 Any questions or comments for the last class, where I remind you we discussed that as far as we are aware theories may have a single scalar field in four dimensions. The only genuine quantum field theory that is a quantum field theory you find is a flow away from a fixed point is a free scalar field theory. Any questions or comments about any of these theories? So actually in the last class we said that we came up with a contradiction to be around the physical mass parameter and not the bare mass parameter and then we added and subtracted m square minus m square. Right. We made the part of the table of the scalar. The free part of the part of the table. Sure. So I wonder if my question is that the other part of the table which you are writing is as a UV scale and may be of any kind to the angular scale. So if you consider the UV scale of the part that consists of the mass parameter which is not of the UV scale but of the IR scale. So if you are writing the free part of the random set that consists of the mass parameter which is the mass parameter not of the UV scale but of the IR scale. We have chosen you now. Yes. Super. Yes. So if you just see that part does not mean that our UV theory already knows what mass parameter our IR theory is going to look like. Yeah. Let me say this in another way. It's a very good question. The question you could ask for instance is what is the correct criterion for the stability of the point. Okay. Now if you write down a classical approach that looks like it has some sort of runaway potential. Here we know you are doing that. Suppose you wrote down. Right. Okay. Right. So our classical approach does look like at least it has an instability. Right. Maybe it will be picked up again by the final point. It looks like it has pretty bad initial instability. Okay. Now the question you could ask is, is that correct? And the answer to that question is that the thing that determines the stability or otherwise of the theory is the shape of the potential of what's called the quantum effective action. Okay. Now the quantum effective action is closely related. In fact, in the theory, the mass of this particle is equal to the quilsome effective action. It's actually a scalar lambda. As we take lambda, the limit lambda goes to zero. As we take the limit lambda goes to zero. Okay. So the thing that will determine, you know, if you've got some parameter and you're, yeah, the issue is precisely the following. It's something that Ronald brought up last time. You see, nightly you might think that having a negative mass squared is problematic. But that's night for many reasons. One of which is that it's problematic only at low enough momenta, so your momenta is smaller than the mass squared. Because otherwise minus m squared less. p squared still is no problem. Okay. So they really choose the volume. You see, what is the net? What is the effective integral you have to do for the low mass modes? Okay. After having integrated out the limits, that really determines whether the low mass modes, the low momenta, are stable or not. And that's what the Winslow mean effective action is. Okay. So you can't just look at the bare action and talk about stability or not. We'll discuss this more. We'll discuss it. We'll prove this theory, that the theory is stable or not depending on the shape of the quantum technology. As people say this, we'll prove this. It's very beautifully done in Walter Feinberg's book, Chapter 16, as of Sunday. Let's not wait for discussion. Okay. But fine. However, so that is the integral, the general equation. However, you could have asked about technical question. And that is the reason we have to do the shippers. The technical integral is that in generating our perturbation theory, we have to do some integrals. In order to get those integrals that we would be over here. Okay. Just to make that, make sure that we shifted and unshifted. Just to make our procedure consistent. So it was not an issue of stability or theory, it was just make our procedure self consistent. Okay. Very good. Other questions about this? Sir, can we take fine tuning as early as today? Fine tuning as early now. Fine tuning is not a very precise statement. That's very good. But now I understand your question. Yes, exactly. Fine tuning and the renormalization. If you make a renormalization condition, saying that M is some fixed thing that you keep fixed. Exactly. That is the same as a fine tuning. Does everyone understand this? What is a fine tuning? A fine tuning is set in the language of renormalization. Suppose we've got, you know, two-dimensional space where we've got one relevant direction. Okay. So all these flows, so this is the relevant direction, this is the relevant direction, the flow is looking at this. The point is that we're going to flow for an infinite amount of time. We're going to flow for an infinite amount of time. We start on a random flow line. You flow for an infinite amount of time, you level out of the ball. How can you avoid that? Well, there is one flow line that if you flow for an infinite amount of time, takes you on the flow line. Maybe this flow line. So how can you flow for an infinite amount of time? You have to stay somewhere fine. You have to be on a flow line that approaches this flow line. As an unknown goes to infinity, you have to approach this flow line in a manner tuned with an unknown, so that after the infinite amount of time, you reach the flow line like this. Okay. This is your renormalization condition. The fact that at scale lambda you are in square. That's exactly the same as a time line. For a random, yeah, for a random time line. So the same point, can we do a serious point? Yes. So the renormalization condition, imposing the renormalization condition is achieved by a time line. Going to a flow line that looks very much like a line that will go straight at infinity. Okay. This statement is basically the same, or it's very tightly related to the statement. The quantum field theories are defined by flows away from fixed points. The quantum field theory of flow line that goes to the fixed point. Or it is. And then it goes. Is this clear? Other questions about it? They will only want to repeat the secretion of the circuit. The scale is not large. They will think, wait a minute. Yeah, back. If the separation of scales is large, then can we have time to use the... We do that. If it's not large. If it's not large. Yeah. If it's not large, you're going to have time to fix yourself this. And then you've got where I would want to cover for you. Suppose you start with a lattice. You put your field in a lattice. The lattice spacing is A. You measure something at a distance of R. Okay. The relevant ratio is R divided by A. If that is not large, your theory will be somewhere in time. And then you will not pick up the bandwidth fixed point. Okay. So, a couple of quantum field theories have an infinite number of parameters. They require you to specify all of the... all of the parameters that are in the neighborhood of the organization. I mean, in the neighborhood of the fixed point. They're all different. I mean, there's a case to say that there's nothing wrong with them. But they're not predictive. They're not predictive. I'm letting you happen to know exactly where you want. Okay. Just specify what area to specify an A is smaller. Because you get approximate predictions by saying we're not too far from this fixed surface. And then if R by A is smaller, you can do better and better by saying that okay, we're not too far from the fixed surface. But let's say there was one dimension 6 operator. That, the coefficient of that will be suppressed by A by R to the power 2 because dimension 6 operators die off like the square. So you'll get better and better approximations. We'll keep operators of higher and higher dimensions. And we'll get a systematic approximation scheme where you're in that in the parameter A right. To that extent they're predictive and there's some definitions. But if A by R is out of 1, then it's basically fundamental is not the right language to use the problem. This is saying if you want to measure a field theory seen from that point of view, one field theory is what happens to physics. There's some fundamental physics. Whatever it is of a tree to come to you. But it's got a lens scale. You're going to learn much larger lens scales than that and the physics is approximately lower. Much larger lens scales than that will be covered by some approximate quantum theory. The exact quantum theory is defined as two things that are autonomously defined and defined as either as a limit or better as long as they're from fixed points. But those these idealizations are the first term in an approximation in A by R and leading on higher terms will be obtained by putting dimensions of successively higher operators of successively high dimension. Is this clear? By the way if you often use that's a new business. You know suppose there's a super you know suppose there's some super symmetric particle attending so we have a sequence. Okay the effect of that so we are working with a standard model. There's this the scale here and that's a result 20. And that's a requirement for the theory which is as just a standard model that's only valid down from here so that's a hardy flow but not very much. So we should expect of new operators to be present dimension 6 operators to be present suppressed by 4 by 20 to the power 2. Now this is of course a tricky game because whether it's 4 by 20 or 4 by 20 by 4 pi square makes a huge difference. But you get by me but there's some other other operators. So the way people parameterize the inverse often is to say let me get standard model plus the most general 5 let me write down the most general 5 dimension operator dimension 5 field I can write down in the standard the standard model as you know is renormalizable but by renormalizable what I mean it is defined as a flow away from the fixed point what is the fixed point of the standard model everyone knows what a standard model is right? ok what is the fixed point? it's the free fixed point what is the standard model? what is the standard model? it's a it's a carry of gauge bosons fermions and the bosons ok the fixed point of the standard model is where the gauge bosons are free the fermions are free the scalars free free fixed point we can ask what are the marginal what are the marginal relevant definitions what are the relevant definitions and the answer the answer is you can tell me what is the answer how many how many what are these definitions maybe that's the gauge company let's go back I don't know excellent excellent so what are first let's look at complete classically relevant the classically relevant there is only one classically relevant parameter what is that no what what about these these what the mass the mass of the the mass of the boson the mass of the boson d5 squared plus m squared y squared that's what the heaps potential classically this is what marginal it's what we've been studying in the last few lectures this is the same theory we've been studying okay classically this is relevant this is this thing dimension 2 operator 5 squared so in the standard model there is exactly one relevant operator about the free fixed point you see about an interactive fixed point we can try down the Lagrangian and know but for the free fixed point we can't so what I'm saying is not a guess it's exactly exactly one classically relevant operator mass of the of the heaps boson okay because it's exactly one classically relevant operator there is exactly one hierarchy problem in the standard model and the one doing the very very sensitively is to get this operator this parameter in the low energy not go off the plan scale that's the hierarchy so this parameter we have to tune very sensitive okay we're coming to the issue that may be operating let's look at the other parameters okay what are the other parameters the other parameters as as was mentioned was this guy okay then there were parameters like the gauge couple some gauge couplings are there in the standard model once you specify those couplings there are no other couplings in the there are no other parameters in the coupling between fermions, bosons and gauges the self couplings of gauge bosons as well as the couplings that are all determined by three parameters this is okay okay these three gauge couplings now of these three gauge couplings we have to we're just going to just ask so these are classically classically marginally marginally and our quantum mechanics are the marginally relevant marginally relevant marginally relevant does anyone know what's the gauge loop standard model sorry this is writing writing writing just so that you connect even before we study just so that you connect what you anyway know to what we study okay so what's the gauge loop standard model SC3 SC3 SC2 SC3 SC3 SC3 SC3 SC3 SC3 SC3 SC3 SC3 SC3 SC3 SC3 SC3 SC3 SC3 SC3 SC3 SC3 SC3 marginally relevant okay so if we were purists and we were doing pure quantum field theory we would not allow this theory except for a free one couple but you know we're not such purists because we do know and this is this is this is something very important that I should have to have noticed this there is a big difference in practice between logarithmic run and the big difference can be illustrated by the following fact that suppose we've got suppose we've got suppose we've got scale separation like we do have in the real world it's a huge scale separation let's imagine that there's no physics between us and the plane scale T and the plane scale so the scale separation is sort of like 10 to the 16 or something like that okay any power loss separation in 10 to the 16 is a huge separation so any running that with a classically marginal logarithm has reached the fixed point push on the other hand what is the log of 10 to the 16 what is log of base of base 10 of 10 to the 16 it's 16 it's not such a huge separation okay what is log base e of 10 to the 16 well let's approximate the square space okay it's 32 you see so these logarithmic suppressions even over such huge orders of mind of people are not very efficient in taking you to the fixed point okay so in QED this flow away in QED from the fixed point that in a pure sense if we're really being loved or not to infinity we would have had no theory except the free you are but infinity as far as logs are concerned which is pretty different from 10 to the 16 and so that is the practical difference for a free fixed point between a classically irrelevant and a marginally a classically irrelevant and a marginally irrelevant classically irrelevant operators are bent away we can forget about them the marginally irrelevant operators are not imperfect okay so even though there is a conceptual difference between these marginally irrelevant and marginally irrelevant directions okay practical applications that's what we have in mind there's no practical and therefore for instance this guy which we spent a lot of lectures discussing is marginally irrelevant is still a parameter in the standard if the standard model were to be defined in a pure sense with scale not on infinity could only happen three are you understanding this? okay but because we will treat classically marginally irrelevant and irrelevant directions more or less on the same footing because they haven't had the time to put the depth out potentially they haven't had the time to repeat it we will count all operators that are either either relevant classically irrelevant marginally irrelevant or marginally irrelevant as parameters this is an extreme version of the thing I was talking to you about that is you remember in the purest sense we will keep only the relevant directions but then if you want powers you want to do better you keep the leading relevant operators now in this case the leading relevant operators are the marginally relevant operators and there's a big gap between how much they come to be and the next in relevant operators because these are lots of less than some hours a bit different is this clear? ok? so the standard model of the model the only classically relevant is the mask parameter the next way the class that's how we're going to group the model again the classically marginal operators whether irrelevant or relevant are 5 for the 4-term the 3-gauge couplets and the zillions of yukawa couplets in the standard model ok? yukawa couplets are also marginally relevant because they're dangerous ok? right so the standard model in the purest sense would be defined as the flow of a prefix point and would have fewer parameters because you wouldn't have the classically irrelevant stuff but then we look at those rg flows that include those operators also because they're the ones that can't do it so we look at a larger manifold than the fixed manifold itself including the classically relevant that's how the standard is this clear? any questions or comments I need to understand your point because you have to find out where the parameter will be delivered ah you have to find out the function that you have to do is by log of 10 to the 16 which is not considered high actually for the Higgs mass is what a 10 to the 16 a 10 to the power of 16 in some way or another way ok? to what position you have to do you get the point ok? other questions? comments? ok ah let us let us let us move on ok you pointed out the fact that rg flows can one is why we can never reduce highly from the already operation? yes but considering if you think these things are really tamper order of 16 should we be able to reduce things in a way like this? yes this ah right right right so there is some information in these modules that is that is true but do you see you get full information suppose you could figure out all of the the number the numbers behind each of the relevant who are in principle allow you to reduce what value let us suppose that there is another let us say we are at some scale 100 and there is another part here with some companies integrating all that particle generates a large set you know some values for all the high-dimensional operators in the same it is very plausible that if you knew all of this infinite amount of information you could figure out what the particle was what its companies were without having to think but here we are left with very the margin you see these marginary irrelevant operators as you say we know something about them in the same spirit we also know something in the relevant because we know the values of the relevant operators so all that we do know give us a few equations constraining what is happening in the unit those few equations are the number of parameters in the stack now that is more parameters than we want something to get away in the moment between the masses okay but it is much fewer so you know it is more parameters than you want for predictive but it is much fewer than what you would need it to figure out the details of quantum gravity to apply the back units like this the information you have about a parameter in the stack okay how the question is not excellent so now we are going to continue our study of scalar field theories today will be the last lecture hopefully I hope so but before this section there is an organization issue how many what is the attendance going to be in class next week if we do have class next week who will still be who will not be here with us so 4 of you are not going to be here yeah 5 of you so I need some advice on this what should we do should we continue have class is that what people would like the other option is to just go ahead and request these people to watch the video but also we have a possibility I other votes is this someone everyone first one have you realized you know we don't want to get in a situation where we don't end up saying much let me have a compromise shall we take one lecture instead of two next week is that okay we don't like that will you guys be back by next Monday no right next week next week you are gone I know week after that Monday nobody is back more than half of that week after next week is Monday will you guys be back by week after that Tuesday yeah yes yes okay so I'll tell you firstly the week after that Monday's class we should be week after that Wednesday okay so that's that's okay you are not going to be back you are not going to be happy you are talking about the next week of the next week yeah next week sorry I am not are any of you going to be back next week I am back one I am there you are back by Monday I am not how about you I am not back so the week after that is basically short okay so I think we are going to have to basically continue some sort of schedule but I'll try to minimize the number of how to do this you may have a clean solution can you guys try watching the new lectures yes do they work where I mean do you feel that you are dead in deeper center okay so how about okay at least for your so how about that we cancel next Monday's class coming Monday's class we'll have it on Friday and we'll have it on the Wednesday subsequent subsequent week and then we try to take let's say what are we doing that sounds like a compromise in usual way which means that reduces one class that you are missing you will miss only one you guys will miss two view the video lectures is this okay yes are you going to have the lectures on Friday no no I have children now really no no we are not going to I mean all of these are you guys are free to do what you want that's not good but but these are the things sorry because otherwise we like to reach November December we wouldn't have talked about that normally we wouldn't have talked about the things excellent let's get it sir by the way one clearing up from last class we computed if you remember the beta function of the 5 to the 4th theory I just wanted to get the number straight okay the number behind that beta function because well it's one it's a number that's sometimes used so you remember we started fighting the 4th or 4th you remember the calculation I'm just going to work out that okay we started fighting the 4th by 4th theory and we got two powers of this term we had a half and then in the diagram we had 4 into 3 by 2 times 4 into 3 by 2 times 2 there's a famous identity so I did this as 6 into 4 and it applied there was 1 by 2 5 4 in the integration measure and then we got the volume of the 3-sphere the unit 3-sphere just 2 by 2 okay so this is the set of numbers this comes with a G-square or G-square okay and then there was some sorry that's a G-square there was some down here okay what are we supposed to compare this with there's a sign which we got straight last time what are we supposed to compare this with what are we supposed to compare this with one of them we're supposed to compare this with we know by so beta function the number that comes behind so if you write dG by d log lambda is equal to some number 10G-square some number times G-square that number is the ratio of these two numbers that's number this number times this okay so there's an additional 24 that cancels this okay what this was 21 this was 21 this is G-square so we calculated this number and we're comparing it against G-0 by 4 so we take the one-by-four vectorial now this number we cancel this 24 so the number behind the beta function is this now let's let's see what we can so 6 cancels 6 2 cancels 2 okay so 4 this becomes 2 cancels that's 3 3 into 2 divided by 16 20 by 6 this is the number behind the beta function it's the sort of famous okay the 3 by 16 times squared is the number behind the beta function provided for there not very important really I mean I also want to actually calculate but just since it's so easy actually today's lecture is going to be about a non-trivial one of you okay it's not about a non-trivial one it's about a non-trivial one of you okay for 3-dimensional so let me get the idea behind that and then we'll start trying to do a few calculations the idea behind it goes as follows we have this G4 operator and we have D by D lambda log lambda 3 by 6 the sign is right because it's log log lambda goes to 0 lambda goes to 0 sorry lambda goes to 0 log lambda goes to 9 centrality and G9 D case after that order G G squared because classically the dimension of the operator was D now why was the dimension of the operator Z the dimension of the operator is 0 because we're between 5, 4 and 4 dimensions how do we judge the dimension of the operator sorry how do we judge the dimension of the operator well we know there are D dimensions these place time dimensions the dimension of the scalar wheel is D minus 2 by 2 okay if we multiply this by 4 because we have 4 times that gives us the next dimension of the 5 to the 4 term by the way can somebody tell you why the dimension exactly can that be 2 into D minus 2 the dimension have been so what number comes here the number that would come here is the dimension of the operator minus 4 so dimension minus minus D this is what you would expect generally let's say the dimension is small we get a negative term which is right because this this is flow to the the way up let's flow to the classical analysis would give this but this is the dimension of the operator okay and now we want to know what they said that evaluates to this case so this is simply minus D times G which is equal to D minus times G write this beta function equation in a fancy the beta function can be written as BG by D log of lambda is equal to okay B by 6 B by square G square so this is flow into D the verge of this formula is just simply so that it's some formula which should only be literally linked seriously when you say B equals 4 the same formula but I've written a slightly stranger suggestive way for the following suppose there is some way of making sense of quantum field theories or more precisely because there will be no full wavelength some calculations renormalization in orbit in a dimension there is not any suppose we could devise some way of making sense of that then we continued then we could ask what would the beta function what would the beta function of the theory be in 4 minus epsilon dimensions when epsilon is small so what can we say these terms completely recovered the dimension so we would get dG by d low lambda minus epsilon 3 by 16 by square into 1 plus some number what was on epsilon then we would ground to know this is true so we could make sense of this idea of the defining quantum field theories in high dimension in non-integer dimensions at least for the point of your renormalization group flow all that this needs to mean by the way is some procedure like a science of renormalization group flow for an arbitrary dimension that agrees with what we mean what we know in every integer dimension some analytic continuation of the formulas of quantum field past integer dimension suppose we could make sense of that then what would the beta function look like in 4 minus would it look like this no just as a little little mathematical physics suppose I'm going to ignore the mass parameter in my formulas imagine that it's 2 inch to be 0 and everybody will look at at least it remember it's much smaller now okay so let us look at what these renormalization group flows look like in 4 minus and now it's better to write dG by dT minus 1 and we put an epsilon G minus 3 by 16 minus square 1 plus over 9 inch G so this is a G just with one-dimensional flow line it's very easy alright what does the flow line look like near G equals 0 and G equals 0 of course there's no flow it's a fixed point set G equals 0 0 but a little away from G equals 0 which direction is the other way this is the segment that this operator is driven in 4 dimensions it's a marginal marginal in 4 dimensions classically marginal has become relevant in 4-dimensional flow because it's classically marginal classically relevant flow lines point out okay but do they always point out to this G squared out makes sense yeah the G squared out makes sense now let's see where this balances where does it balance balances when these two are here the balances when G is equal to epsilon divided by 3 by 16 by square in 1 plus of epsilon or my data expanded 1 plus of epsilon is equal to epsilon 16 by square by 3 epsilon plus order of epsilon squared we've been a little careless because there are also G cubed just at this point you see we've balanced these two terms but this was order epsilon square and this was order epsilon squared this is order epsilon squared because G is epsilon and this is this is this is order epsilon squared because G is order epsilon but what to G cubed that would be epsilon cubed so if we include the correction due to the G cubed term that would give us another order epsilon square correction due to the G 1 order of epsilon further suppressed than what G is so this here is here including all directions so although in order to know exactly where these two terms balance we need to know exactly what's going on when epsilon is small to order epsilon we already have enough information to compute what's happening so let's say this point is 16 squared around by 3 what is the flow line you'd love to see in order where is this diagram reliable this diagram is reliable at small epsilon when we can ignore the G cubed terms compared to compared to either epsilon G or G squared okay but G cubed is negligible compared to compared to both of these in some part range until G comes basically actually not okay so let's see when can where that epsilon G stop being no where it doesn't as long as negligible even one of them it's okay right so we can compare G cubed to G G squared and that's negligible as long as G is yeah reliable G is much more okay is this point within the domain of reliable yes it actually is more because it's this and that's much more easier okay there was a way of continuing what we would tell at least some aspects of it to not in detail damage we would have discovered something very very simple and that is this we would have discovered that the flow line range in four dimensions or higher than four dimensions just about this higher than four dimensions they could that would be classic four dimensions because that would be the beta function we computed that's extra dimensions becomes the flow has become different this one phase point splits into these two this guy continues to have the character of these attractive and there is a refulsive phase point the existence of this that is the phrase here the fact that the fixed point changes from attractive to repulsive is simply the statement that the dimension of five to the four goes from being higher than D at D greater than four to lower than D and D less than four that should always be all right what we have learned in this exercise is that once the dimension of five to the four goes so that it becomes relevant in small enough dimension near four and there is a new fixed point okay and this new fixed point okay now I am going to put in the other dimensions the other dimensions of normal are like this and we will continue to be like this that they always go just universally so now let me try to schematically draw the renormalization group flow lines for such a the renormalization group flow lines for such a diagram would have to go up here this guy is very simple it is not attractive this goes in this goes in this goes out this goes out but this guy is a repulsive fixed point okay but this guy is a repulsive fixed point okay so this continues to go in but this plate repels so the flow lines go like this I still draw like this let me do it again sorry I will operate the other dimensions I am sorry sorry about that let's do it again sorry this guy sorry about that okay this guy is attractive to us but the mass dimension is repulsive it is a renormal okay so every direction goes like this okay I will finish this okay this direction and the exact brand the exact shape does not matter okay okay okay so this will go like this or like this please move it okay so that is some flow lines we will look at this one but which one this one let's look let's look we have dG by d now we have that let's say dT we put a minus extra on G plus a number let's look at the sign of dT by dT so we can ask a question just for this so if I go a little bit away from K only this term and I got the sign I got the sign so if I go only a little bit away from this this guy is positive because and because G is positive it goes this way if I go only a little bit away from here this guy remains positive but G is negative so do that this term is always negative this is okay this is okay this is okay I am in such a situation okay now let's look at this this whole thing in a little bit away we got two fixed points in the sign what are the set of renormalization to root flows that begin their life at our new fixed point only this one dimension two fixed points has how many relative directions one this final four direction as far as the new fixed point is concerned is zero there and we see by this what is the number of what is the set of quantum field theories that start their life as you know that are defined as renormalization from the free theory okay so what we've got is a very interesting situation all of this blackboard defines well defined in the purest sense quantum field theories these are quantum field theories that are defined as flows away from the there is a sub-class of quantum field theories here that can be as well defined as flows away from another fixed point this is an interactive fixed point you see these flows are can be thought of as this flow or this flow a flow that's almost on this line that joins this fixed point to this line so it spends an infinite amount of time near the fixed point that takes off it's very very similar to the way we started by saying that when we go all the way to the UV okay you define you define you know the flows are effectively emanating out of this now we have special theories in this class of quantum field theories that can be as well defined as flows away from this fixed point if you tune the starting point here to be okay this is a very interesting situation in principle also a practice we've been in two minutes about this practical okay but look at this in principle you see suppose you were somehow suppose you were very involved physicist and you discovered this fixed point or you discovered the free physics okay okay you might say well and suppose you knew that the scale of ultramarital physics was very high black scale you might say it is not possible to have a theory that goes away from my one-dimensional space theories that go away from this one-dimensional space are non-prenormalizable to what I've said okay and I don't know how to make sense of them and then you really experiment and you found that what best fit data is going away from the space would be very possible you'd know how to make sense of these these theories but there is a way of making sense of them as well defined quantum field theories but as flows no I mean from this fixed point but from another fixed point so this is one of certain paradigm for making sense of theories with non-prenormalizable time non-prenormalizable in the sense that there is only one relevant direction away from this new fixed point exactly so why do the four theory in four dimensions why do the four theory in four dimensions you know if you had got your that then of course in that case there is a log so that's why we don't care about it so much but if you had taken lambda not to infinity you would not have been able to make sense of a theory with interactions that's what we discussed right there are no scalar field theories with interactions in a pure sense in four dimensions it's the same situation but now it's made up of four dimensions because we're not in four dimensions we're now in four dimensions we've got this new fixed point this now has one classically relevant and one classically irrelevant as we will see here just this thing is I mean the beta function here is not what a g squared, what a g epsilon that's what a g okay so one classically rather than one classically irrelevant dimension about this fixed point you might say that we have just learnt in our quantum field theory course the quantum field theories that orderly will be are defined as as renormalization root flows that come out of fixed points there's only one direction renormalization root flows that's the only thing that we can do if we adhere that will see that will see that will see that will see is this clear however this theory is very defined even when you're out there all the way that you're in but its definition is not as a flow like in this fixed point it comes as a definition this definition is a flow like in this one paradigm one possible paradigm for dealing with non-prinormalizable it's possible it's possible that when you look at a theory and you see that the renormalization interactions the theory is actually well defined all the way to the union and the way it's well defined is that in your space of quantum field theory there is another fixed point which you didn't know about and your theory is defined as a flow away from that this is an example of that is this clear through over the years many people have suggested that this is a possibility of quantum gravity it goes by the name of the possibility of asymptotic safety so when you read papers in the archive and talk about the asymptotic safety paradigm for gravity it's the claim that in the space of integrals over the graph the metric there is a new fixed point quantum field theory is defined by that fixed point there are flows away from that fixed point that take us to you know here to the fixed point and that is the that is how quantum gravity makes sense it's a minority point of view doesn't seem very likely to be true can you? but this is the idea yeah so whether real cross sections can merge or not to get it whether it cross sections which cross sections make the same cross sections whatever whether the integral is converged it's not matter depends on how you define it ideally you are going to go to an integral and add counter terms and say that after this point I won't be able to add counter terms so like how does it matter that you if you don't have to worry about RG it flows everything how does it matter? you see the point is what you are doing is perturbation theory so what you will be so the idea will be that there is her set of integrals that will converge okay but you are not the technique you are using is wrong because you are talking about perturbation theory so we should switch in this particular example this is free and this is interactive in the application of quantum gravity that is suggesting it will be reversed this is the free point and this is the interactive but the idea will be that if you could define this path point is you can't this is an interactive you cannot find that what you get and there is divergence when you do the answer around here okay except for the fact this is reversed yeah it should be very simple what you are doing is trying to define quantum gravity is the claim okay I want to say this sounds very unlikely to be true for various points from various points of view but you know one shouldn't never say never one word for instance keeps writing a review article saying he thinks there is now good evidence but he says something totally nonsense to me but anyway and you know those will be the right answer we don't know so yeah so in fact what you are trying to do is somehow define gravity as an inversely normalization flow loop flow from this fixed point and that definition is problematic leads to all these divergences because if you flow away the many directions in which you can be away that's the half of the divergences you understand that okay but the point is that if you broaden your point of view there is one distinguished point from which to start but this way if you knew about this fixed point you would be fine that is the idea are we saying that that's 4 minus epsilon dimensional because if we just have the 4 minus epsilon dimensional are we having an initial fixed point from where all operators are like irrelevant this one because in 4 minus epsilon there will be some like 5 4 will be an element no 4 minus epsilon 5 4 is relevant that was the calculation we did you see in 4 minus d the dimension of 5 to the 4 is less than 4 by 4 let's take the graph of 3 the scaling dimension of 5 is half in 3 dimensions and therefore dimension of 5 to the 4 is 2 in 3 dimensions okay so the free fixed point is relevant dimensional analysis says 5 to the 4 is relevant but there is a new fixed point because it's non-productive it is not sorry not non-productive but non-free about which this 5 to the 4 term is irrelevant this calculation this calculation of extending quantities directly to non-integer dimension can be seriously and we can do this okay but before you take undertaking this exercise let me tell you why we are doing this why we are undertaking this exercise we are undertaking this exercise because we are interested in a particular dimension less than 4 named in 3 okay now of course you are immediately going to say well epsilon is not small but epsilon is 1 that's true but this is often but for theorists you know you find a problem which you cannot solve you put you enlarge it to one parameter set of problem which you can solve at one end and then systematically correct trying to approach the problem you can solve in perturbation theorem this is a standard tool of the theorist okay now what you do is to take a problem you cannot solve and reduce it to an infinite number of problems that you can solve mainly each order of perturbation theorem okay in this you can try to learn about what happens here sometimes you can see sometimes you don't it's okay but already this is giving you insight because in 4 minus epsilon dimensions this bit point certainly does exist so as you change epsilon you can change epsilon slowly from small to a number to 1 it's plausible that continuation exists in fact smaller clause will be enough accuracy computed a lot of the properties of this thing and equals 3 by going to sufficiently high order in this so-called epsilon expansion and we will discuss what why does it start at 3 why not start at 3 why not start at 3 in 3 because then you have no no no reliable calculations you see this fixed point is not free fixed point okay so we cannot do ordinary perturbation theorem we can at d equals epsilon because although it's not free its coupling constant is epsilon so what justifies perturbation theorem epsilon but at d equals 1 that's not reliable but that's the kind of thing a lot of people do you know new field theorem uncontrolled approximations uncontrolled approximations that are never controlled those are very dangerous things here you've got some approximations given that's very controlled at least one epsilon is small you'll study in detail and then try to see what happens as epsilon is small that's good what yeah I remind you that epsilon is 1 I just remind you that factor in front of the 16, 5, 12, and 3 which is vastly greater than 1 and if you consider that to be a expansion parameter and g is of course an expansion parameter huh huh huh huh yeah now what to say um you know I mean it's not even good good but you see there's you have to be a little bit careful because you found the g is 16, 5, square epsilon by 3 but you need to know what is the right factor that weights perturbation theorem of course perturbation theorem is 25g but that could well be various factors of 4pi square in the denominator okay in fact that are typically better this comes from as you know measure factors okay so the question is whether the truly effective parameter of perturbation theory is less is you know what is what is the parameter it's finding g or are we expanding in g or are we expanding in g times the measure factor I mean how are we guarantee that it's perturbation theory what is the the relevant question is to compute this and then compute the next order and see that and see that the next order is smaller than the first I mean it's still guaranteed to get the series of can we or will we be satisfied to be an perturbation theory perturbation theory is almost almost almost you know if you try to compute where you might I taught a class where I you were not I taught a class in which I discussed computing the in quantum mechanics you got a harmonic oscillator you add the g x to the 4 you compute the change in the energy levels as a as a function in g what is the nature of that series that's what I'm talking about this is well known in intermediate computing I'm saying it's much more much simpler than but it was it was much simpler than a harmonic oscillator almost any problem in quantum mechanical perturbation theory open shift pick out the problem that he works on in quantum mechanical perturbation as for the nature of that perturbation series quantum quantum mechanics harmonic oscillator 5 to the 4 it's not going to budget perturbation theory very very converges we don't care about that Okay, that doesn't affect very much, okay, yeah, other questions? What? In every direction, the oil just zooms to the ground. Let's go, let's discuss first, why they are interested in this factor of 40, okay? Why they are interested in this scale, I think. Of course, you know, it sounds interesting, but there's a very practical reason why they are interested. And I want to discuss that, and so we have a smart slide diagram, a slight digression to understand the criteria in three dimensions, a system of spins in three dimensions, three spatial dimensions, okay? So it's a block, a block of objects that have some spins. And let's suppose that the spins have the property that your Lagrangian has an up-to-down selection. So the typical model for this is the civilizing, okay? Where spins either up or down, they interact with an S1 dot S2, okay? And if it spins either up or down, they interact with an S1 dot S2, and there's a clear selection of problems between flipping on spins from up to down. Now, consider any such model, you know, any such model at finite temperature. As we have discussed, okay? As we have discussed, as we have discussed a quantum system, okay? See, that's also a classic. As we have discussed a quantum system that's three dimensions, a finite temperature is described by some sort of path in tableau on R3 times S1, okay? Now, if we're interested in long-distance physics, long distance of the scale of the inverse temperature of this S1, effectively this is some three-dimensional path in tableau, because we're just colliding on this, on this thing. So it's an infinite number of masses of particles in three dimensions. If we're interested in long-distance physics on the scales much larger than the inverse temperature, let me integrate all those masses. You could ask, you know, it sometimes happens as you change the temperature, as you know, that there is a phase transition. Now, there are two kinds of phase transitions in nature. There are discontinuous phase transitions, sometimes first phase transitions, or the continuous phase transitions, sometimes first phase transitions. It's a feature of, say, you know, all your phase transitions, that in the neighborhood of the phase transition point, physics becomes scale of the inverse. The correlation functions, the correlation functions diverge. Correlation lengths diverge. Physics develops for scale of the inverse. The schematic reason for this is, you can see from a Landau-Ukensberg kind of diagram, a second order phase transition happens when, as you tune the parameter in the next temperature, the effective potential for some scale of the inverse from this to flat to this. When it becomes flat, you've got a massless value. So integrating all of our masses stuff, you've got effectively scale of the inverse. This is a carton value, but that's the end. So this is a carton value of what happens when you have a lot of phase transitions. In the neighborhood of such a phase transition, magnetic particles often have such a phase transition. Magnetic systems, like this spin system we were talking about, often have such a phase transition. You know all about it. You know that this is the case. Magnetic systems often have such a phase transition. This phase transition of spontaneous means magnetization. Higher temperatures, there's no magnetism. Low temperatures, there's spontaneous, but there's some development in the space. And at a critical temperature, you've got the massless physics at some point. You see, if you have a correlation like this, that's finite. It's not scary, but it's this length correlation that goes up the most. No distribution, that's okay. Now, this happens in quantum magnetic systems in three dimensions when you meet it up, or in just a model classical kind of system in three dimensions. Right now, there's an ISIC model, SI.SJ, and you tune some parameter. This happens as a function of that parameter in three dimensions. The question is, try to understand. Okay, and this kind of, in real systems, this happens in magnetic systems. There are several things happen in liquid solid transitions. There's the so-called triple point in the phase diagram of water. So I'll just describe my phase by a scalar variant there. Now, the physics of the scalar variant is very interesting. The physics of the scalar variant point is very interesting. And many interesting things happen in the scalar variant point. Okay, so we'll describe this all in the language of magnetism. And I want to show you, from all of the rest of this lecture, to show you, the role of the rest of this lecture is to show you how very little information of the fixed point that governs this theory relates, computes, many of these things. So let's suppose we've got some magnetic system that has a critical temperature at Cc. Okay, let P minus Cc, that's the final equation, for instance, but what happens to the correlation? So suppose you get two spins in the system, you will compute the correlation. Away from T equals Cc, because there's a mass gap, they die of explanation. Okay, so there's some. Okay, this correlation length goes to 0 as T goes to Tc. That's our infinity as T goes to Tc, because it's getting there. Question, how does it go to Tc? As T approaches Tc? You know, what's happening? How does it go to? True, if it. In the condensed manner, the statistical physicists have a question they have for a long time. Okay, and it's very beautifully answered by this picture of renormalization group flows in quantum field theory, because I now want to explain it. Okay, firstly, how does this connect to the two? How does this connect to the kind of analysis we do? As we said, the spin system, whatever it is, is described by some path integral in three dimensions plus a circle. Now look at distance scales. Because I'm interested in systems where correlation length is divergent, we do it in distance scales much, much larger than a circle. And imagine a path integral that involves only the zero modes of this. After that, we don't know anything about this path integral. Initially, the graph is very complicated. We don't know much about it. But we do know that it's described by a path integral involving spins. And let's say that there is some sense in which you can model the spin by some scalar field. The fact that there was s to the minus s symmetry tells you that there will be 5 to the minus 5 symmetry in this section. So why do we know nothing else about the theory? We believe that it's governed by some path integral of a scalar field with 5 to the minus 5 symmetry in three dimensions. It's an Euclidean path integral. We're proficing completely in the middle. But we know one other phenomenologically fact. We know the fact that this path integral that we do will, of course, be a functional integral. As temperature changes, meaning as we vary along one parameter set of initial path integrals, at one point we reach the space transition. Now let's put this in the language of the renormalization rule flow students. The scale of the inverse temperature is like a lambda. Because at highest, you know, what we've got is a path integral which is on R3 and S1. If we went to lambda knots, larger than the inverse temperature of the circle size. Larger than the temperature. You know, this thing. We would not be justified in doing a three-dimensional path integral. If we're four-dimensional. So at some scale that is set by the circle size, the size of the inverse temperature, that's where our path integral of renormalization rule flow analysis starts. Okay? So that's some lambda. The scale that we're interested in is this photo of the correlation function, zeta. Everything that we will do will only work when zeta tends to this S1. You know, when this is a much longer length than this. So that we've got scale saturation so that it can use the ideas of renormalization. But luckily, the problem of zeta going to infinity is precisely the bridge. We're starting in the UV with a one parameter set of initial coefficients. So we started looking at one parameter set of flow lengths. And what we find is that as we vary along this one parameter set of flow lengths, there's one point on this one parameter set of flow lengths, namely one point on the space of initial temperature at which we find scalar variance. Zeta diverges. We have scalar net. What does that sound like? What does that sound like at that one point where we're floating to a fixed point? Because only fixed points as far as we know have scalar variance in physics. Scalar variance is the beta function foundation. We change. So in the space of UV, if we take some one-dimensional line that is chosen out of the physics of these, these princesses, that we take some one-dimensional line and somewhere on this one-dimensional line we hit the fixed point when we float away. What does that mean? What does that tell us about the fixed point? What? It tells us that the... It tells us... You have a class? No. We have a problem. Oh, God. Okay. It can do analysis, that's all. Oh, God. Sorry, I'm sorry. I'm going to play the analysis. It tells us that the attractive manifold of the fixed point is co-dimension 1. Because on the space of all the... Everything's infinite. The co-dimension is 1. So that if you draw one line, it will hit this attractive manifold. That means in terms of relevant operators, it tells us that this fixed point will be the one relevant one. I'm sorry, I'm going to have to stop here. Where is I going to send this? They have a dog. Yeah, okay. I'm sorry, I'm going to have to stop here. Very unsatisfying. It's like a punchline. It's like this. Next class is now... Next Friday. Next class. Okay, I'll tell you when. Okay, thank you. Yes, sorry, sorry.