Application of Light-Front methods to model theories
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19:04
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Geometric quantizationMassRenormalizationElement (mathematics)Matrix (mathematics)DivisorDivergenceShift operatorLinear mapQuadratic equationLine (geometry)Point (geometry)Basis <Mathematik>VacuumEqualiser (mathematics)Many-sorted logicMultiplication signSquare numberExpected valueKörper <Algebra>CalculationState of matterNear-ringPoint (geometry)MereologyMassDivisorShift operatorDivergenceOrder (biology)Different (Kate Ryan album)Directed graphModulformElement (mathematics)Eigenvalues and eigenvectorsParameter (computer programming)Squeeze theoremQuantum field theoryWechselwirkungsbildNumerical analysisKopplungskonstanteExpressionFree groupTheoryInsertion lossLimit of a functionAnnihilator (ring theory)Term (mathematics)Hamiltonian (quantum mechanics)MomentumGroup representationOperator (mathematics)Heegaard splittingFunctional (mathematics)Complete metric spaceRight angleNegative numberSummierbarkeitModal logicLecture/ConferenceComputer animation
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Shift operatorDivergenceDuality (mathematics)Point (geometry)Quadratic equationLinear mapLine (geometry)Different (Kate Ryan album)Basis <Mathematik>DivisorDensity matrixLattice (group)Matrix (mathematics)Eigenvalues and eigenvectorsProduct (business)Equals signExtrapolationStandard errorMassInvariant (mathematics)State of matterMassMultiplication signShift operatorEqualiser (mathematics)Many-sorted logicCalculationShooting methodSlide rulePoint (geometry)ExpressionBasis <Mathematik>ResultantExtrapolationRegular graphInvariant (mathematics)Connected spaceCurveOrder (biology)PhysicalismApproximationSquare numberEquals signDirected graphLecture/Conference
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Transcript: English(auto-generated)
00:16
this talk would be an application of light-front methods, mostly to the 5-4 theory.
00:23
Overview. First, I will a little bit remind this light-front couple-cluster methods and application this to the 5-4 theory. Then, some light-front
00:40
Fock state expansion results using symmetrical polynomial basis function. Also, we're talking about this difference between light-front and equal-time critical coupling value. Then, some sector dependent calculations result and of
01:09
summary. So, 5-4 theory of just some quick reminder, of course, Lagrangian for the light-front 1 plus 1 Hamiltonian doesn't have this 1 plus 1, there is no
01:32
this kinetic term from the transverse direction, so we have just mass term and interaction term. For the light-front x plus equals 0, mode expansion would be
01:46
like this. Whereas, of course, this x minus is a light cone momentum and p plus is a conjugate, x minus is a longitudinal direction and p plus is a
02:01
light cone momentum. And, of course, creation, annihilation, operator, commutator. For the Hamiltonian, we have these four terms, kinetic energy term, 3 to 1,
02:20
annihilation of three particles creating one particle term, 1 to 3, annihilation of one, creation of three and, of course, 2 to 2. Now, application of this ILFCC to this 5-4 theory. So, we have this light-front eigenvalue
02:42
problem, whereas the light-front Hamiltonian applying to this eigenstate give us this mass squared expression. And without doing the Fock state truncation, we applying this ansatz, so whereas the state is
03:03
really application of this exponentiation of the operator t to this valence state. And these valence states usually include a very small amount of particle. So, for the 5-4, it would be really only, because we will consider
03:20
only odd sector, it would be really only one particle. It differs from this really origin of this coupled cluster methods from the nuclear physics and nuclear chemistry, whereas really valence states are much larger and really for them they did not create more particles, they just created
03:42
excitations. And this operator t is consists of all required quantum number, included light-front momentum, and of course increases particle number. So, if we really multiply both part of the eigenvalue problem by this e to the
04:06
minus t, we will obtain this Baker-Hausdorff expression of the light-front Hamiltonian on the left-hand side applied to the valence state, and at the right-hand side we have this expression and this guy really again
04:20
eigenstate psi. So, eigenvalue becomes this effective light-front Hamiltonian applied to the valence state, and so from both sides right now we have the simple valence state. And now we do projections, we project this on the
04:44
valence state, this equation, effective Hamiltonian, and we also project on the orthogonal to the valence state. And these second so-called auxiliary equations will determine us operator t and that functions in the operator t which
05:04
is really showing the distribution of the momentum and how much we will truncate, we really of course will use truncation right now. Up to this point everything exact, except of course higher Fock state wave function
05:23
connecting to the law of Fock state function through the sum, as soon as we do this truncation operator t, then this connection between allow state and higher state becoming already just some special way, but it's still
05:41
infinite amount of terms. But now we also truncate this projection, so for example if the valence state would be one particle, yeah, for the 5-force theory, here this, if we keep infinite amount of term and we consider only
06:00
odd sector, so it would be term 3, 5, 7, and so on, but because our real term t, we will truncate as many as we need, so if we have, I don't know, 5 function in the t-operator, we will truncate to the term 3, 5, 7, 9, 11. We of course will do much, just do the one equation. And yeah, this effective Hamiltonian
06:28
could be calculated from this Baker-Hausdorff expansion and it would be, of course, we don't keep infinite amount of term, we terminate this when this increase in the particle number matches the truncation of this projection. And in the previous talk, yeah, we talked about the spectator
06:46
dependence on cancel divergences, but we don't have here. Okay, so for the 5-force application of LFCC, the valence state, again, it would be just one particle, one boson state, and we take the simple contribution to the t-operator,
07:06
which is really 1, 2, 3, so, annihilation of one particle, creation of three, so it's really increase particle number by 2 and it's exactly what we want for the lowest order, because we wish, we don't want to
07:21
mix odd and even sectors. And we really will consider odd sector. Function T2 will be symmetric in its argument and this projection on this
07:40
auxiliary state, this would be truncated to just the state of the three bosons, because again, it should be odd sector and we only need one function, so we truncate to only one equation. So, so far you started with this light quantization and then you proposed an
08:03
answer for the vacuum, for the ground state, that's what you've done, right? No, really, we wish not to do a regular Fox-based truncation, so we're still truncating, but we're truncating on the way how this higher Fox state connected to the lower Fox state, but we still keep an infinite amount of the
08:24
force. But you decided, well, you started by deciding that you are quantizing your theory in the light form? Yes, of course, yeah, it's light form. Sure, sure, but did you say why you wanted to do that? Because I'm trying to follow, you know, you can propose an answer for the vacuum then, but... But it's not the vacuum, this is a massive state, the vacuum is trivial.
08:44
Well, I keep hearing that, I keep hearing that, but is that, can you say something about that, why is it trivial? In the light front? Because light front momentum, yeah, light front momentum P+, it's always positive. So we don't have,
09:17
we never have positive, we never have negative momentum, so you cannot
09:22
create vacuum from the, I don't know, the plus five and minus five, yeah, there is no such thing, it's always on the zeros. So the vacuum is empty state and that's it? Yes, of course, no, I mean... And now you're trying to populate it, so now you're trying to obtain... No, no,
09:41
valence states, these five states, it's not vacuum state, it's just this state with a minimum amount of particles. Right, but this is a matrix for what? You are trying to build a wave function? Yeah, we try to do 5-4, this out for space truncation. But you're calculating a
10:03
mass of states, not the vacuum, the vacuum is trivial, so just calculate the mass of states. The one particle excitation. One particle excitation, so the vacuum being trivial, you directly try to identify the excitable states for 5-4?
10:22
Yes, yeah, so again, because we truncated this projection only to the three states, we terminate this Baker-Hausdorff expansion for this effective Hamiltonian, also just one to three particles.
10:56
So creation, yeah, we only keep these terms really who only create two particles, no more. And this is really more efficient
11:08
way than instead to generate all these Baker-Hausdorff expansion terms, just right away, just to look what terms we really need.
11:22
Okay, so how valence state would look. So this kinetic term from the Hamiltonian add to this, and this is projection of the valence state, yeah, we will have only one particle state.
11:43
So it's only two terms contribute kinetic energy and three to one and T2 operators. This is again from the Baker-Hausdorff, because this guy minus two particles and this is plus two particles, so just redirect one to one.
12:01
An auxiliary equation, again, which is cut to the only three states, it has many more terms, but point is that every term just create two particles. Yeah, two particles here, nothing here, but these two particles here, again nothing here, and so on. So we have really only two equations.
12:27
The equations for one function, for this function, what is there known here? The T2 function. We don't know T2, we extract T2 from this auxiliary equation exactly to extract, to find T2.
12:45
And as soon as we know T2, yeah, we can solve our eigenvalue problem. We know T2, we can find physical mass.
13:03
Okay, so this is a valence state equation after integration, so we have this T2 function from the T operator, and it's really rescales. Again, these guys is a fraction of the longitudinal momentum, so we rescale this, and in order to express integral of this fraction.
13:30
And G is right now lambda over 4 pi mu squared, so it's dependent on the bare mass, yeah. Mu is the bare mass, capital M is the physical mass.
13:42
And this leads to the definition of this dimensional mass shift delta, which we'll talk about the meaning of this shift later. And it's, of course, through this shift exactly, you also could say this mass renormalization is, of course, it's physical mass, how physical mass connected with the bare mass.
14:08
Okay, and this auxiliary equation, it has, so again, this is from the 1, 2, 3 term, this is really wave function renormalization term.
14:31
And this is from some other terms in the Hamiltonian, again from the Baker-Hausdorff expression.
14:42
And here we can see this nonlinearity, yeah, T2 tilde squared, second order in T2, so this is so called loop correction, so this is exactly coming from this infinite amount of Fock state here. Because if we really expand this in the series, it would be infinite series over coupling G, starting from the first order in G.
15:12
So it's exactly how this LFCC method implements this correction from all high order state.
15:26
Okay, and without really showing result for this light front coupled cluster methods yet, right, we wish to talk about, just to compare with this Fock state truncation,
15:42
where again, we can see the odd sector and we just truncate up to three particles, so again, we keep one particle in three particle states. Then if we apply this light front eigenvalue problem in the 1 plus 1 direction, then we will have in
16:18
this case only two, it would be coupled integral equation depending on the how many Fock state we keep in here.
16:33
And this is really, yeah, it's already expressed in the ratio of the psi 3 and psi 1.
16:40
And we can right away see that this definitely does not have loop correction in this second equation, which is really coming from the three body sector. It definitely does not have a loop correction, which light front coupled cluster has. So it's simpler, does not include physics from the higher state.
17:03
And there's some description of the structure of the equation. First is the same as a variance equation. So this is exactly the same as the variance state and LFCC. But auxiliary equation is for the LFCC, it contains five terms, kinetic energy,
17:27
2-2-2, time-wave function harmonization, this vertically looks the most important. And in comparison, this Fock state truncation to up to three particles only, it has only three terms and does not have this loops correction.
17:42
Also, kinetic terms are not the same. In the LFCC, physical mass would be for all constituents. But for this Fock state truncation uses bare mass in three body sector. Unless, of course, we apply the sector-dependent renormalization. And I already told that,
18:01
of course, LFCC contains this vertically loops, which is partially summation of all orders. Now, in order to solve really both these Fock state truncated eigenvalue problem and LFCC equations, we use these fully symmetrical polynomials.
18:34
So this is multivariate polynomials of order n and x1 and x2.
18:43
The domain is this triangle for the three particles.
19:03
And of course, because of the conservation of momentum, I remember that this xi is the momentum fraction. So conservation of momentum means that they all sum to one. So we really have only two independent variables. Also, it could be more than one polynomial of the given order.
19:21
So we're using second subscript to differentiate this possibility. And again, of course, we don't take all to the infinity. We truncate to some order n. And for example, this t2 tilde function could be expressed through these polynomials and this coefficient, which we need to find.
19:41
And this is just for the convenience because equations both really LFCC and truncated, they all contain this factor. So we right away got read from this factor. OK, so and these polynomials, we did this work with our students.
20:07
So it could be shown that really all of them could be constructed from the second order. See, guys, right now it's monomials. And some general monomial could be constructed from the second and third order.
20:24
Whereas they look this way. And for example, six order polynomial could be made to two ways. Either second order cube to the cube or this third order polynomial squared.
20:42
And it's of course, we also really, by hand, using Gran Schmidt, we make them orthonormal. For this truncated case or LFCC case. And again, it's much better than, as again was mentioned in the previous talk, much better than TLCQ.
21:07
OK, so now if we take this life from coupled cluster methods auxiliary equation and project them on this basis function expressed through this symmetrical polynomials, we will obtain really matrix equation.
21:28
Whereas what we are looking for is this coefficients A and this delta is again the same shift. It's expressed also could be expressed through this matrices and coefficients.
21:45
I will show this matrices on the next slide. They are completely mostly computed by Gauss Legendre quadriches, which is, of course, if you remember, they give you exact answer up to some order.
22:00
And so really give exact answer for our needs. So this is a mostly really overlap of two polynomials. These matrices. All these matrices, all these numbers are finite. There's no divergence. No, no, no, everything fine. Yeah.
22:23
And now finally result. So again, we find we will find this coefficients A and we will find T2, this function T from the operator T.
22:41
And then we will put to this shift, which is connected physical mass and bare mass. And from here we can find again, we first fix some G and then we will find this ratio and then we take another G and find another ratio. OK, so here really it is out for the LFCC for space truncation up to the three particles and also sector dependent mass.
23:11
We don't talk yet, but it is out here anyway, and we can see that this is doing exactly this for space truncation up to three particles.
23:25
LFCC coming really approximately 1.5 for the G and sector dependent doing better closer to LFCC.
23:41
So how expensive is this to perform these calculations? For example, if you were to include one more term into this, the five particle term in LFCC, would it be feasible or is it becoming quickly?
24:01
LFCC would be because of this non-linearity, because there is this non-linear term T2 squared term. Yeah, it's difficult. So this is the reason why we really did not expand LFCC. We did a Fock state truncation, except instead of just to the three, we did up to nine.
24:20
I will show next. But yeah, LFCC, expensive but doable. So it's exactly really what would be my summary is that because LFCC really we will see later doing worse than truncation to, for example, term 579 state, Fock state truncation.
24:42
Exactly, because we included only one of these T operators, just the annihilation of one particle, creation of three particles, not really, just adding two particles. So be exactly thinking about adding another term to see how much better it will do.
25:08
So summary for LFCC at this point. So, again, it's models relatively simple, but require numerical techniques. In comparison, this Fock state truncation shows, introduce physical mass for kinetic energy terms without use of sector dependent renormalization.
25:30
And again, it's doable. Yeah, particularly using this symmetrical polynomials.
25:41
And these Fock states, this low Fock state truncation after the three particles definitely are doing worse. Now, what about extended Fock state expansion? So here for the odd sector, we will go up to the nine state and for the even sector up to the eight state.
26:03
So now eigenstate of this fully Hamiltonian disinteraction, of course, would be expressed through this Fock state. And so really every physical state will include an ideally infinite amount of the Fock state.
26:27
And of course, it's normalized. And now, just again, this eigenvalue, light-front eigenvalue problem just yield us this coupled system of equation where we really can see that
26:46
wave function of the sector M is connected with itself. Also, these are two up states and two down states, which is, of course, nice and surprising.
27:02
So it's preserve either odd sector or even sector. Okay, and for this kind of to do numeric here, we needed polynomials extended to the not just to the C body, but to the N body.
27:24
And right now, these polynomials, again, it's product of these monomials, whereas these powers, of course, summing up to N. And it started from the second power because, of course, first power because of the constraint is really equal to one.
27:48
Yeah, because again, domain. Okay, so now using these multivariate polynomials, we again obtain just matrix equation, whereas what we're looking for is this coefficient C.
28:03
And we have this different matrices here, like overlap of non-artogonal basis function in the given sector. And here we don't do grand schmid because it's produced so terrible a round-off error. So we really did this using single value decomposition is usually through this U
28:28
matrices and diagonal matrix, matrix of the eigenvectors of the B and diagonal matrix D. And also in U, we only kept column associated with eigenvalues above some positive threshold.
28:42
According to Wilson. And, of course, we can define some new coefficients and new matrices. So finally, we have this again ratio of the physical mass to bare mass squared relative to this critical coupling.
29:04
And this is, of course, interpolation. Again, without sector dependent really, we went only up to the seven particle sector and for the even to the eight.
29:23
And here we also plotting four times odd. Why? Because there is no binding state for the even sectors. Yeah. So really two odd will give us one even.
29:41
So this is some sort of like check and we can see really that even and four odds are relatively close to each other, except near critical coupling, which is not surprising. So here they intersect mass equals zero and a few different points, which is really taken as an error estimation.
30:02
And we take in your critical coupling is two point one plus minus five hundredths. How does this result compare with LFCC? LFCC was going to the one point five.
30:20
Yeah, it was going this way. I mean, a little bit higher than one point five. So this truncation up to the seven and up to the eight for even really doing better. I thought that if the LFCC mass is smaller, it means that it's a variational approach.
30:43
It means that LFCC is better. Is LFCC a variational approach? Yeah, LFCC giving you physical mass. Yes, for the constituents. But because we included only one T-operator, only with this sort, adding two particles T-operator, one, two, three, this is not enough.
31:05
But it's still a variational approach, right? Yes. Yeah, you're reading all this contribution from the high order. So what I'm wondering, my question is, I do a calculation, I see two values for the mass, one from this approach and one from LFCC.
31:22
And I see that the LFCC value is smaller. You mean for? Shouldn't I conclude that I should prefer LFCC wave function because since it's smaller and if the method is variational, whatever is smaller means that I'm doing a better job.
31:42
Why are you saying that this is better than LFCC, can you explain? The LFCC calculation there breaks down close to the critical coupling. Well, I don't understand, what does it mean it breaks down? It gives you some answer for the wave function. No, there are no solutions to the non-linear.
32:01
It's becoming, at this point it's becoming complex. Okay, but let's look at the last point which is not complex. It's lower than the Fock state truncation point. Does it mean that I should prefer the LFCC point or is there some other problem which I'm not aware of?
32:21
I would just add more terms to the LFCC. Why should I add more terms? It's already a calculation, it gave you a wave function. Normally if I'm doing, if there are two different groups which provide two different variational ansatz, I look at whichever ansatz gives a smaller energy and say well that group is doing better, period.
32:42
You can of course add more terms but even without adding more terms they are already doing better. Is this correct logic or not? If you jump ahead to the plot that combines light front coupled cluster and the... But I think this, without combining this question should have an answer.
33:00
Another one. There. So where we have everything light front and Fock space truncation and also sector dependent Fock state truncation. Ah, okay, here is okay. Okay. So, but this is the last point of the light front.
33:21
So up to this point it just, yeah, it's numerically just becoming complex numbers here. Okay, so it's not much different. Up to that point. Really the goal was really to obtain critical coupling value. We did not look for the best wave function, we looked for the critical coupling value,
33:44
so we were doing this Fock state truncation time higher and higher order in order to see this convergence and for example we can see that 5 and 7 almost identical and sector dependent required really 9, but...
34:01
Yeah, I understand, but this whole curve, not just the critical coupling, this whole curve is an interesting observable. We heard in the talk of Marco Serona that he can compute using barrier summation this whole curve. So this whole curve contains a lot of information, not just the critical coupling. So it would be interesting to know eventually not just the critical coupling but the whole curve as a general comment.
34:38
Okay, so in any case, so this light front extended Fock space truncated give us critical coupling at 2.1.
34:50
Now let's compare for the equal time. Again, we really have to do this rescaling because different people using different for the G.
35:03
So now it's all in this G bar and we can see that light front guys always has lower critical coupling than equal time. There is a systematic difference. And so this brought us to the idea...
35:22
I mean, we were talking with some patriarch of the light cone, Macias Burkhart. And in 1993, he did this work which connected bare mass of the light front with the equal time bare mass through these really tadpole contributions.
35:45
If calculated in the equal time. Again, light front, there is no vacuum to vacuum contribution in the light front. But in the equal time, there is this tadpole contributions.
36:04
And of course, clear why it couldn't be in the light front, because we go from zero to four particles and then from four particles to zero, which is impossible unless one of the light front momentum would be negative and we cannot have... Can I ask maybe a stupid question, but I confused a little bit about the sign.
36:22
Maybe it's just a question of convention. Why is the plus on this side of the equation and not on the other side of the equation? Because I would have thought that precisely because you're missing this on the light front, that you should be shifting the light front to give it the equal time and not the other around.
36:41
We better question directly to the mass science, which is much higher. There are two minus signs in there actually. This represents the negative of the tadpole, so... Oh, okay. Maybe there are some. Yeah, at some point it would be... You're missing a piece that's positive. These vacuum expectation values are proportional to the tadpole contribution.
37:03
That's Matias' work to show that correspondence and that's where the minus sign you're worried about. It would be negative. Yeah, it would be negative. So it would add exactly because after adding this stuff to the light front, we will come approximately close to the equal time. Oh, so basically the way that you've defined some convention,
37:21
this thing is actually negative. Yes, yeah, yeah, yeah, yeah. We all agree. Of course. Yeah, yeah, no. We would not present this if it would be... But these verbs are calculated in which theory? A equal time... A equal time. Matias Burkert, he did this in equal time and we did this in the light front.
37:45
And this is some sort of like the... So for the equal time as the tadpoles, for the light front it's C2, P3...
38:02
No. From 3 to 1. So some sort of agreement that it would be the same in both equal time and light front. So equal time... So he was calculating these guys, we were calculating these guys.
38:24
We together. So now the question was how to calculate this. So this is a vacuum expectation value of the square of the field in the interaction, full theory of the interaction and this is in the free theory.
38:42
So in order to calculate this, like in any quantum field theory book, we're doing this point splitting. So we're shifting light from time and light from momentum by the epsilon. So we separate from the zero. And we're also inserting using fullness, completeness of the eigenstate of the Hamiltonian.
39:04
We insert this one here inside. And so now we have to calculate two matrix elements, right one and left one. And of course, adjoint representation for this. Operator shifted by this in this adjoint representation.
39:23
So right, this sandwich could be calculated and of course left also could be calculated. And for the free state. And by the way, because here we have only time creation and annihilation operator.
39:45
We really, because remember this is eigenstate of the full Hamiltonian. So each eigenstate including all infinite number of the Fock state. But it's really for each state only one particle will contribute because of course here only one operator.
40:06
In the five field. And for the one particle free state, because really it would be right now free theory, free Hamiltonian. So we really just have this annihilation operator here and for the left creation operator here.
40:25
So it all could be calculated. And this final expression for these two sandwiches. And we also can insert in this free guy just to have the same.
40:41
Because again, our free theory really can't each only contain one body state. We now can really come to representation of the modified Bessel functions for this difference. And argument, of course, has to go to zero.
41:01
So when it's go to zero, it's really have this logarithmic form and also this earlier factor. And so really this difference equal to this sum. And as you can see, it's negative. Whereas really probability of one body state in every eigenstate.
41:25
And this is, of course, would be logarithmically divergent if we are approaching critical coupling when physical mass goes to zero. And this is how we define this shift. So shift really becomes positive because we're putting this negative sign here.
41:44
So we either can say that light front is equal time minus term shift. Or we can express this as a ratio, or we can express this as a coupling constant for the equal time through the light front coupling constant. Or we can express this as a ratio of the physical and bare masses for the equal time through this ratio in the light front.
42:09
Whatever necessary. Just to see that there is some convergence, we looked at the relative probabilities for this odd sector. Again, it's only up to seven body. But what we see right away is that each next sector becoming ten times, at least order, smaller than previous one.
42:32
So there is convergence, definitely. Contribution from the higher sector, they becoming less and less probable. So this is a good point. But the bad thing is that near critical coupling, 2.1, there is no kick, because there has to be divergence there.
42:50
Because again, physical mass goes to zero. But this guy doesn't go to zero, unfortunately. So we don't see this kick.
43:00
We don't see what? Growth, we would expect that probability near critical coupling, yes, somewhere here, has to suddenly increase. Particular, no, near critical coupling, higher Fock state we expecting would be more and more important.
43:22
But delta should remain finite, even at the critical coupling? Right. Yes, but it's unfortunately, they will, yeah, because when this goes to zero, this guy has to go to zero. And in our calculation, it didn't go to zero, did not go enough to zero.
43:43
So this is the reason why we started to do sector dependent, hoping that invariant mass, I mean, higher Fock state will not be suppressed by this invariant mass. So what is your interpretation of this? I mean, your interpretation is that you just need more states.
44:03
Is that you need higher particle number states to be able to actually perform this procedure. The problem is that you just don't have enough particles. We see that in this regular Fock state truncation, because invariant mass still quite high,
44:21
so this higher Fock state, they still suppressed. Their contribution of this higher, more excited Fock state has to grow near critical coupling. It has to become more and more important, you would expect near critical coupling. And it means that really probability of the one body state has to go to zero.
44:48
I mean, it has to become smaller and smaller near critical coupling. And we don't see this, unfortunately. And so, but from convergence point, at least, this plot of probability, some
45:02
sort of encouragement that each higher Fock sector becomes less and less probable. And now this is a plot of shift relative to the critical coupling, whereas the points obtain extrapolation in the basis side. And these two curves, this is a linear and this is a quadratic fit, which is really view constructed only up to G equal.
45:26
I mean, taking data from up to critical coupling equal one. And I will explain why this into the next slide. But in any case, shift is relatively good because when we add this shift to our light front mass critical coupling,
45:46
we have equal time critical, approximately. So coupling or do you have an agreement in the full mass curve? Because also the full mass curve has to be. Mass, this ratio of physical mass to equal time mass, we can see that there is a problem arise after G equal one.
46:09
So this is the reason why we only used instead to go down, down, it started to grow. And this is the reason why we used for this shift calculation only point up to G equal one.
46:26
But before G equals one, you get good agreement. Yes. Yeah. So this is, again, the suppression of this higher Fock state. They don't exhibit themselves well enough because it's still a probability gain for the one body sector still stay enough high,
46:48
does not allow them to expand to give more probability for the higher Fock state. I don't understand this point, because the M squared has to go to zero and here it goes to 0.84 and then starts shooting up.
47:05
Yes. Yes. Yes. So this is we consider this is why, for example, Fock state truncation is a bad thing to do. Or we have to include much more many state, not just time seven, nine, time eight, but more. Before you showed the plot where the mass was going through zero.
47:25
Before, previously, you had a plot, before you started discussing this mass change, you had a plot which had the mass reaching zero. Yeah, but it was a light front. Yeah. And now it's equal time.
47:40
This is using your correction. This is divided by the bare mass of the light front. So the correction between equal time and light front is diverging once you get to near the critical coupling. And so the connection is breaking down. That's why this swings up here at the larger G. So this is a bare mass for equal time.
48:01
Well, this is straight. We have some unpublished results about how this behaves when you compute the correction in equal time. And there everything's fine. Yeah, there's clearly something wrong in the light front calculation near the critical coupling. That's the bad news in this.
48:23
So because of these. Yeah, so again, this probability of the one body sector in the area, some sort of lower state doesn't go to zero as we wish.
48:40
So then this expression, which is really expression for the shift will diverge. But if we use points up to G equal one to estimate shift, then we would obtain this kind of value. And from there, which is love equal time value for the critical coupling
49:08
with some sort of relatively close to each other, keeping in mind this error. Again, if we don't use our data too close to the critical coupling, which is 2.1.
49:27
And this is why we tried to use this sector dependent scheme in order again to avoid this because sector dependent in the higher.
49:44
So if we truncate up to the three particles, you will have only one body and three bodies. So there is a self energy in the one body, but there is no self energy in the three body sector. And because of this, I mean, every constituent will have physical mass here.
50:07
So it means that higher fork state is not suppressed by this invariant mass invariant mass. Again, this is really a result of the application of the kinetic energy operator on the state. This is what is so-called invariant mass.
50:23
So I will not go to the detail of the sector dependent calculation. In any case, you're given physical masses for this higher state, whatever it am up to three particle or if it's really did this up to nine particles and each previous for that particular truncation.
50:42
We go from here, no calculate their masses. We just use some special scheme from the sector dependent calculation. I think there is this plot is the result. This is a new scheme sector dependent scheme, which supposed to give more room for this higher fork states.
51:02
And again, because convergence is slower here, you have to go for the old case up to the nine body sector. And really, this is a good thing. I mean, this is what is expected. And again, estimation for the critical coupling about 2.1.
51:24
They really only see some sort of a little bit different, but five, seven and nine relatively agree with each other. And now this is a combined result, whereas this open symbols, it just folks, they truncation from three to seven.
51:44
And then close them. Dark figures. It's the sector dependent scheme up to the nine truncation and also LFCC result for this critical coupling. And what is there could be extracted from this.
52:10
So the sector dependent and standard folks, they truncation, they some sort of agree with each other and critical coupling from both cases come into approximately 2.1.
52:22
Sector dependent converge more slowly, and this is really expected. Because we expect that higher fork states should become more important, so it doesn't converge so quickly.
52:41
And also, yes, there is this plot of these probabilities. Again, we were hoping maybe here we will have this increasing of the probabilities near critical coupling again. But at least we see that sector dependent probability is higher, which is again dark figures.
53:03
Higher than the regular space for space truncation. Except, of course, when we go to the highest state. They almost the same, but we still don't see these a growth of probabilities near critical coupling and some sort of summary out of this.
53:33
There is convergence. We definitely see that probability for the highest state is smaller than probability for the lowest state.
53:45
And that also this sector dependent probabilities are higher than regular fork state truncation probabilities, which is again good indication that in the sector dependent approach, higher fork state becoming more important.
54:02
But we expected more rapid increase near critical coupling. We did not see this. So this hypothesis that this high invariant mass suppressing this higher fork state really was some sort of incorrect. So we wanted to go to some coherent state approach or to add another term in the LFCC.
54:28
Just hope it will resolve this issue to see this growth really of this higher fork states near critical coupling. Thank you for your attention.
54:45
Time for questions. I did not understand. It seems to me you were saying that you are expecting to see that as you are approaching the critical coupling, the higher occupation numbers will play more and more important role.
55:09
And you don't see it. Is this a message that I should take? This you can see there is a problem. There is something wrong here because of this.
55:21
Because we expect critical coupling is exactly in all these phase transitions of this higher fork state that has to play more important role. So their probability has to be higher. In equal time this happens. So something wrong with the light front, something we don't take into account.
55:44
One slide we forgot to add is that LFCC does show an increase. But because the calculation breaks down before you reach the critical coupling, it's not clear. Yeah, this was another positive point about LFCC. I just didn't put the slide. LFCC probability would grow, only they started to grow somewhere.
56:07
LFCC we stopped some sort of 1.5, so they grow near this guy. So really we just wanted to add more term to LFCC and see what happened. Because maybe exactly the main point is exactly truncation.
56:22
And LFCC truncated, but just truncation of the way, not the truncation in the states, number of states. One possible problem with the LFCC calculation is that we start with valence state, which is just the one particle state. And that kind of implies you're keeping that as too important. You probably need to expand the valence sector to include more states so that the one particle state can disappear.
56:48
It can't disappear completely from the current onsos because that destroys everything. But if you include a more complicated valence sector, we'd have more freedom. And then presumably we could reach the critical coupling calculation.
57:04
That's one of the things you want to try. Any more questions from the two or three people in the room who actually did do light front? Perhaps some questions? Well, then let's postpone questions to lunch and thank Sophia again.