Sinh-Gordon model and its duality
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Transcript: English(auto-generated)
00:15
Okay, thank you Slava. Thanks for the invitation. I'm very happy to be here in this
00:22
Very interesting meeting. Unfortunately, I cannot say long. But anyway, I try to benefit myself the best I can so what I going to tell you today is a very disturbing fact of quantum field theory and actually how little we know about it and
00:42
The kind of problem we have really to understand better in particular theory related to uncompact bosons so indeed for that reason I decide to Title my talk the problems of the cinch Gordon
01:07
Model so just to have a track what I'm going to say gonna make some generalities
01:20
And you will make generality that let you be very easy with this model, but it's very very I Mean you can be very fooled by this then integrability and exact as matrix
01:43
then I'm going to talk Really on the subject, which is really the root of all the problem of this model, which is its self duality Alas the model seems to be the same for weak and strong coupling
02:02
But this is really where all the nigma come about by this model then I'm gonna tell you about Beta answers and what we can learn about finite volume and we will see that this theory is a matter of fact is
02:23
the mother of very very intriguing renormalization group flow Which lead to the so-called roaming trajectory and?
02:41
minimal conformal field theories Then I'm gonna sketch form factor calculation Simply because I want to show you that the theory is unable to Unambiguously define its ultraviolet behavior. So all the root consists exactly in this problem and
03:07
You be property and Finally, I'm going to present you the problematic formula, which are exact formula, but nevertheless very problematic
03:23
so the exact mass formula of the model and the exact formula of the vacuum expectation value many many actually infinitely many operators and Then we'll discuss the problematic
03:43
Aspect of this model both from Monte Carlo point of view and truncated the conformal space approach Okay, so Let me first present the model. The model is very simple is a
04:02
bosonic field the scalar bosonic field with this kind of Of interaction so this seems really the best field theory you can think of Why well several reasons there is a unique vacuum
04:23
So means that you have not worried about Hidden sector topological sector solitons, whatever whatever is in the theory seems to be under your eyes Second there is a really fast growing potential then
04:41
These Lead you to think that you can simulate this theory very easily. Then is the simplest symmetry just the two and Moreover if you expand to lowest order is 5 to the 4th, which is a repulsive field theory
05:02
So you have not even to bother about bound states so the theory maybe the only content is Whatever once again is under your eyes alias one particle massive particle created by the field itself interacting and
05:24
That's it. Nothing else now Let's dig more in detail So if we expand the cosine we get infinite number of potential term power law So the theory looks like Infinite type
05:41
Landau-Ginzburg theory Remember that the field in one plus one dimension is zero dimension, so there is no problem with normalization all theories trivially normalized if you just took normal order and So you have infinite number of
06:02
coupling even coupling of that type However, the fact that there are infinite number of these vertices change radically the nature of the theory and make it integrable Now this is very very interesting you can do really
06:22
Back envelope calculation. It's extremely instructive Because what you can compute is imagine I assign you a Lagrangian with one bosonic field and Arbitrary Set of
06:41
Even coupling and then you ask yourself What a relation between these coupling constant such that There are No production No production means you have a n particle M
07:03
particle and Different from them and then you want that on shell these amplitude are zero all For any n and any m at all order in coupling constant Well, you can do the systematic way of doing was pioneered by Patrick Dorey
07:24
And then he did let me just first do the three levels And then when I find that condition the tree level I will dress with vertex and this and that And the story is still instructive, so let me just show you the simplest calculation so simplest calculation is to going to four
07:44
so imagine I have just five to the fourth So two to four consists in the following graph Okay, if you have five to the fourth, so you can compute it and the result is M square
08:09
lambda Four square divide 32 I Minus I M square lambda four square
08:21
divide 96 and then if you have five to the fourth, definitely the sum is non-zero So you can have a production to go into four as far as you have enough energy to produce it Now if you want to kill this term
08:40
You imagine that you have the possibility to add the sixth term here so you can add a sixth term here you see two going to four but this is 5-6 term and the result is if you compute it with the proper normalization doing
09:01
This is 48 and All these result end up to be zero. So what I mean is at any given order You can compute some graph which involved the lowest order of vertex and propagator
09:21
And the result is not zero But then if you have possibility to include exactly the vertex which make n goes to M So here there are no propagators you can kill it, okay, so if you do this Why are you not putting any loop diagram?
09:44
You are perfectly right I'm just saying strategically you can find some condition simple condition Just look in the three levels Imagine you find this then you elaborate further Okay, so I will just make a few comment on that
10:01
So if you do at the three level you can do at any order of this you find an amazing Beautiful Recursive equation that As far as The coupling constant of the 2n vertex is proportional lambda to the 4 in this way
10:22
all these theory are Integrable in the sense of the three level. They are not production. Okay, and then if you elaborate further you see that We fight to the fourth sorry with z2 theory so all the even vertex you have only the ambiguity if this term is going to be a
10:43
negative or positive So at the end of the day the three level calculation leave as only possible theory made of a single bosons Just the sine Gordon or the cinch Gordon. So here is
11:02
Hosh G fee and then there is a relative sign and the cosh now Imagine I fix this at the tree level. I told you before that the theory is trivially normalizable So you can is non-trivial a non-trivial, but you can check that all the loops
11:22
Essentially normalize uniformly the mass So doesn't scale differently the coupling Somehow What the change is the overall scaling here non-trivial I'm not I'm not claiming is something that you can see it, but if you do all the calculation properly
11:43
What happens is that you have a finite or infinite? So the loop for instance are a finite here in this theory you make a finite shift, but always the same So doesn't change anything So say differently I can put in a different way is known that sine Gordon cinch Gordon is classically integrable
12:05
Because there are inverse scattering method you can compute all the lux pair and then classically they are integrable Then you have infinite number of conservation law That you can derive from this method. So you have to check
12:20
That the normal order expression this currents Doesn't get anomaly when you quantize it okay, so this is and remain finite So to make this story short cinch Gordon is the simplest Integrable model which is z2
12:41
Even consists of only one particle and the spectrum altogether I'm going to derive in a minutes and therefore looks absolutely Absolutely ideal to try to understand basic things of quantum field theory So if you want epistemologically is the ideal laboratory playground where we can test something
13:01
But as I said that there are very disturbing feature this coming later Okay, so let's assume therefore is quantum integrable as I say I can prove it so this means that The s matrix is elastic and factorizable so any n go to n
13:23
things I can write in term of Two body problem Pictorially something like this so I can concentrate only in two particle scattering then I parameterize my
13:43
my a Dispersion relation in term of rapidities No, no, it's one theory what you call one family I Yeah, sure. I mean there is a coupling constant. No. No, it's going to play an important role. Yeah
14:03
Yeah, okay, if you want from this point of view, yeah is a There is a honest coupling constant here So I parameterize my dispersion energy of my particle in this way you see they automatically satisfy Lorentz dispersion relation and
14:22
Lorentz transformation theta just Additively change therefore the s matrix which has to be Lorentz invariance should depend on rapidity differences So I define my s matrix like this two particle I have two particle or rapidity theta 1 theta 2 and
14:44
Then if I interchange them the amplitude in front Is called s matrix. So this is a convention also Gabor was mentioning before Theta 1 is order a larger than theta 2 and this one is low order differently means the two particles just have scattered together
15:08
And the amplitude is this matrix. So less matrix depend on the difference of rapidities and they satisfy You lethality and
15:23
Crossing now where this equation come from come from the following In s matrix. You have a Mandelstrom variable s Which is p 1 plus p 2 square
15:42
If you write down is NT the Mandelstrom variable T is p 1 minus p 2 This is essentially changing theta 1 theta 2 in I pi minus theta because then this become minus sign
16:06
so in the Mandelstrom variable plane The analytic structure this matrix is like this you have a branch cut At m 1 plus m 2 square
16:21
And m 1 minus m 2 square is the T cut so unitarity means the value of this part is on top of the lip and below has to be 1 and T channel is this analytic continuation So if you say this map is analytic map in the theta plane
16:44
So let me call the upper lip of this branch cut like this So e is mapped here e 2 is mapped here
17:01
e 3 here and e 4 here you can check this is the analytic map I'm using so you see that the unitarity is s s minus theta s b 1 so this is the And Crossing is the same value this matrix here and there
17:22
Okay So these are the general condition that an s matrix has to satisfy and for the s for the cinch Gordon We have an exact solution of them. So the exact s matrix of cinch Gordon is the following
17:55
s beta is equal cinch
18:00
theta minus i sine Pb and b is a function of the coupling constant which has that expression
18:32
Now I'm gonna convince you that all the trouble of the theory Come from this expression
18:41
alias from the duality Let me explain what what I mean Okay, so first of all, this is a really very very simple function So it's periodic 2 pi periodic 2 pi i periodic and as zeros rather than poles
19:01
So here if there are poles in this physical strip physical strip is Between 0 and i pi any poles here correspond to bound states There are no poles there there are just zeros, okay. So where are the zeros use this one?
19:24
So let me show you the analytic structure of this matrix
19:45
So there are two zeros Located in i pi b and i pi 1 minus b Okay, you can see immediately if there is a zero somewhere
20:04
By this relation, there also should be a zero in the i pi minus the same Okay now this s matrix Much perfectly the perturbation theory what I mean is you can take
20:23
You can take this Lagrangian This Lagrangian compute the Feynman diagram Corresponding to the scattering this is very easy actually at least the lowest order For instance, you have something like this plus this plus
20:42
this plus You can compute all this they are finite finite. So this guy for instance give rise to 1 pi minus 1 over cinched theta something like this and Then you can expand This expression term of coupling constant here and matching order by order
21:05
Okay Clear what I'm doing. I Have an s matrix, which is an exact expression of the coupling. This is formula is exact So to all order in the coupling constant so I can expand the order by order in G G square and
21:22
Comparing with the corresponding Feynman diagram which come from the Lagrangian and the match is one-to-one. Okay now you will see why this is a kind of mysterious things now noticed that If here
21:47
Is matching works Yeah, yeah now I'm going I'm going to tell you yeah now you can ask indeed thanks for the question So you can ask where they'll you come with this expression
22:02
I mean you see I mean, it's like NP complete problem in mathematics You don't know where they come from. But if you know the solution, it's easy to check So this is the same if I know this It's easy to check you can do all the Feynman diagram in this but it's very difficult to do vice versa
22:22
Okay, so where they'll come this expression from to start with and so this is precisely the point that this Theory is very much related to sign Gordon theory just by analytic continuation so the analytic continuation
22:41
so sign Gordon you have Cosh, let me use lambda phi minus 1 Now this theory is profoundly different from the other
23:02
Because here the corresponding vertex operator exponential are compact While there the corresponding vertex operator are uncompact, okay So here the theory is very very well studied since long
23:21
The S matrix was known consists of kink and anti-kink the spectrum consists of kink and anti-kink and bound state thereof And the first bound state is the breeder. So kink anti-kink bound state and S matrix of this breeder Was sinh theta plus i sinus, let me call xi
23:48
Sinch theta minus i sinus xi and in this case as a pole Because breeder create a bound state to themselves. Okay, so at this point, this is the famous Coleman
24:05
Fink's which As a validity as far as lambda square is less than 8 pi this is the famous Coleman bound
24:21
Where the bounds come from come from the relevance or irrelevance of this vertex operator When lambda square is bigger than 8 pi this guy is irrelevant So at least from renormalization point of view the theory strictly speaking is free theory although I mean one can discuss but the
24:42
The Fink is is no longer there is no longer a mass gap there So if you want to make sense of sine gordon as theory of kink bound state mass gap and so on so forth Lambda square has to be strictly an 8 pi But when we make this analytic continuation the analytic continuation lambda goes in I lambda which I call G
25:09
This become minus G square and this become plus So what was the pole here
25:20
become my zeros What was a bound there now become completely? invisible bound Okay, so this is where this matters come from. Okay, then once you have it you can check it
25:40
Okay, now these things as it is very nice But is very very positive all the enigma of the model is here So, let me explain why the Fink is So let me cancel now the sine gordon
26:01
So you see that imagine I'm moving I'm moving G and Making G1 bigger than J2. So these guy are gonna move one toward the other Okay, so I increased G and they move in the complex plane
26:23
If I increase G you can plot this function What happened if you send the G? In 8 pi over G if I make weak strong duality What happened is that?
26:43
B Simply going 1 minus B. So what happened is that? The two zeros swap just the position but the analytic structure is the same. Okay?
27:02
Clear So if you make weak strong duality B going to 1 minus G B goes to 1 minus B But for what the analytic structure is conserved you have just swap the position on the zeros. So it's exactly the same
27:20
so the theory is Non-self-dual for what the asmatics are concerned But where the L is written in the Lagrangian. I mean if in the Lagrangian you substitute 1 over G
27:44
Completely different things There is no 1 over G in the Lagrangian Nevertheless the calculation that you get from duality Matching with the Feynman diagram is exact What I mean is
28:01
Once you have a function like this The dependence is not G square. The dependence is G square divided 1 over G square Therefore imagine that you expand this which is already a sinus You expand in coupling constant already at the fourth order term
28:21
The contribution is both from sinus of G square by some term which come from downstairs Okay, so what I want to say is even at the level G4 and farther The number precise number is a combination which come from this function of sinus
28:41
But of this function also this function and much perfectly the perturbation theory of this Lagrangian Written only in exponential of G. There is no 1 over G there, okay so this is a very Puzzling thing, but it's the same for all strong weak strong common qualities. No, you mean for other theories
29:06
Phenomenon that the duality is not not even yeah, but you see what I found here at least I mean Is that the theory looks so simple. There are no hidden sector You see in other theory the structure is so rich. You have monopoles of Kings
29:22
Even a sign order is not self dual is dual to something else to tearing So you have other degrees of freedom you see what I mean is Here seems the theory consists only one bosons and it's able to do all all these things. Okay? Okay
29:44
Now let me also mention something which make the theory particularly appealing also from an application point of view What is really appealing is that? Imagine you restore the velocity of light in all the
30:01
Lagrangian in all the formalism and then you make the double limit C goes to infinity C is the velocity of light and G goes to zero Such that C times G is equal lambda is equal finite
30:20
What you can prove is that the sinh Gordon reduced to the Libellini care model Whose s matrix So the Libellini care model is the one which consists just of Three particles not relativistic because I'm taking C goes to infinity with Delta function
30:47
Interaction This is really the easy model of cold atoms is how you can Discuss the property of one-dimensional bosons and Actually, you can use all the technology of sinh Gordon for instance the form factor by the answers
31:07
I'm going to describe in a minutes to compute property directly in atomic physics So in particular you can derive a recombination rate you can derive correlation function, isn't it? So this has been checked also in in lab
31:21
So I want to say that a part of being an interesting model in itself There is a payoff a by-product That goes directly in experimental physics and atomic physics if you just do the proper not relativistic limit now let me
31:41
introduce new tools To study the theory more detail and the first tool is the thermodynamic by the answers How did you end up with a non-relativistic limit of this as I said you have to restore the velocity of light
32:01
So I usually in field here with this regard, right C equal one H equal one You have to do all this exercise and then you take C goes to infinity limit This is how you realize not relativistic one and then there is a very interesting trick so you have to disentangle fast mode from slow mode and Then the first mode when you integrate it it goes to zero and then what is left out is
32:25
No linear Schrodinger equation. So gross beta is essentially and this matrix much perfectly with the lip linear one and then Once you do all the other form or that you just make this trick simply like that. You can recover
32:42
free all the quantity of lip linear Which is pretty pretty nice For what for why for which reason because in field theory usually you have more constraint that in non relativistic one For instance in field theory you have duality This equation which doesn't hold in no relativistic one
33:03
So the theory is much more constrained and this is the reason why people working in atomic physics were unable to compute easily correlation function lip-linear While if you come from Cinch Gordon you get easily the result
33:20
Okay, so let me tell you Other aspect of the theory which appear in finite volume through the thermodynamic beta answers So let me sketch the idea what it is Thermodynamic better answer you take large volume L You compactified in temperature 1 over R
33:43
You compute the trace of your theory So Here the Hilbert space is made of particle which scatter all the way around But the number of particle is conserved because it's integrable so you can do exact the trace Okay, so I make the story short
34:02
You can compute The ground state energy is a function of the temperature R This is parameterized in this way In terms of what is called the effective central charge
34:22
So this is standard normalization of conformal field theory and then there are a set of equation integral equation Which compute the ground state energy? Gonna write it then I will comment and
35:19
So the formula is a very very clear
35:24
interpretation going to spell out for you So look look this formula here The formula here remind very very much the formula of a free Fermion particle, okay So it's an integral if you write like this you will recognize immediately
35:44
logarithmic 1 plus e to the minus beta e like this is just so the course here is just Change of variable in the rapidity I'm doing The only thing we change is that instead of having the energy of the particle
36:02
We have a function which is called pseudo energy Which is cell consistently? determined by itself So these things satisfy an integral equation Which involved the free part that will be this one?
36:21
But then Bootstrapping through the interaction with all the other particle with the same distributions So this equation is very very nice because you are enforcing the most you can the free Quality of the theory because integrable theory is essentially the closest to free theory you can think of
36:45
but the only difference is that what play the role of energy is not the energy of single particle But is the term in self consistently by the all other particle present in the system Through a kernel which is just derivative of the S matrix
37:03
Okay, so the message is here is that for what? The finite volume is concerned the S matrix determine everything Even the ultraviolet this is the message Okay, so You can in particular study what happens for R goes to 0
37:24
So you can study what happened to the central charge? The effective central charge for R goes to 0
37:44
so you see you can solve numerically this equation as a function of R and then Plot C versus MR and the plot is like this so is
38:02
1 minus some coefficient Logarithmic square MR plus blah blah blah So you get the central charge of a free boson theory Okay, now I will come back to this because once again is very squeezing seems very simple elementary
38:23
But then you will see what is behind this so Now, let me introduce another part of the story Which is the form factor
38:45
Absolutely Absolutely, this is absolutely significant because in other theory for instance in Theory of the minimal models if you take both model or if you take easy magnetic field
39:02
The correction is always power law Never logarithmic never and this the the signal that theory is I mean relate to you Ville is what I'm going to Okay form factor provide another window another view on the model because
39:21
Give you access to the exactest matrix of local operators So form factors are defined of a field I called psi By definition is this matrix element now
39:47
this Is I'm not losing generality Because any matrix element in which I have any number of particle here or even the point
40:02
at X so the field and X I can shift it with the Momentum and This particle are against a of the momentum. So this matrix element if I have X differ from this just by phases
40:23
On the other hand, I can cross any party on the left hand side. I can cross here by Crossing symmetry So what I want to say is that if I know this function precisely disorder vacuum this I have access to the full
40:41
glory Matrix element in any in any configurations now for the cinch Gordon this calculation has been done in full glory and Let me just sketch what it is Is like this
41:24
Okay, so let me tell you what it is. So an s-partics is like the form factor is like this Here is the vacuum is here the field here is the particle so this
41:40
Matrix element as a certain very strong constrain it for instance if I take two particle and I cross it This is gonna be there's matrix So this means if I take F Theta I theta I plus one and the rest on touch Is going to be related to the same function?
42:06
when I cross The two of them Okay, because each time I interchange for the relation s-martificial get a nice matrix So this property of this function is taking care by this function f min
42:21
Which satisfy this functional equation? It's easy to solve it. This function is an infinite number of pole and zeros in the complex plane
42:45
But I mean there's precise expression. We don't we don't need it. I just want to say that Everything once again depend on s-partics. I can find explicitly the solution. It's very simple by By Fourier transform, you see imagine that this matrix is a exponential I
43:05
That T over T FT cinch T that T over pi These are phases. I can always write like this F an F which satisfied that equation Is simple given
43:22
by these Fourier transform you can check immediately and
43:43
then if you make the The infinite product or a presentation you find infinite number of pole and zeros and so on so forth So I want to say this function is well known and determined by this Then what is this factor here? I told you that
44:00
Since Gordon do not have poles So you can never ever a configuration like this in which you take Two particles and go on shell and they become pole but what can have is I can take Three particles and make this configuration where here is the transmission of the s matrix
44:28
Now you see to do that two particles to be head-to-head So this is really the s matrix So this means that the only pole this amplitude can have is when all the rapidity
44:42
Different by any other by I pi Okay, so this kind of amplitude shall have Necessarily pole each time the rapidity difference end up to be I pi and this is taken care by
45:00
this term here In all possible channel, so this is product of this is elementary symmetric polynomial in X, okay so what is left out is a generic symmetric polynomial in X Which is not determined at all by the analytic structure of s matrix
45:25
This is what genuinely depend on the operators because so far I never tell you which operator I'm considering So summary the form factor is a structure Which is fully fixed by the s matrix alone. You cannot do anything about it
45:43
what is left out is a symmetric polynomial in X and different symmetric polynomial characterize different operators, okay, and in the past I've been computing it exactly and I've been found together with kubec an infinite number of solution on that because there are infinite number of operators with very
46:06
remarkable result in term of determinant of symmetric polynomial I'm not going to write for you and just tell you that This symmetric polynomial are Known you ask me any operator you want you say can I give me the form factor of phi square?
46:24
Yes, I can I just go take the formula and you for any number of particle and for any couplings to all order Okay Okay, now this is the things now let's go in more detail
46:40
Yeah, yeah, I am indeed I am going to this guy exactly now Indeed how you determine this? symmetric polynomial You determine exactly? Through the recursive equation That the form factor is to satisfy so this
47:07
n Particle form factor is related to n minus 2 precisely by residue equation
47:22
Which involve the s-matrix? So here is a pole you see there is one particle Here and there is a pole So this determined a recursive equation not going to write for you something like minus x x x 1 x n
47:42
Has to be a very well definite polynomial in symmetric polynomial x q n minus 2 x 1 x n You have to find Solution of this infinite number of recursive equation at once So this is why this form or determinant works because in a way
48:03
Encoded this recursive structure. Okay, and then where is the field? So so far once again is general the field you have to fix me some condition for you some phi square I define as the field which create is different from 0 and level 2 Phi for normal order I define the field which has 0 matrix element up to 4
48:25
So I I can classify the operator in this way And then I can define also vertex operator, which is much better because they find other condition which is called self clustering Alias I can take this potential by definition if you take derivative and this and that always remain the same
48:43
So it's a spatial operator after all I can do it now. Let not me enter in the films Now what I want to point out Leave out all the detail The thing is noticed I have a recursive equation which jump in two
49:02
So Let me denote it by this the number of particle and equal And for me is the number of external particle. So you see I have a way of
49:20
Relating form factor in this way or in this other way But I'm never able To have an operator which link even with odd number of particle They are completely decoupled So let's talk about stress-energy tensor, which is probably one of the most important field of the theory
49:43
For the stress-energy tensor. We know two things by basic stuff We know it's vacuum. So So it's the simplest form factor and this come from beta answers Because I can compute the free energy from that and the free energy is there and the expectation is
50:06
Pi M square the renormalized mass which will be my concern later Sinus the same function B and then what you know is the two particle form factor on a
50:22
On a general basis simply because the stress-energy tensor if you integrate is the energy of the theory This is fixed to be Uniquely this one The trace trace the stress-energy tensor altogether since satisfy conservation law
50:44
If I know the matrix element only the trace I can reconstruct all the other component just simply by conservation So it's enough to add that so you see general theory Allow me to fix a zero expectation value and two particle but never one party never
51:09
No, no, this is the point the point is The stress-energy tensor you can alter with charge at infinity so this is the point so if you have a
51:22
theory with Tim you know I Can alter the theory with Tim you know We for parameters, which I call charge at infinity Filthy I can have always this ambiguity toward
51:46
divergence, okay How about phi squared phi squared at least there's no problem. Oh, no, but there is no problem phi square I mean you have to tell me what you are calling phi square Phi square is the solution of this equation
52:01
which is on the two particle states is normalized to one or and Then on go to phi 4 and go no Yeah, yeah sure sure no no no no no no no no this is the point this is the point The only things where the theory has this ambiguity is the stress-energy tensor
52:24
All the other there is no way you can do it Also here when you add the author All the chain which come from the old term goes in its own way There is no way or relating the two chains. Okay, but when you do that the UV
52:44
Ultraviolet the theory change completely because once you have this TT 2 point function become 1 plus 6 q square divide z minus z 2
53:01
4 you can check using form factor, of course, and then the vertex operator become a different
53:21
we for delta of V alpha delta alpha So V alpha for me is a exponential alpha phi Now is alpha q minus alpha Okay, so what is the important lesson the important lesson is that
53:42
Lagrangian and s matrix can never fix up Okay, this is what I mean Is this much with the content of the theory and this is the problem so I have to take once again the Lagrangian
54:15
Wait, wait, wait This is this is a really wait
54:22
actually have to I Have to be precise. This is the effective So let's take once again the Lagrangian and now let me write like this and now I want to interpret this theory as
54:47
The formation of some UV field theory, but then I have a big ambiguity Because I can take this theory as Gaussian They form it by the symmetric z2 combination the vertex operator
55:05
this make a very precise commitment of the stress and the tensile and choosing central charge equals zero or I can take this as You be this is you will
55:21
But to make sense of you will as conform a field theory have to add charge at infinity Right because this operator now is to be dimension one They form it by this one So the stress and the tensor will change the central charge. Okay, or I can take this as you will and the form by that
55:43
So there is an intrinsic ambiguity what you are doing Have infinity many other possibilities like at some terms subtracted Each of these gives rise to the S matrix. No, this matrix is untouched Untouched because this term is completely invisible. This is a total derivative
56:02
This is the point this is precise the point This matrix which is infrared data is completely invisible 20 perturbation theory Okay It's not obvious maybe some of these non-perturbative definitions could give rise not to an integrable theory No, no, no, this this preserve as I say this preserve any integral be this is build up because it just
56:27
Admitting the presence of author in the stress and the tensor simply like that. This is completely compatible Doesn't affect all the chain of recursive Equation everything is fine
56:40
Now unfortunately, I'm running out of time So let me tell you Two amazing things which coming out from it so from Liouville You know that Liouville is a non compact theory and has a lot of Is a very complicated theory to build I mean to discuss it in particular there are no correspondence between
57:07
Operator and states as it happens in any other conformal field theory in particular are continuous set of states That correspond to vertex operator alpha, but alpha has to be q half
57:22
plus e the momentum p Now here is the point central charge effective is Remember that the Hamiltonian is L 0 plus L 0 bar minus C over 12
57:43
and then the dimension 1 over R in the So you see that if this L 0 again values, it's non-zero You have an effective central charge Which doesn't coincide the central charge, but there are minus 24 Delta mean
58:03
Okay, so if you consider the theory as sine go cinch Gordon saw C is 1 and this guy is 0 So you get exactly this a 1 minus this 1 but if you consider Liouville you get exactly same result
58:21
exactly because Q this alpha remember the formula Delta was alpha Q minus alpha So if you insert this as a minimum States in the theory you compute C effective C effective is 1 minus 24
58:43
P square the momentum of this particle and the momentum is quantized By what is called the Liouville reflection wall So you can take cinch Gordon like to Liouville One far apart of the other
59:01
So particle leaving here momentum P arrive here reflect back arrive here reflect back And so you get a quantization like SP square equal to one essential There is factor in our era which in the term in the story So you can compute P as a function of R and when you substitute you get exactly the same
59:24
so what I mean is Even from the final formula you cannot decide it's completely compatible with both interpretation even cinch Gordon has z2 theory no partial everything or as Liouville
59:41
Uncompact with a hidden sector with a lot of stuff behind The formula at the end is a completely the same, but since I promise you some mystery I have to tell you the mystery Unfortunately, I'm running out of time the mysteries you can compute for this theory cinch Gordon
01:00:00
Exact mass formula is an exact expression I'm going to write for you. Is four pi gamma one plus two plus two i g squared gamma one plus, don't be bored, it is what it is.
01:00:38
I cannot do much, you have to be pleased
01:00:41
that exists an exact formula, I mean, to all order. So this is the formula. Once again, you can check it particularly, but what is the mysterious of this formula? What does this formula define? Define the exact mass gap of the theory
01:01:00
as a function of the cutoff, which you put to define the theory, and as a function of the coupling, to all order in the coupling. Now, what is the starving of this formula? What is disturbing is that if you plot it to fix mu,
01:01:29
vanish and the cell dual point vanish. And so this is a pretty disturbing
01:01:41
because you don't know what the theory is below that. I mean, beyond that. Moreover, this formula is the only formula of the theory which is not cell dual. That's the respect cell duality. So here is the real problem.
01:02:01
What is the theory here for g squared larger than eight pi? What it is, what is behind the air-cooler column? What is behind it? Is that theory which is still massive or here become massless?
01:02:21
Now, I have to tell you little part of the problem and then I close. You remember that there was these two zero that move together when I move the coupling. When I write to the cell dual point, I can put them in the complex plane. I can just make an analytic continuation. I can compute the central charge effectively.
01:02:43
So I'm doing B equal one half plus I theta zero. So I'm putting here theta zero. Now, what happened is that if I plot the central charge like this, the central charge start to get staring case behavior with a quantized value
01:03:05
which match all the minimal models. So the value is one minus six p, p plus one with p integers. So the theory knows all this scale behind.
01:03:22
Amazing. Moreover, if I make the plot real and complex coupling, here is the sign going on the line and here is the column unbound. So here the theory is massless. Here is the cell dual point, eight pi.
01:03:41
Now, if you do the massless roaming, which is massless, the trajectory is just a semicircle like this where the theory is massless as well. So you see what is the puzzling guess. Here the theory is massless. Here is massless.
01:04:01
Well, I mean, you might imagine the scenario that altogether as a generic function of the coupling, the theory is massless wherever even here and the only massive case is here. Unfortunately, we have no way of checking this.
01:04:21
It's a technical point. It implies TSS, truncated conformal space approach. And once again, the origin is what kind of conformal filter are you taking in the UV? Because you have to choose a basis to make the calculation. And the fact that you have non-compact bosons
01:04:42
affect heavily the calculation. So we have been trying a couple of approach. All of them failed in a way or another. One failed trivially in the sense that we choose the basis of compact bosons to compute the matrix element of unbounded and the energy level were pretty, pretty bad.
01:05:02
So the energy level were something like, these are the lowest ones and there is always some highest line which cross all the other and make the story completely out of. Then we have using zero mode technique. I'll have to select out at zero mode
01:05:21
and construct excitation on it. He worked pretty well for small value of G, but he failed miserably when you go G square bigger than one half. Why that? Because you have to acknowledge the following things. Operator for expansion of cosh of 2G on two points
01:05:44
is kind of a trigonometric expression. You get something which goes minus 2G square and this is fine, but then you get cosh 2G phi Z1 minus Z2 2G square.
01:06:03
Since the theory has expectation value, this term become marginally irrelevant as soon as G square become bigger than one half. So you have to take this term in. But once you have this, you have to add the third term, the fourth term, the fifth term. So the theory just explode under your eyes.
01:06:23
And so it's kind of problematic and Monte Carlo doesn't help either for the simple fact that if you want to decide what the mass term here is, you know in Monte Carlo, you have to do the simulation but then you have to scale the lattice side to zero, you have to go to the critical point of the theory
01:06:41
to get the scale. But here means that the mass is already zero or you have to play with the cutoff to be finite because the easy answer from quantum field theory point of view is that look, the mass is not zero as far as you take mu goes to infinity.
01:07:00
Yeah, but in Monte Carlo, you have no, you define mu and then the theory get you what it is. So it is a question and moreover, the fact that the theory grows so fast, make the Monte Carlo worst and worst and worst because you can never span more than the vacuum.
01:07:22
Larger G you got, you're stuck to the vacuum. And so I mean, it's amazing frustration that you have a theory which you know essentially everything, but there are some basic disturbing things which are really at the moment impossible theory
01:07:42
to figure out. So this work has been done with Robert. I mean, we are working on it since long actually. And as I said, it's very challenging. So this is what I can offer at the moment. So just puzzling question
01:08:00
and the poor understanding we have so far. Okay, thank you. So the basic picture is that the naive duality that you had in the S matrix is not correct, you're saying? No, no, no, no.
01:08:20
Well, so you can work it out everything from duality for what S matrix is concerned, okay? So everything, I mean, you're never sensitive to the issue. Now you start be sensitive to the issue when you pose the problem, can I compute the actual mass gap
01:08:41
or the theory as a function, the cutoff and the coupling? And the answer is yes. This formula come once again from better answers, because from better answers, you have on one side the energy expressed in terms of mass, physical mass, on the other term expressed in terms of perturbation theory. You match the two and you extract the mass gap, okay?
01:09:04
But once you write to this formula, you can check once again, perturbativity that is working, you just compute the loops and this and that. But when you go to strong coupling, I mean, the theory vanish. There you, I mean, you don't know at least and never be able to go to extremely higher order.
01:09:22
So you see this and the formula is not self-dual. Is the only formula which is not self-dual. Now the way out, this is what Alyosha's homology which actually computed first in this formula was his way out to this. He said, well, at the end of the day,
01:09:42
mu is an arbitrary parameters. So if I go, g goes in one over g, I will define some new mu. So you see, I can write this as mu, some function g. This function is not self-dual, but then I can make this guy self-dual,
01:10:04
just define mu tilde such that he hold like this. After all, they say, these are arbitrary parameters. I'm not very satisfied with that. I mean, because since at a certain point you are changing the rule of the game. And moreover, the question remain,
01:10:21
is this really a critical point of the theory? I mean, critical point defined must be at zero or is something else? And from a roman trajectory, this is the picture. This point is directly connect to this by roman trajectory.
01:10:42
I don't know. I mean, it's really puzzling. I really don't know, honestly, I have no idea. So it's fair to say that for g less than eight pi, we are pretty confident that here is massive, everything works to the best you can do.
01:11:02
You have out of nothing the self-duality because it just pop up. The self-duality by this analytic continuation of sine gordo, you bite. Seems that in any formula workout, except when you ask the right question, what is the mass gap?
01:11:22
The mass gap is not self-dual. And then as all this property. Is there a lattice version of the field preserves in the- No, no. I mean, in the sense that once again, or better, there is, but it's not efficient as well. Because once again, the problem is that the both is not self, is uncompact.
01:11:41
This causes a lot of problem. This is the real origin of things. Final question. Is there a boundary version of the story that could be used some insight? Sure, there is. You work it out exactly. I didn't talk about the web. The web is exactly the same phenomenology.
01:12:02
Everything is self-dual except term which depend on the mass. In the boundary, there is similar story. There are web as well. Every finger, the mass once again pop up there. So you, I mean, you got exactly the same problem.
01:12:21
There is no way of opening a window and making progress on that. Exactly. So for instance, the web, any expectation value, any, of any operator as mass, the same mass
01:12:40
minus two alpha square time an exact function of the coupling. This guy is self-dual function. So if I do g, going one over g, this function is self-dual. But this term is not because it's exactly the same formula. And this web satisfied amazing property.
01:13:02
If I call g a, g a satisfy the reflection matrices of Liouville of q which depend on q times g q minus a.
01:13:21
And then you might say, but where the hell I'm telling you are using Liouville? I never, never. Because you see, the anomalous dimension is Gaussian. Very puzzling, really very, very mysterious. Very, very mysterious. Is it fair to rephrase this problem
01:13:40
as saying that you're looking for a renormalization scheme that preserves this self-duality? Because all these problems you're seeing is related to the cutoff you're imposing. It's not really a purely quantum field theory. It's not a continuing question, right? How the mass in the container is related with the cutoff.
01:14:01
What do you mean by this? I mean that all the formulas that only talk about quantum field theory observables, they are self-dual. Only when you express things in terms of a UV cutoff, you are seen to have some problems, so. No, there's no UV cutoff in the map. No, there is no.
01:14:21
There's no. No, no, no. This is a, no, cutoff, I mean, is mu, the one you define the bare mass. Yeah, but it's okay. Okay, if you want, yeah. Bare parameter, no. But it's bare parameter, nothing else. Nothing else. I mean, there's nothing. But it's not physical, so to know, that's what I'm saying.
01:14:41
Indeed, I mean, there is a formula itself. Give me your mu, I will define the, I mean, mass has to be expressed in terms of mu. The question is that formula, first formula, does it define the theory or not? Yeah. It does. Which formula? The first formula of the flash work. This one?
01:15:00
Is it the definition of a UV complete theory? Hey, I mean, this. Or is it just some platonic, you know, equation? Because there are some formulas of that type, for example, if the interaction were not exponential but polynomial. Yeah. Then, you know, provided that you normal order it, it's a definition of a UV complete theory
01:15:21
in a mathematical sense. Now, if that formula goes out of that framework, so perhaps there's a problem with that formula, perhaps that formula doesn't make sense. Yeah, I think, I mean, you can rephrase like this. As I said, if I give you this, you don't know what to interpret, because if you interpret, I'll reveal the format,
01:15:42
and by the way. But that's a separate question. Let's interpret in the simplest possible way. No, by simplest possible, no, simply possible way is that this theory, defined like this, a central charge equals zero. Because you are treating them, any expansion in G,
01:16:00
if you do perturbatively in G, is Z2 even. So essentially from the, if you want, my main message is theory, Lagrangian, probably in this case is particularly severe, other is simpler, are unable to fix the UV.
01:16:21
Is somehow a parameters free, even though the formula you can derive are not contradictory. Is the same expression that you can interpret in two different way, they are consistent both. Because the smart does not know anything about UV. So it's a basic things, is not simply to find contradiction.
01:16:41
You see what I mean? You say, well, if I did this, then the formula doesn't match, means no, no, no, everything work perfectly. The only way where you start thinking that some alarming, something's going on weird, is this commitment of this mass. Actually, I probably have to tell you
01:17:01
how this mass format come out. No, probably we have to wrap up and then discuss during lunch. So let's thank Giuseppe again. Thank you.