 So while the slides are prepared, let me apologize for the quality of the slides. That is a little bit suboptimal, but I was not supposed to speak here. And then Roberto told me only at the moment where I was traveling and had no, let's say regular computer and net access anymore. But I think it will never less work out. The PDF we had, yes? Yeah. Okay, so gauge invariant flow equation. Well, we have this background field formalism that is mathematically fully consistent, gives you an exact flow equation. Many aspects are well understood, and many very good results are obtained with this formalism. So why, nevertheless, I want to go a little bit further. Why nevertheless, let me phrase it like that. I was never 100% satisfied with it. There are two reasons. First is if you want technical. When you ever, you have two fields that you have to follow all the time, mathematics get involved. Just algebraically, there are many invariants you can construct, and perhaps even more important, at least I lose a little bit the intuition for the flow equation that you have for many other systems. And let's not forget exactness is a good property, but the fact that it can be, the flow equation can be applied in so many non-properative situations has to do a lot with the fact that we have, we know what other objects are. And that is, for me, a little bit more difficult to understand in this background field formalism. The second one is perhaps even more profound and has to do with what we want the effective action to do. After all, we use a gauge invariant effective action all the time for many purposes. For example, we use the Einstein-Hilbert action, perhaps with a cosmological constant or coupled to a scalar field in order to derive the gravitational field equations, the cosmological equations, the equations that are well tested by precision experiments, all that follows from a gauge invariant effective action. What is effective action? If this effective action involves only one field, so it's a one field effective action and the field equations follow from it. Similar for, let's say, QED. We get the Maxwell equations or corrected by Euler-Heisenberg terms and we can really extract the field equations out of QCD equations to what extent it can be done as well. But at least there are two examples where really we want to solve directly the field equations derived from the effective action in order to do all sorts of classical field theory. This goes on because it's not only the field equations, but it's also the fluctuations that we can compute. For example, in QED, we can compute the full two-point functions, correlation functions from the second variation of a gauge invariant effective action. And you can actually do the same for gravity. That may be a little bit less known. For those not familiar to cosmology, when we compute primordial fluctuations in cosmology, people usually have some mixed thing. They assume some bunch Davis vacuum and then some time evolution of these primordial fluctuations. But actually you can do it in a much simpler way in a much more consistent way. You can just take the Einstein-Hilbert action, take the second functional derivative, concentrate on the physical fluctuations as opposed to the gauge fluctuations, invert on the space, and you get just the, you can determine the propagator, and the propagator contains the fluctuations that we all see in the CMB background. So again, what you do is you extract all this information from a gauge invariant effective action that involves only one field. And you just get it in this case by second functional variation and restriction to physical fluctuations. So this object that we finally want to have in a certain sense in order to work further with it, it's a gauge invariant quantity. It has only one field and it should have the property that the second variation gives you for the physical fluctuations, really the propagator. Well, in the background field formalism, this is not so easily the case. We have two fields and as Jan has told you, the propagator, for example, comes from the variation with respect to one of the fields, the fluctuation fields with the other field fixed. So it's not the same structure yet. So the question is, can you have both? Oops, why does it not answer anymore? So we know the flag flow equations have been formulated in the background field formalism. You have again two fields. So my aim that I want to go for is, is it possible to have a formulation of a gauge invariant effective action flowing action that develops only one field and nevertheless, write closed flow equations for it. So that's the question I want to pose. There's an immediate problem. Jan was telling you, second functional variation has zero modes, so you cannot invert it. And the fact that you add some RK does not really help for that. So this problem, you have, first of all, to overcome. You have to find how do you change your flow equation in order to not be confronted immediately to this problem? So the method or the thought I want to propose you is to divide fluctuations always into physical fluctuations and gauge fluctuations. That is, and then we want to have a flow equation where only the propagator for the physical fluctuation plays a role and the gauge fluctuations give you just some measure term. So that's the basic idea. You do that by a suitable projector. So p squared equal p gives you a projector and it has the property that one minus p projects on the gauge fluctuations and p projects on the physical fluctuations. So gauge fluctuations are easy to understand. If you have some metric g bar and you make a variation of it, then the thermomorphism variation of it, you say, okay, this one minus p applied to them should be the same as the fluctuation. So that projects on the gauge fluctuations and you can do the same of course for young mills theories. In this case, g bar will be some gauge filter. So that is the basic idea. You define this gauge fluctuations and then you can decompose every fluctuation. If you have some given metric g bar, you have another one g minus g and you take some difference, you can decompose it into physical fluctuations and gauge fluctuations where the gauge fluctuations obey this property and the physical fluctuations are then p times f equals 2f. So again, understanding is very simple. If you compute the gauge variation or the thermomorphism variation of the fluctuations, now for a fixed macroscopic gauge field g bar, that is a little bit this macroscopic gauge field will be the role of the background field in a certain sense, except that it will become dynamical later. So if you look at the gauge variation of it, well it gives you in this case a term that just comes from the gauge variation of g bar and then some homogeneous term and if you go now to small fluctuations, you can neglect that and then only this inhomogeneous term remains and you identify the gauge fluctuations with those fluctuations that you get if you just apply a gauge transformation on your macroscopic metric g bar and the rest are physical fluctuations. So that's a clearly defined split. For each series, for a long mill series, it's easy to construct these things explicitly. So if one minus p that I often call p bar that's just given by this combination of covariant derivatives and note that covariant derivatives appear here, so it will be a field dependent projection. That's of course a very important thing that gives you all the difficulties if the projection would be field independent as for let's say a pure billion gauge theory without coupling, then things would be very easy. But projections are field dependent and actually for gravity, it is not such an easy task to really find this projector even for simple spaces like the sitter space. But okay, you work and you know it exists and you can in principle determine it. So just as a side remark, once you have this projection on physical and gauge fluctuations, you can decompose every metric g into a gauge invariant part g hat and a gauge part c hat. This I find interesting by itself. I use it for many other things. You do it just by starting from some given g bar and whenever a physical and then gauge fluctuations go in this direction, physical fluctuations go in this direction and you just integrate physical fluctuations. This gives you then a line of physical metrics if you want and then you have the gauge fluctuations or so on to it. So that's the basic setting. Let me take that out and now let me describe you what type of flow equations I want to get and I actually do get at the end. It has a relatively simple structure. So there is an object, gauge invariant, this gamma bar depends only on one field and its flow has a physical part pi and then some measure parts that I will discuss perhaps a little bit at the end. But the important part is the physical part. So what is the physical part? The flow equation is just a standard flow equation with a little modification. The modification is that in the correlation function here there is this index P, which means we only have the correlation function for the physical fluctuations. We are not interested in the gauge fluctuations here, only the physical fluctuations appear. So this correlation function obeys the projector properties that the projector applied from left or right again produces the Green's function. So that's if you want, we have split our fluctuations into physical and gauge and now we look at the correlation function for the physical fluctuations. So they appear here and a similar projector appears here also in the cutoff term. So this of course, this part here depends again on the exact propagator, but not on the gauge part of it, only on the physical part. The measure contributions, they will not depend on the effective action. So you can just treat them separately. Now in order to close it, you have of course to find an expression of the physical correlation function in terms of this gauge invariant effective action. So what is the connection I want to get? Well, you define also for the second functional variation of the gauge invariant effective action that has zero directions for the gauge modes and non-zero eigenvalues for the physical modes and you define again the projected second functional derivative for it by just using the same projector in order to project out the gauge part. And then you invert on this projected space. Also instead of gamma plus R, gamma two plus R times G equals to one, you modify this equation by restricting it to the projected space on the physical fluctuations. So again, here on this projected space, I mean the second functional derivative is invertible. It will not be invertible if you have zero modes, but for this purpose you have the infrared regulator. So it's really an invertible equation, but of course here's not a one, here's only the projector. This sounds harmless, but in practice, this is not so easy because if you go to gravity, already you have to construct the projector and then you really need to solve this type of projected equation. So there's always some conservation of difficulties in the difficulty in this case is here. You can do it for example on flat space, you can construct easily the exact projector, but then going further is not so easy. So, how to get that? Well, we have somehow to take care of this important property that projector, gauge fixing and infrared cutoff are all formulated with a dynamical background field G bar. So G bar is not fixed anymore, but we want to promote that to a dynamical field. So really the macroscopic field, that is will be the matrix that we really use in experiments and so on and so forth. So this seems at first sight horrible because now all the linearity of the construction of the functional integral is lost. So because this macroscopic field appears in the projector, it appears in the gauge fixing, it appears in the cutoff. Things become formally non-linear and especially the macroscopic gauge field, the direct, the connection between the expectation value of the macroscopic gauge field and the macroscopic field may become more complicated. You have only an implicit definition, but at the end all that will not be, will not matter. So the price to pay is the macroscopic gauge field will not be the expectation value of the macroscopic gauge field, but I will argue with you that this property is never needed, it is not used and the macroscopic gauge field can have a more complicated relation to the macroscopic one. Well, you can construct now all the standard things starting first with two fields, with the expectation value of the macroscopic field that I call here G, this is macroscopic field G bar, you get the standard flow equation and now you ask what can I do in order to object on this physical part? The first thing is you only take, you take a particular gauge fixing, namely you make sure that the gauge fixing only affects the gauge modes and nothing else. So the gauge fixing should be quadratic in the gauge fluctuations with some kernel that is not really important and you let alpha going to zero. So this, if you go back to, you still have at this level an effective action that depends on two fields, on the macroscopic field and on the expectation value, but it has a particular form. If you go back a little bit to this picture here, ups, that is slow. If you go back to this picture here, you will find that the effective action here along this valley, this is the part that does not depend on the gauge fluctuations and then you have a very steep valley in all the gauge direction, it goes infinitely up. So as a result, if you solve the field equations, the field equations are simply that the gauge fluctuations are zero and then if you insert the field equation into the effective action, as you do it whenever you have very heavy degrees of freedom, you end up with the effective action along this physical line. Now I have to go again. So that's the first important property. Now you can convince yourself that of course at this point, this gamma head still depends on two gauge fields and at this point now you can ask yourself how to identify them. The usual identification is you just put G equals to G bar. That's what you do in the background formalism all the time. It's possible, it's consistent, but then you lose the property that the second functional variation gives you the, and its inverse gives you the propagator for the physical Green's function and you lose the closeness of the flow equation. Also, two important properties that you lose. So what you do, you say, okay, I can do other combinations. I can just connect G bar and G by taking the difference to be a physical fluctuation. So this gives you of course a more complicated connection, but then you can fulfill first of all for any connection of this type that is consistent with gauge transformations, which means that G bar and G have the same trans, have the standard transformations on the formalism symmetries. For any of those connections, you get a gauge invariant effective action and then you can use the fact that the freedom in this, in the precise definition of the connection between G and G bar in order to fulfill this important equation that the second functional derivative of gamma two gives you really the inverse of the physical correlation function. That's the important property and that determines the, the form, the detailed connection between G and G bar. So at this point, let me conclude since I only wanted to give ideas, I think it is possible to construct gauge invariant flow equation involving only on gauge field. Doesn't tell me yet that this whole program works because in order to work, you need one more important property and that is that this gauge invariant action stays sufficiently simple. Typically it will have non-localities. The non-localities should not be too untreatable. If truncations can be made that are manageable, then it will be a big advantage but before there is not a simple example for computing non-trivial quantities, you can compute the one loop beta function of young mill series that comes out right but you need more things to be verified. Before that is not done, I would not claim that this problem is solved but I think it shows an avenue that can be followed and if successful it will simplify a lot the formalism and it will at the same time bring back again the physical intuition because then for the physical degrees of freedom the setting is very similar to what we have let's say for standard thing with only scalars or fermions. Okay.