On the enumerative structures in QFT
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Struktur <Mathematik>QuantenfeldtheoriePhysikalische TheorieKörper <Algebra>Algebraische StrukturLipschitz-StetigkeitRelativistische QuantenfeldtheorieStatistikMathematikYukawa-PotenzialFormation <Mathematik>Physikalische TheorieYukawa-PotenzialDiagrammResultantePotenzreiheAsymptotikKategorie <Mathematik>EreignishorizontKartesische KoordinatenMereologieEinfach zusammenhängender RaumComputeranimation
01:07
Divergente ReiheDiagrammLineare AbbildungGruppendarstellungTermZusammenhängender GraphParametersystemWärmeausdehnungEinfach zusammenhängender RaumNichtlineares GleichungssystemExplorative DatenanalyseFunktionalRelation <Mathematik>Aussage <Mathematik>DiagrammKreisflächeEinfach zusammenhängender RaumKnotenmengeNumerische MathematikArithmetischer AusdruckBeweistheorieResultanteAsymptotikPotenzreiheRechter WinkelNichtlineares GleichungssystemParametersystemÄußere Algebra eines ModulsOrdnung <Mathematik>MengenlehreDerivation <Algebra>RelativitätstheorieMultiplikationsoperatorFunktionalMatchingKlassische PhysikTermWärmeausdehnungWurzel <Mathematik>Minkowski-MetrikNichtunterscheidbarkeitDivergente ReiheStatistische HypotheseComputeranimation
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Relation <Mathematik>Aussage <Mathematik>FunktionalDiagrammEinfach zusammenhängender RaumDivergenz <Vektoranalysis>Leistung <Physik>Divergente ReiheZahlensystemZahlentheorieFunktion <Mathematik>GammafunktionOffene MengeStandardabweichungCharakteristisches PolynomDerivation <Algebra>KoeffizientFokalpunktPhysikalische TheorieAsymptoteAnalysisNichtlineares GleichungssystemFolge <Mathematik>TermWärmeausdehnungYukawa-PotenzialSingularität <Mathematik>Graphische DarstellungLoopKonvexe HülleTheoremGleichheitszeichenGammafunktionReelle ZahlPotenzreiheGreen-FunktionDerivation <Algebra>Divergenz <Vektoranalysis>ResultanteKoeffizientGeradeMengenlehreNichtlineares GleichungssystemLeistung <Physik>TermFaktor <Algebra>Singularität <Mathematik>MereologieAnalysisEinfacher RingParametersystemFormation <Mathematik>Numerische MathematikGraphische DarstellungWärmeausdehnungDivergente ReiheZahlensystemArithmetischer AusdruckBetafunktionJensen-MaßBeweistheorieStatistikMultiplikationsoperatorEinsDiagrammEinfach zusammenhängender RaumLoopTabelleKurveApproximationPhysikalische TheorieQuantenfeldtheorieRelativitätstheorieEigentliche AbbildungVakuumBijektionYukawa-PotenzialUmkehrung <Mathematik>Computeranimation
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TheoremLoopYukawa-PotenzialGraphische DarstellungEinfach zusammenhängender RaumDiagrammGleichheitszeichenWurzel <Mathematik>Einfach zusammenhängender RaumWurzel <Mathematik>VakuumDiagrammKnotenmengeProzess <Physik>AsymptotikStruktur <Mathematik>ResultanteRechter WinkelMultiplikationsoperatorMereologieGeradeHelmholtz-ZerlegungNumerische MathematikLoopPhysikalische TheorieGraphische DarstellungOrdnung <Mathematik>Divergente ReiheGrothendieck-TopologieTermRelativitätstheorieTheoremFormation <Mathematik>Algebraische StrukturArithmetisches MittelBijektionTeilmengeWald <Graphentheorie>Gerade ZahlGreen-FunktionFunktionalZusammenhängender GraphMultifunktionSingularität <Mathematik>Lemma <Logik>Stochastische AbhängigkeitDreiecksfreier GraphEindeutigkeitKlasse <Mathematik>Projektive EbeneEinsRichtungDifferenzengleichungNichtlineares GleichungssystemZahlensystemZählenAnalysisStatistikPolstelleSchnitt <Mathematik>TourenplanungComputeranimation
Transkript: Englisch(automatisch erzeugt)
00:16
Thank you Lori and thank you Karen, Dirk and Eric for organizing the event.
00:21
So I will be talking about more combinatorial results. So at the end we should be seeing an application for chord diagrams in interpreting Feynman diagrams in Yukawa theory and quenched QED
00:42
and in a way that will help us to evaluate or estimate the asymptotic behavior of some power series. So let's see. So the first part of the talk will be just recovering what is chord diagrams and some properties of two connected chord diagrams and then we'll go
01:05
into the applications there. So probably you have seen this earlier in the conference but let's just start by defining what is a chord diagram. Imagine you have a circle, you have
01:23
an even number of vertices on it and you have some connections between pairs of vertices. We will be considering rooted chord diagrams. So imagine again the same circuit but this time you will have a distinguished vertex.
01:47
So this will give you a way to represent chord diagrams linearly like that. These spacings are going to be called intervals. The generating series for rooted chord diagrams
02:02
will be denoted by d of x and it's clearly seen that the number of rooted chord diagrams on n chords is 2n minus 1 double factorial just like matchings. If we are going to talk about connected chord diagrams this means something like that where you can't isolate pieces away
02:27
from each other. So for example this one is not connected because we can isolate this piece. It's not hooked with any other chord. One of the classic relations
02:45
between connected chord diagrams and general rooted chord diagrams is this one. You can see where this comes from. So a rooted chord diagram is either empty so you have the one or you can somehow pull up whatever connect whatever is the largest
03:06
connected piece you can pull and next to each vertex in it you will have from the right you will have a chord diagram and from the left you will have another chord diagram. That's why you get the x d of x squared. Another useful identity which is also easily obtained
03:26
is this one. These two equations are useful when you want to study the asymptotic behavior of connected chord diagrams and that's what was done in the work of Michi.
03:43
Especially this equation if we apply the suitable alien derivative to both sides we eventually get the desired asymptotic information for c of x. And that's what we will be interested in doing for two connected chord diagrams. So what should
04:08
be a two connected chord diagrams? You probably can guess. So it will be a chord diagram which in order to get this connected you have to remove at least two chords. So if you
04:25
if you wish to define it formally it will be you have no set of no set s of consecutive end points where with proper size which is paired with the actually this definition is for k connected which is paired with less than k end points out of s. So for example the diagram here is
04:49
is two connected but it's not three connected because we have we can disconnect the diagram by removing these two bold chords but we can't disconnect it with less than that.
05:07
The generating function for two connected chord diagrams will be denoted by c greater than or equal to two. And next we'll be studying the asymptotic behavior of this
05:21
power series. In an older result by Kleitman he gave an argument that there is a proportion for k connected chord diagrams with respect to the to all chord connected chord diagrams is approaching e to the minus k as m goes large.
05:47
We'll be able to extend this result for the case of two connected chord diagrams. This means k is two and we'll see that this value here is going to correspond only to the first
06:04
term in an infinite expansion. So again in the work of Nishi he obtained this value for the alien derivative with parameters two and a half applied to x applied to c of x
06:22
and used that to get the asymptotic information. So the first question do we have an alternative for this relation here? I mean we can dream of something like c of x equaling some expression involving c greater than or equal to two of x and then apply to something
06:46
else so that we can do the same trick. We apply alien derivatives and get the asymptotic information. So that will be the first thing we do. We want something like that involving two connected chord diagrams and that's how it's going to look like.
07:09
Connected chord diagrams can be expressed like this. Pairs of connected chord diagrams divided by x minus two connected chord diagrams applied to pairs of connected chord diagrams again.
07:26
I'm not giving the proof of this result here but you can find it in the PhD thesis. So now what we are going to do is, as I said, we are going to apply alien derivatives
07:42
to this equation to get the desired information for c greater than or equal to two. And our way for doing that is through factorial diversion parses. I guess you already know the definition of factorial diversion parses in the previous talks
08:03
but let's see it again. So a power series is going to be said to be a factorial diversion parses with four parameters alpha and beta if we can express the coefficients of it like that.
08:22
So we have, you can consider this modified gamma functions as expressing the divergence of the factorial. That's why it's called factorial diversion parses and these coefficients are just real numbers. So and the modified gamma functions are for these parameters are just defined through
08:47
the original gamma functions in this way. So our passage from such a such a coefficient to a real to an ordinary power series is going to be through extracting these coefficients
09:05
and that's what the alien derivatives does. An alien derivative is going to be applied for a factorial diversion parses to give you an ordinary power series whose coefficients are these real numbers that were besides the gamma functions. You can think of that as we have
09:24
just excluded the factorial divergence and we are concerned with the remaining information about the series. Turns out this is a derivation as Michi proves in his work and the ring of
09:41
such series is a subring of the ring of power series. We'll call these things alien derivatives and again we have this. So let's apply it to, let's apply the the suitable alien
10:01
derivative to the equation involving two connected core diagrams. We eventually get this expression after some work and after using Lagrange inversion in the middle. Eventually we get that and if we simplify it we we don't have to go into the details here but at the end we are
10:21
going to express the nth coefficient of two connected core diagrams like that and here you can see that Kleitman's result is standing for the first term just the first term here but now we can get better approximations as much as we can by truncating these series.
10:47
So let's see how useful this will become in working in QFT. Just pay attention that we didn't need any singularity analysis to obtain this
11:01
expression, this expansion and you will see that the same happens for many series from Yukawa theory and quenched QVD. So I will start by Yukawa theory.
11:21
So again in the work of Michi he has this table. Let me go quickly through the notation here. So these are the proper green functions of zero-dimensional Yukawa theory and if we have for example the first line is the proper green
11:42
function for graphs with no external legs at all. These are the vacuum diagrams. The second one here is we have one boson edge, one external boson edge and
12:04
zero fermion edges and so on. So these are the proper green functions and we can readily notice here that the coefficients for this one for example which is the most apparent one are exactly the number of connected core diagrams. These are the tadpoles of Yukawa theory.
12:25
So let's see how to prove a bijection between tadpoles and connected core diagrams and that's the main result I want to talk about here and from which all of these other lines will follow. So again for the sake of an example
12:49
the coefficient of h bar to the four in this power series which in its original setting counts tadpole graphs with loop number four is 27. This coefficient is 27 and that's exactly the
13:07
number of these tadpoles but at the same time it's the number of connected core diagrams with four cores. So why is this relation happening? How can we jump from such a tadpole into a
13:21
connected core diagram and for example for these more complicated ones how can we express them in terms of connected core diagrams and would that be useful at all? So that's what we are going to prove that the nth coefficient here which stands for one pi tadpole graphs with loop
13:45
number n is exactly the number of connected core diagrams on n-cores. Let's see the proof. So what we need to do is because we don't see the relation yet we don't see how to move from
14:02
here to the from tadpoles to connected core diagrams so at least we are going to try proving that tadpoles satisfy the same recurrence as connected core diagrams and that's this equation which we have given in the beginning for connected core diagrams. We want to see that
14:21
t of x is going to satisfy that. The first step is to make a stand-up way for drawing tadpoles so like that so remember in this theory these two tadpoles are considered different
14:41
because we have different directions for these for these loops but in our notation let's agree that we are going to forget about the directions but we are going to pay the price we are going to pay the price so for example this one becomes
15:01
that one but this one just for the sake of fixing a direction counterclockwise we are going to flip it like that and we'll have we'll consider we are having this overlapping here by this way we can ignore talking about the directions because they have been encoded
15:23
in another way. Also for the sake of more notation we have so for example this vertex which is the root vertex for the tadpole is going to be denoted by v t
15:40
where t is the tadpole this is the r t edge this is boson edge boson edges are always going to be denoted by a light line and the fermion edges are more bold than it. If we have a vertex b the the the fermion loop it stands on is going to be denoted by loop of b
16:06
and the next edge the next fermion edge counterclockwise to a vertex is the fermion and similarly the boson of it because it's unique. So let's see how to move on
16:24
Big U 1 0 is the it's just the notation from the class of the tadpoles and a first lemma is to prove that the number of independent cycles in such a tadpole or the loop number is the same as the
16:45
number of all boson edges so that's our first step the number of the loop number is exactly the number of the boson edges so we now have to get the bijection between the tadpole and connected chord diagrams taking into account this fact so that's our algorithm to move from
17:10
so again let me let me just remind you we want to prove this relation and this combinatorially this means we want to take we want to take a tadpole
17:24
and another tadpole with a with a distinguished vertex and we prove that from this one we can uniquely get either either for the plus we can either get two tadpoles in a unique way or one tadpole and also in a unique way so let's see how we
17:45
can do that so we start with a tadpole t1 and the tadpole t2 with a with a special vertex or a special fermion edge because they are the same they come the same thing
18:01
and we have we have this one in life so if the if the special vertex is exactly the is exactly the root we'll just return the two tadpoles by forgetting this extra information here if it is not we are going to do the following let me explain that on an example
18:25
it will be more clear so let's see here we have these two tadpoles and we wish to get and this one have an extra information which is this d over here now let's see how we can merge them to get a unique tadpole
18:46
so first of all we notice that d is not u2 so we are in the second case so we are going to do the following we take this whole diagram here and we insert it
19:08
in here in a special way like that since we don't have any internal edges it will follow the following scheme we are going to just put v1 next to d
19:23
over here and we and next we are going to place u2 next to v1 which we have just placed and the root is going to be at v1 it sounds strange but that's how you this way is unique because that's the only case where you you get a two connected graph
19:48
if you so if reversely you you are going to move backwards let's see this later sorry for that but this example will be more clear in representing the
20:03
idea so here we have two bigger tadpoles our algorithm is going to be like that first of all uh we so we move from v1 uh we determine what is the next vertex to it it's w we detach w
20:22
and then we insert w next to d and we take v2 we take u2 and we place it just next to v1 now if we wish to see how we how this produces a unique tadpole let's try to move backwards
20:41
um what i am going to do is i will start by the root that's the only information i know in a tadpole i will check the first vertex next to it counterclockwise and i will detach it i will i i will imagine i cut this edge now the the resulting graph is either
21:04
one connected or it is two connected if it is two connected this this happens in the only case where we have the first tadpole empty from inside like this one because in the second case we always have a bridge somewhere and and that's how we what we mark our way back so
21:24
if i am going to start from this tadpole again i will start by the root the only from information i know i move counterclockwise the count is the first vertex i meet i detach then i check if the graph has bridges this one for example has bridges it has this bridge it
21:44
has this bridge so i will determine one bridge and that's possible uh in polynomial time uh yeah i will determine one bridge and then again i will check the the remaining graph which contains the the root i will again check if it has more bridges and i will keep
22:04
repeating this process searching for bridges till i find the last bridge and this will mark the end of the algorithm i'll just detach there and i will put the w again next to the root of the second that hole and i will recover in the other case if it's too connected i will know i have
22:25
this situation where there is a simple tadpole and a more complicated tadpole and to recover them i just need to remove uh and i just need to detach the vertex next to the root like that and then i will uh i will remove the the root forever and create a simple
22:47
diagram um yeah i actually i have the drawings here so like here i start by this one i detach the first vertex like that i search for bridges imagine i determine this bridge first
23:05
then i will keep searching for more bridges here i find b1 and if i search again for bridges i will not find any so that's the last bridge i detach it like that and now i have the uh
23:20
the t2 is it's the extra information in here that's where i detach w and then all i have to do is to place w next to the root which is the only information i i ever know in a tadpole so that's how we prove that the recurrence is satisfied by tadpoles but can we have some
23:43
means of jumping directly from connected core diagrams to tadpoles and vice versa yes and that's that's what we do in this theory um so uh it it only takes the advantage of the root
24:02
shape decomposition and the map psi which we have which is some kind of an order defined of the tadpoles so so let me let me explain this on this diagram here so if i start with
24:20
a tadpole like this one please ignore these red numbers because they should be information you get at the end if i start by such a tadpole i just apply the uh the function big psi we have defined in the algorithm i split it into two with a distinguished
24:41
vertex in the second tadpole and then i keep doing that all the time till the end and these are just corresponding to simple core diagrams one core one core one core then i will move backwards again using the core diagrams and the extra information i had here and my way in
25:02
inserting the core diagrams is by reversing the the work of the ratio of the root shape decomposition and eventually i get there that's the uh that's the corresponding core diagram
25:20
in the middle of doing that we need an order on the vertices of the tadpole and that's what i that's what i have defined uh using this map psi but i i will i don't have the time to uh to write it down here but it is simple it's just imitating the root share decomposition
25:42
on tadpoles making use of the internal structures that we have discovered here so that's it we have tadpoles corresponding to connected core diagrams it follows from that that the vacuum diagrams can be expressed like this pairs of connected core diagrams of 2x
26:05
if we have two external boson edges it will be expressed like that and here we see that two connected core diagrams come into the picture you can see it here if we have two boson edges like that first of all i will move from this
26:27
fixed vertex here i will search for the last bridge uh actually i will first detach this vertex and i will search for the last bridge which in this case is this one similarly from starting from here i will detach this one and i will search for the first
26:45
bridge which is this one in that case and these will mark where i should cut i will get that thing and this thing which again is going to be isolated like that so you get one of these and two tadpoles and we can express this one also in terms of connected core
27:08
diagrams uh that's that's how we can express it it turns out that it is expressible in terms of two connected um i don't think i i have the time for the other uh for the other parts but
27:25
just to say all the lines uh or all the uh the graphs with the different external structure turned out to be expressible in terms of connected and two connected core diagrams and they all follow more more or less they follow from the uh the result of connecting
27:45
four diagrams and tadpoles what is the advantage of doing that we can get the asymptotic behavior of such a green function without singularity analysis because we already have we already know everything asymptotically about connected and
28:03
two connected core diagrams um a similar result is for uh quenched qed uh i could prove that the counter terms uh these special counter terms of quenched qed
28:24
are counted by two connected core diagrams as well um and that's it we see advantage of all of that is to evaluate the asymptotic behavior more easily thank you thank you
28:46
all right are there questions for ali david has his hand up yes congratulations on this wonderful bijection so i have an obvious question in unquench qed what happens if you
29:00
restrict your tadpoles by forest theorem to those in which fermion loops contain only an odd number of vertices what can you say about the subset of core diagrams that arises from your bijection um so uh you mean you are allowing fermion loops to exist in uh in that setting you mean uh no i i imposed for his theorem
29:24
you went from all tadpoles to all chord diagrams but your some of your tadpoles would not exist in qed because on your fermion loops you had both odd and even number of vertices so if i restrict it by forest theorem that says that we only have an even number
29:41
of vertices on a fermion loop then that identifies by your bijection a subset of chord diagrams so what is what is the character how on the chord diagram side do you describe that subset of tadpoles which satisfies forest theorem um yeah but it's not necessary that
30:02
we have an even number of vertices on these tadpoles it can be uh arbitrary if i if i understand the question uh properly so for example this simple one is having just this vertex um also this one have like three vertices that it's not that we are just taking uh
30:23
uh taking this and putting it as a chord uh it's it's it's different it's more or less different but again we can have odd number of vertices but the thing ali is if you imposed if you looked at just the subset of diagrams that had
30:45
this oddness restriction then you would get a subset of the chord diagrams and david's asking which ones you would get okay i see um i i'm not sure david um i i didn't think about that but i think it would be interesting it's a very interesting question because implicit in your
31:03
work is that subset of chord diagrams that corresponds by your explicit bijection to forest theorem so you call them for a sales diagram you know write a paper about about for his chord diagrams yeah anyway great work thank you david yeah great work
31:23
thank you are there any other questions for ali i just have an obvious question maybe briefly um what about the higher order cases like the three four five connected
31:42
um i i have thought about that mishi uh but i didn't go anywhere i don't know if i have told you earlier but yeah i i didn't know uh what happens with k connected core diagrams generally um it doesn't look easy and i think just as a case of k connected graphs we we were not able
32:08
to uh to move forward i mean even for the graphs not not chord diagrams we we only can express the generating series for two connected graphs and the same happened here but i hope that
32:21
the result by cleatment is an evidence for uh that we can do something uh i mean it's it's giving you a fixed pattern for the uh proportion of k connected chord diagrams so maybe there is a way to express k connected chord diagrams in terms of the previous ones i i but i i don't see it now happy to see you
32:47
right yeah thanks for the great talk ali good to see you too thank you all right let's thank ellie again