 quantum field theory in the sitter space and one what can we do about these problems by resorting to number two treatments so okay a quick outline of my talk I will just skip this so why are we interested in studying quantum field theory in the sitter space well as probably most of you already know the sitter space is like the leading order approximation to the inflationary stage of the early universe and I could also add that in the to the current stage of the accelerating universe and so going back to the early universe we are interesting in the evolution of fluctuation of quantum fields and to be able to say how these fluctuations behave at late times in the inflationary stage now in particular spectator fields which are the ones that do not drive inflation are the ones that we'll see at the sitter background which is fixed and they just evolve in this fixed back so as I will tell you in a few moments there are several sorry several unsettled questions with regard to interacting fields in this in this the sitter space so just to briefly a brief reminder the sitter space can be written in this in this way I mean the metric can be expressed in this way in what we call the cosmological patch which is only the expanding half of the sitter space and what we want to do is to add a scalar field a minimally coupled and with a self-interaction so here I put a mass but in a few moments we will see that what happens what different values of the mass so since we want to calculate flag I mean I studied the fluctuations at late times what we need to do is calculate late time correlators so things things like this kind of of quantities so in order to do these we need the so-called closed time path formalism or also called the in in formalism which allows us to compute expectation values rather than for example scattering amplitude this is a necessary inconvenience of the evolving background time-dependent background okay so what kind of problems do we get into when we try to do this so consider a free field with mass equal to zero when we compute this late time two-point correlator function we see that it grows with cosmic time so if time goes on for long enough these fluctuations eventually grow at the point that we don't longer control what what happens with the background we cannot ignore other effects this can be traced to the fact that for a massless free field there is no the sitter invariant vacuum for the quantum field theory however okay for it there's a lot of things we can relax here let's assume for a moment that we have a small mass but different from zero now we see that the correlator function is no longer growing with time it's just a constant but it has this sort of behavior with the mass so for small masses compared to the hover rate this will be very big so okay for free field it might be not very important but once we turn on interactions we see that the perturbation theory has this loop factor in which you we don't only have the in the coupling constant but also this enhancement with the mass of the field so let's say for masses below more or less this this amount we are in problems because our loop factor might be very large so the perturbation theory seems to break down in these cases so what can we do about it we can resort to non perturbative treatments these are some a few of them there are many more and I will briefly talk about this these three cases so in the first case this is a an unperturbative approach for the in-in theory I mean the usual quantum field theory in the sitter space what we do is to generalize the standard effective action to a new effective action which we call the 2pi effective action this new quantity has not only it not only depends of the mean value of the field but also on the exact propagator so we treat exact propagator as a new degree of freedom independent from the other so formally we can write down this this thing in this way where this last bit is a sum over a lot of diagrams with this particular property of being too particularly reductable but more importantly they will build them using the exact propagator so to give you an example in our land of fight to the fourth theory we get these diagrams at the 2 and 3 loops but remember that now these internal lines get have the exact propagator inside them so all of these diagrams in particular let's look only at the to look part we see that even at the leading order these are very technically difficult to treat in particular this this diagram over here is non-local and since with the it depends on the exact propagator is not much we can do about it so what we we do as a guest is to keep only the local part of the of the the 2p effective action this is called the heart approximation and under this approximation what we are doing since again we have the exact propagator inside here is doing an infinite resumption of the so-called super they see diagrams of the standard perturbation theory so it's as if we were summing all those diagrams from the standard perturbation theory so these diagrams have the property as I said they are local so they address the propagator and we get a number to add the correction to the mass this approximation becomes exact in the land large and limit but for a different value of n I mean n being the number of fields if we were to put an n fields like this one there are all sorts of subtleties regarding the renormalization of this effective action and I don't have time to go through that now but if you're interested we can talk about it later in the coffee break but we have done some some research in this regarding these problems okay so moving forward now I have two equations of motion with a very respect with the field and with respect to the propagator and these equations of motions show that for example the propagator picks up this mass and this mass is not the three level mass of the of the field but rather the the one that is given by this self-consistent gap equation and this is because I use the exact propagator inside here to compute what this mass should be so this gap equation has has solutions that allow me to set the the three level mass to zero something I wasn't able to do in the in the standard perturbation theory and I get a dynamical mass that goes like this suddenly in ordering in the coupling constant so what is interesting here is that now even if I try to put the mass to zero the interactions give me a dynamical mass and it has a non analytical lambda dependence which is a signal of a non perturbative result so well so so much for the the to be effective action now I will briefly tell you about a different approach which is called the stochastic inflation here the idea is to okay so since them the modes of the field are continuously exceeding the horizon and going classical from the point of view of the large modes they see as as if there was a source of stochastic noise I mean these modes that continuously go out of their horizon give them random kicks so we can write down a lunch of an equation for the long modes with certain properties at least in order the the noise is caution and then from here we can write down a focal point equation for a PDF for the field in a function of time and in particular let's late times it looks like this so we are able to calculate late time expectation values in this in this way using this PDF and we we arrived at this sort of results when we consider the massless case for our lambda factor for theory and from here in comparison to the free field result we identified this dynamical mass and found it to be like this so as you can see the lambda dependence is also non analytical it depends on the square root of lambda but the numerical factor is different from the one that we found in the previous from the 2pi effective action okay so so far we have these two approaches there are no there is no clear way how to improve systematically this calculation to go beyond this order in lambda so we can try the different different approaches so finally I would like to talk about the Euclidean the Cedar space approach what we do here is so instead of considering the expanding patch of the Cedar now we consider the global coordinates discovered the whole the Cedar space we perform an analytical continuation to imaginary time and then we need to compactified this imaginary time dimension in order to get a regular metric and the result of this whole process is that we end up in a Euclidean sphere here I put it these the number of dimensions but usually four okay so now that we are in this sphere we can do quantum field theory what we do is we expand everything in the spherical harmonics or the generalization of spherical harmonics to d dimensions for example the Euclidean action now looks like this where here is the interaction part and this is the free part and from here we can build our free propagator and it looks like this so the interesting thing is that now this is a sum over a discrete set of modes this is spherical harmonics so it's easy to identify where lies the problem when the mass is very small or it goes to zero the culprit is the zero mode the one that related to L equal to zero and it's the only part that diverges when the mass goes to zero so there is only one mode responsible we weren't able to do this in the original in in theory so now that we identified the problem we can treat it number two but the B so what we do is we split we the field in the zero mode plus the rest and we can write down and generate in functional but now since five not is a constant there is no kinetic term for for five not and the path integral because just an ordinary integral so for example for a massless field it looks like this this is something you can put inside Mathematica and it will give you a result so we can compute it exactly okay so moving forward from this generating function we can compute the two point correlation function again read the mass from there and we arrive to this result which if you if you recall is the same as the stochastic procedure so very nice it seems like we are getting a to to to reproduce a result that was done in in the Lorenzi and the sitter space but the advantage here of the Euclidean formalism is that now we can go beyond that and compute corrections that come from the rest of the modes the non-zero zero modes so what we do is split the interaction part of the action in the part that only depends on the five not and the rest this is the part that we compute exactly nonpartuitively and the rest is what we used to expand and and define our new perturbation theory so the new in the generating function looks like this and now we expand only on this s tilde so by doing this we get a new perturbation theory in which phi not scales as lambda to the minus one fourth and phi hat scales as one so for example a term in an interaction term like this will scale as square root of lambda so that's good because we we know that our result has this non-analytical dependence so we compute next to linear corrections to the dynamic mass and in particular we do it for the ON symmetric model so we we can then check for n equal one or n equal infinite different known results computed in the in the in informalism and we did we did it in two ways we can compute it from the two-point correlation function of the zero mode but also we can look at the effective potential and and check that the quadratic part is the same gives the same result so the result looks like this here n is the number of fields again and g hat is the propagator of the non-zero modes and it has the property of being regular for any value of m at least if it's for smaller m than the Hubble constant but in particular we can set m to zero here without any problem since we separated the zero mode and this quantities here are just correction functions of the zero modes computed with the zero the the CTA not generating functional so this can be computed exactly so now we can compare the our results with the the in in quantum field theory for example for mass equal to zero it looks like this so this was the linear result now this is the the correction and also we can for example set the tree level mass to a negative value and see what happens so what we get is that the dynamical mass picks up this this form and it's it's rather easy to see that in general this will be positive so what we have is that we started with a negative tree level mass but we ended up with a positive dynamical mass so we can interpret this as a symmetry restoration okay so in any case these these are competing the large and limit so we can we trust the in in computation and we see that they agree with the Euclidean approach okay so just a brief summary so the 2p effective action provides a resumption of an infinite subset of Feynman diagrams in the in in quantum field theory but it's technically difficult to go beyond the local approximation and this approximation only is justified in the large and limit so other number two particular approaches in particular the number to a treatment of the zero mode in Euclidean the center space allows for a systematic way of computing corrections and we don't need to put the number of fields to infinity we trust the result for any value of n so it would be more general and okay so at the leading infrared order the Euclidean and stochastic calculations also agree and this is just a consistency check okay so just to finish some open questions relating this sort of approaches we actually are not sure whether the Euclidean quantum field theory is equivalent to the in informalism in a strict sense because we are only comparing one one thing here the two point correlation function in the ER limit there is some nice discussion in the in the massive case they compute diagrams so at the dioramatic level they see that this agree we have several questions we can go beyond the effective potential to compute the full effective action if we can do that then we we will trust that the two approaches are actually equivalent and to me that one of the most interesting questions would be if I go to the Indian theory so I have Lorentzian the center space I have a continuum of modes and this continuum doesn't allow me to pick up only one and say this is the zero mode that is causing me the the problems so can I identify someone similar to the serial mode and treat it non-perturatively in a systematic way well we don't know okay also some questions about the stochastic picture and its equivalence to the Indian QFT but okay we can talk if anyone is interesting we can talk about it a little bit and finally if we can make progress only from the side of the Indian quantum field theory and go be on the local contributions so that's all thank you very much