 Thank you very much, Jean-Paul, and thanks for giving me the opportunity to present something here. So I was asked to give an introduction, and I was not asked to give solutions, so I will avoid giving solutions, I will talk about problems. But if you are interested in solutions, I mean there are some talks and post-hours related to what I talk about and then address please the people who have the solutions. So it's actually not the problem or the topic of the background approach to the FRG or in general the background approach, but what we are after is to discuss background independence, which is very crucial in quantum gravity, but of course you also have this problem around in other theories if you then introduce, because there it's a luxury, a background field, but in gravity I hope I can convince you, I mean it's not the luxuries just built into the theory. So I will talk about this from the perspective of the FRG, because we have the FRG conference here, but of course there are also many other approaches where you can discuss background independence of quantum field theories and I can already because I know his slides point to the talk of Daniel Becker, he has a very nice physics introduction to this setting in his transparencies. So if you are not satisfied with my technical introduction, which may not be pedagogical, please wait for the talk of Daniel. So I think I don't have to explain the setting here. So we want to solve the quantum field theory, we use the flow equation and the flow equation is an equation for the effective action which depends on the mean field, so the expectation value of the field. So let's assume we have one field or several fields, it could be a super field, but now we try to solve this theory with an expansion about some background. So for the time being you can just think about a simple scalar field theory, let's take the easing model, there you would have a real scalar field and just for some reason I will explain some of the reasons you could have, I want to split the full field into a background and then a fluctuation field which I integrate out. So this means I have an expansion of the effective action, so I have my effective action here at scale dependent and it depends on the full field, phi, but now I split it in phi bar plus phi and I expand it. So I get the effective action of the mean field, so of my background field and then I have just the expansion coefficients here which are just the correlation functions and they are here the 1pi correlation function in this given background. And just notation wise the derivatives here are just those with respect to the fluctuation field and of course you can also take derivatives with respect to the background field and then you get these mixed derivatives. And of course in the general context, I'm not talking about gravity at the moment, you do this in order to facilitate the computation. Maybe you have some information about the background, I mean about the physical background and then you expand your free energy or effective action about this background. Okay, so as I said you could take a solution of the equations of motion, you could take some topological configuration or defect because you think these configurations are important for the physics you discuss. That might not be a solution to the equations of motion or you just have a non-trivial numerical problem and you just take a background which simplifies your numerics. So there are several physics and technical reasons to choose a background. But as I said here take a scalar theory so it's your luxury. We just do this in order to get better results or to improve your convergence whatever but it's the luxury. Okay, so now because you start with an effective action which is just depending on the sum of these two fields you have because of translation invariance and shifts in the field you have a non-trivial identity namely, well here it's trivial, the derivatives with respect to the background are just derivatives with respect to the fluctuation. Okay, and then you can just work out relations for the correlation functions because at the very end what we usually do in our approach is we work out relations for the correlation functions and then these correlation functions satisfy the following identity. I take the end correlation function so that's the end derivative with respect to the fluctuation field. I take one derivative with respect to the background and it's the end plus first derivative with respect to the fluctuation field which again comes from the fact that I have just an effective action which depends on one field. Okay, that tells you that your effective action which you write down in two fields depends only on one field. Okay, and if you make an approximation to the flow equation maybe this identity is not satisfied even though it was so trivial, right? Now this will be called in the more let's say intricate context of gauge theories or gravity. This is the split-water identity or the Nielsen identity and also is related to the modified slough of tail identities which we have in water, which we have in gauge theories. Now if you put in a regulator which itself depends on the background field then this identity is broken because you put in a function which only depends on the background then of course this translation invariance is broken and you get cut of dependent terms here and then this is a modified split-water identity and a modified Nielsen identity such as we discussed modified slough of tail identities in gauge theories and there's a tight relation. So that is a set up I mean this is a trivial example but these are the equations we will deal with and as I said here I mean choosing a background is the luxury we do it for technical or physical reasons but of course physics will not depend on the choice of the background. That is the background independence on this very trivial level we are discussing. Okay so I think I have said a bit some technical terms and explained what we mean with background independence and if this was to technical please wait for Daniel. Okay let's go to gravity. Okay and the phi is just the metric field we may have other fields if you have metacontent so let's write down the Wetterich equation for pure gravity. Nobody complains but you should complain of course this is not working this would be great we would have one field everything is nice so background independence is no problem. Of course the second derivative of gamma with respect to the metric has zero directions we have a gauge theory we have different morphism invariance so we have to gauge fix. Okay so we do a gauge fixing and since there are also different morphism invariant approaches I will also later if I have time say something why in my opinion they are as good as gauge fixing. Okay I mean I have to put in some provocative statements every now and then otherwise we won't have a discussion so that's my first one. Okay so we put a gauge fixing here I took a linear gauge fixing there are other gauge fixing I'm very general here so you have some function some gauge fixing function so some operator apply to the metric field you square it so you do the standard thing you know from quantum electrodynamics or QCD or whatever in the gauge theory so if you put alpha to zero you have a sharp implementation of the gauge fixing if alpha is not zero you just average over gauge fixing conditions. Okay now you see already here if this gauge fixing should be quadratic in the gauge field so the gauge fixing term you have to choose a background metric in gravity you can't avoid this. Okay you can say well why not why not drop this term here square root g bar and I don't have need the dependence on square root g bar here in the gauge fixing condition yeah but then you choose a flat background so it's unavoidable in gravity if you gauge fix that you have a background and since I said I mean what about gauge fixing and gauge or different fission invariant approaches gauge fixing means I choose a coordinate system in my configuration space and I have not seen any computation in the flow equation otherwise which does not choose a coordinate system so if you choose a coordinate systems as good as choosing a gauge so even different different fission invariant or gauge invariant formulations in some implicit sense choose a gauge. Okay so now we are in business so now we can invert the second derivative with respect to the fields of the effective action we have now a split they have to be needed to introduce a background so we expand about this background this is a linear split so we have a fluctuation field as I explained before and the derivative here in the flow equation is that with respect to the fluctuations not that with respect to the background of course we have now an upgrade of this trivial identity I've showed you before I drop if you wish explicit expressions for the non-trivial terms so this would be the standard identity you take the derivative with the h and it's just the derivative with respect to the g bar and then there are terms which come from the gauge fixing which now also introduces a difference between the background and the fluctuation field and in our approach of course also inevitably there are terms which depend on the cutoff function in gravity again I mean if you choose a flat background in your cutoff function it means to choose a background so also Rk will depend on this background okay so you have these two terms okay so you have a non-trivial identity now to solve and in a gauge theory this non-trivial identity which tells you that fluctuation derivatives are related to background derivatives also encodes the non-trivial BST symmetries or gauge symmetries in gravity differential symmetry of your theory so it's a highly non-trivial identity it correlates it relates fluctuation correlation functions and background correlation functions the background correlation functions have covariance or covariance but the fluctuation ones don't have this they have non-trivial symmetry identities the sloughn of tail identities and going from the trivial symmetry correlation functions to the non-trivial one is encoded in this identity here so the L term encodes the non-trivial implementation of gauge or differential symmetry for the fluctuation fields okay but nevertheless what this identity tells you it's the same as before at the very end we only have one field so it encodes the background field invariance or independence of gravity so that's the equation if you wish which we have to solve okay so of course now what you can do is to say okay I'm in the regulator dependence fair enough if I put the regulator to zero I mean it's gone so let's drop the regulator terms in this non-trivial identity as a first approximation because at the very end we have to compute something right and it's getting a bit difficult to solve these equations and then we go on since this was a nice argument we say okay let's drop also the L terms right it's gauge fixing terms I mean the physics is in the gauge invariant or different fissure invariant part so let's drop also those okay then you you are back at the trivial identity right which tells you that the theory if you solve this in this identity in this approximation is a function of the background field plus the fluctuation field and that's the background approximation and it works very well however I mean now we are in a state since in gravity I would say since five years where we overcome the background approximation so now I can speak loudly about I mean what is the problem with the background approximation and why you should be really careful okay so now let's look at the Wetterich equation in the background approximation you have one field simple so you have a field of one background field and here it's just a second derivative with respect to the background field because the second derivative with respect to the fluctuation field in the background field degree very simple equation background independence seemingly is restored in this approximation I say seemingly because I tried to convince you now that this is not the case okay so it looks like a simple equation you have one field but now the dynamical field and the background field are identical and the cutoff function depends on the dynamical field it depends of course on the background field I mean you cannot put the dynamical field in the cutoff but now you mix them up so it depends on the dynamical field so you put in dynamics via your cutoff function into your theory so let's see what you can do okay now I'll make a regulator bootstrap since Daniel bought me already I'm not saying that you should do this I'm telling this because you fail if you do this so now a very simple cutoff functions the following you take the regulator to be proportional to gamma 2 because the only non-trivial operator which comes from physics is gamma 2 so maybe we can get rid of it okay and then there is some cutoff function little r it's dimensionless and you can choose it whatever okay I leave it free at the moment now let's look at the cut let's look at the flow equation now you have one term which does not depend on gamma 2 it's one loop exact you can immediately integrate it and it gives you one loop effective action okay and then there's another term where you have here the let's say the infrared regularization that's the r over 1 plus r it gives you an ultraviolet finite and infrared finite flow and then there's something you could interpret as a generalized anomalous dimension so it's a functional anomalous dimension now you have a representation now because of your cutoff choice in this approximation where all the non-trivial physics is in the anomalous dimension I think it won't work in general okay but I mean if you're not convinced let's go on now I do the following because I have this large freedom in the cutoff function I like my flow now I call the right hand side of the flow equation f and I choose my f as at which and I try to find a regulator which gives you this flow and I can ensure you I have played around with this and you see here I mean this is referring to work with Daniel Littem and there's a one in gravity with Sarah Volkatz a student at the time in Heidelberg you can do quite a lot so if you give me you want to have some flow in some theory which has a certain form give it to me I give you the cutoff function which produces this so now you should be alarmed because I'm telling you you give me a theory you want to have something coming out and I can produce it okay and why this is so I mean you cannot find solutions to this differential equation for any f but for quite a large variety why is this so well I mean we have identified the fluctuation field with the background field and which means we are bringing in additional physics via the regulator right and of course we shouldn't do this okay so if you're still not convinced I will present an example and I know I'm in the asymptotic safety community I think I'm loved for this I like to make QCD examples and as examples go they never really work right so you should be very careful to bring the information from one theory to the other but nevertheless I will go ahead with this example this is again work with Daniel from 2002 you go to young mill theory there's a universal beta function because you're in a critical dimension four dimensions you're in a critical dimension of the theory so the one loop beta function young mill theory is as universal as it gets you cannot you can do everything you want with your regularization scheme it always comes out the same the two loop beta function which is also universal where you need math independent schemes but with the one loop beta function in a massless theory like young mill theory there's no way that you can change it and since I say this I will now tell you how to change it I go to the background approximation I take a regulator a class of regulators which again is proportional to the covariant derivative I'm doing a one loop thing so the second derivative of the effective action is just the Laplacian with some spin term so it's minus the covariant derivatives squared plus spin contributions for example for the spin one for the gauge field the spin contribution is just sigma mu nu times F mu nu sorry it's proportional to F mu nu so and then I allow for a regulator function here which is infrared divergence there's nothing bad about infrared divergent regulator functions they just make your infrared suppression even stronger okay you compute the one loop beta function in this background approximation that's the result so I have written it the way there you see here's the universal result and then you see that I mean depending on which divergence you choose here in the infrared you change it that's not good now you can use the split water identity in order to distinguish the field dependence which comes from the background dependence of the regulator term you see I mean this is now background dependent and the one which comes from really the dynamics of the field and you can cancel these terms and you get this correct one loop beta function if you would straight away look at fluctuation correlation functions of course you for all cut-offs get the correct one loop beta function so this term comes only because I identify dynamic fields in the background fields in the cut-off with dynamical fields right so in young myth you have to choose an infrared divergent regulator because this implicitly implicitly brings in an additional scale in gravity we have this additional scale because the Newton constant is already a scale dependent quantity so you expect this also for a convergent cut-off functions so again alarming thing okay but background days are over right so we have fluctuation approaches so we can now very well distinguish between fluctuation fields and background fields in gravity and also in other approaches and I think this is one of the topics here of the remaining talks to show what are the ways to overcome this problem I will now also very at the very end of my talk briefly say what we are doing but let's first of all go on okay so I hope I have convinced you that background independence has to do something with effect to distinguish between the fluctuation field and the background field in particular the one you introduce via the cut-off function carefully okay so now you have the approach you have a fluctuation field H bar a background field G bar and now also let's say improve or allow for an improved setting there's a geometrical approach which is gauge invariant so then even the or different fissure invariance then even the correlation functions of the fluctuation fields are actually different fissure invariance but what you buy with this and that's in all the the gauge or different fissure invariant approaches the case you buy a sort of non-locality and we'll come to this later because there you have to be very careful there's no free lunch whatever you do there's a lower bound for problems okay you make your expansion and then I hope I have convinced you the there will be non-trivial identities which are satisfied by the fluctuation correlation functions right in those guys you have to take into account okay and on top of this because they have non-trivial dependencies on the well they satisfy the non-trivial slough of tail identities there is nothing like a gauge invariant approximation to the effective action in terms of the fluctuation field which simply does not exist it's not bad we know this from QCD and other gauge theories the effective action is just given by the set of correlation functions and if you want to extract physics the best is to go to the zeroes order correlation function which is the effective action of the background field that one is straight away different fissure invariance and for example in QCD you can show that this is directly related to scattering elements of the s matrix okay so at the very end we are interested in this but in order to compute this we need all these guys and they have non-trivial identities okay I made this joke already in Stockholm one and a half years ago and my number of jokes is limited so I make it again so those Italians here in the audience certainly can translate it to you if your Italian is not that good so what we want to have is this guy here which is background independent this correlation function the zero orders correlation function I hope I have convinced you starts the background independent information so if you change the background here nothing will change okay and the other things if you change the background so the other correlation functions the fluctuation correlation functions which are those on the right hand side of the flow equation so those in the diagrams they change the background okay so you have one thing which shouldn't be background dependent in a sense that the equations of motion you get from this in the correlation functions are background independent but those guys are okay so I nevertheless now give you the English translation I find it a bit boring even like the German one so if you want things to stay as they are things have to change so if you want to have background independence for this guy here those should be background dependent you shouldn't even look for truncation where they are not because then you break by having something which is background independence you break background independence okay so it's a bit convoluted but I think it's correct okay so now we have our effective action we look for expansion schemes and I again mention there is no different fism invariant expansion scheme part of the effective action if you look at the meta part there actually we know also from QCD that you can expand in gauge invariant or BST invariant objects but for the quantum gravity part pure quantum gravity part there is nothing like this to satisfy non-trivial identities it's very difficult to write something down in a closed form which satisfies these identities okay again what is at stake well if you identify or if you use background if you use different fism invariant truncations I go again to QCD there we know that the background correlation function has a trivial scaling it scales like p2 so the propagator is 1 over p2 it looks like a QED propagator I'm now talking about Landau gauge but that applies to essentially all linear gauges the fluctuation propagator actually shows the mass gap of QCD so it is a non-trivial correlation function and if you don't have this mass gap in the correlation function you won't find confinement better very accurately said you lose the confining property of the order parameter of QCD you don't want to work in this approximation in QCD it's again hard to translate this in gravity but this is essentially the last warning sign I give you and I'm finished so resolving these differences is getting if you wish less and less important the less important your power counting for the operators is so it's very important to resolve the difference between the fluctuation field and the background field for the cosmological constant versus the graviton mass parameter and then it goes on I mean the ordering here is of course my liking I cannot prove anything here I can only tell you please be careful okay does it matter yes it matters we have examples by now in metagravity systems where the sign of diagrams is different if you look at the background correlation functions and the fluctuation correlation functions okay so the rest was what I was supposed to talk about but I knew already that I wouldn't have time and I come to the conclusion so I hope I have convinced you that we get background independence in a general setting from background dependence and this is crucial for me so if you try to make a shortcut here you potentially will fail I'm not saying you fail but you have to argue a lot the flows for the fluctuation field which do not involve any correlation functions of the background field are closed so you can work out the flow equation system for the fluctuation fields for the gamma 0n I've showed you and then on top of these flows compute the flow of the background field so that's how you should do it that's actually even related to also the locality aspects I showed and of course applications not only in our framework where we make a vertex expansion about flat but now also curved background of course are plenty okay thank you