Glenn Barnich - Massless scalar field partition functions & real analytic Eisenstein series

Transcript

pointing out it's only for d is equal to three that you have this direct connection to photons and gravitons so i will use periodic boundary conditions to start with so first of all if so if d is equal to three and alpha is equal to zero i have the partition function of the casimir effect as i tried to show yesterday and i then i have to tell you what this additional observable here is its linear momentum in the direction which is going to be the compact direction so pd when it will be relevant will be minus integral d minus 1 x pi d i of phi and if you express everything in terms of modes so pd will be something like so let me forget this prime because it's technical it's uh at this stage i take periodic conditions in all directions and then p3 is really it's d3 so it drops a k3 if you want and in terms of the periodic modes is this it's this kind of object here then remember that there was this map which told you what it was in terms of the modes of the photonic e and h modes in terms of the photonic e and individuals [Music] this is just xd because in the end i'm only going to cover of course we can do more general things but i'm only going to cut linear momentum in the direction that i'm taking smaller and copper but of course from the very start you could look at all of them and then it would be giving but the integral for the hamiltonian you wrote d to d minus one this at this stage should still think okay like this and here as well this of course depends if you are denoting the space time of this patient and of course later on as you will see when i take two directions which will be small continuous one will be d minus one and the discrete ones will be there will be two discrete ones one with time and one with space so then it will be really d minus one that's kind of confusing okay what i was trying to say is there was this map on the level of oscillators between so if you express this in terms of the oscillators for e and h modes that we discussed yesterday in the in the three-dimensional case it turns out that so there was this map between these oscillators so you can re-express it in terms of creation of e-mode and h modes and it is really a kind of felicity operator like daniel was saying so you see that this guy destroys and destroys an emote and creates an h mode and vice versa and it's kind of dressed up with the k3 or that you need in this direction but this turns out to be the good observable for modular invariants if you're just interested in temperature inversion symmetry you don't need it but if you want modular invariants like in two dimensions you need an additional observable and that turns out to be the good one but the question is what it is in terms of space-time electromagnetic fields because you would think naively we were thinking that this was also linear momentum in for the for the electromagnetic field but if you re-express it in terms of the electromagnetic field it turns out that there is one observable which is almost close to it which is called spin angular momentum of light and which is basically it's the following object so you look in any optic books like mandel and wolf or something like that you look at spin angular momentum of light and here transverse means that i am on the constrained surface so i already throw away the parallel the like yesterday the a parallel and e parallel so this guy if you write it out in terms of modes turns out to be almost this guy but it's it will be this one divided by k so it's not quite the right one that you need for the mathematical stuff that you want to do but of course it's easy to fix because okay this guy i should say this guy if you choose this component it's the third component of spin angular momentum of light that will fit so the one that will fit on the nose with this mode expansion is the following the following guy so you can continue to call it p3 even though it is not p3 so it will be a kind of non-local observable which is related to this spin third component of spin angular momentum of light just by this non-local expression and as we discussed yesterday in fourier space this is really like dividing by by k if you hit okay again about the notation so the perpendicular symbol for you oh it means many things [Music] to zero and putting this to zero i'm not using this this i will not use this consistent in a consistent way because there's many [Music] kinds of meanings to transfers but here it's really really like in the sense of families decomposition you have this is and there is this tension between the the two things if you have boundary conditions this what i was saying yesterday was electric and transverse magnetic was with respect to the boundary and this is with respect to degrees of freedom in in this sense but okay but it's very easy to implement it is really putting this a parallel and this pi parallel equal to zero and in reduced phase by means really solving the constraints and getting rid of that stuff of course probably it means that if you would do this in the language there would be a closed extension to this which cancels the stuff that you have too much it's just to know what this precise partition function would mean if you would do it with photons i mean it's this observable that you need of course this is reverse engineering to some extent okay so now i will do a computation that probably you have all seen in one version or another in a masters course but i think it's useful to do it here because it's the simple computation in the large volume limit of the partition function and the way i present it it will be relatively easy to see how this changes if i take an additional direction to be small so now i will be interested in canonical quantization at this stage i will so there will be nothing about modular invariance so canonical quantization i will take the chemical potential to be zero and i will look in the end what happens in the large volume limit so the easiest computation that you can do in which in fact you have no divergences at all if you put the system in a box by putting periodic boundary conditions in all dimensions is to compute this object here where you take the normal order hamiltonian so you forget about the zero point energy just by choosing the normal ordering and then you just compute that in a box with periodic boundary conditions and then the hamiltonian is is of course in that case i mean i already said it but in that case the hamiltonian is just this discrete sum so it's really putting it in the box means that you don't have this continuous this continuum of oscillators but you have discrete oscillators for each eye and i if you want and you have this standard thing of quantum field theory that your your field appears as the superposition decoupled superposition of harmonic oscillators with frequencies given is it okay then you know that the partition function will be the product because it's decoupled the product of partition functions of individual harmonic oscillators each coming with their frequency so you really can use the simplest quantum mechanics thing so because then you have this basis this tensor product basis in which everything is decoupled and in which you have a basis which diagonalizes each individual hamiltonian and then you just have to do the product of harmonic oscillators for all these ends is it okay so in formulas what i'm saying is that this guy is going to be in the product of n i belonging to z d the sum of the occupation number for each of this system and you get to do the geometric sum for each of these objects here but i mean these are what i tell the students usually is do two decoupled harmonic oscillators and try to do the partition function of that you find it's the product of the individual bond and here it's just a product for all of these ends did okay so and then you sum the geometric series of course you know that this guy is going to be one over minus epsilon beta omega k i right and then comes the step so then you say ah but i really want i don't want to do this product because doing products i take the lock and i want to do sums so i do ln of z d beta is going to be minus the sum of this n i's minus this discrete sum of the log of 1 minus e beta omega k i and then comes the step of so you have regularized the system by putting it in the box and by putting periodic boundary conditions but then you don't want to do sums because you are better at doing integrals so you take the large volume limit okay and large volume so you take l i large which basically technically means that you go from the sum over the ni's to you have to take this thing that you get from the change of variables so the sum becomes the volume over 2 pi to the power d and you get the integrals over the k is it okay and then so that that's the step where you go from something which was defined on a torus if you want with periodic boundary conditions to add back because you just use this so you can do hubble planner or something you use this just technically that's what you exactly do okay and you get this volume factor so this is uh the this should be yes vd out of out of this just because the the decay when you think how you go from the case to the ends there is a there's a jacobian if you want okay so and then you go to spherical coordinates because you go you can go to spherical coordinates because uh because this only depends on the modulus of k and not on any angles and then you get to use this formula that that you teach your students when they have to do dimensional regularization which is what is the volume of the sphere in d minus 1 dimensions right because the result that you get when you go to spherical coordinates is v d over 2 pi to the power d the volume of the of the s d minus one and then you have this integral to perform which is a single integral you get a kg minus 1 from the jacobian going to spherical coordinates and then it's really beta k the sum has become this integral and the only integral that is left is the radial the integral over the radius and this one so in dimensional regularization at least or in i mean you don't need this is the standard formula in terms of gamma function okay that's the volume of the d minus one sphere so this is the result that you have and then you do a couple of changes of integration variables and one integration by parts to play with this formula here and it turns out that if you do that this one becomes 1 over d beta to the power d gamma d plus one zeta d plus one you don't recognize it immediately but you get an integral after a change of variables which you recognize as the right zeta function which appears up to some powers of gamma okay and then you have to massage this result a bit probably all of you have seen this kind of computation in three dimensions exactly where you get zeta 4 because this is the standard black body result then you have to massage this formula so you have this two gammas sitting here so you have to use something which is called the reduplication formula to simplify that one so that's one that you have to use a duplication formula that's one that you have to use and then it's useful but not essential to define what people in the literature in the mathematics literature call call the completed zeta function so this is a zeta function that gets that is accompanied by uh some factors of pi and some gamma in order and this is the good combination and you can also see it's a good combination because all identities especially ones that we are going to use later it's much easier in terms of this completed zeta function than in terms of the standard theta function because of lots of factors of gamma and pi go away if you want so and people use this but it tells you so if you then express your partition function in the large volume limit you find that result is going to be just this thing so the volume that we have pulled out beta to the power d which comes from here and all the rest were numerical factors and you get so you get this result okay so you get this completed zeta function times the volume over beta d this is exactly what it needs for this to be without dimensions and you're happy and then you look at what is this so this is the scalar black body result in any dimension and you look what it is in certain dimensions to recognize to see if you recognize your standard thing so for d is equal to one which would be the result in conformal field theory it is this thing because x psi of two is then exactly this thing okay you have in in two spatial dimension you have zeta three over two pi and then you have the area over beta squared and then the one that you recognize is the psi 4 which is pi squared over 90 which is the scalar blackbody result l1 l2 l3 over beta3 so this result times two is the black body the standard black body in the large volume limit because photons you have twice this result okay okay and then there's another nice computation that you can do which is you repeat the whole thing and you are not using periodic boundary conditions but you are going to use directly conditions just to see what it gives and what you think is the result if you do very clear conditions for that in the large volume limit mr chairman is going to be independent of the boundary condition in the large volume limit so you think ah there is twice as many oscillators if you do if you do periodic boundary conditions than if you do directly but if you go through it carefully you will see that there is a cancellation of this two from another two which comes because the volume is also smaller so it is exactly as it should be and then this fits with i'm not an expert in that but it fits with a result from statistical physics which says that in the large volume limit things should not depend on how you choose your boundary condition so at least in this small example you can check that explicitly okay so with directly it's exactly the same and this is what's no longer going to be true if we choose some dimensions to be small then everything will depend exactly on the boundary conditions that you're choosing it's directly boundary condition same result okay so that was the exercise now how this appears from the path integral point of view how would you redo exactly this computation if you do a path integral well first you would start with something like that you would time slice so if you really wouldn't know anything you would time slice this or rather temperature slice this expression here like you do when you do the evolution operator in in you in lorenz in quantum filter where you time slice here you have to slice the temperature and you get the path integral representation for the partition function so that's and what you get in when you do it directly is of course you you get first the the the hamiltonian version so the products over all of these points e x a over two pi exponential minus s euclidean hamiltonian and this action is like that beat the x d plus one integral dx minus i p pi the euclidean ended in the time derivative and if you do everything you have plus plus h minus i mu p d kind of here okay so where should i put brackets probably well there's one bracket like this and this integral only goes over this part because here the spatial integral is already understood so it will be something like this okay okay and now just sometimes it's more useful to use so beta times mu is what i called alpha before sometimes one parametrization is more interesting than the other this depends on what you are just computing and what is interesting is that if you compute the partition function [Music] became t plus 1 that's similar to the question no and it is very nice why is it so really why is it nice because that's the next point if you're computing the partition function automatically the partition function requires in the path integral there is a prescription for what path you have to take and it's periodic in your medium type so what is nice is that now all your paths are chaotic in space time and no longer just in space and the periodicity in space time is beta the inverse temperature so as usual when you do this path integral now suddenly space and time which in this canonical thing were completely decoupled they get come on the same footing in a sense and especially if you integrate out of the momentum so we had periodic boundary conditions in all spatial directions but now we also have a periodic boundary conditions in the time direction and you'll be you begin to see that there can be a symmetry when you exchange space and time these circles in fact and this is kind of hard to see on the canonical quantization but in the path integral you suddenly see that all boundary conditions are periodic both in space and in time and then you go lagrangian so how do you go lagrangian you integrate out the momentum because it's quadratic in the momentum and then the action that you get when you integrate out the momentum okay and maybe let me so the volume in euclidean time is going to be beta times the spatial volume so there is a space time volume and if you want to think of them in a unified way if you had these periodic boundary conditions in all time direction you also can think of beta as ld plus 1 because it's the periodic condition in in this additional direction okay so if you integrate out the momentum what you are going to get is this euclidean lagrangian action which is going to be one half d d plus one x euclidean and then d plus one phi minus mu d d phi so this comes from this chemical potential so the fact that you have this chemical potential all it tells you is that rather than having the standard laplacian in euclidean space it's a little bit skewed the laplacian because you have this chemical potential sitting there but basically you have the scalar field to compute just the laplacian euclidean with only periodic boundary conditions is it okay and so this skewed laplacian in d plus one euclidean dimension this just means that you replace dd plus 1 by this combination here and it's due to the chemical potential but then what are the eigen functions so in this case the eigen functions are 1 over square root of v e exponential a k x a and now my a for a change of course before now runs from one to d plus one if you want okay all the small dimensions so these are the eigen functions and in this case you are lucky that you know exactly the eigenvalues of the problem because the two rows is simple and you have all of them completely so the eigenvalues lambda and a are just 2 pi over beta n d plus 1 minus mu 2 pi over l d squared plus sum over i 2 pi and i over l i squared and then if you know of course the eigen the eigen functions and the eigenvalues you know that basically this lagrangian path integral requires you to compute one over square root of the functional determinant that you have here okay and this you can do because so you know that in this approach the partition function that you are computing is supposed to be if you think of it discretized the product over all of this all of these ends square root of 2 pi over lambda because you i mean it's a quadratic integral and you know that this is supposed to be the determinant of this kernel that you have in your quadratic lagrangian and for bosons the right power is minus one half okay okay and then you want to compute that and in order to compute that you define the zeta function of this problem so if you want to do it correctly which most often you don't want to do you have to divide by something like in dimensional regularization to make the thing dimensionless okay so you have to introduce a kind of mu which is like the mu of dimensional regularization so that you have a dimensionless determinant and then you define the zeta function of this operator to be just the sum of all of these objects of this eigenvalues the non-vanishing ones and you also have this 2 pi mu squared to the power sitting okay and then you have so you have to make sense of this product here and how do you make sense of this product and now i am at stage okay maybe let me keep let me keep this one up um so then the next step is to in to check if you can reproduce the blackbody result like that and this was actually done in a paper in the first paper by hawking on zeta function regularization he did the scalar field exactly like that and it is really extremely readable right it's all everybody's after that using this kind of presentation for for this computation and so so i'm taking mu or alpha which is the same thing equal to zero and i take the large volume limit again and then i look what happens to my zeta function and the zeta function i will so mu is zero and then becomes something like 2 pi nu to the power s volume d over 2 pi to the power d i again get this this if you want this integra the the sums in the spatial directions become integrals and there's only one sum remaining which is the matsubara sum over the over the time frequencies if you want okay so the only thing that remains is some nd plus one only the only time remains small if you compute the partition function if you compute a partition function time is small you're not so you're not like in scattering theory doing also t goes that this coordinate sending to infinity so it's not zero temperature this one stays small so a partition function already means that you always have the time circle if you want this is a small direction in your problem okay only the time direction is small and that's why this one stays discretized then you have 2 pi over nd plus 1 divided by beta squared and the other ones become continuous and you have the whole thing to the power minus s then there is one step that you have to do so there's in this case there is one divergence which is related to what happens when the matsubara frequency is zero because then this integral just diverges and you cannot do a lot so you just remove that you put a prime and then that's done you do dimensional regularization or something like that for that so you just remove the guy nd plus one equal to zero and then everything is pretty good if you want and then you do a change of variables and you continue and in the end because you see if you do that you can put the 2 and make the sum over over n star only and you begin to see the zeta function appearing already again the standard one so what is going to happen if you do that is you do a change of variable you massage the the thing and you are going to get so vd and then the same kind of volume of the sphere discussion that you had before two pi to the power d lots of you also get rid of a 2 pi of beta which you pull out of everything and then you change variables integration variables so in the end you end up with something similar than we had before not quite the same thing over gamma of s okay so this is what the zeta function turns out to be and then you have to use that in s there is so one over gamma of s you know exactly how it behaves so it's s plus order of s squared and remember that in this formula for the partition function you're supposed to compute two things zeta at zero i hope i put it up i put it up maybe i forgot to put one formula there so let me add it now so partition function explicitly in terms of the zeta function is it's basically one half of the derivative of the zeta function at s is equal to zero and then you have to check if zeta is in zero is zero in order for the result not depend not to depend on this regularization that you have chosen that's the formula usually you only give this part but somehow you also have to check that zeta of zero is zero so it doesn't depend on the regularization and indeed if it does i don't know so what usually happens at least as far as i can tell for instance there was this long discussion so all this is in front space so it's easy and there was this long discussion in the physics reports by david where he was arguing that for the scalar field you should use the conformally coupled scalar field in order to do something i guess this is related to that because then you went back to that situation and things behaves much better i don't know i'm not such an expert on that i'm just telling you that in the case is that you can look on this too high this is always true you don't work so the result is kind of of this regularization independent from that regularization at least i don't know it's not my i mean you should ask somebody who is a zita function expert and not somebody who pretends to be yes so this is the new this is the chemical potential because in this case of only large volume i don't need any [Music] [Music] prefer to use mu for the chemical potential i prefer alpha but for some reasons mu is sometimes better but then you are not usually the parameter that you use in dimensional regularization to put to make everything a dimensional [Music] so this guy depends on you but this guy okay depends on you and you and i'm putting new so it's this guy which is zero and here this is still depending on very good so then it turns out that and that's what i wanted to say now this data function which you can do by elementary methods because you are in one of these rare cases where you know all the eigenvalues exactly and you don't need to use any approximation methods so then you check because of that because of this at zero it is zero and if you compute the derivative and you you get this finite result okay the derivative at s is equal to zero you can see that all the others have no problems in s the only okay so at okay the end result is that the guy at zero is zero and that the partition function turns out to be if you massage everything turns out to be psi minus t so you have to compute this derivative with respect to s put s is equal to zero put one half you only have to put the derivative of s here because otherwise all the rest vanishes so it's basically everything that is there at s is equal to zero except for this guy that you must have okay i can show you the detail i mean of course it's if you do it as a course you have to okay so the result is this result which is very similar to the result that we had by canonical quantization so you're kind of happy the only thing that is different is that it's psi minus t and not psi of d plus one but then the whole trick about this uh about zeta function computation is that when you do it like that you end up in the wrong part of the complex plane where your zeta function is not defined by the series you think it is defined but you know that the analytic continuation of the zeta function has very nice properties and one is this inversion formula you have to use this inversion formula which the for the zeta function which brings you back to the parts or to the good part of the complex plane and the whole magic of analytic continuation of this data function stuff is so somebody has to give you these properties of the data function to bring you back to the good part of the complex brain this is the usual thing some of the integers being minus 1 over 12 you know the the standard stuff so this sum is not well defined but then you regularize it like that and you bring it back and then you get the minus one over twelve it's the same thing here okay this there are of course experts on this but this is how it works when you do this is what you have to do this and what is nice about this completed zeta function is that this reflection formula which is the key in analytic continuation is very simple in terms of the completed zeta function and much simpler than it is in terms of the standard theta function where there's lots of factors of pi and gamma floating around so this is the guy that you have to use at z is equal to d plus one and then you have shown that your path integral gives in a very nice and covariant way uh the same result that you had by canonical quantization but of course you have to the price to pay is this zeta function technology is it okay okay oh well you will not see any gravitons but okay then there is something more that we need which is the casimir energy on r d minus 1 cross s1 ld okay so i mean what i've shown you in this lecture is nothing i have shown you the blackbody result in two ways how to compute it but it's of course a complete standard result but it's interesting to see it in that way because then the generalization two more small dimensions is just almost technical you just have two sums and only n d minus one integrals but it's the same zeta function that you compute and you have a double sum in terms of a single sum only matsuba okay for the casimir energy the problem so you have this hamiltonian and if you do not choose normal ordering so you have this kind of things to compute and you see that the zero point energy for d spatial dimension is something like so you have to compute this thing at zero temperature and you see that you have a very similar problem where instead of having the products of these eigenvalues now you have the sum of these eigenvalues with one half to compute that's the kind of divergent expression that you have to compute and either people have done really they have put the system in a box and compared what happens if you take minkowski space and then the space between the plates and subtract one result from the other and then stay with a finite piece so that's one way of getting a finite result for this for this object here and the other way is again to do zeta functional normalization or regularization of this sum so that's the analog of the one plus two being minus 112. and that's the one i'm going to use and by very similar so you define instead of defining the sum of the one half of omega you put the power minus s to the thing so that it that it converges and then you do analytic continuation of the result that you are going to get let me not go so in fact if you do the zeta fung let me just say what happened the zeta function regularization of this sum you will see as compared to the computation before it's the same computation than before basically up to some details the only thing that is changing is that d goes to d minus one because you only have now you have one small spatial dimension and all the rest large because to get the casimir effect you have you have to have boundary conditions in space so that's one d goes to d minus one beta which was ld plus one you have to replace it by ld the length of the small dimensions and the difference is that s you have to go as close to s minus one half because that's what you have because you have to to do the square root of k squared that's what what's omega is so that's related to that if you do that computation in the same way than before the casimir energy for periodic boundary conditions of a scalar field with periodic boundary conditions in only one dimension has exactly the same numerical factor than the black body result and there is on top of that just the thing that you need to make everything dimensionless so then you compare and that's what lots of people have done so this is a complete low low temperature thing that you do this is the leading contribution to the partition function at low temperature you see that up to something which is fixed by dimensional reasons the factor here all the pi's and the two and the zetas if there are is exactly the same than in the black body result if you compare these two situations and this is kind of at the heart of this cardi formula where you relate what happens at low temperature to what happens at high temperature because in some sense so the same numbers that i had put up up here exactly here for the cosmic energy so i started at 10 right so i have a bit more time but but this of course is slightly different right you can see that okay and then you can do two small two small dimensions if you do two small dimensions you are now looking at your scalar field in space time where you take only the d minus one dimensions latch you take one spatial dimension small and if you are interested in computing the partition function the temperature is always small okay it's always a circle as i've seen so you have two circles so you have a torus here and you begin to see that there can be a symmetry in this system when you exchange the role of the small spatial dimension with the with the small time dimension and modular invariance has to do with that playing with the two circles the two periodic boundary conditions that you have and and switching these things around and at the same time switching low and high temperature okay so here is the volume of the large spatial dimension that remains minus one okay ah you're right it's wrong of course the power because this should be d and because the energy it's not dimensionless really people are careful it's great i wish i could say the usual stuff that i'm putting this arrow in to check if you are but it's of course not true okay so two small dimensions is in space time you are looking at a testing here right and you do the zeta function again and the only thing that is going to change for this and now look you keep the mu because the mu is a kind of non-trivial twisting of this s ones so if you don't have mu then you really have that if you have the mu then you have a more general torus not really the one which is like that in a way that one can make completely precise and as nikola was saying there also should be a new but now i don't write the new because nobody ever does the new in fact you just so well only the people who are doing it vigorously so now the thing that factorizes in term in front of everything is the d minus one dimensional volume in the large direction so that's the one that you pull out which would be area in the case when you do d is equal to three and then you have two so the this is because this d minus 1 sums become integrals and now there is really a d d minus 1 k over nothing and you have a double sum nd plus 1 and d you have to do again the prime and the eigenvalues so you don't only have the matsubara frequencies but you also have discrete frequencies in the small spatial direction oh i ld over ld and the squared plus the large remaining directions and i runs from 1 to d minus 1 the whole thing to the power s then you repeat the computation of the partition function and in that case you are going to find the following thing so it's interesting to organize your alpha and so alpha and beta you can reorganize them in the way that people parameterize the skewed torus which is you define this modular parameter to be tau plus i beta over ld and you can just at the moment think of this as just a complex change of variables from your alpha and mu from your sorry from your beta and mu to these variables and then you write the partition function and you do the same computation in this case zeta function by using the correct reflection formula and the most compact way to write everything is the following expression so it will be over two l d d minus one now it has really to be d minus one because it has to be without dimensions and then you have a 1 over tau 2 to the power d minus 1 over 2 so this tau 1 and tau 2 are just real and the imaginary part of the modular parameter and then you get this completed eisenstein series if d is bigger than one and the result so this completed eisenstein series is almost like a zeta function except that it is a completed zeta function so it's the same decoration factors of gamma and pi which makes everything easy and then there is a double series and this is a real analytic eisenstein series which is really given like that it's a double sum where you are not summing over the thing when both integers are zero or sometimes you denote this by that minus so that's the same thing then the zero that we deleted before there is a certain power of tau two uh in the denominator in the numerator and then it's nd plus one plus tau and d modulus squared okay that's a real analytic eisenstein series and why people like it is because it's modular invariant what does modular invariance mean it means that this object has is invariant under under the symmetry tau goes to a tau plus b over c tau plus d where a until d are real are integers sorry and you have this determinant condition that this guy needs to be one okay this object here is invariant but the partition function is not just this object here but it has an extra power of the imaginary part of the modular parameter here and that's why the result will be only modular covaria and a not modular invariant but that's still good enough for you to relate low and high temperature to the power d minus one ln z d of tau okay so that's the modular invariant covariant partition function that you are going to compute and then you can have so this result is valid for d strictly bigger than one and also i should say that this modular invariant real analytic eisenstein series is not the ones that you find directly in most books on modular forms because you find usually a holomorphic version of that this is not holomorphic it depends on tau and tauba because there's the modulus squared which is there there is a version of that which is really holomorphic and that's what appears in most of the discussions on the mathematical side but of course this has also been studied in great great details real analytic not holomorphic in that case and [Music] then maybe my last remark is that you would think that for d is equal to one you just use this and find the modular invariant partition function of the free boson in one plus one dimensions it's a bit more complicated because in the this case for d is equal to one this series does not converge and it's not a good series so you have to do a longish discussion and this is the discussion that is done in it's own zuba or something or also in polcinsky to show how it's called i think the first chronic limit formula to show how if d is equal to one the real thing that you are getting if you are really careful is so you would think naively that you get this if you put d is equal to one but that's not true you have to do it carefully and what you get when you do it carefully you can do it by hand also it's but you can also look in all of these books but it's kind of hard sometimes to understand the logic you really get this thing here which is the two-dimensional result if you do it clearly but it's not direct to get it as a limit of the higher dimensional result some extent the higher dimensional result is easier than the two-dimensional result and as i was saying so this is the the kinteta function and this is the stuff that you get from the particle on the and you have periodic boundary conditions okay maybe let me stop here [Applause] yeah so i mean in the gravity an interpretation of what it means to [Music] take modular dimensions of your partition function it would be interpreted as some instance like things that that have an interpretation of security is there something similar in this context but this is just a boundary picture you see this is just the scale of field sure sure but i mean are you again uh there is one way to see this which is one method that i have not yet talked about you can also derive this result in a third way so now we can do canonical quantization and there is a third way to do this stuff which is this uh work line for a showing a proper time [Music] if you do it in this approach which is in fact the most powerful approach and which is the one which is of course the mini version of string theory if you do it there you rephrase your problem as the partition function of a particle or not the partition function has something as the heat kernel of a particle in d plus one euclidean dimension and it turns out that if you compute this heat kernel by path integral methods and you have a circle you have to take into account the path that wind around the circle yeah and this gives exactly this other sum and this other thing that you have and well this would already be if you do the black body result by this method you have the particle that y is around the surface so if that's the intercept so you do have this picture but you have to do work life and i'm not sure that i have time to talk about that okay everything takes time but this is the method where modular inverse is the clearest in some sense because in some sense you can see without computation because here you first do the computation and then you look at the result and you see that there is this limit but sometimes simply you want to understand before you do the computation not after and in this world when you can see it if you treat the torus but this is nevertheless it's not gravitation it's purely it's in this approach that you have at least something similar questions um [Music] so again so you mean that you remember for this zero you mean that you prefer to take some informal [Music] there has been this well i can show you the the plates where they are discussing that so in his physics reports the old one on quantum fields in curved space the width he knows the boundary conditions for electromagnetism we're discussing perfectly conducting ones then he's trying to understand why the scalar field is working or not working and why is it depending on the regularization in the discussion he says now the special feature is what but if we want the same kind of good results for the scalar field we have to take the conformally coupled scale this is much better and here's this discussion of the improved energy momentum cancer that is by jackie and these people everything is conformal so even era feels in curved space if you look at it people usually do the conformally coupled scale maybe and now it's periodic boundary condition in this small direction and so you said that for large volume it doesn't matter but yeah no no yeah absolutely i take periodic but i take periodic for in three dimensions for a special reason because i've shown that perfectly conducting because in that case is equivalent to periodic for the scalar in the one small dimension so with that i am discussing photons in the cosmic so you can check the only thing that you have you put d is equivalently you put ld is equal to 2d which was the separation of the plates and then the partition function that you are getting is exactly the ones of photons in these kinds of box which the partition function of the finite temperature has been effective and also of gravitons if you choose adaptive boundary conditions somehow what was nice is you bring back the problem to scalar field only in this particular case i would say otherwise but we were kind of surprised that it was this is the easiest way so you you look at these big books on the casimir effect final temperature cosmetic effect they give the partition function in all details and kind of first mapping into a periodic scale gives you the result much faster i would say but you have to do this trick on the oscillators to map to systems this is probably our own our only contribution to the field because this field is studied he all did also he cuddled me with books and books on that