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Introduction to Feynman Amplitudes

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Introduction to Feynman Amplitudes
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In this talk, aimed at master's and Ph.D students, I will explain how to assign integrals to certain graphs representing physical processes. After discussing the standard integral representations and their underlying geometry, I will give an overview of what is presently known and not known about Feynman integrals, and indicate why they are of interest to mathematicians.
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the thing In and so 1st of all I should say that almost everything in in the 1st half of this is in the standard textbooks and quantum field theory I would explain that I have to sign into pools to grass In the 2nd half of the give the sort informal survey all of what is known and what is not known about these interests the parliament and in this subject is knowing where to start with typically ,comma filled course will talk about the final profit to grow up to 7 points and then by analogy the makes noises about which expansion expansion graphs and then sort of start again but up to the and there a sort of leap of faith that connects the 2 parts and so at what I would do use the start the story then with perturbation theory seems as good a place as any so I just said that we want to cross at the center of the storm so input tentative quantum field theory the the interactions between grasp are represented the money markets an interactions had just signed in the morning interactions between particles are represented photographs they always feel slightly awkward as a mathematician talking about these things and is an example of a graph representing 2 particle scattering and exchanging phone at all His resolved another graft the same particles exchange of photons responding leaving becomes an electron post on there and later which made to an article for public so paragraph that will draw like this represents some kind of interaction between fundamental particles and the game is that the primary such craft that physicists Associates an amplitude I G which is given by some integral and it's going to be a function Clinton data which are masses and mentor of particles the function but I and the final answer the compiled experiment but of tanks by signing of all and of course Anthony diagrams of this type which represents the same outcomes and observed outcomes and and so 1 has to do something you can only in practice some of the 490 many grass and as it is when you cross your fingers and hope and that's as somehow this expansion converges but it doesn't so I just don't understand about this this very difficult unsolved problems relating to this but we have to we want some divergent series and um we don't worry too much about the this all seems to work OK so today and to make sense of invocations and Jean a look at this game theory and work in Euclidean as opposed to the cost based so far and space-time already the the where and dimension will be is a positive even integer so it's important it's even and so it will work with the Euclidean norm suspects X squared is the son of the square of its components effects but and they could be so the the challenge that have to bear in mind throughout this talk and these howled how we think about opportunity How do you we make sense although the zoo calls attitudes hygiene so this was Vermeer or sense number once when received some examples I want to think about it and will that the problem is the challenges to trying find some order in this huge morass of examples so course I should say that I am calculating these ideas from attitude is big business nearly all the predictions for particle collider experiments are obtained by calculating amplitudes RG and comparing them with experiment so there are I don't know how many people but many teams around the world the banks of supercomputers running nite and day calculating
opportunist this is and if Europe many the particle physics phenomenology uh uh paper and journals they'll be filled with huge expressions for former opportunities so this is a very important problem and we're very far from having a mathematical theory of property market so that it should begin and a final draft but for me then will be a graph so Graf has a certain number 7 set of vertices 7 sentence internal please and a set of external this is the case of the ad this is a graph that satisfies the usual the usual rules and entire ages I am each edge is connected to 2 vertices "quotation mark this is just a subset of pairs of vertices the external edges on the other hand talk connected to a single but Texas reps and that the maps from external pages to the vertex to which they are attached so here is an example of the phonograph the triangle 1 as the vertices the 3 internal edges 3 external edges and for forgeries on number the edges for later use 1 2 3 Times the external edges represents particles which which I will typically represent as in common but not always but his own from Graf also has some extra data the namely the date of this year's among all groups of people who know where she was going on in the I know .period question 1 about 90 this was this the program the graph is not necessarily as warranted no that I'm good orientated or didn't just a minute to write down integral From identical they would depend on your orientation I'm so the gruff also has an extra data all mass particle masks Mt which is a a real number if you're paranoid you can take it to be positive but as will see it's only the squares of the masses that come into the formulae so there is a master for every internal edge the answer to every external edge we have momentum Q I which is at the vector I'm for every external so here and expert on Q 2 here I Q 1 I and so am the convention will be almost always sometimes emerges as the conventional always be to have momenta coming in with if you reverse the side if you reverse the If you want represent an outgoing particle that's the same as in going particle with the negative momentum so we can always assume that all particles incoming if we allow the Q eyes to be negative informed directors yes that's a very small detail they had the next remark is that these these data subject to Clementon conservation madam conservation assault Q I was 0 for all external injuries working right 91 to write down an integral associated to such a fireman Goss a former golf is the is the graph with this this data but and to do that we need some discretion variables so to every age internal hedge and yet the retirement age we assign any momentum variable arrival chair In the vector the announcement temporarily and a despondent choose an orientation and grass just down into but it would depend on this it's going to Panama's foreign and city's orientation was the 1 a minute and then he writes M P able to be find carried if these internal alleged and the incoming momentum if the accelerates and then the
pharmaceutical going in momentum space for it use our so this depends on on the external momenta Q I and particle masses and we such an integral part of our real space City dimensions products and internal edges falls it's good ceramic is a D vector to integrate this is component by component and in the denominator air Katie scrappers and credit times for optimal vertices and here delta function clearly there where the summer's over all edges both internal and external uh which meet the verdicts the so I am there this makes sense because because it doesn't orientation by the meaning of this this time the useful for our use of all of Anchorage 2 on the image of our name Ng thank you because a number of years I thank you so this is a R D times and but some so this factor here is I I'm represents if you like momentum conservation and every vote would the somewhat of a mentor it but at every verdict sums era because of this of course sponsors is the momentum space fine integral as this most commonly seen and of course it is necessary makes sense In May documents so a small the UN's adult Delta Delta functions the he is living here on the various lies in the lack of ideas ideas in Delta is that it is the decked out the functions OK so yeah method assistants prefer the of the U.S. and the defendants your comments and said that the woman of the noggin anger chew while you could do that you could do that but I wasn't 1 of them ashore unit example estimate but I said to choose an orientation on G and then when we signed a momentum to every edge so will have protesters who have some K 1 K 2 and K 3 and sorry this Varteks will get the equations K-2 this case 3 because Q 1 I that it is Frontex will go K-1 prosecutors 3 because characters and this verdicts will get OK want to skate 3 because capital so there is doubt function as a means to integrate over only this region itself lost much of that was due to last yeah right as clearly means I get televisions and he the deal was the result of sees that this additional doesn't make art status and call doesn't make a huge amount of sense In the 1st place and that's when the 1st thing I'm going to change it maintenance cost as sense so this is what you see in in physics textbooks have kept it exactly as it's written I'm and exactly 100 essays that that's fighting the immigrant all days it was common practice to use permitted representations as classical and effective after a long hiatus it's not coming back into fashion so classical and modern approaches 1 USA the parametric for the parliamentary representation which is much more satisfactory after many points of view so I'm not gonna drive up from there the motion stop makes sense yeah the meeting each use cutting costs which were planning to use for cutting costs I but yeah writer I'm working right in principle yes absolutely out but I working a simplified this is a simplified the theory where I can just take it to the 1 woman I could put a parameter beginning yes I'm really gonna consider 1 rougher times so we know how many votes is announced Of course in a a genuine now quantum field theory that the the true finals much much more complicated but at the end of said something very briefly about how the changes just regarding a review of the whole the idea of the qualifications for young viewers a former prosecutor I'm OK said to get from this defamatory forms and the 1st all rejection I'm so this is the identity 1 of the X he calls into the goal of the demands of the idea of this is valid for 6 posted I'm so the idea is to introduce a new variable after for every page every time and then writes I'm in this integral and replace them this factor here where integral from 0 at infinity Peter minus the say something it looks as if
we're going in the wrong direction instead of doing into intervals adding more into gold but we'll see where there's a lot to be gained from this it was unclear and so what happened to this integral it becomes I will now be the integral a bunch of integral sincerity infinity for every after parameter assigned every edge part of the D & G experts minus some of the Kerry Wood's case square semi squared off and then the same stuff on the society's before marched Afghan and sperm products the day after right there so made in school more complicated the apparently by introducing new variables now the point is that they had that we can actually do the but momentum integral so this involves gasses formula for this integral Over the real it's minus Pyrex squared the X equals 1 so you can multiply into pools of this thing together and To get a high-dimensional and then in general d dimensional analog of gas integral it is as follows if we have Q X is minus X transfers a X there has to be why so here is not the time please asymmetric positive definite a matrix be as any come vector and on the day so this is a most quadratic form then the integral over oddity the 2 the Q X X the bus I thank you very much be scared at the at the beach yes thank you Of course every other constant I missile ,comma later we have a constant here it's just going to factor straight out of the integral because doesn't the thank you for that and so the upshot is that this is a party to the DDE over to Over the square root of the determinants and times he too that the transporters at prices
so I'm in this can easily be deduced from that appears surgical by died analyzing the nature and it'll it'll it'll break into a product of D copies of this and go after changing variables right now the idea is due to be applied his identity 2 of the injured were the top plans to actually do the attack came into roles in this expression In above and the upshot will be there we will obtain I into a new integral only then I don't know number Middle Libby any carries the lonely Alford these the masses and but the office the engine accused yeah OK so this can be done in general where is quite a tedious so I'll just do is on 1 example and then states general also OK so good still very simple example next stage this craft and we assigned a mentor K 1 and K 2 to these 2 intelligence so if you prefer weekend official statement communications landscape and to the delta function but in integral is momentum conservation of these vortex that reduces the both the same conditions so reduces the equation K plus OK to call Q so if you like that starts automated integration and so now I'm gonna write down the exponent so far so the exponents in an integral service are minors case ready to send square after so what is it in this case it is a powerful 1 careworn squares someone squared has offered to OK so the 1st thing to do it's 2 hours substitute in the the mental conservation conditions so this is our 4 1 K 1 squares has offered to Q minus K 1 squares the Stalinist term and then the masterminds apparent accumulates here so the 1st thing you have to do that is to take this part I'm completly square course you can always do this In this case it gives they but the make efficient 1 plus suffer too and then the times K 1 minus Q the to is and I need to add to cue the square so I think this is this is the process that we like it the minus but but over performed his offered to OK so it can be to describe I'm so let's write this I so the next thing is to have a shift change variables the shift shift rivals I'm I call this K Salesforce K the calls minus Q after 2 of 1 suffer too and this is just off along the south too Kerry squared plus the rest 2 and that changing variable as of course the the way integrating over and over Arthur D. so the DH quotas translation invariant and so on From this I am representation recommend plug-in gasses formula and right on the answer so we get to this so that this is the the timing exponential we have exponential of minus theirs integrated over care and all the stuff on the rights just does not depend on K all it's just going to factor out of the integral and we just have to integrate the 2 the miners offer 1 has offered to carry square and that's easy using this form of Tata's right on the answer with we applied to the deal too integral From certain facilities the 1 the offer the often want suffer 2 comes downstairs and I'm analysts stuff over here this unchanged factored out the integral in the 1st place project so in the general case speaking others Watson general you can complete the square and put in musicals as that of bookkeeping to look at what the answer is at the end you get a hit the determinant of a sudden matrix and and then you have to buy something called the Matrix treated 2 2 press
but that determinant in terms of some profit quantities but and that's rather straightforward and finances that the we gets added into going some some trivial factors which amended law empowers clients times the integral 0 2 infinity so here product over all internal legends the a sudden polynomial "quotation mark and then you get exponential minus foraging queue of obsession -minus songs so that is 1 of the parametric form of the firemen integral Sunday and I to explain to you what these side and 5 so G and far GQ on what are called 29 meals end do it OK Chrysler animals so 1st of all I saw I subsides the it is a polynomial 1st discovered by kickoff I think in 1853 so it's a polynomial in at just the an alpha parameters and depends in no way on the external lamented all the masses no dependence on Q I am it's an important fact that it has integer coefficients that 1 play any role in what I today so his a formula for its action the many different ways to to interpret this Crawford just give me 1 and it is the sum of overspending trees In the grass and then you take the product Over the Edge variables not any spanning tree 30 offices spanning tree it is a spanning tree that means some tea is connected simply connected so that's the best news that it's a tree and spanning means that's it meets every vertex of the graph G but passages that I should die she said "quotation mark qualities here G is and synergies connected throughout the summer synergies connected by itself model this this is going permission and so misunderstood example 4 I support the best-supported to do these operated on so students to this example I want to this race simple there only 2 spanning trees as 1 2 of the Treaty which meets every vertex on this too and so the graph .period million it is the product of all the is not spanning tree anyone in this case and we have to himself and other to South 1 is indeed the term we found and in only 2 years before his death he doesn't think next server or yes yes I have the right to promote dependence on the Q eyes um yeah spanning tree as it sought to think of tea as a subset some of the contagion intelligence yet external momenta some likes playing a role in this at all we just ignore the just as you do another example and this graph which we had earlier spanning trees are 1 1 3 Boston Virginia against specific examples the the analysis example for later on the spanning trees has the same 1 too and 3 and so outside it is 1 of 2 after 3 years of 1 himself 1 of to so that's 1 of the times in the integral and then we want the 2nd semantic polynomial FIG nine-hour does depend on the external momenta that I represent just by that accuse so it's a polynomial again integer coefficients in the office and in fact in adult products scale products falls all the excitement um Prince of the formula is a that's minus the summer so part of responding to trees Adams Spencer 90 take hits the T 1 so the top product is is the Euclidean shouldn't or product I'm and then you have the products of the edges not until the office so what is aspiring to a tree the T it's fine to treat if ammonia so it's a forest 2
components the tea has exactly 2 components she wanted to and simply connected which reported method and its bonds so spans means that meets every verdicts of the craft I and so what is due to the T 1 rigidity to secure to the TRI equals totaled but the sum of all momenta entering the Treaty on totaled entering Tr an example of today let's say example we did earlier a so spanning 2 treaties and well only 1 it's for this spanning 2 tree to 182 and so we get minus Q . minus Q times the product of all the absent edges that stuff 1 of 2 and that is exactly accuse squared 1 of the 2 which was exactly I have the uh the time up their minds of 1 of 2 the he's going to have this important legislation it's all these months here on the because the here yet about the minus sign him millions in here there's minus sign here yeah but group less than 2 months this Council plot it is the more interesting example and the triangle graph we had earlier in the day this on general support that is fine not that it is from this the definition of 0 5 but this is this is minus hit so here so this is minus minus-5 minus after 1 of the 2 Q squared From the blackboard over which is off after stuff too executives the the loneliness compared to the minuses behind so I think it's fair Simitis said of itself don't this in the formula right differently -minus here and I think it'll work the Simitis here and then there's an A-minus here 3 months of this year has been there I of Minnesota what with his dog product to the store brought always produces a minus so there is some minus because you've got to use is less than what you haven't yet most of us said that security was prosecuted 2 0 Parliament and conservation so the way of money right this is plus Q T 1 squared I'm that equals prosecuted to scratch but I wish you write it this way it's not clear symmetric into 1 entity right this way overseas systematic but the price you pay as you once but is not this sign is not this is completely polls this is entirely positive coefficients this minus gets absorbed in stock all absolutely yes what otherwise you would have a single command rejected and so a warrant ,comma behind labeled this up to rotation it's this so what aspiring to trees but here they also and enable them to warn the Saskatchewan Q 3 to Aswan spanning tree the key to to think you want and the last 1 Q :colon at this event and so what do we do we take the um the dollar product called the moment entering the 1st treaty but minors she won Q to rescue 3 and we take the product of the edges which are not present that offer 2 of 3 same that's a minus Q 2 don't you want to 3 for 1 of the 3 In the last 1 gives minus Q 3 1 prosecuted of 1 of the 2 and then of course by momentum conservation the little right up here like you want due to rescue 3 0 so this is just a quarter Q 1 square after 2 3 she 2 squared off for one-half of 3 rescue 3 squared a forfeited because in in the final months of all coefficients of positive such a very important us because working in Euclidean nuclear and space and and that's what guarantees that these into intervals makes sense somewhere but some values of Q and so these changes a viable formula and so on will be justified In some region of Q & amp and for some value not necessary which may be real number as an experiment this wasn't ramp on this that these good polynomials should have walked with her case so have to been then writes and this is not a standard notation but I like to 2 package this give give symbol and I consider this polynomial Figeac you plus summer the company squared off Sussman the standards annotations and now
1 I thank you very much Thank you thank thank you very much and a subside it is a is homogeneous of degree I'm H G so rewrite H. is what this is called the number of rigs the grass so it's it's the dimension of H 1 Of the grass and on the other hand so far GQ and hence sorry GQ ,comma all boast homogeneous cause in in in the office dependent on things as well but of degree exactly 1 more HT press 1 that follows from some ,comma Toro properties spanning trees trees to "quotation mark right now we want to get to a projected into gold you want to get acceded to that 2 things you can do at this stage and we have these firemen in these former that as an exponential integral unbeaten either to what's called a minimal subtraction I which somehow preferred but it can also do by dimensional regularization and as the latter's is slightly quicker and gets straight to the office 100 do that and so that the basic idea it is we want to get rid of this this exponential factor so there is 1 that can be done by doing 1 more integration so quality had I integral urged infinity so let me right at the new notation the next day "quotation mark achieved the times I'm not change variables Lazarides Alpha B P equals lavender BET where to some that BT is equal to 1 land as opposed to remember and then this differential form here products the office the can be written on recall that Ng is the number of internal Indiana where under Jr work on the calls some minus 1 to the other side the consensus of all manipulation the and so this can be rewritten In the former while in the region where the the beaches all positive at a cost of forget the sound you go to war of Omega GA but over side G so now I viewed side use it as a function of of the beaters um it's homogeneous so but I'm often arise BT and generics comfortably integral this case means the only it is revered as a function of the Peterson said in the office mn here yet eaten minus a lander so from here of it because of the degree of the CSI 29 men exactly 1 more than the degree of side when you plug in when you although offer prompted by land it will produce exactly 1 coming out because the degree here is exactly what the degree there and yet landed to the minors the mg -minus duty over to each chief the number of animals To this end go here we can do is the final the final step funnel into go we need to do and I produces 80 I'm a gamma gamma values Jammeh calls political stg times sigh over sigh GQ and To the minus stg well set S D G it is essentially this quantity which pops out just minus the number edges lost D that too times number lives and this is called is is going to measure something about the convergence of this integral and so it is called the superficial degree all of this diverges from the ground I think that it can be positive or negative society when it's uh um positive distance calls go to diverge from its negative is going to have His the graph is called logarithmic Oval Office divergent when it equals 0 the Union of Concerned the says this approach the Beta on right Peter and everywhere it will become break cluttered from now on the mirror the next into Golubev thing a function of Peter the conclusions the policy it is that we can write off I'm integral after maybe some trivial factors In the form this can acquire a gamma function the integral all over this type of plane but in fact you notice that its arms uh the Intergraph and the comes out is actually homogeneous a degree 0 so it's something slightly nicer to as an integral overprotective face is the site of the Battle of someone who was fond of all those involved in the is correct this is superficially have divergent so this is very positive on the graph is divergent so this should be negative yes it is the military will use the European office Oh sorry I thank you yeah this is great that is not you think yesterday it it demands how much it diverges 1 it's very big it's a means it's it's we're going get poll here that support them so we Soviets IG dearer to
over CSI GQ and to the power of sergeant puts sufficient good and here are because I want to start at the moment so this is a and a projected integral look the bias I think sync signal and so on Signal equals the sets the 2 1 thank you and so this is a projective integral many makes sense that if you applied in these degrees here this is the into grant is homogeneous of degree 0 "quotation mark but is another problem is of course and that's this can have a poll so this scam a factor and can have a poll In particular when the superficial degree of divergence His is no I'm so very briefly that the idea of dimensional regularization the idea is that you replace and and this is what is done the most common in practice no 1 followed a huge fan myself and is replaced d with the minus epsilon some small positive Epsilon and before a Taylor expansion international I'm so offices during practice is the computer the coefficients all it's onto ice but I can be both positive positive and negative the likelihood of review you would have a division of the disease alone is really could use yeah I guess so so you can you can view it as as an integral of this plane if you prefer but the point is that you could have chosen any other hyperplasia have to choose wherever it is that the the sum of the the equals 1 could use only about buying it still works on this if you like is a nice way 10 codify that so was a political meaning the restricted are fine shots and calculate into intercom on that front that's exactly the same as choosing a hyper plane at infinity and integrating over I find real space so for physicist this is a way to 2 to at the Finca business ,comma off to the energy with delta function given by this equation I need it contains the fact that it is that you can replace this with any depending on OK so that's the I'm end of the the 1st half Pfizer said most everything is in textbooks was informed that instead of having a pause I should just press on and in 2nd half hour to give us a little panorama all of what we know about these intercourse In this regard Angelo regular the very fancy juries not absolutely not enough Thompson no that's the point of key .period and then there there is do so um OK so what do we know about that fine it was of course it's impossible for me to give a completely exhausted the list let me try to give a picture of the landscape so the 1st remark is that the general I want to the diagram a site amid the fall remark but before I do remarks so and all these this procedure and this rhetoric Ghassan into growth analysts works by fall it much greater generality works for gage theory from and the upshot is um there we can always writes and form an integral In the general form and so will happen is if you do this is to get higher maybe get different powers of sigh and sigh in the moment of areas integers and Alpha is some polynomial In the office maybe it takes that coefficient and some Clifford algebra but demand holding 2 Grammys homogeneous such as this is homogeneous and increasing so that's the that's the general form of a financial economist at 4 what was the peak of the extremely competent and going to answer but back to the zone skated diagrams the 1 diagram can be number can expressed using a single function "quotation mark to function rather than the dialog with the tunnel with them as defined by the following summer CNN and squares this converges with more takes less than equal to 1 and hasn't analytic continuation 2 a to C To Monteverdi's functions and the lower them so am I. any any grass at this you can always express the amplitude using just the 2 analog where the X will have a big a complicated expression is with different arguments acts but with the X will be some functions all of the masses in mentor and some possibly quite complicated way villagers as a single function essentially the described about the team 4 of the nation's School of Rock the government believes that the problems of the world environmentalists and we can discuss this and so the general the general diagram the attitude is not known as so there isn't a functioning economic up
in the canon of special functions army means this also means that the the singularities and the analytic behavior of the general tune about 2 is not on is where and so so yeah and it has affirmed that um if you take as an arbitary large number external legs close or reduced to most the seven-member reflects I'm into the corresponding result is not known but I don't think it should follow a the media age when you something about some series of weight should follow but the use of the based the you the original allotment of minutes on this year yet it typically involve Square square roots a quadratic form in the masses moment so even a simple example like the other trying on its itself in general it's it's going along formula that shows to them in it's quite complicated "quotation mark something and so too did manage it is not known in general I there is a special case has become very fashionable at the moment and that I should be studied for many years so it's called the sunset support for optimists it's the Sunrise diagram so this is a particular to diagram so of course face these difficulties will people typically do is make some assumptions the 7 masses a 0 and so on and so forth and this has a long history of use recently sold for the general case with all non-zero masses and the key names but some are brought in from all over and the and that Adams governor combined still the have very different approaches but these have that so the final word on the status of this the two-goal I'd recommend if you want to know about history conducted inspections I'm sorry I forgot to say something that dialog with them dialog is an iterated integral I can write it in a very simple integral form 0 0 listen to 1 lesson of the Tunis impacts let me take X and between 0 and 1 for the sake of arguments then it can be written cousin into gold just involve included differential form and so this is an example of an iterated integral Paul C. -minus 0 1 some bizarre reasons fully understood this space that seems to the functioned MySpace describes a vast the number of foreign attitudes Friesinger completely clear and is in fact an exception but what it does involve I'm also iterated twice something toss a treated elected integral I want some from Europe to curves and 1 way to do it right down to get the now pursued is by average Is this function and NZC Q has nothing to do with this queue here so secure it is a mannequin Acuna QQ nothing has nothing to do with the queue here and this is mean some function at a dozen makes sense it's it's for many reasons 1st of all we're averaging among Multivalued function arm and it's going to have a singularities win it is it was a very brave over a small depending on this I'm so you have to regularize in some way but if you do that you get some function and this function essentially describes the the altitude uh here completely like you know what it will be some function of the masses and acute someone wanting more than that I'm next there is a sense is the 1st sort of a difficult case a huge class although amplitudes also on the and on expressible I'm using the former across a functions "quotation mark multiple plenty of about which I think will hear a lot more this afternoon to part of my job is to set out not in speakers and so here's the definition the multiple play longer than some generalization of the dialog within the answer depends on all variables that it is the sum 1 lesson the came this 1 to the K 1 6 to the chaos this so that defines found some analytic functions on the region where the excise that's strictly guessing wrong I'm by continuation it extends To a Multivalued function and on see to the game Paul -minus on the Net stake out the a certain trusted diagnosis given by the success of products privately that he sees the stocks C-minus C minus the origin the AA-minus this cross bottles and this is nothing other than the modernize place all minus 3 points on a sphere we think ominously particles on and it's not entirely clear as a priority in the the CIA all yes thank you very much on this is a success this is constructed this is just consecutive where half of the thank you certainly of appoints a 0 1 and that monkey but for some reason this particular model space which in the case are equals 1 it's just minus 0 1 it plays a special role that is not new understood what that the analog should be to describe all higher-order fundamentals and is such a thing but this is something I am which is used a lot in physics but which also emphasized that it is not finally it in general told and scientific explain what it is and then said he wiped 1 should shouldn't extrapolate beyond its range of legitimacy is it's 1st of all there are iterated intervals there's no reason to
think that general financial growth should be interested controls and there's something where special that they have a way to braiding so what that means is that the weight all of good test awaits to such a function that's the sum of the indices and I the opera property out of this weight is that if you differentiate them the multiple over there all of a sudden whites and there's assemblers explicit formula for this such as Morgan watched all that he can write this as a linear combination all functions of no wait when and of multiple putting voting logarithms all strictly lowered In fact it wait 1 less so this summer across structure in its functions and um this weights to keep differentiating in a certain way you get wait 0 and that affects the fact that it traditional away whatever emphasized because this is offering expect infected it said that this holding journalists it's not true this is a very special properties all this particular class of functions OK I neck so that the sum not of this type it's something else and that should exist across the In the interview about falling in love you I will leave that to this because this afternoon and I believe that that will be discussed in so I'm youngest reset setting up the often and there should exist so that there is a well-defined class we we understand if the farm is a gross of this type that should be on that there should exist class a class of multiple elliptic types and such that OIG His multiple elliptic this is the sort of nonsense fairy quality papers defining what such functions are and this is not known as mishopin worked out a all so the first-class beyond the point of rhythmic is completely unknown Saiful 1 or 2 examples like this maybe that in 1 of the and I think it's a good thing to have no so would mean something like this the to we we sort of forced it to be a lipstick on on the Jacoby uniform isolation of the 2 girls by averaging over acute you can do the same thing for multiple .period logarithms it's like more subtle and that's a good start to a perfectly valid definition of a multiple elliptic put it over them what's entities look at its rated in schools on the last as a whole class of functions this is something the genus 1 a analog of this story and they should describe another notch trusted diagrams the net cost of not less and so the boundary and then the boundary where the body of a regime ends is not something is known about it and it is not known precisely focus on site most specific examples 2 job interest number theory often rapidly so typically these examples a very difficult because there are lots of masses Memento and the altitude is a function of many complex variables and that's where none of the difficulty comes from but often in practice you don't need such generality and to be um things get easier if you make some restrictions on the movement of animals and so the other extreme we have 1 single scalar processors and so these are always by GQ and 1 is a function of a single variable so there could be for example all the masses of 0 and there's just 1 incoming momentum for it could be we just have 1 nontrivial mass and what typically happens but is that the dependence on that variable is completely trivial and it will distract us out of the integral to the underground so what you're saying is that the amplitude to giving numbers so function the fact is out and the coefficient is an interesting number and then the question is what one-out walk we set up the numbers that come out of quantum field theory so his one-family re-sampled and which on some sense the most exotic says the analog of the sunrise it's um uh diametrically opposed to the please logarithmic class where we feel we understand or and so he's all the Bernama graphs 7 symmetry and all very interesting and these were 1st study by the difficult so many uh do 1 example in the complete detail so we have this is the same picture like this and and what we're going to do is impose I am 1st of all let me set all the mass is equal to the same masks and and you know what is going to be just 1 here so I'm going to put almost as equal and accorded equal to any of the other way to see all qualified by isolated said the bananas yes I think the bananas will be this family of grass that's a ban on I don't know what's going on here looks nothing like a banana but you know you are summarized the money the I'm so the gruff putting on real is this off 1 of the 2 1 of 3 itself 2 of the 3 I the 2nd graf 49 you is all good exercises to revise the definitions of aircraft with musical value this is queues great off 1 of the 2 of the 3 um and this point monoxide and gives this equation Q squared off for 1 of the 2 after 3 classes and squared off Warne was offered to 1st of after 3 after 1 of 2 as of 1 of the 3 at 2 of the 3 which is very nice summer and defines a beautiful From if Cuba what is the
next thing now and the board you yes I'm so the next thing summer border says OK we let 2 9 at work and equal to dimensions come and see and puts M squared so this is not quite the Euclidean space 1 them slightly dishonest but will put em squared equals Q squared equals so science critique was minus Q squared equals 1 and and now so now we said so using board has notation saying look at his patron .period compartments His conjectures and the integral it is edgy so if you if you look at the the number of reported number of edges what happens is that sigh drops out of the integral and we get excited and so em this em here in his notation is this graph where am -minus 1 is the number of intelligence what annoying shifted indices drunken stick with indexing to make it consistent so if you if you do stick out he squared equals minus and squared cost the the the the the M Square just factors out of the integral so we can just set and ends when he was the 1 of the Act media time is not on the readiness of the armed forces yeah cost and I'm counting them alleges is not listen and is that members that circuit the as powerful that is what we just computed it's this integral and its omega-3 this projected integral not for want of 3 offer to the south of the red peaceful integral and to work it out its projected group so to work it out we just work on some are fine shots salesclerks and after 3 equal to 1 and that's just the offer warranty offered to platform suffer too 1 South 1 1 2 so as a very and my son to be convergent Lynch Glenn and his A-list of what's known about these integral as part 3 is of 3 square at 3 as far forward as this 1 it is Place squared of 4 this qualifies there is conjecture much has been fruitlessly since under the and that it's should be for Pioneer Square 215 times the old function while the model for so this is some pretty serious amounted to a number of arrests as
boss 6 is experimentally given by 48 remains each of 2 times special value of the old functional model form where full it is um await for uniform it's explicit products plundered attend the 2 values so this is something I won't dwell on this too long but it's something that number fares can vary much about and it's not at all understood how and if at all this pattern should continue again an agenda it's not known we offer this it's not there's no conjecture for what these intervals should be so we see that Embraceable funded diagrams is producing some numbers relating to many of an active topics in number theory and we really do not understand had to make sense these events are in some sense the worst possible from here examples I don't know let me turned to a completely different from the of a single examples of these will be residues In massless fight the 4th and these at these examples the government of asked about petitions interested these for again on and all masses will be 0 I'm so also that Groff is in infighter full I am it turns out to be equivalent to the fact that all the vertices all the grass all degree for last summer his arm example and so what we have in this situation is that the number of edges is twice the new number so the Super superficial give Devon's is 0 and win the divergent case so the this quantity we had earlier has a poll here never in trouble the but we don't think is winning at the coefficients of 1 of the Epsilon game the fund integral Indian and it turns out that this does not dependent on the external momenta it's just a number Q yet so that executives and sets the residue called the residue and vital to abuse of notation under undetermined underqualified again that is money out of letters so it is quoted by the actual pharmaceutical residues and it's just given by the integral Of hunger of upside squared so previous cases of the integral was special in the PCI dropped out of the ground is especially because the CSI and grand OK so on these in schools .period always no so I pretended to Vineburg is that this integral convergence but if and only if the is what is called primitives but he did say use of language printed refers to something some of destructive came much later but the condition is the football so strict subgraphs of duty the superficial degree of divergence of the 7th graf negative and that means that the troops should be bigger and twice that number of edges should be at least twice the number of and so the 1st example we we saw already so I leave you to work it out in the Definitions this is an art form after careful 1 1 the growth in nominal squared with his for all of the yet does that go into the streets of Jerry Lewis fought summonses onto it was a force emerges as a consequence of all of this and this orders from you upset and so on to work this out we will be working half uncharted say um I for 1 equals 1 so that's offered to customs at Infiniti G offered 2 you want to suffer too squares and that is the number 1 so there is you this crosses 1 is enough mn after that they get a lot more difficult as look for him more about this afternoon let me just give the 1st few examples some of the 1st period of the next graph which satisfies all these conditions is this graph so residue is 6 times as until 3 this gives 20 isn't as follows the Syracuse to this gives her a 36 C 2 3 squares square of this amplitude the sum identity looking several identities looking grim scenes and finally "quotation mark this graph 6 minutes too many years 2 1st computer numerically saddam but what crime this this external until leaky cannot be done symbolically position on exactly what it is 27 over 5 fired the 3 45 fault OK so these numbers here so there's ETA 10
an hour the values of 1 of the more people .period albums and they're given by next some this converges with an last critical to the musical multiples isn't this this graph and this is 62 3 tonnes up in many different quantum field theory and assess if the reason that was on this board before that you can always writes a general opportunity I'm but the enumerators in and the sort of numbers you need to get a finite family of numbers coming out I man it's the same 6 C 2 3 that shows up in "quotation mark increment Rx articles for it shows up again and again so there's nothing particularly specific this theory seems very specific but actually I gives a good indication of um quantities the describing more general "quotation mark of fuel difficult I'm OK so I'm multiple the values have a kind of weights have away but now it's what I'm saying is totally abusive the waterside awakes to a number I you can't do that so it wouldn't it doesn't necessary making sense and in fact we don't know whether the signing this integer to use real numbers and survey's makes any sense at all let's conjectural but these more positive values operated by the way that means a donor relations know how to break the law relations With rational coefficients between them the Tremont positive of different lengths cancel committment and influences that way it is hard to all Yao also got a lift as well and so this officers understand the notion of weight and I it's used a not in I'm in more but as a sort of guides to various things and physics teacher so it is known that there is some infinite class and photographs With such graphs In 5 full primitive such that I Gede residue it is mostly divided while linear combinations of more so as a criterion for them and so on there's a common at all "quotation mark wants a much farther and they're all 8 news I'm all so-called modular grass um which means amplitudes we residues we do not expect to be of this type to some of these numbers described of vast class all these altitudes and something goes badly wrong at a and we get some grass which of these residues are expected To be something else I don't know what to call them the number for want of a better name they should be multiple modular values certain types of numbers relating to monitor for the absolutely not discussed I'm another comment was making um Is that not every month for the 2 value seems to occur that means if you take the vector space generated by the residues of these graphs they don't seem to fill the space multiple believed their some very specific sup space with some very interesting properties and finally and there's only 1 family across which we actually know the amplitudes and his his exact graphs so you take a bunch of triangles like this then you connect them from head to toe and so on and so is zigzag fight in this case the index counts number of I'm so there appeared to his hand on emissions which was um conjecture by what has climbed for a long time give the foreman of this opportunity so when we write it down the thing that's most inquiries I'm so depends on a parity of and so this family of graphs gives every the old value of the remedy function so we note in particular that all the auditor values definitely because of you fax from this you can to deduce that the products of quality to violence but unfortunately there there is no other farmers across his opportunism known to all orders in history and there is no family his opportunities even conjecture so there is no way to know much is known the government is the fact that the whole thing was the residue here so that from now on I was means say in this section as I have been a bit sloppy here in this section when I'm talking about residues and calling residue of which is unfortunate because of the station no but it has no cute and innocent civilians and or Deke was for yes did equals foreigners who don't want to talk to I am I'm being sloppy and they don't play any role because because of this comment here the coefficient of 1 of its alone does not depend on the external momenta so I can just ignore it it's got a full list analysts convinced that this bold lost his removal from the tolerance for those who think that the is he that 2 things you can do it said or you can also look at this as a tutor functions like this in 5 4 3 so he would 0 coming in here for example and you have just a single moment of in and the altitude of this is some trivial dependence on cue times the integral way you we close up external lacks the vacuum .period these families of tentacles actually computing you take any integral like this and engrossing break a leg this computing the amplitude of graph this it is also completing the residue of the cross 4 . 4 services these cross encoding off ventricles the center so it is a conjecture that I am and that's an interesting challenge for young person I'm it's in my paper with stats and so next year they need primitive across the Tiber indicating that they had to fight the fall I suppose that satisfies and he was to achieve this section and Page chief news that we conjecture Friday that the opportunity is strictly less than his exact role so the other
zigzag is on the most it is the biggest possible aptitude in the fairy somehow reminiscent of devotees in from a tree where you medicine complexity is a pure numbers so this a positive number I'm so we conjecture that the disk offers the biggest and if this were true it would imply I abound the replied they now the g over 40 the age there is at most asymptotic .period form to this to overrule by on page G commanders fear to produce some bounds on the size of former attitudes and cause this is the 1st step in trying to understand whether you have the convergence properties for the recent ability properties of your divergent series I think conjectures is far better far stronger than what's a walk and approved at the moment the because you need to understand all the divergent grass and renormalization but that a lesson they have a curious feature of theory 1 tempted to try to show that during elementary operations on the cross some always increases the opportunity something that maybe some to explore them the story what women these long people and then he would you want to understand if you haven't got this the sum of all graphs have some divergent series you need to know very precisely how come across the and how big how foster growth and that usually means that something mine Newfoundland it's not the that's it is these residues up for some of our contributions the beta functions which are independent of all the normalization scheme so these numbers no matter how you realize they will always be in there Of course other graft which have Sutherland says the need to be minimized and then I do not know the goods I mean better choices so but I don't know of another this sort of thing that is if you if you guessed all graphs of this nature will tell you something about the the radius of convergence but this has some final remarks and in the last 5 minutes there are open-ended I'm so a lot to talk about this this notion of weight that's been creeping in um has been creeping into the picture so the majority of the known amplitudes and which so for example the pointy logs and then the which is from the bulk of what but we know about at the moment they had a grading emotional weight customers grading is highly conjectural in the case of multiple vitamins and we can ask 1 of his weight comes from and if it's got an instant physics recall that the of the I'm a vector space other Q is graded but if ammonia in its section of a group where some so Noranda Inc you start alarmed acting on VAT His number to the end if and only if really of degree then so we can think of is grating as a group action and um this suggests some of more other things as well suggests some the possible action involving group on attitudes February doesn't make much sense this a hazard upon comment in general this should be weight as well but the weight cannot be degrading there is a notion of weight on all of the teams it's sunny ,comma grading we should be careful this and will be infiltration will be increased infiltration and if we stick with this but and that the normalization of the awaits we've used up to now to the physicists 1 then we will we must expect half integer weights this is a factor about Wal-Mart Outerbridge homages but I don't think it's been a fully assimilated into the physics community soul in athletics we tend to do is multiply always by 2 this county pointed out earlier so was a conjecture that I'm so contractually this is a highly conjectural I there should exist a very large amount business the algebraic a group of major cities jurors Prada breaks is like an infinite a group of SMH I'm safe which acts on this place all of generalized amplitudes Justice bases only contains a scoreless after half after finding was something so this is the vector space I've always field I would say firms of regularized because he's and cause a possibly divergent president identical all of the form the general shape some numerator India integers and polynomial it is homogeneous such that the the holding to Grannis homogeneous degrees here some sense this is like a vector space all possible opportunities you'd ever want to consider any quantum field theory yet thank you yes absolutely rhetoric apart many fixed and 50 year life I think this should be something similar all donations as well so this is the spice of all amplitudes arm and that should be a wake filtration the gradient this is a space of properties of weight that most something and
since Axel F and that the action of the should preserve presented by the action of this group but refused the percentages acting
on all possible operatives around so see is would be called a cosmic group you know who came up through with the who was of special this virus move across integral and that the world was surprising for a collection of jewels absolutely you concerning do that alone situation since the original designs or what innovative yet appraisal on actual numbers is a transcendent here shows what they show this is fueled by the you can do this under the points uhm which I would have time to mention but at the point is that that's um properties such as the defining of closed on the group start extraordinary yeah absolutely women there's no reason why the the the the the gavel hundreds of some some some crazy number like this should still be up at the scene so in some sense it wants it is somehow is somehow the periods of some operate in the Catskill Mountains very strong conditions that emitted from a group of the pitiful of a group in the country mountains has a very strong so that right special formally for the action and he had a sense of operates structure that material and so on this this is a useful concept it really tells you something about amplitudes says the action of this action of this group on amplitude will be expressed in terms of small aptitudes and some control where the ISO says it has said it's in it's an empty statement if it set up appropriately but it will be a very powerful tool to study precisely this because structure so this is notion of cosmic Gallic Eucharist is uh due to capture so and when I say this to you because there the abstract of 1 of the talks those mentions of coal product so let me explain why this comes from the so on so we should have this group fighting on the space of 5 minutes away and someone Fareham for example is that I this space the space is fundamental this is extraordinary strong facts have infinitely many grass and infinitely many possible into grants but this expresses finite dimensions look has some magical feature of these intervals the Corundum family that's just not true I'm so greatly that the duel Paulson action it is a actions so if ROC is the ring of functions finish here on this group then the group is equivalent to a Coke products on the functions so I think this may come up again this afternoon and this action here start is equivalent it is equivalent To Play Co actions goes to far Cancer policy and the game is to 2 computers collection on grass and to use that to get information about kids and that's 1 way I in this challenge I mentioned at the beginning of how do we think about all aptitudes and happy but some structure in there In this was information 1 way to do that is to use this action group all I am a collection as a single hear more about this afternoon so over the world ones .period this is 1 we In the field of I am yesterday this doesn't make sense on numbers because you run into transcendence conjectures we don't know that as he 2 5 could be rational so we don't know that it makes sense bites and indeed in this am I'm in this family you the conjecture due to pound their chests is that this this space is closed under the the spaces on this fiscal action so so that the the group will send this amplitude to combination of business so it it will will willing map so this will generate a representation of the group which involved z to 5 and C 2 3 which are too small and diagrams so the fact that you're allowed to have the 2 5 3 you know that you're allowed to have visitors 5 3 is allowed to appear at this place and only in this place because you previously had 3 amnesty to 5 in the right place before you're not allowed to have visited 3 times 2 2 because that would give as too and there no grafted so it somehow Roger defies you can you consume these examples exactly how the conjecture the question of the ground yesterday were he was moved to the University of Houston financing need to yeah you it it died diverges in you have to have 3 normalizes you had enough time to the nursery we didn't a written overall regularization for the overall um the divergent speakers at the superficiality of diamonds 0 such a regularization but every all the subgraphs which have superficial to be registering need to be ready rice and such as well and then the speed and consistently that's renormalization so a way to do that I don't have time to but unfortunately the question right the few