1/3 Topics in Quantum Field Theory and String Theory

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Holographic View of Singularities in General Relativity I will discuss new features which emerge when one studies several types of singularities present in General Relativity using methods stemming from the AdS/CFT correspondence. Some of the issues involved are the black hole information "paradox", complementarity and the nature and properties of space like singularities. I will attempt to present in each of the lectures problems which I feel need further study.
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how been what they this a you will not land it was a man the on the other hand the ground being an handed from what I know of that this is the 1 of talks on topics in quantum field theory instincts I'm very happy and honored to be here too appointed 4 Lindley Michelle chair as a graduate student I of course knew the name and 1 of the seems which you researched was face structures of various series so this will be part of my life of my talks I'm also very happy to however the is show together with my colleague from my neighbor from northern country from Lebanon Ali jumps of the OK now it but I'm my my general talk in this 1st step 2 lectures I'm going to be about very as a sinks so I would like to start by pointing out a letter which In line after he had been appointed as as a professor In stock got from his friend a very good friend the voted policy and this is part of what it says in case you can't read it says so this is to congratulate him and he says I'm not of the opinion that finding new laws of nature and indicating new directions of research is 1 of your great strengths it was you've always developed a certain ambitions in that direction I find much more beautiful than those of your papers which deal with applications of knowledge the series such as for example and at the start the paper with machine I about the new scattering formulate satellites that are OK so isn't it good to have such a nice friends who will tell you exactly what they think about you that you should not lead to any of the limitations you should stay within your limitations and the new obligations now I think I should mention that there I found this In nearby in a biography show biography which mentions incline and 1 should mention that did work on Colusa climbed series so probably just sort of applying maybe gotten by accident and really it's the Shoshone who should work on them but maybe that's worse when mentioning OK now
for the last few years nature is sending us also a similar letter and what appears here of various bounds I'm I'm afraid only what could be possible masses of all kind of particles which fantasy of series salt when they should be there and we find that if there that heavier then what expected no right now there is some local or maybe to be global excitement about 1 particle a etc. and which in a few months when all if it if it's there and not in any case it doesn't immediately fit into any of the fantasies anyhow so this is the letter we got from nature and they're so supersymmetry strains extra dimensions OK now
however scientist reacted so we we can say that we have just not listening even tho nature is trying to tell us or we can say like many for many years we just ignore and we have our own convictions on how things should be and we hope that eventually Our convictions would be listened to and we will continue to work on those problems independent if we get this or not and that's of course a risking their situation but that's a tee ball which just was very honored to get the prize I sing expected for decades something which Einstein in the beginning refused the talk to accept and nevertheless other people ignored both what Einstein thought about black hole is and what he sought Aboul Khair radiation and then both of them together somehow did give some so they the Lexus maybe should be taken in that point of view or maybe not I will
continue and the 1st set of works I'm going to describe the 1st topic I'm going to describe issues related to geometry and space-time singularities and trying to understand trying to understand word is done mainly with holes above so this is and I will discuss the long series of works on the subject and some sinks into ignorant and I would talk a lot about things that we don't really understand because I didn't want just to present here closed works servicing the Dominican just find in the literature but more and this is a scene which will appear small works which are still open ends in them which need to to be fair and this is all of them no matter when they were done work in progress so the basic their questions that I will try To answer here is what does geometry capture when 1 studies quantum gravity the diagnostic tool in which I will spend some time very long time correlation so they can be very and this goes under names like quantum noise and complexity and I would describe it the defined in detail as we go along and these issues I'm going to mainly discuss in holographic set up In the key single photographic up says that when you have when you are in the conditions of a S CFT this series is really defined by the sea of the and in this year's team you can get the final number Taubert of results the way how to interview a geometrical description to them when is it possible to to what extent is possible it is already an approximation and the definition actually comes from the quantum field theory this is Moniz said it you know it would be nice if 1 could calculate in this duality on both sides independently but it's not obvious we don't have that definitely we don't have the tools to do long perturb of calculations on the string serious side while in the quantum field theory we do have much more acknowledgment many more developed techniques so really the point of view is Z date of information To the extent that it exists is secure on their see if desired and it's less secure when you talk about the geometry and the bulk physics so if I would say there is some relationship hero it's a responsible adult in this relationship is a quantum field theory because of the Singson really In what in many cases clear-cut wife when 1 goes to the other side you will receive now approximations which is exactly the question to what extent geometry capture "quotation mark when the geometry and Senate classical geometry solutions of equations of motion to what extent they capture what is going on and this will be related to issues like record information and also to the structure of space flight a singularities so the bottom line of a large part of the talk is going to be that geometry and this would be shown by concrete calculations geometry is very good in giving averages even averages of vary not perturb attentive smaller quantities however they will be sinks which it misses and we weren't sure what commences and what it captures and this will be related to bolster the black on information so-called black on information paradox and to the claim that the Einstein wasn't the quiz PIR on which I would say a few words I won't go into great detail but I would show the details Of the will they drink when the agreement is
discussed OK so but I'm not going to give you ideas 50 going to assume that you already heard something about it this condensed slide captures the basic claim the claim is that an equally form super young series with which has SUN gage group and is in 4 dimensions captures describes the CIA festering moving on the background I would say which includes ADS 5 process 5 and a black hole at the very least in ATS 5 process 5 and that specific ways to relate the parameters I've written them here anyhow the case when I'm going to use them I wouldn't bring them up again OK know 1 of
the most striking sings and now will discuss and discussing phase structure of the series this is a very let's say surprising spectrum that any accord for supersymmetry it's supposed to give us so the result I'm showing you hear is the result of an imaginary graduate student which the appeared let's say 25 years ago in the office of his Ph.D. adviser and told them what I have just succeeded today are there for respectable all any quota for young males super conforming an equal forests when young is skates hearing for dimensions and I have isolated I've calculated the density of states and may find the following bent structure of the series and the band structure of the Serie a contains at peace where the entropy goes like a power of the energy which is near to 1 but smaller than 1 9 of 10 then it has a part which goes slightly energy then has the part which goes like the energy to the aid of 7 and from a certain point onward it behaves like the entropy goes like the energy to the powers report while the region as graduate student is unknown to us is because his adviser kicked him out of his office and told him there's no way this can happen and that every team he everything you right here is counter to how we how we usually single more field series for example let let's consider now Appaloosa client series when you have composers planned series are supposed to be extra dimensions but we don't see those extra dimensions we live in our four-dimensional work so when we promised to see the extra dimensions when the energies would be large enough to excite these extra small dimensions and we're going to see the particle not just as 0 modes but the particle exceed patients there and that is when we should this is begin to see the 10 dimensional if it was a supersymmetric series aspects of string theory OK but until that is the case and the height of the energy when you have this competition mentions the larger the dimension looks like the nor the energy like we're at low energy is the smaller of the dimension of flight No we're going to see that this behavior is totally the opposite this behavior at low energy indicates a 10 dimensional system at high energy it reduces the dimension and there that's not why widely graduates who was kicked out there it makes sense it's a four-dimensional series so I'm going to show it to you how these different things come and there are going to pay play a role in hours checking of what role geometry can play so 1st the claim the claim is but this parts here signifies a 10 dimensional series this is characteristic of energy expectations of attend the mention of his era this is characteristic of A 3 string theory this is the Hoggard on spectral and remind you then comes a period which is characteristic of a 10 dimensional short shorts and black coal and their short should black hole in flat space and then comes indeed the four-dimensional series so let's go to these various components OK here is what would just what I said before that but in general if I have a few series and I was sure would demonstrate to you the entropy at high energies the equal means it it goes in high energies like the energy to a power which is smaller than 1 and it goes like the dimension of space-time minus 1 over the dimension of space-time for free strings 1 expects that the entropy goes slightly energy for black colors in flats pace this is in general a number of dimensions you expect to have the power of the energy which is larger than 1 and and in general we show again remind you how you get there but of course power larger than 1 creates have suspicion because you have a negative specific heat for this system so it's really unstable it's you can't make thermodynamics in such a system where black holes supposed to be related to thermodynamics and there are many other issues to which which come about once you can have such a high density of states so let's take 1st or education or a few series so Wearing a box always but but I always put I'm Modinomics in a books on and for the black hole we
need the book's anyhow and in 4 other systems the boxes for convenience for a putting an infrared couple and will do so I can define the system would be sold in all of series which our conformal aura as a syntactically free or I would characterize them in all series which have a finite number of thresholds that means that as you go to higher and higher energy you do not produce a rest lots of new particles but that the number of particles so constituents at a Serie has is finite then you can if you go to a very high energy above all the skaters and their numbers finite by the way unlike string series This is the RuPaul at fostering series where there are always new particles then you can get you get their relations I said in the following manner you assume that in a field theory there's there's no gravity involved black also not created so everything is extensive the only state you have in your problem are the temperature and the books so if you count the number of states at very high energy it must be equal to the number of constituents times the vote the volume of the books and there because only skaters available are the temperature and the length or are of the box this exactly would be that their amount of entropy the amount of energy is calculated in the same way except its energies so when it's to multiplied by 1 extra scale to get energy and that it is the temperature remove the temperature from these equations and you will get that the entropy goes like the number of degrees of freedom at high energy to the power won over deep and then you get energy to the power D minus 1 overdue so this is a general arguments why at high energy you should always get this dependence now in recent years and there were arguments by Cassini it is really trying to obtain a more rigorous way various type of bounds in particular what was called the beckoned Stein bound but in all those cases you are not really looking at the high energy spectrum of the system just studying something else and the bound there was just 1 and from the point of your field theory it's obvious that the bound is going to be 1 there but there are issues about short-distance behavior and so on which I'm not entering in here so take this this is a derivation it gives the correct result but but that there needs their subtleties which 1 has to take care of so this vindicates before you go back and look at them with the student calculated this is characteristic for high energy intend dimensions and this is characteristic of high energy in 4 dimensions no you realize that such a system a tells you the energy people can be made little money because they believe they live In a 10 dimensional world well actually the number of dimensions is much smaller by 6 small and of course we could also be in such a framework with little money we think we live in 4 dimensions but from such a point of view may be at higher energies on a number of dimensions would be actually smaller is that would somehow be the case this there then would solve many ability properties many properties we have and think about at short distances In many cases but this is there and what I mean that same reason in this year I'm looking also for example in solid-state physics because my experience in the past the outside general relativity usually most Florida most phenomena that we found in particle physics had their earlier counterpart in condensed matter physics so I'm looking for a 1st Solid State Systems which had such crossover phenomenon and maybe that there was like trying to see what characterizing there but that's part of an open problem to try and find systems which are not part not GIR light systems unnecessarily maybe they do have to have the door and I don't know but when they stopped Donald systems which has a property that the infrared dimension is larger than the UV dimension and I remind you again from normal competitive occasions zeal the dimensions His mother is a higher 1 here it's the opposite it's the III which is the higher the visa smaller
OK here I doing a fast calculation to remind you why you watch black-on-white the entropy goes with the power of the energy which is larger than 1 so I use it beckons Stein hawking the idea that the black hole entropy goes like their area I need this is just to translate the Newton coupling in whatever dimensions to the string coupling and then when you you if you change the language and Rome to the language of strings so strings give you instead of the box you get sustained which is a string you will find that the watching Rangers is related to the energy by this power you plug it back in and you find that the entropy goes like the power all of a D minus 1 over the -minus tool and let's see here did here is the number of space dimensions you see from the calculation already from Horace Andy minus 1 the deed is not space-time here so I'm sorry in that way in their transparencies before space-time here they just signifies space no but I
make Europe aside the comment which many people have emphasized we know that it we are lucky in order to understand physics which we measured up to now not including quantum gravity the skaters are separated so the fact that we are ignorant on what happens at high energy did not disturb us From calculating G minus-2 was an astounding precision calculating the weak interactions with fractions of a per cent calculating the strong interactions with per cent accuracy for some phenomena in each of these was this high accuracy even tho we don't really know what happens higher escape and knew we may never know sold the fact that's scales are separated it is very helpful show when we look at the Galaxy's all we look at atoms we have and that we have a big separation of scales and this enables us To work no nobody owes it to us Of course we could just be lucky lucky so that we can use renormalization group effective with Fichaud ideas and so on but it is a very helpful saying but
somehow when gravity comes in we have to take into account that the stops being the case and I show will remind you in 2 ways you can see that there was a point of view is usually we expect that there if you take a master of particle and its Compton wavelength then the higher the mask the shorter the larger the masters the smaller the Compton wavelength and that is true until we reach thinks of their which of the coalition or plant legs and plunk and we see that in this case is actually the must increases and the object ranges the equivalent of Compton Rangers actually increases as well so the separation of scales stocks to be so clear and they remind you that the same happens also for strings if outcome competent dimensions strings have winding numbers and have momentum so there must is related to the length of the competent dimension bye momentum most going like inverse rages squared but winding modes going like the Rangers so for them objects which are known Futurity but which are like go like winding molds again the heavier the object of the larger than it actually the ranges of it is so this idea that we have separation of scary becomes more complex when gravity is involved and this is something which time and again people think about in any case we see that for black holes the register short should increases was the master's was the energy drinks they are Mark had a random walk in occupation number I just write this down and looked entering no until and how how this exactly goes and when you add up to even those these objects are different than what we expect you we find that their entropy and interactive here is actually goes like the energy suffered free string theory without taking interactions into account that the afterward have to baking directions and found the entropy goes like the energy In particular this means that the system has a limiting temperature no date I don't know maybe in 1 of the talks which will come I will discuss in more detail if you cross that temperature a little bit or not but asymptotic lead you definitely for free strings you have a limited temperature no In if what I told you you about a DSC safety is
correct let's go here
then you see there is no limiting temperature actually in that system because when you reach High energy the entropy goes right the claim is it goes right into the 3 quarters that has a positive specific 8 you can increase pumping energy the temperature would increase so there would be no limiting temperature so here is a case where In it would seem that inflexible you have a Hagadorn maximum temperature but when you in a system that you can study in detail which is ADS this bound on the temperature disappears and you can increase the temperature at which OK so we're back here
to this day Graf which explained clearly to the nine-tenths and each the 3 quarters In that respect namely how what I would identify from the students with the claim was here what's claim was there then the adviser probably also threw out because of how come the field theory why should there be strings and why shouldn't this fields here which has no gravity there should be objects which behave like black holes from the term of the entropy so these are all the reasons that the student disappear however later on once a DSC 50 I was accepted even tho it was never proven but there it's a lot of indications that many aspects of it are correct and we will see here how nontrivial these things are it was this actually was argued in the opposite way namely from the point of view all the phases of gravity this is what you would expect you would expect to have a phase of graphic Thomas intending mentions because a string serious 10 dimensions you would expect a correspondence .period where the highly excited gravelly tones actually turn In the strings and this is why you see that then there is another correspondence .period where the string begins to form characteristic length which are like the short rangers related to the mask of these objects and that is that correspondence .period relates strings into black holes and then as you go along with energy because you're ADS NO ADS is important the black colors and I will show them we did in the beginning there small and they don't realize that there is a curvature of the system but when they become as large as the curvature of the began to realize that living in a ADS and in Gibbons and hoping at these are the transition points by the way if you you there are calculated in a very naive way but which is very powerful you just say these are the effective degrees of freedom to build the cocktail say the cocktail contains in its brother tones the Coptic contains in its strengths the cocktail contain small black holes and the cocktail contains large records we just watching for each 1 of them you know how the entropy should behave and you ask who dominates in this cocktail at a certain and energy space and the words I told you before are related to this comparison of entropy I no this was very useful indeed series when 1 studies the face structure of dates during the degrees of freedom in the most general case would be diamonds objects which have both electric and magnetic fields and make again reach that later on but the simpler objects let's say have just electric and magnetic charging that that all that and you can find out which face the system is by doing again entropy a argument so the key issue is the key assumptions you identified the relevant degrees of freedom In afterward you just do simplistic thermodynamic the arguments and actually they will borne out by a very sophisticated calculations going under the name Zyban Whitten and what happened after that so this but the idea is that you use the same type of argument which worked so well in Gates series Newsom also in gravity and then these other different transition .period this is when graduation ceases to be this is when string to become Blacks smaller records and this is 1 small the colors become should codes inside a genius well I think it's a small world situation when you the EU as you see you have here there is and the transition that a point would be 1 of a G-string square so this is what you would need that not to be pushed too much from 2 0 then than you would not see well engineered but GS times and Islam .period Islam OK With replied yes but not now yes sir yes you you have to parameters if you have 1 parameter which is the toast coupling and in say because it's part of the axis of legacy of the the liturgy only 3 properties become more apparent when this coupling is very it is very large and then you do perturbations in string theory which is of a company genius OK now I want to
take this the guy we had
here and I wanted to write the temperature of the system so if I write the temperature of the system
you see that I have positive specifically for that Robert constant Hagadorn temperature for the strength negative specific the dishwashing black colds and flus space and then positive specifically for black callers in a which was again the big surprise this was found by Gibbons and hawking long ago and somehow it escaped its significance escaped from many years that these are black holes which have now have that property no the bonds and hawking use them in a actually is an example of confinement and I will discuss with the weather comes in OK so let's say we have this type of behavior so if you when you do your costs and statistical mechanics this invites a Maxwell construction namely you see that if you have a micro canonical analysis which is what I've done I've shown you the full information but if you want to go to a canonical ensemble you actually jump and have a first-order phase transition you will go from the system here which is a system of private homes for education it too the system where you already see only the four-dimensional properties of the system and that the party just strained and black shorts and black hole in flat space up transient they would disappear you these phases don't materialize In the canonical ensemble if Europe temperatures you control parameters you will go from 1 phase to the other and this goes under the name hawking page transition and I will discuss it now from the side all of them
gravity so the idea metric I'm going back after I told you what would be the consequences of the work of the student which didn't exist we go to the ballot outside on the ballot side we have an ADS 5 matches and you see there are squared which appears in the metric up there and what actually Gibbons and hawking were doing they wanted to build the a Faraday cage for energy it is important to 1 study such problems to be able to say there is clear where wall where you put the system and energy cannot pass through but energy can pass through anything so their way of building a Faraday cage which in electrodynamics which accounting gravity was to put a system in 8 years where the 1st component you can relate to buy this is you Euclidean when you go to the region who can relate to a temperature and you see that the temperature goes to 0 as are the distance in goes to infinity so in a sense you're arrange the system lives in a box by using this matter and that was their direct of confining the energy in this in a certain range is you look here effective temperature the effective temperature goes like this 1 was a strudel G 0 0 and it's therefore that this behavior then if you use if all the data you give us is that the system has a boundary which has a certain structure then another object which has the same boundary is a black hole sitting in India which I've written down here and you see that when you take to infinity the boundaries of the saying they differ when power is of order related to the mosque here so this are thermodynamic auto configurations which are solutions of Einstein's equation of motion 1 Is it can be viewed as a term of ideas In the other can be viewed as a black hole in 8 knows these up to solutions of the equations of motion there is no general analysis so we don't know what the most generous solution of the equations of motion there could be others and you will see that actually would be nice if there were if there were others even tho nobody owes us you would see it as you would see that the final description is like that so this through different solutions are going to be
the competitors Of these 2 different phases which I've shown here what sits here is terminated here and what sits here Is is a black hole and the way you do weights Alibek and hawking and the bones you know you calculate from general relativity what are the
free energies of these 2 different objects and you ask now as I told you from the picture we would expect a first-order phase transition which is what was found by hawking and Gibbons it for this configuration and what what actually happens if opened formal boundary is a sphere S 3 times the circle which gives the temperature then it's lot more temperatures there is actually no black hole the black singular you could not Close the black hole with a cap it becomes singular the black hole only begins to appear as a configuration so as a candidate once the temperature of the order of the curvature so there is a phase which is just German ideas because it has no competitors then comes the phase when the temperatures of the order of the curvature when there is a competitor but in the beginning the competitor loses the the black hole has low doesn't win the free energy competition but this is for an order 1 change in curvature and eventually when the temperature Richard scales which are larger than the curvature about some specific amount it's a black hole which takes over and you find out that the black hole it is the dominant configuration so the 2 phases which we so
here just might this
construction actually materialized in the thermodynamics of general relativity that you do here so you have the black hole which Iberdrola and you have a cylinder which is no what once you look exists you you already find itself involved in another question it's something that people last without even hand having good handle general activity it is Is there a client in general relativity what do I mean you we know that when people realize but you have to write down an atavistic the they realize that the number of particles is a good number as long as there are no interactions once you have interactions you have to have superposition here there is the question is about topology namely is dawdle allow yourself to make apologies which are different or don't you and it may be that basis a post-election rule which says it wants to live in a certain topology there is no need for you to go to another topology as I said inclined paradox the question was can you keep yourself content working with a fixed number of particles or will you have to there is no supra selection rural won their interactions and you would have to have a superposition of a fixed number of particles you started series OK what 1 finds here is that if ABS 50 should work you have to have and you have to have the permission of the licensed to mixed apology as late as I said there is very strong indication that it is correct I I don't know of any proof that there is a hole in that so if you take that as your basic point of view you have learned that topologies have to me why hot this come about when a you look at the fear the series on the boundary at short distances it's entropy goes like and square when where and was the end of this year when Gates you when you look at the dictionary and calculate the entropy of the black hold it goes like an square on the other hand In the book ADS spot there is no end square Everything is of order 1 so if you start and say this series lives only in a and I know I should not allow myself to mix of black coal because it has a different apology is classically doesn't have a pie 1 if it ends here so classically I don't see a play 1 if I would not allow the black would get a contradiction and I would have to give up idiocy of the OK I mean you may want to give it up but that idea you was I this Edward Witten actually suggested he said OK no let's take allowed to change the topology we know that in gravity it with the black hole will dominate and that black hole gifts contributions of order and square and the question which is really left open and not soften the and even the people conjectured about it is OK we have here a behavior which we have shown in gravity In the future there was this imaginary student which they have analyzed but the imaginary students left so we don't actually have a vandalization there and here is a prediction that there should be a when you are a necessary for very large and you should have a phased transition but that phase transition there there were suggestions on what it is I'm not yet familiar of approved in this series that that it was there but in any case this was a case where actually gravity topless on the Gates hearings OK so here I repeat when the temperatures were much smaller than the curvature was only a years anyhow when the temperatures of the order of the curvature you had both Germany DS & EG & black :colon In be and when the temperature actually got to black hole is the 1 with positive specific unit 1 was negative specifically In any case it's the Tourmalet years which dominates and eventually it's a black hole which which dominates and these are the 2 phases of this year's so people listen mine we learned something about 8 years as safety years a total of some open issues they're involved and now let's relates this information to the original question which was what can geometry capture and in particular let us go to the case of there in black on information in an apparent OK so here the original and work on this was buying and the tools he used was ABS a shifty and the idea was to punch a hole in the arguments of still the calculation that you can do in principle Is the following you asking if there is a black hole information paradox of not OK take some initial bout state the so if you would know the dictionary which we don't know put it to be an initial 50 state you know the Hambletonian of the CIA 52 feet series let it evolve you will learn what the final status go back with a dictionary which doesn't exist to the bottom of it and see what happens so you can answer the question of how exactly does a State involved by going to the future but as I said this direct approach is still beyond us 1 doesn't know how to the dictionary is not developed enough to be able to do that so instead 1 looks at a different way To address the same issue and the different ways you take a system at term of the equilibrium now the quantum field theory sure we think also about the black hole like this and will come to but you take the quantum field theory and you perturbed and you ask how does it react to this perturbation what happens to it as time increases so when considering at thermodynamic on somebody with some density matrix role and 1 calculates the correlation function role 80 A of 0 In 1 looks at very long timescales and I want to see what that does so if you
look this operator of
any of the you can develop it by it to the IHT on 1 side and into the minus IHT on the other side insert as we usually do in elementary
quantum mechanics a complete set of states and you will see that this function the of the is a sum Of the density matrix by matrix elements times many phases and the way this system would behave well eventually it does depend on the I think it's now appreciated more than it was appreciated 10 of 15 years ago it does depend on the details of the spectacle it does depend on the details of the operators 1 inserts and I wish it we show you what other classes of results but 1 gets now
what does 1 know about such a correlation function no there your in the classical K is in the quantum cases that I don't really know how strong the year are many people In the audience normal strong this year are that there things which are done in quantum mechanics well some sings in quantum field theory and some holiday common wisdom and intuitive I don't know at what basis of rigor really have so if 1 looks classically and 1 tries to answer the question how what are the characteristics of this average time average correlation function than
without entering 1st into details which 1 should How the spectrum looks like how the matrix elements look like 1 says the following if the phase space is complex and if Lee over this year and is true maybe you have conservation of volume in phase space if you are under the conditions which allow that then if you calculate some you are or were let's say you are in some element of face-to-face then there would always be at time t for which was and for any Epsilon you would be In a distance epsilon from that volume To do that in quantum mechanics instead of the compote face space you need a discrete spectrum in cell volume Conservation Union June territory the instead of looking where you are in phase space you calculate some correlation function at time 0 and the claim is that any correlation function you've calculated at the time 0 there would always be a timetable for P 4 . com for which the correlation function at time t of peak when the difference in value than its own initial value by Epsilon it will always exist this is a claim so let's say people we were very conservative and say that they have seen it all it could be true because you see it all for a Doniphan friendship of Humpty Dumpty it is part of the French culture Missouri In equivalent Humpty-Dumpty so in which falls and breaks I will come back together again contrary to what you teach in thermodynamics so what is the way note to to make this it is useful for us so if you look at the quantity GFT overdue of 0 of some not too large correlation function you take its absolute values square and you take it time average so why would you take this quantity GFT overdue 0 absolute venues where you integrated from 0 to capital you divide by 1 over the and sent to infinity not what what was this measure the claim was it this inquiry would return essentially to what it was so that means that there are things which recur the function cannot just go to 0 now you want to measure was what weight this recurrence appearance and it turns out it's enough to put 1 over tea you don't need to put a higher power of tea in order to be on lower power 50 in order to see that actually the same does survive and I show you the
claim houses goes but the claim is that this quantity is bounded from below right 2 -minus a number at times the entropy it's not just mine entropy this but it's a number which sits before the entropy so this is a very strong statement and you will see that in the context of a dynasty of the it's an amounted but a statement it would be a statement which is the lower bound would be 0 to all orders in 1 over but it's actually not 0 because it has a electric part which is which give this country but business contribution so what does this
tell us it tells us that Always Usually intuition that's system just caramelized which would be just this exponential drop isn't correct because if that would be the case there wouldn't be a lower bound so that means things have to recur and the question is look at what happens then ,comma How does it manifests itself so right now I was just that we keep in mind we are speaking about gravity OK but now this is a a lecturer in quantum field theory gravity doesn't appear it will reappear same-store but for the time being it's not here OK so now you
define a quantity which is the noise which is its average this fruit of this average value that Chile will normalize it
this is again I remind you the quantity so let's go and calculated so in general if we take this object tickets at the G 2 and a half this was a you call to know it was called but the arrival of the nation guests and see the you you have the right and this change this letter changed your novel the same just as it became the relationship of the 2 0 0 0 1 McNall without using what you you will appear right now was noticing Digital will appear in the slider 2 but thank you and I don't know how to change it so that that's so this subtle different letters which have been identified so we take this quantity and we averages and let's take there to make things simple let's take the to have I have to get a feeling no diagonal elements in the letter B a matrix where we have removed is that only elements if you want you had an element of a person before it was called you know it's going to be I could removed from it write a new object were removed everything from the they so in in that case it is clear that if the diagonal matrix elements of 0 than that when I take the average it would enforce the and equally E R & D and equally as because all all of the things that yes this is the line above is evident respect so in that case but 1 I would have gotten that safety is that some there world is the density matrix so is positive and the square of matrix elements is positive so in that case he is actually positive and you see From there comes to know about it in the case of General they a B that the things would fluctuate around 0 but this captures a feature that there is a lot more about OK so now you do an estimated and I would show you a rough estimate and the estimated gives you that you get here into the minus as but also actually in here they can appear at another number so the main singers the decay stopped you cannot just go down because you have to satisfy a lower bound here is a rough estimate is that you take them there the matrix this is say on the maximum was actually see it Teakwood 0 and this is the normalization factory lost about and you find by analyzing the numerator and denominator that their ratio you get here is of the not the noise which would take a square root would be easy to the the only was an exercise in a given year or definition of entropy this something In this year of little OK so there's comes out of the density that is let's say it would be a micro canonical ensemble than it is in the normalization of the Michael canonical ensemble as would come to the same in a canonical ensemble it would be the same I didn't it completely here the approved for invited to look at that in the paper and it but I will show you a an example where s appears also saw the object to be around here In this in this here but I told you seems to depend on being sold be actually reintroduces and I was shown later on the lives of 2 next time I would show would be which is related to the In A. I did state thermalization hypothesis with the matrix elements of being contained in them and need to so can wear what where do where does In other words doesn't appear it appears in role it appears in the sun because we're way summing states and the number of states we have is the to the S and it can also appear in the matrix elements so this Sunday places where the entropy images In the final outcome is that it goes the estimated in in certain cases likely to the minuses and that's the end of
the month the lifetime to of time he's been with us electorate that we haven't reached that you hear it here this was just a state of what is the average what is the average noise "quotation mark then we have to see the belt recurrences of nowhere into averages OK so 1st time at the 1st thing is when you look at these graphs of of how
this would behave the tournament usual tell modernization show that I I would expect that the coordination function drops exponentially end it's expected when there are a large number of conservation laws 1st you go to games generally you 1st removed stuff and then after you go further you you get power losses but in any case the thing goes down but let's say that not too many conservation also losing just caramelizes so this would go depending on the operator definitely that Universal would go like each of the minus some government the and because you need to which an average of 3 to the U.S. at the time when you reach the average is afforded us for the 1st time scaling the problem as formulated here Is is that that is the time when the system begins to reach the average and it cannot continue its behavior know how we intuitively do you look in the summer you say that as long as the smaller than at all much smaller than then the spectrum for you which continue the highs above unprincipled uncertainty principle for time and energy you don't yet feel the very discreet nature of the energy levels so the exponential decay part is a time when everything looks to continue and that stopped when you reach times of now I use the truce but this is what I've written here argued Duncan calculations that Of the horse that I have not yet seen somebody really even during the computer simulations showing that all these intuitive ideas of the ones which are accepted other ones which actually occur so now once you reach this value over into the minus says what happens so there is the
1st time scale which we call the Heisenberg timescale where the system begins to feel that it is the sum of many different let's clocks in single feed to the ion file difference of energies dynasty each 1 each phase some circle and the point on the circle for the clocks have deviated and you begin to feel their each its own frequency How do you estimate the time that time you take you look at a certain band you have you say that the average 1st characteristics time of that would be related to 1 over sorry to live at the top of the at the there beverage energy difference because you have a sum over energy differences so that average differences in energy would be related to 1 over the number of states in this district bank which means the time goes like the number of states so this time goes right to the U.S. this is not yet the time which is particular to you because you can be much more particularly than me once we decide which epsilon you need you may want to have much smaller abstinence than I want to so this number here it to the US does not depend on how picky we are it just the statement that I'm going to feel the fact that I have some average difference in energy use of the system so I I don't have thinks of order 0 I have things which are now into the minus now in order to be able to to want to reach an estimated so
this goes on and this is the highs and the at times and those likely to be honest now you can ask yourself OK I want to be look at this set of clocks and I want let's say all of them together and look at the global shift to be was in Desert of angle where they were originally at time t . 0 which was era and I want to see how long that how long that takes so how do West and now comes your precision because then
codified is related to how much precision you so you have this motion on tolls each clock defines a certain competition area the problem at the time it takes should be won over the probability which is related to the volume the volume would be dead time for Over 2 apply 2 the number of degrees of freedom there should be a power attorney multiplication so the volume to which you want to return is dead time for over 2 2 the number of crops that you have and you want to return in all of them so that means that the estimated time goes right to the 2 number the degrees of freedom times and although times have been culled from and this is going to give you the plant collector which should be exponent of exponent of times the mobile Dupuy over overdone conference so if you don't care very much and you make that call for 2 your back to the highs and the times but gives you a very particular and you want to really took to the dead for significantly smaller than 2 then you will have you will have times of exponent of exponent of us and then would be really returning not just that you are not 0 and that you was there which would be returning 2 of the original value that you had so this is
when despite will become a four-to-one 1 so these last 3 times the time of drop stop of the drop of the thermalization or whatever behavior then there is the time when you begin to feel there is a difference in the average difference in the distance of the energies Of course when the then smallest adaptive use the longest time scale than others get it instructors the timescale you look at the average and then exponent of exponent of less you begin to return to where you once were so here is how this how's this would do would look under this repetitions in time no
it's interesting that all these times go egg but they export each is an exponent of the other 2 can ask What is the love because we are destined to the yes indeed to the to the and the love affairs is related to something that did not describe it would not describe here which is called the scrambling time which is more or less the time it takes you to spread the information until where rising before black hole using your calculations said in the past so the that's another time but I'm not using it here but it's also related by log OK but I'm sitting here before to Boston can't look at numbers In that you want to see what OK what does is give so I'm afraid the natural unit is universe lifetime we call a duet and let's look at St not that the 2 the an oddity to me to be so the page time this also goes for other reasons things of order as go under the name of page times so look at page times of for a black hole the size of a problem it's time to the tenuous the page time of the black hole which is tend to the 9th solar masses Quiles over with 3 km radius is tend to the 87 U.S. and this is just as so what is talking here about something which science fiction doesn't there touching in what what 1 is going to epochs which are enormous so if so why but at what a justification of doing such a thing so that is that I would go back to phenomenology In many years ago when John Illiopolis entries grenades there was no 1 looking that's it's something which was of no significance to most people which were flavor changing neutral currents at time there was no nice picture of the Standard Model to most people in mind soul why concentrate on flavor changing neutral currents in their absence who cares nevertheless from that they derived a deep understanding of what is the structure of the universe in particular but you need to have at least 1 more quark to complete the double was strange for so here we are in a situation where we have a very consistent and in a way I would say that any hold any hole you can find in this fortress all of course work of bastion of consistency because that's not what tourists come to it the experiments as we discussed His words why so even if there would be a problem after III to the to the let's identified the problem and once there is a whole maybe 1 can inflate it and that it had lived there for what track but maybe you will find that once again the 4th reactor it doesn't allow it you have a problem so the justification to going to these very long timescales is 1 is looking somewhere for a hole in the argument in documentation OK so the summary that timescales related by that there is a scrambling good time A. there is a page titled The End of the decay there is a Heisenberg time when you begin to feel the average distance and then there is the Ponca at time when you weaker To what you really started so how is
this related and here we have to the original problem with discussing OK so the way it is related it is by doing that EDS 50 and as I told you the responsible adult in the relation is a safety so we know that if we do any quota for superior series 1 is fear the series unitary we have no reason to suspect the series not unitary no evidence of that and the spectrum is decked because I'm sitting honestly so 1 is sitting in the conditions of a general conformal field here and there is a lower bound To the object at hand it's OK this is a spectrum of states and the point is there is a low above the law abound in I'm talking not about the generous series of this series hand goes like this the 2 the miners this S's goes like minus and squared because we're in the joint for presentation that could be numbers before but it's and square this goes like into the minus 1 of Virginia town that appeared in their slide where I showed you the dictionary between the field series and their ingenuity and that means that this is not from the gravity point of view would slump attributed to any finite ordering Jeanne and Spanish but for an louche and if any is incinerated also finishes so you have to think of an end which is extremely large but not in thin and you find from the CIA 50 you know that there is that lower bound and now OK you found it let's just find it also In the body let's go to a temperature which is well below the hulking page transition so we have the black hole and let's calculate a typical correlation function inside the black hole everything in the background of the black hole and let's see ,comma the lower bound and he was appointed by the Senate emphasized that there is a power Europe you go when you calculate the correlated off 2 scalar fields in the background of a black hole you go to the territory's coordinates so you shift the horizon to minus infinity and you find that the proper data can be calculated by solving a non relativistic quantum mechanical problems sitting in a certain effective pretension and effective potential characteristic is that it's little the light as it approaches the arising minus infinity for such the perspective it doesn't again so EUR ground to be sure the teams would behave as you expect and that you get the lower bound you lost that you do the calculation and you see the reason that it was important to have adept spectrum and indeed you find that the black hole gives the result which is 0 so you can't drop a DSC IV the because you get 0 From the black hole when you know from the responsible adult there is a lot about it is the time to move on to the end of the year as the crossing Norman molds whatever words you want to use but as I used here because I used in the assumption the fact that the suspect was adept this assumption 170 here the spectrum is not get so you lost ground and indeed you don't get you don't get that you don't get that so you get if you wish what you know anyhow the because normal mode behavior so you get an exponential drop exponential drop of we did for the lower bound 0 finished a contradiction OK but you you Godzilla so you you can dance conservation laws or not but it is you would you at the end just strollers throwaway Diaz 50 Senate's proving that when the we didn't all of but it's no it's not the same as the law so that is not from using U.S. so we took a on U.S. attitude and said Well that's not concentrate only on the winner we have to configurations and I remind you that there can be many more which we don't know about you but we have to configurations as long as the leading configuration delivered to them we follow it but it is a leader configuration gives us 0 we should go to the other side of points and the other saddle point is terminated so 1st if 1 does it once again just like before just complementing the entropy you see that you need the change to have several topology you cannot live in a Super selection of OK so in the length politically correct language if topological diversity you have to allow the different apologies if solves the problem so you get out of the black hole and you go to terminate the S which was highly among leading configuration what characterizes In total coordinate how does the effective potential I described to you before there is a box and that's that characteristic of it's a box so you get the potential where you have a box In the box appears near also near where was the horizon it's moved away we have the box and exist stage you have the background in which the spectrum it is director and the gap is determined by the curvature of the DS so now when you calculate the object going to find that there it to respect the lower bound so you see it you see that a configuration which was down by the to the miners Esq he was called to help when the leader failed and it delivers it gives the right answer but it delivered for what it delivered for the average and I told you in the slogans in the beginning sinks geometry works for averages so you we had here an average which is extremely small exponent of minors as I think you got an idea of how small it is so it's a very small effect and nevertheless this small effect can be reproduced by geometry but this is the average what happens when you ask OK here is a real I will end in 5 minutes and by the way the elections will be an hour and a half and not 2 hours that's the good news from from today is the sole there if we look 1 way to think about this we had the black hole we had a black hole and then it suddenly becomes as it moves in phase under becomes terminal some of browbeaten and laugh yes everything is lies only by the I'm way beyond yes I am way beyond the hulking page transition I'm not discovered we can discuss was what happens below but might I'm taking a terrible average way above the hulking page transition so everything is a large black holes here he has the U.S. Justice trunk would not the way we like him OK in some sense instant on tunneling the superposition whatever your front as 2 tries and tries to build and then the dissipation instead of helping in the happening in the presence of the black hole the dissipation happens In the presence of a DS and this is somehow in type of attempt to see how this recurrences come again and again however if you look at the exclusive quantity not the average but you ask what CEO 50 is that every time then you see that this is cannot explain the problem the geometry fans here this way because we knew it from the start we know that Blanca occurrences order 1 because we had to resort to to the rules of configuration to give us the peak In the zeros it would be multiplied by to the miners and squared because the probability to be in it and not invade leading configuration is suppressed by into the mind-set so what happens is you get an average by having Short Kwok-hung highs and the times the recurrences happily happened very frequent at the height of each recurrence it is minimal at to the mind-set where the Cheung behavior should be larger occurrences which have been extremely rare and that we know from there quantum field theory and both gave it happens and it's very nice to see in detail that gives the same up to coefficients in the numerator however they don't give the same exclusive behavior so we sell here an example where geometry gave an extremely small quantities correctly when you were talking about the average it's fate together they correct behavior when we talked about this very minor the year so we see here the limitations of this summit classical description it cannot as far as we know opted to date it cannot it doesn't mimic the detailed behavior and I think it's nice to seeing that cause bringing is given by black colors more details are specifically once on Notre produced and I think it gives some meat to the general feeling which I reviewed by another example in the next lecture that indeed in there is some cause draining when we talk about geometry and when the questions are very very detailed we have to resort to other stuff except classical geometry is not giving us the the very special rights up to year you generic wanted corresponds to "quotation mark so maybe the top of the measure something geometrical and half of the race officials to see if he got it put on hold no know and there were the many proved Richard 7 children since I have not improved show but I would say that the idea if geometries that had a description it should be generic but that that there never is a claim I mean also averages if you wish something on generic and we decided to pick an average and the geometry works so there is no doubt that that can be quantities for which geometry and work like but the idea when you use geometry is not that you have to look out of the properties find 1 property for which it works they idea is what happens generically choice icing the idea here would be that for generic objects averages do but it a more detailed explosives missus and you can see by how much it misses now what what was 1 way which suggested out and maybe this will be the end the with that OK so that this reminded us like QC the people were trying to find what is a configuration which we need to confinement it didn't happen at the end and it's a more complicated phenomena so maybe that looking for the average for the ideas and having also the black hole is just the tip of the iceberg of many many lots and lots and lots of stories you could tell geometry and when you some over all of them maybe they look like toasts brick wall because if you don't put a brick wall of whatever what you want to call it you can you can arrange so that you get also exclusive C team for everybody you can do it the problem is that and this we shown on paper but the problem is you don't know what the only From the strain point of view what is the width of the brick wall you just impose there exists and there for the series there exists a recall for which were produced because of course it produces a gap spectrum but what what what it is it in order to have a real understanding if that's the right way you need to get it from the strings he resigned you have to get on with and then have to show that that would actually corresponds To and what you get from the QF so there is a way that there would be many many configurations and the sum of all those configurations would effectively be the black hole was a brick wall so that you get the spectrum which was again the idea of 250 million others how to get the gap spectrum OK it is not a OK but the but cleared the main thing is to get the because that's without that there nowhere to go then we do that in the next over details the 1st thing to make the Infiniti the continuous infinity into industry that's that the kid the kissing him to do OK so the conclusions off what was done her mother Senate punched a hole in the Hawking's argument he shoulder because this will be his arguments evoking did not include the possibility of changing topology some of the burden shifted and other people may have picked it up or not and I sing 1 is still discussing this issue topological diversity is required this without it it won't work I also would say that we in the for the fact that the even it works for the average for such a minute quantity shows you how difficult it is to punch breach this hole in the wall of the consistency of string theory we have not yet succeeded In in doing this and that these are the slogans and for next time I will stop and continue because 1 question which would come out would say OK what did we have here we had a very strange situation that the longtime correlation functions behavior was dictated actually buy that dynamical this favorite configuration this is generic or did it just happened by accident in this case and we would start by addressing this and relating that to the hour equal EP but OK I will stop here and there and