 There is this question why we add epsilon term proportional to phi squared and not to any other operator and the answer is of course, we are trying to add the simplest possible thing that gets our job done and it is just convenient to add a quadratic term which will get in which can be integrated easily by Gaussian integral and so, it will just modify the kinetic terms and therefore, it modifies only the free propagator and does not enter into interactions of course, eventually you are taking epsilon to 0. So, you just want to put it in the most convenient place and such that it dumps the integral also. So, actually it is quite a fortuitous thing because originally epsilon was put by hand meaning that i epsilon prescription is one out of many you can have right I hope you know this. So, prescription so basically the propagator is inverse of this operator we have box plus m square phi and in the notation of where it has some external function J. So, we convert it into a Green's function problem the Green's function problem says and since we are going to call it that let me or let us put G at first some Green's function of x x prime and now we set it equal to minus delta for x x prime right. So, you are familiar with the Green's function technology right. If you solve for this Green's function then any if we for some boundary conditions then the solutions then solutions of same boundary condition can be found for any J simply as phi x equal to integral G x x prime J of x prime probably with a minus sign yeah because there is this minus sign there I do not know why, but these are conventions. So, you can always find anything like this. So, the question is of putting the boundary condition right and formally G is simply equal to G x x prime is just the inverse of this so whatever acts on this right. So, it is so formally it is inverse of this operator this can now be defined in momentum space it becomes simpler because we can write it simply as equal to 1 over p square plus minus m square because minus p square plus m square because we will put e raise to i p x. So, the box operator well I guess it to just worry about the sign overall minus sign that does not matter the relative minus sign is relative sign becomes minus n plus ok. Now, this thing which is of course, in detail equal to p 0 square minus p square minus minus p 0 square plus p square plus m square which we write as equal to 1 over and I take out a minus sign p 0 minus omega p and p 0 plus omega p where omega p I simply defined to be equal to the positive square root of mod p square plus m square. So, now there are these 2 poles in this and the answer for G is the Fourier inverse transform of this I guess 2 pi square in Ramon's notation and e raise to minus i p p mu d. So, it is p dot x minus x prime for I mean Minkowski inner product times this minus p square plus m square this is the formal Green's function and you have to give it some meaning because of these poles how will you do the integration across these poles. So, it is split up into d p 0 and then integral d 3 p over 2 pi square, but then this thing with an overall minus sign. So, now the question is of how you negotiate the poles in the p 0 axis and there are several prescriptions you can give. So, the question is of the prescription to provide to identify the Green's function that you need for your boundary condition and if you are in classical electrodynamics as you will see in Jackson's book for classical radiation you need retarded boundary condition which means that whatever G is is equal to 0 if t is less than t prime. So, the order of the arguments will then matter and it should propagate from t prime to t, but if t is less than t prime it will be 0 and is equal to something non-trivial. So, for t greater than t prime. Now, whatever so you have to give this boundary you have to create it like this you have to specify some prescription for negotiating poles so that you get this and advance is defined by this equal to non something non-trivial and for t greater than t prime, but is equal to 0 for t less than t prime sorry. So, advance means it is coming from the future it will propagate things only backward in time. It is quite a curious and interesting history that Feynman was actually fooling with these boundary conditions along with his advisor John Wheeler as a student because of a sorry this is taking us on a historical trail, but if you will see Jackson's book towards the end there is a whole topic of radiation reaction. So, radiation reaction is actually a crisis of classical field theory because what happens is that if you oscillate a charge it emits radiation. So, it has to slow down, but can you derive the slowing down from the equations of motion and there is a problem because so if I wiggle a point charge radiation goes out I can of course, calculate all the radiation going out and therefore, I can calculate the rate of change of energy d e by d t I can calculate. So, d e of charge by d t should be equal to d radiation or just say energy carried away by power in the radiation minus of the power carried away by radiation. This equation you can easily write, but can you derive it from equation of motion of the charge because equation of the motion of the charge is x dot dot equal to q v cross q e plus v cross b right. Can you integrate this to get and we know that energy is equal to half m v square for non-relativistic charge. Can you do a first integral of this motion to get this and the get its rate of change to match this rate of change. There is a serious problem because e is a point charge or q is a point charge and what is the value of the electric and magnetic field at the site of the charge. There are no other charges in the universe this is the only charge oscillating. So, it has to be its own field, but if you put its own field it is infinite. So, there is a conceptual difficulty trying to get this equation to be derived from what in the old literature was called ponderomotive equations for the charge. Ponderomotive means the dynamical equations. So, not only that this radiation formula is well known. So, this equation is well known what this has to be. So, can you reverse engineer to somehow get an expression for x dot dot yes. So, the radiation zone is only the 1 over r zone which will be realized at infinity. All the remaining energy will sort of remain its in the fields which remains confined yes it remains confined to some near zone, but it will not be lost it will not be radiated to infinity. Yeah that is what we so think of think instead of continuously forcing suppose with an impulse you and you have a you should have a spring of course, so that it oscillates. So, you are putting some external force into the problem and if the spring is conservative spring the radiation will take away the energy. So, you just pull the charge and let go. So, it has to come to rest. So, it has to undergo some rate of change of the other stuff will just kind it is called reactive fields and they are not dissipative fields they will not take away the energy. They will keep reinforcing the motion in some way probably, but the big problem is that this is even one used to know the dependence. Oh no not this dependence, but therefore, what happens is then that the only way you can get the answer is if you put. So, it actually looks like you have to it determines. So, these conditions determine x dot x triple dot. So, this to recall I will not be able to remember everything now, but the reconciliation constraints x triple dot which is called the radiation reaction force. I can actually even remember why it would become triple dot if we remember the pole formula, but you can see it is in it is given in Jackson. Now, that is beyond Newtonian scheme because it is now x triple dot. So, there were various rather exotic proposals one of them was to say that actually there is some action at a distance between charges and that it is not dissipated in this detailed way and so on. Dirac being a clever guy observes that the x triple dot formula comes out exactly correct if you use this equation, but in the fields you do not put the self field trivially, but you put retard and minus advance ok. So, Dirac prescription and now not to be mixed with that prescription, but Dirac proposal E B obtained from G retarded minus advance reproduce behavior correctly because only G by itself would be divergent at the source, but if you do G retarded minus advance then it exactly cancels out the local fields and retains only the radiation zone fields ok. So, that correctly has in it behavior of the radiation zone fields. Now, this tickled everybody's mind while it was retarded minus advance and so on. So, Wheeler and Feynman tried to create a system for this which is a abandoned theory it appears in reviews of modern physics sometime may be 1948 or something like this. They come up with a completely crazy idea which says that actually in the classical physics all influences should be taken as retarded plus advance that is at least symmetric. So, you may ask why is it that you are forced to take G retarded and there is no answer except that that is what you observe you only see cause. If you just take Maxwell's equations at time symmetric. So, why should you be taking only retarded? There is no answer from within the theory you have to put it from observation. What they now did was to grab on this opportunity to say that actually it should be time symmetric and they claim to derive the retarded minus advance prescription by a very crazy process. They say that all the charges in the universe are emitting retarded plus advance, but now what happens is that if this guy is going to oscillate at some point its advance potentials are advancing upon it you have to picture what advance potential means like retarded means a plane wave going I mean a wave front going out advance means a wave front closing in. So, if you are going to do it now 3 million light years ago from Andromeda galaxy a wave front is closing in on you. So, that it comes here right now only if that happens will you. So, what the show is that if you take all the charges in the universe and consider the advance and. So, this advance potential is hitting all the other charges who then react to it and whose reaction then comes and converges on this charge. If you do this drama correctly which I do not fully believe, but they claim that you exactly get retarded minus advance at the site of the charge. So, the charge is trying to emit retarded plus advance, but because its advance potential influenced everybody else in the universe finally, the net result looks like it is retarded minus advance at the site of the charge in such a way that effectively on all the charges then you only have retarded happening ok. Now they consider this a great triumph also of solving the arrow of time problem. So, you may you have a cosmological arrow of time expanding universe you have entropic arrow of time thermodynamic arrow of time, but why should microscopic electromagnetism even when you do not have any entropic considerations you do a very simple dipole experiment that should there is only retarded potentials. So, they claim that they solved the arrow of time problem of classical electrodynamics by the absorber theory, but then this absorber theory was not found sorry Wheeler absorber theory was rather gobbledygook and. So, Feynman was playing with these combinations even as a student and later on however, to do the relativistic electrodynamics and its greens functions he eventually had to pick it from Stuckelberg because the prescription was even more weird than retarded minus advance. It split positive frequencies and negative frequencies making positive frequencies go forward in time and negative frequencies go backward in time. So, there was one more prescription which had been missing from all of these and that is the prescription we use. So, there is a very simple. So, to get retarded all you have to do is to make sure that both the poles are below. So, when you close in the upper half plane you get sorry that would be the advance. So, if you do it like this then when time is positive you get contribution of the poles, but if time is negative then you do not get contribution of the poles which can be arranged simply by shifting the poles down. So, to both of these you. So, this minus omega, but minus i epsilon and plus omega, but also minus i in both cases it becomes plus i epsilon because you will subtract minus i epsilon from both. So, that is one prescription, but if you cross multiply it you will get an answer proportional to p 0 times epsilon i epsilon, but p 0 is not relativistically covariant. So, you will be start with a non-covariant propagator and so on. Mr. Stuckelberg for whatever reason he used figured out that the correct thing to do is what we know now which amounts to putting one up and one down. So, it amounts to doing this. So, for positive time you want let us say this to for positive frequencies to contribute. So, this is correct, but for negative frequencies it should contribute when it is negative time. So, at the negative pole it is also minus. That also makes the answer it now multiplies omega p I think which is at least positive number. See p 0 is being integrated, but omega p is positive square root. So, it comes out epsilon times a positive number and that uniquely fixes the. So, it is a prescription that one has to put to get that propagator. So, you have to interpret the Green's function in some way. The formal expression is simply inverse of the differential operator which you can find in momentum space like you can do in electrical engineering where you have both quadratic and first. So, this is all second derivatives, but if you had first derivatives as well then you need a Laplace transform instead of Fourier transform and you know that problems become algebraic instead of having to solve differential equations. Here just to I may have forgotten to emphasize, but the point of this machinery is that it is like a master key solution because it converts a partial differential equation problem to an integral equation with the boundary condition already built in. So, you do not have to careful remember to put in any boundary condition. You just feed it into this machine and integration is always the psychologically easier than difference then solving the partial differential equation because integration just requires you to put some you know do a Simpson integral even on multidimensional space. It is not a conceptually difficult thing and you can implement it on computers etcetera very easily. So, the philosophy of Green's function method is that coronal as we have been calling it is that it converts partial differential equations into integral equations except that you also have to remember to pack in the boundary conditions in that Green's function. Green's function essentially comes from differential equation theory. You did not have any quantum field theory at the Feynman did not do any quantum field theory at all. He just put the propagator there and integrated. So, but the how to get the propagator from quantized field theory that is where this time ordered product of quantum fields and vacuum to vacuum thing comes in quantum field theory it becomes this. This was elucidated by Dyson. This is due to Dyson that only if you do time ordered product of the quantized fields you will automatically get the stucco work propagator, but then that you needs to recover propagator simply has to do with having a covariant propagator. The other boundary conditions will give you non covariant propagators. If you do purely retarded then you have a particular choice of time slice and if you do Lorentz transformations it will not remain covariant. And so, you can see the explicit co-variantness of it in the expression as well because it will be p square minus m square plus i epsilon and it everything is Lorentz co-variant. So, what is quite miraculous is that that time ordering which in Dyson's prescription look something artificial because you have to specify it. In part integral or the functional method is automatic because of the interpretation of that functional integral as a time sliced integral. And on each slice you integrate over the whole configuration space of the scalar field or whichever field it is and that is what it is. So, now just to reorient you again what let me write our emblemic formula we have been writing for a few days now and we have this formula w of j is equal to w of j equal to 0 times e raise to minus i over 2 and this what we have been writing isn't it integral dt dt prime dt 1 dt 2 f of t 2 d of t 2 minus t 1 f of t 1 where dt t 2 minus t 1 is equal to integral d e over square root 2 pi probably e raise to and then theta of t 2 minus t 1 times e raise to minus i e t i e t plus theta of t 1 minus t 2 e to the plus i e t. So, I am sorry this is w of f right now it is for the oscillator, but in field theory this becomes w of j equal to w of j equal to 0 let us write it like this and e raise to minus i over 2 integral d 4 x 1 d 4 x 2 of j and then there is this delta Feynman. And we will learn this technology that you can obtain g n point functions which are defined to be the average value. So, this is actually just vacuum to vacuum and time ordered product of that is right. So, this object then would be calculated from the this formalism by taking variations with respect to j because each variation would bring down a phi and put it in the. So, this then basically just acts sorry going back the original action function acts as a weight e raise to i times action acts as the weight function with respect to which you take averages and they can be computed like this. Now, it so this is why this is called the generating function. So, w is called the generating function of Green's functions and evaluated at j equal to 0 ok. So, you vary and then said j equal to 0. So, the w of j is called generating functionals.