Courant Events Right: to tell us, critical Review of Constructive QT, and here, there's, I think, some titles missing, particularly.
Courant Events Right: Thanks a lot for the invitation. So, the title was given by the organizer, so I did not change it, so I didn't dare to change the title, so it remains as close, also because I like…
Courant Events Right: So, but when I was preparing the talk, I realized that it's a bit difficult to speak about quantity theory without saying anything about
Courant Events Right: about constructing quantum field theory, without saying anything about quantum field theory, so I will present… I will give a short introduction to quantum field theory, which is, of course, a sort of impossible task, but I think it's necessary, and…
Courant Events Right: Also, because the constructive quantum field theory at the time was very much embedded with physics and quantum field theory, so it's a bit difficult to separate the story. So also, so my talk is divided in three parts, I mean, I will read you some facts of quantum field theory, yet then I…
Courant Events Right: That would… I don't know here.
Courant Events Right: So then I will speak about the golden age of quantum field theory, sorry, of constructive quantum field theory, and then I will tell you some more recent results. First, I want to stress that I will focus about high-energy physics, but the community of a lot of people working in constructive quantum field theory
Courant Events Right: You know, around the 90s, shift to condensed matter.
Courant Events Right: So, I mean, I've seen that a number of papers quoted in this conference topic in the 90s, but the point is that this community went in the direction of condensed matter, with the idea that… with the reasonable idea that condensed matter was simpler
Courant Events Right: And there are more models. So, I mean, there were two papers, Fatim Galavoty and Federman Trubovitz in the 90s, I remember when, because…
Courant Events Right: was just in the period that started my research. Then all this com… I got a number of people went in this direction.
Courant Events Right: Unfortunately, the idea was that condensed matter is much simpler than quantum fertility. It is.
Courant Events Right: But it's also very much difficult. In fact, the idea was that to understand ITC superconductivity, which is still open, I mean, since 40 years and so, so the idea was, I mean, let's do a similar program, ITC superconductivity. Then, of course.
Courant Events Right: I mean, still, I mean, there was a conference in Rome just about accessibility situation, even if physics is the same as in the 90s. So, this is the reason why I will stop in the 90s, but from the 90s on, there is a lot of research about extended family surface, but models. The idea was to prove that
Courant Events Right: or IPC support utility, or even lower match capability.
Courant Events Right: none of this goal was acute, and… so this is the reason why we stopped about that. But let's… let's start about quantum field theory, which is the sort of basis of our…
Courant Events Right: of modern physics, of course, it's the language of physics. It's applied in high-energy physics, in the condensed matter, in statistical physics, and so on. And I cannot resist to put some numbers, because I like numbers, so this is a famous example, so shown to every student in physics to motivate
Courant Events Right: them, I mean, the Germanic factor, in which this is the theoretical prediction, used standard model, and this is the electron Germanic matter factor. Measure, this measure, I think, refers to Samarwa group.
Courant Events Right: in Arva, and by Rabia, and then you see, it's impressive, the precision of this prediction, so it is a sort of spectacular Rezzaldo standard more that cannot be forgotten. In fact, there is this famous sentence by Feynman.
Courant Events Right: And… but now the tradition is much better than that. So, I mean, this is a number that cannot be… cannot be forgotten, okay? It's incredible if all these numbers are the same. There is more deviation for the more, for data, for now, the firming up, with the…
Courant Events Right: We want to understand this small discrepancy for this one, okay? Of course, I mean, these are obtained by series. These series are now convergent, and even not asymptotic, so this location is unclear how it's found. So, I mean, there is a very nice problem for everyone, not only material, but…
Courant Events Right: all humanity, to understand how it's possible that we get this sort of spectacular prediction with mathematically should be clear. And I would say the situation is very much the same as 300 years ago, in which Newton, I mean.
Courant Events Right: Doesn't use, of course, derivatives, use infinitesimals and fructions and strains objects, which are not very well infinitesimals, which were very not well defined.
Courant Events Right: Mathematically, and then Bertley, who was more mathematically oriented, said something like, these are goals of the factor quantities, cannot understand what are they in this much, but it was very, very much aware that these things were…
Courant Events Right: I mean, the prediction of Newton were correct. So, I mean, after 100 years of war, at the end, we understand the mathematics of Newton, and then I think 100 years
Courant Events Right: works. We will understand quantum field theory. Okay, so, but then this is an introduction. That's a basic concept of quantum field theory.
Courant Events Right: Okay, as we know, as everyone knows, quantifiable theory of several graphs written in the probabilistic averages, so fields are typically defined on the lattice, so… and that, which is called the inverse of the lattice, is called the private top-off.
Courant Events Right: Then, there is a finite volume, which is called also infrared cutoff. Then, the inference problem is to remove the volume, and this is typically of interest in congested matter. If you want to consider, instead, the continual limit, that is the limit of crucial limit for quantum theory.
Courant Events Right: So, don't forget that the lattice is a regularization for high-energy physics, but instead, for condensed matter, is the physical ionic lattice, so you don't want to remove it.
Courant Events Right: Okay, sorry, go ahead.
Courant Events Right: But the nationality is very similar, right? You put an artist, and then you want to see what you have.
Courant Events Right: So, also, it's important to recall that the limited cyste is believed to exist on the strong sector, as we heard, it's asymptotically free, but not for the other two sectors of the standard model directly.
Courant Events Right: What the butt.
Courant Events Right: Yeah. Okay.
Courant Events Right: So, what will keep this effect in point of view? So, you don't want, really, to remove the lattice, but you want to keep it large enough so you don't see its effect. This is a possibility. So you consider that the standard model holds up to a certain energy space, and the mathematical problem, I will tell you something about it, is to get some values.
Courant Events Right: Okay, so… as everyone said, I mean, the basic objects are the Schwinger functions, which are
Courant Events Right: Written as probabilistic averages, so they are written as averages of objects with a probability measure.
Courant Events Right: So, it is also interesting to introduce the, to introduce the genetic functions and the effective potential.
Courant Events Right: So, let's say something. So, in the bosonic… so one put… I put myself on a lattice, so I will avoid problems of definition, so… so I put on a lattice, then in the bosonic case.
Courant Events Right: you have, the phi, this big phi R just, belong to R, are Gaussian variables. Then this is the famous phi-4 model.
Courant Events Right: Then, of course, you can write the expectation in terms of sample product of covariances, what is called the Wick Rule, in terms of propagator.
Courant Events Right: And then you can use these expectations to write, formally a sum of Feynman graphs, or if you consider the law, connected Feynman graphs. Let me recall that,
Courant Events Right: I mean, this is just a way to represent. In fact, for instance, Schwinger, which is one of the heroes of quantum field theory, didn't like very much these Feynman graphs. So, Feynman graphs are not something really intrinsic to quantum field theory. In fact, Schwinger said something like, I mean.
Courant Events Right: now everyone can understand quantum field theory, but it was not positive about that. It said, okay, everyone, now it's like… it said something like, it's like small calculators that everyone can make a square root of, I don't know, 100 instead of using 9. So he didn't like to use them, that everyone could make computation using graphs. So that's…
Courant Events Right: Okay, so, so pure Bosionic theory, EX models, pure Younger Mills, easy models in D, retent entry, and so on.
Courant Events Right: Then there are the fermions. The fermions are, instead, was introduced at some talk. The fermions are given in terms of Glassman variables. There is this representation very similar to the other ones, but it's deeply different. Deeply different. The other… the bosonic one are generalization of…
Courant Events Right: Gaussian integral, this is our generalization of determinants, essentially.
Courant Events Right: So, Fermi model, which was the first quantum field theory, models for conduction electron, PCS, ABART, easy models, dimers, 6 vertex, 8 vertex, all of that. So, for instance, this is the Fermi model in D124. This one is the…
Courant Events Right: It's a formal way to write it.
Courant Events Right: then if you want to write it on a lattice, you put yourself on a lattice, and then you also… you have to add, for instance, this temp, which is called the Wilson tab. This is a wave… there is this annoying fact that if you put on a lattice, you get too many balls.
Courant Events Right: So you keep the fake ports, you have to add more with sometimes our other method. Okay, I issue using the pointer, because we don't see when you… when you point, we don't see anything.
Courant Events Right: I see a green foam.
Courant Events Right: Oh, cool, yeah, welcome.
Courant Events Right: Okay, so, let's go on.
Courant Events Right: So again, now you can write in terms of graphs, using a sum of… again, you have, again, the weak rule, just the students of Perry, and then you can write with the same with this, with this, science, okay?
Courant Events Right: Ben… You have bosons and failures, quantilateral dynamics, electro-with theory.
Courant Events Right: all the standard model, protocol dynamics, and so on. So this is the massive quantity for dimension, you put the mass, which is simpler than on the lattice, so you're looking at the black hole, I mean, you have a good integration with respect to the A, which are bosons, and the psi, which are Fermions, okay? And then you put on the lattice.
Courant Events Right: The interaction is… can be written in this way. So formally, you have this again, this something like that, F squared plus M plus the mass, and so on. So now, there are some interesting points to say, that there is… formally, this is as invariance under this phase transformation, and this implies by
Courant Events Right: classically, by Netter theorem, the conservation of the current. If there is no mass, there is invariance also with respect to the current.
Courant Events Right: Okay, then if there is no mass of the photon, you have also this gauging variants. Also, it's important that physical observables are gauging variants. Okay, so there is no… you have this side doesn't… you don't change the thing. So this is formally. The problem is that if you want to put everything on a lattice, then
Courant Events Right: Some of the such properties are lost.
Courant Events Right: So you can… we can add the… in the case of massive U1, you can add excite N also in the lattice. Then you… the longitudinal part of the… of the boson propagator, this one just GA, becomes this one.
Courant Events Right: And then, instead of the… of the…
Courant Events Right: observation of the current, you have these sorted entities, which are exact in this model, but they could be not exact if you consider other organizations. For instance, the mental one, okay? Also, it's important that
Courant Events Right: there is independence on xi, because then you can put xi equal to 1 and eliminate the longitudinal part. So this would be not true with other normalization.
Courant Events Right: Now, instead, the color talent is not… not conserved anymore, instead, so this later becomes urated, and then you have this…
Courant Events Right: This famous formula that I will comment later.
Courant Events Right: If you consider that a week is even more difficult, because now the interaction is chiral, so you need… you cannot put the mass of the fermions, then there is the famous cascade of problems
Courant Events Right: which was solved by the heroes in the 70s, then you have to put the eggs. The eggs generate the mass, but there are the anomalies, the anomalies must cancel out, and so on. So there is a cascade of problems that can be… you can solve just looking at. Okay, so the point is that in the electroweak sector, you have… the interaction is… is… is… you have chiral currents, and this chiral currents is not…
Courant Events Right: you don't have any more PC invariance with respect to Xi and so on. But this is just… I will come back to this. There, you could not even write a model.
Courant Events Right: So in… Yes, in fact, yes, yes. There is not even a model that you could write. Effective did not write.
Courant Events Right: That's exactly the reason why I didn't write it. It was a cat, so yeah, that's the point.
Courant Events Right: But on the other hand, I mean, must exist something. So, okay, anyway. Yes, in fact, this is the reason why.
Courant Events Right: So, let's see the history of… the glorious history of randomization. So, let's… let's now proceed as it was done. So, we remove all cat-offs, and then we use the root rule, formulate, then we remove all catals. So, we get to see this. And then, it was this discovery of the first people doing computation that you get infinite.
Courant Events Right: Okay, so the idea is that you have these diverging internals, then you can make regularizations, which are not just done order by order, for instance, momentum regularization, and then this was the regularization of beta, dizone, phi, and so on. You can say, okay, also the Bayer company is changed.
Courant Events Right: Then I… then I invert formally, I put the cutoff, I invert this relation, I write the dressed coupling in terms of… pair coupling in terms of the dressed one, and then I get the series in the dresser.
Courant Events Right: And then the minor columnist piece is that in certain theories, this P lambda is financed, so that was the big discovery for the parameter piece happens.
Courant Events Right: Then you can say, okay, epsilon, I put in a number, okay, completed by experiments, and then I truncate the series, and then I get predictions, okay?
Courant Events Right: So now there is the famous criteria of normalizability, which is obtained. It's very, very simple, because you obtain this criterion just making this power counting, and then, for instance, for the… for phi 4, this is the formula. Then, if there is that N is the order, and then D is the dimension, N is the number of paramil fields, and psi and so on.
Courant Events Right: So you will see the Ferry model is not a normalizable and equal to 4, and so on. Instead, 5, 4 is normalizable and equal to 4, and so on. But of course, I mean, this is just a very rough criterion, but to prove that to all orders.
Courant Events Right: you can absorb the divergences, it's not easy at all. In fact, there was this famous proof by Bobolubov, and Borvolubo and Parasyuk, which was able to prove to all orders, because it's not obvious at all, because we have this problem that diagrams
Courant Events Right: at any order contains any possible kinds of diagrams, it's not easy to do. So even the proof of that was a very hard piece of antibiotics.
Courant Events Right: so.
Courant Events Right: But then, unfortunately, for quantum electronics, or for gauge theories, things are much more difficult, because
Courant Events Right: It's not enough to consider a power counting to understand if the model is randomizable. You need more. The power counting is not good. You see immediately, you need a reduction of degree of divergences, so if you remember, the…
Courant Events Right: The boson propagator has this part here. This goes bad when you… so… Let's say it this way.
Courant Events Right: basically go to one, so it does not grease. So if you use… if you use the…
Courant Events Right: the propagator as it is, you will get terms which are not decay, so the theory at planetary is non-normalizable. So you have to prove that this piece, the longitudinal constitutional, is zero, otherwise you get a non-normalizable theory.
Courant Events Right: And then, as we've seen, that depends on what identity. As I was saying, in the electroweak body, there is even no way to… to write that, and so on. So, in addition, you… the…
Courant Events Right: The running capital constasants are not independent, there are relation, even by the word debate, but these world editors are broken by cutoff, and so on. So there is a number of problems when you… it's not simply power counting, if you want to apply this to… to gauge theory. And then in the chiral case, there is even in other problems that you… you can… there is, for instance, in the electroweak, there is… there are
Courant Events Right: possibilities of doing that, only if
Courant Events Right: the anomalies, we have seen, they canceled. In fact, it was shown that they canceled in 74 by Eliopoulos and others.
Courant Events Right: It is… I think it's something very, very nice that You get revisability only
Courant Events Right: If you have leptons and plaques.
Courant Events Right: And then only the charges of that verify this relation, which again imply that the problem and the error front has opposite charges, okay? So it's very nice that it's that without randomization, all our work immediately disappears.
Courant Events Right: Because the atoms would be not neutral anymore. But it's nice that, mathematically, we get that condition that I find very, very nice. Okay, so it's not normalizable if you don't have the proton and the electron at the opposite charge. Okay, so it's not anthropic, it's just that material is not normalized.
Courant Events Right: Okay, so at the end of this story, so it was very short, but…
Courant Events Right: You get the ability. So, for instance, the geomagnetic factor is a series, and then all the coefficients are finite. You can truncate.
Courant Events Right: And then you get this,
Courant Events Right: For instance, the Germanic effect. So the first computation is the 48 computation of Schwinger, actually in the tomb of Schwinger, basically.
Courant Events Right: Schwinger put this forward in this installment. And then, then these other people saw my feet. I mean, there was a mistake, and maybe no time to describe this. Now, instead there was a… it was a mistake in the computation, right? A20, a diagram was missed, and then…
Courant Events Right: That puts a law that makes a mistake, and then it was corrected later, and so on.
Courant Events Right: And then, also, I want to remind this JK environment, 72, which they compute the… and then the best result is this Aoyama, then it's done numerically. Okay, so they compute all these numbers, and then you get this number that… but, of course, you can say, as mathematicians, about this series, we know that it's not convergent, so how can be, okay?
Courant Events Right: So, let me say now, very quickly, that the more modern way to see realization, but it's not obvious that it's completely equivalent to the other one. So, I think people in the INH visits make computation, but yes. Sorry, since you were talking about perturbative normal visibility, what is the status for the full standard? Is it true in retros?
Courant Events Right: We'll step in.
Courant Events Right: I wouldn't count to them. I would say not.
Courant Events Right: to your team.
Courant Events Right: Just shouldn't.
Courant Events Right: Even pocket annuities.
Courant Events Right: I don't know what that is, but I did it.
Courant Events Right: I, I, I, I, I said, sorry.
Courant Events Right: So, that is the famous top of the proof, that I will follow the problem.
Courant Events Right: So he bought the Nobel Prize for that, so of course, it's a very important result. Let's see what we do. Just the same idea right there.
Courant Events Right: So, the history of humanity, and talk… brief history of humanity, I mean, from here amidst to all. Okay.
Courant Events Right: So then, then we arrive to the settings, the organization group. The magician group is another point of view. You iterate scale by scale, as you will see in some talk, and then you get the sequence appears, which are, after using this,
Courant Events Right: using the fact that the sum of Gaussian variables is still a Gaussian variable, so you decompose. You can write the… you make a second equality, you integrate, and then you get the series of effect potentials. Now, the main idea is scaling variance, so you…
Courant Events Right: You notice that the fields with a certain cutoff is distributed almost in a similar way, patterned with the scale factor in front, and then you realize that the monomials behave as the monomials
Courant Events Right: with the NFL up to a factor. This factor is given by this. And again, you already discovered the dimension found using perturbation theory.
Courant Events Right: Okay, so now, so there are the terminology of remission group. If this object decreases, it is called irrelevant. If this increase, it's called relevant. If it remains more or less the same, it is called marginal.
Courant Events Right: So, the integration is done separating the relevant terms from the others, so you… so the others are called random coupling constants, and then are… or are… if the quadratic terms are put in the… put in the…
Courant Events Right: in the free measures, then change the mass, the wave functionalization, so on. The others are sort of called the effective coupling, so they change the coupling, and then, as was said before, this coupling can decrease.
Courant Events Right: Or increase.
Courant Events Right: So now, in high-energy physics, one chooses the parameters at scale n to affixed values, and then to scale 0. In condensed matter, one does the opposite. One starts from 0 and then go in the direction of the DMP infrared.
Courant Events Right: So, let's consider quantum field theory in high energy physics. So, in a non-renabisable theory, for instance, then you… the coupling decreases.
Courant Events Right: Then lambda address… but then you want the lambda address that is order 1, so this means that you cannot increase too much. So this means that there is an intrinsic cutoff scale of the order of the inverse of the… of the company.
Courant Events Right: Okay, so in a normalizer theory, it means that you… there is a maximum… so it goes well in the direction of the infrared, so you… you cannot go too high in the ultraviolet. So, normalizer theory is very good in the infrared and very bad in ultraviolet. It's the opposite for the supernormalizer models.
Courant Events Right: So now, let's consider renormalizable models, in which you have… you have some… some dimension, is it zero? Then, again, you distinguish between three cases, marginal relevant, marginally relevant, and marginal marginal, okay? So these three cases are the three cases we have to consider, okay?
Courant Events Right: So QAD, quantum electrodynamics, which is the most beautiful theory, according to MINI, or electroweak marginally only theory, are marginally relevant, so there is a maximal cutoff, but this cutoff is very high, so physically it would be good to get bounds up to the scale, into the water cell.
Courant Events Right: But still, it's marginally,
Courant Events Right: It's marginally relevant. Instead, as was said before, in QCD is marginally irrelevant, and then you have this asymptotic freedom. So it's very bad in the infrared. In the infrared, you have this mass generation.
Courant Events Right: So, let me make a short recap.
Courant Events Right: So, in theory, it requires international cutoff to be removed, but there is a relation between normalizability and cutoff. In a globalizability theory, the cutoff could be exponentially high, while in order could be a power.
Courant Events Right: So, in fact, in Noria, Saboltier is,
Courant Events Right: Not bad at all. It is perfectly consistent. In fact, Fermi theory, the famous tentative ordinary derogi Beta of the 34, is not normalizable, but it works very well up to cut off of the time, and in fact, it's used to for the prediction of the child, for instance, so it's good. Even… sorry, it's very… I looked this paper, and then they assumed that
Courant Events Right: in which they predict the chart, and they assume that the theory is analytic. In fact, it is analytic, but I had no occasion to tell this to Marianne, but…
Courant Events Right: It is real analytic, but really analytic in a mathematical sense, okay? But then it works only up to scales 1 over E squared, so it was good for the experiment of the 50s, but now it's very bad in the… now it's a very, very low scale, so they were very well aware that
Courant Events Right: So, 123 is renormalizable, this is a big discovery of diazone, Tomonava, Schwin, and refinement, then one could reach asymptotically high cutoffs. However, dermativity, as I said, require world identities, and then you need… you need world identities, they are broken by an additional loop, and so on.
Courant Events Right: Even more difficult electroweak theory.
Courant Events Right: You don't have to put the mass of the electron, because otherwise you break Carl symmetry, but then the mass will be generated by the X, even by… so there is this famous normal simplicity proof.
Courant Events Right: In which he assumed the validity of work in the evidence, okay? So the proof, the normalizability… sorry, if you read the Weinberg paper, not two-page theory the paper is written, I guess that this theory is a normalizing one.
Courant Events Right: Then, to prove to all orders that the technology to this attack is zero, so improve the ability, assuming that the word indexes are true. But then, these voltages are true only if the
Courant Events Right: And nobody starts.
Courant Events Right: So, if the… so the model of vibrate by itself is not normalizable. You have to put quark electron so that it fits up.
Courant Events Right: Not dead.
Courant Events Right: But as it was saying, you need for these anomalies to cancel what is called Adler Bartin theory, which says that anomalies are given only by the lower order.
Courant Events Right: And even worse, the big problem is that for the electromix sector, there are no regularization ensuring
Courant Events Right: As in the case, that there is independent.
Courant Events Right: This realization could exist only for the values of the charges, so it's not… it's not obvious at all.
Courant Events Right: But if there is no organization, how do people do calculations?
Courant Events Right: Well, it's just using dimensional realization.
Courant Events Right: I don't think they use, really, the… as far as I ask to a lot of people if they make computation with the Monte Carlo itself, I don't think they do, right, for that set of people to set. I think they use Monte Carlo itself just for a strong set, not for the wind. They just use crafts.
Courant Events Right: Yes, but that's what we usually mean in physics, perturb a different invisibility, you know, you use Timoreg.
Courant Events Right: Does it work? If it works, you're happy. You're saying, at that level, are you happy or not? The level of dimensionalization, yes. Okay.
Courant Events Right: The point is that I'm not taking it as a matter of finisher.
Courant Events Right: And then there is this asymptotic freedom, browsing, wheel check, and so on dignified mass generation, which is the topic of this conference.
Courant Events Right: Okay, also, don't remember that there are a lot of prediction in condensed matter, and so on. Okay, so anyway, so finally, this ends the first part, so it should be…
Courant Events Right: on time, okay, 10 minutes, I spent all 252. So, now, let's just consider the classical result in mathematical physics. So I…
Courant Events Right: So, Gallavotti and Galavote and Nicolot tried to write the… understood the Wilson group in terms of, in a mathematical physics way, finding bound… bounds for the…
Courant Events Right: For the good feature, so you get bounce.
Courant Events Right: And… but then these are not still per tool, but at least we have the same factorials, and then you… you have this tree representation in terms of trees that you get iterating by… step by step.
Courant Events Right: What I… what was very nice is that if you use these trees, you… you get the same… so the re-solarization group produce the same
Courant Events Right: Forests, which were used by Heckton, Borolubov, and so on, to prove normal as a…
Courant Events Right: So, I mean, you can get the structure of clusters, or using the graphs, and then separating the overlapping emergencies, as was done by itself, but directly, without thinking at all, using the, using the
Courant Events Right: prevention group of weeks. It is the same. This is an example of class principle. Then, this 3A cluster will use it in a famous book by Feldman, Hart, Rosen, Wright of the 88.
Courant Events Right: For QV, for that… in this paper, they proved n-factorial bounds. The point is that they use momentum regularization, so the coordinators are all related, and then, in fact, one needs to introduce non-vaging variant count attempts, A4 and A2, and so on.
Courant Events Right: So the point is that at the end, when you remove or cut off order by order, you can impose the word identities are verified, but we know that we cannot really remove… we cannot really remove the cutoff with… at the non-performatic level, okay? But…
Courant Events Right: Another approach was based on randomization group, is given by Pochinski, and there is a very nice review of Porchinski, 92, about the randomization group, which I suggest is very, very clear.
Courant Events Right: It's based on flow equation, and this method was used by Keller and Copper in, to prove order by order method of query 4 again, but again, one needs this A4 and A2. This is a problem of an emission group with Wilson cut off to get that.
Courant Events Right: to get… you get this broken term, this damned broken gauge symmetry. And then this was extended to young mills, but without matter. So the proof of pair is for young mills, but without matter. For the full electro-week theory, I think no one has done.
Courant Events Right: So now let's go to the no productivity results.
Courant Events Right: So a crucial idea for fermions is that there are cancellations, an idea which was in Cayaniello, Novochimiento, which, as it was Italian, I was supposed to know, but I learned this fact from the paper of Gavieskopiane. Then I asked the people, but do you know this Canelo? Then I discovered it was very famous in Italy, but I didn't… I just read in your paper that, you know, I said, who is Cainello?
Courant Events Right: Then I discovered it was the Professor Guerra, for instance, but they didn't know anything about it.
Courant Events Right: So, anyway, this, this was, this was the first, apparently, according to them, that used this idea of the Berlin and ground bounds for,
Courant Events Right: So the idea is that Fermion has this cancellation, so you can prove convergence using this determinants bounds.
Courant Events Right: And then, this can be combined with Bridges Battle Featherbush formula, which is the trophies in Lesus.
Courant Events Right: I guess this… if I understand well, there was a mistake in the proof, which was corrected by
Courant Events Right: by the salam, if I understand well, yes, it's correct or not. I read that in some paper. Yeah, you gave an alternative, so it was not the proof. Anyway, so you… you did also preparation. No, you did… you changed the proof in a place. Not in the form, but there was a sort of hole in the proof, as far as I understand. Anyway…
Courant Events Right: No, no, no, that's his later. Okay, anyway.
Courant Events Right: So this, this, this formula is very nice, because, you see, the problem of graphs is that they have too many. Okay, so that's this, for instance, this common. I mean, when you use common graphs, you have an n factorial square graphs.
Courant Events Right: So with the 1 over n factorials, you get n factorials, so even if you prove that they are bounded, then they are too many, then you have… you cannot prove validity. But this formula is very nice, because you extract
Courant Events Right: different languages, so you use this G, using,
Courant Events Right: So, it tells you to make the iteration over the coordinates, and then you can bound the determinants with the maximum, and then in this way, you eliminate the n factorial. So, in a sense, you are regrouping enormous class of graphs in a very simple way. So, you eliminate the n factorial
Courant Events Right: So I would say that the first proof of remissability of a theory would be easier if you
Courant Events Right: lose this forward, but instead history went differently, and the first proof of a renormalizable theory, I will skip about the normalizable theories for lack of time, so I'll start from day 5, was done by Gavisky and Kupian, in which they construct the gross debit model for N is greater than Q.
Courant Events Right: And then, simultaneously, Feldman, Manier, reversal, and Seneho. But they didn't use… they didn't use this formula, but they used the face-set, the composition, and Maya, and cluster expression.
Courant Events Right: Later, Lesniewski considered instead uses that formula for the Yukava model, which is supernormalizable, and then… then he has other problems, because you have to extract lines, but then it is the first place in which I think it was used at the ground bounce.
Courant Events Right: Then
Courant Events Right: The genuine model, so it starts, as you see, this paper are, 92, no, 99, not so long.
Courant Events Right: Is that statement? 1920 sentence range? Okay, anyway,
Courant Events Right: But anyways, yes. What about this failed one also while their flight in order to be more? This is fair, that's true.
Courant Events Right: Huh.
Courant Events Right: It says, history of… put an order.
Courant Events Right: Okay? So then, this genuaries, we sort of grass delivery with n equal to 1. This theory has,
Courant Events Right: It is 2002. This theory has,
Courant Events Right: Not really a fixed point. It is a marginal marginal case. Instead, the case of the… so, the Yokawa case is supernomalizable. The Grossed case with n greater to 1 is marginally irrelevant.
Courant Events Right: The genuine case, which is essentially grossing away with n equal to 1, is marginally marginal. So you have a line of fixed supports. So it is more difficult because, of course, if you have a similar theory, it then is enough to go to the second row.
Courant Events Right: So you go to the second order, then you bound the rest, so you know it's never. But if it is marginally marginal, then you have to hold all the orders to prove them. Otherwise, you never finish, you completely know. The fourth order, then one can say, well, what about the fifth? So you can…
Courant Events Right: put you in the forehead, okay? So this was done by… at the beginning, by doing direct argument, and then we gave, I will tell you how we did this, a direct proof of this venture beta function,
Courant Events Right: Much later.
Courant Events Right: Okay.
Courant Events Right: No, no, it is two-dimensional.
Courant Events Right: Yeah, yeah, see, this is two-dimension. Then there is this mass… it is the pros and the way in two dimensions with n equal to 1, essentially. Infrared.
Courant Events Right: Okay. This is the JLU model. I mean, it's a model for non-relativistic fair position.
Courant Events Right: So in this 99 was the period in which started this interest for condensed matter. Then there is paper, Mass Generation in the Infrared Gross Nevatium, which is very similar to the problem of young meals. So, I think this is very interesting, but there is a gap
Courant Events Right: between… it is related with the thesis, I think PhD thesis of Kupian, which is for the nonlinear sigma model. I think it was very much connected, but
Courant Events Right: In this paper, I'll compare, there is a gap between the infrared and the ultraviolet, so you cannot flow… you cannot follow all the flow, okay? This problem of the gaps of the two regions, it appears everywhere.
Courant Events Right: what was done in the BGL, okay?
Courant Events Right: In a nonlinear sigma model.
Courant Events Right: what was done for the new leader by asking.
Courant Events Right: It must prove the gap, the generation of the gap.
Courant Events Right: We're gonna fix about?
Courant Events Right: with a free circuit off. I mean, if you want to do really begin to do…
Courant Events Right: Like this one, we have to remove water provided at the infrared water point.
Courant Events Right: Yes. So Copar had two papers, one on the Sigma, one of which came… Yes, yes, yes. The one on the Grossover was with Eva Manu's.
Courant Events Right: You are right. Yeah. Yes, yes, sorry, I thought about it. This is the one of these…
Courant Events Right: Yes, French. I don't know how… yeah, of course.
Courant Events Right: This one was the same, I guess, the sigma model of a land.
Courant Events Right: Then, Kavisky Kupjani did a later expansion for Ferriers. Then, more recently, Julian Masvidovich Professor applied similar ideas to,
Courant Events Right: to consider simplicity in the… what is interesting? We get an intensity.
Courant Events Right: And then I have not no space in the slides, but then here starts these enormous developments of incandescent matter, models, and so on, but I will not speak about them. But then, since the 90s, there was a lot of interest in this community of people to go in the direction of condensed matter.
Courant Events Right: Maybe, but maybe I could say something as I… Bye.
Courant Events Right: The problem in condensed matter in two dimensions is that the singularity is not a point, like in quantum field theory, but it's a surface. But that's changed completely in the problem. So it's a surface, which is called fairly surface, and then you have a lot of strange geometical problems. Okay, now bosons.
Courant Events Right: So I put some results, so, part of what it may be very old result of native, being 5 or 3.
Courant Events Right: Infrared 544, Even part of it?
Courant Events Right: again, 85, and then Magnens and the order.
Courant Events Right: So, Nate 6, then it is the book of Roland Briggas' late, probably the adult for the book. Then there is this episode of Spanish by Brigis, then the Serum also built this complete floor, and so on.
Courant Events Right: So, pure young Japanese, which is really the core, yeah, yeah. Yeah, in fact, in fact, you are right, in fact, you are definitely right, you know,
Courant Events Right: Okay, so I decided to put the cutoff in both directions. Now you say that, of course, yeah, since we are talking about cost tracking.
Courant Events Right: I decided to speak about this. Oh, yes, yes, that's all. I have the answer. I remember the answer.
Courant Events Right: I will speak about only the reharmonizable model.
Courant Events Right: accept of the one we talked about. I want to say, otherwise, perfect footwork, Joanne, you can.
Courant Events Right: But I… the point is that I will speak about only about,
Courant Events Right: I'm sorry, now it's better. Remination group and the normalizable models.
Courant Events Right: There were a lot of non-super nonizible models, whereas I'm also expensive in its own. I did not find it.
Courant Events Right: But also because I want to almost finishing time.
Courant Events Right: Yeah, but because you want to arrive to these younger meats.
Courant Events Right: So, Agamemnisi, this is his famous paper of Balaban, 84, 89.
Courant Events Right: with infinite forward. Here, this model is with the infrared bundle cutoff, and the lattice revolverization, which preserves the gauging variance, block spin, and so on. This method we're using quantum recently for quantum driving information. Then also, Nanya Ricassov did this paper.
Courant Events Right: This is very different, now I have not much time to discuss the difference, but in this case of manual universal, they use momentum realization, so you get, again, these countens, which are no gauge imbalanced, and then they use a lot the positivity of this lambda, where instead, the Taliban use some kind of complex correlations, but
Courant Events Right: But doesn't have this thing. The Magnier and the Marseuse, they're not saying it's a proof, right? Well, okay.
Courant Events Right: Yes, they, they say that this is… okay. Well, okay. I'm just trying to… Yes, yes. But also Balleman says the movie's very long, so the point is.
Courant Events Right: What is a proof? So also, the human being is important. If you have a… so, suppose you have a proof.
Courant Events Right: this unit in an island, and no one can understand the reef. Is it the proof or not? It's like the problem of it being falling in the forest, no one has it. Is it proof or not? Who knows? I mean, it's also important that the human beings read, understand, or reproduce, otherwise,
Courant Events Right: What is it?
Courant Events Right: So how many folks here fit into the category? Also, then… okay, so the… here I answer with another job thing, which I always say is that
Courant Events Right: For everyone, the level of rigor is insert.
Courant Events Right: There is no… even everyone, even if someone, even if Lumber, even everyone thinks that other people are less rigorous, and they are more… too much rigorous. Every… for everyone, the level of rival is off.
Courant Events Right: So that's, my answer. There is a state for everyone. There is no, no one can say, that's… it's enough. If you ask to, you know, in logic, say, oh, no, okay, you don't understand. Now you're making me…
Courant Events Right: Lot of time to think. It's important. So then people look at… so this is just the ultraviolet, so the easy part. So what about the infrared part, which is really the main aim of this collaboration? So again.
Courant Events Right: There was that hope.
Courant Events Right: in the 90s to say, this is too difficult to study a simpler problem. Mass generation for VCSDO in two dimension, or Bose-Einstein condensation.
Courant Events Right: using the Balaban ideas, this was a big rule. So people, again, never completed, because there are still this problem of connecting the two regimes, the larger planet and the ability to fight. So in between, it's difficult to connect.
Courant Events Right: There was a big enterprise, a lot of nice ideas, still incomplete history.
Courant Events Right: Then, of course, I will not speak about the recent… I will not speak about the older result, the link in JFC, wherever assigned, so then I will not speak about the more recent result of this collaboration, because tomorrow we'll see them, so…
Courant Events Right: Okay.
Courant Events Right: Yes, so… Let's see… so here… well, now, I remember the sometimes of my result.
Courant Events Right: So here, the main problem is to remove the collaval cutoff and improve the mass chain motion. But of course.
Courant Events Right: What can say, what about the literal receptor? I'm interested in that. So, in the electron receptor, we know that we cannot make this… cannot remove the cutoff. It is proven for 5-4, by trade, canines and domino cocaine. They proved that it's important, it's triggering the theory. Despite this fact, the theory, of course, has a meaning, you want to arrive
Courant Events Right: to exponentially high cut off. And then it's very important, also, that mathematicians remember
Courant Events Right: the numbers, for instance, the mass of the electron must be much smaller than the mass of the Z.
Courant Events Right: So if you integrate the mass of the… if you integrate the fermions.
Courant Events Right: then you get… I mean, if the mass of the frame must be greater, then the charge is completely useless, I would say. The mass of the electron is almost zero. So it's very important that also mathematicians remember the scales of the number. So this is 10 minus 6, very small. So the mass of the electron cannot act via laser at all.
Courant Events Right: I say that because a lot of papers iterate the fabric.
Courant Events Right: And… but that's… if you enter the family, you need a strong hand.
Courant Events Right: So now, the main problem is to control the flow. One is to prove relation between that side of it said.
Courant Events Right: Okay, so now, finally, let's arrive some… some… some of my results. So let's consider this massive security tool, which is, I want to start by with the basic problem. It's like we did before, but in two dimensions.
Courant Events Right: So, without masks, So, you can make computation of a perturbation theory.
Courant Events Right: This was done, for example, GOJ, and in 71, they proved… they used this, they proved that J5 is equal to 1 over pi. There are normally 1 over 5.
Courant Events Right: And then there is sine dependence, so that's… so… and then also there is a decrease of power time. Remember that in fourth dimension, you have the theory which is non-normalizable, which must be renormalizable. In two-dimensional, the theory should go from renormalizable to supernormal.
Courant Events Right: Okay, so immediately people, when CDs, say, okay, but then there are solvable models.
Courant Events Right: Some of our models are Sommerfield, Hagen, and so forth, but they get differences.
Courant Events Right: So if you consider solvable, you get any value for the anomaly, and you get an infinite wave functionalization, which instead should be not true. Why you get that? It's not… it's so different from the presumption.
Courant Events Right: Is this a mystery? No, because these Excel solutions are very formal, but when you consider formal integrals, you have to regularize in some way. They regularize these integrals using momentum regularization, so typical brain.
Courant Events Right: So now, let's instead consider the full, thatization.
Courant Events Right: Again, the model is the same as before.
Courant Events Right: Then, for instance, you can prove this theory, that there is no infinite wavefunctionalization.
Courant Events Right: the limit exists, and then there are nobody's one of their power. So exactly what you're getting perturbation.
Courant Events Right: So the results are… I don't take the limit A equal to 0 in this case, but I don't want, because I want to consider uniform in A, because in fourth dimension, you don't have, like, a good thing.
Courant Events Right: So there is no way-fashioned radiation. Okay, this was done in vector. So let me set two words how this proof.
Courant Events Right: So now there are two regimes, so I put the mass of the boson, so we have in force trioviolet and infrared regime. And in the ultraviolet, it is super normalizable, so I can…
Courant Events Right: Let's do this.
Courant Events Right: In the infrared, you have all this reliable, then you have all these kind of random constants, so again, as important metrod dynamics, you need a relation between the random couple constants.
Courant Events Right: So, at the end, the magic is that you have to prove the quantity is proportional to the wavefunction, to the square, okay? And then the zeta Y, so it's the J, zeta 1, the… the current… you have a relation between the rounding of the coefficients. They are identical to the quantum quantum electrodynamics, which is the one in particular, 0.
Courant Events Right: So how we prove that? Because if you consider a major group, it breaks the vulnerabilities, okay?
Courant Events Right: And then, also, you have to prove the anomaly. So, how is… it can be that the anomaly is a single number, so that's… maybe I should have stressed it more this fair. It's very strange, this result, maybe I should say more.
Courant Events Right: Typically, the physical objects are a series of terms, but now it is only single term for them.
Courant Events Right: Okay? But why there are all the other entities at PS?
Courant Events Right: Okay?
Courant Events Right: So now, both problems are started, take 8 minutes. Both problems are faced just introducing the reference model.
Courant Events Right: In which, in the infrared, is similar. So this reference model is a non-local theory model.
Courant Events Right: Which is the idea. If you put on the lattice, you have.
Courant Events Right: Gauging value the sort of volatility you shall exert, but you lose the current seal.
Courant Events Right: Okay, so you cannot have a… some delta. But then if you consider this model, you break
Courant Events Right: because you put the momentum cut off. So you combine… then the idea is that we can tune the parameters of this reference model so that the fixer point is the same.
Courant Events Right: Because it's a line of fixed pole. So you can tune the parameters so that they go to the same pole. So I use a bit of the property of 1 and a bit of the property of the 2.
Courant Events Right: Okay? So now we… so the infrared of this model is very similar to the other one.
Courant Events Right: So now… but then I put an infrared at all. So I have a sequence of models, and then I compute the physical observables at zero, then I get information for all the cost, because I can change the… I can change the cap.
Courant Events Right: Then, now, if you consider the world identities, you will get this data, which is the cutoff effort, which breaks the word identity.
Courant Events Right: So, naively, you could say, well, when they send the observable F to infinite, then it's zero, but it's not. The point is that this data does not fetch, because there are the anomalies. But if you extract the anomalies, then it will be small.
Courant Events Right: So you can prove that, you remove that off, it's given by number. But if you fix R, H, it's a small projection, so you have to make tricks like that, that you consider the composition, and then at the end, you have to show that there's all these graphs, except this one, this golden zero column to infinitive.
Courant Events Right: Just because we have Just because this chi minus 1
Courant Events Right: It's the fact that you have one line which is handed to those deeper parallel.
Courant Events Right: For all these other terms, it appears that A.
Courant Events Right: So, at the end, in this way, combining, you will get relation between randomized.
Courant Events Right: Now we have to prove the… so that's it for control for the flow.
Courant Events Right: Now, how you… you prove that? Normally, then, again, you can… you see, these are relations that are in a bit quicker than normal.
Courant Events Right: Santa.
Courant Events Right: Then you have relation… you see, this gamma mu is there.
Courant Events Right: It's the G5, so you have anomaly spot for the current and the clear current and the car, because if you put momentum, the current is not conserved.
Courant Events Right: But then, so you have all these numbers are set, but then you have
Courant Events Right: Now, or the tweak, but we did with this event.
Courant Events Right: Well, so…
Courant Events Right: All this, as I said, I have to fix the bare cutting to require the same fixed point.
Courant Events Right: But they are not independent. Again, there are water that is connecting all this bare tapping of the reference model, so there is a lot of algebra. At the end, all the dependence on London disappears, and then, by a completely different method, you get the anomaly, which is not normalized. It still depends on that, so there are some…
Courant Events Right: So, let me say quickly at the top that that's also where it was very important.
Courant Events Right: there is a very big universality class of this model. Among them, let me stress these diamond models, in which this is the same class of this vertex models, the model interacts solvable one model.
Courant Events Right: And then you can prove some relation between the amplitude and the… you know, using some ideas from Kenyon or Concord, discrete analog, convenient with that, with this approach, we can prove some… some relation.
Courant Events Right: Okay, let me finish.
Courant Events Right: Okay, so let me say two words in 3 minutes.
Courant Events Right: Sorry, you are not protesting, same thing as though much time.
Courant Events Right: You have 4 minutes. Only 4 minutes. You are not very generous. That was my route, was hoping for more. So what do we have in more dimension? More dimension with 4 dimensions.
Courant Events Right: We have the same model as before, so fermion vector model. Of course, in this case, we cannot do better than that. You see, you have
Courant Events Right: the cutoff, A, would be 1 over T squared. It's not arbitrary, because it's not normal, so on. But still, it is very interesting that we get the other variables exactly that, without any dermatization.
Courant Events Right: the idea is quite simple. Polygonization group tells you that,
Courant Events Right: 3-point function is equal to the free one plus the meditation plus attempts, which is… which is regular, and using regularity. And what identity?
Courant Events Right: And the regularity, you can get.
Courant Events Right: You'll get accepted.
Courant Events Right: We have… it is nice that we use the same… the same ideas in completely different fields, for instance, for the… for graphene, because graphene can be considered a regularization of quantum field theory.
Courant Events Right: And then, you see the same argument produces you this… this universe. This is the same as anomaly or the transport group.
Courant Events Right: Okay, now let's ask to this. Okay, here, this is interesting. How to get higher cutoff?
Courant Events Right: To get a magnet off, we have to do as before, so we have to consider the skates greater than the name, they compose the bosons.
Courant Events Right: use the word identities, but then you have to… I don't think there are conceptual problems, but you have worked through, you have to compose fair symposiums, put the word identities, put them together, and that should work.
Courant Events Right: So that's, I think, is a very strange that no one has done. We have done only quarterbados. So, final result.
Courant Events Right: So, I want to return where I started, to this magic Gman Institute, which is south of them.
Courant Events Right: I think it's one of the most nice results in physics.
Courant Events Right: So, can we pull a bound for the rest of the G minus 2? So, we can do that only for the Z sector. That is the easiest one, because it's U1, and there's a mass, so that is the easiest scale. So, Jacob and Beinberg, and then other people compute this. This is the second order for the compression.
Courant Events Right: And then what we want to prove… and then, now I don't have time to explain that, it's not even not obvious to give a definition of the geomagnetic factor in a positive sense. It's not complicated, because if you look to the books, the geomagnetic factor is the same terms of four factors. We have to
Courant Events Right: Produce the definition of books in,
Courant Events Right: In terms of correlation function, but you can do that.
Courant Events Right: Another very important point is the following. The G minus 2 is related to an irrelevant one. So while the anomaly is marginal, the G minus 2 is irrelevant.
Courant Events Right: In this negative dimension is the derivative of the three-point function, which is zero dimension.
Courant Events Right: Please explain why you could ask that. So you know that the alpha, the anomaly was universal.
Courant Events Right: In the transport between the universal, they're all… one step, the gym, is not… is a series.
Courant Events Right: Okay, so why is a Jesus series that the novel is not? Because
Courant Events Right: positive, because you will see. One is that it's universal, and one is irrelevant, and the other one. So I think it's the last slide at the time. But that would be…
Courant Events Right: The result, which I like, because this says to you that you consider a model of the U1 sector, Zeta, with a cutoff.
Courant Events Right: Latin.
Courant Events Right: You want to say, this model is well-defined, the limited infilt volume, then the contribution to the geomagnetic factor is equal to the result of Weinberger and Jacket, plus an R, and these are the sum of three terms, the second one is the difference between PATOF and
Courant Events Right: You know, and the third term is the sum of all the series. So the series can be bounded.
Courant Events Right: What is also important to note is that M is very small. It's a… it's N squared over M large N squared, so you have to pull the same factor in the second order in order to see this.
Courant Events Right: Okay, and it is what you achieve. I mean, there are cancellations that you can expand, and then also new usage of that. Of course, the dream would be to put here a log, seems easy to put just a log, but the log would be physically meaningful, but it's difficult because we have to live on both
Courant Events Right: So then also you have a computation, your observation of anomalies, you have problems. I'll show you the imprints for nuclear.
Courant Events Right: Okay, so this is sort of two worlds of hope. So, from Newton to Cauchy, there were 100… yeah, so,
Courant Events Right: That's, you know, scared.
Courant Events Right: So that's…
Courant Events Right: Well, yeah, so let's start in session, I think.
Courant Events Right: So, I have a question. So, in some cases, like, I think for the solar system's humidity, people were actually able to put…
Courant Events Right: The actual numbers corresponding to the solar system, and see some things worked out as they should.
Courant Events Right: In urine that is gluten.
Courant Events Right: Because we are using a small company. Do you think that one candidate for one or one candidate for 7 is going to suffice today?
Courant Events Right: This is the sentence I always write in the paper as you understand to physics group. So, the result is say that if in the first stage, I mean, as you know very well, in the first stage, direct by
Courant Events Right: That's the problem for Lorov, about the solar system, then the assumption was that the mass of the head was a fly, right? And then it improved, improved, and the riser to asteroids.
Courant Events Right: So I'm almost… I'm completely sure that this is doable here.
Courant Events Right: Again, it works, because I only need to go from the slide.
Courant Events Right: First of all, what you need are not the model which improved. The problem is there was a lot of people working on numerics, but I don't see any reason why this could be not done. But let me add that now, I mean.
Courant Events Right: Yes, I think so, because these methods are unuser loans from the NF.
Courant Events Right: Let me say even more. For the KM problem, that is the serious study by the initial group.
Courant Events Right: Which is exactly this method. And using wasted series.
Courant Events Right: I don't know if everyone is familiar with this topic.
Courant Events Right: When it's a series of decisions for the stability of the solar system.
Courant Events Right: Which was validate its own bill.
Courant Events Right: To send to the table, and then…
Courant Events Right: then it was proven to be convergent in certain cases, and from that, some stability of solar system can be derived. But the first proof of Polarov assumes here, the constants, then…
Courant Events Right: the mass of the hand is flat, so that's worse in the human is the mass of the flat. But then now, it works for asteroids. So asteroids and light is a bigger… doesn't work for the solar system, but it's going to be unstable, but works for big asteroids, which is much better tomorrow.
Courant Events Right: So I'm not sure we'd arrive to 1 or move into the 137, but maybe I don't know.
Courant Events Right: I never really look through the numbers, because they're not good.
Courant Events Right: But maybe not so bad. Questions?
Courant Events Right: Or, just appear again.