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2/3 Topics in Quantum Field Theory and String Theory

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In the Eye this you will not mind it a man In the on the Net thank you what you I will repeat some sings which I said that last time to give a general framework it 1st of all I'm the discussing the problems which interest me no it is but 1 of the features of them is that they are not all of them the prepared and packaged in a in a well defined way they are loose ends and their problems which I will some what we've already seen in some more that we're going to address as we go along the general seating area which interested me here it is what does send a classical geometry capture what type of phenomenon and what is the fate of singularities of various types in string theory their generosity toward 1 users is there they CFT framework where In this duality a pipeline arrangement actually the responsible adult is the field series and we try to extract as much information as we can from the quantum field theory In order to be able to extract various interesting qualities about what happens on the string series side description but in a sense the security is the definition of the monk attributed the Phoenician the strange series side up to to maybe some loopholes which appear but this is a general point of view and I will repeat that most of the worker described here was done with Jose above 1 from modeling OK so we spent quite a lot of time a discussing fields fury it and correlations Sinfield's fury at lunchtime and I didn't enter I wouldn't know what it did not enter into the details of what this precise tool .period correlation functions I am calculated in the future series and it would turn out that the result depends very importantly on what these things what these operators stop now in retrospect it will come in a few slides we would see that for Celtic operators and I would try to define later what I mean by the Celtic operators if 1 calculates that two-point function in this field theory which has been discreet gapped spectral and his unit terrain then there's some general features 1 can say the 1st feature is that depending on which precise operators such shows and there is Is there a power like or generically an exponential decay In the beginning however a certain correlation function which we defined which was the time average all of these things written about integrated time-averaged taking that values squared normalizing it by its value would 0 has a property that while it starts order 1 by definition here it reaches a certain no abound around which it can also be legally so I would say it's not a precise law a abound but it is it gives you it's a story it is precisely for the average of the square it's not precise lower bound for the same before being squared but before being spread it would fluctuate around that average nor abound because it would have to reproduce so 1 has a decay 1 begins to feel the fact that the suspect was not continues after a time which we call the highs and the time which is the order of the exponent of the entropy of this system and then after a time of exponent of exponent of phase 1 approaches which is called upon come time 1 approaches the of values to whatever precision you actually want no the fact that we took this particular operator and this keyboard asked me later they OK so you took this operator it's an operator to local operator on their boundaries series What does this correspond to in the body so that general idea but it was never proven it is part of that let's say don't mind the field which people are trying to probe and understand is that the behavior of this Caltech operator corresponds to the behavior of the local two-point function in the background of under determinant in this sector and in the presence of a black hole in on distant so the cook the claim is that the way 1 captures the let's say Blair coordination black chorus properties of the system comes by this operator being co-opted and the precise details of how localities matched are not known and are not taken into account here no we found out that if 1 looks at this correlation function which is the correct answer calculated on the boundary and compares it with the ballot results at a temperature in which is a temperature where black hole dominates let's call it the statistical ensemble that is technically called Beyond the hulking page transition which I described last time and in some detail how we've to construct a couple months constructed etc. then there is a paradox because in the presence of the black hole calculating such as singing about would not shown no abound but this and the average quantity would go to 0 instead of being bounded from below now the remedy for that for the average was To suggest that 1 has to remain requirement has to recall the other stationary point and the problem which was the ADS terminating his background and I explained why In calculating in the background which is terminated Diaz the correlations the average correlation function behaved exactly as it was expected that with the same numerical values as what 1 got from the future so when is relieved that the ADS CFT correspondence work and 1 learned on possible that you have to take in the bunch series at least 2 different topologies which was always a question that doesn't take the same topology are you there is supra selection Road which sticks to his 1 topology or should you take several ones and they also remarked that actually before this this is due to meet the there was in the it then look at the higher level it was shown that if you don't take the black hole when you take Tourmalet ADS is a background and you forget the black which has the 0 story which has a 0 . 1 unlike terminate guests that you would again get the paradox of the Sept 2 AFP places independently where you needed to defines the rulers of the field series so that you should atleast state all fields which are solutions of equations of motion which have the same conformal boundaries even if that apology when you look at the whole body is different so these were the various lessons and that we learned and then we learned that if 1 asks what happens at any particular time t we totally missed picture namely we cannot produced by the term of 8 years which does reproduce the average it does not reproduce more exclusive property which is how the correlation function behaves at each time so we learned that geometry can produce very small
and to the minors as effects when there are averaged in in 1 particular example and on the other hand we have shown we have seen that it did fail 2 get more exclusive properties the simple geometry of just taking the 2 solutions didn't work we discussed what happens if 1 takes tries to take more solutions won't repeat it it was the sense it correct that for the case time has reasonable note because governors totally not universal government really depends on the details of the opera the key to the S and lead to the arrest of the song these are universal they only depend on the fact that this is every Celtic operator on the other hand the particular value of gunmen does depend on which operate 1 puts on and as 1 does not know the exact correspondence between the bank and the boundaries this exercise means is you can do it try and do it case by case but if you don't have a general picture like you have here and people look at some specific cases but that the picture is not gender and we ended up with a question which I would on Saturday was the failure of the thermodynamic the dominant contributions to reproduce the average we defined quantum noise was the average value of the square of this operator was just an accident or is this something which would occur In general in a certain certain general under certain assumptions as usual so again the conclusions was what I wrote before and that there at that stage that the burden of proof that a well-defined information products exist shift to the claim that the biological diversity is required instant series quite a formidable bastion of consistency because we didn't succeed to extract into the minus as effects which I'm not very easy to extract so this is a summary of what we did in there in the last stop no the burden of the fact that the burden of the proof what shifted back to the claim didn't mean that somebody won't lift the ping-pong ball and and shoot it again and indeed after this set of works have been done in another
attempts to claim that the series isn't well formulated came a by it sure claims that for various reasons which would require a set of talks on at all on its own which are not here is that actually whoever volunteered many years ago 2 0 parachute into a black hole just out to see how he crosses so she crosses horizon and enjoys experience it would actually never enjoyed the experience would be boys and that it would meet when they reach the area over the horizon would reach a firewall where the temperature would be In defined would be infinite and the object would just boys this has led to many lots of exchanges and in the literature and from my point of view the problem has not yet been formulated sharply enough to be able to decide 1 way they or another even tho there are many people which would not share this opinion but this is my own opinion so I will concentrate on 1 particular round in this fight is there a firewall is the horizon there's something special or is the horizon like we would think from GI something that you could just crossed using the equivalence principle as insurance policy and nothing would happen or is this fallacious and I would take 1 particular aspect which follows the scene which we discussed here what does geometry capture and what does geometry Miss and this case it goes under the name are equal EPR IRA's Einstein was an EPR Podolski is added to this and the claim is the various paradoxes several of the paradoxes which were brought up which led some to conclude that the simplest solution would be a firewall other solutions would be more radical even gets the problem gets resolved because entanglement issues it can be related to the existence of fines tunnels and bridges again it's very nice to to have to discuss it I won't do
it here I would just discussed the particular element in the controversy which relates to the scene when discussing so there was a paper by sentence asking which said the following let's take the arguments that a group of people I'm highly emotive Kaczynski and maybe the younger guys would remind me I'm missing somebody starting was located on the OK Sally maybe empire but the paradox Is is is just not there so they say let's consider an eternal black hole In an attorney with the very good indications that an eternal black coordinate Diersen its dynamics can be described by the product of 2 disconnected conformal field series each living on 1 boundary and writing a maximally and pounded state which would produce the geometry and expectations of the geometry of the return of black hole and the state would be the following you hero by the Wingfield Suri 100 on and time damage but they have the same opponents in 1 time froze this direction the other times frozen that direction and each state here would correspond to some story in the body so the state they take here is the Maximilian time states said they organized the chairman Don Young then they take the product of I state of the Hambletonian and efficiency aid which is identical to widen state and field Serie B and you multiplied by the to the minors the energy the a level appropriate for the state and and a divided by some temperature T with this temperature T will define what is the temperature of the tournament so if you want to build a black hole of terror of of temperature T E This is where you put the parameter to and the 6 claim is that all the physics here would be the physics here and in particular if you look at this object it has no firewall so this is a situation which in principle would obey would be 1 of the examples used by people who claimed the firewalls and they say Look Here is a counterexample so the debate did not follow the above the counter example but about the generosity Of the counterexample the claim was this is a very opted because case and then my elephant poaching scheme suggested OK here I just remind you that it if you look at a one-sided black hole that is you integrate over the other the interior you will get the density matrix which would be appropriately describing the the biblical
Plano and world so the arguments of flavor of Britain's Sky and Maliphant but just below the
following day OK you took a very special state which was a maximally intended state why don't you take a state where you have a general an structure here with some general matrix GNN and not insisting that this is delta the function look at this stage and now calculate the two-point correlation function which we discussed before and then they said OK we learned in the past how the two-point function behaves in the presence of of geometry and this is what I described to starts being the 1 in decays and then it reaches its average and jumps around the average which from time to time big jumps and mostly just very small fluctuations and they said that if you calculate the noise with scholars in this configuration you will not get this
behavior which we saw before that was a clear you would not and emphases was here they claimed that you wouldn't get that there ever the height of this correlation function it is of the order into the minus mind-set you never reach a 1 so it's not something which at least is known
be reached by a geometry because when you look in there if the black hole of terminology is you get a totally different behavior this will give you a different behavior sold for a generic state in the quantum field theory you would not have a geometry and therefore you cannot argue that that everything is smooth you just don't know what is going on there you cannot describe at some particular configuration and say look here is a very I'm singular but a near the horizon geometric configuration because the court noise calculated N on the quantum field theory side does not correspond to anything that we know about geometry and maybe there is something new about geometry but then surely but it's not that what our experiences in geometry OK so this led to a 2nd round of looking in more detail in all the questions of the behavior of the long-time behavior of the correlation functions and in particular at point with maybe it was known to people it but let's say when I spoke about it here many years ago definitely was not appreciate it enough buying by me Is is that all observables you choose and the way that it will police and that if they are for the dynamics has a great influence on the results so the same social duty Florida emphasis of
emphasized work for Celtic operators are not true in general 1 should really revisits this and studies is for each type of
operators now in particular as I told you the common wisdom From what would represent successfully and get give you features of black hole physics and about the claim was optical operators in on the boundary and the Celtic operators will defined in the following way they're supposed to be the representative operators again doesn't mean they're not other operators but that's supposed to be not represented so how will the Celtic operators a defined so this is taken from sinks which will done many many years ago in which fundamental quantum mechanics and statistical physics and it went under the I I value to harmonization hypothesis I that story eigenvalues demanded by India against state optimization hypothesis and the argument goes the following they say there the general operator which would be somehow emission operator Of does not commute was 102 on young and more all over if you look at the Eigen functions of this operator there highly uncorrelated with those of the Hambletonian age and these are the operators which you should measure the time dependence endit to be more specific that claim was that if you look at that unit during matrix which takes operator the generic operator if the basis of energy I did states and you die of analyzing Sobhi is a capital B isn't I again States of energy days there's been small B is being very organized into the basis where it's all like functions that the matrix which does it is excelled random matrix so if you do this if you make this assumption so it's a strong assumption where you say you have the feeling on what is the right fit and as I said among the open the loose ends and people aren't working it from any point of view is to give more and more basis for this foundation I would to foundations to this claim I will just take it and see show you just what are the consequences on the structure of representative likely to be from this definition from its definition so the claim is and I was showing that the typical matrix or at the representative matrix has day of another form this is again in in Britain and the basis of energy items states on which there is nothing special to say and then it has an off day another firm 4 where you have shown a small number smaller B which multiplies in this city this being it will be relates of course the eigenvalues of this bill is written between United State of energy and indium and this how you define the bar and and on gun so this is some number which has no special characteristics then you get the typical the banks are magnitude is of the order of the to the minus entropy over to which is a strong statement so these are very small of diagonal matrix elements but they exist everywhere because this object here is supposed to be a random matrix of absolute value 1 so it's something to the I picked up where Tadic random so this type of matrices are supposed to be the ones when 1 does the calculation so how does 1 get to this result so in a very crude demand there let's look at the matrix you must be a unitary and by the assumptions that it essentially all Oregon functions of the participate in 1 I can function of the beat you get that each matrix element there into to the estimated supplements so each matrix elements must be of the order it to the minors as Over 2 so this is not the you the although they have little part of me is you some number beat you bye so this would be of some of the previous elements of some random because of their character the product would be to the minors and because each is to the miners as of a tool but then you don't get into the act of them really something together coherently but because of the random nature of use it some random walk and you get into the S over to so that's how you end with this being being of the order the minus over to and multiplied by some other stuff before the 1 which is rendered so this is a picture of 1 has another told you this where ideas bored by Usher purveyors and the Deutsche suddenly from many many years ago so now 1 does given that this is the matrix which is characteristic for the black hole of what you should ask what you should use when you ask about systems where black code should be there then they're all for the system of an equal for its acclaimed it you do the calculations with the mother of many type of calculations you could do and I would I want to all year I won't show you all there are some way you do the noise when it's one-sided the EPR noise is of course when you have that you're looking at noise or as they showed you for states ban em was a major GNN between them what's that what noise do you get there then you have what happens to be discussed when that several bands and also when you have is a system which is causing terrible OK this is just to show you know that they are you can do various things you get various results which depend on many considerations I'm not sure I'm not going to enter into all the details and is going to give you the results that we are interested here but I just want to know to show you because it was a lesson for us the jurors a lot of structure involved in which operator you choose to take and what state do you take it as a state you're interested in the system which is a conformal field theory is it that they have another visit entangled is a pure Is it mixed or all of these things change and each 1 gives different results for their
release so now let's go back to the original playing so much the Senate's asking to diagonal state they said that is the state you can go ahead and calculate the noise in that state and of course you there is no horizon at all a soaring always singularity refer no sign of singularity on Verizon and you get the same result as I discussed last lecture so this order looked very geometrical and very on problematic so we have
the value of the correlate to return itself before being normalizes of all the while it characterized by the geometry and the noise it gives is of the order into the assessed fines some numbers which I didn't discuss any now let's
take this state which is Monday and I must ask what happens there so you can ask what is the value at equals 0 or in general and what is the maximum value that you can expect this to get To give but taking the lessons of last lecture you can also ask what is the average noise that this produces so if 1 looks
at the 1st question the 1st question was the 1 in which it was addressed by Meyer of the Raczynski look at the correlation function and try to identify its maximum when the matrix were the ones that tight we have discussed in a reminder that our M and many of these random thought the random there was into the mind-set over 2 which was multiplied by the order 1 but random phase which appear so when you write look at this correlation function you get this normalization structure then you have to go just something over and then you have these number of terms you have these which have faces and you have this stalls that's 82 the S dimensional tolls which we can build from E N Indiana no it is clear that each of the phrases he was overloaded so you can try and use part of the it's telling me the face structure the random part of our mn which you have for the general state however there is only an exponent S of such independent phases you can use and you need in order to make this of the order into the U.S. it is to make all of this some coherent you need into the tallest terms and you can't do it so this won't help you and what you end up with is that when you look at a random walk structure here you have only to the S elements which means that the maximum that this will ever give you is to the mine and therefore for a generic stating that in the conformal field theory if you try and calculate what does it give you don't get the geometrical picture which is at least some the source of order 1 and it repeated itself this order 1 many times very sparsely but it was there here you cannot get it because again the argument Uvira Torres all phases but these are not enough phases would be to give to get rid of the random structure of R & amp however if you calculate the noise that is the average value of the correlated you don't get exactly the same result as you got from the day of another kind of state so this is another example old geometry in this case that a eternal ADS which was really corresponding only to them March intended state giving you the correct the exact and correct sorry In giving you the correct average quantity the average quantity was calculated by the geometry very good the moment you want to change and looked into more detailed structure like what is the maximum this hazard certainty what I mean is there at the for which this BB order 1 the answer is no indeed there Ma often punches Keys stated correctly this doesn't look like a geometry when you ask very exclusive question but if you ask an integrated question you get the right result from the geometry so this is example number tools and they said you can can we show that it does not depend the noise the average noise does not depend on the entanglement then this is just a remark to people which were more concerned with this problem and aspects of it that if you want to get in order 1 amplitude which I showed you is not what you would get to get into the mine assesses the maximum you need to bear have a state dependent conditions and indeed some solutions of the problems of the firewall came to the direction of changing a little bit how the rules of the game and say that some of their their particular since you measure Our depend on the state where you are doing the job you're doing the measurements so indeed never mind what's coming you actually know that this is what would they idolize In future reais assault what will was the Serie B this is a random parked on mattresses a and B. if you have at that particular relations between the matrix elements then for this specific operators which is again type of question you asked me to bow before but this our operators which are at the more interesting because airspace dependent state dependent the depended on the matrices Omega you needed to use for the victimization if you take if that is what happens then you can arrange for good but that's
not the generic situation so to conclude what we got here up to now we had 2 examples where geometry gave a number which was very small and the noise it was it the minuses says it's not perturb active from the point of view of this gravity series should be 0 to all orders in perturbation theory and nevertheless geometry can capture but if you ask a more exclusive question like how does it behave at any time to geometry thank cannot doesn't give the right answer maybe eventually away would be found but as of now it we don't know and in the question could be so murky I was using his analogy when people 1st sought today and you knew you could solve confinement by instant dominance in 4 dimensions didn't work and then people began to look at various configurations 1 of them were called Maryland and 1 started to invent more and more words to try and get the results you want but it ended up as a very murky subjects which actually did not lead to any concrete conclusions and it could well be but their attempts to get exclusive information out of detainees summer classical geometry will also end up being very murky maybe some nice structure will emerge and dimension let's say the issue of the brick wall of the stretched Verizon which may be by some way can effectively described the sum of all kind of things which are murky but that where that was the start of his today beverages of very subtle and small quantities are given by geometry more details geometry does not want to give them but you know the result from the field theory but you don't know how to reproduce From the Bottom please sir knows clear there's not a clear calculations which produces now I want to go back to 2 more details about this in particular I'm interested in the details wasn't accidental that the low energy band that was terminated yes dominated the behavior of the system and gave us the result we wanted so here we take care the matrix which has stood do totally that's a different structures in terms of number of states which is characteristic of an equal for I showed you that actually fall bands let's start to most of the moment just look at to events so there is 1 where there is a lot of States very high and there is 1 band with the number of states is very small and the claim is that
generically it would be that the low energy band will dominate and how does this work so so there are a few bands OK that's the sense that it could be the system doesn't have a total ban structure breaking it into pieces doesn't make sense I tried to argue last time and I would show again the graph in a few minutes that in this system at hand it does make sense to look at several events so the calculation I'm giving you the estimates I'm giving here or in the case when that it makes sense to look at the band structure so if there are several bands there is a probability that you will land in that band and then there is the type of noise that you would produce if you are in this band for example when they were told when we go back not to the Futurity was the band's but we think about the body picture there were 2 configurations that was a black hole With a certain temperature and there was terminating and each 1 was given with the weight of the free energy corresponding 2 it's it to the Gibbons said Hawking calculation of the free energy of that system so for a general object cover "quotation mark correlation function you would give the probability that you are there multiplied by what happens if you are in that band and what type of behavior you will get and when you do a this analysis you expect as they said the probabilities will be related to the free energies end the noise we estimated pound noise should behave and noise goes like to the my necessity for the time being well looking about Celtic operators so if we look at that this system we see that actually if we take the noise and we multiply it by then probability which is a free energy we get exactly the board's Monday distribution which will tell you that the low energy band will always dictate the level of the noise the average noise so that it it's a simple argument that was missed before and it explains that you will the fact that the thermodynamic dominating Ben did not give you the noise is not accidental it's always their lowest energy which will give you then the maximum weight therefore the noise that statement Number 1 no let's look at the case where the system is causing integrity is assisting system is causing terrible the relevant operators to calculate I'm not these ETH operators but in that case that clearly operators which are if you feel that the stations that means there's some combination of a and a bigger if you have some quasiparticles picture for that so you can calculate what is the noise not often ETH operator but an operator which has this structure where the coefficients here are given by what you would expect the Bulls distribution in a system of which is it's a quasiparticles expectations and the
result which let's forget you the calculations of important for me now the result is the following you would get that the noise is much larger it goes like 1 of us screwed of the entropy so it's not the to the miners SRS over to affect just 1 over period having a much larger and noise so this is the spirit again to realize that when you do any th system and you look at it's like a long-time behavior when that is accomplished the cooperator you get a very small noise but if you're in a system which is nearly Integra below has lots of conservation laws where quasiparticles frequency particle picture makes sense you will get a much larger the amount of fluctuations in much larger knowledge and that goes like when I was growing fast OK 1 can
do the same for the EPR
noise and ii I emphasize again that the dynamic representative dynamic dynamics representative observables and there and has a big change on how the result OK they would come back to it here I can
just say this is the type of future theory that which we discussed in the ADS
see 52 listen to see if decide the several
bands I totally served Robert Johnston dimensional Jesus during this short should black Codes and flat space and these are sure to black coats and ideas and there is this for bands that it are expected from the bulk side of K would be nice to get it of course from the boundary
side and here is what will we discussed that depends ETH black chords and strings if you take the th and for the guys you take operators of the form that you discussed which like was particle so now you can build the calculations in these 4 bands and here I would just give the final result so if you look at the noise at a temperature which is much larger there a booking page transition so the black or dominates the term of economics each of the band's gives a certain answer from here you get from the government on Gaza Get 1 of us grew too fast and the Nunavut you get the various pieces and you find that again which is the leading 1 and you get the same type of answers that we go before it's low energy in which
determines in general you can say that the average noise is determined by the lowest band the fastest variations are also determined by the low energy band on the other hand the height and the long-time variations which I'm sure hear the long-time variations and the total height they're actually determined by the leading to Modinomics configuration said she would be listening to yes you're part of an you listen to ideas you know which other pieces which you can associate with the low-energy part which dictates that average noise and which are associated with the various times timescale in the
problem so the conclusion of that of the party is John Major produces correctly the average property which I would say now we have to average properties given directly by geometry it produced a very small number operative result it does not produce finer details of the non-productive at the behavior of the time-dependent is currently so that's where we stand here slogans I repeat diversity crowns geometry captures what I said I think this is just repeating his itself and thus making the slogans clear OK now out you wanted to inducing a 10 minute break would be good enough Our goal what do you say OK
so from now on you're here the your sponsor permitting them to legal OK so now I want to take a bear different as a diagnostic tool on this for the system before I have that they had discussed longtime correlation functions as a diagnostic tool no I want to discuss complexity no I would define complexity and will discuss classical Quantel manner calculated various properties of complexity of singularity who try from them to learn what do singularities behave like and in particular I'm telling you already know that some singularities will have a property which reminds you of sink which to born like to to work in a way that is of systems where when you approach singularity you'll find it at the coupling of different points in space time and you actually have like the strong coupling lettuce system that is a very low degree of entanglement and you just have to each object to bring its on quantum mechanics and similar things will or ideas a lot like that will come as a result of this analysis but 1st I would I need to define the complexity whatever it expected properties and then what happens in the presence of singularities now by emphasized in the beginning of the 1st on the end of today stalled that there has to be a responsible adult N In the analysis namely that 1 needs to understand how the quantum field theory works and 1 has to have precise quantities and well defined quantities in the quantum field theory for the long-time behavior this was the case here this will not be the case that is what even the quantities defined on the quantum field theory side need have loose ends and there since 1 doesn't really understand in the definition of Of this template and it would become apparent where where the loose ends irony hopefully eventually these would be closed I think it's a very interesting a line of soldiers will develop I don't know go by it that's a dentist by Daniel hollow by lenders asking the FBI's stiff Shankar many people letting the Stanford Group at 1 time or another is spend a lot of trying to push this this thing forward they concentrated on among analysis of records here and concentrating on I wouldn't contemplate and analysis of singularities and we will have various type of singularities to analyze so let's start with the definition Howard went how does 1 classically defined complexity so here is a definition and I think there are no precise when we go to the quantum theory is that no precise a data on how universal the statements are was I'm going to but that statement would depend on various inputs and that it could be that some features of the results depend on the people I would tell you what is not expected to depend but I don't know if this is the last theorem switch validate so 1st you're supposed to start with a simple initially state a simple initial statements not highly intended state so you take some physical state and this is a reference to different for him and once again the question how universal the answers would be and how much they will depend on what you think it is all I think to some extent of then comes the notion of simple operations I will give examples of simple operation what we have we are in reality some quite well doing some classical system in the beginning and then it was the examples would become politicized you'll see How to comfortable so simple operations I would give an example of a simple operation simple operations are supposed to be if you have a state it's given to you then you make operations which of changes easily at 1 point of this which defines the state all right at a small neighborhood of that point and you have a small number of these simple operations could be 1 before but a small number much smaller than the number of degrees of freedom you have in problem and then you define the complexity of the state as the minimum number of operations you need to construct that states From the initial stage so here is a concrete example let's say
that the state is defined bye and it if it has the best points in the state and it's defined by being 0 0 1 and if you want to identify you can have a up overlords the 2 identification or not is up to you a simple operation is is that at 1 moment you take Israel's you want to 1 0 0 1 2 0 and you have a site OK is this would be like the entropy because you have essentially due to the as different states in this Hilbert space so the long before it goes like this so it's essentially as it you who should think of the entropy no the maximal complex classical complexity for this particular problem where this was the initial this where the operations this was a system is proportional to the number of sites is it OK to boys at precise enough of them OK so this is is their classical entropy of the system when
you don't quantum mechanics seems become much more complex because the state now it is a some with complex coefficients both states this is not the most generous state which which you want to calculate the complexity a general state in the Hilbert space and remember we want now to give each state and the Hilbert space the number which would be its complexity it is true To their SO 2 the K whatever number you want lettuce .period is you have here end you have you have to tell all coefficients 5 OK so you can be happy that there is 1 phase and 1 normalization but this is irrelevant relative to the large number of information you have to give about the system in order to know what it is all so the many
forms of pure states requires a the a but in the definition requires an absolute solution because this is a continuous many this is the money for of possible states so there is no no meaning of what often minimal number of states because that's a discrete motion so what you need is 1st 2 already are being issued on the field serious side imposes and then you can ask what is the minimum number of steps and you will define what steps they can be let's say if fittest spin system this could be 4 types of Sigma matrices that you allow signup collector that the 2 sides Sigma miners strike and another 2 sites Sigma Street this to measure you pick a set of operators and you but more on their initial state and you ask what is the minimum number of operations you have to do In order to bring it into a region all there state that you want to determine its complexity and that's the best you can do from the start so that's why I told you that the that the problem of definition starts in the beginning but people do it and that's that the people try to do this scary and it is the answer is and if some intuition to how you get there is that there is abound on the complexity the company this was shown abided by the of and collaborator whose name now escaped me maybe you who was more but at the supports a toss of paper and ended the day the claim is that the complexity goes right into the air times not before 1 Nova absent and this is important that that the solution epsilon fucked arises out so you can identify piece here which is epsilon independent which is easy to the entropy so if you the claim is if the total number of states of the system taste of the order the this is very complete the maximum complexity estate can have and the proceeds synopsis trip amend the answer may look trivial but the consequences of not sold not so trivial and remember it's very important that to remember but this is not let's in we also think of the Hambletonian as some of the transfer matrix as something which takes us takes a stake in times when it's not I state than any moves it in the Hilbert spaces and tries to cover the whole inhibits based in this motion but that it would not necessarily give the minimal number the time it would take for 100 ton enough assistance to get 2 the state we want to measure of the complexity of is not necessarily the should the best shot up and this is 1 of the issues here office systems What is the difference between the minimum number of steps what is this time it will take to the Hambletonian to do it Is there another Tonya where the time to get to reach the state minimally would actually be the time it takes to state From tour to go from initial to finance and thought that this structure of the of of this is not is a study the please can you believe it we sensitive political and what is going on I last for example could be rising model that's making the international tourist cases To the what is this as would be an answer to OK Nova not the Hilbert spaces would be all the dimensions of the number of dead the dimension of the Hilbert space is the dimension of this object here because you have all of them remember Idaho which called it a blotch
spaces Inc I'm sorry we should
remember it has all of the fights and the and embattled dimensions with the dimensions so that so that they mention you should take care
of this it is easy to the times the 2 Bs minus 1 over to weigh up your hands as he sees fit to sum it up in the that's OK but the project gives you only in the case which reduces by 1 that's because world as we know that is a total phase which is not important and then there the normalization which is the other 1 which comes here but basically the large part comes from this morning but this this cannot change of the 2 biggest times into the air Over to this is really the enormous dimensionality of that space OK so if we
have we look at that pace and we put sales this is the number of if you wish that's an estimated to help you what you were asking we have 1 of rap's along the 2 basis sense in that space we have at each step all the order as choices to make so when and steps 1 can reach as of the end I act once twice for this the number of states assuming no repetitions head and it's stuck at a fixed point was something of that nature so the
complexity is bound from this you see that the complexity is bound by it to the US and that is the result is that 1 gets
here OK now let's relates this is that their statements which I told you I'm not sharp enough because we don't know all the dependence on the initial state we don't know how universal it is the hope is that if the state is simple and also the most simple states would give a very similar definition of to complexity of the state it's not clear that exactly which operations I would do and what when and how many operations I'm allowed to do simultaneously how large if I have the states on how many elements should they work simultaneously In in quantum computing these called Gates and the various dates of the system again but the belief is that there is if the state is simple enough that is has a lower origin of their original state is low demand and the operators that you act on local operators and their number is is much smaller than the number of sites of this like this then the results are mourners universal this this is a feeding but as I told you it's not much sharper now what how would complexity go for a black hole and I will discuss and shouldn't show we will discuss that so the idea was like this if you have a black hole its complexity in the beginning grows no other unlike the case we discussed before actually here nobody knows in the bottle why the complexities stopped growing but we know there is a 1 of the people of the part that is found in that I will show you the pictures he said the young so this that let
militarily show yielding Jim geometry here that state
the results of it and then I was shown the geometry which corresponds you calculate various much similar volumes and I would show you what it cost would see exactly what response to so In the black hole case there is a period of increase of complexities ability it reaches was because of the bound which you know from the boundaries should reach abound at that stage it's not clear how actually what objects that exactly you will be because we don't know the mechanism which would give that bomb and then because of the Pong cover recurrences of the system the system will repeat itself it will go down go up again goal down so I didn't show you were tuition for this is that there were showing a moment the timescales here are similar to the ones we discussed last time it reaches the maximum the time of the Order of exponent affairs because the assumption is that monitors the complexity goes linearly was dying so the it takes for this would to reach such a situation is it to be honest and therefore the complexity of this would fit the bill and then it would go down again no this reminds us very much
to this thing we had before but in the end of before what we had is we had something which dropped down here we have something which increases it reached abound here it reaches an upper bound there it reached a mobile and then attributed its here we want us understand how the lower bound what is the source of the lower bound and what is the source of the continuance structure in the case of complexity at least for me up to now I don't
I'm not from I don't know what the mechanisms are supposed to be now from what I understand from continent from nations series it's expected but there are regions where such a system we indeed but as I said that the CGT it should be Konstantin Tszyu should increase premiums times is the temperature at of the time and this is more or less given by dimensional arguments and the claim is and this is now a claim which 1 has to check the claim is that the way to calculate the complexity it is by taking is the maximum volume and I will show you off of their system would show you what is the maximum volume how you define it at a given time so here is
not related to your question so you start with a black hole the In a DSS is all of this thing otherwise there's no connection to the Sea of 50 so it's easier for us to laugh at this stage a large black hole OK this is Loud Records and the temperatures above hoping page it always so it is it's a temple that down all small effects which occur at temperatures which are lauding the curvature of the system you this will it's complicated enough like this so you take it where the system is supposed to be simpler that is look at a temperature above the page then you start to this is the way diagram which is currently hold and you define the complexity in the following way you take time when both boundaries to be the same you could take them different but it doesn't matter you could get 1 1 of them fixed and that the other increase that's up to the wouldn't change the characteristic of the results so given that say the same time here and there even tho the calculations of unlawful also for independent times we would have a TB year-end 38 here so this is anchored now calculate the volume of various sections which are a character understood .period so that many of them shows the maximum 1 that will give you the complexity this is that the claim of off what is the right thing to do and the reason it goes to infinity yes that's a problem so I told you 1 does not understand how unlike the case that we had before where we understood the lower bound and its source In this case at least as far as I'm concerned may be of some of the people figured out it's not clear how a semi classical picture will describe them In such their saturation know why did this come how did this come about what were the motives of so that I in proposing in looking into that and proposing going a proposal the it's not something that the army would be great if 1 could really put prove that and do the exact mapping from the boundary to the bottom of the listed in state the holders of the people Molly always remove cases they infect other Tory asked about ideas it's you always remove that the Infiniti and you're interested in the increase you what you on your measuring the increase so you take the reference volume you throw it away sorry admission of center so you all throw away the reference volume and then there is something which increases and that's what you want to focus on your not interested in the constant you're interested in the 1st derivative you comes from and it comes from being in the body from the fact that as time moves up the extremist surface becomes larger and larger so here it was like that the opposite OK so what was the motivations where did they come from why this geometrical object which you know anyhow at a certain stage would violate abound it can't work all the way but why is it that way do that where there were several arguments 1 argument came from the ideas of using networks to build space-time and when you use which is another subject which actually I'm not denying an expert at all if you say if you don't believe you see that the time progression of building increases the complexity of the system in relation to its volume so that this was 1 of the motivations of white people I wanted to associate this with that with that Was there increase with story was complexity the other reason is this is more or less here when you cut it this is the length of the Einstein wasn't bridge and people wanted to understand how the Einstein Wilson Bridge increase in time it's originally not its volume but people thought about the links no it was clear that the timescales involved here with totally different than the thermodynamic of timescale because as we Salter modernization His of order as In the system or maybe even August definitely not higher than this the page to describe why The Times associated with that Einstein wasn't bridge increasing into the US so that was another motivation to to try and check if the volume would follow and the volume did follow No I don't want to I cannot they argue a
accepted this intuitive claimants see what follows up from it why is this dictum is that is correct especially because the authors several months ago it changed dictum and they said that instead of having here a volume they want to calculate the action on the classical what they call the will of the week wedge of the classic collection which had the same actually for black holds had the same very similar properties as a volume itself so as I told you we have your on territory for which we don't have the established and clean definition on the field serious side because we don't know all details of universality of this definition we don't know where for example the quantity Epsilon or again I don't quite know where the quantity epsilon would appear here exactly In the calculation that fuel was crucial in order to make the cells on the field serious side where this option appears in the body is is not not very clear but to remember that the whole point of view of occupy bond was that the entropy part can be isolated independent of the Epson so it's probably sitting in some coefficient multiplying but it doesn't it's not important from the point of view of the US dependence on the dependence was like to the 2 on the undefined future inside the building blocks of the we are where we are calling about the complexity and we have a temperature and the stench to take to the black hole of time than me obviously you should know it does it does it does but would say this season it's like honey the law is OK OK so it may be that I jumped you 1 job that you know that that made you ask the question I give you the answer so forget another black or Singh of the fields here OK so now the future we want to let the whole Hilbert space there are many ways you can map the whole Olympic space you can go and pick your trailer and going in but 1 way to map the Hilbert space is 100 and develop the system so take a state which is a simple state and that each time the department not if you I don't want my kids think of them as Tonya efforts I consider them Tony and I wanted it's not a good way to be a ways you just fuel but not I can't think of them as get the well-defined state I elected evolving time 1st I afterward do it term leverage of that's not the point .period is I isolated it they're developing time as a developing time I get I reach every time a new state I ask what is the complexity of that state before that so that's that's where you get to this this thing but you have to remember but the complexity of the state could reflect a shorter and a smaller number of operations than the Hambletonian that you there so you let the Hambletonian evolve it so you 1st jump would like to say other complexity just related to the time after a change the unit's appropriately it's the time it took me to reach but it's not because that is not necessarily the minimum number of steps that you have you would have to do to reach the state and the minimum number of steps is not defined by the Hambletonian is defined by some other quantity which is simpler than that so this is how you how how the question of tying dependence of complexity comes in and this is why you know there is a bomb because eventually once you cover the whole Hilbert space it's over you're not going to get that 2 takes whatever you get the highest complexity took to reach a state of the EU covered the whole but space was your time you're not going to get anything more than that but you also have the issue of porn :colon occurrences so that will follow you would go to states which have lower complexity as you as you move along with the existence of the buyout is clear "quotation mark there isn't there is the most complicated state given a certain absence Is that attempt finally I didn't think it was sent you you will find you know you say you want it you would do it with the operates sailing asked how many this is much more of much better from a tomb imitations you would have some matrix and let's unitary matrix and you would ask by sub by small and unitary or admission matrices a how many times you have to multiply to get the matrix which Heisenberg gave me so both pictures should give the means you can ask the question involves pictures it's a question of if you want to act on the state along to act on the OK so as I said in the bulk it's conjecture to be the
maximum in value and it comes actually was 1 more in 1 Moss said the statement and the statement is apparatchik could see here and that's why I
became more and more and more interesting the state was like this that and it was borne by calculations done when you add perturbations To the Justice development described here artificially if you wish but too well defined in the in the presence of the black hole you can also add shock waves and see what happens when you were chopped waves into the system and the claim was like this if you have a system which is sought to be the black hole where complexity increases then there would be no problem at the horizon the Verizon should be no single if on the other hand you have complexity decreasing then the horizon I would be singular and very probably they show that in this case there is no problem because the complexity the time increases and therefore there is no singularity is on the horizon and that they did a up examples by adding a shock wave so they they could see where you could have a singularity but then was of a small German folk you would get rid of the singularity and they because it would shift enough the horizons of the system to develop without any singularity involved so they gave a lot of examples in the context of black or physics that it makes sense to think that indeed when complexity increases there is no singularity and which she was to say there is no firewall because in this situation which has some generic properties you actually have complexity increasing and as I said the things which are unclear what exactly corresponds to the complexity on the from refused to reside in the body what Is this mechanism which stops the complexity increasing because it must reach a maximum so if the Senate classical picture may be the reason but is there a something classic a picture which was stopped the advantages the 2 introduced here in extra time scale an extra object which happens and extra mechanism which happens the increase in time of limestone walls in bridge at length of volume which is not atomization scan all this happens way after optimization has been reached so if you made a perturbation you caramelize all this happens way afterward because you have time scales of the order into the US it was like this this is seaside signatories so of the myself about please feel free to everyone in the 1 what would say here that you reach a singularity so this is I but he'll what they would say by the time reversed is seen here that's for them an example of wine the presence of a singularity Hawaii complexity decreases when singularities yacht that's for that 1 but there is a Whitehall some somewhere here and if I felt like I was supposed to be reassigned but remember we didn't even leaching of the otherwise and forget the singularity slices that the state will always have the same kind of which we look normal did you know what again what you need is you can do and they didn't take a general TB and video and underlies how seems to be as a function of body maybe I was same they said if you use Boston variance so OK but that's a very particular cases but this was not the 1 mn went on the wide and was what were boost gays in the Busquets is the 1 we receive variants you want a lot of seizing the whole point is not to say that so it's right you can range you will get a function of TB minus J and you can endanger usable both can arrange whatever you want with the genetic a stake TB and just let them very each 1 independently you see the complex increase keep 1 fixed to the other fixed some of them the company said again the issue is not finding things which won't do the dishes to find something which we measure and they suggest they want to measure the quantity they want to measure and they suggest that this is a quantity which will measure your suggesting quantities which were not measuring OK there they know that complexity changes in time on the bond assessing this is clear independent of universality issues is to leave the state monitoring it is the the state of licenses and everything said this is something of that coincided with the time of the blast this question here so now I need to start making selections in the middle of this year the creates started this is also states that the universe is something complicated symbolizes that he will be a whole lot of this individual will be something else there will be a little more worried that that's what they said they don't perturbations they can do perturbations robbers that this is the 1st of a long before the fire before most if not all of these things the only thing we would like to discuss afterward so I don't think maybe don't understand exactly what you what to say let's discuss what mobilize I fired Minnesota discussed after which it would be wonderful converge OK so so that the well was 5 minutes it could be also moment you
unilaterally OK so they give the that they give their motivation on why they want this subject and as I told you and how it got changed but I think the motivation is not empty the it's very it's interesting enough and they're interesting enough questions to pursue which are not clear and the problem looks very similar as 1 over the problem this dual problem over there Of the longtime correlation functions to of the lower bound to of an upper bound very similar type and behaviors they calculated according to the definition what happened from black holes and what we calculate is what interests us In 4 this would discuss the next time what happens in the presence of single singularities and will do the same calculations but we would do them for very singular various singular backgrounds including the Cousineau case including other cases and I would describe what types of singularities this 1 is interested in and maybe I will I will so and and of course we can discuss next time before if you want more of the motivation of of the group clearly they sing it's inappropriate sing for complexity what I want to end no is actually something which I would use the blackboard just to amuse ourselves with Glenn going to use this for next time it but there's must do to release but it would be OK so we all along collected in quantum mechanics at how the potential expert here and there is no perturbations that you can do you familiar with the fact that if you have this In system you can always OK that this is a chauffeur ground state before the problem I'm going to mention it could be anything that you can always turn into a problem .period you have the energy scale sitting here multiplying this bike tour operators which comes with an equal weight so that means perturbations hearing means nothing for this problem it you know the harmonic oscillator and here you would get age but only got as a function of the hook coupling and the mass of the particles and this is general for any and all animals so that such problems in quantum mechanics don't have any perturbation you cannot make perturbation all the relative couplings due to you is give you the relevant energy scale like another many arguments you can say where perturbations in the summit bounded needs operator is bounded so if you take em to infinity doesn't mean that expectation values of p won't grow to infinity is on on the it's so maybe it's not just the classical problem and flounder goes to 0 you Donald F. expectation value x to the end when not to contract there this simple and the dependence and maybe but without all this you just do a simple rescaling and you get that this is the case no there 2 cases where this is wrong so the let let's say 1 case where my statement was wrong in 1 way you can do a trick so the case when my statement is wrong I want to I want to discuss that is Everything I told you if you do the exercise of free skating you would find it works and longer is and is not mines if any equals minus-2 what you cannot do the trick namely the coupling which would appear here for an equal minus still is a real coupling in the problem this can also if you want to be an example of an anomalous dimensions in quantum mechanics because things will depend behaviors which you due by skating arguments would someday be depend on the coupling and the air a this particular problem which is of the form be squared over 2 m classes and let me take in general if if G is positive this system is I will describe its properties but it's well defined if G 0 2 free particle if GE negative I don't remember the normalization but gene in absolute value Wenger's negative in absolute value it must be sink it's 1 quarter but maybe it's 1 half of them don't remember each 1 of these numbers 1 good and beyond that this is not itself a joint operator anymore and you really introduced by hand a singularity of the origin but as long as long as this is not happening In none of these cases will have about state and the system would behave in the way that before in the following ways on that stage a positive In the case where Jesus positive you have the potential to actually use it if it's one-dimensional than the space broke into if it's small dimensions then it's just a regular part of of it but let's think one-dimensional so this system you can solve exactly Of course like in a free particle case you won't have anything in there too but you will have for any energy E which is larger than 0 you would have states which are playing wave normalizable but for equal 0 you would not have a plane wave normalizable state the state would be its it's normalization would be worse than playing way and therefore you have a very interesting system it is the specter Miss continues bounded from below but does not have a brownstone if you wish time translation invariant is broken in this quantum-mechanical case because you don't have the most energy state of the system and it's interesting what properties the system has and I'm going to use it in the next and the next lecture so I wanted to introduce you to to it also mention mentioned again there is no something more fashionable this is something which was discussed from my point of view but soldier for being a doubt following fall on many years ago it did the supersymmetric Keyes was for being but the main structure is already there before you do the supersymmetry and in this mode this modern sport coat from former quantum mechanics because it has a cemetery which contains you have the operators H D E and K I will discuss this topic the operator H translations and imparting the emerges the scale and carries a special conformal transformations and this is just XP Plus X and this is X quiet and they close Minnesota ,comma 1 symmetry which is exactly the conformal invariance so it's a very interesting since no recently in recent days not months the people are looking for another type of system which is close to 2 related to the main could dive again from a different point of view and that trying instead of having this model having given shortchanged derivative you you have just textbooks squared do when they're trying to use the short Jim derivative and get another type of conformal model of the site will not discuss that's now part of a Fabius 52 minutes a hot topic of research and I will use this model in order to discuss possible How singularities behaving in streams Suri using idea sifted so pricing by this exhausted Is this lecture and would we
remain I would maybe prepared for 2 bowl for
the next lecture I can leave you in more detail the motivations of Saskin about fuel it said the
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Objekt <Kategorie>
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Mathematik
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Computeranimation
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Körper <Physik>
Addition
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Bilinearform
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Term
Menge
Physikalische Theorie
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Randwert
Energiedichte
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Gruppe <Mathematik>
Superstringtheorie
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Total <Mathematik>
Kategorie <Mathematik>
Gruppe <Mathematik>
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Punkt
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Kategorie <Mathematik>
Validität
Klassische Physik
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Objekt <Kategorie>
Singularität <Mathematik>
Minimalgrad
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Total <Mathematik>
Auflösung <Mathematik>
Extrempunkt
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Mathematik
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Bilinearform
Komplex <Algebra>
Quantenmechanik
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Nichtlinearer Operator
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Grothendieck-Topologie
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Objekt <Kategorie>
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Körper <Physik>
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Mathematik
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Element <Mathematik>
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Ordnung <Mathematik>
Geometrie
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Komplex <Algebra>
Analytische Fortsetzung
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Gerichteter Graph
Computeranimation
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Prozess <Physik>
Punkt
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Gesetz <Physik>
Komplex <Algebra>
Gerichteter Graph
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Einheit <Mathematik>
Existenzsatz
Minimum
Unitäre Gruppe
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Nichtlinearer Operator
Dicke
Verschlingung
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Kategorie <Mathematik>
Gebäude <Mathematik>
Klassische Physik
p-Block
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Frequenz
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Randwert
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Körper <Physik>
Garbentheorie
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Ordnung <Mathematik>
Aggregatzustand
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Zahlenbereich
Derivation <Algebra>
Term
Arithmetische Folge
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Spezifisches Volumen
Grundraum
Einfach zusammenhängender Raum
Matrizenring
Mathematik
Relativitätstheorie
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Unendlichkeit
Objekt <Kategorie>
Diagramm
Mereologie
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Punkt
Momentenproblem
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Wellenlehre
Minimierung
Physikalismus
Besprechung/Interview
Komplex <Algebra>
Trennschärfe <Statistik>
Spezifisches Volumen
Grundraum
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Zentrische Streckung
Lineares Funktional
Dicke
Mathematik
Kategorie <Mathematik>
Klassische Physik
Störungstheorie
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Rechnen
Objekt <Kategorie>
Singularität <Mathematik>
Rechter Winkel
Horizontale
Ordnung <Mathematik>
Mechanismus-Design-Theorie
Aggregatzustand
Subtraktion
Vektorpotenzial
Punkt
Invarianz
Hausdorff-Dimension
Klasse <Mathematik>
Gruppenkeim
Zahlenbereich
Derivation <Algebra>
Bilinearform
Komplex <Algebra>
Quantenmechanik
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Übergang
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Algebraische Struktur
Negative Zahl
Harmonischer Oszillator
Symmetrie
Freie Gruppe
Translation <Mathematik>
Vorlesung/Konferenz
Modelltheorie
Kontraktion <Mathematik>
Korrelationsfunktion
Zentrische Streckung
Parametersystem
Nichtlinearer Operator
Lineares Funktional
Grothendieck-Topologie
Kategorie <Mathematik>
Ruhmasse
Ähnlichkeitsgeometrie
Physikalisches System
Störungstheorie
Rechnen
Unendlichkeit
Konforme Abbildung
Arithmetisches Mittel
Energiedichte
Singularität <Mathematik>
Betrag <Mathematik>
Mereologie
Ordnung <Mathematik>
Normalvektor
Ebene Welle
Riemannsche Fläche
Aggregatzustand
Besprechung/Interview
Hadamard, Jacques

Metadaten

Formale Metadaten

Titel 2/3 Topics in Quantum Field Theory and String Theory
Serientitel Eliezer Rabinovici - Topics in Quantum Field Theory and String Theory
Teil 02
Anzahl der Teile 03
Autor Rabinovici, Eliezer
Lizenz CC-Namensnennung 3.0 Unported:
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DOI 10.5446/20255
Herausgeber Institut des Hautes Études Scientifiques (IHÉS)
Erscheinungsjahr 2016
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Mathematik
Abstract 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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