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3/3 Topics in Quantum Field Theory and String Theory

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been the there's a new you will not find it where it a man the I want you to think of the issues and on the news OK but as real are lucky that the ball is actually it here and as the ball was complaining that there are not enough calculations on the backward blackboard and just ideas and that he wants to see the calculation part so I have that in and in anticipation that he made Be here nevertheless today so I I actually the lecture is going to be in 2 parts the 1st half hour I would do very fast and I will highlight points which I would have done in greater detail To conclude their series of 3 lectures but the other part 1 is going just to be about anomalies I'm going to discuss anomalies how you have global anomalies how you have local anomalies and do what specific calculations which I think are instructive and there has been in Neversink as they told you that open problems also there but but the area is much better studied much better known than he is so what what did we what were the things that we passed through in in the tool formal lectures we discussed that send classical geometry captures 2 0 averages which are very small and of the order of the to the minor says 1 of was when both of them were related actually to noise 1 of them was related just to the behavior of the long-time behavior of a two-point correlation function and we discussed Celtic case into global send me into global cases we saw a large variety of possibilities for the value of the noise as a function of the entropy and we sold that semi classical geometry gives correctly these very small numbers we show however that Sony Classical geometry frames To give more exclusive information in particular how the correlation functions behave at any given time and not just averaging them this appeared once In the about 10 years ago in the context of a black hole information paradox and reappeared in the case of the E are equally PIR paradox where once again it seems geometry may not play a role but actually we have shared with seen here the geometry can capture a very small quantity which in this case was related to the EPI structure various objects in particular an eternal black hole and then deviations from that and I want to go into any more then we said I said that I would like to see this study more about singularities of course are also the black hole and his assistant has a singularity inside it and of course also the firewall is a singularity but I wanted also to discuss other type of singularities now we know that in streams fury singularities which are timeline actually I resolved from time to time there are examples in streams series where such singularities are resolved and anyway and now a model of the camera can do it but you should close Europe your eyes and I will tell you because I don't want you to see is a flips through everything cannot tell you when you can open them when I reach my target so my advice is
close your eyes if you insist on keeping them open
take your problems OK so I
wanted to there to reach here actually that's true but I want wants to reach them and we know that in string series any concept that we know as unambiguous 1 1 probes nature bye .period particles has an example of where it becomes ambiguous or if you wish symmetric that's a nice award when you probe nature probes which are extended the most the earliest and most famous 1 is just simple distance you would think that if we have 1 dimension which is compact defied and reduce our and 1 the Rangers 1 0 where are the 2 2 are totally different if you quantum mechanics of particles on these objects the spectrum is completely different when you do string series that is not the case and as you know that is a tip of the iceberg off a class of symmetry known as the duality said there is a whole groups very long structure and Lockeford penetrated into mathematics under mirror symmetry and there are a lot of ambiguities of that nature an infinite number discrete infinity of of then we have examples of series were the target space has different apology nevertheless Strange fury would give the same answer we have that for small cases and we have it when there when the sinks a smaller I want strolled examples we don't have here at the time but just remember that if a write-down Nawzad that w modeled on 11 1 on a certain group manifold that I can describe it is about the group many firms whose dimension would be the dimension of the group Oregon define it on the contents of algebra which dimension is the rank of the social have here an example of a the topology being different because the topology of you want to they is different than the topology of the different groups and the dimension is different because it's dimension gene 1 case and I are in the other case it's so we have this concept of topology unambiguous the number of dimensions we know is ambiguous largest small in the 1st lecture I spend time about how 4 dimensions with a certain set of degrees of freedom can describe 10 dimensions with another set of degrees of freedom then you can go on and do commutativity we know that we have examples where 1st certain degree set of degrees of freedom we can consider the system on a manifold which is commutative and for another set of degrees of freedom the many does not commuted to a you can look at as a associate dividend the example in the presence of Fluxus where the money for maybe associative or not and you would get the same answers and then you go to singularities so the simplest cases in streams fury when it's small you take a sequel 1 more than you right look at a circle of a certain ranges or you look at on or before Defar different regions Dolby circle is just that Of the hearing don't have to what you can use your imagination just I think you just let the targets place container piece in it which is a circle where you have identified the angles that are in mind so by doing that you have objects which on a circle were open strings if you attach them to these points they become close to and they still produce a twisted sector you can study the whole model but the main thing is that this object has of course to fix .period said 0 ended by so this is actually the lined with these 2 fixed point and this object has a times like singularity in quantum mechanics however you know that when it has a certain range used it isn't exactly equivalent to a certain circle with another Rangers so that is already 1 example than there are other examples of which are more sophisticated which I won't entering where you go into certain points to generate extra massless particles and then you have a different type of topology a different type of singularity structure so actually I don't know of any concept in mathematics which has not become ambiguous or symmetric 1 1 study when there are at least examples of where become ambiguous or symmetric in the presence of extended objects as probes and I think that's very interesting and I think it's a great pity that we don't have a unified picture of that but in each case we solve on its own while I would imagine that eventually mathematicians what do you or we will teach them or they will teach us how to have a unified picture which explains how come all these something you it is ,comma now as part of these ambiguities I would like now to look into cosmology and I would make here of the claim 1 Kendall ,comma cosmology which seems to have no singularities time goes from minus infinity two-plus infinity and you just lying on Europe How do quality hemlock and that's it nothing ever happens to you and on the other hand you have
another door frame in which you have a drastic end of time and in which there is a catastrophe everything explodes so you have 2 different pictures on the same phenomena in the body so this is a very interesting Singh and we we study that in there with holes above born in quite a lot of detail to try and understand how come and it is possible to have both these pictures now we know that in general when we do singularities when when we study singularities in an almost eerie the fact that the appearing almost series is a reflection of our ignorance we don't know that there is quantum mechanics we think about classical physics and we miss the there's to again we miss 60 patients but went with increase knowledge this singularity some of their anymore general relativity is a trickier object because it hasn't horizons and under the spirits of horizons maybe something can we there are those as you know under the general principles which are not again proven from anywhere but you impose very general censorship issues in hopes that such things all such singularities always protected by horizons but does it have to happen does it not happen this is not obvious and there are a set of examples where this actually does not happened namely you can show that using different a DSC of tears at all but there is no need to say
that the singularity be there so there's where distributes a half hours I want some seems to remain I it was you so once again I want to remain with you is this picture of the more achievers so there you have here various islands in this case the islands where we can live and 1 minimum could be better than the other and we try somehow to jump from 1 to the other and in this case I also captured the idea that OK you also want to live on the other minimum no In general when you have a few series which has To
minimum With no gravity then we know but there is a mechanism by which you go From this minimum to this minimum and this was analyzed by many people and by the Russian group by by the USA U.S. groups and the cook their mechanism actually comes a little bit even been tuition from statistical mechanics is OK let's say you are sitting in this year way function of which is concentrated around this state you try and stay there no this is not an ideal state of the humming onion you must if you just right here the way functional was no components here it would not be the way function was denied state of the poignant and then Nigeria's it's suddenly a bubble all the children
vacuum would appear and the question is will this vacuum with the revolution succeed always a revolution face so this is an attempt to make a revolution if it will of the bubble would increase the revolution would be successful if it would shrink it would fail I know how does analysis go we know that if you Bill you create such a bobble Of the right phrase in the wrong phase and I'm not talking no brevity involved then you are in a situation where you gain energy proportional to the volume of the system but materials energy proportional 2 the derivatives that you have to pay for if you go from 1 phase to the other things and here I took what is called a sin 1 approximation of where the where it's very drastic and all the all the derivatives are concentrated here you can also make them more blurred and do analysis on what happens in any case 1 affected goes like the volume and 1 effect goes like this the surface so this means that there would be a revolution which will succeed it could be that this if the bubble which forms is too small and it will contract again but eventually there will be a bubble for which no matter what the parameters of the problem are let's some 5 4 seriously blunder to wind up the 4 no matter what the parameters are this is going to there would be a fluctuation which is large enough and it will take over and everybody would be the whole system would move to the other voucher so this is a situation of tunneling from 1 vacuum and to
the other is what happens in the series and it's not unexpected because both of these actually live in the same Gilbert Spencer was never a doubt that the state here is just some excited states of what exists here now what Pullman and in which she has studied what happens when you add gravity and that there when at the time the cosmological constant was supposed to be 0 so the question was what happened when the vacuum energy was larger than 0 know actually what is interesting for us is what
happens when you tunnel between 2 fuck you are both of which have let's say negative cosmological constant because we are using a Diaz as a tool In that case what you see is not what you get and the main
reason is let's say you sitting here sorry even here and you ask and stable or not state so In ADS there is a major feature as you know well that the surface and their area and the volume stayed in the same manner so once you know what the parameters of the modern are you can't have a victory by having it very large going to a very large scale the scale the issue of if this would work or not work is just determined by the parameters so you may sit in a vacuum here but in that vacuum you would nevertheless not go here because the parameters of such a doesn't allow it there is also what is called the Brighton Friedman bond even if you sit at the point here reducing you have the attack tuna and unstable direction is unstable the Tucker doesn't have a large enough must you would also be stable but what is important for the discussion here what happens is that generically unless you do a very fine tuning and this was shown in the paper of commander-in-chief was calculations that if you go from here to here you crunch it's not that you will now go to a new vacuum no matter how many victims you had that at least you settle in the new know you crunch so crunch means that that if you do the semi classical calculations to the region of validity of your calculations it seems like your forming a curvature singularities and afterward you outside the validity of the calculation and you don't know what happens so ADS is a very good place to try and study what happens when you have singularities inside and what does the horrible rough exterior tell you about it because of photographic This is the words I said because of a gruff exterior contains a total number things here I mention that in flat space and Nagin of bubble will always be to surface in not it depends on the parameters of the problem sometimes it works and sometimes it will and then we went again I advise you to Close your eyes we we made a set there in which 1 can study those in 1 particular case is you can take the space and make a deceptive formulation for and when you do that you have inside here somewhere a crunch and the rents in part I don't of Indian time to go into this diagram in detail it just you start with zeal part then you built tear the heart and walking away function and then you move from here to the rents in part by doing the analytic continuation and again I invite you to look at the papers for all of that the bottom line and in the time I'm dedicated to it is this system which it has ADS and and has genetically a singularity in here it consists of gravity Mehta and Meta has an old D ,comma 1 symmetry in the Lawrence part and all D plus 1 symmetry in the so if you have that symmetry then generically you have a crunch solution in sight History if you demonstrated by looking at this company system of equations then you go to the boundary and you have some type of fury the bond and this series on the boundary to study is easier than to study black holes because you find out that the singularity reaches the boundary or if you wish to use the degrees of freedom of the fields fury in a finite time so there is a representation of the problem on the day the serious side and you try to see what is representation and the lesson are the following you can have a series on the boundaries which has a relevant operator if the relevant operator has a positive sign so let's think of a mosque where let's say that you have underscored 5 square Zycie Yuri now when you go back into the body when have a singularity in it but the singularity would not be a crunch but would be that as you know what is the I armament office Yuri which had the must you go just for the vacuum if you have some interesting topology OK you will get more than just a fraction state but generically you just end up with the vacuum state which means that in the battle you have drained out a piece of the geometry and to create what is called the bubble of finesse so this is a situation when you add the positive relevant operator when you add a negative relevant operators you may succeed it in this new vacuum you still have many degrees of freedom if you have there many degrees of freedom you have a nontrivial geometry also here and it is this geometry which in the balance has a singularity I'm not showing here again the details and getting to the phone just the fact so you have here on the sitter space but no gravity just an expanding universe you find yourself in a situation where you have a well defined ceiling so the boundary Serie tells you there is no problem everything goes well the bank has in it a singularity How do you see In the on the boundary that there is a problem you change coordinates and I think I gave a talk about this for several years ago here and you can expose that singularity but if you use this conformity laundering this is the bread and butter spontaneous and all and let's call it some of the spontaneous global symmetries here and it looks perfectly well on the boundary it looks because singularity in the body so that means that you can try and reconstructed details of the what happens in the bank and we take a quantum-mechanical example in the paper and facing the seem to remember about that you would imagine that the term are 5 square which appears every time you have the conformal field series and then you perturbed but you have the term spent you would imagine that in quantum mechanics and disappears but actually to our surprise wouldn't see the notice before it does not because the curvature goes to 0 at the same rate as a conformal coupling which multiplies it goes to infinity and the product is finite usually in 4 dimensions you used to have a 1 6 multiplying the off-ice score but there is a general dimensions structure of that and Wendy goes to 1 that is quantum mechanics there coupling explodes the curvature goes to 0 and the public to finance so you can do play with this and see How you doing just diagnostics of a searing which on the boundary has no problems while in there which means it doesn't have problems within the Balkan and resist the OK then I would say in 1 word you can also have a Serie where the series has the air operator for example marginal relevant which is unstable if it's unstable also the bulk is not curable so would have a problem on the boundary and you have a problem of the bout there an interesting numbers Europe problem which arises if you allow this sink to oscillate butterfly effect like the Kapitza problem of the inverted Bender Rome and actually that has very interesting structure number Suri I'm not I won't discuss it what is the point which I wanted to discuss for further reference that the point for further reference here I just don't have the fact that for further reference to the point I wanted to discuss it is
this OK I told you about duality on all but about the stories a circle it's are all 1 over No 10 me how did we actually How brains because after all of the universe it could be in 1 direction a circle we have no evidence that it's not that we don't have any evidence that it is so let me announce now that the discovery would just made In some new experiments that you never heard about and they just discovered that there is a circle so how do we decide if it's our 1 of our How do How do we not to say that in my opinion that the reason is the way we have bitten by looking at local observables This is our structure so for us the momentum modes and the winding moans of the natural probe of the system so we which we I would distinction of R & D 1 over our goes by momentum we saved through localized wave packets of local playing or plane waves winding modes are not built into us no I was feeling is that the same thing happens when you have that would do well descriptions Of this singularity and about that is if you have probe which then
you will not know let me call do here again I wouldn't jump on all this I would just use this if you have probes which are local you will and you live in the city you were not known but there is a crunch you can't go on your hammock forever if you build probes which are very large and increase inside together as the Hubble constant as the been constant growth and you word volume evangelical gravitationally coupled but it just grows if you use such large observables we believe that you would have doomed people saying that they would know about the so this would be the knowledge into which an open problem to identifies its observables what other winding like modes which would tell us that indeed a catastrophe is going to happen In the coordinate systems in which a step apostrophe they always traces which would not see it would be object which would shrink much faster than the Hubble constant on the scale so they would never notice because of shrinking they would not notice the imminent catastrophe so To me that's an important question to understand Is it true that indeed the boundaries series says of the singularity can be cured without anything extra happening as it would seem or not and I think this is 1 of the keys to look at OK no In the last
day of campaigning in 5
minutes I would I would at all that part I discovered the Shoff complexity a defined complexity last time and I'm glad the private too give people again tell them what complexity is and it was supposed to be a measure if it or what happens What is the structure of the singularity and there was a conjecture which was borne by looking at black for a long time that if complexity increases then Everything is OK no singularities of formed in the process however if complexity decreases then for a black hole you would expect the horizon to be like a and this is what I told you last time so we ask ourselves what happens for various types of singularities and what we found the
general structure again
please close your eyes if you don't want to get in what we
found that in the case we study of singularities and that's 3 type of singularities that the complexity decreases Starwood's a singularity of course all the calculations can and cannot continue when your very when your string scared because then you'd Europe toward some of them but did you reach skaters which US drinks games and curvature which of the Order of string scales appropriate 1 over then you find that the complexity decreases and in 1 case a
is for example the cars now
in which a universe you in that case that's the most trivial example because you just take the word volume and shrink at at equals 0 so here are when I say decrees increase its stalwart equals 0 so you have to go back from positivity to Teakwood 0 This is the magic of the problems you this is a magical the boundary you can extended the coverage streak Richie flat to the bottom of this is 1 extension which has the same boundary you don't analysis of calculating the complexity I told you last time you used to the idea that you have an extreme volume surface never mind that hello the
final results of of this I think it is that as
you approach the equals 0 this system the complexity decreases and this is the volume of the CIA 50 times the number of degrees of freedom times a cut off you put because you don't want to reach the singularity itself so it you are approaching it but you're doing the calculations before the singularity and this reminders of when trying to think about it reminded us of the structures in the models which to buoy used to play With and those are the structures of which we don't know again this works of the Raczynski sheets and wouldn't forget How much legal reform of income and income article of these the as is their works not we never news of generic or not but they had the property that the system as you went toward the singularity developed much larger time derivatives and spatial derivatives so if you have a lattice picture the system looked like in the strong coupling on the letters each problem lives on its own a quantum mechanics of the letters and this as the state is a very simple states such complexity is indeed very low so so somehow the fact that we got it for cars there may be less surprising but we also got it for the cases I describe it before which were the singularities sitting inside a BS in all these cases we found the singularities to decrees the reason that the singularity decrees if if you wish for the a namely in the computations themselves and I will some
believe that was the last thing
again Clijsters shut your eyes OK this is the end of the show
the results of the
calculation so the results were that if you looked at the extreme low volume you found that the contributions you get From the part which is the use the party Our of what I've written here what you get From the IIR plans sales decreases with time these increases I was dying when you summed them up together the reason that the sum decreases it's because the number of degrees of freedom in the infrared is smaller than the number of degrees of freedom in the UV and the process has in at a constant for all of degrees of freedom from the movie to the idea so this is the final answer that you get 20 decreases so With the was a prescription that the volume is what is supposed to be the complexity we found 4 straight type of will fall off types of singularities they all had 1 common feature the complexity decrees which somehow gives you the picture of a system which is the capital into many different space-time .period so you may be 1 step In into getting the feeling of what singularities look like using the towers which actually have tours on the boundary between here and with this I actually haven't fulfilled my promise and the and led in
private to discuss more but this concludes that this part of of the singularity I would would maybe just say again to me was very surprising that we can live where the series which in the balance has a singularity this is not what I would have expected usually we wait for the nite on the white horse to come and save us quantum mechanics something strange series here it seems that you can live without it however as they told there is the issue of what probes we use and it could well be that was probes which we are not used we have not yet developed in this series which long extended probes 1 would actually see a the singularity even when you are on your have sinking you would never see the singularity OK now I go to the onto discussed a number so when people ask me originally asked me to take essentially made whispers ,comma lectures so this was part of this was part of 1 of my best biscotti lectures and it was more to educate and foreign advanced audience so anomalies King kinase whatever pays plays a role In our physics you physics understanding and In streams fury as you know there is the central charge minus 26 and the central charge minus 15 which come out of an anomaly play a very important role in how we construct streams here and the open question I want to discuss would be related to that Gates series it took many more decades to appreciate the fact that you can't write down any Yates Yuri you want there are some restrictions on what you can do In 1 of the restrictions goes under the name of gage anomalies and I would like him to discuss simple examples all of this and it gives an understanding of what happens because his his hate Florida so if somebody tells you OK you can't quantise a series they say why not I want to quantized so what punishment will you suffer In the situation where will ignore these anomalies now in general how do you get to a nominees so you have a certain on the way we are maybe maybe this will change Over what a true or hear from him so this year sold 1 set of very interesting lectures been should we really pay continue and right the dungeons and actions this has been a very successful In a way of doing physics but as you know we are very ungrateful and everything that succeeds we afterward say are we maybe we should have done maybe we should have done it differently and we ignore it so 1 point of view which is 2 of her probably many times recently is maybe 8 1 doesn't need Lagrangian or not that you don't need them but the circumstances were even if you want them there aren't there you have no idea how to write them down note the usual way we write 1st the classical Lagrangian we identified symmetry OK now we can be pleased by it having symmetry or we may not care if it has symmetry but it seems that it took in order to understand at least the basic interactions which we observe in nature assuming the symmetry which many call now redundant which is great symmetry was a very useful way To do computations and to confront them was experiments so we I have spent a lot of time in writing down the dungeons of systems which have engaged which are gage invariant and they contain spin-one particles and they contained spend half particles and they contain spin 0 particles now the cemetery played a very important role especially when 1 tried to take into account the from quantum-mechanical structure of the series that is during normalization when 1 does renormalization 1 finds out that's that's what is called upon to the dungeon which is a true 1 cannot preserve the symmetries of the classical Lagrangian now sometimes maybe it cannot preserve the toll sometimes you may be in a situation where this Lagrangian has several cemeteries and by preserving 1 you can do it but you cannot preserve others and then are faced with a choice know I don't maybe I don't give a damn about all of them maybe I want to preserve 1 and not the other and what would be the consequences of doing that so I want to exemplify these In simple cases I would start with the quantum-mechanical carries knowing quantum-mechanical case you can say you talked about the renormalization you talked about the how could I mean How would you see any type of phenomenally in quantum mechanics so actually but when 1 discovered anomalies they gage anomalies they came in 2 parts 1 road down some petition function and the question was this could be some regularized partition function the original 1 had the symmetry of 4 symmetry and you ask what happens to that when you apply to to it's a gage transformation so How is z of any of GE related to the original 0 anyway now the gage group as a gage group is of course a local symmetry but it said has a local structure and the global structure and I would I would show here in the example in quantum mechanics will actually discussed their gates the local gage symmetry but its global part global not in the sense of fair that it doesn't change was space-time coordinator but it's part of the global structure of the group because a group of dates from the transformation of some groups and this group has local structure same that entity but it can also have some non-trivial topology and you can ask how does this object behave under large gage transformations and this is an anomaly I wanted to discuss no the system I'm going to take would 1st have a quantized Yager and then the gage will not be the gage fields 1st would be not quantized and then would be quantized so it's a very given the timing and I think a very simple system and make it a billion you can discuss the same from non abelian systems and the reference for this is the work of the I'd done together which would and it's still we talked fishermen and bottom Remele some of them say it's too trivial to be worthwhile even to mention I will I will mention so that dungeon was going to get going to take will have a complex will have a fair would fermion sigh of tea and it would be coupled so there's only time because quantum mechanics was a key component of Phase 2 PCI affecting this would be the Lagrangian because I am in the proper number of dimensions I don't need to worry about gum a 5 governed Gummer 0 and so unjust they're all defined to be wants know this system is invariant under certain cemeteries it has a gage symmetry and it has a global symmetry and the gage symmetry would construct so it has a global part and the local park so what is the gage symmetry so there is a U 1 local gage symmetry and that is if I think sigh of tea and then multiplying as 1 is used the to them under the but would be take sides to 2 and multiplied by that phrase takes sidebar multiplied this phase and if they were team into a key class the city for flounder team this Lagrangian as written is invariant under the symmetry it is also invariant under a global symmetry this is a Sumatran were presented to clash with and that symmetry is taken and interchange signed and sidebar and take a and movement into minus that is what you would expect from charge conjugation select scholars charge conjugation this is a global symmetry I do that symmetry at 1 point I think site to me sidebars to minus now in order to give this interesting structure I know because I want to this the case where there is an enormous so I have to give some structure and the structure is I'm going to put gene on Sunday that's when some region from 0 to capital T and I'm going to impose periodic boundary "quotation mark conditions on the gage here so I have to goes to the the goes from 0 to obtain were from the point of view of the gage field I want to impose their periodic boundary conditions OK so no I won't show you but it's a trivial exercises you can do for yourself just interchange client side bar put a 2 -minus say and then try to get back to your original Burgundian remember the anti commutation relations of the 2 sides and sidebar and remember that signed the imitation relation is 1 you will retrieve back this modern after doing this so therefore you know you have to cemeteries and the classical level OK no just note for a moment that when you look at the system you write down which is also actually modern today you write down all line into gross also thinks that the 6 In that system so for example let us consider the winners so I can write and all of in this language some people sometimes call it the portico law because it goes in the temperament direction let's look at this object from 0 to to any of the the data now let us check its gage invariance properties so if I substitute in here I substitute this I have the 1st it is a term so long as do the substitution so I'm going to get to the original material which is exponent of I have a duty from 0 to 2 years and then going the 2nd to the terms or it will be a product in the 2nd term because I have a total of derivative here is going to be eyes times London 28 -minus longer than 0 now here years you will see that the structure of the gage were coming in Tokyo I could say that I wanted the gage can be such but that's the edges both long-ago fatigue and long-billed 0 should resume namely I'm interested only in local gage transformations which knew that they start and the end at the same point for example the identity I don't allow them To change in that case for those transformations in the class of Orlando's which obey this this would be of this would be the local part of the local gage invariance the 1 salute which is a quantity here is invariant under that another quantity however which would be in another 2 set of gates transformations which would be invariant under then would be the largest gage transformations and they could allow myself there London Of the blindness Colombo 0 not to be the same but being pulled .period this would also keep the series in very no w and this action here is invariant under both so I have the 4 engaged the transformations here OK In however there is only 1 lesson to learn usually and I would do it in the 2nd example when we go to 2 dimensions usually when you quantized fury or a nice way to upon series by going to the gauging there you can have goes small and you have to have a poignant and then you dabble belies you're series by submitting your wife functional to dolls small and by this you reduce the number of degrees of freedom From day 2 D minus 1 was a knot and by applying dialysis lawyer repeated on the state's you get to be minus 2 degrees of freedom OK but in this system is it possible to always go to the gage optic was 0 the answer is no because 1 of the things which defines the system is the value of the Wilson loop you have no what is possible and you can't take any configuration safety and turn it into a kind independent Gates turned into a time independent feed so anything that I did you as a background or afterward integrate over any of the can always be reduced to a constant and the value of the constant this can always be taken by an illegal gage transformation by local 1 would be the 1 of our key if if you wish the average of a of the details from 0 to 2 there's another place was a savage comes in but for different reasons and this this object bar nuclear by definition vanishes because I've taken out all the 2 dependents and now How will a bar change under the residual gage transformations have which obligates transformations which have this property so I can take for example can local gage transformation which could start from 0 let's say and end it would buy it How would a of the change under this digital information so this as you see is part of the local part of the global this is part of the global part of the local gage transformation because at times 0 and at time t because different values cannot be deformed to their identity so under this you will see that they a goes to bar class the plan then so unless a book on you are in a situation where already this object here has been arranged to be approved by you cannot take go back to 0 so you are in a situation where you have a remnant of the gage group what you have to maintain is a cemetery and has to remain a symmetry of your partition function and that is the 1 where a bar get shifted in such a way the whistle and this will have different values depending on what was what background I put I cannot go to a bar equals 0 unless and I'm in a very special configuration OK so this is the start of the constraints that we have now we have to go ahead and calculate the partition function now there are 2 ways to do it and this is part of the question like afterward to ask you if there is 1 part which is very complicated on some scale and that is at least for someone on the phone to mechanics would be complicated and that just do the function doll the integral Over PCI and sidebar and see how this thing depends on it using the Lagrangian I would show how to do that but then the same scene can be done with the Hambletonian and the Hambletonian gives it a dancer and 2 lines where there just as you will see if 1 likes loan calculations it is along calculation soul no let's do the functional integral over PCI and sidebar if is wobbles only degrees of freedom we have a Dow ship the the the dungeon in his gumption In the field side we would have a determinant to the power minus a half safe it would be run really because these are the last 1 variables to get the opposite side so what you would have here when you the functional integral you would get z z with a bar remember I am I if I used gage freedom as much as I could they fix this down To the integer apart the appropriate that a captive story the opposite I kept a fractional part Of the value Of the background fields which I cannot modify any more big gage transformations into gage invariant and I call the partition functions depends on that and this was the result by integrating the the PCI bond with an exponent of life as forward tomorrow and we just I went ahead and did it so this result is the determination of the linear derivative minus the plus but OK and if you want to there some infinity in quantum mechanics got it here because of this object here needs liberalization it's an infinite products is determined so you need to regularize so the question is comes know-how to regularize and now we keep in mind that there were 2 cemeteries would suggest stage we treat the same we don't prefer 1 to the other we would like to maintain bits a buyer engaged under the the global part of the local gage the equal to the original 0 phase that is under the 2 Part II and over tea I want symmetry and they also warned that my regular partition function the of minus the question is can I keep list of things together this was a global symmetry this is part of the global part of local symmetry the global again and the 1st from the point of view the topology not in the sense that it's a global the transformation and can we do it OK that's a question you would see the Hambletonian were calculation it's very easy immediately you see what is possible and what is not possible here you need to work so there is 1 way in which 1 does the calculation in higher dimensions that's called upon the loss what you do with you take the system at 10 you add to the system that hand can muster the master by definition all bases no matter global-local part of the of the symmetry of the gage group it's always there end if you look at this object you have changed the series but then what you do is you take the mask a to infinity and when you take the master infinity you can't do that which means physically of the couple that particle now we were seeing them what we do in the short part of the 2nd part of the 2nd part is that you can't always be couple massive particles when there is an anomaly present it's at their revenge will come at you but here there is no anomaly because they took care of that all the symmetries a Gates images be preserved so I can just take to infinity pool and I can see what happens so now in order to be able to do that 1 EDS will actually need to add a few Polly the lot I would just be right to your houses goes on and see if there is some combination which will give you a finite resource whatever that finite regularize result is not sure it's going to be invariant under on all types of gage transformations and you are it's it's mercy if it's also invariant under a ghost to mind-set this in this way you put you do it you don't prove that there is no other regularization which would preserve bowls but on the other hand you respect the old tradition of using probably the last which is an indicator that that's the case as I told you when we go to the Hambletonian formulation you would see immediately what is possible and what is not that bond you won't have to go through the phase space of all possible realizations OK so what do you have in the case here so the partition function as the both of which is regularised using Polyovy the Lars you actually have here 2 you need can order to get a finite resigned so you said you take both to infinity and what you have here is determined amendments to the list of borrower multiplied .period as a determinant of willingness to move on class and 1 which would be taken to infinity to the power of and if you this tells you if you wish out for 1 here tells you how many species of Polly of Polly the last Spartacus you want to take but this is a mathematical regularization so he doesn't have to be an integer but in principle it would tell you that the number of specious because the determinant would go up by by the number of species many would have hit determinant of minus the side the bomb plus tools clauses to the power of 2 now according to what I it said this inside the a If I had knowing I I needed hear the truth is I don't remember now but there is this as far as I remember this is a correct singular would have to justify what the have to do the right to do that this turns out to be finite when 1 person for warned last month of 2 vanishes In went out for a more morning classic out 4 2 and 2 0 2 this is it if I have here and I but in the conditions it would fall no 1 can show it in detail and what you get here it is the kisses for you know just 1 feels so there is a limit before 1 goes to infinity limit of 2 goes to infinity and then you have a product for and going from minors and 1 who class then tool where and 1 go to infinity and to Infinity I mean nobody tried to play here double skating I don't there are several ways you would do it but at this stage it looks like it doesn't matter how they play maybe there would be a little more structure 1 would relate the various ways of going to infinity in any case you get this product of so you have a free field so you know you have to to apply these affair on so when you do functional intervals not at all we have something going from 0 to take the gage field emphasize its periodic but if you want to preserve the country's for the firm have to do on the grill so that's why you have you then and plus a half plus a and this comes from the 1st thing and then you have this multiplied by the time over to him closer class a buyer and here is what you ask so maybe I just omitted the eye there and here it the England here Of course I remember at IDT which is so be against this we have to go through it all from 1 so you take this products In this now becomes the function so if you define no he also OK so now this becomes it is the policy the denials of their becomes to be the limit all of them my goes to infinity limits all of them goes to infinity where am I was defined to be MRI see over to apply and a which will come in a moment the priority over to try and you get this limit and you have here the gamma function of and told closely plus a divided by the number from version of minus 1 plus a half and then you have your ratios and to class suharto because I am 1 Clinton have something similar to the power of government to can you tend to mostly plus they can of tool this is for the Government will continue to have the kind of minus and were task ahead of us today concern to the firm locations they said nothing is the event known hereabouts that skating Limited is needed or not in any case this now goes down and becomes the limit if them goes to infinity "quotation mark scientists your cost of plus-size tool for the the post and the tools and from here you get the conditions which over written there in the final 1 should take them into account what do you get when you get 1 plus exponent of I gave 2 what a long way to get such a simple results that's the way in history was 1 more obsessed with the use of the words of 1 of the longest shot to use this hit him in the area to all along got 1 of the European it correct police here the 1 so this is the answer to that 1 gets this this has 2 properties it is invariant under the transformations that that today remained from a bar so remind you of today bottle and had to do there we were checking is is invariant under a viable story about listed by any the it is invariant under that however it is not invariant in general for aid goes to minus so you can be sure that that Polly the Laos liberalization gives you the regularization which is gage invariant but doesn't work for a equally table if you do not invariably under a ghost minus about OK away from the usual 1 of time available to you both ways this is the signatories outside of the volatile we will not you would say viewers see whether all the you will see where the ordering and tricks of ordering coming but not not as elegant Jr the the financial liability in the interior of the rule I think this is a stumble Bjorklund dungeons of these it's are displeasure it's the best thing a history armed OK we can discuss playing that game but I'm with be well that this morning the I chose that once it has the right to the has the cemeteries I wanted so I can do that then we can see what happens if you want to change so 1 thing is you can ask How do I make this invariant under a ghost Wednesday OK that's a legitimate question is he is a very it's no actually to do that I need to multiply by term which is not to gage invariant I need to add to it term which is if I would multiply by an exponent of minus on the Obama this year it will also receive our capital city overtones this would do the job but this terror it not a gage invariant in general under the transformations we have discussed before it will depend the 5th and is even so far so good I can't I can't do this OK I can try and this is actually looks like a chore assignments during the few I beds because in this model the church sermons Siemens like and I add this term I can correct and now heads EAT cuisine minus said but I have lost for a general value friends I have lost the symmetry so in that case I have OK that again it doesn't say that this is the only way to do it maybe there are other ways but this particular way left us was 1 symmetry and not with the other you can play around the leasing this is the easiest way to see it that in order to be able to make it invariant turn it into a "quotation mark you had to multiply by a function which itself is not in general invariant under the global part of the local gage transformations not the most general once they are something for which it is it's a half for which it is and how for which it's not so this is the result it that 1 of obtained if 1 starts again given the time I want know show you that but you can't shut you can start in a way which I've been issue is natural you are guaranteed at each step that you are going to have charge conjugation you don't care about gage invariance but you do a charge conjugation regularization it's a little different and then you indeed get gets a result Judea by multiplying these so you would get that you are as you are guaranteed at every step you were also in the limit will charge in conjugation variant but you're not invariant under the general they were was not under this general transformation because of this factor to which enters OK so this was an example of how to weighs 1 which guarantees gage invariant which is probably the last which I know the defined comes back to missing the eye is what is special about all the last selection you put it that not natural fermions because you put the eye that was what policy that was a trick of quality and the I'm sorry the 5 dropped after 1 and a on the other hand if you do it didn't show you the natural regularization which obeys charge conjugation you're going to find out that you will day charge conjugation guaranteed majority miss OK so the 1st question in and it's always terrible Solomon's choice was to choose to sacrifice and maybe sacrifice Balsam who cares about the mixture you would choose 1 of the the other this decision cannot be made immediately it has to be made Wednesday's quantized if it is not quantized it's left to the arbitrarily when a few you can keep 1 you can give the other the other ways to regularize the and then you will have now let me show you before we make that Solomon's decision which in this case is again you choose 1 and not the other it is How would just look in the Hambletonian formulation no again I should emphasize that this type of phenomenally for decades was missed and actually was discovered by Witten and in the mid-80's maybe or something because for decades nobody realized that this extreme position is and it actually gives you all a limit on the number of on the old necessary evenness of neutrinos for example which are allowed to have in your in your system so it has consequences it's here just the quantum-mechanical example but it has more serious consequences soccer here T-ball I will do what you want I will take ambiguity in ordering into account so we know what the Classico from Estonian is it's a system which is 1st time derivative so you know that when you do that is under comes from you would actually not to succumb to the momentum because it's the 1st time derivative so he drops and all you are going to seize back to the term you had itself so there Hambletonian of the system is going to be on a bar OK but what how in what order so now that I'm going to quantum mechanics I know that I can have their say I can write it like this which is a little bit what you wanted to do and write it like virtue :colon will look at well it depends as you know on how we decided to look at how we do that so let's not quarrel about that no it I agree that a lot of phone numbers 2 ties are less important eyes are crucial to the sun would agree so under the condition that from minus better it 1 I don't have any type of combination they're they all have the same classical so the problem here which was a problem off organization I had a determinant it's an infinite products have to define transmute into a much simpler problem I just have the issue of ordering there's nothing else there's nothing infinite here but I don't have that ambiguity in how I want to auditors and the results will depend on I ordered because it multiplies this will multiply a bar and we saw that the gage dependents will depend on that Seoul 2 to looked at such a hundred Tony a in the general this case I'm hanging in there OK it even in the regularization you can already see that In order to have once charge conjugation symmetry for example that would require that they clearly better peak was minus a house In order to have the symmetry that you change size and size sigh DeGuerin PCI and a it was minus now in general you can now go ahead and calculate the petition functions and the partition function this operators side data PCI the number operators to eigenvalues 0 1 so when writing down the partition function is of the firmly on states is that you have a firm and this you just some overseas 2 options and when you do that that's the way very simple this think you are going to get the exponent of if I removed undefined remain was better than going to get here exponent of minus a bar tee times better times we want which I get when the state is empty and then I have won the state is full of wooden exponent of minus so this is my partition function so here we only ambiguity a had was the ordering and I have to say is there a value of better for which most cemeteries can be preserved we so I showed you were told you that when you do this transformation should get better equal minus half which serves you already that it's not going to be the case but you can't cables so here unlike the Lagrangian where I really had to prove to you that there is no organization which exists which can preserve both here you see it depends what better you're going to take you can arrange to get both a sorry both violated or you can have 1 of them for example if you don't better equals 0 you would have won if you'd do better equal and you could have the gage transformations if you if you take that they "quotation mark minus hot you book charge conjugation but you did not get the other symmetry OK so this shows you an example where the Hambletonian is so much simpler then they on gender and I wonder I don't remember seeing Lendl suggested to worry the Lagrangian was it was almost a deserved was having 1 of 100 1 of them was suggested to be buried with honors and deserves a single-cylinder engine but could be the Hambletonian in any case you see that in this particular example it's really much easier to look at 112 formulation I'm wondering if there are other systems where you have very complicated calculations doing a Lagrangian if things could simplify when you look at the subsector just landed on him OK now let's make so here we sold it for gage invariance there should be and we had a case where an equal 0 the other integer ends would work as well and the minor staff doesn't work now what about the quantum and mechanics so in the quantum-mechanical case led the way this manifests itself is what is goes now goes slow all the the state's is obtained by the derivatives With respect to the variable which is not strictly All-Star which is cyclic which is a knocked or a body so goes slow would tell you that when acting on the stage at the college the operator G which is decided euro sign plus better so when GE effects on the states side a on that minus I should get 0 Mexican that's claim let's go so slow no In general we started his example of better equal 0 and that was it OK but if I played better equal minus a half there is no way I can add ferrying arms and the basis condition if I take better equal and it means my system wouldn't have won firmly under but I would have ended and it could work but for better equal miners have there's nothing they can do which means my Hilbert space is empty and this is a manifestation in the Hambletonian formulation this is true in higher dimensions the way that she goes to slow comes in the way you see that in the quantum series you cannot give up the symmetry of global the transformations is that you will end with an empty Gilbert In the dungeon formulation you would integrate what I've written which was was but they cuisine was this was a partition function if you integrated over DA which bar which you should if it's a functional integral it would vanish 2 0 so that 2 ways to see it in the dungeon formulation where you have a global anomalies you integrate obligation when its quantum you get 0 the series you could build a partition function you can define innocent it's cleaner to mean the Hambletonian record but space is empty so I put too much the system is over constraint to bed so I would have to sacrifice was great regret charge conjugation so the system even tho classically it looked like it had charge conjugation the consistent quantum series cannot have charge conjugation this is an example for the global environment now can I give an example of the local anomaly in this I will do win the 5 minutes which remains I would because I can tell you the consequences so this I cannot delinquent we don't know how to do in quantum mechanics but in their work which began with Adam should mail Ian Holliday and Mike China awaits we tried to take a model which can be solved exactly even tho it has let and that is issuing modern if To the dimensions just go ahead and sold it and see what to do now the problem doping problem I have in mind which I think has not yet been solved and I wish I would've sold but if not someone else it is due no political strings fury but don't do the trick of making it critical by adding the appropriate coupling leave it long critical as it is and quantized it's insolvent and see what series visitors because all the way up to now where people did what they call critical strained series but actually completing counseling nominated by adding the piece to the system and then solving which is interesting here very nice results but it doesn't face the problem of what happens if you disregard the rules the authority integers it and bad things should happen so let's see how hollow however general 1 would look and then look at the two-dimensional case so what is the general structure which would have just like you can do is aware that the white equation you can do Paul was done much earlier accounting degrees of freedom in the cage of data so you know you have engaged object they knew which has been degrees of freedom you can go and forget about the global structural issues which had discussed during a during Optiqua 0 so the problem became the minus 1 you right down 100 Tonya but you find that the generator of gage transformations which keep across 0 which means applying independent they formations that generate just small and let's say the Hambletonian is the squared Lusby G for the free cases would just be difficulty in general would also have minus some charge density he the and B don't commute because he and a R conjugate variables and that these Current Affair but nevertheless when you do the calculation because this is the generator of the residual gage transformations of the system this will venture into the calculation and you find that H in Geneva which means that you have the Super selection rule freedom today underlies the tools simultaneously and the way you get the correct number of degrees of freedom you say OK a diagonal lines a generic but all I can function of page however I pick on results by Indian values sorry only those Eigen functions which are annihilated would according to Dellslow is better diagnosis this would be this sector where for example ransom variants will be resuscitated because here made an explicit breaking of Florence variance but because the series is gage invariant then if I work at stops set of space of states which are annihilated by Jane I look at the spectrum it is a dispersion of relativistic dispersion relation I have resuscitated and the number of degrees of freedom went to what it should be a D minus 2 and you can do the same with the will of the wit equation and goal from and the gravity From the number of degrees of freedom embodied in gym you know which is deep times deeper swung over to reduce it by the same type of operations go to synchronous gave imposed the deep type of dialysis laws and you will get this encounter so this works very nice Is there any loophole in argument again yes there is 1 group .period I have assumed but I can simultaneously they underlies all the genes which means I have assumed the G of facts and your wine vanished for every X and Y if they don't then I can't believe I stop him because I cannot bear the license the same time and the point of anomalies which appear not to the topological once they've discussed but the simpler ones which were known from many indicates the ones which are related to Gates transformations which are near the identity is that this doesn't happen when you have serious where this doesn't vanish you have an anomaly where does it come from it doesn't come from the part of ability it comes from the pot broke know it's true that in non abelian gage series like in the winner the await equation you don't get the constraints commute but you get on this side another constraints Saunders sub space where the constraints vanish in dust from you and therefore it works well so if you have a pure gage that's just without work there is no problem there are indeed the reside in Brazil J and they will be here G K and would be some FIG whatever appropriate structure constants about on the sub space where the gene vanishes they commute so there's no problem however the world part this is where the matter comes in and doesn't have that property there you have derivatives depending on what they you are but you have dealt the functions of derivatives of the functions depending again how many derivatives appear depend on the year dimensionality and in particular in 2 dimensions we can solve the problem when you add Cairo matter Carol meta means that the hour is different than the last you're going to get here some that said something which is he r-squared minus he has squared multiplied by a derivative of a delta functions so that would be a situation where you have an anomaly so in general the lesson to take away independent of the details of the modern the way that there are normally manifests in 120 and formulation if it is the local part of the local groups the age group it's affected Gallas's laws don't believe there is as the number Of course look at what commute there is a scene number on the right-hand side on the other hand when you have a global pact of the local gage group as a it was an anomaly there Hilbert spaces empty you cannot you've overcome strangers system you would find no state which ablaze but also you have to Have a Suri where this anomalies council and counseling In string series when we have the conformal anomaly we speak rough very roughly we speak about 26 dimensions with about 10 dimensions which is applicable to other ways of doing that that's the way we counsel them anomaly in string series the way you Yukon that normally in its U 5 is by having a 5 hour and 10 that the way you consulate in it was a gravitational anomalies is having relations between electrons and quarks that many in crucial ways in modern building where the anomalies they come in so to conclude that puts in this would have been lectured on the 5 between what happened so where do it if I would solve this model exactly what can be done in 2 dimensions which in the gauging articles so the 1st thing is should expect a Lorentz invariant dispersion relation I see no reason is a Serie has an anomaly if left is different than are there is no reason why the DJ not equal 0 would be Lorentz invariant I picked a direction and the answer is not Lorentz invariant you get an expressive dispersion relation you have in it he related to take to the force and took a square so it's not the relativistic dispersion relation but you can solve the Serie exactly then just like here you had both of you suggested multiplied by this there is a way here to where there was the main to counsel than nominee but adding this was zooming up there is always equivalent To having issued another set of fermions which canceled the nominees and integrating them out and integrating gives the was a miniature there there is an ambiguity and you're talking that it to the Super experts there there is ambiguity of their own gage invariant terms the whistling term canceling the nominee unique but in the interval they can also be gage invariant terms what do they represent there are infinite number of ways to cancel an anomaly if right is different than the left I can have an infinite number of distribution of charges which I ate away series each of which would end up exactly counseling the nominee case for example let's say that the hour -minus is left was 1 OK so I need to add an left which is 1 but I can add the left which is sorry then I can add any right to which to work in a truly rights which are 1 of the an infinite number of ways to cancel the anomaly by adding extra staff which is not a you just pick up the single part which is and that's what the gage invariant terms reflecting it's interesting seemed to know and then the final surprise I have on the subject is they are in conformal field series what unknown cost this concept mode concept models our mothers which are very interesting because In in string theory they can replace geometry biological and you can do exact calculations using the algebra instead of living was a semi classical picture which is geometry so they have had and I think will have and are having a lot of of users know if you take the model which which has there was model G it has a subgroup of and you right Sogo our construction of the difference of the genes in the ages this model itself is not a WCW model but still conformity and its conformity charges the difference of these 2 and we have shown in the past was culpable but that this model can be written as a gage as a strong coupling limit off engaged with Whitten mode and you can see exactly how this model constructs now the funny part is that if we have a group you over age I 4 profits for that's a strong coupling limit that's a deep infrared of the problem we found that actually and then you can gage I sought for years they cannot engage in normal symmetry In this case as you can get journalist symmetry and that you find a relativistic dispersion operation which was to be a surprise so again in these models you take by writing down some 50 gauging that CFT but not ending in terms of squared the term F-Squared into dimensions has conformal weight so it's conformal invariance is not spontaneously broken it will not be created and indeed it's not creative but you could think also of the model was their square and then just take accounting which is dimension for like the last 2 infinity it's the same reason consequence and in that case as I told you somehow this on numerous Schwager model remains with a relativistic spectrum which to me is a little bit of a puzzle how the how that could have happened so I seen could buy giving deposits which are concrete and in the beginning telling you about an area which I find really very interesting and I believe will develop where it would be interesting results but in which things are not as well defined as we would like and 1 needs much better definitions which is a quantum information series related to gage series and gravity would begin and give you something of that and there was this finished my duties here on that and the next day new low-emission professor would have to do the same 4 who kicked the 1st by the
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Pseudo-Riemannscher Raum
Physikalisches System
Rechnen
Frequenz
Quader
Randwert
Singularität <Mathematik>
Physikalische Theorie
Mereologie
Distributionstheorie
Einfügungsdämpfung
Relativistische Quantenfeldtheorie
Momentenproblem
Gleichungssystem
Gesetz <Physik>
Raum-Zeit
Richtung
Phasenraum
Vorzeichen <Mathematik>
Trennschärfe <Statistik>
Zustandssumme
Vorlesung/Konferenz
Substitution
Phasenumwandlung
Gerade
Auswahlaxiom
Superstringtheorie
Abelsche Gruppe
Grothendieck-Topologie
Kategorie <Mathematik>
Tabelle
Gebäude <Mathematik>
Auswahlregel
Biprodukt
Rechnen
Ereignishorizont
Entscheidungstheorie
Randwert
Forcing
Menge
Rechter Winkel
Konditionszahl
Prozessfähigkeit <Qualitätsmanagement>
Ordnung <Mathematik>
Riemannsche Fläche
Eigenwertproblem
Subtraktion
Renormierung
Allgemeine Relativitätstheorie
Klasse <Mathematik>
Freiheitsgrad
Loop
Variable
Algebraische Struktur
Spieltheorie
Affiner Raum
Indexberechnung
Konfigurationsraum
Varianz
Thermodynamisches System
Determinante
Stochastische Abhängigkeit
Kombinator
Schlussregel
Unendlichkeit
Summengleichung
Symplektische Mannigfaltigkeit
Eigentliche Abbildung
Resultante
Impuls
Punkt
Prozess <Physik>
Natürliche Zahl
Gruppenkeim
Kardinalzahl
Technische Optik
Komplex <Algebra>
Konforme Feldtheorie
Übergang
Eins
Arithmetischer Ausdruck
Regulärer Graph
Nichtunterscheidbarkeit
Minimum
Stützpunkt <Mathematik>
Zentrische Streckung
Nichtlinearer Operator
Lineares Funktional
Parametersystem
Deltafunktion
Exponent
Reihe
Stellenring
Klassische Physik
Frequenz
Kommutator <Quantentheorie>
Teilbarkeit
Konforme Abbildung
Zusammengesetzte Verteilung
Geschlecht <Mathematik>
Ganze Zahl
Nuklearer Raum
Körper <Physik>
Gammafunktion
Geometrie
Koordinaten
Mechanismus-Design-Theorie
Aggregatzustand
Nebenbedingung
Ortsoperator
Invarianz
Hausdorff-Dimension
Stab
Physikalismus
Gruppenoperation
Zahlenbereich
Derivation <Algebra>
Transformation <Mathematik>
Quantenmechanik
Punktspektrum
Term
Topologie
Multiplikation
Gravitationstheorie
Symmetrie
Mittelwert
Inverser Limes
Zusammenhängender Graph
Widerspruchsfreiheit
Minkowski-Metrik
Leistung <Physik>
Kanonische Vertauschungsrelation
Mathematik
Finitismus
Dispersionsrelation
Physikalisches System
Objekt <Kategorie>
Singularität <Mathematik>
Modallogik
Quadratzahl
Flächeninhalt
Residuum
Mereologie
Basisvektor
Normalvektor
Innerer Automorphismus
Innerer Punkt
Numerisches Modell

Metadaten

Formale Metadaten

Titel 3/3 Topics in Quantum Field Theory and String Theory
Serientitel Eliezer Rabinovici - Topics in Quantum Field Theory and String Theory
Teil 03
Anzahl der Teile 03
Autor Rabinovici, Eliezer
Lizenz CC-Namensnennung 3.0 Unported:
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DOI 10.5446/20256
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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