Quantum Field Theory and Gravitation

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Quantum Field Theory and Gravitation
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The incorporation of gravity into quantum physics is still an essentially open problem. Quantum field theory under the influence of an external gravitational field, on the other side, is by now well understood. I is remarkable that, nevertheless, its consistent treatment required a careful revision of traditional quantum field theory in the spirit of algebraic quantum field theory. Moreover, it allows a background independent perturbative construction of quantum gravity as an effective theory.
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after the top and the time and and I I think you very much for the production and also for I would like to think although organizers from light in meters is nice event I want to speak about the relation between current of usury gravitation this is based on joint work is only about 180 images bitch and that of no but we
all know that's the problem of relating quantum theory and gravitation to long-standing issue all around 100 years old and there's not a definite solution under the null In spite of the fact that Trenton physics as well established the universe of lot of a nice verifications In particular although in elementary particle physics da Costa mathematical status is not as good as any number relativistic quantum mechanics OK Bloomberg mix at the LAT nevertheless this applied this great success toward experiments and elementary particle physics and I was on the other side of the gravitation as nowadays well established as a classical theory the lot of experimental and confirmations but we have not yet consistent theory which combines both CO no here in my talk I want to approach this problem from rather conservative point of view me from the point of view of current and future readings and so the 1st step is then tool uh look at kind of history under the influence of external gravitational field which mathematically means that you have to study confused Serie on generic ransom many intervals and I will then discussed but In this sense as this can be also used for including procurement of quantum gravity so when you start to to define kind of users have backgrounds in the and the 1st moment but might think that this is a very simple problem because in classical field series everything as defined in terms of and geometric terms insults rather easy to of former late a classic of usury on generic manifold Swanson electrodynamics just I need to notion of different forms exterior differential and need see watched to define the core differential and if you on the right side so and
associated you need only a very very few things just that together differential manifold and that you have metric which is then used to the of watched work no menu and trying to do was something similar was kind of feuds series this season to stamp out to the formulation of quantum field theory uses a lot of other structure namely you define the concept of particles in terms of marriages were representations of sea comparator group you assume that there exists a distinguished states a vacuum that just defined as a state all particle size 7 we have as a main physical observables the thinks describes the transition from incoming to old going part of the country the relations and we use the technical means Azizi momentum representation just possibly due to the translation variance and as another tool in the USA transition to imaginary time so that we can the theory on Euclidean space and when Christalle booths to put kind of history prophesied that none of these features a strong presence sold this image than the generic case the group of space-time cemeteries as trivial so there is no useful concept of particles on a crypt space therefore there is also not the notion of a state visit particles so the vacuum is no longer valid concert the transition to imaginary time so that you can replace the low-rent since many followed by a Riemannian manifold is not you as possible in general the cannot calculations on momentum space and if you just close your eyes and you will write Fineman grasp the new meets a problem that has no unique definition of a fine propagates 4 and they're actually theoretical obstacles are then seen and some some effects but lost the 1st observed was that is a phenomenon of particle operations on free theories you might find that the definition would you call a particle but then you see that this depends on the choice of cautious service so when you change your pushy service and you get a different particle number so you have parted the creation from cautious service to cautious service and actually even infinite amount of popping immigration so this is shows that the particle concept is not meaningful and if this is phenomenon of Hawking radiation so that if you look at the collapsing style and you start from state measures sufficiently regular then you see that asymptotic estate develops from radiation and even on because spaceships this problem if you look at an accelerated observer and then receive some some of properties of the state the so-called onward effect so all in all it to the fine print of future and curved spacetime it's very important to the couple those features which are genetically from features which are among geometrical somewhat as a dramatic if just these are few decorations these are commutation relations so general the idea bright properties of the theory and this forces want to use the formalism of lighter brighter interviewed serious developed by heart tests Of course this is so you have your physical quantities as elements of an abstract idea problem of course is admit representations on Hilbert space but you don't use a specific representation the day however 1 ingredient of Hilbert space representation France's very important was a structure in front of his series is a suspect on conditions a condition that energy is positive and so what you need in order to poses this program is you have to find some local version of the from ship OK so this is the plan of the lecture so will give sure remarks on runs in geometry and chewed equations that I will discuss condensation in the sense of information conversation the local version of the spectrum condition this year impulse buy using some tools for Michael of analysis and then never discussed how renormalization can be done as this framework was the notion of appropriate the notion of covariant and I'm discusses on the example of a scalar field but then never indicate how this has to be generous to include gage series and even gravity OK so let me
start with some basic notions of smooth manifold we've got space time and space science and my ear talk always surrounds since this themselves the Mexican Sorensen signature if calls occurrences which tension factors which after all this time acolyte like we have time orientation which is imposed by choosing some Nova vanishing Time-Life vector field and so we can save the Curtis future directive and then
the defines it as the future of some point X set of points Hawaii which can be reached from ex-spy future directive calls and in the analysts maybe can also defined the past before .period knowledge of what is
important for the following structure is the notion of law will have publicity so space semis called globally have folly if it does not contain close calls ,comma end if any 2 points x and y the intersection of surpassed of Hawaii and the future of X is components and the nicest feature his globally hyperbolic space-times is that they have she's services so hyper services with except the 1 for each non-extendable cause of crime and they even have affiliation by surfaces so this is you have a rather simple structures of the different figure to work product of so many my times the real access particularly globally hyperbolic manifolds and never come back
and the fastest and most important features that normally hyperbolic linear partial Q the head of polls Bush's problems In particular the unique retarded at advanced Green's functions actually these the properties of globally hyperbolic space times are forged quite some time but approves or not so all the the uh to compute whose future and sanctions that you find it already order books the same statements the knowledge that struck at a simple examples of Llangollen equation and that they have seen retarded and advance propagators s MEPs forms a space of the dysfunction compact support crew spaces of compact of twosome was function to the use space of dysfunctions ,comma supported the use of space of Small's functions and stay are uniquely determined and the Moses of the of the tangled operator and by support conditions it's support of dared our F contained as if you chose to support of S and by some of the advantage this advanced propagated know what is crucial for the structure is that these tool navigate at different so we have a difference of these often called the calls of profligate and and commentator function this will be a crucial are Dupri construction namely commentator function you can define a binding uniform which is under symmetric and this by uniform on the tests functions spaces and adjust their generator but those addition aeration is of a very simple fall namely that if you'll these these form vanishes 1 of the entries is the image of declined gone operated applied to test structure the sniffing the session of the not underestimated because it action Aussie Aussie retarded propagators advance forward just if few takes action and just the role of advance of retarded changes and because it was a difference of changes aside this was not a lot last year was not only that the boys noted that last about all old I'm sorry is not select last operator I'm sorry this is a convention in assessor commentator functions none of OK not know this fact that this is the anti-Semitic um it can be used to define the possum bracket of classical field serious just the following that the commentator is easy possum bracket of figures at Point X and Y it's just sees the integral ,comma losses operators was justice distribution Delta of 6 1 this this is going to convince the world that it is in the works just like that can make 1 I where did actually
get into this without doubt as an operator which MEPs attests functioning toward was most function as a result is again a function of X cancel but but here I write note of X Y which is integral ,comma operate focus North come
to condensation this space reduce also was functions not only solutions Olson's functions ends we define observables areas functions on the space of configurations yes also the this option was a so-called offcial formalism which is of course put respect oneself toward toward a solution space but this is usually more complicated and it's for the conversation turns out to be much more convenient to use the offshore from use to those using "quotation mark export let a I don't want to would discuss the measurement problems yeah it's just I call it it's just the name of the moment yet but but I think it's actually the observers are constructed in terms of these objects kind of breach of hold they can be observed as a different person I want someone is sold these artists of MEPs on this on this space of configurations and course and rather than who runs a huge set so let us look at special cases once the white feud serious you're was to use linear functions so it would just be the right to field fire this summer the intestines ETS and you can then of course might apply these functions In this sense of .period vise modifications are you get polynomials engines cited a reservation is to look at polynomials of the fields that In this way but you have to fear that different points of view intervention this dead test density in and variables and in another
class of functions which is very important because the interactions of formulated in terms of these the functions of the local functions this suggests defined in terms of a function on the Jets space of and as a technical tools the need to differentials the furniture ability properties of functions so the notion of the derivative is very simple namely that ends derivatives justified as a directional derivative and because a function differential if suspense derivative defines a symmetrical distribution density and there's some further conditional independence on the part of the few configuration 5 cheerful wanted discussed but I think this is a notice of well-established framework of infinite dimensional analysis example this it science and this case and its most functions somewhat not necessarily compote support but I assume that the the functional derivative as distribution was compatible OK I know I can extend per se pull some bracket to the space of 4 of its larger good a set of functions but by the following formula for the post of records to such functions capital S & of and he just given the terms of the commentator function in the credit as a function derivatives of F and G this is all properties of the people some bracket and most originally believed defined by payouts for of sparked you know what you want to quantise structure overuse use of concept of defamation condensation which means that we look for US assertive product which depends on the form of a beach ball and which tends cozy the classical .period 1st product limit age but to 0 as a commentator divided by Irish bar uh approaches to and there's a simple solution for this December in this in this case there is a violent yet conversations that defines a start from marked by this formula here so you fly a formal policy areas of by differential operators to the product functions at different points and then at the end you set fire 1 equal to fight and to fight too and this is To be understood as a formal policies and each Bob but of course in case you function as a polynomial This is a fine example yeah social because of the polynomials this is an exact definition buttons a general cases only a formal policies solely to assist them with his on the case of linear functional then the stock fraud artist just using classical product class In 1st file the term the right times half of commentator you know what this batch disease by money out from abducting print of the it's serious to batch feature is that if you will we want to extend these products from the linear functions to nonlinear local functional then you get problems namely that discuss it the simplest case of priced well so far I scratched and agreed it was a test and that the efforts of all different functions of differential everything is fine but when you applies this formula you get the thumbs the terms because of derivatives is just the classical product that gives the film was 1 of which is OK but then you get the terms as the 2nd derivatives and then you have to scrap cease distribution Delta but Delta is single on and in general products of distributions are not well defined and actually in the case of Dr. you cannot help but there is no reasonable way of defining the square as a distribution yeah but not all the positivity offer energy helps us as in all from ordinary contributes to and we can also do this is his last from about namely the news effect since his condition and information condensation Kunitz Texas stop product B can replace the commentator function is this exponent by any distribution age was anti-Semitic path this 5 times a commentator function because on his anti-Semitic part is tested the the problem and if you do there then you get a stop
products which are called here on the slides inside each product which is equivalent to the previous month namely let us look at the operated Gomaa index H which is defined as this is a exponential-of of H 5 true times the presence distribution capital at a perhaps was the 2nd function derivatives and then so this is again defined sense of formal policy areas easily check that the 2 of them stop progress the origin of by Miguel product and to start each product are related by this operator and is operators is invertible sources it's written equivalents of some products but we would like tool find distribution age such that we can extend to stop further to more singular functional so we have a certain vision of for for function age so the 1st thing require that the buyer solution of the planned on occasion this is just nice because then dysfunctional it's ever but to vanish on solutions from an ideal Kansas problem the is disrespect his property as in the classical theorem which is very nice because the entry would like to go to the onshore formalism that the divide alter ideal of functions vanishing on solution then in order to have a well-defined product for local function as wants that .period 1st products of this distribution success furthermore we would like to have something which is a definite deal wrong falls out product which which can be required here may real requires that age is the distribution of positive type which means that the integrated age this the test and the TF and these complex conjugate and test density of bile then you get to of a non-negative number of h is not so much economic aid but the word is there the the uh actually would end as CSC is emetic part of age because you have the same function both sides so so that's really positive that this is where the requirement and modest seek consequence if you use Section H product and you have a nice feature that you immediately get a lot of states states on the five-year of functions which a positive in the sense that there assume positive values on products of that if by and if you have seized properties and you can easily show that that every evaluation of dysfunction and a few configuration gives the status of just a coherent states so this is very nice she chose to use a year away from here it's related to the killer structure that that's true and furthermore I have this requirement and positive energies namely by I wanted this function age select sort locally the positive frequencies and "quotation mark it's just what we already know from the two-point function and McCusker space so we just want to save as much as possible from the situation on because is there as a white two-point function exactly fulfills all of these conditions and actually Aziz transition to this new stop product as the amounts to be called just because of the heroes of the late locally the Cleveland some kind of local kms from the source of the problem is that there and actually see came as states was positive temperature all satisfies these conditions so instead of the vitamin of whether he could also use it to two-point function of the Camus's fulfilled all these conditions so locally there's no difference between Zia a positive energy and the energy condition came as it is positive temperature
yeah OK so no I can do this and I to discuss this condition and uses techniques from Michael local analysis and particularly when uses the concept of away from the Bayfront Center is a subset of the cold tensions based the commentator function Delta this distribution on the product of so many felt itself and Mayfield Edison a subset of seek alternative space of producer manifold itself so are tool points of the manifold and ex-prime and at each point you have to call that don't case respectively K Prime and the reference said of Delta just because Delta's a solution of the tango arbitration can be shown to be of a fully formed there exists the knowledge it is a connecting X and X prime and cease call that case this Copa religious tangent vector of his duties it at the starting point and they take apart the transport of you collect to the other end of the current and edits to the other victims must give 0 so is some kind of translation invariance or momentum conversation conservation of fuel this is the very from of Delta and if you look at the benefits of the positive frequency part it's just half of Sister positive half of Sicily of so you just have the additional conditions that the the momentum was a covert .period X this future children and actually it's clear because the sum if you have a sum of 2 distributions of vary from such is contained in the Union of his tool of a front sets so solid skin the smaller it because when you have this condition that 2 times the imaginary part of the classicist take function then this is the smallest possible Bayfront Center of course you put all the thing of negative sign of this relation this would be the deminers functions but deminers because of his minus signs of positive so what you see is that the strong positivity concept positivity in the sense of mechanical probability and the positivity is the sense of of of the spectrum conditions the positivity of energy coincide for the scale of you I know of no the fundamental reason why this must be so at uses as a great problem in defining the free on the cuffs space-time this half integer speed higher than 1 so that you can get a lot of problems because it's positivity issue is usually not there seemed to be no reasons that successfully led for the scalar view that this was the 1st over the Arctic region further deregulation also works 5 occured know all this considers the reference that can then be used to define what is called a half mile function this is you talk about Tokofsky and so good at about 6 cups had function as a solution by solution of supplying chronic patients such as 2 times imaginary part as a commentator function and the bayfront sitters Justice positive half of the difference in the sea the different uses the call to Michael Lopez spectrum conditions and ages of positive ties the rate of disease along with former of virtue it is not a guy with a "quotation mark it's it's for the for the 1st entry for the 1st entry I think is the movies that they no space like it is light like light like but future the actually Aziz this concept of had malfunctions all or dozens of other far Tokofsky and when you can just try to find more explicit fall for that amount of resources for almost 50 years this is the form you divided present not pleasure the Times not like Marcus W. There You'll Dublin Arsenal's functions as Sigma's Justice scrapped kiddies at distance between 2 points and what about Sikorsky proved is that his definition is equivalent to this definition actually this definition has to be made more precise because you have to on the discusses single rarity in light likened to time directions you have to say what happens if you are old side of synergy of rigidity is equally convicts neighborhood and so far so this is a very complicated definition which most fully spelled in favor of K walls and it's a major technical advance said that as this can be reduced to this simple property of civic from so this actually it is a common definition of outermost solution becomes of Michael local analysis makes the uh uh use a lot of things which could immediately be done so 1st since this was the construction of the polynomials uncovered space Summers operator valued distributions so because you just computer product correlation version of that product everything can be done in terms of distribution H and its derivatives and you just have to see that all the products which arise of well-defined and this can be done in terms of of a fund set a condition is that the some of of the references of affect never should hit the 0 section of the bundle this can be cut this just a consequence of this positivity condition on the forms the core of activists for instance In this way you can prove that in the Harlem must not only is the expectation value of synergy momentum terms as well defined but the visit correlations of modified so so so if you want to understand the Beck reaction by looking as expectation areas-energy amended desert have of course you have to also to look for the fluctuations because otherwise the expectation value would be not of much much use and other aspects of the Clinton energy inequalities originally proposed by default but you're few faltered sees Michael local spectrum condition is very the important generalization many of weapon finds a city's C expectation values to energy density and harder mustard is not necessarily positive this is already true for the energy momentum turns and cost space but from the news media the energy and the energy density graph of real value of tests from then it becomes bombing from below and this can be shown in terms of size differences and maybe the most important thing is that you can do all uses techniques see ultraviolet renormalization of kind of USA Inc of space-time this will stand by Bonetti myself and then completed by Poland's people who work work there infrared is a different issue this is a different issue OK so
hold this minimization North Sea method based on concept of calls of perturbation theory as originally proposed wish go back but with wealth then look old and all the terms by which time Plaza so the basic idea is that you define the time of the products of its products of refuses operator voted distributions of Fox space and requires at this time products satisfy few is the most important is that the time of product coincides with the operator product if the arguments of time all so what you would usually require and then famous paper of options lather from 76 lead with succeeded in proving that solution satisfies Ex-Im's exist and that the ambiguities labored wasn't known renormalization conditions and then you can construct a solution either directly but you some money cases somewhat complicated all by a sum of Sonoma thoughts BPH that apology Lomond cut of dimensional regularization of no he
wanted to apply this it took us space-time this required to reform relations the only uses local calls it for this for this wasn't flat Specht said no individual which are now on the cuffs by some Soviet cannot use all the techniques of have to journalists that that we have cannot have noticed distinguished distribute the representations of Fox the which space free of this sort of norovirus miss out any reference of expressed we don't have cancelations and 3 and the need to new concept of study about a claimant because just makes no sense to takes integral of all space-time if you don't know what the situation at the end of specimens and you need renormalization conditions for China sense humor involves of of discussed with this formally but you do all this to construct and operate a call the time ordering operation which is formally given but this former authority you take the final propagator associated to capital H shares age class multiple Pelosi retarded propagated and you you have disparity of this distribution age and is a 2nd derivatives and opium omitted Utrecht rage following us of thing of affected each part of the effect of one-half and to know the Christmas how this is of course rather full moon Sunday to pursue but this can be made with size and it can be made precise as a man from the set of mighty local functional to the so-called Michael calls of functions with briefly explain what this means that we have local functions we respect the local functions those with vanish at 5 0 so a constant functional has not concluded that happened in the mighty local functions stresses that this is just a unit Gerard generated buzz local functions and the Michael Kors the functions I was also functions which reservations satisfies a certain conditions Of course it's expenses condition Adidas maybe a little bit too complicated at the moment just remember that as a condition just made in such a way that the products which arise of about it does is make samples of the applies is operated tool a symbol mighty local functional which assesses scrap of true the functions of the from 5 square integrated visitors functions and then we have players operators of biggest 3 times just see 1st Summerchase unit and the 1st derivative and the 2nd and the problem is again in the 2nd derivative because we have since square of this final propagated H. and that's the final propagator it's not not a solution of homogeneous guy called migration it's the propagator salt contains references of the delta function petitioner said diagonally for the from some of the other functions just as a dying alone and x x so too .period should coincide and co-workers have to do to the but all this means that not even cake can be arbitrary all case I admitted to all known vanishing case can be made and then of course there's no positivity condition and so when you defines the square and you can to call vectors and you could see it and this is just uh and shows that z in general and this this and these these scrap is not well-defined distribution but this holds only at coinciding points said Norman coinciding points you always have the situation and that you can use these positivity condition so this product is well-defined from Vermont coinciding .period but not for points of people and this holds a very general and so what remains at the end of this is already in the framework of the action and blouse and and the general case can be reduced to this it is zoom following mathematical problem you have a distribution which defined also submanifolds and you want to extend it to the full manifold and you use something corresponding to translation "quotation mark variants bank of space images of picking the problem but this fall and uh the society and you would you get a distribution which is defined everywhere outside of point and then you can discusses extension in terms of the scaling degrees will eventually find a 9 for this is the following if nation you just gave distribution by some positive effect alarmed that you might present a solution bias of Paul of Londoner and this sequence of distribution converges 2 0 means a sense of distributions and slander goes to 0 and that Islam opposed to the year and so you would test the disease's singer rarity as the origin and then the theorem is that if the skeleton Greece fine and then extending successor and moral the extent of his ascent skating degree and to such extent as defined by the derivative of the other functions of all skating degree minus and some particular scanning category is smaller than and the distribution is unique and in this sustaining degree of infinite then AT T cannot be extended so this theorem I used to
Bloomberg yeah OK I'm not sure what it 1st make gave complete proof forces of even not able to find a computer so it would be nice to know it has have a look at all the government many I'd like to focus solely its head so this theory ,comma replaces completely is son of a regularization but In order to get a specific extension there often useful and just explain how this is related to some of regularization techniques that the following considerations soul um let's assume a fine getting degree but logically good president of Saddam engine and then but can show that the distribution communicate extended to all of test function of its vanish at the origin of the sea although lasted really degree of divergence was as integer largess indicia smaller the front of the getting the then which was a projection on the complementarity substrates of test function space ends then every ex-tennis given by these homeless so we just take the composition of the distribution was there is a projection 1 minus W W projections of demand substrates and want to you go for it and all exchanges of successful note such a projection is a simple form that it can be written into the pockets inflation the sum of all WIC daytime In the days data so the functions w either form a basis of these fundamental such space they just just Doolittle's there to the basis of the distributions arising from the from derivatives of the delta function so this means just that the derivative of W at 0 or this as a chronic other times some signs no assume they have some regularization techniques of you places to use by some sequence TN and we assume that he and the convergence to keep on the substance of functions which manages the overture then TWO was defined as the composition of the this 1 Monestime you not have replaced Steve ITN and then I can apply IT and to both terms so 1st flight to this unit and then I like projection of annual and what we see is that we can construct its distribution is the limit of the end of the subtract certain divergent multiples of that other functional the shelves hold divergent competence occurrences so it's just because you have this projection but you cannot submitted 40 itself but only for the approximate sequenced in areas there 2 games meetings all I'm sorry a yacht here this is the inverse of the city was not said I'm sorry this is not all of it is something I OK so also the ends are not related to we have a few days time is no way to none of that so this was just some of the General Command of poses some taking a problem to reduce this problem of extending distributions on manifolds which management which are not defined set manifold to tool this situation where have just 1 point so in a sense you use sometimes calling its new to the submanifolds Lazio's yet and that's the way it can be done but this is technically technically demanding but can the at done who analyze this from the mathematical point of view very carefully but it's is the end you arrive at this problem but this of course requires some exercise and Michael often analysis OK you know we have all the means to construct this time operators Soviet look at systematic Fox space of space of local functions so Fox space not just in the bikes since soldiers were considered to be said there some of Semitic tens of policy and on this basis we have modification meant that MEPs space and was a space of mighty local functionality the defined the end of the semantic Fox various tools displays of might Michael calls functions just by using the seas by differential operator in terms of Savannah propaganda and because you go to several affect us they have to use allotments Rawlence's creates these indices i j this gives all the common Ettrick final rules and so on ends the bot is important is that these modification turns all-too-brief I checked this is little with surprising because you the folks Paris you have functions of different views considerations and and the functions of 1 configuration but due to the smallest most of the local functional so the local functions are single in on the diagonal but moves along the side but the singular sense wasn't as a diagonal cuts most belong to and in this day and then lost to construct the inverse of em and at the end you can find the time or operated just present .period some of these MEPs G and composed this season because of so this is definition of teams and if you like
pass can a former Lotus is a pass into goal memories just convolutions to measure induced by the from property but there is no need for the definition this does not help because you have to do the same regulations so that the people but the but to formalize as they also use season relation estimates suggest different different formally no we can define the time of the product name is the time of the product is now just a Cleveland tools a classical product by the time of the and operator was formed with the and so this this and this statement of his theorem which of course could raise as an essential point that these indication that was a notable this was done because of his cappuccino has so particularly the time-honored product can be defined as a binary product of functional enter this product turns out to be as deceptive and commutative because it's equivalent to losing just then they can define the
time-honored exponentially it is just as the exponential serious composers these MEPs teams and by definition 88 strawberry on local functional this means that the final functions in the normal although form related to age so if you change age they have also to change our function then they can defines interacting if you if we have some interruption elves and this is something 1 might call to Milan met From the priest was interrupting the array and this is just Bobby who was formula in applause interrupting fuel so that takes Best Mate thanks to the minus 1 prior to the time of the product of the incident to the function and is well defined it teachers a minus 1 of us in this set of with function Hassan generalization move that could be divergences budget for this said it's wonderful OK many of hazardous said there's some ambiguity in the in the definition of time of products and this ambiguity can be discussed in terms of saving modernization but a sense of stability that Peter Munk is not exactly the same as the minimization of realism so it's really a group subgroup of all the analytic by ejections off the face of local functions and crucial relation is that this is added to his dysfunctional this John support In the 1st derivatives should be should preserve identity and then there is something which was called to the May theorem of renormalization an unpublished paper 1st professional and the book is a different versions of the series was different generalities and so the statement is that if I have formed such a formal asthmatics and any other form as as head is obtained from the original 1 by composing at this element of the schools so all Christian of renormalization can then be deformed and properties of the school so fans could anomalies and so on can be discussed in terms of support OK
I know all the things you go to bed using and that this isn't also undercut space out there this is a very general of cause and flat space Aziz's smallest what was known but that can be generalized to people to work no note this but there remains 1 of the problems namely the inventive and busses renormalization 1 it has to do with it for every point of the space-time independently because there is no symmetry is general so hold to compare this realization procedures at different points of space-time and of course there are still some feeling of houses should be done so if you want to have operation which is locally defined but to makes his precise is rather complicated but there Of course this as a problem would not it's does not exist on the cost is best there have sufficiently large symmetry can have other space of last cemeteries they have similar advantages but indigenous generic case you already has a problem that you will so is that it seems that you have independent conditions at each space-time .period picture of course would be very happy not there is a solution of this problem is to construct theory not on 1 specific space impact on all space stands in a coherent way and again so the generalized the pockets that sums in the useful when soldier the idea is that we look at sub-regions space Simpson's own right courses such regions should be shown only a short vision globally hyperbolic and the generalization is a full length of the Aceh saying that every globally hyperbolic and constructive will re-enter the and the manifold of a given dimensions certain seaside evolved in perturbation theory really restrict ourselves to to the we the Heffernan injective homomorphism of the of the region and physiological officers specimen and this if you have to subsequent then this homomorphism just can be composed and then we have some condition of local community activities substance embeddings the Met into space like separated sub-regions sends corresponding was commute and there's some dynamical L'Auberge just outside the city the image of the space and em a kite contains a cautious surface then all isomorphism in the
language of having raised serious just Tulsa's itch to kind viewed series of Francophone incredible space-times is admissible and that exists more officers to the category of the unit of stodgy this injective homomorphisms a small pheasants this is a concept that was called locally cover and credit series and the paper by together this kind of fashioned homey opponent end In this framework 1 can define a Fiat independent of the space-time namely put field feel is the just natural transformation which is a founder of tests functions faces as the kind of future we function from so this means you have the field fight a family of maps fight em From the test such as both of them for the IT Rosser said to em and which transforms in the appropriate way on disease under and medics and so if you will this is more suggestive notation here you see immediately what happens if NEC could em than Kyung pressure preserves anti-Semitic subsidize on mattresses symmetry of the space-time is just as usual notion of Quebec over and feel of like space so this condition that contains the usual notion of covariance but it's a valid business marginal no we can use this forum amortization many of defines the time although operator is a natural transformation and so this means Emison admissable separation of and than the restrictions Of the time already operator the 2 T M we must coincide with the restriction of tea enters a smaller separation must be must inside there's a practical obstruction to produces this effect that there is no natural hot about functions this is related to this problem that there's no vacuum state and in but this can be solved his problem and the solution was a series of paper by Holland's and only of the only none of the others did not and give it to use them but you have to and to isolate C is easier as the disease known ambiguous those ambiguity is there about you the the singularity so if you look at the Formosa had a malfunction that some parts which are which known ambiguous and this ambiguous by and you have to show that what you do is does not depend on that because now actually this is a quite demanding is not so much not an easy exercise on the other part of the reason for his there had a malfunction just means that you have a certain date from set off and then but then that gives his explicit formal offers a Harlem of function which involves this Smalls functions you V and W and it turns out you want the I'd uniquely determined by geometry and W it's free and you have to show up at 2 2 2 0 0 6 2 did discuss a dependence on W that's that's CEO of what you have to to to do OK no I don't have to know what I discuss this on the level of nobody can generalize it to dear the only thing that you mean the that's what this would is statement a leading actor usually there remains some some ambiguities France's you have to cut because a cover charge that just can't be tested of course only on this basis nomination curvature of forming his start from across the space that you cannot determine this constant concern too but this is a finite number of parameters in every normalizable it's also the site of the others would be a constant again OK so what do you recommend no Your Honor fields can be discussed this gives no fundamentally new problem but you can make a lot of our roster making wrong signs and factors and I think it's difficult to find paper the altered signs are correct and I but engage Susan Grant or by a firearm eventually had 5 I tossed the ball wide at the yeah that's OK so engaged citizen gravity of minus new problems tosses the Koshy problems not posted Trezeguet's hearings and in the classic history would just fix the gage in front of history you'll cannot do this directly recalls things which if you would like to vanish have nomination commentators opossum brackets and formalism of all canonical duty this year for the IOC OK that's um the problem rests with Derek practices is that these scenes you the single largest structures on a very difficult so so I think this I'm in not any rigorous discussion of of security brackets and the uses the festival for us all year practice to eventually ,comma home with those in the area that a problem is OK I just long so I have not found any place where this was created and say it's a sufficient morality the problem is the following that all these incredible feud theory and that isn't for the measure confusion older sister smear of space and time you cannot pursue only in space this is create singularities but it the concept of Iraq brackets is usually defined as 6 times and as yeah so so so so for the linear view themselves as is OK the problem is if you go to nonunion fields was and you get additional singer writers and I just don't know whether this can be done but come in young this series we know what to do with using for the pot of gold and undergoes has BST transformation and you just at the end defines the idea of observers as the cohomology of CBS and we can do try to do the same for gravity but in gravity they have a problem namely that was just as the same as young men's defines its and all the montages trivial because normal groups of this is a major problem it can in principle be solved in the following way namely the use dynamic if you've themselves is called this of course is not always possible this depends on the age of the gravitational field but for generic few cooperation is possible and so the idea is to "quotation mark recently they're not necessarily on show just a generic in the sense that there are no special cemeteries so so that you have enough independent say safe covetous scale and then gestures such as generally globally hyperbolic matchmaker and Unix expands and the extended action in the around this after 2nd all of us so you get a free serious uncontrolled dismissed Hamas states and there than on at CUC remaining Thomas but perturbation theory and constructs everything and this has to be done in such a way that the BST invariance holes and at the end you proved that if you change background infinitesimally then the theory does not change this is due to the principle of tablet of agreement approved by parliament and and states can be constructed at the end by it the if you start from a solution of the classic integration and use his concept of a coherent states of worlds OK so I come to the end Seoul have tried to explain all this from functorial approach to contribute blocks this was originally developed for the purposes of criminalization of spacetime but In principle it allows also the framework for background independent approach to Clinton gravity Of course this does not change his problem of longer anomalous ability which means that you can perform everything step by step but you get you all around you terms is a branch in every although so in this sense is normally normalizable but I think there's it's probably possible nevertheless to interprets this time as a series a sense of effective history 2 and a half actually the have problems to see any effect of the country position of gravity in experiments so things as supply hope that these effects are very small souls but actually this is supported by overcomes renormalization group and a group of white and on the other end it's at the moment seized formalism side bridge too far from each other but the of things can be compared proces approach of wider there is 1 major major like this all of procedure there's no after no application to physical phenomena and this is you 2 1 happily feature namely the restriction to generic backgrounds maybe if you want to have an explicit example huge typical it was not a generic but in some respect from purchasing additional symmetries and then you cannot directly uses this concept so in this case you have to use other methods of 1 way I would be to add certain fields like for instance a vast fields of Bronwood could show a vicious being used just for the same reason in the a front of gravity and some of that but this is Bachrach has to be done which was not on now thank you very much for that and to it was used there is less than this year's opposes military muscle in the a loss of knowledge in the world view of the fact that won't use the of long used very much in in the lately history the war and what you want is useful only 1 will last only actually to practice currently yeah that earned his source on an object In it's so so when would do it on on on costs things that I think you can compare on and the cost is best but on profits and there's not much which was from another formalism actually a single complete pools this is the only proof focused specimens and the strength of the framework but there but in the but in the on across the space that sunglasses stress equivalent to other methods so as to your user methods which is most useful for you so that there are certain cases rather than on diseases of showing blossom it's to shorter calculations but there are also cases was more complicated so I I think I would actually what we analyzed false hope to use dimensional regularization in the framework of a string of losses this gives us his gives a position space version of Saddam two-dimensional regularization similar to what was already there and not not to raise the same but there is a spirit of some broke of body Gambia but but you but you can also use other regularization themselves so that sort thing the question is how sound the last ball of the unadjusted data on the US this use any kind of these usually is its head in states that it is more political the opposite of principle you can actually use the word to what you can do is you can replace they called his time ordering operator you could replace places by some more regular object just by the approximating as a final propagated by something regular and then you get a guy just insists pushes conversion of Stevens and was actually this has already been analyzed in the paper so you really get seed flow equation Is this framework but you can also so it but but you can use any any approximations of course momentum space makes no sense but you can use just something yeah that's just Mich as the air think that might be a problem for interpretations because you know you are free to use any offenses you can just look at the sequence of tests functions which converges the sense of distribution the final propagated yield "quotation mark the theory of a unit of the area in your area he was young but connection is not as close to ourselves also so what you can do is to and you can the moon you can discuss it in terms of sees these good news good the term endit specific so you what you can show in a certain sense Azizi Villazon and renormalization group converges toward toward to the right thing yes so you if you have a new hewing to use some regularization you get you get you on your in the rigorous time ordering operator we have your rice from estimate like then you come fueled an element of Sea Shepherd Peterman renormalization group related to this transformation and then you can show that in a certain sense this disease so products converge but I think the exists the statements which exists at the moment I'm not very strong it's related to this it's as if a customer that this pattern of active calculation of the coefficient is around the minimization called it's a right answer for the villains onion renormalization group I think that usually there is a calculation usually neglect certain town that you consider to be a rule of and I think that's the question how far this is justified and I think this answer has been given to my knowledge only in very special cases particularly for renormalization here is the nation's only this time it will take system relation but I think the problem is the as far as I know all of them are not very strong some similarity but it's not exactly but the division of what would you call back reaction "quotation mark worries that action here is that the folks that's of course all hope I think not so far so Soviet behold the formalism and principle should give it but the event there is something wrong somewhat but should compare artists much of compact a itself semi-custom ranch integration the every Plaza guides said equation has a problem that consistency problems if they letter the this is the B


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