 So, let me write down the sequence of things we have been doing and then you can see. So, so far is that we defined the quantum mechanics through path integral which actually meant through transition amplitudes, but we basically checked that this gives correct kernel for the free particle. Now, from this point on we introduce this method of external current or forcing function and this is a method there is no real forcing function. So, this is method of forcing function which will later become external current this is Schwinger's way of thinking of it. We also made a transition from transition amplitude to vacuum to vacuum amplitude ok. Again a concept essentially due to Schwinger vacuum to vacuum amplitude is apparently a fake thing because why would you what would you learn from going from vacuum to vacuum, it is like being back to square 1, but the point is that in fact, the vacuum to vacuum amplitude is done in the presence of the forcing function. So, you get a vacuum to vacuum amplitude in the presence as a function of this auxiliary variable and then by varying this variable you can obtain all the information back ok. So, but this is as a function of functional of the forcing function and that is the key thing that you obtain the vacuum to vacuum amplitude in terms of the forcing function and then this force w of j becomes the generating function functional of endpoint functions or green functions. So, that is what we have done so, far what we so, we can say what happens is that we do omega plus infinity omega minus infinity to be equal to integral over d q f d q i of omega plus so, omega infinity q f t f times the more physically transparent transition amplitude right. This is obvious getting and in the presence of a current j with this what did we call it f over there here f. So, vacuum to vacuum amplitude in the presence of f basically takes this form and then this becomes we saw in the limit that we take t and this here we take t f t i to also infinity infinity and minus infinity. We basically recover plus i the basic path integral is with i integral minus infinity to infinity d t of whatever the action is we have been writing in terms of q q dot t, but plus i times integral q t j of t f of t. Now, at this point itself we can observe that if we define correlation functions as not phi I am sorry q t 2 q t 1 as equal to 1 over i d by d f t 2 1 over i d by d f t 1 of that transition amplitude and evaluated at f equal to 0. This is all we did so far effectively except for the calculation method of doing quadratic integrals to recast the path integral in various ways and of course, we will be using it again and again. We also use the stationary phase method to check that the path integral has to become stationary on the classical path. So, but other than that this is all there is and if you have n points then you put n of those and then you can recover the answer. There is an illusion among people that path integral is a good thing to do quantum mechanics this is completely wrong. The main use of path integral is only after you make transition to quantum field theory and then to derive relations between greens functions. So, q f t again has been used primarily as an S matrix theory the only thing we do. So, we tell everybody to get them excited that we are calculating n point functions, but what we really have in mind is calculating scattering of n particles. So, all we do is there is a very formal procedure which then converts the n point function into the n point S matrix n particle S matrix. So, we we calculate those transition amplitudes the S matrix and not necessarily energy stationary states. So, in fact, q f t fails completely I shouldn't say completely, but q f t has not proved to be very useful to compute any bound states nor is path integral very useful. I know that there is there are several textbooks entire textbooks written on how path integral is very useful in quantum mechanics. Well you can read them for their own value whatever they have, but I have never read them and I can vouch that no chemists will need them to calculate the many electron bound states. The chemists do use however, the greens function ideas because they want. So, greens function ideas make it a little bit more formal and peg it on a slightly different level, but path integral is not going to be. So, the only computation you can really do with path integral is the Gaussian integral and later we will see that it helps you to derive the so called diagrams. So, so called weak contraction at two point function set of time, but there is the kind of power that you have in a partial differential equation which allows you to solve I mean hope for getting exact solutions for many different potentials that does not exist in quantum field theory. And in quantum field theory so primary use of this method has been to calculate S matrix elements. If you want to calculate so I can even write it down here how should I say becomes most useful. So, this beta Salpeter functional equation has been studied by a lot of people and lot of work exists, but I do not think I mean we were never taught that it so calculates very and if it was then we would be not doing lattice gauge theory. So, that is the status, but we can also say the other use of the functional formalism is in fact, to implement this theory on the lattice you can implement quantum theory on the lattice in this functional formalism. So, for lattice gauge theory also it is a useful thing, but lattice gauge theory is then just a completely numerical calculation it is Monte Carlo calculation of that functional integral because there is no at the only approximation schemes or the only exact calculation schemes available is the Gaussian integral or the so called steepest descent method in some approximation you can it is like the stationary phase we will see it. So, there are very few tricks available at the functional level that allow you any kind of quote answer, but this trick does allow you to obtain functional relationships. The trick of partition function is trick of your partition function in thermo in statistical mechanics also and use is something similar. So, that is really all there is to it, but it is extremely powerful for the purpose for which we are going to do it. The conversion from Green's function to S matrix is itself quite a formal statement, but once you get over it you get used to the idea it is not all that difficult. Hopefully I will be able to do it if I have the time. So, we have to interpret this as product over all E where you have to order the E's the energy spectrum because path integrals are always ordered and that was one thing I was going to comment here I forgot where we wrote this Green's function we actually end up calculating only the time ordered product. So, it is automatic in path integral that you will get the functional method that you will get the time ordered product. So, this is the W 0 and the W which is not of much interest anymore and the other part in detail is well let me see if I can say it without. The only thing is the sign in D and it is not square root because so T 1 minus T 2 there is a minus sign. The square root is in the Fourier transform definition, but here it is not there and E squared minus omega squared plus i epsilon right. So, everyone knows all this and you know that this boils down to theta where should we write i E will be equal to omega. So, i omega t and plus theta of T 2 minus T 1 times E raise to plus i omega t think there is a 1 over 2 i omega from the poles right. So, because it is square you will be taking E plus omega E minus omega and each one is a pole. So, from each of the poles you get each of the pole is relevant depending on whether this is greater that is greater than 0 and the value of the pole is that the 2 pi goes in the integration in the contour integral. Thinking and it was a great discovery when Feynman used this propagator that positive frequency particles go forward in time. So, omega is a positive number it is positive square root of the omega squared. So, it gets a minus sign which we which is the correct time evolution according to Schrodinger convention of setting energy operator to be equal to plus i d by d t. So, with the minus i it gives correct omega. So, this is going forward in time, but this would give negative energy or would go backward in time and that is the interpretation. I if this has not been told you before I might as well spend a little time here telling you about is going forward and backward because this is at the heart of causality in quantum field theory. So, is the or it should be there in. So, this is a very general argument and the way Weinberg puts it particularly is that. So, look at the uncertainty principles uncertainty principle there would be 4 statements, but look at say d e by d e d t uncertainty principle. If you try to, but in relativity it is not delta t and delta x that really matter because 1 persons t is another persons mixture of t and x and actually Weinberg writes a so called uncertainty principle which is written like this. So, it basically says that the space time interval between 2 observations has to remain greater than the Compton wavelength ok. So, you can think of this you can put the M on this side and treat it as if it is well not really, but this is the form in which it is written in his book and you can see that what this is saying is that you can have single particle interpretation only provided you do not probe the object in space time intervals that are smaller than the Compton wavelength ok. So, delta e delta p no. So, we I meant to suppose a delta t square minus delta x square here, but now in quantum mechanics you could you could always probe the system more closely and then you will lose the single particle interpretation as you have been taught probably in its most relativistic quantum mechanics course is emphasized that you will create particles. If you probe at this length scales shorter than this then the value of delta e and delta e will have to be larger than the mass scale of the particle and you will end up creating more particles and the single particle interpretation will be lost. But we also have a more specific into more specific statement suppose that I have creation of a particle. So, now we draw this space time diagram and the light cone normally if you create a particle here it will be later found here right. So, this is T 1 and T 2. So, it will propagate from this to this, but quantum mechanics only tells you some inequalities it does not say delta x cannot be less than something or delta t all that you have to do is maintain this. But if this happens you could also have a situation where T 1 is here and T 2 is here. So, T 1 prime I do not want. So, I will remove the old T 1 T 2. So, suppose it is T 1 and T 2 like this, this may not be forbidden by this relation because all I have to do is adjust that the delta t square is bigger than minus delta x square remains bigger than this m square and well actually it will be negative in that case. But because I do not have control in quantum mechanics it may very well happen that I create a particle here, but destroy over there. This is the also say bell inequality thing you do something here and it gets also it determines something over there. So, the point is that we recover the causality correctly in this case because it is possible for you to hear the events as space like separated. So, it is always possible to at least reverse the time for space like separated events it is possible to rearrange this is a little let us see how does one recover I would have to really tilt it a lot until the projection on to that axis reverses the directions of T 1 and T 2. If it is a space like separated interval then I can and if I choose my new axis to be like this then now I have to do a projection parallel to this axis. So, this will go so I have to make the slope of this bigger than this. So, I do this and this. So, this is T 1 and this is T 2 it clear T 1 T 2, but now and this has a slope like this. So, I do draw a new choice of axis which is highly relativistic. So, it is approaching the light cone very thin close to light cone. If I now project these by drawing lines parallel to this they will go and hit the time axis in the reverse order intersected it already right. So, the time and this one from here to draw like this until we hit this axis already there. So, if you will do this carefully in your notebook you will see that the projections which are this is the new x prime axis. The projections projection lines parallel to x prime axis which go and hit T prime axis the T 1 and T 2 are reversed and the well known result and you can find the algebraic expression for the Lorentz boost required for this to happen. So, now there is a question of causality that you emit a particle here, but absorb it here which is at a later time in this frame of reference. In the other frame of reference it will look as if it got emitted first and got absorbed later. This problem is solved by a quantum field theory because of this because in the other frame of reference it will look like an opposite charge particle went backward in time that is what the interpretation is. So, for a charge system Q plus Q created at T 1 and absorbed at T 2 is equivalent to minus Q created at T 2 and absorbed at T 1. So, this amounts to the T 1 location reducing its charge in both the things. What is same is thus both the charge of 1 is reduced and that of 2 is increased. In the person who is observing frame of reference as is clock sticking he considers time sequence as going forward in time strictly and in if the two events due to the this uncertainty in some this is what I meant to suggest. In some space time region well actually it will not be a circle, but some kind of hyperboloid when so long as this is of the order of m anything can happen here. In particular particle can get produced and annihilated at space like separated points and if that happens with a particular time sequence and if it is space like separated in another one it will look like creation event is happening after the destruction event well that is not true because it will be in his frame in the other observer's frame of reference. It will be interpreted as minus Q sequentially got created at T 2 prime which in this frame of reference occurs before T 1 prime. We will find that if you compute the for a massive field if. So, for massless fields we will find that the support is only on the light cone. So, if you are sitting at you start with the origin as the reference the greens function the Feynman propagator will have support only on the light cone only on these points outside of that it vanishes, but if you do it for a massive particle then you find some slightly different function which has of course, higher support here, but it also has a little tail outside it is not strictly on the light cone. It is a exponentially dying tail just like in a barrier problem barrier problem the wave function penetrates under the barrier. So, in quantum field theory the two point greens function will actually protrude into the classical relativity forbidden zone, but exactly of the order of the Compton wavelength and not more and if things happen in that region. So, if you see creation destruction events in that region they will be resolved by this explanation.