 OK, excellent. So in the last class, we discussed these exact renormalization group equations. We explained how the renormalization group in before got to the flow of the number of parameters. Both of these parameters are irrelevant. Our parameters of the solve that would just flow to 0 under classical escape transformations. And the quantum version of that statement we understood plausibly that there would be that the renormalization group equations tend to have some sort of fixed plane, or some fixed hyper-surface, or some kind of n-tionality. The irrelevant parameters were not 0 on the hyper-surface, but were determined by the n-tionary parameters. So the net conclusion of that was that a lot of things there would be predictors, because it allowed a few parameters to mask our ignorance of what was happening in the UV. And to completely characterize all the measurements below a certain introvert scale. Any questions or comments about the general philosophy, the general structure of the equations? Yes, at least in perturbation theory, that's true. It's always true once we look at the relevant model operators around the interventions. But these are the same as the relevant model operators around the ultraviolet fixed point in perturbation theory. Because in perturbation theory, nothing can change very much. Non-perturbatively, outside perturbation theory these might be quite different things. We'll see. Again? There is a true perturbative renormalization criterion, which is nothing but always a very good parameter to be ultimately at a particular stage in the combination of the two relevant parameters. But how about non-renormalization theory of the criterion? Does the non-convergence to a specific surface in a way to find out and totally denote the structures of the high state of the level of refinement? You want to know what does mean for a theory to be non-convergence? Exactly. Good. So see. Suppose we had a fixed point. Again, such that there were, I'll give you an example. Suppose we had a fixed point, such that it had no relevant operators, no relevant marginal operators. All operators were irrelevant. So it flows to the fixed point. Now you see the problem. You see that suppose you started at any point here and you left the renormalization loop flow for an infinite amount. As long as we are in the basin of attraction of this fixed point, you will always reach here. This is the kind of situation we have. Now, in simple examples, this theory is just a free theory. A simple example of this is the theory of gravity. Where in the theory of gravity, every operator that you add, every operator that you add around the free fixed fixed point of just del h and del h, gets high dimension. And so at least plausibly about fixed point, certainly perturbation theory about fixed point is relevant like this. Now the issue is the problem. You see, now suppose since there is no this, there is a fixed point, which is a distinguished point. And then there's this infinite dimensional space. So if you want to make a theory, so the theory of gravity was just an example, there's a question. So I suppose you want to make a theory of this fixed point, that's easy. You flow for an infinite amount of time and you get to the fixed point. However, suppose you wanted to make a theory that was deformed away from this fixed point. If you had ultimate, you know, explicit control, you could perhaps, this depends on the data, but you could perhaps engineer some pre-normalization workflow that would end up, let's say here, after a very large amount of flow. Technically after an infinite amount of flow, if you, you know, keep going up that trajectory. So you're starting a scale lambda naught, you are at scale lambda, and if there is a flow that does not degenerate in some way, if you go to the UV, you can go for an infinite amount of distance, then you could take lambda naught infinity, moving along that curve. So then you always end up here. But the point being here, that if you want to get to any particular point here, you have to do an infinite amount of fine theory in the UV. Because there is nothing particularly distinguished about this point, if you have to land at this point, or that point, or any other point. Even that there are an infinite number of irrelevant directions away from the fixed point. But if you started at some random point in the UV, you could perhaps try to engineer things so that as you were in the UV, you moved further and further to the UV as you took lambda naught infinity. You could perhaps try to engineer things so that you don't end up right at the fixed point. But then, unless you had exquisite control over what was going on, you wouldn't know where you were going to land at. You could perhaps engineer things so that you end up in the neighborhood of this fixed point. But you wouldn't know where in the neighborhood you were going to land at, unless you had fantastic control over what was going on. Let me distinguish this from the situation where there is some sort of fixed point. If you started in the UV, no matter where you started in the UV, you would end up on this plane. They have a lot of fat here but no matter where you started in the UV, unless you did something very crazy, you would end up on this point. The point is not difficult to get to. But if you want to get some theory that is in the neighborhood of the point, there is an infinite number of such theories, none of which are particularly distinguished. You will go to the center. But suppose you try to arrange so that suppose we have got some gene on here and this is lambda. Such that this flow is arranged so that you reach here, at scale lambda. Now you could ask, can I instead try to get a normalization proof flow that will reach the same point as lambda not goes to infinity? Totally impossible. You are going to imagine knowing this by taking, you know, but as I take lambda not I move around this trajectory. Do you understand? Imagine trying that. Actually it could be that this curve will degenerate at infinity at some point. So it is not possible. But it could be that it would work. Let's leave open the possibility that it would work. Then you could imagine some sort of renormalization proof flow. Then we would end up here, even though this lambda not goes to infinity. Okay? But that would require extreme fine tuning of where you started out in the universe. If you started out at some other point, which is here because you were just in here, but if you somehow tuned the thing so that it didn't, you were near but not in here, you would end up somewhere else. Do you understand? No, all the way. In this particular case we have to add all kinds of operators. You want to stay around on this particular. You want to be some crazy UV. This is what a non-verbal, this is the statement of non-verbal. You see, when you have a fixed point that apparently has no marginal or element deformations around it. Then flows to that fixed point are of course easy to work. Very easy to work with. You start off with anywhere you take that fixed point. But sometimes that fixed point itself is trivial. This is like the case of gravity. That fixed point is trivial. Okay? Now, it's easy to get that fixed point but we're not interested in that fixed point because we don't live in a theory with free gravity. So what you want to do is to get to the neighbor over to the fixed point. Where some some deformations are done. And the key point about non-renormalizable is that there is no distinction. There's an infinite number of directions in the neighbor of that fixed point. There's no distinguish point at which way. The statement that neighborhood of that fixed point is not in any way distinguished. That's the key point. Is this clear? So suppose you wanted to try to make you know, suppose it made sense to do you know, so gravity has many certain things. But let's suppose we do cycle for theory. For for theory. You've got a theory of some number of other things. That's a good one. Okay? And you want to make a theory cycle cycle cycle. At least in perturbation theory this is an irrelevant interaction. Okay? Because in four dimensions the operator psi as of field psi is classical dimension three house. This is classical dimension six. So this is like the six coupling that we talked about in our discussion with the exact answer. Okay? So classically this just lamps up. So there is of course a free fixed point in theory. So every operator around it is irrelevant. There is in that the flow diagram in the neighbor of this point looks like this. This is in fact the exact statement because the fixed point is free. So we know that it's free. It's not some statement in perturbation theory. It's just a true statement. Because once you reach this point itself it's going to break the area. So it constructs a normalization loop flow equation Now this is the issue. Now suppose you wanted to make this theory whatever it means. Suppose you started with this particular so what you might want to do is you might want to start with a clinic and cut out the amount and some g0 times psi plus psi is psi plus psi. You can do this and try to choose g0 to be function of lambda. But because there is no fixed split if you do this in a way to hold g at lambda times fixed where g is the coefficient of the same operator at scale of lambda. Like we did for the pipeline. If you do it and you choose d0 or lambda n so as to hold g of lambda times fixed perhaps you can achieve this. But the second part of the theorem that we proved namely that the coefficient of all other operators will have a smooth lambda not close to infinity limit but not even. You see because what you are doing is this you are starting from different along different trajectories and there is no since there is no convergence it is converging only to the point but not. So suppose we got a flow and that looks like this and another flow that looks like this. Say we choose our g-flow our g-skill and this guy is fixed. So we start from here we we start from here we did enough flow time so we got here. Now here how we flow is enough so that we get here. These two theories this is like let's say this was g6 this was g8 These two theories were in the in the exact renormalization sense and equal g6 by construction because let's talk about g6 according to g6 it was a 6th energy like equal g6 by construction but they were different values of g8 and different values of g10 and different values of g12 there is no reason suppose that they will not because we do not have this fact that all flows tend to a constant flow size you understand this but in this case also we have this mass parameter and not a popular one so can't be artistic to a one parameter we could but that is still a free theory a free massive formula of course it is like a one parameter set a free theorem of one parameter but it is not quite fixed points because you see the mass parameter is massive so the model precisely was going on here there is a one dimensional relevant definition which is the mass parameter it is easy to handle this surface but there is one dimension of relevant definition one dimension of relevant definition of free theorem suppose we are doing chiral formula so that we wouldn't have the mass parameter just keep things simple just to keep things consensual then it is exactly the same is this very clear yes very good you see the point here is that we can adjust where we started from here in order to set G4 fixed which means that if we compute 4 points cabinet by this choice by construction we get something like okay on the other hand we have not managed to set G6 fixed in terms of the software infinity in general this G6 quantity as lambda not goes to infinity now we will try to fix that how do we try to fix that we might try to say well instead of starting with the bare the ground G6 and G6 not let me choose G6 and G8 whatever 6 points cabinet whatever that is 6 into 3 by 2 okay so we could try to choose these bare values so that we will fix G6 to a particular value G9 to a particular value in the I1 great we can do that but then G 12 will die and then we will have to do the same thing and we will go on forever you see so if we write a bare measurement we are writing it in the UV light so we already have all the notes fixed no so the philosophy we have for renormalization was let's remember what the Pulsyn scheme proves that we could choose G0 in that case for phytophore theory we could choose G0 as function of lambda not so we choose G0 as a function of lambda not G4 at lambda stays fixed in perturbation theory that we can do this in more or less obvious because the leading order G0 is G4 and then there are small corrections it's more or less obvious that you can do this the point about flowing to the fixed surface was that once you do this the full Lagrangian at scale lambda not is determined in the limit of lambda not that was renormalization renormalization that was the fact that it was possible to choose G0 as a function of lambda so that G4 is held fixed and once that's done everything else is also held fixed that's a good limit so then you can clearly label what your G4 is that's parameter but once you fix that parameter is this clear? on the other hand in this situation when we choose G6 not as a function of lambda to try to hold G6 fixed we presume we can do this but G9 is not fixed in fact we diverge as we took lambda not G9 at scale lambda would in general diverge as we took lambda not completely clear so so when you're kissing these basics are you thinking it's an American evolution? you see we're looking at the 4.0 yeah in terms of two free parameters in terms of in this case one free parameter namely G4 at scale lambda then to know how that theory fits to a particular experimental situation that's what G4 at scale lambda to match its purpose right but before we do that let's just see how we can produce parameter sets of consistent theories and then we can choose those parameters to match the instruction let me say this again because it's a very important question let me say this again first reminding you it looks like there are 3 or 4 lectures to be normalized this is not correct this is very good so let me remind you what we showed in the case of the function of this equation where we started with the Lagrangian which is delta phi of x squared plus m not squared as function of lambda not G not to the 4 as constant we started with this delta not or we started with the renormalization with the Lagrangian at scale lambda not we can write this okay then we wanted okay we wanted to study we wanted to look at theories we want to look at theories such that at scale lambda the phi to the 4 coupling was some G of 1 no momentum phi to the 4 was G of 1 at scale lambda we got a very complicated Lagrangian it has all kinds of momentum dependencies for 4 or 5 uplifts it has 6 5 uplifts a large infinity of number of parameters at scale lambda we put one constraint that at some I fixed lambda G of lambda was something fixed let's call it G suppose this equation was true when we started with this theory we started with this theory and we also put the other conditions it was fixed let's call it A suppose this condition was true for some I was for some M not of lambda not and G not of lambda now we want to change lambda we want to increase lambda so as we change lambda not if we did not if we did not change M not we get M not and G not fixed then these two conditions would no longer be true but we have two conditions and two things to adjust the two things to adjust I am not of lambda not and G not since we are taking to the test it is a linear equation so we can always do that adjustment so we can always find trajectories M not of lambda not and G not of lambda these trajectories are tuned so that these two conditions are always met as a fixed scale not let's call this lambda because it is just healthy fixed but changing lambda not keeping lambda fixed then because of this convergence problem because of this renormalization root flow at least in perturbation theory at this structure this convergence structure where we have G say 4 once we do this and we take this off to infinity if G once we met these conditions all the other paradigms actually in lambda also are not changed as we change the key point here was the convergence of these two and at least we can always make these two conditions these conditions could always have been true but the conclusion, the result G say, G say and so on are all fixed and what lambda not going to infinity would not be true so now suppose we are in that situation we are in this cycle of all the other because G6 is not true we will do the analog of this let me write it down because G6 is not fixed we got this G6 of side we got G6 of the G9 of the side to the 6 and so on and just for simplicity G8 which is two derivatives side does not matter we could try to choose this as function of lambda and increase our our scope now we got one more way to play this we can rate for for instance G9 that is clear lambda it will be help fixed what have we done we have insured the nine point scattering zero moment about nine point scattering will be finite we have not insured that nine point scattering will be finite six point scattering will be finite six point scattering will be finite but we have not insured that eight point scattering to do that we will also have to do G12 as function of lambda and so on so we will have to put an infinite number of parameters in our Baylor branch now that in itself is not a disaster the point is each time you put a new parameter you also put a new renormalizable renormalization condition which is you also add a new parameter in your theory because you say here our theories were labeled by two parameters all this was spoken in this is how we constructed the final parameter in the new theory would be G9 if in addition to this we also choose this guy as function of we will have to say put another condition we will have to scale it down now the six point six side now for some value so we will get a new parameter in the theory it is a general fact that for each adjustment you do with the other parameter you have a new parameter in the internet since you did not have to do an infinite number of adjustments in the automatic you have an infinite number of parameters you can see and this now is simply this guy you can be away from this fixed point but there is an infinite number of directions in which you can be away and there is nothing distinguished over any of the developments so the statement that you are a little bit away from the fixed point carries very little information because you have to say in which direction and that in which direction because the flow is not at this point you see what is the flow like the flow is like this you start you go like this you go like this so if you fix G0 you fix G6 you don't fix G0 then you are fine is this clear so statements like 1 6 3 4 yes they are meaning in this context suppose you see what is that statement that statement means let's first talk about today's talk let's restrict ourselves to perturbation so suppose we had you see this is 5 6 8 4 dimension is the renormalization of those we studied with potency so that is this one before and there is the G6 is convergent now what do we this A if reasonable we do in the after value always leads us to this fix but you might by in the last class we discussed one take that sounds reasonable but it is actually unreasonable is it clear that one can do that but not take it to this fix that is to scale our A6 as lambda not to square you see because the convergence was like like a 1 over lambda square so if you scale X6 in the inverse of that and there isn't enough time in the flow for you to reach the fix surface so then if you do that you will not reach the fix surface but once again the point is the point why the fix surface is distinguished deviations are not always in machine deviations so if you try to you know this is an attempt 5 to 6 is non-renewalism what does it mean what it means is we will try to adjust the coupling constant of the 5 to 6 order in such a way that it will affect the 5 to 6 coupling at scale in order to do that we need to choose X6 not to scale like lambda not to square but if I didn't nobody stops free world nobody can stop you from doing that so you do it that's great so the point is this does not uniquely define it meaning that if you now do this and try to take the lambda not goes to infinity level you will not have a good definition of the theory because by deviating you deviating from direction now you force that direction to have the X6 of a particular angle but you don't control the X8 or the X10 and then you take lambda not to infinity those in general are the same the general point is this you know a control deviations away from it are widely quantum theory has an infinite number of directions in which you can go but a finite number of marginal levels is fixed here ok so if you are willing to do the natural thing and flow to this fixed area you are fine try to go away from it all that makes sense have you understood this the really important point is not that is not the claim that there is no consistent theory that is not the claim the claim is that while there is a consistent theory while there might well be a consistent ok you will not be able to reach it reach a particular point without specifying an infinite number of parameters doing an infinite amount of parameters is this totally clear we have fixed of this where g6, g8 all are fixed in terms of d1, g4 yes are these values of g6, g8 is it in terms of our old final diagram is it saying that we can just find the 6 point function by taking 4 point vertices and nothing else it is saying that if you work in the theory in the ultraviolet theory you take 4 point vertices and nothing else and you get the right answer to the same but it is saying one other thing saying that the problem with that is that you get a uniquely determined 6 point function that is obvious what is not obvious is that it converges it has a good lambda not goes to infinity at statement the fact that the ultraviolet theory completely determines the 6 point function of the ultraviolet is the content of this you see let's say what the content field here is defined as limit of lambda not goes to infinity for any particular lambda not of course you have a uniquely determined 6 point function but the question is how do you know the lambda not goes to infinity and that it exists and this is not a vacuous worry because if you look in this diagram it does not exist of course it is unique so these two statements are equivalent the statements that flow lines converge and the statements that all all correlators have good lambda not goes to infinity are equivalent statements for a different game you never have to worry about the fact that e6, g8 are all non-zero on the fixed surface yes we never have to worry about that because you see how there is a fixed surface we want to get to the space surface we start in the UV where the fixed surface has two parameters m and g if we start in the UV with a Lagrangian that has any two parameters two renormalization conditions we will get to the same point on the fixed surface okay so a convenient choice of two parameters of the UV Lagrangian is m not and g not because we need to adjust two parameters to get to the fixed surface okay that is why we never but you know suppose you are quantity nobody could have stopped you from and you would be fine to start with l not, g not and g6 determine there is some arbitrary function that is important x6 x6 is determined by some arbitrary function of x2 not and x1 that would also help you to get to the same because of the convergence so there is nothing particularly sacrosanct about starting with that UV Lagrangian but there is nothing wrong with it just because there is the right number of parameters is this completely clear very good question so okay excellent question so the question is that we are starting with quantum field theory in this mathematical operation that demands lambda not with g and this is very nice but the question is is this mathematics with the physics you see because in any real physical situation you would expect some new physics at some finance and then you can't declare that this is exactly correct so quantum field theory is useful only when there is a significant separation of scales between the physics that you want to measure and the scale of new physics okay everything that we do will be corrected because you will not have if you don't quite start with lambda not with g you will not quite have time to reach the fixer so there will be corrections to understand but these corrections you see will be suppressed by lambda not lambda by lambda not to some part in our case lambda by lambda not square so if lambda by lambda not is large enough what we do can be a pretty good approximation to what happens this is one of the reasons why so far quantum field theory has been an extremely useful tool in particle physics because just experimentally has been the case that there have been very large separations of scales so for instance suppose we looked at QED was an excellent excellent theory for many processes what is the mass scale for QED in the mass that we like or maybe if you applied atomic physics in the mass of the proton whatever some scale that involves electron balls or maybe a thousand okay actually it's not a complete theory again it's embedded in the standard model but where does this embedding happen it happens at the weak scale at the scale of the W protons which is considerably higher by several orders of magnitude to the scale of the mass of the electron that separation of scales is what underlies the excellent functioning of QED as an autonomous theory without worrying about it's embedded in the standard model okay and so on sir you see as we as we emphasized this separation of scales is both a blessing it occurs it's a blessing because it allows you to do things part by part but it occurs because once you've understood your low energy like QED now you want to understand the next level because the input of what's happening at high energies is very low at low energies that's very difficult so the way we actually did it was building accelerators not by studying small subtle deviations this lambda not by lambdas okay and this is what is happening in the search for new physics in the LHC as all of you know we're all expertly hoping that something truly interesting no serious theorist comes to the table interesting and shows up at LHC and it's very interesting but it's very interesting 40 years ago it's not new physics you know it's great that it was found some great triumph of human intellect blah blah blah blah what did it teach you that you really in some sense very deeply suspect of it you know when I was a PhD student I learned about the Higgs particle I was not stated as a very flaky objective model is presented as the truth okay now what we found at the LHC is we found mass of Higgs that's great having explicit numbers but fine it's not really new we want new physics new physics would be something we did not know or that we perhaps suspected but it was too flaky a suspicion to go to textbooks so we're really hoping that people see new physics at the LHC but really nice that we could see the new physics without having to go to very high LHCs okay but it's very difficult because of these suppressions you think because experiments in the MPS the backgrounds and so on you put out small deviations in a clean way and stuff good excellent question another question why is there's no new physics apparently is it between community I'm saying between two scales in between there's nothing no there's just a normalization of the flow of that same thing you see what do I mean by new physics what I mean by new physics is generally speaking since of course you do an RG flow the kind of RG flow I didn't provide in the past clearly that breaks down if there is a new massive particle in some sense that is not captured in this RG flow it's the new massive particle is not there okay so this kind of RG flow cannot describe the physics until below the mass scale of this new one there so this kind of analysis can only work in a regime where there's no new physics meaning no new particles it's all described by the same RG flow basically okay so if the scales between new physics are well separated then quantum fields theory is a good description for each one of these new scales until the batch gets a fundamental okay of course there's another possibility there's the other possibility that you know you will have new things happening top top top there's no separation of scales in which case quantum field theory as I set it up becomes much less useful when they positively set much more interesting this is good other questions come up good so as we discussed in the previous class why so probably you think this is a car job calculation good so they only pay that these RG flows I don't see a way in which these RG flows can actually predict new physics from the RG flow yeah right so if you see if you take the RG flow try to run to infinity try to do the UV that's the wrong direction you should really be running the way suppose you do that and you find that at some point you get some inconsistency in the flows that's a good question yeah in the flows that if you take this flow run to infinity and then you get some theory which has a problem I would tell you that if you actually see the physics of this theory low energies something has to intervene before you get to higher that would be a clue ok let's stop for the non-retomalizable interactions it's a little more the same there are these lectures by Wilschitzky on effectivity with damages in the early 1990s we address questions our questions so so you see one of our analysis gives the question in the end our strong suspicion the structure of the exact normalization was not there ok so we have his strong suspicion that's what he's saying all about since we go along that the r t flows always have the following structure that a distinguished point called the fixed point this point is adjustment is defined so that the flow equations leave all the couplings at that at this scale couplings do not flow this is the fixed point of the Poisson stream if you have a question at this point d by d lambda is c and let me discuss if this was the case you remember we said that there was x x n lambda which is equal to some beta n of x n in general this was the structure of the equations and then we linearize these equations around we linearize these equations around fixed points making certain assumptions we argued that we could diagonalize this action ok and the eigenvalues of the diagonalization are called the scaling dimensions of operators around this fixed point ok that's the definition scaling dimensions of operators around this fixed point are the eigenvalues of the diagonalization of these flow equations linearized over the fixed point the assumption we made is that you know based on upper table analysis that of the eigenvalues only a finite number would be either positive or zero the rest would all be negative which is basically the statement of having most of your directions and then in the last class we studied flows around such a big spot so for instance if the number of relevant directions was zero without this value let's for a moment forget about the possibility of marginality if there was one irrelevant planning the x axis was the relevant axis the y axis was any other axis any of the infinite number of relevant very irrelevant the y axis was the relevant direction and the x axis was any of the other irrelevant actions thank you ok now you see in such a situation in this more abstract situation that we were dealing with is there are there fixed planes ok are there fixed planes and the answer is the analog of these fixed planes are the fixed planes are basically these planes you see because suppose you have got some renormalization group flow that leads you to the neighborhood then these irrelevant directions in the neighborhood was pointed below that point leaving you to lie somewhere on this axis so this is the non-perturbative version of what we tried to say the fixed point of the renormalization group flow equations there is a fixed plane fixed surface generated by the flows that lead away from this fixed plane in relevant directions just stay on that surface there is no claim about what the value of these irrelevant directions in some space this could be happening as a non-trivial value of g6 non-trivial value of g8 it's not to say that x6 and x8 are 0 at this point so whatever it is but there is this distinguished fixed surface quantum field continuous quantum field theories are theories along with this distributed surface ok how do we define these theories we define these theories by doing the kind of analysis we did that is taking as many parameters in the UV as there are relevant operators around the fixed point including these parameters in the UV so that we get to some particular value parameters around these fixed points and we are guaranteed by the convergent nature of the flows to the fixed surface ok that all other parameters and our theories will be determined is this here the catch phrase that goes with these words is let me repeat it it is quantum field theories are defined by the normalization root flows away from fixed points in marginal or relevant theories is the statement completely clear if you have understood this you have understood something very important about quantum field theories there are very distinguished quantum theories there are very distinguished quantum theories and those that live at fixed points of the renormalization these theories are a name they are called scale and variant or at least four dimensions of the proof conformity very rare clear knowledge conformity but they are very distinguished logically very distinguished based on quantum field theories because they are the mother of all quantum field theories quantum field theories are defined as renormalization root flows away from these distributions if you were into the game of classifying quantum field theories try to understand the structure of all quantum field theories very good thing to do is to first understand all the formulas and then try to characterize all the flows conformal field theories and the fixed theories on the fifth point of renormalization are extremely important theories is this clear? so that means when we are constructing a field theory we are starting with the lambda not tending to infinity limit but at the fixed point no, why there is nothing said again? if I have a point at some scale lambda that is in some relevant direction there on that bi-axis yes and I start taking lambda not tending to infinity limit while holding this fixed point then I will this point this fixed point and this is the point that I said please so the lambda not going to infinity limit would be the fixed point there for this infinity limit no, if we really started at the fixed point we would never move suppose at scale lambda not tending to infinity limit so that at scale lambda not tending whatever it was you would have the fixed point first is the way unlikely because of fixed point it is not of your distinguished form say you started the theories at scale lambda not tending to infinity limit and not tending to infinity limit the fixed point in general will have all the things so it is very unlikely that you would actually start at scale lambda at exactly at the fixed point that would require very precious precious unless it was a fixed point but if you did start at scale lambda not at exactly the fixed point then you would never leave it at any scale if you started at the fixed point you would never leave it what do you have to do where you will in general start is infinite number of flows somewhere we have exactly one there is infinite number of flows all of which leads to the fixed line now what you have to do is to do one tube the tuning is that once you hit it at scale lambda you would be at this point sometimes it is quite a serious tuning because if you just started at a generic point when you take lambda not to infinity you would go infinitely far away on this axis very often interesting getting theories not infinitely far away but a finite distance so you have to do one tuning that is the one parameter that you would have to choose that is what at the scale of an ordinary tuner it is not at the fixed point when you do one tuning so that you will have a renormalization root flow that will reach near the fixed point before diverging so that it does not go too far away I said this is a qualitative way because one day it will be exactly where we set you choose the uv parameter so that the higher parameter at scale lambda is in that field that takes care of it is this clear unless you start at the fixed point on that line no matter what you do you are only going to be asymptotically approach to y axis that is what is defined defined as the limit lambda not to infinity look at any flow when it is a little arbitrarily close to whatever point you pick it would not be zero distance it would not be zero distance unless you take the limit and you have an infinite amount of flow suppose we got one renormalization let us solve the beta function it will be simple so let us suppose the relevant direction is lambda r the function p is lambda lambda r whereas e to the power a r positive ideal density and lambda r and that is the lambda which is e to the power minus a i r that is lambda i r which if you take infinite time basically on the right now which is look if you take infinite time this is not to zero these two things go hand in hand because one of them is blank now that is why we need one tuning the tuning is that you have to keep changing lambda r of zero you have to tune that so that you will this guy find it that is the tuning of the one parameter the unit but once you have done that this guy is guaranteed to have is this clear good question not necessary you see if the fixed point happens to be a free fixed point then being close to the fixed point is equivalent to being able to put it in but there are many fixed points we have been studying one of them in this class there are not free fixed points meaning the physics of the fixed point is not that of free fixed point in which case being close to the far away from fixed point has nothing to do with ordinary perturbation ok you want to be close to the fixed point just so that you don't have any loss here you know this parameter is an asset you don't want to be very much so let me say there are two equivalent things that I want you to understand carefully that they are the first thing is that quantum field fields are defined by the grandeur specified in the degree with a few renormalization conditions L fixed and the UV scale they can take here the second statement is that quantum field fields are defined by flows away from fixed points in marginal or random conditions and your question was basically how are these physics and you see the point was that if we start with a flow from the UV and we get the UV scale in infinity and all deviation from this fixed surface the surface generated by marginal and random direction around the fixed point have gone away therefore these two things are equivalent let me say that again in the sense of full renormalization blue flows there are many flows that don't start at the fixed point and go in random directions the approach to fixed point you know how to continue in a random direction and the point will go back this is such a flow nothing wrong with it but if you take the lambda not close to infinity you may not be able to end up on such a flow you will only end up on the fixed surface what is the fixed surface in non-predictable terms ok what is the fixed surface in non-predictable terms it is the surface generated by marginal and relevant flows away from conformal that in the end is what quantum field theory is the set of theory is generated by but continue quantity this is my definition of continuity a set of renormalization blue flows generated from fixed points along marginal and random directions it is a very simple thing but it is something most people don't understand and it is very important to get this idea how is a marginal taking this away from our fixed point ok we have to study marginal I have to study marginal flows to be finding this job I have not yet talked about it because marginal flows are two possibilities marginal flows should be either exactly marginal in which case we do not have to fix the point we have to fix the line ok in the case that marginal flows are exactly marginal then any point of this marginal on the fixed surface is as good as the limits are defined as where you start on the fixed surface and which flow away from it in terms of right now where a marginal direction is not exactly but it just is a direction such that when you take the data function and expand it so linear order is 0 but a quadratic order or some higher order is not 0 then we have more possibilities in fact these possibilities are realized in simple free fixed points so we can study that with that can be out any other questions sir can we hide lambda r0 so that that exponential is like the fixed point exactly coincides like the upper point exactly coincides we can so one that is a particular example of the blue point that can be done can be done but does it no problems at all that is the best if idea it is conformity is there conformity is there is a beautiful it flows away from that how I added layers of complexity ok no problem at all that is not the most general for lambda not going to infinity then those two equations irrelevant directions that is guaranteed for it will I mean decrease the distance with fixed surface and that the upper equation will go along the relevant direction but for one condition I need to tune at the ultraviolet scale to stay close to a fixed point but that how does that guarantee that the first equation will not hamper the property of the second equation I mean the same I mean so basically my point is like if I am going to lambda not r lambda r not at the scale so that will somehow influencing the e to the power area right yes if I if I am looking at it is tuning this guy so you mean the same guy of course it could be that there is coincidence could be that you choose some line in your initial points so this will happen but it is generically very unlike so that is the kind of flow so you see let us look at these things what what what do we want you see so for these linearized flows about this fixed point we will just do it so let us just try to do it so suppose we got some scale lambda r and some scale suppose it was true but suppose it was true for a moment the whole flow is in the linearized region around the fixed point and there were only two such there were only two operators okay now what is our flow defining okay our flow is defined by lambda not let us call it lambda not IR and lambda not UP in the okay so suppose that at scale lambda r we have lambda IR and lambda relevant suppose this was true now what we want to know we got some other scale lambda of interest we want that at point at scale lambda of interest lambda IR is not so our relation is lambda of interest lambda of interest this is our condition okay so we arranged this we just arranged it by solving because this quantity here is lambda of IR of lambda IR is relevant to lambda of interest lambda not by lambda but A times lambda not interest by lambda not is that lambda IR lambda is irrelevant we will arrange this to be held fixed the tuning is to choose this so that this is equal to this it says nothing about lambda relevant this is held at order of 1 this is what just happened you see what we actually do is something like this you see because what we actually do like we did in our final fourth year is choose only as many parameters of UV theory choose only as many parameters of UV theory and as they were relevant like this I got the other way relevant like this sorry we are just as many parameters so that is equivalent to tuning these guys generically on our surface okay does not give some generic values to these guys to have these guys not go to 0 but require extra tuning which we do not do is this clear? the generic behavior is that this steps out and this goes up we do what you need to prevent this from learning there is no reason to expect this to not happen different numbers so on the one family we just said we start with in order for both to happen neither happens okay we will need in general two tunings but we will do only one so now we are going to generically but it is very close and if I tune only the relevant one then the rest is defined not by the condition by the surface of the surface we started with the one surface so that surface is a very complicated thing in terms of I pretended here that these flows were always in the neighborhood that is not true unless we go to the UV we are far away from this so what this actually is once we start from this particular surface that will lead to some lambda IR and some lambda UV effectively so if we tune our one pilot that adjusts this to be so that does not go up this in general will not also happen it will have to be quite some conspiracy now I suppose you should keep your mind open for such conspiracies that is a very unlikely thing you understand right okay other questions go ahead lambda IR is lambda relevant at IR and the right hand is saying sorry sorry sorry if you have relevant and irrelevant operators okay you have to do as many functions as there are relevant operators irrelevant operators will take care of themselves relevant operators are fine is this clear let us turn to the case now that we have got the slightly more sophisticated understanding of renormalization okay now that we have got the slightly more sophisticated understanding of renormalization let us once again revisit the fighting within this bulletin scheme blah blah blah business what is that stuff in terms of this fixed spot what is the fixed point what are the relevant directions and in that case since we did refutation refutation theory what is pertinence pertinence theory is constructing quantum field theories in the neighborhood of a fixed point there is something very interesting you see suppose we have got a quantum field theory because quantum field theory is a neighbor of fixed point that is free okay then we very easily understand the space of relevant operators okay so now we have we are fighting for theory because scale of theory we are trying to construct a fixed point so far we are going to do better than you we will sell efficient we are going to do better we are going to construct not free theories but so far we are just looking at the free theory okay so whether the operator is around the free point whether this is the mass of it that is x by y and that is why you look for everything else is right that is a easy hand we have understood the quantum field theories are described by flows away from the fixed point in marginal or relevant directions okay and we want to understand that statement in this case we also want to understand the marginal property okay because here we are going to marginal maybe it is this height of the form so what do the normalization flows look like on the space of these two axes so now we are not drawing the irrelevant directions either relevant directions damp out we understand that very well quantum field theories are defined by flows away from the fixed point marginal relevant directions what are these flows okay so on the x axis let's move the master this is the damage to operator clearly it is a factor in this level it is a linearized theory in the linearized in the linearized level because this is the marginal of it to leading order in this company just the beta function and linearized level so in order to understand what these flows look like in the upper direction in order to understand what these flows look like in the upper direction we need to go beyond linearization okay I am going to do a calculation for you I thought I could do in this class maybe that would be next class to compute what is called beta function this thing for the next two minutes just to proceed I am going to give you the answer and the answer is that this beta form is equal to minus sum number times the x term of the square remember where the flow is towards the interact okay so p by dt of x4 is equal to minus sum number times the x4 square we compute the number in a moment I will give you a bit to have this class next class we compute this number but the what is interesting about this thing what is interesting about this thing is that if you look I have got it the master goes away the public statement is that and to draw it it is attractive it is basically like that the flows will look this guy damps out it doesn't damp out as fast as an exponential let's see how it works let's replace this number this is the sum number sum number it is just complicated suppose this number was 1 by scaling x4 what does that equate to dx4 by d by x4 square that is d of minus 1 by x that is either 1 by x equal to t plus 1 suppose this constant was 4 x4 to 0 x4 is equal to 0 as d goes to infinity marks lower than exponential it is like a power law decay so schematically we normalize these two flows now in this situation although the approach is not exponential there is basically no real difference between this operator and an irrelevant because it is going towards differences from which a different rate of approach from the event so in the end that this was marginal rather than irrelevant it is a bit of a detail it is an important detail for many businesses because the rate can sometimes be very important we will discuss the same details but in the end this is a bit of a detail so what is quantum field is very defined by in this case in this case quantum field theory is just defined by this but it is longer than region but in the larger lambda not goes to infinity so in the end quantum field theory is defined by this now what is this line just mass deprivation around the free field free field point which is the free field which is the free field this is the statement of the fact 5 to the 4th theory in perturbation theory this approach they approach to this fixed surface it is very slow perturbation theory it is called very slow it is one power of x for further than what you mean further perturbation theory the approach is very slow so you don't easily see this there is a lot to say about it and that far from proved that triviality is actually it is true but you see we have already made a pretty plausible case that 5 to the 4th theory we thought of as a quantum field theory really in a straight sense only defines a free field so the Chinese key is RG flow equations to cast to this plane the plane labeled by x for and the mass and in perturbation theory you could very clearly separate the other irrelevant directions from the x because those map are infinite these maps are only powerful so in perturbation theory it was able to generate that that lived on this plane but if we look at it now a little more go away from this perturbation point we look at it from above we see that actually one of its directions was about as bad as in your life so the theory is that we worked so hard to define if we really satisfied the non perturbative way would be only free quantum which is of course huge this free quantum theory is just need chapter 3 input hahahaha but now just suppose that there was a theory such that somehow or the other when you did the calculation in you're equals suppose that was the case then this direction, as well as this direction you think to me like that the minus sign made it effectively irrelevant, even the margin later, the technical terms margin later. The other sign will make it marginally relevant. And you remember that our rule was a constant between theories and defined by flows away from fixed points in relevant directions. So these would now allow us to define a two parameter set, labeled by the mass, but also by the coupling constant. This would allow us to define non-trivial quantum field, the quantum field theories as flows away from fixed, flows away from free points. I want to emphasize again that it happens to be a factor in foreign emotions. All classically relevant operators are just masters. So suppose we are in foreign, this is not true 6 dimension, but if we were in foreign dimensions, if you take the statement that quantum field theories are defined as flows away from the fixed points by relevant operators, and you looked only for classically relevant operators away from free fixed points, you would only get free theories. It is only masters that are classically relevant. However, in foreign dimensions, there are theories, there are operators and a margin, classically mentioned. Then it is a very important question about whether these operators, these classically marginal operators are classically relevant or classically relevant. If they are classically relevant, that happens to be the case as we were shown for the fight before. We only, all our work in this theory, if we are purists, we really want to define quantum field theories, if we are purists, all our work has been in vain, there is no such thing, so we are in the free fixed point. On the other hand, if they are classically relevant, then we get new theories, theories labelled by coupling concepts that could do something interesting. It turns out, as we will see, it turns out that 5 to the 4 theory, the marginal operator is classically irrelevant. In QED, the marginal operator is classically irrelevant. It was this understanding in some words, none of these words, but this understanding in some words that led the deep thinkers of the time like Landau. To conclude, that classically operators being, operators being classically irrelevant was a generic feature of quantum field theory. You have a physical argument, by the way, from this little screen, we discuss this as a physical argument. So, quantum field theory was there, they had to be there of the pyranas. So, widely held view of the 1960s. At least in some parts of the world. What changed, but really resurrected quantum field theory, certainly as a description of quantum field theory, was the discovery of one free fixed point about which the operator was marginally relevant. And that is non-linear engagement. In the non-linear engagement, I'm going to describe various, and this G here is a marginal classically dimensionless, therefore classically marginal operator. And tells us that when you do the calculation to this, when you do the calculation to this particular parameter, you find that the operator is marginally relevant. Our theory is defined by where, where on this flow we are at some scale. We can be quite far away on this flow at some scale. Which means we are quite strong to operate. So, these QCDA, non-linear engagement theories are paradigmatic examples. A paradigmatic examples. The head of, well defined, non-linear quantum field theories, defined as flows away from free fixed points. And because it's a flow away from free fixed point, free fixed, flows away from free fixed point can be constructed. At least the name of it. In perturbation. You see, perturbation takes place when you go far enough away. Because this parameter is a quantum constant. You go far enough away, perturbation takes place. But the name of it, you can always, around a free fixed point, you can always construct the flows in perturbation. You can control the flows. So, okay, right. So, we try to understand in general the structure of flows by fixed points. We specialize to free fixed points. When we look at free fixed points, just simple dimension analysis tells us that we often have classically marginal operators. Okay? These operators could turn out to be either marginally relevant or marginally irrelevant. Okay? And they're, well, in many cases they turn out to be marginally irrelevant and therefore do not define them. Some cases they turn out to be marginally irrelevant. Do define them. And that is the case of ordinary UCD. Okay? So, in some sense, not being able to define a series of four dimensions, only when I define a quantity of series, then we can define starting with the culture of perturbation. Okay? Then we set two or three more things. The first thing I wanted to say. You have a class. I'll stop in two minutes. The first thing I wanted to say is zero, the linear operator in general, zeroed in some other higher order. We could have, we could have such a thing. We would of course have to find a theory which is, but in general, this is possible. But this is physical for many cases. In perturbation theory, firstly, even having a marginal operator is a very exceptional thing. Why should an operator have dimension? Exactly. We find that in a number of theories. We find that in a number of free theories. Even these are also free theories. You see, all these theories are defined around the fixed point, is a free theory. Now, free theories aren't, of course, very exceptional theories. Operators are given by dimension analysis, or either integers or half integers. It's not so surprising that if it were four, it's always a nice integer. However, if you look at interactive theories, interactive theories, we will study some of them. There is no, it would be quite a coincidence to have a dimension for an operator in that. See, in general, dimension of operators would be some irrational number. For it to be exactly four, it's quite a coincidence. So, an example of this, although we will study it in three dimensions, is the Wilson-Vichon experiment, the fixed point that governs parameters. We'll study it immediately. The next item on my list is this. Okay? Yes. If we are close to the fixed point, then I can approximate the equation with expansion in X4, and with a sign with positive or negative, or that depends on the situation, I mean. The equation I have written GX4 dt is equal to something proportional to X4. The sign in front of it depends on the, that we determine whether it will be marginally relevant or irrelevant. But if I take the exact equation for marginal case, and try to, I mean, what will be the same? I mean, what will be the sense of renoneration? If I do it, what will do it exactly? Not by expanding in force 1, and if I really have, I mean, at a distance from the fixed point, let's say. You're really choosing non-linear one. I mean, non-partner, where do you sense? But what will be my... No, no, this is not even non-partner. It's just different, you see. We want, whether in perturbation area, non-partner or outside perturbation, we are defining quantum field theories by taking a UV cutoff to infinity. In order for that to be controlled, we want to be nearer to some fixed point. So, let us go back. Quantum field theories are defined. Quantum field theories are defined by flow to some fixed surface, but in a UV cutoff to infinity. And the way we get that is by being nearer to fixed point, and it's set of relevant flows. So, it's part of our basic definition of quantum field theory, that no matter where we start off in the UV, we will always end up near some fixed point, or near the fixed surface of fixed point. So, this is not about perturbation. For that reason, understanding the surface... So, what we want is to understand the surface of theories generated by flows away from fixed points. And for that, understanding is very important as in an absolute mathematical sense to expand around the fixed point. Because the space of flows around the fixed point is characterized by the tidal influence. So, there's no approximation. It's the scale. There isn't one or two more things. This, of course, is very unreliable but it gives some intuition also. Suppose we have the kind of flow for X flow. And that was like the 5 to the 4. So, you remember what this part was, the X form is equal to 1 over T plus 1. On the other hand, this was in the case of, let's say, 5. On the other hand, when we look at the other side of the video, let's say the schematic form is 1 over minus T plus 1. Now, let's study this. You know, we're only doing this to one loop. The actual beta function as we go along the surface changes. It's not reliable but it gives you some feeling for what might be. Let's look at what happens to this. Suppose here we've got some oscillations. Let's take T to plus infinity. And we take T to plus infinity. As we take T to plus infinity, our, this one, we just go to zero. On the other hand, here, we've got some oscillations on the other side. Then we can't really take T to plus infinity. Because when T gets C, this line already grows up. We get some oscillations like this. Both blocks are equal. Both blocks are equal. Exactly, exactly. But in the flow to the IR, what's happening here, is that as you keep flowing to the IR, according to this one loop equation, it's not reliable. At some fixed point, we've got a blow-up of current. We can set T equals minus infinity. And if you go towards the IR, hang on, I'm going to just talk about this one. And you know, deeper and deeper in the IR, there's some lower scale, some scale that is low enough, so that the coupling constant grows up. The scale at which this happens is a distinguished scale. It's called lambda QCD. It's the scale at which, according to the one loop beta function equation, it shouldn't be used. But according to one loop beta function equations, the coupling will happen. It's an estimate of the scale at which physics becomes strongly coupled. And it's clear that physics is getting more and more strongly coupled as we go to lower and lower energies, as we're going away from the experiment. And when we get to scales of our lambda QCD, that's an estimate of the scale at which physics becomes strongly coupled. That's the lambda QCD which you see becomes non-productive. Now, the whole thing gets more close up. I mean, it still just becomes more than 1 out of 8. It takes itself non-productive. You've got to be patient. If you want to present, this is one precise definition. Now, of course, if you look lambda QCD by 2, it will be about a 1. You see, in the one loop approximation, it actually grows up like so. Yeah, true. Now, if you want to say when QCD is order 1, it will differ from the scale by a number already. It won't make much of a difference. I'm saying that definitions are different. You can define it as lambda QCD as a scale at which X4 blows up. You have to be finding it. Or these things, lambda QCD as a scale at which X4 becomes greater than 1 or... I mean, it exceeds the... It exceeds the perturbation region. It's not a very precise definition. I agree. So, suppose you want to assign a particular number. What I've said is something precise. But you said something not very precise. But these two numbers will differ by order 1. So, if you want an estimate of your scale, these are equal. You see, suppose you want this. Suppose you want this to be order, I don't know, 1. Then you would be solving the equation minus E plus C is equal to 1. I would be solving the equation minus E plus C is equal to C. What is the difference in D? It's 1. So, your lambda QCD and mine will differ by E. But 1 is arbitrary while infinity is seamless. I'm saying that if you're interested on the quality, then it's the same. On the other hand, if you look at this guy, there is also a blow up. But when D goes in the opposite direction. So, as you go to the ultraviolet, as you go to the ultraviolet, the coupling constant blows up at some finite scale. If you work according to the formulas of one loop in the opposite direction, you would find that it is impossible. You would find that it's impossible to maintain the condition that you stay at a fixed coupling constant while taking lambda naught to infinity. It's impossible to maintain the condition that you stay at a fixed coupling constant by taking lambda naught to infinity, even if you allow yourself to do something like that. Because once there is a certain separation of scales, if you are at a fixed coupling constant at some scale lambda, there is some other larger coupling constants here. Finite but larger, at which the coupling constant blows up. And then the equations don't make sense at higher than that. So, QED, or the spiteful theory is an example of a theory in which you cannot continue, at least in this one loop. It's an example of theory in which you cannot continue the renormalization loop flow equations all the way to infinity if you start with some fixed coupling at a finite. This is another indication of the triviality of these things. You can continue to infinity, but only if you start with the coupling equation. And then it will all just stay the same. The last, the statement said that in the last five minutes, are just meant to be intuitive. Because they are not precise because they are based on using one loop equations. And they are not valid. Because while a coupling constant blows up, you should be using higher order terms. Obviously. But I thought I would just say to give you some rough sense of what's going on. Yes. How would you be in saying that things blowing up in the river are a sign of non-triviality and there are higher probabilities? It's the sign that the theory is not autonomously defined. You need to use this experiment. So, this is not by itself a well-defendant. In fact, the theory is not a well-defendant. In fact, it's up to you. Okay, let's stop the class. We need an extra class this week. Does Wednesday work for you people? Wednesday at the usual time. In the afternoon? Terror. Okay, let's. So Wednesday, what we will try to do is to get a little more serious about this stuff, we will try now to construct we will first check the beta-function equations for 5 to 4 theory, and then we will try to construct one non-trivial fixed point. Okay. Is it very good? Very good. Unless it's all illegal. I'm actually going to turn it off.