega prime equal to capital R e to the i theta. So d omega prime is going to produce it R times d i d theta and so on. And in the denominator you have the same thing omega prime minus omega is going to produce an R. So you will end up with a chi infinity as a constant in the integral. So fine, you will still get dispersion relations but you get an extra contribution here plus something time pi times chi infinity or something like that. So you can still write it down, still compute it but now suppose you say well I do not know chi infinity I just know that it does not go to 0 but I do not know or suppose it is a function of the angle. Suppose the limit that it goes to does not go to 0 but suppose it is a function of which way you go to infinity you have no handle on that. I have assumed here that all through uniformly it goes to the same constant but if it depends on the angle then you cannot do the angular integration, right. Then this is the trick you do in that case. What you do is to modify the function whose analytic behaviour you are looking at. So the omega plane, omega prime plane here is omega, choose some value of the frequency at which you know the value of the susceptibility. So let us suppose there is a point omega naught and let us suppose that chi of omega naught is known or measured so suppose so you know the value at this point and now I am trying to derive dispersion relations. So the function I am going to look at is not chi of omega prime over omega minus omega prime minus omega. This is not good enough because it does not converge fast enough but if I multiply it by this omega prime minus omega naught then I am in good shape because there is an extra omega prime capital R sitting from here and the d omega prime will cancel against this. There is a 1 over capital R here and there is this fellow who goes to a constant, some constant at infinity so there is a 1 over R and this contribution will still vanish. But now you will say aha that is not good because there is a singularity at this point now. So what you will have to do is to consider the contour this way and then an indentation here and then another indentation there and then all the way to infinity. So that is one way to do this. Another way to do this so what would that amount to doing? What would this contribution be finally? It would be i pi or minus i pi whatever it is chi at that value sitting there and then omega naught minus omega would sit there. You can save yourself some trouble by considering not this function but this function. This is a constant. This function if this is got a simple pole at omega naught then this function is well behaved at omega naught. So I write this version relation for this function keeping track of the fact that this is a complex number in general. It is a yes I convert this singularity into a removable singularity by subtracting this. So since I have subtracted this this is called a subtracted dispersion relation and this is called the point of subtraction. So there are several ways of fixing this problem of chi not going to 0 at infinity. Subtracted once. This assumes that chi is such that it goes to it does not go to 0 at infinity but goes to a constant. Suppose it goes like omega prime itself what would you do? Then I would have to subtract at 2 points and so on. So each time you add a denominator it improves the convergence okay and then it is doubly subtracted dispersion relation and so on. As long as it does not have an essential singularity at that point at infinity you are in good shape as no there is no. So if it goes if it blows up like some polynomial like some power of omega prime as omega prime goes to infinity you are okay. Blows up exponentially you cannot write dispersion relation okay. So these are techniques for just techniques for getting rid of singularities but this tells you what the basic idea is. There is one more very important case where you have to deal with this situation which is as I mentioned it happens so happens in many cases that the DC susceptibility is actually divergent where it blows up namely you apply a steady force to the system forever and the response becomes it diverges okay. What do you do then? That implies that chi of omega has a pole or singularity at the origin and in the simplest case it has a simple pole okay. So suppose that happens suppose so this part is taken care of now suppose chi of omega has a simple pole at the origin so it is of the form some residue A over omega plus regular part in the neighbourhood of omega equal to 0. So suppose it has a simple pole at this point at the origin here is my omega then the thing to do is to go back and consider this contour and consider the original function itself. So you have chi of omega prime over omega prime minus omega and you look at it over this contour so indent this in the upper half plane so as to avoid the pole and stay in the region of analytic behaviour of this function and include the behaviour of include the contribution from here in the contour integral. This is going to give you this indentation is going to give you minus i pi chi of omega what is this going to give you? What is the contribution going to be? We are going to have to integrate remember that on this contour omega prime is just equal to epsilon e to the i theta plus 0 because it is at the origin so this thing here integral d omega prime over this little semicircle so let me call this little semicircle little gamma little gamma this fellow here is going to become equal to this is going to go like a over omega prime plus the rest of it right so the leading term is going to be a and then an integral from pi to 0 epsilon e to the i theta i d theta that is d omega prime divided by omega but omega is epsilon e to the i theta so epsilon e to the i theta that is the chi the behaviour of chi with an a here and then there is this factor which is harmless so that factor is 1 over epsilon e to the i theta minus omega so the pole contributed a 1 over omega prime which cancels gives you this epsilon to cancel this so this fellow goes away and this is i pi or minus i pi times a and the rest of it will follow with a minus omega so the whole integral will have a contribution which is essentially a over omega which it should because you are writing a representation for chi and if it is got a singularity with residue a at omega equal to 0 there better be an explicit term a over omega that is what is happening here so tacitly the point of subtraction has been the origin in that sense so in all these cases we know how to deal with this situation but the physics of it is that causality leads to dispersion relations from us chronic relations for the susceptibility generalized susceptibility and that is a general statement and you can write it in terms of physical frequencies using this which is then usable usable for numerical evaluations okay now let us come to terms with what this susceptibility actually is we have to look at the response function a little more carefully so let us do that we will not attach some physics to the whole thing so given for instance a quantum mechanical system can I say what the structure of this response function is what does it really look like etc in particular I want to be able to write things in terms of this spectral function if you recall I pointed out that just to write these formulas out again chi a b of omega is integral 0 to infinity d tau e to the i omega tau phi a b of tau this guy here we showed was in the canonical ensemble it is equal to beta times the equilibrium the canonical correlation between a dot of 0 and b of tau we already defined this quantity the classical case is subsumed in this as a special case and this was analytic in the upper half plane etc etc we also had defined a Fourier transform of this quantity of omega and I call this the spectral function for reasons which will become clear now this was the Fourier transform of phi the response function so this was equal to integral d tau minus infinity to infinity e to the i omega tau phi a b we still not talked very much about what is this quantity for tau less than 0 because the susceptibility just involves this guy here I will come back to this we will deal with the question of how to define it for negative values of tau but notice that this spectral function this guy here was also related to the green function we found out what the Fourier transform of the green function was and the Fourier transform of the green function which is this multiplied by a theta function of tau if I called it g a b of tau that quantity it is Fourier transform was the susceptibility exactly the susceptibility we also found a relation between this fellow and this fellow by writing a theta Fourier representation for the theta function and if you recall that was phi a b of omega was equal to an integral from minus infinity to infinity d omega prime phi a b tilde of omega prime over omega prime minus omega minus i epsilon in the limit in which epsilon goes to 0 from above and there was some i factor somewhere here would you check and let me know there was an i dependent on the Fourier transform convention which I fixed and once and for all but I am pretty sure there was an i here somewhere i or minus i or something like that minus i I just want to keep this straight so now let us see what the content of this response function is in general what would it imply we will take a specific case and I will do this in the simplest notation possible and then we can add frills to it later so we will look at a quantum mechanical system to start with we take a and b to be Hermitian operators their physical observables or operators corresponding to physical observables and then I am going to assume that there is a discrete spectrum of the Hamiltonian H naught in this system just so that the notation becomes simple. So let us suppose that this Hamiltonian H naught has a complete set of states labelled by a quantum number or set of quantum numbers let me call n for collectively so I have H naught on a normalized Eigen function phi n is En on phi n and n runs say 0 1 2 3 whatever a discrete spectrum just for notational convenience and then this is an orthonormal basis so let us say phi n phi n equal to delta nm and similarly sum over n phi n phi n equal to the identity of it so it is a complete set of states and it satisfies orthonormality are you familiar with the terms completeness and orthonormality? So given any state vector of the system you can always write it as an expansion in terms of these phi n's here. This summation here is supposed to sum over states labelled by this quantum set of quantum numbers n and not the energy levels because they could be degeneracy in general so every time I write a sum like this it is not over the energy levels per say but over the states of the system. Then this response function phi ab of tau if you recall this quantity was equal to the equilibrium expectation value of A of 0 B of tau equilibrium that was one of the formulas we had for the response function and in the quantum case this stands for 1 over ih cross A of 0 comma B of tau. So let us calculate this in this basis we need to compute a trace that is what this thing here means. So let us calculate that in the basis phi n you can compute a trace in any basis you like but let us do it in the basis of Eigen states of the unperturbed Hamiltonian H naught pardon me no it is still a commutator no no no the response function is not a dot after I compute right then it becomes dots and so on what happens then you get rid of the commutator and you get this cobalt transform or whatever it is and then it becomes a dot but before that it is just a commutator. So let us calculate this let us calculate A of 0 B of tau equilibrium this is equal to that is the first term in the commutator we just compute this number this is equal to a trace which means a summation over n phi n e to the minus beta H naught that is the equilibrium density operator A of 0 B of tau phi n which is equal to a summation over n e to the minus beta H naught on phi n is e to the minus beta E n because it is an Eigen state so e to the minus beta E n phi n A of 0 and now for this B of tau let us write this as e to the power i H naught tau over h cross B of 0 e to the minus i H naught tau over h cross that is the meaning of B of tau so that is the Heisenberg picture operator but that B of 0 is the Schrodinger picture operator because that is at t equal to 0 that is the way we define the Schrodinger picture okay. So this fellow here I have to put this in but what I could do is in between I put a phi m phi m and sum over m that is the identity operator and then I put this in e to the minus i H naught tau over h cross B of 0 e to the of plus i and minus i H naught tau over h cross times phi n this is equal to a summation over n a summation over m e to the minus beta E n on this side what would you call this? This is a Schrodinger picture operator the time independent Schrodinger picture operator and if I represent it in the basis formed by the Eigen states of the Hamiltonian unperturbed Hamiltonian that is the n mth matrix element okay. So this is just a n m there is no time dependence that is just a number some complex number in general does it have to be real number a is a Hermitian operator does this have to be real number no not true in general does the diagonal element of a have to be real number okay alright. So this thing here a n m and then here I can pull out an e to the i this becomes e sub m and this fellow becomes e sub n and you can take it outside and what is left is B m n and then in e to the power i over h cross tau and then you have an e m minus e n that is it. So let us give this a name let us write this as e to the power minus e n minus e m what does that give you that says and let us now it is natural to do the following this is the energy difference between the n and states let us call that omega n m times h cross. So I define h cross omega n m by definition equal to e n minus e n it says a of 0 b of tau in equilibrium can be written in the compact form summation n comma m a n m b m n times e to the minus beta a n n and then e to the minus i omega n m tau. So it can be written in quite a compact form. The next step is to do it for the other part of the commutator right. So let us write the other way B of tau a of 0 and this is now going to have B of tau a of 0 and since I want to match with that let us take the sum here to be over m and the sum here in between to be over n and this fellow becomes B out here we have to work this out properly. So it is just a of 0 this fellow here and now introduce a complete set of states here by putting i equal to summation over n phi n phi n therefore you can write down directly what this quantity is B of tau a of 0 equilibrium equal to summation over n comma m you are still going to get B m n and then you are going to get a n m which is the same as this. So this is still a n m b n n but now this is going to hit this and give you e to the minus beta em. So this is going to be e to the minus beta em and the rest is going to be exactly the same as before. So therefore what does the commutator do and let us divide by 1 over ih cross because that is what the response function is. All these portions are common but this becomes e to the minus beta em minus e to the minus beta em and this goes away. So this is equal to 1 over ih cross summation over n summation over m a n m B m n times this product that is it and this is equal to phi a B of tau. So we have now explicit equations in terms of physical quantities the matrix elements of these operators you have a representation for these operators then all that rigmarole about going to the interaction picture and so on and so on finally ended up with just these quantities the tau dependence is sitting here and the temperature dependence is sitting here and this is a set of exponentials that is it. Now it is trivial to calculate what the correlation function is what the spectral function is what is going to happen you are going to do this integral and there is a minus out there exactly. So what will this be and we will stop with that today phi a B tilde of omega therefore equal to 1 over ih cross summation over n summation over m a n m B m n e to the minus beta em minus e to the minus beta em times e to the i omega minus omega n m integrated over all values of tau that is a delta function 2 pi times delta function. So this is 2 pi so the spectral function is a function of frequency has peaks at all these points if the number of the levels are very close to each other it is a large system with many levels then it is going to look like a continuous spectrum but this is these are the characteristic frequencies the next thing we will do is to write down the susceptibility based on this representation. So it will give you a explicit omega dependent thing involving the transition frequencies of the system these are the transition frequencies of the system between which love between which transitions can occur due to a perturbation okay. So and then we will interpret this so let me stop here today.