 So I'll give you a brief summary, so today we'll hopefully get beyond the general discussion. But the general discussion is very important. So I'll give you a brief summary once again of our main ideas, more for questions, and then we'll start trying to understand the scale of doing this more specifically. So let me give you a summary from a slightly different words than what we've used. We have these renormalization group equations, and we believe the structure of these renormalization group equations is that they can be controlled in the infrared by fixed points, in the many fixed points, in a set of, in some of the space where the gauntlet itself is. The thing that's really important for any given fixed point is the set of relevant definitions. Marginal definitions that are marginally irrelevant come to just like relevant definitions. Marginal definitions that are marginally irrelevant come to like irrelevant definitions, and irrelevant definitions like that. Now, suppose we've got, for instance, we've got a fixed point with two relevant directions. The flows come out of the board, and the board is the space of the relevant. The flows from far away come out of the board, and sort of come in, are quickly adapted to the board. However, they equally well quickly retell from the fixed point onto the board. So if you could just do something totally random, you'd land up on the board. The board of course is a curved map. It's not an example of that. You land up on the board where infinity, once you float in an infinite amount of time. That would be the j-hex. However, we're not interested, you see, these things, for instance, if this was mass variation, landing up over here at infinity would be turning on a very large mass compared to the energy scale of interest rate problem. But typically, it would be an infinity. We're not interested in such things. So in this space of such renormalization flows, what we do is tune two parameters in the UV. We start in the UV, not completely randomly. We start with a two-parameter center. We tune these two parameters so that when we get it to come very near the board, we also approach very near the fixed point. So at some given scale lambda, as we take lambda, not infinity. We're at a finite point of the board. This was the procedure we implement. Changing lambda, not changing the bar parameters. So as to be at a finite point. Again, at some finite scale. Now, let's think about this a little bit. We were at a finite point at some finite scale. What does that mean for a wave will be at scales one million times? So there's lambda not which we take infinity. There's lambda which is finite. And now let's make some large number. That's somewhere between 1 and infinity. So some scale lambda prime. That's much, much larger than lambda. But much, much less than lambda. Because lambda prime is much, much less than lambda not, we get the board. The irrelevance reformations have done that. We're on this model sheet. But because lambda prime is much, much larger than lambda, in order to know where we were, we do the inverse flow on the board. For time, log of lambda prime by lambda. But how does an inverse flow on the board look like here? It looks like this. So now we do the direct flow. What does it say? It's saying that in this process, at scales lambda prime, as the ratio lambda prime by lambda goes to infinity, we are arbitrarily near the fixed point. So this limited procedure, engineers flows that in the actual limit go and set it almost on the fixed point. And then go. Do you understand? The limited procedure that we use to defend quantum field theories tunes our flows. So that they basically are the critical flows. The critical flows are those that go at end up. So it tunes it. So that's basically the critical flow with a very small deviation. That deviation goes to zero as lambda not goes to infinity. But it's tuned so that exactly at scale lambda will be in the spine. This is who I become as for any intermediate scale between lambda not, you know, let's say square root at some lambda prime such that lambda prime by lambda square root of lambda not by lambda. We are in the limit of lambda not goes to infinity. We are already at the fixed point. It is for this reason that we achieved the statement that I told you about last time. Namely that quantum field theories are defined by renormalization of root flows away from fixed point. This is an extremely important, not very, not completely whether we see the statement. But the definition of a quantum field theory. We constructed by this limiting procedure. But all that this limiting procedure achieves is constructing at finite scales. The scales you've got to allow is constructing a flow away from fixed point. So this limiting procedure is one way of constructing but it's all it's doing is achieving the following statement that we want to construct a flow away from fixed point. Quantum field theories are defined by renormalization of root flows away from fixed point. Okay, renormalization. Now it's a renormalization of flow away from fixed point. See every fixed point has an infinite number of directions in which they form. But definitions in most of these directions take you back to fixed point. These are the irrelevant. Those do not define quantum field theories. Quantum field theories are defined by flows away from fixed point. Continued quantum field theories. Okay, commencement of people often use the word quantum field theories include cut-off word. This is not our usage of it. We want quantum field theories that are defined all the way through the after-mine in the procedure that we've talked about at the moment. Okay, and such theories are defined by flows away from, is this completely utterly true? The space of quantum field theories associated with any fixed point has a dimensionality specified by the sum of the parameters. The number of parameters is equal to the number of irrelevant or exactly minor operators about fixed point. Quantum field theory. Quantum field theory. What you are saying seems to be there that you end up with an infinite mass of theory. What we are saying is that? Quantum field theory. I think on the cut-off, it seems there that you end up with a theory of infinite mass. No, if you start with a quantum field theory then you don't necessarily have to, you won't necessarily have to go to the back of this. You see, we float to the backboard because we start with a lambda not equal to infinity. Okay? So you start with a quantum field theory, you put something at some finite scale, then, and you want to be exact, then you find an infinite number of possibilities. So, quantum field theories, or at least me, and quantum field theories are just a couple of field theories. But to the extent that we want to look at them in the neighborhood of these fixed points, they are, they can be thought of as flows that go both away and towards, to take from them. The irrelevant automations are not in that domain. Okay? We are interested in defining pristine, pure quantum field theories. Okay. How can you have a pure quantum field theory? How can you have a pure quantum field theory? Can I have a pristine quantum field theory? Okay, so now there are many, many things in your question. Firstly, it's easy to have a quantum field theory as a practical incident. A non-relativistic quantum field theory is a practical incident. That's not by itself an issue. Now, there are many aspects of your question which are very interesting. One of which is non-locality in space. Breaking it. This cup opposite here. India says non-locality. Non-locality in space is usually not an issue. Non-locality in time. Sometimes conflicts with unity. Sometimes conflicts with unity. Okay? So, there are many aspects to your question which we will not try to analyze. Or a bit at the moment. And it's a good question, but let's... Ah! Okay. Any other questions? In the space of all couplings, there is a surface. On that surface, there is an extinguishing point. There is one flow line that goes exactly to that extinguished point and stays there forever. That's why I speak. Okay? Now there are flow lines in the neighborhood of that. That go very nearer to the fixed point. Stay there for a very long time. And that's that moment. So, one of the theories which does that redefine are defined by flow lines in the infinitesimal meaning of the attractor flow line. Is this clear? Because they have to stay in the neighborhood of the fixed point for an infinite amount of time before moving away. Okay? This is exactly what we want to say in the infinitesimal definitions of the fixed point in the relevant relations. Is that clear? What we're going to try to do is what we're going to try to do is to is to Firstly, I'm really sorry about this problem which you won't be promising. Yugi, should you come sit with me one day, we'll make a problem setting. So, this one is because it's pain of neck. But, okay. So, now what I want you to do is to understand these beta function equations in perturbation theory both for the scale. Okay? In the problem set, I'm going to ask you to do the straightforward procedure. Now, just to take conscience ease, re-normalization of the equations and solve it to leaving dr. del order in perturbation theory. Okay? Just to have a little variety in class, I'm going to work out this problem in slightly different pages of the equation. It's the way that we listen with every single key. Okay? There are many different views of this subject. Okay? So, the problem that we're going to address is the following. It's the same problem we looked at. But now we're going to get the sign. That's not the problem. Okay? So, we take this actually, the bare truth d5 divided by d2 m0 squared d2 2 plus, let's call it g0 divided by 2. This is our bare at scale up. In a nice sophisticated analysis in a nice sophisticated analysis such as re-normalization group flows we use this nice smooth cut-off. But in this crude approach to what I'm going to be doing as opposed to the most sophisticated thing you do. Okay? I will just imagine that the cut-off was a shock. So, imagine that the scale function instead of being like this as we assume in the class was just this. Okay? So, when I've got an action up to scale lambda 0 all modes that have moment to bless the lambda 0 okay? And what I'm going to try to know is how to compute the effective action at scale lambda 0. Starting with the effective action at scale lambda 0 to this one should be okay. So, basically in this g0 as we will see later it becomes very cheap. You'll see that. Okay? So, this is our book. I'm going to do this in this crude way just to give you more intuition in different ways. The way you look at problems is more systematic. But this adds a new layer of intuition so it's not very visible. Okay? So, what do we have? We've got the action. We've got a new body an exponential of minus x we've got a new gradient space. We took so much trouble to understand how to go to a new gradient space. So, we've got this action and we even remember that because of the hard nature of our counter-off. In this action at scale lambda 0 the modes that propagate are all fine modes that have momentum less than or equal to modules of momentum less than or equal to. Okay? Now, I want to find the path in there that is equal to as far as generating correlation functions are concerned. Okay? So, for that as Kolychinski does I put a J P 5 B 5 P I want to find the action that is equivalent as far as calculating correlation functions are concerned. Therefore, it's equivalent as far as Jacobins. Okay? To this action but with momentum at scale lambda 0 I'm working in momentum space because I want to make rest of it. Rest of it? Rest of it is in position. You can write the rest of it in momentum space. Why am I working in momentum space? I'm working in momentum space because there's something that I want to say. J is non-zero because non-zero support only for momentum. That's why I do it. Okay? So, what I want to do is to get an action which maybe has a different dungeon which is a path integral over 5 with momentum only of the scale lambda that will generate the same correlation. Now, how do I do that? What you have to do is very clear. What you have to do is to do the path integral. Over the months with momentum between lambda and lambda of those path integrals will involve only terms with momentum between lambda and lambda. That will just give you an overall normalization. To the old action. Normalization should be okay because we only care about normalization. Okay? So, it's only those terms in the Lagrangian that involve some fields with momentum between lambda and lambda and other fields with momentum below lambda. That would be important. Okay? Let's start. Because we're working in collaboration here okay? with the sculpting constant. In order to do the path integral we first expand the exponential with the sculpting constant in the data series. That's what graduation theory means. So, what we get, we get e to the power minus del phi is the whole thing square root of 2 that's m0 square root of 1 square root of 2 okay? And then the first term is minus, the first term is 1 oh, and then there's this plus minus here is what I'm going to give you in class today. Now, d phi this is what I was saying, this is this. And then we have 1 plus then there's this term just g0 phi to the power of 1 square root plus g0 squared there's a g0 phi to the power integral that's what we're doing. So, this is 1 minus it's e to the power minus s then there's plus 1 by del phi that's all. For all phi there's momentum lies between lambda and lambda. Now, the point is that in the exponential different phi and also different momentum don't talk to each other it's completely decoupled. And the action is just momentum. Okay? So, if we didn't have this we could just do the same thing. And we would have concluded that since we don't care about normalization. We would have concluded that there's no flow. And we do have this. What do these terms do? So, let's look at these terms term by term. You see, let's first look at the one is like not having this. That's not too real terms this guy. This is the g0 phi to the power by 4 factorials by 4 factorials. Why? Because we want to keep track of whether the momentum is below or under. And we see before there are many possibilities. One possibility is that all 4 phi's have momentum between lambda and lambda. These guys, you just do the integral. You see you're not going to normalize it. Forget about 4 phi's have momentum below lambda. Feels with momentum below lambda. I just spectators and however they try to they background the sources. We don't do anything like this. That's just saying that there is a part of the action that goes along to the right. The downside of importance are when at least one field lies in the upper momentum range and at least one field lies in the low. Now we're going to do this path in bed. And as we've discussed as you know very well the path integral with Gaussian explanations follows week's time. And in particular property of that is obvious is that if you put an odd number of phi's in the high momentum range just because we integral same statement as integral x e to the power minus x squared by h squared is equal to 0. So the only possibility is that we will have even number of phi's in the upper momentum range and even number of phi's in the lower momentum range. Yes. Because in the basic zero the integral over d to the power of phi. Yes. Okay. The terms of interest are those in which we've got two phi's in the upper momentum range and two phi's in the lower momentum range. And when we use week's here so let's say that these are the two phi's they have to be contracted against each other. The contraction gives us a fact from the property. The property runs from one momentum to another. So this was a momentum p some upper scale momentum p this is a upper scale momentum minus p. The whole thing is a momentum conserving delta out. So this was a some p prime some lower scale momentum p prime this would be a minus lower scale momentum. This whole thing is represented by this prime when these are the momentum this is p prime the momentum at the lower scale momentum that come in and go out of the problem these guys. These are the upper scale momentum and have to be integrated. So what we get is we have to choose which two guys which two guys to choose as the upper scale momentum. Clearly this is four into three ways. Okay. Four into three divided by power of two it's half. Fancy people will tell you what kind of factor. Okay. So other people think that what is the integral that we have to do the integral that we have to do is let's say we have four guys we form three by two pi one of these five No, no. Six ways. I agree with you just one. You are clear. It's by four. But you know this is what you should thank you. But this is going to contribute to an effective action which will change the mass square which has a by two. So it's a factor of less than it's contribution to the effective action for a bare mass. So this is the right way to do it. Another thing is that if we come to yeah, you have got to do it. Okay, great. Okay, thank you. Okay. And that's it. And then of course what remains is five p, five minus p, but we can really add that to the position space. That's just fine. But this graph does is give you a shifted effective mass. The lambda naught is the largest scale in this space. If lambda naught is the largest scale in this problem, what is the leading order behavior that's large lambda naught of this intensity? Well, it's before p by p squared. So it's lambda naught squared and some number that you can see in the view. We'll have the volume of the three-square and so on. We'll keep track of that. Okay, sir. So we get delta. So we get some number which we can get. So we get minus G naught which we can compute by four times log of lambda naught and then depending on whether lambda is larger than or less than this mass. Okay. Let's assume that this mass was very small. This would be a by lambda. If the mass was comparable to lambda, this could be a more complicated function. We do a function of n by lambda. So if we put this thing in there, we have to do the integral. That does matter. That's the lambda non-dependent section. That's clear. And this is clear. Is this clear? What do you have for p-square? P-square? No. This p is a coming in. Okay. So what do you have for p-square? So what do you have for p-square? Oh, I'm trying. Let's see. Why is this a red circle? We should take more than three. We've got the Lorentz transformations of rotation in this Euclidean space. Lorentz transformations are rotations. Something is below 1p, a rotation of it remains the red circle. So this is Lorentz and Euclidean space. You might have thought of red transformations in the moment in space. No, no, no. This is not three-moment. It's four-moment. That's why we've got an Euclidean space. So what do we do in Lorentz? Is this clear? What is this saying? This is saying that when you do the integral over the intermediate point of, you pick up an additional contribution to the mass and additional contribution to the mass. Okay? In addition to what we actually have in your in your version of the grand general. Okay? So this is the then mass square. We say that this is the additional contribution to the mass. Okay? Because now we can re-explanation this. Okay? So it would in fact be e to the power minus m0 squared to us. Whatever this works. Number by 2, the whole thing by 2. Number by 2, lambda naught. I was mistakenly thinking Yeah. For the leading term, this is nothing. Oh yeah, you were going to assume that lambda naught is... No, you see. If m was a what? If this denominator was something. Oh yeah. Even if m was already lambda naught, to change the number. That's not possible. What? m is the same as n. m is the same as n. So it doesn't appear in the leading divergence as lambda naught. So when you do this seriously, of course, there will be lambda naught. This is all integral over n. It depends on m, it depends on lambda naught. It depends on lambda naught. Something integral which you can evaluate. But when lambda naught is very large, this goes like that. What this says is that the mass term gets re-normalized because of coupling events. This is a familiar statement for many of you about this problem with n actually. You know, what this is going to do is say that the net effective mass term at scale lambda to be re-organized is a sum of two things. There is the m naught squared term. Then there is a large term of order lambda naught squared. Now you remember when we did our analysis of the normalization. We measured these coefficients by non-dimensionalizing with respect to the scalar between the line. Okay? So this quantity here is with respect to lambda. It is an enormous mass if lambda naught is much larger than lambda. Okay? So what we are saying is that even if we happen to start in the Lagrangian with n naught equal to 0 or some very small mass what that we do is in the Lagrangian at scale lambda generate an enormous mass with respect to the scalar physics that we are interested in. Okay? If we are interested in a theory whose mass squared is held fixed at scale lambda what we are going to have to do what we are going to have to do what we are going to have to do is to adjust m naught squared so that it very sensitively cancels this leaving behind only a finite this is the fine-tuning we talked about the fact that we have to fine-tune our starting point to start very near the critical the critical renormalization that for every relevant operator in the game there will be a fine-tuning we will have. Okay? It is a crazy fine-tuning if you think that the theory is just built at random if you think that at this high scale that we are not doing mathematical constructions of the theory but we are actually looking at some real world there is a theory at blind scale at some scale effective blind scale lambda naught which does some physics and then happens to give you whatever it will you might think that well it will choose some effect of it whatever it is and then it is effected the mass that we will get down at a low scale will be some m naught squared which typically was all m naught squared because that was the mass scale at every physics minus some number times m naught squared no particular reason for these two numbers to cancel sensitively in which case we can get a field of huge mass in which case we will conclude that every scale of field in the theory by the way for fermions, chiral fermions there are protection mechanisms chiral fermions cannot be given masses but for scalars there are no protection without superspeed so that we know so you might think therefore that if physics at some high scale was totally generic and simple that every scale of field in the problem at a much lower scale would end up having a mass of order in the high scale even if that physics was somewhat special so that it chose mass equals zero at that high scale even so at some lower scale we would end up having a mass of order in the high scale now this conclusion this very simple calculation conclusion appears to apply to the real world because in the real world this has been experimentally confirmed last year there is a scale of particle and the heat goes on the heat goes on has a mass of about a TV TV and this scale is far separated from the other a fan scale and physics mainly the black scale the unification scale more or less the same and it remains an uncomfortable though not sharp person of nature that this mass is so heavily separated from that scale because it seems to be saying that the physics of this higher scale work that I think that is a mechanism to stabilize such mechanism exists in super symmetry but we have not yet seen that work in the real world or there was a conspiracy the conspiracy was that the high scale physics was such that it is M0 squared that the M0 squared for the scalar field was chosen precisely so that it would cancel the M0 that you got by denomination of flow running and this is much more impressive than this calculation makes itself because as you imagine there are contributions to this mass at this order one loop to this curve point there are also motivations two loops to go x to g squared so while each of these contributions is suppressed compared to the other by a coupling constant the coupling constant is let's say 0.1 0.2 here so the suppression is not very much whereas these masses are very large so this cancellation would have to be very impressive cancellation up to very high order in perturbation theory in order for it to give us what is the result so this leads to what people sometimes call the naturalness the thing is that if you had high scale physics that was somehow agonistic about what happens in the in the eye eye whether it does something natural for it in the in the UV maybe something that sets M0 to some nice value but nice value that is local in scalar space that doesn't know what will happen once you flow down then it sounds totally absurd that this cancellation is happening to some such high precision this is the puzzle not a very sharp one because we know it's not like a mathematical contradiction UV is everywhere it could have chosen this but it seems somehow very odd that it knows what was going to happen in the eye especially given the coupling so this I don't want to spend too much time thinking about it's not a very sharp puzzle unsharp puzzles often lead to a lot of waste of time because I am serious spend a lot of time thinking about a puzzle that actually when you see correctly it evaporates it's not a puzzle when the puzzle is not very sharp what's really sharp is to have a mathematical contradiction two formulas that are same different things a blind category contradiction he was under something I got an idea it was infinite yeah I can complete it's fine that's not a contradiction these other things are a bit easy black hole information paradoxes you might be a paradox in this second I don't know if you're adding exactly what the paradox is maybe there's no paradox that kind of what should I do with suspicious paradoxes we cannot be clearly stating in terms of formulas but anyway this is what people talk about when they talk about the naturalism since it's talked about so much special we've got types of things I thought I would mention it for 5 or 10 minutes before we move on to more serious why put that number on us when we're at Planck's scale I mean just because the news is above it I mean you just have to compare it with all the people that actually talk to the Planck's scale but let's say that whatever it is at Planck's scale we start that that's unlikely to be some wonderful field theory about Planck's scale because from all indications it doesn't matter but whatever it is let's say that we believe that it's likely to be a quantum field theory below Planck's scale at least much better than Planck's scale so the framework of renormalization root flow should apply from let's say 10 to the 14 G at least that should be that's enough to build your balance does it help if you perhaps say some beautiful let's say 10 to E 10 to E means it makes the paradox less of a paradox but still quite quite a paradox 10 orders of magnitude is not that much easier to now what would it help is if you had new physics and order one new physics was of the sort that stabilized the book gozans okay super symmetry actually is great for that super symmetry is a symmetry of the gozans so if you've got a formula that's kind of so chirality protects its mass to zero that same thing protects the gozanic masses okay but this linking of formula and mass to gozanic mass happens only when super symmetry is a symmetry now in many super symmetric scenarios super symmetry is spontaneous we'll discuss things like what it is what should be known super symmetry is spontaneously broken so at and below the scale of the spontaneous breakthrough this protection ceases to occur and so it's very natural then for a scalar to gain a mass of order the scale of of this spontaneous breakthrough super symmetry so if you had super symmetry when broken at scale of order at the end that would beautifully explain this but if you're taking just to the companies the companies around that is the scalar of understanding if you take it at the end because it's broken only the mass but it's broken it doesn't matter it doesn't matter whether it's broken the masses or the couples it doesn't matter because in the intermediate you see once you feel interloops the fact that the masses everything will get okay so that's the kind of thing that will explain it but we have not the most comfortable explanation of that sort we have seen super partners by now and anything we have not seen them maybe we will in the next one but one shouldn't be suspicious because often things that are going to happen just happen postponing the next one okay okay okay yeah 90 degree is the expression M not square has to be negative of another square quantity so does that contradict the fact that in the Lagrangian there the mass term becomes negative yeah it becomes a negative term their mass term becomes negative but that doesn't matter because you see effectively there is terms coming from interactions they are canceling so you see the path integral they are doing yeah now you might ask what are we actually you might ask does this make the path integral unspeakable let me say that I wrote it as N not square it just needs some number to explain and the number you agree okay at least give at least the G not is positive that's fine but you may ask you might worry that our whole procedure makes sense okay so what you say is true and the body might be that our whole procedure makes sense okay and your body might be familiar as far as look this integral is nice and well defined is positive since we are doing perturbation theory we are doing an integral with this M not so if you actually have to start with M not square negative your procedure is this might be over okay and it's a good one so actually what we would do to deal with this body is this you know actually what we would do is to write this quantity here as M squared and then put it in M not square minus M squared now remember that M not square minus M squared it starts at origin so it's also a constant defect in that sense it's M squared is the mass square at the physical scale that we would keep it there that's actually consistent because it's other things M not square minus M squared all basically cancel each other once we get to those things so this is the fine-tune AS this is the fine-tune AS the fine-tune AS is that you have to start with M not square that cancels this object here and some very sensitive but now there is a question of just consistency of procedure okay you see we are doing some path integral and we don't want to use a bare path integral we are doing some path integral and we are doing the path integral by dividing it up into free and interactions now it would be inconsistent procedure for that free path integral to be different and the free path integral would be as you say if M not square plus M you see because the path integral the integral dx bar A x squared where A is positive is not negative so let's say you raise a legitimate body but it can easily be preferred okay so let me deal with that in the following okay so in order to perform this integral let me so now of course I have to know how do I choose my M not square I am going to choose my M not square to satisfy a renormalization condition my renormalization condition is that at scale lambda by mass is some fixed M square in this quantity whatever it is okay plus M not square let's be more accurate what I will now do and this is a good point whatever I will now do is to say the following let me say that I am going to modify my procedure I am going to modify my procedure to write this as d phi squared this is the renormalized mass the mass that I want to establish at scale phi squared by 2 plus G not phi to the form of a term okay plus M not squared minus M squared by 2 yet no one M not squared minus M squared by 2 is but I know that it is order G M not squared would be X okay and then the procedure see this part remains unaffected but in addition I have to put it down a minus M not squared minus M squared by 2 that was I want no mass coming except this so I have to demand that this is G what I want to tell is that the mass at low scales is what determines the stability of the problem the simple solution is that M not squared we have P squared and M not squared but we have P inches of all values including 0 we have to do the path integral over all phi do you want those modes as unstable as unstable should I write them all that because this what is it written M not squared plus some number G not squared M squared from M not squared you are getting a quantity which is not of order G but order L not squared so that thing is still L not squared what thing is L not squared M not squared minus M squared is L not squared in fact it is exactly what this is yes so how does it solve the problem no it just solves the problem in the bare part in the free part of the path integral there is still a phi squared but this part we are taking to the evaluation we are just doing pulling this down one by one you see what we are doing is path integral I just want to make sure my procedure is consistent that is what I do yes the thing is you first do a module review you are showing that the M squared or rather the M squared I will just you are getting the M squared I will just and then you are putting it back and then you are doing the first module integral again it does not seem very easy let me just work the procedure let me start the procedure by saying that I will take my action before doing any vectors write it as L phi squared plus M squared phi squared by 2 plus J not 5 to the power of M squared plus M not squared minus M squared by 2 I have done nothing this is just true I don't yet know what M squared is this is true for any M squared ok now as we will see this quantity whatever it is that is fine G not will be the smallest smallest number lambda not is some scale you see perturbation theory in the end will be perturbation theory in the normalize time I am not for 2 minutes you see we will get everything fine but if you want justification of this procedure take lambda it is something famous and make G not small enough so that G not tends to any power of lambda I am sorry for being so insistent no no no you could not have done this in the previous step because we would conclude that M not squared was negative so in the previous step you would have achieved a self inconsistency you would have said that the procedure that you adopted was not self consistent because it was doing integral to the 4 meters by x you should be insistent no no it is great in class many often say wrong things you know correct I mean that is how good class goes please don't apologize yeah yes but you see it is not that it is order lambda not squared that is not the finding the finding is the fact that the number behind the lambda not squared has to be exactly the same as what you get the order is not finding order is what you expect it is that the number has to be exactly the right and not just this one you see and one way we could get this and two way we could get one more term three way we could get one term and this would have to cancel all the numbers you see that is a huge thing huge conspiracy what no no M not squared is not M not squared is not M not squared is not M not squared is not it is term of the Lagrangian it is the real term of Lagrangian you see you could have a tachyonic mass in the way as long as the mass in the physical once it goes to those scales it is not tachyonic however we do care for the procedure this is the good point that you would question money runs away money runs away is a good point even if there is some deep physical reason we only care about the mass in the I R procedure was self inconsistent because I should be doing Gaussian integrals with negative M not squared this is an important point and that is how you could care just by shifting your in perturbation theory let me say that when you do perturbation theory perturbation theory is an automatic procedure once you know what to start perturbing from the question of what to start perturbing around is an arc rather than a science and it involves intuition it involves guessing where you are near the right answer so if you try to do perturbation theory around the wrong starting point you can trouble we are getting into trouble because we are starting the very perturbation theory around the wrong starting point once we shift to the right starting point then this is the right answer it is a good point okay let me have to do the function if you are obviously hiding how much of acceleration do you want to make because we have no refinery of the value of M in this theory we just want to cancel to M it is in the real world that you can estimate and there is huge quantity so that comes from this value that comes from this value sir he was saying what is the fine-tuning it is true that you have to cancel this to this but then what you are left with is some M so how do we estimate the canceling where are the fine-tuning because what is M and in theory it could be anything M is finite so a fine-tuning is fine-tuning once you are in a situation in the real world situation we are in this huge fine-tuning because M0 should have been ordered up to the right scale but it is already 16 that is the whereas G is just you do what if we started with the effect then we would have no issues then you know the M but we don't know the effect you can do the RG learning yes you can so if you started with what we started with and then the RG learning to get the effect of the RG learning you still have the following thing you see you have to know what is not to start so in order to define a quantum theory we define the quantum theory saying that we start with some M0 in lambda so that we float to Lagrangian scale lambda where the power mass parameter is so we have to do our adjustment however you thought this Pulchinski renormalization equation you can use it to solve that equation in perturbation theory as we will see as we will see as we go along the parameters G act at the low scale so we will address your question really clearly in a while probably at the end of this lecture this question is what was the true burden in your parameter we will address that very clearly and what other questions about this what is the mass now let's look at what happens in two loops when we take this guy when we take that guy this guy and this guy I am representing the four five to the fourth in a line once again what could we have but like this and this but this is already accounted for because if you take the diagram and compute it and exponentiate that diagram like we did we just added that up in the mass we exponentiate the diagram that generates this product of these two diagrams that's nothing new the genuinely new diagram that we have in this order and that we could have like extra mastery of our decisions or we could have added extra mastery more often like this so when everything in the browser loops to high scale and everything that is outside almost this thing complicated additional contribution to this thing of order g-square would change nothing conceptually it would be pain to calculate and that's all but change is nothing particular in our way of thinking the right diagram is the g-square that is the g-view what? the third diagram you are right so the parenthesis of the order g-square is like anything else up down has been a subset of course you could have this but that also is taken into account by I today just adding this a subset what? by the way this is not possible you know that I'll come to you but just to say this is not possible no no no there is no one PR here we just do a positive but why is it not possible so it's not possible what? who has told you you are talented without surgeons why is it not possible you see this was low scale and everything in here should be high scale but between here and here we have delta option so this was low scale and this was outside what? all external lines in these five diagrams are low scale all internal lines are high scale such a diagram is impossible so you see with Sonia's renormalization group automatically tells you non-amputed diagrams are not yet this we will see from a different point of view in the example of quantum interaction from the transform of the transform the only procedure is automatic simple fact about low and high scale some set diagrams yes maybe you are right you are saying this is an anger you are right this is a new, genuine mass renormalization group in fact it would be more than mass it could also generate some weight function renormalization I am going to put that aside for a moment but the much more interesting fact is the screen that is the one I am going to focus on this is okay what is this diagram and here we do it there is a sign we really want to get this is the sign of beta motion what is this diagram this diagram well firstly there are two there is minus g0 there is minus g0 so we get a plus g0 so we have a 4 factorial square 2 factorial from exponentization 5 5 5 5 5 5 5 2 is to take each of these 5 and group them up with 2 groups of 2 that will be low scale momentum and the 2 that will be high scale how many ways of grouping this up in the groups of 2 that will be this is the thing you corrected me on so this is 6 once we group to the groups of 2 the 2 that we are going to contact up have two different ways of contacting let us check what that is let's check what that is 3 4 3 8 16 let's see the 6 cancels 24 leaving for 4 so there was the 8 left over 2 cancels this 2 so there is just a 4 left over 4 into 6 ok let me check this is 24 into 1 by 24 into 6 into 6 into 2 by 2 which one I have got 2 5 in the first 5 in the first in each of them I have to choose 2 that I will contract with contract and 2 that will not see what you are thinking of is that suppose I had I am going to contract this with some external P1 P2 P3 P4 then the guys with P1 and P2 can choose to contract with this guy that is not what we are doing ok it is a different counting for what you are doing yes why you remember it was 4 minus N in this case is 4 so it would be 1 by 12 this is a cascading marginal the dimension of the coefficient was 4 minus yeah so let me just check what this thing was 16 ok great so what we get is ok and then we get what this guy is now because of money brother's collection ok we put an in square of it ok so we will get B by 4 over R squared ok now let's suppose that the momenta that go into this these lower scale momenta whatever they are and go into this graph P2 P3 P4 ok delta function tells us P1 plus P2 is equal to P3 plus P4 minus plus P3 plus P4 let us for P1 plus P2 then this graph will be P plus R over P squared ah I am sorry I am sorry because let's suppose that this momenta is R this momenta P minus R these are the two propagators in the graph what see because this is actually how we this graph is when we started our discussion do you remember we started our discussion saying we often renormalize we should say we conclude the exact direction and we found that this is not the other way about it ok we have to do this integral from momenta to momenta not should be this P3 P4 thank you we have to do this integral from momenta to momenta not so what will begin we get g not squared and the volume of the sphere over here it is 2 by squared here volume of the sphere volume of the unit of the sphere ok times by 16 times 2 dR this is by 2 pi 4 ok by R squared per m squared just to give you an example that this mass here was much smaller than even time we go down to finite scale but much larger than a mass so we can ignore that mass then we can compute this diagram we are interested in the so this diagram is now a function of moment P1 plus P2 but let's say we are interested only in the term in the 5 to the 4 theory that is independent because the coefficient that would be g4 because g4 is the term when you take 5 P1 5 P2 5 P3 5 P4 there is no factor of momenta ok so I am going to compute g4 ok for that term we get P equals 0 do you understand this let me say this again the whole expression here is some analytic function of capital P ok so we get some effective 5 to the 4 company which we can tell the series expand in capital P that will give us various values of the coefficients that we call g4 in our discussion of the of the coefficients given in organization group terms of higher and higher derivatives suppose we are interested in the term g4 0 for that we just set P equals 0 ok so we set P equals 0 and what do we have then that is very easy we have got g4 squared 2 by squared by 16 into 2 by 2 to the power of 4 times now because m is effectively 0 compared to a couple of systems so 0 there is no scale in the problem ok and you can easily see that this thing is basically just log of this is log of because it is just d i at g4 let's let's let's let's exponentiate this value we are going to have to you can take the same trick that I am going to go minus play if we exponentiate this back again what we see we have got a plus here but if the exponentiation because we take it into the pump minus s ok and so what we can do is that g4 and scale down that is equal to g0 g0 squared 2 pi by all these numbers what is the relative set you see I have taken this up there is a relative minus sign between g0 and this however you say that is what we want to see if you think of the effect of the formula text it is minus g0 plus minus g0 that is the relative set I am just choosing to do this I am just choosing to do this if I choose this then everywhere in this integral since r only runs between lambda and lambda not since r only runs between lambda and lambda not then everywhere in the integral r is much larger than x that is just the simplification so that we can see the answer it is not essential just on how you can see ok so this answer will be read in many ways what is g4 lambda what is g4 lambda g4 lambda is remember what we are doing is this we are trying to do this so in the effective action at scale lambda we have a 4-5 coupling the 4-5 coupling has an action in all momenta and the 5-5 we are taking even the term which has no momenta ok the coefficient of that is g4 this is g4 0 what g4 0 should I call it that it is that g4 0 that is what I meant ok yeah yeah we are going to have the same problem but we will soon come to similar problems we will always keep the physical couplings in the thing we will expand around and shift to the other things but isn't it a little inconvenient to do it every time for each order because we get 1s plus 1s minus 1s plus 1s 1s plus 1s 1s plus 1s 1s plus 1s 1s plus 1s minus 1s yeah so you know what you do is just start with this thing with g2 plus other things actually we don't even have the same problem technically because you see in this, in the Lagrangian way expanding around the thing we are using to generate perturbations in g4 so in this case you don't need to we didn't do it because we needed to generate gaussian path detectors though it had to be well defined that's the problem if there was a plus sign here then g0 would have to be in the negative and the issues of pathological diverges are negative to g0 the issues of pathological diverges are negative to g0 we cannot judge whether the pathological diverges are you say because it's a non-linear path detector there are all kinds of things the point was that each integrally actually performs perturbation theory has to be done and g0 is irrelevant okay okay now let me look at this formula this is a very important formula this is very important formula we will try to understand it in very different ways the first way to try to understand it okay the first way to try to understand it is as follows if we want to keep g fixed if we want to keep g fixed at scale lump if we want to keep g fixed at some scale lump if we want to keep g fixed at some scale lump how do we have to how do we have to change g0 how do we have to change g0 okay as a function of as a function of lump okay in order to keep g fixed that's one question you could ask and another question you could ask is starting with some value of g0 and we go to different values of lambda how can g of lambda let's address that second question first the question we are going to ask is does it increase or decrease okay so as we go as we go lower and lower values of lambda this guy becomes larger but it's a larger negative okay so basically it looks like it decreases but this is not very satisfactory for many reasons for this in our general analysis of the Poinsettian normalization you might remember that we had we had the general statement that dg by d lambda was a function only of g we take dg by d lambda we get a function only of I mean we have something which which shows about g0 that's not very satisfactory and so we had a general expectation that has not been borne out by this calculation but the point is that the general expectation is a general expectation for the exact result okay and could perhaps be borne out once we take into account all perturbative corrections that a class of perturbative corrections do what we want so let me do that first for interpreting this formula no higher order of the exponential would be higher order of the exponential would be like this and this it's not very different actually connecting these terms in each of these bubbles is exactly the same can you see that because each of the integral is only due to the net momentum running through these clouds and the net momentum running through clouds is the same okay so up to some factor up to some factor what we are going to get for this suppose we had suppose we have n such problems we get g0 to the power n we get g0 to the power n and times this integral that integral whatever it was i of p lambda okay that to the power this would be n plus 1 this to the power n okay and then we are going to get perhaps some factor so I will leave it as an exercise for you to get that that factor is just trivial but this factor is the form that takes this guy and makes it a derivative series I am just summing up class of graphs this is one of the sets that I am going to do summing up class of graphs to get a better answer okay and effectively what the renormalization group does is makes me some classes of graphs bit by bit hang on so you see instead of getting the net effect is the following instead of getting g0 minus this thing what this turns into is 1 plus so g of lambda 4 of lambda is equal to g0 into 1 plus g0 into this i of p which is some some number okay positive number that is log of lambda formula can be written as follows let's write this formula as 1 by g4 at this here I will divide it through by this guy alright this is 1 by gt that is the same thing now I take this to the left hand side 1 by g4 of lambda is equal to 1 by g0 okay plus positive number that is log of lambda formula this here is simply the integrated form of the relationship here is simply the integrated form of the relationship d by d here once we sum this subclass of graphs does have the property that we expected namely dg by d lambda was a function only achievable it's a function of dg of lambda in the way so that as log of lambda is made smaller as you go to the IR okay this guy decreases that correctly captures the future correctly honestly captures the future that as log of lambda is made smaller this guy decreases going off to 0 on the other hand as lambda is made larger g increases okay this of course is the statement that this coupling g was marginally even in that as you go to the IR as make lambda smaller g goes to 0 not at first order but at second order by the way log of lambda is minus the t that we had this t was flowed it was flowed towards the IR so its d by d t is equal to minus so it's again like this now we have established by this computation we have established by this computation that the 5 to the 4 coupling that the 5 that the 5 to the 4 coupling in the in the fixed in the free fixed point of the scale of here is a marginally irrelevant coupling we chose those graph factor we chose those graph factor we chose those graph factor okay you see what you were going to tell us to say the following you could you clearly have said that suppose you just took the 1,2,5 then you could have taken the g of lambda whatever we had g0 minus g0 squared that's numbered and log of then you could have computed d t by d t okay and what you could have got is equal to minus so plus we got lambda plus g0 squared number exactly like we got except we got g0 squared and then you could have said we know from general analysis that the right answer has to be a function only of g0 now to leaning order perturbation theory g0 agrees with g0 so what function of g agrees would at leaning order perturbation theory give you g0 well it has to be g0 so you could have just done that the right one was satisfying to actually sum the graphs to show you that if you didn't do the perturbation theory it will actually happen it tells you that you see what it means that we get plus g0 that we got from the 1,2 calculations that is plus plus g also plus number times the general analysis we had said that if you worked it out to all of us you see general analysis says that we got to get sum function of g you know an error function of g such that at leaning order it reduces to g0 that's a unique function that's g0 so it has to be the right answer g0 but what does it mean that the right answer g0 it means that if you computed all the higher order connection you would get many other terms one effect to those many other terms would be to replace this g0 squared by g0 I identified the graphs so what we are going to get what we will actually get we got this plus g0 squared so how do you do let's say 2 loop connections one of the things you will get is this another thing you will get is some genuine new term so you get this thing plus some other g0s g0s let's say q but once again that would be a function only of g0 so that would be replacing this guy also by g0 you can group graphs up so that one effect is only completing the sins and in a region you get new terms so if you went to 4 loops then you would have to have the term that you get by completing this guy the third term that you get by completing this guy plus something else the point is that in multiplication theory you can actually identify these graphs and then you can identify the set of graphs that completely change this into g all the other the remaining graphs have 2 loops we have to get to the new thing similarly there are 3 loops I have to identify the guy that completes this guy and then the remaining graphs will be the new thing then the renormalization group effectively organizes perturbation theory in a slightly different way speaking to your in a perturbation in g not in g0 that's the point we'll understand this point better so the correction to g is equal to g0 plus higher order exactly so this g0 squared does a great interleaving order with g squared but the only thing that could happen was g squared this we argue from what else it could be but at least I found it more more satisfying actually identify the graphs that can burn this into g squared actually every order you can do that you see the great thing about the renormalization group is that it's a differential so when you iterate a differential even though the differential is only generated by like a basic graph like this the iteration of that differential that same term generates all these groups so if you compute graphs to compute the small scale renormalization group the iteration of that differential generates all orders graphs in the that's basically what's happening we will to say it again quantum field theory has a perturbation theory as perturbations in the physical coupling constant not in the bare coupling constant and that's what this resubmission of graphs is showing and actually everything is organized as function only of the physical we will understand this point much more clearly very important that the bare coupling actually plays no role apart from some auxiliary device that constructed the theory plays no role in any physical quantity and then in the end we will answer the question that you asked we will see what the use is so what have we established what we have established is this equation that's called this one and this equation that's called okay now the equation one can be ready in a minute we put g0 cube we have to put g0 minus g0 squared for g we have to actually solve g in terms of g0 and second order and put that in g0 cube which is high whatever the higher order in g no but this is the point the point is that you will be able to always identify sums of graphs that do that if you get a lower or lowest order you get a g0 there will be sums of graphs that effectively convert that g0 you see that that's what's happening we got a g0 squared when we looked at just one graph but there was a whole series of graphs that we could just identify which turned that into a g squared this is a generic thing okay so the fact that perturbation theory is perturbation theory engine it's a reorganization of the graphs of perturbation theory you can reorganize the graphs to read some such things to make a perturbation theory explicit that's the point it's a gas all order all order perturbation theory doing graphs to all orders in perturbation theory in some sense it's a reorganization of perturbation theory you don't do all graphs you don't need to do all the graphs you organize an infinite set of graphs at all orders in perturbation theory another infinite set at all orders a simple infinite set so this arranging is equivalent to that theory it's automatic if you didn't want to do this you could have used the argument that you get g0 squared just to compute the function we know that it has to be a function only of 3 squared this you remember right there is a structural feature that features the beta function equivalence that dg by d lambda is a function only of g so you could ask well we know our general graphs that this thing has to be a function only of g squared okay that doesn't depend on where you started doing it in the realm this is not a it's not an issue of of accounting it's a serious issue it means that you don't need to have solved the renormalization root flow to know what the beta function is it's a local tradition dg by d lambda is determined purely by what what your g is there not how you got there that is a physical thing and I'm saying you can see this if you want to at the graphical you just need to accept it for the purpose of computing beta functions forget about it we need some experience we will have more to say okay I'm not a limit it's a very important part of renormalization theory the perturbation theory is perturbation theory in the renormalized okay and we will have more to say about this but before we get there take this equation and read it in another way let us read it as of lambda as we change lambda in order to not change g forward g not of lambda not in order not to change not to change g forward and we get the same equation you just differentiate it square equal to first minus sign for the fact that this is our answer the second minus sign that this side can change so yeah it's actually the same equation so we see, that the two things are the same answer the answer to two questions attributes the same differential the first question is how do we have to change our bad coupling as we change lambda not or we have to change our bad coupling as we change lambda not okay how we have to change our bad coupling as we change lambda not in order that in order that physical theory has killed lambda against fixed to this equation and G naught must be developed. The second question, once we have chosen the backup, if we change the scale of our physical couple, how does this G of lambda change? That also is given by this equation. G of lambda goes to 0 as lambda goes to the 0, I will provide it. Conversely G naught becomes larger and larger as lambda naught becomes larger. If we take this equation seriously, there is a limit to how large it can become because you see the money is at 1 by 2. So, when lambda naught by lambda becomes so large that becomes equal to 1 over G 4, then this equation, this must go to 0 and therefore G naught is broken. So, if we plot two things, how does G of lambda as we go to the I r minus lambda behaves as we go to the I r? We see that, so the answer to that question is that so let us do two plots. The other plot thing we are going to plot is how does G naught lambda, lambda naught goes from 0 to lambda, as we go to the u v. So, let us first plot this. How does G naught lambda naught goes from lambda to I r? As we go to the u v, we are going to plot a particular scale, that is some particular scale, that scale is given by, solved with the equation, which solved as a particular lambda that is called a lambda. And lambda lambda was given by solution to the equation, number that is log of lambda lambda by lambda is equal to 1 over G 4, if we differentiate both sides lambda lambda is equal to lambda times e to the power 1 over number times G 4. Scale here is the scale at which G naught goes. So, we plot in log, let us plot in, let us plot lambda naught, G naught rises and then this is an example of a renormalization group flow to the extent that they have one loop approximate distance. So, an example of a renormalization group flow to the extent that they have one loop is reliable, that cannot be continued all the way to the end. It seems to tell us that this theory with fixed G of lambda and some fixed scale lambda is not a well defined quantum plane there. It is of course, we already knew from before because it is simply the statement that this coupling is marginally irrelevant rather than marginally irrelevant. And we cannot define quantum field theories, but doubt away from a fixed point in a marginally irrelevant. This is this older manifestation of that statement. The combination was made by Landau in context of QED, something very similar happens. This the place where this happens is called a Landau pole. And it was taken by Landau correctly, it turns out to signal that those quantum field theories were not really well defined. We have a much better understanding of this. It is just simply the statement that is marginally irrelevant direction. This is still useful from the point of view of seeing how you get a problem as you flow towards the IR, sometimes. Because you really know this theory is a fine effective theory, but it needs view physically for the program. Okay, great. Yeah, right. So this is not a reliable conclusion. To the extent that you could, this is an indication from our view. It is the one new manifestation of the fact that we are dealing with the marginally irrelevant. Okay. Now, that is also not G of lambda versus log lambda, it must be minus log lambda to the IR. Okay, so how does G of lambda look? G of lambda, that's fine. Let's, we got that here. It's G of lambda to the IR, this guy is becoming infinity. So it goes to zero then. The fact that G is damping out, okay. The fact that G is damping out to its fixed point value. That's fixed point is just a fixed point. Okay. And it is basically the fact that G of lambda is marginally irrelevant. So this concludes my first brush with the study of the renormalization of flows from a very simple theory, the scale of your theory in four dimensions. Okay. And here we see that the next summary of that, you know, that is fixed point, that is fixed point, which was just a free field theory. Okay. And it has only one marginal direction, which is the master, sorry, that is the direction of the master, continues to be free. We have not succeeded in our attempt to build a long-trivial note. Okay. Not a field theory, that's not so far. Okay. Now, let me say, I've done less than I hope to today. But let me say one quick thing. Okay. Before we start, one quick thing. You see, you have these, we found that this phi of lambda was marginally irrelevant. So that suggests that there is no quantum field theory defined as a flow away from the free fixed point. That is anything but free. But does it tell you that there is no phi to the fourth area, no scale of field theory with phi coming up, with phi coming up? No. We have to be careful about being like this. Because in the space of renormalization loop flows, they could have been another fixed point. It was not inconceivable that there is a second fixed point. So let me, you know the master direction and write just the G4 direction. What we are saying is that the G4 in flow towards the IR in this fixed point is this way. But it's not inconceivable that there was a second fixed point. Yeah. Such that G4 would be a relevant direction for that fixed point. And quantum field theory here could be relevant. Not defined as a flow away from this free fixed point, which we just concluded, does not have interesting enough relevant directions. But defined as a flow away from this fixed point. Do you understand? There could have been just in the space possibilities. A new fixed point. Such that quantum field theory is defined by renormalization loop flows and how do these go there and in the IR go to the fixed point. Then if this was the case, this fixed point could have just been an accident. Something that expresses the fact that this field theory wherever it is becomes free at the moment. But the theory would have been defined by this fixed point. Such a possibility is the question of is fine to the four theory in four dimensions, a trivial theory. Is effectively the question of does there exist a new fixed point of this one. Now you see because this fixed point wherever it is does not occur in zero G4. It's very hard to answer using this fixed point. Because the flow away from the fixed point, the existence of the fixed point flows away from that side. If it existed, it would not be put up. So you might think it's very hard to rule out. However, there have been extensive lattice searches for such a fixed point. And nobody's ever found it. And I don't really know what the situation is now. When I was a Ph.D. student, David Gross saying there's an almost proof that it doesn't exist. It's probably an almost completely complete proof. So this theory is going to be in a concept of existence. But as far as we know it does not exist. We are not aware of any fixed points in the case of the single scale of the other than the free fixed point. Unfortunately the free fixed point is true. So for dimensions, we completely understand the set of one degree theories with one scale of eight. We completely understand it because it's true. We strongly believe that for a single scale of eight there's nothing more to do with one degree theories for dimensions. You can go and say that everything that anyone will ever know about one degree theory for dimensions is for the single scale of eight. Because you study which presentation that I took. Why one degree theory? Because all the caveats, I don't come to cut off one degree theories one thing. I count them only flows away from fixed points. Contrary to the theory. That's why the next level is the degree theory. But from this pure point of view there's nothing left. However it turns out that the situation is different in three dimensions. And that's what I'm going to tell you about in this class. Three dimensions is very interesting because the one degree theory of three dimensions governs the phase transition of the input to solid of ferromagnet, the isopheromagnet. We understand why all these are true. Very interesting. And there it turns out that there is a not really a lot to think about. You can define five to the fourth in theory and in three dimensions. That you can define five to the fourth in three dimensions along this. Why? Just because in three dimensions what's the dimension of the operator five to the fourth? Five is the dimension of, therefore five to the fourth, about three to the third is the dimension of two to the fourth. And that's what's really interesting. So in three dimensions five to the fourth, not as a matter of subtlety, but as a matter of dimension analysis about the three fixed points is a renovatory. So there are two dimensions of phase of field theories in three dimensions. Clearly you can define interactive field theories in the single scale of three dimensions. What's interesting is that what the situation turns out to be, as we will see in the reverse of this, the reverse of the one degree. And you take this three fixed point, define the quantum field theories obtained as the set of renormalization flows away. And if you flow in a particular quote unquote master's direction, the infrared limit of phase is another fixed point, which has only one renovatory. Which is like, I'm not at the master's definition. So it's the inverse of what I want to do. The three fixed point has two renovations. The new fixed point however, which has only one relevant revolution. The relevant revolution you should think of as n squared minus 1. This is very interesting. Let me tell you why. I'll tell you why in just one minute. You see, as I said, as I explained in my 20-day, suppose you did a three-dimensional ising. As you know, it undergoes a phase transition. And as I will explain, the theory at the phase transition point is an effect of three-dimensional quantum field theories, defined as a plane. But you say, in order to get this phase transition, as you change temperature, you hit it. This tells you that in the space of renormalization load flows, if you take a one-parameter set of initial conditions, you somewhere hit the phase transition. This tells you that the number of relevant operators of this conformity theory must be one. And because if you're starting the slice of renormalization load flows, and there were two relevant operators, you need to choose two conditions in the ultraviolet in order to hit that one. Generally, you choose length one. You'll miss this. The set of renormalization load flows that end up in the fixed point would be co-dimension two of the two relevant operators. But since you chose change to only one parameter in here, maybe the temperature, and hit it, it strongly suggests that the set of renormalization load flows that hit the fixed point is co-dimension one, meaning there's one element. So this fixed point here, this free fixed point is not a candidate. It's from a scaling bearing behavior of the ison model at the space transition. There's just too many relevant operators. There are two of them. On the other hand, this one, because there's exactly one reference to it, he's a candidate. It's called the Wilson Fisher fixed point. It's what got Wilson and Fisher at a better price. The Fisher gets the better price. I suppose they both are right. Some people got Richard Ray in the space. And then, the scaling bearing behavior of the magnetism as well as the solid treatment. Beautiful, very interesting kind of field theory which we'll discuss in the next class. If you want to read about it in the words, roughly the terms I said it, you will find as you grow up that you will find no quantum field theory textbook satisfactory.