Courant Events Right: So we're happy to have, Zhao Kozlowski from my own, who will talk about the Asmatics bootstrap itself.
Courant Events Right: Thank you very much.
Courant Events Right: Let me also thank the organizers for having set up this event, the invitation and possibility to
Courant Events Right: speak about this, here. So, what I will aim to do over the next hour is to review one possible approach in low dimensions to constructing certain specific class of quantum field theories called integrable, through,
Courant Events Right: what one could call the affiliation, program.
Courant Events Right: So, in the talk, what I would try to, show you is the…
Courant Events Right: basis of the construction, what are typically the kind of objects which are involved in the construction, what are maybe a few techniques one can use in the constructions, and what are the existing, results being so far, and what are the open,
Courant Events Right: Questions. Is it working?
Courant Events Right: Nope.
Courant Events Right: So let's make food, we're gonna make those.
Courant Events Right: Okay. Okay.
Courant Events Right: So, so let me start by, discussing the overall setting, and since I will focus on this particular model later on, so what one wants to do, one wants to start, start from some, classical, partial differential equation, nonlinear one.
Courant Events Right: So here I focus on the St. Gordon, equation, which is known to be integrable in 1 plus 1 dimensions. For me, the,
Courant Events Right: spacetime variables will live in R11, when signature 1 minus 1, so here we call the, Minkowski, norm of the, of the vector. And then, what one wants to do is to, find some way to quantize
Courant Events Right: the,
Courant Events Right: this wave equation, so it would lift phi into some object, B phi, which would take functions in, say, the Schwartz space on one, and produce, operators
Courant Events Right: Unbounded and some here with space, and more generally, one wants to do it for other classical expressions, like exponential of the field, or maybe it is exponential preceded by a certain amount of
Courant Events Right: finite order derivatives, spacetime derivatives of the field, and list them to some operators on this space. Again, a distribution, that
Courant Events Right: So, in fact, those are not per se, the quantities which one wants to study in the theory, those which are of
Courant Events Right: main interest in the physics of these, quantum field theories are the correlation functions. So correlation functions would be, you take some of the objects you manage to quantize in this way, you, take of all the, of them, evaluate it at some,
Courant Events Right: functions F1, FK, and you average them in the vacuum, and actually B, so this is, this is a distribution, so you can represent it in terms of its
Courant Events Right: generalized function, symbol integrated versus, those products of those, SKs, and typically, the objects which would be measured in experiments are those, correlation, functions, those generalized correlation functions.
Courant Events Right: So, one path to, now, there's, there's one way to, to, to construct this, this huge theory, those, white one. So, the idea is to, turn the thing a bit upside down, and to say, well, let's…
Courant Events Right: So both, we have some, generalized functions, W of one variable to two spacetime variables, and two infinity, which satisfy a certain number of properties, which I will not detail here, which are compatible with the presence of,
Courant Events Right: Okay, symmetry, and,
Courant Events Right: and other properties, and also which are compatible with the presence of causality in the theory, particularly local communicativities, states that such functions, when you exchange two spacetime variables which are neighboring, and which have a negative.
Courant Events Right: Minkowski.
Courant Events Right: If all this list of properties is satisfied by your primary functions.
Courant Events Right: a generalized function instead, you're automatically guaranteeing the existence of a certain limited space, and this, capital flying, object.
Courant Events Right: Such that the correlation functions would compute for this,
Courant Events Right: functions would exactly coincide with the W case. So this leads to a way, a tentative way to construct the quantum field theory, by just constructing those generalized functions, and a bit allows one to remove a bit this whole operator, setting in the construction, and so,
Courant Events Right: Focus on the most important objects, which are the, gradation functions.
Courant Events Right: Okay, so, first steps to compute such,
Courant Events Right: Y-band functions, were done within the form of path integration, and you can see it has some form of algorithm which
Courant Events Right: allows you to compute perturbative expansions of those Weitmann functions in theories, which can be seen as perturbations of free theories. It was developed by many people. And then, well, there were many mathematical imprecisions in this.
Courant Events Right: Constructions of this led to the development of constructive quantum field theory, first in… was formulated in the Euclidean setting, because it was easier to, set up certain, justification of certain…
Courant Events Right: bounds and, existence of, per se, limits. And basically, within that setting, what one was able to do is to produce a certain
Courant Events Right: Taking finite volume regularizations and taking the finite volume limits to infinity, maybe other regularizations to infinity, which were producing, correlation functions,
Courant Events Right: As such, which were then good candidates to produce,
Courant Events Right: the white one functions by first checking that they satisfy certain amount of properties, which are called the Osterwater-Schrader axiom, and this was basically guaranteeing that one was constructing some quantum field theory in this, way.
Courant Events Right: There are alternative characterizations of, measures new, which would lead to producing such correlations, first to stochastic, quantization, and, also with many, many people involved in the development, and, also, you know.
Courant Events Right: this one plus one-dimensional settings, through the use of Gaussian multiplicative, chaos, which produces Some may be rather…
Courant Events Right: explicitly controlled expression for the measure. In particular, within this approach, it was possible to produce, correlate certain informal field theories, but also of the Sinch-Jordan model, which I've shown before, in finite form.
Courant Events Right: And as I said, well, one checks that such objects construct in this way, so such distributions synthesize a set of axions, which are also about the shadow axions, and then this produces the quantum field theory,
Courant Events Right: as before. But, so those are existence results. But what I would like to go beyond that, and to have a much better understanding of these correlation functions, simply because the ultimate goal is to produce objects which are fully controlled.
Courant Events Right: So, and which one can describe good enough to compare with measurable data? So this whole,
Courant Events Right: Structure works quite well in the perturbative regime, and comparison with experiments works quite well, but going beyond the perturbative regime is,
Courant Events Right: more complicated, at least in the terms of precise controls and deferments. In 1 plus 1 dimensions, what one expects is that there is a
Courant Events Right: universal behavior in the short distance regime of such correlation functions. It will possibly go up with a certain scaling dimension, delta, which depends on the operators you put in the correlation function, and then there will be some term which is independent of the scaling parameter delta, which depends on all the indices of these operators.
Courant Events Right: Correlation function one computes, and the expectation is that this should be a correlation function, or actually a theory which is invariant and a conformal to information, so a conformal to the theory.
Courant Events Right: then immediately the question which I would like to answer is what kind of conformal field theory there is. If one is able to identify the conformal field theory which captures the UV regime of a given quantum field theory, which
Courant Events Right: CFD correlators would graph a given correlator on the
Courant Events Right: on the QST side, what would be the formula for the scaling dimensions? And one way to be able to address this for very specific theories is to take the integral paths based on the S-matrix program, which I will explain
Courant Events Right: Now? Sorry, could you, explain all the alpha hides, and why the features are you depending on alpha came?
Courant Events Right: Yes, so the alpha K, in principle, you can consider correlation functions as many operators. So here, for instance, you could take the correlation functions of the field operators, but you could consider correlation functions of quantizations of exponents of the fields, or some descendants thereof where you
Courant Events Right: multiply the exponential by some products, and what I consider by alpha case, it's just indices which would… the choice of a given alpha K would choose what type of operators from whom you compute the correlation function, and then the scaling dimension as those delta of alpha case, they, in principle, depend on the operators you date here, and this is something you would like also to
Courant Events Right: to, to determine. Yeah.
Courant Events Right: Okay, so the story of the bootstrap approach takes the truth in the early days of constructing quantum field theories, where people observed already that there were various divergences
Courant Events Right: appearing in the construction, and the idea was that maybe let's try to describe the theory by only objects which are measurable eventually in experiments, and in particular, the S matrix. Why? Because those objects should be
Courant Events Right: By construction, 3 or 4 possible divergencies.
Courant Events Right: And this was what was proposed by Wheeler and Heisenberg in the late 30s, early 40s.
Courant Events Right: there were various attempts to set up a theory, one-to-field theory, just based on the S matrix, some progress was made, some properties of this.
Courant Events Right: In estimates were framed through formal observative extensions, but then the problem was that people did not manage, really, to find out enough… to find out any viable
Courant Events Right: as matrices, which would then help to go, push the thing further.
Courant Events Right: And somehow, then the whole activity of the S-matrix approach was slowly dying out, until 70… in 76, Granik and Werkides started to… computers order by order to actually order for…
Courant Events Right: the S-matrix of the Cinch-Gordon model, which I've shown you before, so the quantization of the PD, and then they observed that actually they can reabsorb all the first force order of perturbation theory into one single expression, so just kind of,
Courant Events Right: function, and that this function was actually satisfying other probabilities one would like to have for an S matrix, so unitarity and crossing symmetry.
Courant Events Right: And then, they, they argued that actually this, this, this matrix should be really an exact, formula for the Sinch-Gordon S matrix, and that, scattering in the Sinch-Gordon model should be given, when one has n-body scattering, should be given by a factorization of
Courant Events Right: scatterings of just two particle processes, and the order in which one does this factorization should not matter. This idea was taken very quickly by Samo Logzikov and Karovsky-Tun, who managed to produce another explicit expression for an S matrix, this time for the sign-modern model. It was already a much more involved structure, basically because the particle content of that model
Courant Events Right: was… Richard,
Courant Events Right: In particular, this was argued that for such a model, the model, the S-matrix also has to satisfy what is called the Yang-Baxter equation, which translates the fact that the order in which the scattering, two-body scattering takes place doesn't matter, and then many people proposed to constructed various other S matrices by solving the Jen-Baxter equation.
Courant Events Right: So then, this led to the possibility to implement the S-Matrix program. And so the…
Courant Events Right: integral bootstrap approach is consistent starting from the data S matrix of a given total, so this is… you can think of it as some object which solves unitarity, crossing, and
Courant Events Right: the Young-Baxter equation, and from that, construct a input space and a set of,
Courant Events Right: quantum fields on that Hilbert space, which would then realize your theory.
Courant Events Right: And this was, this program was set forth by Herovsky Weiss and then, Penos Mirnov, who managed to, eventually axiomatize for this, this approach into a certain amount of
Courant Events Right: equations, which I will write later on, which are supposed to fully characterize the operator content of the field. Now, what does this approach actually allows one to do?
Courant Events Right: Well, it allows one, in principle, to compute explicitly, and I'll show examples later on, endpoint functions for the quantum field theory associated to that S matrix in Yinkovsky 1 plus 1 dimensional spacetime.
Courant Events Right: And also, because there is some integrability behind, it leads to the possibility to produce
Courant Events Right: a certain amount of explicit conjectures, which were not proven so far, to some of the UV behavior of such theories. For instance, the results of the 2000, who predicted that the correlation functions of exponential fields in the short spacetime
Courant Events Right: distance between the TX going to zero blows up, like, this Minkowski spacetime difference to some power minus 4 alpha squared, with an amplitude which is explicitly, which is given explicitly in terms of the
Courant Events Right: trying to… some, when it… it grows.
Courant Events Right: There's also explicit conjectures, relating, applicable to much more general class of models, integrable models, which predict, which allow you at least
Courant Events Right: identified which conformational theory should grasp the unique behavior of which model, and what would be the type of scaling dimensions, possibly describing this behavior. And this was initiated by terminological by these people in particular.
Courant Events Right: So, so now, in the remainder of the talk, I would like to,
Courant Events Right: get a bit more into the detail of the construction and show you how it would work. So the starting point, and from now on, so I will just focus on the Cinch-Gordon quantum field theory, because it is the simplest… it has the simplest non-trivial
Courant Events Right: S matrix, to particle S matrix, and, the Hilbert space is also the simplest one that one can think of, so this allows to give the flavor of the construction without going through
Courant Events Right: technical things. So the stories, as I said, started with the work of Granig and Bergerez.
Courant Events Right: who argued that actually the theory should contain one type of asymptotic particle, and that the starting point for describing the Hilbert space on which one would realize the golden quantum fields can be given by a Fox space.
Courant Events Right: There are some of elsewhere spaces of all those rapidities.
Courant Events Right: Which would describe the disasymptotic states of those, bonds.
Courant Events Right: So, then, well, in that space, well, a vector will be given by its component in each of the false spaces, and you can think of the nth component as some incoming asymptotic end-particle, wave packing.
Courant Events Right: So, this theory has a purely diagonal scattering, and the,
Courant Events Right: scattering factor is just a scalar. In beta, it's the replenity of two,
Courant Events Right: particles, it takes the… the difference of rapidities of particles to stay in this explicit form. So this is the form which was originally recognized by Granning and burgliness, who calls all the perturbation calculation. And there is a parameter little B, which is a rewriting of the original Kaplan constants appearing in the
Courant Events Right: classical than PDME, which stays value between 0 and a half when G ranges from 0 to plus infinity.
Courant Events Right: So, once one has the Hilbert space, now, I will introduce the…
Courant Events Right: operators which can act on this building space. So the first operator one wants to have, because it's a… one wants essentially to build,
Courant Events Right: quantum field theory on this pivot space, so one wants to introduce operators which represent the basic quantity symmetry of your model, so that starts with the translation operator.
Courant Events Right: So, we take some spacetime point Y, and the action of the translation by Y on the vector of the reverse space, it acts diagonally on each of the
Courant Events Right: force-based components by multiplying the nth component by a plane wave… product of plane wave factors that can each evaluated at one of the n rapidities.
Courant Events Right: The plain old factors involve Minkowski scalar product, which I recall here, between the spacetime vector Y and the
Courant Events Right: Bear.
Courant Events Right: and the momentum of the, two momentum of the asymptotic particle. Here there arises the mass M of this asymptotic particle, and this is one of the parameters of the, of the instruction. So this is step one. Now.
Courant Events Right: Step two, how one would realize more general operators in the space, and in particular, pure operators.
Courant Events Right: So,
Courant Events Right: So, one can think of the field operator. It will act on a vector, and then it will produce something in the zero space component, n space component, and etc. And, because it… so for the moment, I'm not focusing on the
Courant Events Right: distributional and the spacetime variable nature of this operator, just keep the spacetime variable as it is. I will discuss that a bit later.
Courant Events Right: So, this all affects… Produces another component by, linearly acting on
Courant Events Right: All the components at zero up to infinity building up.
Courant Events Right: F?
Courant Events Right: And taking dysfunction as M.
Courant Events Right: nth component in n-square of RM, and producing a functional n square of RM, and then summing up over all these contributions. Again, for the moment, let's stop focus on any convergence problem, this construction here, or simply act on functions which have finitely many.
Courant Events Right: entries. Now, to have, In the realization of,
Courant Events Right: symmetry on the level of your field, one would ask that such x dependence of the field will be factorized by the joint action of the translation operator.
Courant Events Right: All of some reference operator at the given point.
Courant Events Right: And then, the bootstrap program, what it does, so, in order to construct the field operators, you need to… one needs to construct this
Courant Events Right: Oh, and every square of RN to RN to L square of RN. And this is the heart of the bookstore program, how to construct them as developed by, by these people.
Courant Events Right: So, how one does that? So, let us first focus on the zero particle component.
Courant Events Right: So the zero particle component is just linear form acting on a vector of your liquid space. It sums up all components of your,
Courant Events Right: vector, F, and maps them to scams. Now, Boom.
Courant Events Right: Basically, one sums up, linear forms on the square of the
Courant Events Right: are M. So how long will you represent them? Well, first of all, one could think of them as
Courant Events Right: Integral operators, which…
Courant Events Right: whose x dependence is already fixed by this joint action of the translation operator, so I have written it down here with this product of playing wave factors. They, act on the function, input function at h, and then one integrates with an integral
Courant Events Right: Which is actually one of the first postulates one would make in this construction.
Courant Events Right: One wants this kernel to be a plus boundary value of a certain geomorphic function in the variables theta1
Courant Events Right: which belongs to a physical strip, where each of the imaginary parts of the beta ranges from 0 to 5. So it will be a neuromorphic function on this.
Courant Events Right: this strip, and the reason one would pick such a postulate stems from some formative calculations one can make in the, SB agent formalism, which basically dictates who do that.
Courant Events Right: So because these are plus boundary values of, neuromorphic, functions, basically it means that this will produce… this integral kernel might be a generalized function, which would produce, distributions of the direct masses and principal values… the value, functional…
Courant Events Right: generalized now, once one has set up the formula 14, and for the…
Courant Events Right: zeros component to set up all the other ones. Well, one just generalizes this construction by adding up a dependence on the
Courant Events Right: exits variable profile N.
Courant Events Right: So now the…
Courant Events Right: building blocks of the action… of the projection of the action of O of X on the n squ space, are these OMMs, and they are on function of… of the square of RMM.
Courant Events Right: By adding up some products of plane wave factors, which are against them just from this general action of the station operator, and a certain integral code, which, again, would be a generalized function issuing from certain boundary values, polymorphic functions.
Courant Events Right: Many variables, and that's it.
Courant Events Right: So now, to characterize your operators on this space within this formalism, what one needs to do, one needs to fix the generalized functions FM
Courant Events Right: plus boundary values, and more generally, those integral scales. And if one finds a way to set up those functions, then one has constructed the PIN operators for…
Courant Events Right: And the bootstrap program sets out a set of equator, axiomatizes the set of equations which are satisfied by those functions, and which leads to that.
Courant Events Right: And they are construction.
Courant Events Right: So, so I will now list the axioms, which one postulates that these integral satisfies. And first of all, I focus on the simplest, functions, FM of beta n, which are called prone factors. Carol? Yes?
Courant Events Right: So these fun factors,
Courant Events Right: These form factors would correspond to matrix elements of your operators between, vacuum and asymptotic states, and if you, you, you, you can kind of formalize this, this, this, this,
Courant Events Right: matrix elements in terms of a pass integral, then you produce those form factors. And actually, so this is a good question, because one could start from such a form of…
Courant Events Right: Set up, and then using a formal path integral,
Courant Events Right: justify that such the axioms which I will write exist, but then one just turns the cats upside down and says, okay, so I start from the theory which has those axioms, and then I roll out the thing.
Courant Events Right: So, the first action is the rapidity exchange of actions. It tells you that if you take a form factor in n variables, and you swap two neighboring variabilities, then, you multiply this form factor by a S,
Courant Events Right: by this S-matrix, evaluated as the difference of the neighboring rapidities, to
Courant Events Right: So this is very closely related to the definition of the S matrix in your theory, up to some analytic continuations from the
Courant Events Right: Ingoing and outgoing states to all possible orderings of the rapidities.
Courant Events Right: There is a second fact… there's two other axioms, which are the monotrony and the nomatical axioms. They come together, and they translate the fact that,
Courant Events Right: the theory is causal. So if one could kind of formally derive them by imposing certain, causality, so commutativity of, operators and space… local operators and space-like, separations, and do some
Courant Events Right: Handlings, formal handlings seem to finally justify that such
Courant Events Right: equation holds. So, one equation is the monotomy. It tells you that if you… so you postulate that those FMs should be meromorphic in a certain strip, meaning both with 2 pi in respect to each of the rapidities, and if you shift one rapidity by 2i pi.
Courant Events Right: What you do, you move this rapidity to the other side, and just to the end, and the third axiom is a kinematic pole action, so it tells you that those
Courant Events Right: form factors, they have both in the complex plane when two of their entries, approach each other by, with distance i pi, and it gives you an explicit formula for the rescue. So it relates form factor in n plus two variables to form factor in n variables.
Courant Events Right: And finally, there is the boost axioms, which
Courant Events Right: translates how operators transform themselves through, Lorentz boosts, and it, just produces, overall shift in their abilities, produces a prefactor, which contains the spin of the operator times the magnitude of the boost.
Courant Events Right: Okay, and then the idea is that
Courant Events Right: Those equations, first of all, they are not uniquely solvable. There are plenty of solutions to those equations. The equation… the solutions can grow with a certain power of e to the K. K can grow from 0 to infinity whenever A goes to infinity.
Courant Events Right: So they have an exponential growth in each of the betas, and the idea is that all mild growing solutions to those equations will lead you to the whole operator content of your theory. So, one solution would, for instance, produce you the field operator, another one the exponential of the field, another one all these descendants, which I've shown.
Courant Events Right: And, and…
Courant Events Right: So now there is a very long history in the licensing solutions, which is already a bit astonishing, so this kind of axiom state actually produce the calculations of your form factors to solving this kind of reminis problem for the functions at M.
Courant Events Right: And it turns out that this treatment problem can be solved, explicitly, or this one can produce
Courant Events Right: numerous solution, right? In various forms, as it was developed by all these teams. Maybe I, just show you one structural
Courant Events Right: form of, solutions to which is the most, efficient one. So the idea is to solve by steps.
Courant Events Right: One looks for solution… one produces solutions defense in the following form. First one, one factors out a certain function of the difference of rapidities times another function, Kn, which will be a certain transform of a simpler set of functions, PN of O.
Courant Events Right: The role of this function f, which appears as a prefactor in a double-ordered product, so first of all, it has an existence expression in terms of certain, the Long-Fourier transform.
Courant Events Right: And this function plays the role, too, of solving the first axiom, so it's, and the second axiom. It,
Courant Events Right: It allows one to kill the presence of this S once one multiplies by this function, so it solves the S symmetry axiom and the 2I by bilateral axioms. So what it tells you is that the form factor has this prefactor here, up to a symmetric and twi-by-periodic function Kn of the ns.
Courant Events Right: And then, the scale is a certain transform, which I grabbed.
Courant Events Right: Here, so, it has some, one act on the function p of the rapidity variables beta1 to beta n, and some auxiliary set of variables. Here they are distributed in the case of the Central model. They just run through 01 to power n.
Courant Events Right: And, and then there is an explicit kernel for this.
Courant Events Right: transform, whatever it is. And…
Courant Events Right: What this transform does is that it transforms the relatively complicated pull action of your,
Courant Events Right: contractor equation to some simpler set of equations for the N of O, which then you can Oop.
Courant Events Right: On which you can already attack the problem of solving and producing various solutions.
Courant Events Right: And in particular, Barack Oblukianov and Babuchan Karovsky, they produced this solution for the P function, which would be associated with the exponential of the field.
Courant Events Right: operator.
Courant Events Right: Here, so you see this is a p function which does not depend on p. It has a non-trivial dependence on the else, which is of this form, and the gamma, which appears e to the gamma phi, appears, just in the exponent here, with some normalization prefactor, also being explicit in terms of the reparameterization of the models component.
Courant Events Right: And so, again, the idea is that one has a system of equations for the PS that will not rank, and once you have
Courant Events Right: you have plenty of solutions for the PNs, and all solutions of the PNs
Courant Events Right: Allow me to answer who's all the operator content you would like to have.
Courant Events Right: So… So… Possibly.
Courant Events Right: subject.
Courant Events Right: But after all, it's only a certified patients.
Courant Events Right: I don't know. I, I, I didn't ask questions, question, so…
Courant Events Right: So I've explained to you how to construct the first kernel which was appearing in the linear forms
Courant Events Right: Using the zeros component of the action of the field, but then one needs to produce all the other components, those Ms of alphas and betas. And that one postulates the fifth action of the Bunch of approach, which is the reduction civility action.
Courant Events Right: So it's an axiom which allows you to start from one kernel M and N, and move the first entry of the coordinate out from N into the others.
Courant Events Right: one by one, but of course, one has to specify in which space of generalized functions one wants to solve this equation. And basically, what it tells you is that you can move one of the coordinates of alpha n the other side.
Courant Events Right: So reducing, basically, the number of coordinates, in the outer variables.
Courant Events Right: plus adding up some terms which involve direct masses, rather than suggest matrices, and
Courant Events Right: Matrix and kernels involving less variables, again, in the Out of…
Courant Events Right: variable and a variable on which one integrates. So actually, this recursive relatability, you can think of it as really a way to rewrite just the LSS reduction on the level of these matrix elements.
Courant Events Right: But on the level of these integral kernels, and you just take it as a starting postulum of yours.
Courant Events Right: And you couple it to the initialization axiom, namely that when you have… you don't have any more… any outlets in the integral kernel, then you just produce the contactors which were already constructed.
Courant Events Right: So, it's a…
Courant Events Right: Well, first of all, one has to message it a bit to see that it's a well-defined distribution of induction, in particular because
Courant Events Right: You see that there are presence of direct masses here, and moreover, because you move some of the outer rapidities to the other side, shifting them by I pi. And because you have a pole at two coinciding rapidities, which differ by I pi, this in some sense has to be given to them, but this can be done.
Courant Events Right: And the bottom line is that this is a well-posed distribution of induction, which we can solve in a combinatorial form. First solutions were given by Belov Smirnoff, which Simon would have a bit systematize this, this,
Courant Events Right: This construction, and basically now discloses the first step of the construction, because it produces an explicit realization of the field, as certain operators and elsewhere on this forecast.
Courant Events Right: Good. So, once you have this, and there's some way to identify which solution gives you which operator, one can affect the problem of computing correlation functions, I guess.
Courant Events Right: approach.
Courant Events Right: Yes?
Courant Events Right: Can you tell us the phone that you are?
Courant Events Right: One-to-one correspondence with a free movement.
Courant Events Right: Not in the general case. I think in some cases, it could have been, been done for other integral quantum field theories,
Courant Events Right: Let's finish to… minimal models.
Courant Events Right: In some cases, it was done, but in general, this is absolutely an open problem.
Courant Events Right: Because this means that there are also a bunch of accidents, that's consuming.
Courant Events Right: And also, yes, yes, yes, or some, some, some, some sort of,
Courant Events Right: deformation of it should be seen, and this should pop up in the UV limit of the theory.
Courant Events Right: But so far, this is… this is not… not known in general, maybe being just some…
Courant Events Right: Already actually classifying all solutions to the bootstrap equations, or to the P function, is also still an open, open question.
Courant Events Right: So, going back to correlation functions, well, one now, now understand the… so, so far, actually, the construction of the operators, one could just keep the spacetime index,
Courant Events Right: As it was, we're still making heads.
Courant Events Right: provided when I have some sufficiently regular functions, which have maybe finitely many components.
Courant Events Right: But if one wants to compute correlation functions, one really has to go to the distribution nature of the operators, so one sees this O of X of some generalized operator-valued function.
Courant Events Right: And then, to set up the calculation of correlation functions first for well-defined operators, I will introduce projection operators which project the little space to the Earth
Courant Events Right: top space component, okay? And I will introduce truncated smeared operators, which are just projections of a field operator smeared by G, and consider first the regularized correlation functions, which are just products of such
Courant Events Right: projected, operators, and such objects can be,
Courant Events Right: Well… well defined through the procedure, and in principle, by summing over all values of the,
Courant Events Right: projection indices, so through all of NK-1, one, recall, one, one, in principle would get the correlation function.
Courant Events Right: So how… how will the calculation goes for… for just the simplest possible two-point function?
Courant Events Right: So, this one takes two, test functions whose supports are mutually spaceline.
Courant Events Right: And, computes this correlation function first of the
Courant Events Right: projected operator, the operator O1 is just projected on the zero space because of the presence of the, vacuum state, and the operator O2 is projected on the, n space. Then, it is a generalized function of
Courant Events Right: variables X and Y integrated versus F, and the…
Courant Events Right: Yeah, informally, so this generalized function is this matrix element. To make sense of this matrix element, one really needs to build on the distributional matrix, because it allows one to regularize the multiple integral, right, below just
Courant Events Right: By going to the definition of how such operators are.
Courant Events Right: So, how this would go? Well, I have just a product of two operators, think of them as integral operators. This one connects
Courant Events Right: zero, particle space to n particle space, and then we go from n-particle space to zero space, so this is the following of those two integral versions. That phase factor, which was appearing, previously, which I spit down explicitly. And then by symmetrizing the integral and deforming it to, some
Courant Events Right: integral over the wheel axis shifted by i pi over 2, depending on the sign of
Courant Events Right: spacetime parameter at X, and doing some Rouge transformation, this two-point function, is written as a
Courant Events Right: This time, well-defined and is urgent.
Courant Events Right: And for the integral, involving the product of two form factors, and an exponentially, exponentially decaying factor, when the speed of the decay controlled by the spacetime difference between XT and zero, and this
Courant Events Right: Chain of transformation is justified by Recalling that
Courant Events Right: This is a… all of this should be taken at the center of this.
Courant Events Right: And this distributional interpretation allows one in particular to… because you see, those integrals a priori, they will grow up, like, e to the L, e to the K, and this is the oscillatory factor, so those integrals, per se, are not convergent, but you can understand them appropriately as to them.
Courant Events Right: But after events, they become stronger in the boat.
Courant Events Right: Now, so as I said, the true correlation function is then recovered by summing it up over N,
Courant Events Right: So considering such a sum, each sum is a manifold integral, which, after massaging a bit with the formula I've shown you, it takes the following form. So one integrates over RN. There's a one body confining potential,
Courant Events Right: became exponentially, exponentially fast. There's a two-body interaction, which form I have plotted here, which has an explicit,
Courant Events Right: And… 4 years ago.
Courant Events Right: Presentation.
Courant Events Right: And there's a many-body interaction coming from the scan transform.
Courant Events Right: And if one wants to make sense of this two-point function, then one needs to estimate how this kind of integrals grow with n.
Courant Events Right: And, so…
Courant Events Right: This can be done, actually. The problem in this form has a sort of a structural similarity with some n-fold integrals.
Courant Events Right: Which arise in the non-trivial part of, the partition function of
Courant Events Right: permission, or more generally better, ensembles. And using,
Courant Events Right: Some refinement of concentration of measure, techniques.
Courant Events Right: One is able to provide an estimate of the N…
Courant Events Right: n-fold, integral building the previous series, up to some hypothesis on the growth of the solution's PM, which are natural from the point of view of the bootstrap construction. And basically, the summit of the… this two-point function in the n-particle sector decays almost like a Gaussian.
Courant Events Right: So, the first thing what one needs to say is that the control and the reminder here in the upper bound is not uniform in the space-like separation between the variables.
Courant Events Right: This is… it should be so, because one expects that in the short-distance regime, short space-like regime, so the new regime, there's a certain blow-up of the point functions, so this should stem from the fact that it seems to be convergent.
Courant Events Right: The decay is, of course, slightly… is just slightly subgaussian, so the fact that there is a 1 over n factorial doesn't change anything in convergence. And this time about, it explains why, at least on a numerical level, people were observing the such for a practice series. You could just take a few of them, and this was already giving a very nice description of them.
Courant Events Right: Of the triple-traditional evidence.
Courant Events Right: Now, in order to build a theory, well, as I said in the beginning, one needs to construct not only two-point functions, but actually the whole table of multipoint functions.
Courant Events Right: And this was already much less studied in the literature. There were some partial results, where some pieces of the summons, which will build
Courant Events Right: 3- and 4-point functions were obtained by these people, but nothing systematic. And then we'll see more, we had the problem of computing those regularized, projected, multipoint functions, and we managed to produce some closed expressions, for them.
Courant Events Right: So, if we focus on test functions which have…
Courant Events Right: whose supports are all pure space, like, separated, then actually, the…
Courant Events Right: projected… the K-point correlation functions of projected operators can be written in this form, where the W, R's are
Courant Events Right: well-defined, affiliate functions, so they do not need to be taken in the distributional sense, but only for such
Courant Events Right: hypothesis on the support of the test functions. And I want to show you just the lengths of the kind of formula, explicit formula one obtains. It will be a generalization of the formula which appears for the two-point functions. So for that, I need to introduce some notations, if I have to.
Courant Events Right: sets of two vectors, one n-dimensional and then one M-dimensional. You can concatenate them and produce an dimensional vector. If you come up with a vector, you can reflect it and take its coordinates in the opposite sense.
Courant Events Right: And then finally, the factors, they were satisfying this as symmetry when one was exchanging two neighboring variables.
Courant Events Right: So I can introduce general factors, which allow me to swap two concatenation of vectors in a form factor, and this produces a sort of generalized S-factor, which you can compute, explicitly if one wants to.
Courant Events Right: And finally, I denote the macroscopic momentum
Courant Events Right: Associated to a vector gamma as the sum of two more metals, which are the ordinance of
Courant Events Right: a vector gamma, and the uniform vector with just one entries by P. And I do not write dimensionality of the form factors. Here, it is justified by the arguments of the form factors. Then, the closed expressions are the form factor
Courant Events Right: 100K point functions states the following form. So it's an explicit multiple integral representation, or actually summed over various multiple integrals for the contours, which are
Courant Events Right: So, first of all, when sums over, integration in respect to various variables, gamma and EA,
Courant Events Right: Of various numbers, which were all gathers in a single vector n. There is some constraint on the entries of this vector, N, depending on the regularization parameters RP, which would reduce it. Then there is a product of plane wave factors, a bit like the function, a product S of gamma, which,
Courant Events Right: involves,
Courant Events Right: just drawing from these exchange S functions, which I introduced before. And then an ordered product of form factors, which couple in a different way the various gamma variables on which one integrates over a certain counter, which is explicit with a slight shift of the real axis by some, well, slightly misplaced
Courant Events Right: imaginary, parameters. So, that's it. This gives the closed formula for the K mode function. And then, in principle, one can get the K mode function just by summing over the regularization parameters. This,
Courant Events Right: And assuming that everything is convergent, then this produces such a closed-form series of multiple integral representation for K-point functions. But the convergence is a problem.
Courant Events Right: So, in principle, the techniques used for the convergence of two-point functions could be applicable here, but as I said, for two-point functions, there's a refinement of
Courant Events Right: Concentration of measure techniques arising in random matrix theory, but those concentration of measure techniques in random matrix theory, they are based on
Courant Events Right: The real variable setting, and actually, it was a bit lucky that the two-point function place was a real variable setting, but this is a genuinely complex value integral, and you want to study how the summit of those
Courant Events Right: embedded integrals grows when the number of each of the integrals, or the dimensionality of each of the gamma PAs grows to infinity. So for this one, needs to develop a concentration of measure techniques for
Courant Events Right: For complex valued, integrates.
Courant Events Right: to control how it grows with N, so the first step in this direction was taken
Courant Events Right: in my work with LSDN and Alexander Little, where we managed to set up the concentration of measure techniques to describe the large number of integration behavior of betain samples and the relevance of complexity, potential. And this is the first step, but there's still some
Courant Events Right: A few steps.
Courant Events Right: You would plan to be able to overcome to be able to attack these kinds of incidentals.
Courant Events Right: And then, maybe, as we come to the last point.
Courant Events Right: Which is the checking of the Weitman axiom, because so far, what one does is that one develops some axiomatic framework, which produces new operators, actually to check that those operators dissatisfy all the operator properties.
Courant Events Right: And in particular, characterize their domains is extremely complicated from the formula, but what one can do? One can compute the correlation functions, and just check that they satisfy the white man axioms, and then this guarantees you that all this construction gives rise to the Weitman.
Courant Events Right: And actually, the way the bootstrap axioms are written now, most of the Weitman axioms are really built in and easy to check on the expressions we obtained for the multipoint functions. The only one which is actually quite difficult to check is the local community.
Courant Events Right: This step in this direction were taken by Kirilov and Smyrnav already in the late 80s, where they managed to prove something like a weak local community by assuming that they could multiply the operators in the commutator.
Courant Events Right: and the O series involved in this product were well defined, they could show that if one takes two smooth and completely supported functions, then the commentator ranch. So the truth was just checking the algebraic manipulation of all
Courant Events Right: And actually, with the concentration of measured techniques, which work for the convergence of two-point functions, this
Courant Events Right: can apply this result, and it gives such a weak local commutativity. But this is not enough to prove local communicativity of the Y-band functions, because in that case, well, we need to remove the complexly supported hypothesis.
Courant Events Right: And what we managed to do with CMO is to prove the… start from the hypothesis that the series for multiple functions are convergent, system with which improved in severity.
Courant Events Right: But if this is true, then the multipoint keypoint functions which characterize, say, the product of field operator, they will satisfy all the Weitman atoms. In particular, this local community has to prove. So, this says that if one takes two
Courant Events Right: test functions, which are space-like separated, then one can swap the field operators in this pronation functions, and thus, up to the convergence hypothesis.
Courant Events Right: multipoint functions with the number of points bigger or equal to 3. This shows that this group construction, at least for a central model, it gives rise to a point in field 3.
Courant Events Right: Now, what can be said beyond the cinch?
Courant Events Right: Golden Model, and what are the other challenges of this approach?
Courant Events Right: Well, actually, so as I said, the S matrix within the bootstrap approach is…
Courant Events Right: sort of as a solution to the Young-Baxter equation. This Young-Baxter equation that takes its roots in, the,
Courant Events Right: representation theory of quantum groups, and using this representation theory, one can build a lot of solutions to the Baxter equation. Each solution, provided it also satisfies unitarity and processing, will produce you a viable candidate for a S-matrix of a Degrebo quantum field theory. Of course, one should then identify which integrable quantum field theory is saved by going to the semi-classical limit, but in particular, it's known for sine
Courant Events Right: golden model or other field theories. And from the S matrix, one can deduce the particle content of the theory, or if you want, the structure of what Hilbert space you should put as a force space on which you would build the operators.
Courant Events Right: And then once you know that, there are techniques. Again, building a representation theory of quantum groups and the so-called off-shell Becky ancents, which allow you to produce form factors, so analogs of the form factors I showed before, which would then allow me to produce the expression for the field.
Courant Events Right: And then the open problems in the field, well, it's definitely already for the Finch-Gordon model to prove the convergence of series describing multiple point values.
Courant Events Right: Then, actually, because the only case where multipoint functions were computed is the Sinch-Gordon model case to produce some expressions for multipoint functions in the more complex model, at least in the same golden model, to see how it works. This is technically a bit
Courant Events Right: more involved, because combinatorics involved in the computation of multiple functions become much more complicated due to the fact that the sign-old theory has a much more complicated
Courant Events Right: Political structure.
Courant Events Right: And then what is interesting is that these, kind of integral
Courant Events Right: The description of the identity theories, they provide you closed expressions for the two-point functions.
Courant Events Right: Once they are closed, you can think of attacking the problem of short-distance asymptotic analysis and extracting this UV behavior out of the closed format. So far, this is still… the techniques for being able to do so are in its infancy, but in certain cases, it's possible to do it. For instance, in the Sein Borden model, there is the free Fermion point. In the free Fermion point.
Courant Events Right: The two-point function reduced to a fit-rome determinant, which is actually of, integrable.
Courant Events Right: Integral operator time, and for those
Courant Events Right: Perform determinants. You have some dependence in the integral kernel of this platform determinants from the
Courant Events Right: short UV distances are, and you can analyze the asymptotics of this straight-home determinant using some Riemann-Herman technique that sooner limits this descent. But this is just the case of the Frieferman point. But away from the Frieferman point.
Courant Events Right: you do not have any more reductions of return, and something else has to be, invented. And what would be also extremely interesting is to be able to set up a bridge between this bootstrap construction, which just starts from S matrix and some postulates, some actions on
Courant Events Right: how this estimates allows you to build the field operators, and connect it with more traditional constructions of quantum field theory to constructive, pass integrals, or through…
Courant Events Right: And so I still feel.
Courant Events Right: Jocelyn.
Courant Events Right: Can you get the field equation in this? Yes, so for Cinch-Gorten, you can get something, like, you would start from a P function, which is supposed to describe you the
Courant Events Right: A field operator, and you could get something like the…
Courant Events Right: quantized version of this golden equation, where you would have the dalumation acting on the field operator, times a term which you built up from operators, which builds the exponential of the field, and e to the minus exponential of the field, and all this falls in the weak sense. And then you have a
Courant Events Right: Small, kind of dressing of the mass parameter, which appears in the equation, which appears explicitly.
Courant Events Right: And you can get it also in the Sanctum case.
Courant Events Right: Yeah, the question's… Zoom.
Courant Events Right: But I assume that what they mean?
Courant Events Right: generalized community through the… states.
Courant Events Right: Can you…
Courant Events Right: This I don't know, because, like, the simplest case where this would appear would be the Singorden model, I think, for exponentials of the field, and then there…
Courant Events Right: Well, they are proving, yeah, local commutativeness is still…
Courant Events Right: Very, very combinatorial, so it's still an open question, I think.
Courant Events Right: Or some modification of this local commute activity to encompass for the tail of the operand.
Courant Events Right: Are there any developments on the idea that form factors are, like, generalizations of conformed blocks?
Courant Events Right: go to infinity, they should not.
Courant Events Right: I think it needs to…
Courant Events Right: there should be some relation between conformal blocks and form factors, because the form factors know something about the UV limit of the theory, but this would not be so… just to limit beta goes to infinity of the form factors. It should be some resumation of a whole tower of form factors.
Courant Events Right: And altogether, they should go to the conformal blocks, but this has not been so much exploded. Another example, on the side of conform blocks, sometimes there is probably clear
Courant Events Right: To the formation of control blocks.
Courant Events Right: look at the new book of Roblox, for example, they give it by some…
Courant Events Right: given by the sums of the partitions, or the L diagrams, so those kind of straightforward Q deformation, which is motivated by the connections with age theory.
Courant Events Right: Do you know of any meaning of these two deformed conformed blocks?
Courant Events Right: Passes from the global union.
Courant Events Right: Oh, good on.
Courant Events Right: No, no.
Courant Events Right: Yeah, so, what do you get on the level… do you get something on the level of operators, like self-addoin, or maybe… do you get that the exponential of the field is actually exponential of…
Courant Events Right: I'll bet one.
Courant Events Right: Yes, yes, yes, so you can… so, for instance, Gordon, you showed that the field operator is self-adjoint, and that the exponential of gamma the field is gamma derivative is the field, and you can get this for also various other models. So you… you can get such kinds.
Courant Events Right: Relations Perfect.
Courant Events Right: I didn't appear to see.
Courant Events Right: Okay, cool.
Courant Events Right: And… so… Yes.
Courant Events Right: So, crossing symmetry on the level of Sinch-Gorden model, it's… is,
Courant Events Right: well, the particle is its own antiparticle, so it's just some relation for the S matrix.
Courant Events Right: it would be maybe more seeable for sine cotton model where the acid solid, anti, anti-solidoms.
Courant Events Right: No, no, no, monotronomy is really something that enforces… that is enforced by causality of your, of your theorem.
Courant Events Right: Yes, yes, because it's some form of annihilation pole of particle with antiparticle. So maybe the name Bootstrap will come from the fact that you have a… that N plus 2 residues reduce to
Courant Events Right: residue of a form factor with n plus 2 reduces to a form factor of n, and then you… what you do, you actually solve this residue equation backwards. You solve first for two particles, then you deduce the general form for 4, and you go up. That would be the idea of the question.
Courant Events Right: So, and indeed, those are annihilation pose of, particulars within category.
Courant Events Right: Very hot.
Courant Events Right: purple.
Courant Events Right: You can run into all kinds of trouble that folks I had yesterday, when I… There's an agile factorization.
Courant Events Right: Is it a theory I can ask you for this, or is it something that…
Courant Events Right: So, for the Cinch-Gordon model, such problems, I don't think, would appear. They would probably appear if you would consider more complicated test matrices. So, for instance, things could happen if you would,
Courant Events Right: even consider Cinch Gordon with what is called CDD factors, which would press the S matrix, this could cause various kinds of problems. Because it could cause problems of blow-up at too strong blow-up at infinity on your phone factors, which you cannot then
Courant Events Right: regularize properly, using distributional nature, then you have to do something with something else.
Courant Events Right: things like… well, it's not… Okay, honey.
Courant Events Right: Yes,
Courant Events Right: But they will probably be related to different theories which have, like, more… which have bound states, which can be formed in different ways, and then coherence of all. That may be.
Courant Events Right: The assumptions are sort of clear, but they throw out… Agent.
Courant Events Right: So…
Courant Events Right: I think those complications would come if you start to put… you look at… you… you start from a class of S matrices, which have several
Courant Events Right: pose in the physical strip, so it will have, bound states, and then the Hilbert space is more complicated, and it allows for more volume of complications somehow.
Courant Events Right: Are there perspectives on building value?
Courant Events Right: Yes, yes, yes, yes, yes, so for… I would state that for sign-Gordon, it would have all these… all these aspects, but it's much more heavier to state.
Courant Events Right: What kind of theories can this be imagined to be developed for, say, a nonlinear sigma model?
Courant Events Right: It's like a complete deal.
Courant Events Right: how to reach
Courant Events Right: The general class of theories, I wouldn't be able to tell. Like, for all theories which would be a S matrix would solve the Young-Baxter equation, in principle, one could do that.
Courant Events Right: all of this, because the road is settled, so I think you would have some solutions which would correspond between
Courant Events Right: Any old classical system that already has the expected
Courant Events Right: Like the nonlinear sigma model, classically, it wouldn't have a mass, right? Quantum mechanically, Should.
Courant Events Right: So this wouldn't possibly be applicable, but in such Gordon is classically, there's a mass. Yeah, I'm not sure it would be applicable to nonlinear Sigma motors.
Courant Events Right: Yeah, yeah.
Courant Events Right: Whoa.
Courant Events Right: parts of abuse.
Courant Events Right: The fact that classically they didn't for the last people don't get there doesn't matter. So that doesn't matter, that classically… It just matters that classically, one is very different from the other. We just assume that here.
Courant Events Right: Thank you, sir.
Courant Events Right: It's called acceptable.
Courant Events Right: Excuse me.
Courant Events Right: outside building.
Courant Events Right: Okay, I suggest that we continue discussions in social break. Thank you.
Courant Events Right: Do you need your.