 Yeah, thank you very much for the introduction and also for I would like to thank all the organizers for inviting me to this nice event. I want to speak about the relation between quantum field theory and gravitation. So this is based on joint work with Romeo Brunetti, Michael Dötzsch and Katarina Reisner. Now we all know that the problem of relating quantum theory and gravitation is a long-standing issue around 100 years old and without a definite solution until now. In spite of the fact that quantum physics is a well-established theory with lot of nice verifications, in particular also in elementary particle physics, where of course the mathematical status is not as good as in non-relativistic quantum mechanics. Oh, yeah. Okay, but nevertheless it's applied with great success to experiments in elementary particle physics and on the other side also gravitation is nowadays well-established as a classical theory with a lot of experimental confirmations, but we have not yet a consistent theory which combines both theories. Now here in my talk I want to approach this problem from a rather conservative point of view, namely from the point of view of quantum field theory. And so the first step is then to look at quantum field theory under the influence of external gravitational field, which mathematically means that you have to study quantum field theory on generic Lorentzian manifolds. And I will then discuss in which sense this can be also used for including perturbative quantum gravity. So when you start to define quantum field theory and curve backgrounds, in the first moment one might think that this is a very simple problem because in classical field theory everything is defined in terms, in geometric terms, and so it's rather easy to formulate classical field theory on generic manifolds. For instance in electrodynamics you just need the notion of differential forms, exterior differential, and you need the hot stool to define the core differential on the right. So you see you need only very, very few things just that you have a differential manifold and that you have a metric which is then used to define the hot stool. Now when you try to do something similar with quantum field theory you see that the standard formulation of quantum field theory uses a lot of other structures, namely you define the concept of particles in terms of irreducible representations of the Poincare group. You assume that there exists a distinguished state, the vacuum, which is defined as a state where all particles are absent. We have as a main physical observable the S-matrix, which describes the transition from incoming to outgoing particle configurations. And we use as a technical mean the momentum representation, which is due to, which is possible due to the translation invariance. And as an other tool we use the transition to imaginary times so that we get a theory on Euclidean space. And when one tries to put quantum field theory on a curved background you see that none of these features is present. So there is in the generic case the group of spacetime symmetries is trivial. So there is no useful concept of particles on a curved spacetime. Therefore there is also not a notion of a state without particles. So the vacuum is no longer a valid concept. The transition to imaginary times so that you can replace your Lorentzian manifold by a Riemannian manifold is not possible in general. You cannot do calculations on momentum space and if you just close your eyes and you write down Feynman graphs then you meet the problem that there is no unique definition of a Feynman propagator. And actually these theoretical obstacles are then seen in some some effects. What was first observed was that there is a phenomenon of particle creation. So in free theories you might find the definition but you call a particle. But then you see that this depends on the choice of a Cauchy surface. So when you change your Cauchy surface and you get different particle numbers so you have particle creation from Cauchy surface to Cauchy surface and actually even infinite amounts of particle creation. So this shows that the particle concept is not meaningful. And there's this phenomenon of Hawking radiation so that if you look at a collapsing star and you start from a state which is sufficiently irregular then you see that asymptotically the state develops thermal radiation. And even on Minkowski space you have this problem if you look at an accelerated observer and then you see some thermal properties of the state the so-called unruh effect. So in order to define quantum field theory in a curved space time it's important to decouple those features which are geometrical from features which are non-geometrical. So what are the geometrical features? These are field equations. These are commutation relations. So in general these are the algebraic properties of the theory. And this forces one to use the formalism of algebraic quantum field theory as developed by Haag, Araki and Kastler. Of course these, so you have your physical quantities as elements of an abstract algebra. Of course these algebra should admit representations on Hilbert space but you don't use a specific representation. There is however one ingredient of Hilbert space representation which is very important for the structure and quantum field theory. This is spectrum conditions, the conditions that the energy is positive. And so what you need in order to do this program is you have to find some local version of the spectrum condition. Okay, so this is the plan of the lecture. So I will give a few remarks on Lorentz and geometry and field equations. Then I will discuss quantization in the sense of deformation quantization. The local version of the spectrum condition is imposed by using some tools for micro-local analysis. And then I will discuss how renormalization can be done in this framework, what the notion, appropriate notion of covariance is. And I discuss this on the example of a scalar field but then I will indicate how this has to be generalized to include gauge theories and even gravity. Okay, so let me start with some basic notions. So we have a smooth manifold, which we call a spacetime. And spacetimes in my talk are always Lorentzian spacetimes, so the metric has Lorentzian signature. We have causal curves, which are tangent vectors, which are always time like or light like. We have time orientation, which is imposed by choosing some nowhere vanishing time like vector field. And so we can say when a curve is future directed. And then we define the future of some point x as a set of points y, which can be reached from x by a future directed causal curve. And in the analogous way we can also define the path of a point. Now what is important for the following structure is the notion of global hyperbolicity. So spacetime is called globally hyperbolic if it does not contain close causal curves. And if for any two points x and y, the intersection of the past of y and the future of x is a compact set. And the nice feature of these globally hyperbolic spacetimes is that they have Cauchy surfaces. So hyper surfaces, which are hit exactly once by each non-extendable causal curve. And they even have a foliation by Cauchy surfaces. So they have a rather simple structure. So they are diffeomorphic to a product of some manifold sigma times the real axis. Particularly globally hyperbolic manifolds are never compact. And the, for us most important feature is that normally hyperbolic linear partial differential equations have a well-posed Cauchy problem. In particular they have unique retarded and advanced greens functions. Actually these properties of globally hyperbolic spacetimes are around for quite some time. But the proofs are not so old. The complete proofs are due to Bernal and Sanchez. But you find it already in older books the same statements. Now let's look at a simple example the Klein-Gordon equation. And there we have the retarded and advanced propagators as maps from the space of test function with compact support to spaces of compact, of two smooth functions. So D is the space of test function with compact support and E is the space of smooth functions. And they are uniquely determined as the inverses of the Klein-Gordon operator. And by the support conditions that's the support of delta RF contained in the future of the support of F and vice versa with the advance propagator. Now what is crucial for the structure is that these two propagators are different. So we have a difference of these often called the causal propagator and later named the commutator function. And this will be crucial for the algebraic construction. Namely in terms of these commutator function you can define a bilinear form which is anti-symmetric, anti-bilinear form on the test function space and which is degenerate but the degeneration is of a very simple form. Namely if you these form vanishes, if one of the entries is in the image of the Klein-Gordon operator applied to a test function. Sorry doesn't it look symmetric the left section or the all the no no it's anti-symmetric because the adjoint of the of the retarder propagator is the advance propagator. So just if you take the adjoint just the role of advance and retarder changes and because it was a difference it changes the sign. Oh it's not the Laplace operator I'm sorry this is a convention yeah it's just a commutator function no no okay now now this fact that this is the anti-symmetric can be used to define the Poisson bracket of classical field theory which is just the following that's the commutator the Poisson bracket of fields at points x and y it's just these the integral kernel of these operators so it's just these distribution delta of x y. No no delta okay the convention is that here oh I'm sorry. Delta is an operator which maps the test function to a smooth function and the result is again a function of x yeah so it's but but here I write delta of x y which is the integral kernel of this operator. Okay now I come to quantization so we have the space of field configurations for a scalar field and as this space we use all smooth functions not only solutions all smooth functions and we define observables as functions on the space of configurations yeah so so it's off shell it's off shell it's a so-called off shell formalism which is of course one could restrict oneself to a solution space but this is usually more complicated and it's for the quantization turns out to be much more convenient to use the off shell formalism yeah. Maybe observables with inclination marks. Okay I don't want to discuss a measurement problem yeah it's just I call it it's just a name at the moment yeah but but I think it's actually the observables are constructed in terms of these objects yeah and which how they can be observed is a different question I want don't want to discuss so these are just maps on these on these space of configurations and of course I rather rather huge set so let's us look at special cases so for instance in white man field theory you are used to use linear functionals so you just integrate the field phi with some test density F and you can then of course multiply these functionals in the sense of point wise multiplication so you get polynomials and slight generalization is to look at polynomials of the field in this way that you have the field at different points that you integrated within then test density in n variables and another class of functions which is very important because the interactions are formulated in terms of these functions are the local functionals these are just defined in terms of a function on the jet space of m and as a technical tool we need differential differentiability properties of these functionals so the notion of a derivative is very simple namely the nth derivative is just defined as a directional derivative and we call the function differentiable if this nth derivative defines a symmetric distributional density and there is some further condition on the dependence on the on the field configuration phi which I don't want to discuss but I think this is a nowadays well established framework of infinite dimensional analysis psi is another test function psi is in this case a smooth function it's not not necessarily compact support but I assume that the the functional derivative is a distribution with compact support okay now I can extend the the Poisson bracket to the space of of this larger set of functionals by the following formula so the Poisson bracket of two such functionals capital f and capital g is just given in terms of this commutator function integrated with the functional derivatives of f and g and this has all properties of a Poisson bracket and was originally be defined by Payals so called Payals bracket now we want to quantize this structure so we use a concept of deformation quantization which means that we look for an associative product which depends on the parameter h bar and which tends to the classical point wise product the limit h bar to zero and the commutator divided by ih bar approaches the Poisson bracket and there's a simple solution for this in this simple in this in this case name is a very model quantization where you define the star product by this formula here so you apply a formal power series of bideferential operators to the product of functions at different points and then at the end you set phi one equal to phi and two phi two and this is to be understood as a formal power series in h bar but of course in case your functionals are polynomial this is a finite sum yeah so for the polynomials this is an exact definition but in the general case this is only a formal power series so let me describe it on a case of a linear functional then the star product is just the classical product plus first order in h bar a term i times half of the commutator now what is bad with these bimoyal product in quantum field theory the bad feature is that if you want to extend these product from the linear functions to non-linear local functionals then you get problems namely let me discuss it in the simplest case of phi squared so phi squared integrated with a test density f is a well-defined functional it's also differentiable everything is fine but when you apply this formula you'll get three terms the terms without derivatives it's just a classical product then you get the term with one derivative which is okay but then you get the terms with the second derivatives and there you have to square these distribution delta but delta is singular and general products of distributions are not well defined and actually in the case of delta you cannot help there's no reasonable way of defining the square as a distribution but now the positivity of energy helps us as we know from ordinary quantum field theory and we can also do this in this more abstract framework namely the use effects that these condition on deformation quantization do not fix the star product we can replace the commutator function is this exponent by any distribution h whose anti-semitic part is i times the commutator function because only the anti-semitic part is tested in the commutator and if we do this then we get a star product which we are called here on the slide the star h product which is equivalent to the previous one namely let us look at the operator gamma index h which is defined as the exponential of h bar over two times the the distribution capital H paired with the second functional derivative and then so this is again defined in sense of formal power series and you can then easily check that the two star products the original by model product and the star h product are related by this operator and also this operator is a is invertible so this is really an equivalence of star products but we would like to find distribution h such that we can extend the star product to more singular functionals so we have a certain wish list for for this function h so we first we require that it's a by solution of the Klein Gordon equation this is just nice because then the functionals f which vanish on solutions form an ideal in this product with respect to this product as in the classical theory this is very nice because at the end we would like to go to the on-shell formalism where we divide all the ideal or functions vanishing on solutions then in order to have a well-defined product also for local functionals we want that pointless products of these distributions exist furthermore we would like to have something which is definitely wrong for the by model product but which can be required here namely we require that h is the distribution of positive type which means that if I integrate h this the test density f and these complex conjugate and test density f bar then you get always a non-negative number actually what enters here is the symmetric part of h because you have the same function on both sides yeah so that's really positive this is a requirement and what is the consequence if you use such an h product then you have the nice feature that you immediately get a lot of states states on this algebra are functionals which are positive in the sense that they assume positive values on products of f bar f and if you have this property then you can easily show that every evaluation of this function at a field configuration yields the states these are just the coherent states so this is a very nice feature of this one it's a little bit like when you pass to a caler to a caler structure it's related to the caler structure yeah that's true and furthermore I have this requirement on positive energies namely I want that this function h selects locally the positive frequencies and okay these conditions don't come from the sky it's just what we already know from the two-point function and minkowski space so we just want to save as much as possible from the situation we know minkowski is there's a white man two-point function exactly fulfills all these conditions and actually this transition to this new star product just amounts to wick ordering it's just the wick ordered version yeah could you also select locally equilibrium some kind of local kms condition rather than the positivity of energy actually the kms states with positive temperature also satisfies this condition so instead of the white man two-point function you could also use the two-point function of a kms state they have fulfilled all these conditions so locally there is no difference between the positive energy and the energy condition on a kms state with positive temperature yeah okay so now I can this I to discuss this condition when uses techniques from microlocal analysis in particular when uses the concept of wavefront sets so the wavefront set is a subset of the cotangent space so the the commutator function delta is a distribution on the product of the manifold with itself and the wavefront set is then a subset of the cotangent space of the product of the manifold with itself so there are two points of the manifold x and x prime and at each point you have a covector k respectively k prime and the wavefront set of delta just because delta is a solution of the clangordian equation can be shown to be of the following form there exists a null geodesic connecting x and x prime and this covector k is co-parallel to the tangent vector of this geodesic at the starting point and then you take a parallel transport of your covector to the other end of the curve and you add it to the other vector and this must give zero so this is some kind of translation invariance or momentum conversation conservation if you want this is the wavefront set of delta and if you look at the wavefront set of the positive frequency part it's just half of this the positive half of this wavefront set so you just have the additional condition that the momentum or the covector at the point x is future directed and actually it's clear because the sum if you have a sum of two distributions the wavefront set is contained in the union of these two wavefront sets so it's cannot be smaller yeah because when you have this condition that two times the imaginary part of delta plus is just the coveotator function then this is the smallest possible wavefront set of course you could also think of the negative sign in this relation this would give the delta minus function but delta minus because of this minus sign is not of positive type so what you see is that these two positivity concept positivity in the sense of quantum mechanical probability and positivity in the sense of of the spectrum condition the positivity of energy coincide for the scalar field i know of no fundamental reason why this must be so actually this is a great problem in defining three fields on kerf spacetime this half integer spin higher than one half so there you get get a lot of problems because this positivity issue is usually not there seem to be no reason that this is related but for the scalar field it is related for the direct equation for the direct equation it also works fine fortunately okay now this concept of the wavefront set can then be used to define what is called the had a malfunction this is due to ratsikovsky and so kodik to ratsikovsky had a malfunction is a solution by solution of the clangordan equation such that the two times the imaginary part is the commutator function and the wavefront set is just this positive half of the wavefront set of the delta functions this is then called the micro local spectrum condition and ages of positive type don't wavefront set is always pointing out in the future and not yeah for it's it's for the for the first entry for the first entry yeah but it is no spacelike vector no spacelike yeah yeah yeah it's actually it's light like it's light like but future directed actually this concept of the had a malfunction is older than the work of ratsikovsky and you can just try to find more explicit form for the had a malfunction so this form is given here this is the form u divided by sigma plus v times log sigma plus w where u v and w are smooth functions and sigma is just the squared geodesic distance between two points and what ratsikovsky proved is that his definition is equivalent to this definition actually this definition has to be made more precise because you have to to discuss the singularity in light like and time like directions you have to say what happens if you are outside of the geodesically convex neighborhood and so far so this is a very complicated definition which was fully spelled out in a paper by k and walt and it's a major technical advance that that this can be reduced to this simple property of the wave front set so this actually this definition of had a mass solution in terms of microlocal analysis makes it use a lot of things which could immediately be done so first this was a construction of wick polynomials on curved spacetime as operator value distributions so because you just compute correlation functions of wick product everything can be done in terms of these distribution age and its derivatives and you just have to see that all the products which arise are well defined and this can be done in terms of the wave front set the condition is that the sum of the wave front set of the factors never should hit the zero section of the cotangent bundle and this can be a cut it's just a consequence of this positivity condition on the on the covectors for instance in this way you can prove that in a had a mass set not only the expectation value of the energy momentum tensor is well defined but even the correlations are well defined so so so if you want to understand the back reaction by looking at the expectation value of the energy momentum tensor you're of course you have to also to look for the fluctuations because otherwise the expectation value would be not of much much use other aspects are the quantum energy inequalities originally proposed by a fault but fuser found that these micro local spectrum condition gives a very important generalization namely what one finds is the expectation value of the energy density in the had a mass set is not necessarily positive this is already true for the energy momentum tensor in minkowski space but when you smear the energy the energy density with the square of a real value test function then it becomes bounded from below and this can be shown in terms of these wave front sets and maybe the most important thing is that you can do using these techniques see ultraviolet renormalization of quantum field theory and curve spacetime and this was done by bonetti myself and then completed by pollens and waltz infrared is a different issue this is a different issue okay so how is renormalization done now the method is based on the concept of causal perturbation theory is originally proposed by strickelberg and bogel juboff then worked out in all details by ebbstein and glaser so the basic idea is that you define the time-ordered products of wick products of free fields as operator value distributions on fox space and you require that these time-ordered products satisfy a few axioms and the most important is that the time-ordered product coincides with the operator product if the arguments are time-ordered so what you would usually require and then in the famous paper of ebbstein glaser from 1973 they succeeded improving that solutions satisfying the axioms exist and that the ambiguity is labeled by the known renormalization conditions and then you can construct the solution either directly which is in many cases somewhat complicated or via some of the known methods bphz polyvilla momentum cutoff dimensional regularization of what's else now we wanted to apply this to curved spacetime this required a reformulation which only uses local concepts before it was in the flat space this for this was in a flat space yeah now if we do it on a curved spacetime so we cannot use all the techniques so we have to generalize this a little bit so we have cannot have not this distinguish this representation on the fox space so we do it space free yeah wizard no refer without any reference to fox space we don't have translation symmetry and we need a new concept of the adiabatic limit because just makes no sense to take the integral over all spacetime if you don't know what the situation at the end of spacetime is and you need renormalization conditions which are in a sense universal i will discuss what this means formally what you do is you construct an operator called the time ordering operator which is formally given by this formula so you take the final propagator associated to capital H which is just H plus a multiple of the retarded propagator and you you have this pairing of this distribution hf with a second derivative and okay i omitted here factor h bar yeah so you should think of a factor h bar and the factor of one half and now the question is how this is of course rather formal so you have to to see whether this can be made precise and it can be made precise as a map from the set of multi local functionals to the so-called microcausal functionals let me briefly explain what this means so we have the local functionals we restrict the local functions to those which vanish at five to zero so the constant functional is not included there and then the multi local functionals is just this is just a unit with algebra generated by the local functionals and the microcausal functionals are those functionals which where the wave from set satisfies a certain condition of course this explains this condition in detail is maybe a little bit too complicated at the moment just remember there is a condition just made in such a way that the products which arise above the fact just as an example so we apply this operator to a simple multi local functional which is just a square of two functionals of the form five squared integrative as a test function and then we have apply this operator so we get three terms just the first term which is a unit then the first derivative and the second derivative and the problem is again in the second derivative because there we have the square of this Feynman propagator hf and but the Feynman propagator is no not a solution of the homogeneous Klein Gordon equation it's a propagator so it contains the wave from set of the delta function which is just the diagonal so the the wave from set of the delta functions just the diagonal xx so two points should coincide and the co-vectors have to add to to zero but k can be arbitrary all all k's are admitted all non vanishing k's can be admitted and then of course there is no positivity condition and so when you define the square you can add two co-vectors and you get zero and this is just shows that the in general the square is not a well-defined distribution but this holds only at coinciding points that non coinciding points you are always in this situation then that you can use these positivity conditions so these products is well defined for non coinciding points but not for coinciding points and this holds very general and so what remains at the end this is already in the framework of ebtschen and glaser and in the general case can be reduced to this is the following mathematical problem you have a distribution which is defined outside of a sub manifold and you want to extend it to the full manifold and you use something corresponding to translation covariance on kerf spacetime which is a technical problem but can be solved and so at the end you get a distribution which is defined everywhere outside of a single point and then you can discuss this extension in terms of the scaling degree which was originally defined by steinmann so this is the following definition you just scale your distribution by some positive factor lambda you multiply the distribution by a certain power of lambda and you ask that this sequence of distribution converges to zero in the sense of distributions as lambda goes to zero and as lambda goes to zero yeah and so you test the singularity as the origin and and the infirmum of these numbers is called the scaling degree and then the theorem is that if the scaling degree is finite then extensions exist and moreover extension is the same scaling degree and two such extensions differ by a derivative of the delta function of order scaling degree minus n so in particular the scaling degree is smaller than n than the distribution is unique and if the scaling degree is infinite then t cannot be extended so this theorem i said this is due to today yeah okay i'm not sure who first gave a complete proof of this so we were not able to find a complete proof so it would be nice to know it i think i'll do that i yeah okay also for many faults i yeah okay okay so that's so this theory replaces completely the standard regularization techniques but in order to get a specific extension they are often useful and just to explain how this is related to standard regularization techniques uh the following considerations so let's assume we have a finite scaling degree but larger equal to the of the dimension and then one can show that the distribution can be uniquely extended to all test functions which vanish at the origin of the order of the degree of divergence which is just the integer largest integer smaller than the difference of the scaling degree and the dimension then we choose a projection on the complementary subspace of the test function space and then every extension is given by these formulas so we just take the composition of the distribution with the the projection one minus w w is this projection the finite dimensional subspace and one is the unit operator and all extensions are of this form now such a projection has a simple form that it's can be written in this virac bracket notation as a sum over w alpha delta alpha the del alpha delta so the functions w alpha form a basis of this finite dimensional subspace which is just dual to the to the basis of the distributions arising from derivatives of the delta function so this means just that the derivative of w at zero is the chronicle delta times some sign now assume we have some regularization techniques so we replace t by some sequence tn and we assume that tn converges to t on the subspace of test functions which vanish at the origin then tw was defined as this composition of t this one minus w now i replace t by tn and then i can apply tn to both terms so first i apply to the unit and then i apply to this projection w and what we see is that we can construct this distribution as a limit of tn where we subtract certain divergent multiples of the delta function or its derivatives this shows how divergent counter terms occur in this formalism so it's just because you have this projection but you cannot split it for t itself but only for the approximate in sequence tn so i guess yeah there are two ends on your slide with different meanings oh i'm sorry yeah here this is the index of the thing this is not the i'm sorry and this is now all on our end okay so so these ends are not related yeah but your space time is now an rn or what no no so this was just the general comment of course there's some technical problem to reduce this problem of extending distributions on manifolds which vanish on a which are not defined on a sub manifold to the situation where you have just one point so in a sense you use some transversal coordinates near to the sub manifold yeah that's the way it can be done but this is technically demanding but i can so this was done in this paper by Bonetti myself and there's a recent PhD thesis by viet Dung who analyzed this from the mathematical point of view very carefully but at the end you arrive at this problem but this of course requires some exercise in micro-local analysis okay now we have all the means to construct this time-ordering operator so we look at the symmetric fox space over the space of local functionals so fox space not just in the algebraic sense so it's not considered to be a herbert space just the sum of symmetric tensor powers and on the space we have the multiplication map which maps the space into the space of multi-local functionals we define the n-linear maps tn on the symmetric fox space to the space of multi-microcosal functionals just by using the these bideferential operator in terms of the Feynman propagator and because we go to several factors we have to use the Leibniz rule and this creates these indices ij here so this gives all the combinatoric of Feynman rules and so on and what is important is that these multiplication map turns out to be bijective this is a little bit surprising because the the fox space you have functions of different field configurations and at the end you have functions of one field configuration but due to the smoothness of the local functionals so the local functions are singular in on the diagonal but smooth along the diagonal so they're singular transversal to the diagonal but smooth along the diagonal and and this then then allows to construct the inverse of m and at the end you can define this time ordering operator just by the direct sum of these maps tn composed with the inverse of m so this is a definition of t and if you like path integrals you can also formulate this as a path integral namely it's just a convolution with the Gaussian measure induced by the Feynman propagator but for the definition this does not help because you have to do the same calculations but the formula is there so you see the relation yeah so it's just a different different way of formulating this now we can define the time ordered product namely the time ordered product is now just equivalent to the classical product by the time ordering operator there's this formula written there and so this is then this statement of this theorem which of course contains as an essential point that this multiplication map is invertible this was done together with Katashina Reisner so particularly the time ordered product can be defined as a binary product of functionals and this product turns out to be associative and commutative because it's equivalent to the classical product then we can define the time ordered exponential which is just the exponential series composed with this map t and by definition t extravially on local functionals this means that we define our functionals in the normal ordered form related to H so if we change H then we have also to change our functions then we can define the interacting fields if we have some interaction L and this is something one might call the Moeller map from the freeze to the interacting theory and this is just Bogey-Huber's formula for the for the interacting field so we take the s matrix to the minus one product the time ordered product of the s matrix with the functional f and this is well defined if t to the minus one of f is in this set of multi local functionals yeah so in general there could be divergences but for this set it's well defined okay then as I said there's some ambiguity in the in the definition of time ordered products and this ambiguity can be discussed in terms of the renormalization group in the sense of Stuckelberg and Petermann this is not exactly the same as the renormalization group of Wilson so it's really a group it's a group of all analytic bijections of the space of local functionals and the crucial relation is that this is additive if these functionals have disjoint support and the first derivative should be should be the identity and then there is something which was called the main theorem of renormalization in an unpublished paper by Stura and Popinot and okay there are different versions of the theorems with different generality and the statement is that if I have found such a formal s matrix then any other formal s matrix as had is obtained from the original one by composing it with an element of this group so all questions of renormalization can then be deformed in properties of this group so for instance anomalies and so on can be discussed in terms of the group okay now this is in the curve space everything is in the also in the curve space yeah yeah this is a very general of course in flat space this is more or less what was known but but can be generalized to to curve space now there is but there remains one ugly problem namely when one does this renormalization one has to do it for every point of the spacetime independently because there is no symmetry in general so how to compare these renormalization procedures at different points of spacetime and of course there is some feeling how this should be done so if you want to have a operation which is locally defined but to make this precise is rather complicated but of course this problem would not exist does not exist on minkowski space there you have a sufficiently large symmetry you can have other spacetime which large symmetries where you have similar advantages but in the general generic case you really have the problem that you that it seems that you have independent conditions at each spacetime point which of course would be very ugly now the solution of this problem is to construct the theory not on one specific spacetime but on all spacetimes in a coherent way and this can again be formulated in the language of algebraic quantum field theory so we generalize the hakasla axioms in a useful way so the idea is that we look at sub regions as spacetimes in their own right of course these sub regions should be generally generally should be globally hyperbolic and the generalization is the following so we associate to every globally hyperbolic and okay contractable orientable and time oriented manifold of a given dimension a certain c star algebra or perturbation theory we restrict ourselves to star algebras for every isometric causality at orientation preserving embedding we we have an injective homomorphism of the algebra of the region m to the algebra of the spacetime n and these if you have two subsequent embeddings then these homomorphisms just can be composed and then we have some condition of local commutativity so if these embeddings map into space like separated sub regions and the corresponding algebras commute and there is some dynamical law which just tells if the image of the spacetime m under kai contains the kushi surface then akai is even an isomorphism so in the language of category theories just tells you that the quantum field theory is the functor from the category of spacetimes is admissible embeddings as morphisms to the category of unital star algebras with injective homomorphisms as morphisms this is a concept which was called locally covariant quantum field theory and in the paper by together this reiner verschen homioponetti and in this framework one can define a field independent of the spacetime namely what is the field the field is just a natural transformation between the functor of test function spaces and the quantum field theory functor so this means you have a field phi as a family of maps phi m from the test function space of m to the algebra associated to m and which transforms in the appropriate way under these under these embeddings and so if you use these more suggestive notation here you see immediately what happens if n is equal m then kai kai should preserve the symmetric sorts and isometry so symmetry of the spacetime is just as a usual notion of a covariant field on a given spacetime so this condition contains the usual notion of covariance but it's valid in this more general framework now we can use this for renormalization namely we define the time ordering operator as a natural transformation and so this means if m is an admissible subregion of n then the restriction of the time ordering operator to tm must coincide with the restriction of tn to the smaller subregion must be must coincide with tm there is a practical obstruction to do this there's the fact that there is no natural Hadamard function this is related to this problem that there is no vacuum state and but this can be solved this problem and the solution was in the series of paper by holland's and what how do you mean it's solved I mean there is an ambiguity no no the non-ambiguity is there but you have to to isolate the the the non-ambiguous so the ambiguity is there but you the the singular so if you look at the form of the Hadamard function there are some parts which are which are non-ambiguous and there's an ambiguous part and you have to show that what you do is does not depend on the ambiguous part yeah actually this is quite demanding it's not not not an easy exercise the Hadamard function just means that you have a certain wave front set and then but then I gave this explicit formula for the Hadamard function which involved these smooth functions u v and w and it turns out that u and v are uniquely determined by geometry and w is free and you have to show to say to discuss the dependence on w that's that's what you have to do okay now I up to now I discuss this on the level of scalar theories now we can generalize it to Dirac yeah so removing these ambiguities does that mean we can say what do we mean by when we say that the coupling constants of a quantum field theory are everywhere the same yeah so if you if you have a dimensionless coupling constant this is everywhere the same yes this would this would be the statement yeah actually usually there remains some some ambiguities for instance you have the coupling to the curvature which is can be tested of course only on a space with non-vendishing curvature so from if you start from minkowski space you cannot determine this constant yeah so you have to but this is a finite number of parameters in a renormalizable field but there also there would be a constant sign yeah there would be a constant again yeah okay so Dirac and Joana fields can be discussed this gives no fundamentally new problem but you can make a lot of errors here by making wrong signs and factors and I think it's difficult to find paper where all these signs are correct and but engage theories and grab oh my I would mention if I might never consider power of by yeah okay so engage theories and gravity one has new problems because the koshi problem is not well posed due to the gauge theory and in the classical field theory you would just fix the gauge in quantum field theory you cannot do this directly because these things which you would like to vanish have non-vendishing commutators or Poisson brackets in the formalism of canonical field but can you just pass the dirac brackets yeah okay that's the problem is with dirac brackets is that these the singularity structures are very difficult so so I think I have no not any rigorous discussion of the dirac brackets so you say that in passing from Poisson to dirac brackets you eventually potentially encounter problems with singularities the problem is okay I just don't know so I have not found any place where this was treated and say say a sufficient generality the problem is the following all these in quantum field theory and at least in four-dimensional quantum field theory you always has to smear over space and time you cannot do smear only in space this is create singularities but the the concept of dirac brackets is usually defined at a fixed time and I mean it's no really standard thing for example when we pass from dirac to Majorana fermions that gives you just the factor of one half and that's yeah so so so for the linear fields themselves this is okay the problem is if you go to non-linear fields because then you get additional singularities and I just don't know whether this can be done okay but in your mill series we know how to do it using for the puff of ghosts and anti-ghost if you have the BST transformation and just at the end we define the algebra of observance as the cohomology of the BS operator and we can do try to do the same for gravity but in gravity we have a problem namely if we do this just in the same way as in young mills we find that there are no the cohomology is trivial because there are no local observables this is a major problem it can in principle be solved in the following way namely we use the dynamical fields themselves as coordinates this of course is not always possible this depends on the gauge on the gravitational field but for generic field configuration this is possible and so the idea is to not necessarily on char just generically in the sense that there are no special symmetries and so so that you have enough independent say say curvature scalars for instance and then you just choose such a generally globally hyperbolic metric and you expand then the extended action in around this up to second order so you get a free field theory one can show that this has had a mass states and then one adds the the remaining term of the action but perturbation theory and constructs everything and this has to be done in such a way that the bst invariance holds and at the end you prove that if you change the background infinitesimally then the theory does not change this is due to the principle of perturbative agreement proved by holland's and waltz and states can be constructed at the end by if you start from a solution of the classical field equation and use this concept of coherent states around these okay so i come to the end so i've tried to explain how this functorial approach to quantum field theory works this was originally developed for the purposes of renormalization on curved spacetime but in principle it allows also a framework for a background independent approach to quantum gravity of course this does not change this problem of non-renormalizability which means that you can perform everything step by step but you get new terms in the Lagrangian in every order so in this sense it's non-renormalizable but i think there's it's probably possible nevertheless to interpret this this theory in the sense of effective field theory at the end actually we have problems to see any effect of the quantization of gravity in experiments so i think there's hope that these effects are very small so actually this is supported by work on the renormalization group in the group of Reuter and Un at others and it's at the moment these formalisms are a little bit too far from each other but principle i think this can be compared to this approach of Reuter there is one major lack in this all procedure there's no up to no no application to physical phenomena and this is due to one ugly feature namely the restriction to generic backgrounds namely if you want to have an explicit example you typically use not a generic background but you use some of this background which has additional symmetries and then you cannot directly use these these concepts so in this case you have to use other methods so one way out would be to add certain fields like for instance the dust fields of brown and kutscher which have been used just for the same reason in loop quantum gravity and so that but this is work which has to be done which was not done up to now thank you very much for your attention yeah i have a question i mean this as far as i understand your approach is very nice very much on ebb strike class yeah now in in quantum field theory in practical quantum field theory ebb strike class is not used very much for people who use dimensional realisation and more i mean what would you advise with this somehow related to the one but last point if i actually want to do practical calculation how should i proceed yeah um so so um okay in uh so so when you do it on on on minkowski space yeah i think you can compare it only on minkowski space but on kerft space time there's not much what was done in other formalism actually i think no complete proof i think the only proof for kerft space time is in the ebb strike glass framework but but in the but in the um on minkowski space ebb strike glass is just equivalent to other methods so um you use the method which is most useful for you so there are certain cases where where these ebb strike glass method leads to shorter calculations but there are also cases where it's more complicated so i i think i would actually what we analyzed was how to use dimensional regularization in the framework of ebb strike glass this gives us gives a position space version of the dimensional regularization similar to what was already um not really the same but but it's a spirit of some work of bolini and gambiachi but um but you but you can also use other regularization methods so that's not yeah so i have two questions i don't understand the last problem but yeah maybe that's too technical i don't know the other question is is there any kind of wilsonian version of the remormalization group that i could use to analyze these field theories of course the naivri doesn't work because usually wilson needs momentum space but maybe there is a more clever yeah so in principle you can actually uh what what you can do is you can replace what i call this time ordering operator you could replace this by some more regular object for instance just by uh approximating the final propagator by something regular and then you get you're just in this uh polchinsky version of the wilson renormalization group actually this has already been analyzed in the paper bars so you really get the flow equation in this framework but you can so so but uh you can use any any approximation so of course momentum space makes no sense but you can use just something yeah that's uh just uh which i think uh might be a problem for the interpretation because you you are free to use any for instance you can just look at the sequence of test functions which converges in the sense of distribution to the final propagator and then you yeah i mean so related question in a sense uh because for me it's not a view that in general the renormalization group and i will send wilson renormalization group will be related to the renormalization group in the sense of renormalization of the theory of addiction of ambiguity which are the three parameters yeah so in your case what you define as renormalization group is clearly the group acting on ambiguities yes yes is there any uh why you could see that it has something to do with wilson renormalization group and change of scale yeah you can see it but the connection is not as close yeah so so so um so what you can do is you you can um you can discuss it in terms of these um these um okay um so what you can show is that in a certain sense the wilson renormalization group converges to to to the right thing yes so you you have uh you use some regularization you get you get your your um regularized time-ordering operator you have your regularized formal s matrix then you compute an element of the stuckelberg- peterman renormalization group related to this transformation and then you can show that in a certain sense this uh these products converge but i think these the statements which exist at the moment are not very strong it's related to the sort of question whether this perturbative calculation of the coefficients of the renormalization group gives the right answer for the wilsonian renormalization group i think that usually in the calculation you usually neglect certain term which you consider to be irrelevant and i think that's the question how far this is justified i um i think this answer has been given to my knowledge only in very special cases particularly only for renormalization there exists a relation but i think the problem is the relation as far as i know them are not very strong there's some similarity but it's not exactly the same in last question in the case of quantum gravity does your approach give a better definition of what would be a quantum back reaction than your g0 you said g0 is offshore but at some point you want a good g0 which incorporates back action yeah yeah yeah is there does it help or not okay frankly that's of course our hope i think we are not so far so so we hope the formalism in principle should give it but when we solve so what one should compare this one should compare with the so-called semi-classic Einstein equation the semi-classic Einstein equation has a problem that consistency problem this experimentally should be wrong it is a proof experiment okay