String Theory and M-Theory | Lecture 9


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String Theory and M-Theory | Lecture 9
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Susskind, Leonard
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(November 23, 2010) Leonard Susskind gives a lecture on the constraints of string theory and gives a few examples that show how these work. String theory (with its close relative, M-theory) is the basis for the most ambitious theories of the physical world. It has profoundly influenced our understanding of gravity, cosmology, and particle physics. In this course we will develop the basic theoretical and mathematical ideas, including the string-theoretic origin of gravity, the theory of extra dimensions of space, the connection between strings and black holes, the "landscape" of string theory, and the holographic principle.
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start let's try to get to heart summary aspects of string theory which I are really very central we haven't talked about them but we set up many of the questions we set up during ideas arms however show you 1st something which has the duel with white string theory has gravity and that it has to do with why there was such unusual constraints on the allowable kinds of string theories that you can have a I'm not going to actually get into 26 10 mentions firmly give you an example which shows just how tightly constrained the idea of string theory is compared to the idea of point particles on point particles can move incurred faces a very very simple example would just be this is something you can't if you can look up by suspected elementary mechanics textbooks ordinary particles moving on a cursor surface serve as Kabila's as some sort of curves deficit of positive Rickard negatively curved if in fact it doesn't have to be two-dimensional but I'm drawing it is two-dimensional that's a two-dimensional surface and particle on up particles is described by a point on that basis a point velocity if you want to know where the year future history of it and was on a trajectory at was on a trajectory and satisfies an equation motion the equation of motion as some form of Newton's equations are projected onto the surface warehouse point particle moved thinking now about ordinary nonrelativistic particles moving on a curved surface without any forces as it did the forces of course are better the hold it In the surface but other than the forces needed the holder the surface nor the forces has a point particle will wilderness was another way of saying Moses trade possible that doesn't deviate from left doesn't deviate right and moves along with you guys that's right if their forces on the foreign kinder another hour then it may move more complicated trajectory yet but your could imagine that this would be consistent you could imagine that might not be consist house failed to be consistent love write down sister differential equation just type differential equation here is a way that it could be that which is of course is not consistent perfectly consistent eyes always a surface more voters a surface and smooth differential below all good things and my surfaces but what he really mean by a motion of a particle satisfying a differential equation as always you mean break up the trajectory Inter pieces replaced derivatives by distances and think of a discrete version of it at 1st that derivatives are really differences all are and then take the limit take the limit In the appropriate way of more and more little time intervals question how I know that the limit exists and I know that the limit exist could be better as put more and more points in that the limit of what good is a calculated the trajectory itself has no limit exist well . 4 Reza just described by ordinary differential equations people have been studying differential equations the mathematician differential was a young man and we know that the solutions of these things are smooth regular and have good limits of value an Eskimo complicated question you could say when I'm really interested in quantum mechanics and quantum mechanics we have a more complicated kind of question instead of just looking for a trajectory we ask questions for example what amplitude of particle starts here at a later time we will discovered he had we find again there is an action principle but action principle is not defined the trajectory of least action which gives a differential equation but to use a path integral patentable some all trajectories including wildly varying once again a trajectory being defined in terms of breaking it up into pieces and then taking a limit now how roaring the problem exists this result yet alone said Mr. Young a meeting takes place inside could point gap we're talking about physics even it is surfaced on a surface when of court those features which are completely independent of how the surface that depend on the geometry of the surface but not in anything outside now but had only know that is very very complicated had integral structure exists in the sense that it has eliminated a limit as you take the grain structure finer and finer they are the answer is much much more complicated but it's equivalent to solving it would nearly Schroedinger equation on arms and again that's a more complicated can require equation of partial differential equation partial differential equations are more difficulty but partial wasn't very smart dined and figured out how partial differential equations work and we know from 19th century physics and so forth but the limits do exists to make strange theory is more supple infinitely more supple in string theory we're not talking about a point particle we're talking about an extended object now that's the string itself imagine a string which is constrained the move In a surface and this service does not have to be two-dimensional but as I said I'm going to think about the dimensions is a strained
that's a quantum mechanical strain which means it subject to a lot of the vibrations in fact just to just take an example let's talk about string theory where the space that the string moves on to real space the space of string moves on his here happens to be a spirit With radius all are it's a field radius are all still point particle 1st backwards to a classical porn particle 1st point particle was atmosphere what is its Connecticut and uracil grungy and as always is just 1 half and squared Mass times velocity squared which is also the momentum squared divided by twice man of peace squared provided by twice the mess but now as a constraint and the constraint is a particle has to stay on his feet nevertheless this velocity that what that means is velocity vector is In the surface of the sphere the momentum back is also on the surface of the sphere but other than that it's just the standard classical mechanics now it would delete quantum mechanics then on a sphere spheres the motion of a particle atmosphere was a circle of course a great circle critical of on the momentum is quantized had ITC that the momentum is quantized woes various ways to see it but is a simple way if the momentum is P E and the radius of the sphere is are the angular momentum he'll they are angular momentum l physical the peak kind in magnitude anyway and angular momentum we know is quantized that's quantized in units Of the on a bar so quantized angular momentum here and above red angular momentum is is fundamental because quantized so it pays to rewrite the energy in terms of the angular momentum but tried P as l are sticking here then it becomes L squared provided by twice and work twice in math and now are squared right squared you know what they my squares called moment of inertia the moment of inertia and this case of a point particle that year and the symbol for it Marty elsewhere and all twice the moment of inertia so what characterizes the motion of this particle His 1st of all its angular momentum but the moment of inertia moment of inertia plays a central role in these EU energy levels in all of the properties the trajectories but particularly the energy levels a rumor doing quantum mechanics and that's important for good solar classical mechanics of point particles as described this way and the quantum mechanics appoint particles is also described sway LB angular momentum we know how to do with angular momentum quantized OK so a straightforward now will talk about a string of strained movies on this Speir periods Eds constrained be in this field is fear the spherical surfaces of O'Leary there is no outside the medical services inside only this article surface itself however resisted the spherical surface could be higher but for the moment as two-dimensional then it's going to move around How will I know that in the limit that I think of the strain as a collection of little points How although I know that the properties of the little string have good limits the answer is visitors we had we not that we don't know the answer is that the limits don't exist show you why and what happened the 1st thing is I understand a little more about the signs In shape of strings on little more about them so let's calculate how big statistically the string 0 . oscillations oscillators the Stringer constantly vibrating even in the ground state because they're vibrating they give the string a little society is this trio size and I'm interested in trying to figure out how big that string is when the 0 . oscillations are included so how I do that I go back sure formula that we have on the blackboard may be the 1st or 2nd lecture but still open strings just for simplicity remember this formula position the point on the string was Sigma Sigma is the parameter along string is equal to the sum of all over oscillators but each oscillator has a frequency associated with that sworn in and we have scored 8 plus and plus 18 in cosine of stigma and is 1 of the things the square root of In downstairs is everybody remember that probably not perjure go back in you know maybe factors as to where I don't remember details are where factor to lose From of amount but that's not important visitor structure of it now let alone a calculated want calculated in the ground state of string Brown state of the string is a state with no oscillations I want to calculate the mean size of the average size of cell the average size of rivets calculate the average square size we will square X square
X X squared at any point guard Richard point or give also deserves some are and some are in much ever have a plus in the forest a minus a tire 8 plus any plus a miners and or square and square root and cosine in Cigna cosigned and stigma I've just written down this expression twice once as a somewhat in wants as a summary and but nothing special now calculate is its average value in the ground state average value in the ground state in quantum mechanics we represent by always stands for the grand stage they were calculated severed value and that means we are calculated average of this in a simultaneous groom state of all of the oscillation they might think it's 0 right well that's not quite true every oscillator ground state quantum mechanical oscillator has a little bit of 0 . energy that will put a bit of 0 . energy gives it an average fluctuation in position or in this case in the in X itself and how we calculated let's look at the various terms calculate the ground-state expectations your old old important pieces are these creation and annihilation operators what happens if I take piece with a plus plus cannot have an expectation value from in the New year Grand state will not like can't reason it is too late pluses when they act right create a state with true with extra units of energy you've raised the isolated in twice and that has no overlap with the ground state all so chose With 2 a pluses backed a bad can't contribute to this Alberta Teramoto Was where they miners do it hits the ground state kills it right it's also true insolently when a plus extra vote left it kills lives so the terror of with 8 plus a plus and a minus a minor given nothing where bounce 8 plus times a minuses a Major said right so that's not there but what about the 1080 minus a part of good works works it a minus times a plus that still there 0 so be it but I don't see any reason why it should be 0 OK wears a say it says we create 1 unit of excitation for and oscillate this as we remove 1 unit of expectations for the end Our so can you removed a unit other than the 1 you put in a call so this can only be none 0 if In is the same as ever but if it is the same as them where you get off the Konica go up 1 miserable discreet continuous variables 5 you get 1 if any equals M and 0 otherwise another word this summer all that and it is only a single sum it's the Sami and a minus and a class and all over square root of end sorry not in in square in Times Square demand just and the cosine squared of ends said no call sign squared is always positive and have reduced to about a half call event Sigma is a function of this square of course it's a the axis because the square was positive and averages to about a ahead that's good enough for our purposes it's it's never negative you can't get any cancelations and it kind of isolates around the half sorrow for most purposes sufficient just cause a hair the nuts about right Bo all 0 I want this thing this thing is just 1 you create a particle or you create expectation of me and parcel later and then you could get out that average value is just 1 we now have an answer the and it yet about X square or they would exclude means in moment it's equal the 1 half was up important thing here jazz Solomonic and 1 over In the 1st always X square was his acts I wrote here Texas say signal but I left out 1 turned but turned the founders the center mass motion is what turned which corresponds to 0 frequency which is just a center maps so really the thing that calculated was average of the position of the string at points Sigma might miss the position of the center maps think about that for moment what it it means to have a strained this case in open string said don't mass is Naomi not beyond the stray but different points on the strings are separated from the center may be average average separation from the center of my guess is a measure of the size of the thing the center of mass is where December masters and if the point of the strain on not at the end of the rest me the spread out as a non-zero
means a strain uses extended inside and Eli captivating it now we find out that X Querida's this sum of 1 already in trouble trouble because while their wives of trouble simple but now most of these oscillations are incredibly rapid the logic in the oscillations are very very rapid oscillations maybe we should just say really all those very very rapid oscillations on physical come off all week take the 1st in the 1st 25 oscillations indoor account and see what happens have a lot of 25 in Largo 25 is not routine number because large 25 Waterbury CT 25 not a small number he moderate number but them were not doing string theory where essentially doing a string of 25 particles were not doing string theory our we've got to go to limit and equals 0 now it's an extremely remarkable thing that this causes no problems for the scattering of strings In flat space flats pace attack it's a real magical very very delicate cancelations that make sure that this strain which is essentially infinitely big with rapid fluctuations taking place all over the place doesn't bombard other strings far away even extended out a very rapid fluctuations it's amazing said of cancelations which makes sure that all of this works out but that those cancelations don't always work they work and some geometries another problem show you what was wrong on the sphere and it doesn't go wrong in flat 1st of all the message all of this is that 1st step terminate his somber at some maximum immunity In Mount maximum frequency later on women have to repair the damage by letting a maximum frequency get larger now I can ask how does this string be here as a function of an Max in flat space LOU his way to think about it as a function of then Max the string occupies a certain regional it's the region basis X squared off are was are squared is equal the logarithm of and the high Ely Max the bigger which seems to spread out but very very slowly but nevertheless it seemed to spread out well if this is moving in flat space it has and energy which is just 1 half its man's 5 times P square it doesn't matter whether it's that our the motion similar massive a billiard ball is big knows exactly the same way as emotional arrest bows of the center of mass Europe was the same way it is different maps but in flat space the motion of a particle Santomassimo particle separator off the fact that it's and extend the object you don't have to remember the fact that a extended just interested center mass motion but that's not true in curved space could space don't separate off so easily so let's see the euro Tapani him yes astray but but the strain up at the North Pole here the a string is up at the North Pole and if I makers my 1st approximation that is only 1 or 2 oscillating modes Baerga start with no oscillating modes if you're not oscillating modes and the string would be a point it wouldn't be spread out over all the via our case is just a point that strained with just a point and I would be over here and it would move as an ordinary point particle has a point particle moved in most of that energy which is the square of the angular momentum over me moment of inertia what's the moment of inertia it's just proportional command on the radius squared times Arabia square this year we send and the 1 there would just be the square of the radius of the sphere of point would move around this yeah I'm great circles and I would move with energy which would be controlled by the radius of the sphere the radius of the sphere go into this calculation comes at a moment of inertia OK now let's take into account that as we a had more modes of oscillation the strings starts to spread out it starts to get a sign is grows that grows with time but grows with more and more aroma oscillations thought to get bigger
fills up some regions which looks about like that incidentally if you ask what that look like the average it's simply looks like a tangle the more HMOs that you and more than the bigger the kind it's fluctuations on top of fluctuations on top of fluctuations in this thing starts to get bigger now think about it motion around the center of mass of a think of the center of mass of it moving on the same great circle the center of mass was still move on the great circle for about its moment of inertia moment of inertia about an axis user axes moving its broad around that acts the relevant quantities it is the moment of inertia about that axis standards governing body the square of the distance to the axis the moment of inertia about an axis is proportional to solemn in the world over the mass distribution each little piece of the mass distribution you take the massive that little piece and you multiply by the square of the distance to be the axis that you rotating about that artists that this point over here is closer to the axis than his point appeared fractured 2 extremes but suppose we include so many modes that Her euros better troops down to about here it's a rotating around the sphere this way it's are although this year it's not inside the on this year the point on the sphere here that is a distant are from the year from From the axis what this point earlier out from the axis is less than a more normal it will slow send over it but as deceptive because adopted this year is really a smear is really here I'm not it's like mowing point damping here when the axis going in this direction it would be the same this this this point over here which is out here somewhere is the same distance from the axis is the point but have not this point every year it really is closer to the axis the point they appear so what don't happen where I calculate Our when I calculate peak moment of inertia the moment is a bigger loss smaller are exactly the same as it was when I pretended although the string was at a single point it's gonna be smaller rank as good a small we are a mess of the string is the same incidentally that does not change would you wear more more molds just oscillate differently but the rest reprimand there's just some of the masses of point particles that make it up the man's stays the same but the moment of inertia gets smaller meaning of that is that the quantum mechanics behaves I say to you could say the strain is spread out and like that and therefore closer to the axis or you could just say it's hands if the string moved our smallest beer it's an therefore all of the quantum mechanics essentially takes behaves as if the strain was a point but was appointed by Suarez figure but the forum always city where the big gets eventually UN enough modes it's gonna go grow 0 so that almost as big as the entire sphere another words well it's got some extremely wary fills up for good fraction of the sphere in that case the effective moment of inertia is due to be much smaller and bending moment of inertia it would have been if it would've been a point part but beta lawyers there's no limit least adds in its larger and larger the string begins the cover the holes for years and the effect of moment of the militias actually 0 the average position of it is right at the center of him that the moment of inertia 0 0 well that's a 0 and the denominator here that's not a good thing Ferrer emotion of anything on your tools around is easy if they show further edits it if the dollar that's now 0 it's not 0 but OK yeah are you have to beat it did does go to 0 but Jacoby of the think about different way what you do is you say the string behaves if you had a few only take into account would say a dozen normal then this spring behaves as if a lawyer
point in our closest circle but now instead of a dozen normal modes let's take 2 dozen normal then what it does Is it starts to behave like careened on this inner surface here this is not obvious this is what is called renormalization group normalization group is a way of following the system as you any more and more degrees of freedom which of the following honestly is for a small change in and Max but once followed for small change in and you find out that behaving as if the war lump but say this big now you edit more structure it begins to behave like a like a collection of lumps but also form string really just do the whole process over again they with a dozen would it behaves as if it were a more or less a point on in our region now and more of them and it begins make itself into a string of these same lumps that has the effect of pushing in again and eventually it just push is 0 so there is no limit there is no limit the limit of a string moving on a curved surface and this is not curvature of the warrant due to the curvature you wouldn't have this effect due to the curvature of the sphere this happens in any number of dimensions of that as you a had more and more modes behavior does not 2 of finite it just tends to behave eventually as the most similar strain on an extremely small infinite has small sphere which means in the denominator here 0 and that's not a good theory that Lapidus so the answer is strings on spheres are not good strings on by makes sense what it is the conditions for a surface surface I'm not using turning In a general sense that could be it doesn't have to be two-dimensional could be 3 4 Can 26 dimensions wherever wherever number dimensions space my surface is a geometry with metric geometry with the Lee money in geometry where the rules what you really want to is a set of rules which approach a limit if you near limit in another word by a limit on the limit as in Max gets bigger and bigger you don't want to be answers the change as an next goes From a 100 billion 100 billion plus 1 of that matter 400 billion world 200 you want against stabilized called a fixed point your audience at the stabilize and not to change stalwart you would like is a situation where the geometry that when you do exactly the same thing the geometry does not seem to change the could define ineffective geometry restrict Whitacre geometry and now we sort of average over the properties of an extended object and that defines the motion on a different surface different surface being effective surfaced that the spread out outstripped moves on what is it keeps changing onerous as we please send Max they will not not be aware live it would mean that some point as you and more and more structure of string the effective geometry stops changing intends to someone that limit will be of good geometry for further string theory to make OK so I'll tell you what 1 can build calculation very precisely for an arbitrary geometry and you can ask when you would end additional not he said you got to some and Max annually 1 more mold how does effective geometry the restrained moves on was Cheney while you describe In Riemannian geometry what's the what's the variables describe a Riemannian geometry the metric right the rector but the description of Europe Jean you Alexis this don't think of it now was the gravitational field although it is a gravitational field of symmetric varies he knew X we have a geometry could be sphere it could be anything else could be a hyperboloid it could be thing with wiggle wasn't it might be space time geometry itself the strangling on such a geometry the effective geometry will change a little bit why because the stringers is spread out a little bit the center of it they will move in a way that responds to or slightly different geometry and so we can guess what the change in the effect geometry when you 1 more mold OK the answer is 1st of all that depends on the curvature of the field warrant curved this would not have happened this has to do with the fact that the viewers so whatever the change in the geometry if it were flat you would not change but curve does change in general so there must be on the right hand side something which involves the curvature tensor the left-hand side is a tent with a mule and annuity a second-ranked cancer symmetric just so the right hand side has to be a cancer Eds not you itself with some of the cancer that has to do with the curvature if in Korea nothing will happen so must be the overriding inside something involving curvature amateurs called capital or are full curvature tensor has for indices thing with 4 indices are over said New England go that court new window for the moment was escorted out of beta gamma you can't put off downstairs upstairs room often are written with 1 index upstairs From this she's downstairs OK this glazing those makes the indices don't match but is there a thing that you could make out the curvature tensor which also has only 2 in the season which is symmetric anybody the Ricci tensor Ricci is simply the
contraction Our Frau from you know that's called Orem you know the Ricci tensor sold when a careful calculation misuse plus some that's Martinez miners but the Ricci cancer it corresponds spirit getting smaller you structure were remind that this is the equation that governs how effective geometry of the string changes as you a more and more of our fluctuations restrict mathematicians have remained for this equation yeah it's called Ricci flow Richie flows a tremendously important part of mathematics and its 18 equation that if you start with a geometry and then you change it an accord with this equation what happens is it starts to deform and flow to geometry float call Ritchie floor has very important applications in mathematics I don't know whether it was discovered by string fears of 1st discovered by mathematicians and but that's workers is it's a fun thing out of the geometry because the points of that closed doubt that kind of diffusion of the geometry which is buzzing about points that follows them Her fluctuations of threatens and effective germ is the history books should not listen to change when you act the change when you a Ibarra it's a logarithmic change we you change the course of by a factor of armor no change in the library of car of the exact but it doesn't matter if it's the changes you change the cut off now overlook keen follow is geometries which don't change when you change the color we want something stable we want to say we're talking about string theory on this or that space we don't want the description of it to keep the wandering away as we add more so we really want on geometries which are stable and don't change would change their reaching change car was that that means geometries we're the Ricci tensor is hero of the Ricci tensor is 0 as I say it's uh well it's not like saying it's flat flat place is does have Ricci tensor 0 of course as all of its components of curvature equal 0 so flat space as an example of this question are there other examples of spaces this is called Ricci flat space has vanishing Ricci considered it called Richie flat Ricci flatten this is essentially the conditions that strings propagate sensibly Ahmed geometry have good limits as a number of mode goes to infinity aren't so this less a condition on the geometry is called Ricci flatness it could be the space geometry of the string is moving in fact there that's the real think about for the moment a spacetime geometry space directions time directions but anybody know what that equation equation has another meaning yet it's essentially Einstein field equations for the gravitational field Einstein field equations bird geometry it's the vacuum Einstein field equations on stunning the vacuum no matter just gravity curvature all right out the Einstein equations for you and then I'll show you have the same of this equation anybody remember this is oracle's T some might back some a as tensor called capital Jean you know it's called Einstein concerned reasons called capital genes got some new gravity but it's also a call foreign you know minus 1 half Genia on well I was are on is are alfalfa or are alfalfa and that's equal to team you know where senior newly is the energy movement energy momentum tensor means it is momentum of cookies ENG momentum over a cup of coffee but not the energy momentum of the gravitational field everything is in here moving around look at the vacuum Einstein field equations means the field equations for the gravitational field when there's nothing on the right-hand side 0 that's called Einstein vacuum equations are any interesting solutions to this acquired but other the solutions of the font others 1 what out Mr. singularity in its its it's nor love Roberts Walker also just singularities but there the the number of visitors face requires a cosmological constant is gravitational away here gravitational waves the effects of gravitational rate it said a similar to Maxwell's equations will sources Maxwell's equations without charges for various solutions not just a plane or just electromagnetic radiation of gravitational radiation also exist solutions of this kind of equation here in four-dimensional space time four-dimensional space-time one-time direction 3 spaced directions the solutions of this include all kinds of things like gravitational waves interacting gravitational waves gravitational waves can do complicated things in Einstein gravity but there's a whole family at infinite parameter family of complicated solutions of these equations right now we're interested in reaching flat this doesn't say arm you know equal 0 but let's take let's take what Apple 1 index upstairs 1 that's called trace when some 0 alpha offer by Webster so let's raise 1 index let's right are new new minus 1 half new who are how fat far
equals 0 what's dealers this thing he warned annual orgy with 1 new downstairs 1 upstairs but a Crocker
belt Frank Seretse just call a quirk of Oprah filled with you With new announced that newly
equal canoe and members outflow all alpha elsewhere equals 1 half my was is equal to 1 half what happens this when you said you would both pickled the media for right 4 4 before the erection of space service becomes plus or minus a hair prepares to rule are alfalfa it does that it does not say what they term through it to altogether the applause to a minor stoic every member but it's quite inconsistent a Masora 0 write this equation implies that this is called b Ricci scalar curvature scalar curvature scalar must vanish if Einstein's field equations with correct they're satisfied so this is not the Einstein field equations an empty space is just on know equals Euro where we learn we've learned that he only acceptable geometry which makes stables and which the geometry doesn't change in changing changing changes you and more degrees of freedom are the solutions of Einstein's field equations Ucon quantize strings in background geometries which satisfy Einstein's equations and you can knock formulate them correctly in spaces which don't accept foreign stars equation as pretty impressive that out of consistency condition 1st things that have a well-defined geometry in the limit member degrees of freedom becomes infinite you derived Einstein's field equations Ferrara for extremely impressive back there is a fear it he said that that is very
closely connected to calm formal invariant but not conformal invariant based on current form variant of the world very closely connected with that but it is actually the conditions for come for more variants but they they are connected but so do the simplest way to think about that in general because the string spreads out the effective geometry that it sees is different than the geometry you started with and in general doesn't approach a limit as approach because at the limit Ricci tensor 0 Ricci tend to 0 or special OK now I wanna come subject always and wait I ever chief cheated I haven't told you the whole story the whole story is really does do work except if you're in the right number of dimensions our subtleties of this kind of thing where there are quite supple and really only works that the answers become independent the Max in the geometries Ricci flat in the right number of beverage cans the super strength theory 26 so that's a basic principle the answers should have finite limits as an ax goes to infinity and that is enough to tell you the geometries that you move it must be solutions of Einstein's field equations was a gravitational waves and other things was now I wanna come to the problem of Weatherly duel with pain or 26 dimension however we make sense out of and the answers is called compact efficacious compact applications is simply a process of taking however many dimensions we want to get rid of by getting rid of them will really get rid of them but we make them small enough but they become invisible except the very very small are rolled them up in April manifolds that coarse grains the apparatuses don't but they may affect and they will affect I'll have real serious effect on structure particles so for destructive elementary particles type of elementary particles into could have but they won't be visible is directions that that you couldn't move around quite so let's talk about compact education we have let's take case of us superstring theory where there are came dimensions of space time which is sick too many and 1 good thing is that it can never mentions a space time have only 1 time and 6 spades and a 9 spaced and we would have we would not know what to do with 2 kind of West Palm physics was some extra space dimensions but that's no problem existed more but will mean that more than 1 time beats me here there probably as anything so superstring theory is a theory with 9 spatial dimension sites jets such as these In isn't RightWriter like others well over only 6 party can more easily a ditch is not a case for war dimensionality of your work and that was looking at colony those mess that being in intervention years more so here earthly turned out an electron design you can have multiple clocks so there are more full time within a design an essentially singles out is taking a single as a clock said is that it synchronizes a lot of others signals ties things together next was mins imbued with only 1 a mail
order clerks but collapse is not the same this is real life usually last is due in which they are painted of this theory harm I know now with 8 times renewed every point the spend their time region where there will one-time regional them but we are following if the amazing thing about the worst thing is that these geometries based on geometry everywhere he said there was a long time it also has a C C and the type the plan I don't think did never rock there are people who appeared memberships however it is never room never become a part of mainstream thinking thinking it 1 more time more one-time direction while the warned Clark OK what's it what's gone on yacht so we have these 6 extra dimensions of space how we get rid of them of course not not your role a mop the simplest model would be a world in which on the large scale it was one-dimensional space time is there but let's just freeze time one-dimensional space a long line because imagined particles on the line articles up and down the line baby buying together maybe maybe they form molecules maybe they do interesting things our most maybe even conform life because signal each other by launching a particle from 1 group toward other group could Newcastle possibly interesting physics these people live on this line 1 made up of point particles but I was like to say they have a rather boring life each 1 has a friend the left may be a friend the right but my job is that they don't have a Social Circle because you can't have a circle and mentioned calf high but right and also they can't feel they can't pass through each other they get the big bang into each other born but then they look power microscope and they discover the line In this really service now they're too big feel so under great big things which were burned down but that doesn't mean that they are smaller things and small things now move both along a line and perpendicular lines so their new degrees of freedom there a new distinctions between particles particles can move this way they can move this way they can even move this way and as a home game this is the most elementary and simple example of compact vacation another way you could think about it let's try think about higher dimensions let's suppose ordinary space was told that but that when the creatures who lived in this ordinary space magnified if they discovered a trainee 3rd dimension what might that look like OK and about a 3rd dimension same sense that that circle here provided the venture his work might look like there's the world that they think they live on but now they look under a microscope and they discover that there's a direction perpendicular Service some thickness some sectors there will call a cop cleaned the ceiling the bottom thing floor and this summer bulk stuff namely space in between OK now it is something else having ages like this makes trouble for string was the strings stop of fluctuate eventually fluctuate out to be given me a medium the distance between the straight the something the same thing we do when we well we're going to do was identified the floor with
ceiling we could have done this back to the store the cylinder is really a in which you identify the floor with the ceiling say that if you go how few have you come back here another words a written the cylinder resulting from the rebound by identifying warned age with the other when you go out here you come back in here same thing here the floor in the ceiling identified so that when you go out the ceiling here you reappear on the floor now this direction there is no way to no edge anymore and is perpendicular direction here has apology of a circle is periodically identified this would be an example of Compaq the final 1 out of 3 dimensions could you compact fight to let 3 beverage comedy big dimensions big ones would be leftovers to to compact 5 2 at 3 1 so I would be back the world a one-dimensional was something new to compact dimensions that I want to show you that would look like a particular way of doing it this isn't particularly of dawn your cake aII bone in a debate that as kind of a prism infinitely long preserve will you call a prison at prison I guess no prison the wrong word arm rectangular something other rectangular work only apparel applied except that considered the extent of right now it has ages voted the where the edges is we're going to identify this edged with this edge and this edge with this said much as the edges of point over here is identified with a point from a flawed point on the ceiling is identified with the point the floor on a point on on the back wall is identified with the a point on the front wall his back wall news from so anything the goes out through an edge reappears on the opposite face at the same spot that no as compact of Ireland's knowledge is where I want to make these identification you really are no ages anymore feud to draw out there to draw a plane freely plane you take a cylinder you slipped and your open up our it has a floor-to-ceiling but they keep in mind that the real geometry is identified and has no edge saying there are no edges anybody who walks out 1 side reappears together and doesn't even notice so this is compact defined to level 3 dimension about 2 out of 10 for 6 out of 10 young you could do the same thing and exactly the same thing and this process is called tore Rydell compact vacation look it wasn't have to do with Taurus with talk about 4 because for I the simplest geometries to you but compact application at the edge of this year this face over here is a Taurus to them like for the murder Burma topologically so let's talk about 4 the simplest chorus is a circle of a one-dimensional is at the circle yeah circled on like a chorus well it is a story about a return defined the does look like a chorus at stake a Taurus as an when we talk about Torres we're not talking about a shade embedded in three-dimensional two-dimensional 3 dimensions what we're talking about His topology were talking about the topology of the boundary not the interior of the forests were not talking about a solid terms were not talking about going with no warning inside we're talking about a mathematical surface a report carried forward talking about when we see the word Taurus is the topology of the surface OK a tourist and let me now do something to it I'm going to cut it Oh yes come over here take a scissor slice of around here remember
we're not talking about the stuff that's inside the torus will only talking
about surface cut it an open it up for was a look-alike looks like a finite piece of a story but I have to remove with yellow cake you open that I it's not the same as the Taurus but it's topologically the same as the Taurus as long as we remember at this point is identified with this point this point is identified with this point anybody Ho walks along here and across is this edge Osoria comes back over here that's or here I saw the missing game except Whitacre cylinder now and identified the left cut here with the right cut through the storm over here that is now mathematically a Torres but we Kingo another step we can now take our scissors cargo on here and now open up and becomes a rectangle so a Torres is topologically the same as a rectangle but who have to remember that the rectangle as identification at this point is called AD and this is also an 80 at this point is called and this is also called beat let's follow the trajectory of a particle as it moves on Taurus hypothetical particle or fake particle moving on her a mathematical Taurus like that's what happens to it follower Esparza starts over here moving this way and assume that it was a straight line OK no forces on what happens when it gets to here reappears on the bottom rank our colors at the same angle because it has been accelerated the gets Dorelia Soria reappears are here now what Thursday he ate with reappear back over here depending on the angle and depending on the ratio of the sides of the Taurus it will either in the ratio of the sides of the forests are rational repairing our it will simply repeat itself and eventually find itself from the original line Egan 5 so that it will be a finite number of bands like theirs and then repeat itself if the ratios of cider incommensurate producer in commensurate angle then it would just keep going and going and going into it filling up the entire crew are getting arbitrarily close to any point but this is not important thing important thing is that had 8 Torres is a way of compact defying space that's exactly what we did over here we this and identify with this this and identifier so this would be called to that compact application of tool they mentioned keeping the 3rd warned on compact 5 but the compact application would be a Taurus car I are especially simple geometries compact application I know there are other geometries that you could try it and there people I can draw up quite the same worry but you could take all lying times as aligned columns of two-dimensional spheres of it every point line a little slice moving up and that did that insect moving up and down the line would be able to move along a line or arc with fear attached to each point on her the just as Torres attached to each point here you could do was ask for reasons outcome whose views are not good things show compact quite so this is the toroidal nappy do the same thing with some higher dimensions but suppose we have where the body of the freedom ancient Fioritto 3 dimensions we want to invent a thing called the Greek chorus of three-dimensional Taurus Ruddock 3 the dimensions and make a mint or Taurus yes it's not hard to do exactly the same thing you take a cue dozen that had killed 80 Pirro applied court amount to the same kind of identification this face already 80 is identified with his face over here a is for our here and be with us will be and the backside of our see This is on the back is identified with C front that becomes what called the week Taurus because the apology of 3 Torres again not edges although Trevor brought to draw the edges and that would effectively shrink we directions if you made for a small case you made physically small Hua hide 3 dimensions Greece from your coarse-grained their measurements of of you doing string theory in 10 dimensions and you compact of like 3 care and spacetime dimensions compact tried 3 you you would have loved 7 inches left over 1 time 6 basis that's also not what we 1 guess we sex dimension so we make a sex Doris 6 Taurus I can draw on the blackboard for you suppose I could draw for tourists but they're Gary clutter up the black terribly got to watch the figure before Q called there were that there and then identified faces of that but on the whole volumes over the fire volumes of but John mathematically while you doing is ending taxis and keeping and making identification a toss of darts the early stage of the Korean man of yacht of the couldn't afford the if periodically
identified now the combat forces at the close the old wellhead Yukon have sects 19 where dimensions which don't come back and sells it just wouldn't be our world U.S. wiII good-natured 2 were to compact applied some dimensions and others that we haven't got the biggest idea but we do not we live in a world of the is is safe string
theory foreign nor the theory that I know quantum field theory none of them makes sense with more than 1 time but we only have 1 real-time with is no more room for a more for reasons that remain state what else would I'm not so that's ideas that might like was it just because we get but they are a they are spatial curled as close as something that is done because we can be bit it doesn't mean it's the real world remains at a thing weakened study find out properties see how it behaves and compare it with the real world you just say maybe it's something we don't know what it is the case that was stopped with dead are this is easy and straightforward it's not enormously supple rounding up these Tora Tora are pretty easy not enormously supple this solvable we can work our work particle physics like quicker work out what with the world on such a setup as like we can ask how close that have similar Zucco the real world that's the only reason it's easier to set up that we know and understand it as gravity has quantum mechanics of particles has both under orders from Milan's study it also has black holes so study black holes and find out if they do this so bad that this a would let a toe to symmetric supersymmetry membership trees religious to some things relax or something tumor job on the use if they villagers thought but if you look at this is what don't think about this look right Rozas these opener wearing a mathematician speaks about forests or a physicist for the most part he's talking about apology not talking about geometry mail In fact the Taurus when it presented as a rectangle all let's let's talk a little bit about how many how many in equal for either are as we were learners of Torres but it's a rectangle solo 1st of all has an overall area all or equivalently and has a length and IT with them a height as a with an eye height were parameters were are those parameters Is the size of it 2 Kattegatt the BET area call as signs parameter the other is the road and that's a product of the UAE times a year with times the height was also the race yield of the with times a high quality with aspect ratio or the shape I'm so is a here is a broad at chorus but these are the same really the same chorus bad geometrically identical geometrically the identical every point on he and is equivalent of point here this maps this maps of its clear just turned the picture on side we haven't made a new kind of Taurus self 1 thing we can say is the racy or side but the radio of the side does not go from Lyle 0 to infinity it goes from 1 to infinity why 1 to infinity because Woods less than 1 the ratio let's say of a horizontal vertical this is bigger than 1 this is less than 1 but the same Torres so we ratios besides can vary from from basically from 120 what while some 0 1 away believe as a factor usually pick from 0 4 1 go from 0 to 1 anything else anything else that but you can do to change the Taurus muted yet it is a game that we haven't discussed let's go back water piece of saga we're going to identify the left
aids left circle over here with the right circle here but before we do it let's imagine that twist twist by an angle so that you don't identify this point with this 1 up here but you identify with any displaced angular point far here is identified with here this worry here identify with that your imagine in your mind before making the identification making a little twist and then making identification that something new and what it looks like let's put stood vertically the drug is way megalopolis steamy forward make identification that's slides things along with carpet if we did Billy twist we would have a rectangle if we don't waste then we don't identify this point with this 1 we identify with that point this point with that point and we might as well draw I'd like this from Idaho draw as 2 of the 3 weak that's not quite right now write everything on the bottom gets it identified with a point above it and really left gets identifiable with this offset with angle so what the programmer tries all right there is the aspect ratio deserve to be sigh therefore we take the overall area then getting rid of the area let's say unit things viewing area the aspect ratio and angle we things call my dear wife is of my delight of the forest 3 of them the 3 three-month alive the Torres of the overall size ratio of the height of the way and the angle the movie the angle of leading set technical Maine the twisters called being twist THAN Fischer support and vain shock be overall volume this call the killer modulus we raise your grasp Esper Gracia was called a complex structure bad technical meanings and see them you know roughly what a 5th flow of Liberty may reflect that they make Avenue Klein bottle works different apology made a few hours if you take if you take this point he error at identified at the point in the front this point with that point this point toward that pouring in this warned them back that Klein bottle Upsala Upsala Robert Murray arm approach now it's none will reenable man crew the not a young young the you can have a best right that you can absorb that kind of thing into the metric from for a you're right there are other ways of performing the Taurus but those can be absorbed into the net for long but these Towry quarrel flat they have flat geometries or they can be given flat geometries faces they can be drawn on the blackboard without stretching and I don't mean the real I don't mean the real Donna real doughnut is a truly curved surface it has curvature on How the boundary has positive curvature I'm in near the whole here as negative curvature I'm talking about it's there's a mathematical structure that's been mapped onto the plane in this way it's completely flat triangles on it have a 180 degrees of some of their angles even on this 1 and in particular its flat richer Ricci flat flat and flat as a string theory on these geometries is well-defined too good a good mathematical structure of string theory to exist on the by far the easiest way hide other dimensions and it's called Arroyo compact vacation it's not good it's not good for the real world real world is more complex arm but I thought I would show you what the simplest form of compact now let's take a five-minute break somebody wanted to make an announcement can remember the Yao for I
role of flat so awry on all flat arise is the plural of forestry tore I all flat and any number of dimensions and therefore there also Ricci flat because their Ricci flat string theory strings moving on chloride good things they have a sensible well-defined mathematics there or other Richie flat spaces are Ucon takes 6 dimensions that you don't like to to get rid of them by compact of them not on Taurus Buddy replacing them by Taurus but by replacing them by other Ricci flat manifold on and particularly their class of Richie flat manifolds which are as Columbia manifolds which have very special properties and where string theory good theory these are incredibly complicated far beyond the year of the scope of this course but there also kind of things string fears her were tell enough complexity and lack of symmetry that they do look more like the real world but they don't really involving really new concepts that some mathematical concepts sure Avery mathematically difficult objects but no really new principles come out of it is look more like the real world the number them try to get into em Torres's grew for us we can see some very interesting things will we start to explore string theory of OK but let's go let's take on marble but personally just infinite are one-dimensional world with an extra dimension that goes around her little you see what I call a circle of one-dimensional this is a line element periodically identified with stuff that I we cannot particle which HMOs around here can have a component the motion horizontally nothing special but that that's just media such as the momentum along with directional would have been there even if you hadn't Compaq well even if you didn't have the extra dimension you have momentum along with direction but you also have momentum along the other direction you could have both of them at the same time a particle moving so obliquely the spiraling around saying we have a component of momentum horizontally and a component of momentum along the circled the components of momentum along along the circle of quantized momenta eyes momentum on periodic spaces are always quantized the easiest way to see it is just decisive pausing gun supposing that the circumference of the circle that's just cause a conference comply are adjusted that's just calling the circumference of I take the momentum of along we are direction the momentum of long sorry along with the conference let's call that key along the circumference PC's P along the circumference and I'm multiply by all are which is the circumference divided Tupaiidae I get something which looks an angular momentum and that it is a kind of angular momentum it was an angle associated with this those circled the triangular position along here and Peter and are as as I get it is like angular momentum that's quantized in integer multiples of punks constant sorrow the momentum along the circumference storm and got along the circumference is quantized in units if we set talks constant a warning unit of momentum is 1 lot our let's suppose the particle with talking about his restless and happens to be a massless particle then its momentum for a magnitude of the momentum is the the same as the magnitude of it energy energy from massless particle it is equal to momentum of course is a factor of the speed of light apart from the fact that the speed of light energy is equal to the magnitude of the momentum this component of a momentum Kory positive or negative it's in integer the implication is that is that there is a quantized amount of energy even if the particles not moving along a horizontal direction and it's moving in these circular Deborah direction it has an energy riches In units divided by all that energy it from Mark point of view we don't see the circular direction is just a thing standing still a thing standing still but the manifestation of the fact that it's moving that has some energy has some next next the now both
particle and it comes in a digital photos of 1 over on the smaller Arias the bigger this they between best levels the bigger the spacing between massed levels by knowing the spacing costs we we don't measure any of this but a principle by measuring spacing between the masses Of these different particles how are they different bit different from a macroscopic from our point of view we look different different messages from the point of view of a small object that we are a small detector which could see inside saying it is particles which have different components of moment the misdirection armed with cool will call on different particles because they have different masses and the spectrum of them namely the spacing between tells us how big the internal spaces the bigger the internal space the closer the levels stroll level diagram energy Energy equals Zero on maps vesicles zeros right away he then we can have a plus 1 unit on rotational momentum around here that would give us a particle of mass 1 or heart and soul for about is negative isn't it has negative energy not BOB energy is the the absolute value of Bavaria state energy is always positive so is a particle of mass while Marty as a particle form are yet to Wal-Mart minus auto are the spacing between the Mets levels is a direct reflection of how big this the small the circle the larger the space this article is vary very big then the levels of vary very close together so if we discovered excited states of particles which were very very very densely space we would say or would look sectors in extra Big Brother Big dimension it we find the spacing Dudas kind of emotion is very very sparse we say that the into small right OK so easily energy levels for b components massive particles where the particle issues simply of particle moving around the direct make and the something else in a compact space you can have a different kind of particle which remain outside meaning from the big dimension just looks like a particle but wife has an entirely different structure only celebrity if all particles posturing if particles really that I might want to draw this was a little string over here but just think of it as an almost kind you a particle let's forget the fact that spreads out of tiny particles and moves around the whole set a moves around and gives it a series of energies in this way but the something else you could do nothing you can do with a string and 8 compact space like this is 1 the strain on around like that here this band around here is 1 swarmed around a cylinder of the political rubber band and that can also will move up-and-down reactors 1st of all it's localized it's a definite place in the big dimensions to it as a location from our point of view it can move to move up and down a particle but also particle it's a particle which is around the extra dimension was it amassed 1 the mayor's it's may us now is not due to its motion in this direction still look to the potential energy of stretching string around how much energy would you expect there to be in a strained stretched around here the answer is clearly gonna be proportional to the length of the string of certain tension and energy per unit Lane firm restrained Let's centered equaled the 1 Lorimar units tonight the matter nor units produced work in units with the string tension is 1 of a string stretched out to a certain distance then the man that that string hazard proportional drooling overstretched How about a string which is wrapped around his direction toward you expect rest be proportional the circumference far yard radius so this 1 will have equal all proportional all but you Duluth you have around several times but I can't do with the euro for the coffee cup but Icahn Roddick Thank you know number can I get back together again about how I know I know I got it the hour of crosses here and connects back here if I had a rubber
band that could draw a rapid get beaten up a rubber band twice around here you're twisted but I have not this 1 what what's the massive it how much energy it have 2 are prices much much better it's all we have on our Arab energy way of energy to want 3 or 4 or if stream has an orientation there is if you think of the string is having a direction the newcomer rapid positively Yukon rapid negatively 1 wrapped positively is 1 wrapped negatively so wrapping winding number called winding number a winding number to be positive and you have a rubber band on the rubber band you draw little arrows to give a sense of
direction and he got a wraparound arrest you could wrap around the rest of the arrows of this way or so the winding number can be positive or negative or whatever the winding number it is the massive that strings is equal to the absolute all are kind absolute value the winding number OK so there's another spectrum of particles here and that's spectrum of particles has energy levels all whose disappearance is not proportional 1 over our was proportioned are itself was a puzzle are a very small but suppose that Aura's very very small then this spacing is large and spacing is very small it takes very little energy to around a small circle but for the particles which are not laugh but which are not worn around but which are moving around literally moving in have momentum the spacing between levels is large forest so I have these do specter of 2 different kinds of particles what is called these once you're called collude Klein particles this idea has been around vote particles having momentum extra Compaq erection goes back to beginnings of committee LUs collude 1st had the idea 1917 think it's grown so great deals and stand but string theory introduced a new idea of 1 particles particle rarity and they have very complementary kind of Specter so if you see particles with a large space in you must also see particles with small place both combinations together would tell you something about the nature of the compact direction but not notice that there's no way way the make the spacing arbitrarily large here without having a very small space in a small basic of course means that it just takes a little bit of energy to create 1 of these things on the other hand what happens if you take are very big now imagine taking are very big take on bigger than the spacing of these codes Klein particles is very small these are very close together now offers very big 1 0 Marjorie small he's become very closely spaced and half these they vary widely spaced why because it just takes out a lot of energy to wrapped around a very very big so here these 2 complementary kinds of particles is almost symmetry was almost the symmetry of the spectrum way you replaced all are by 1 over replaced are by 1 overhaul and you replace momentum by winding number what happens you just interchange fees 2 and the spectrum stays the same that seems to be a symmetry of the spectrum of pore of of string theory is only applies to the extra dimension is no wait rapper string around the X Villarreal x-axis but applied to the extra dimensions this seems to be a cemetery where you replaced all of the winding by momentum around that direction at and the system comes back to itself that no no no no it's another miracle on misses the duality between warring number and momentum and cool to climb momentum now from what I've told you hardly clear that this is an exact equivalents in every possible respect it's just the equivalent of the energy spectrum reversed on of the particles in fact it's an exact symmetry if you take a compact correction and you start shrinking and smaller and smaller lots some point incidentally via the spectra will be equal some some radius this summer radius where the spacing between the polluter Klein particles and the winding particles will be the same we call out Oracle's wanted and Munich at Oracle's 1 spectrum otherwise they a cross on it he tried to stop trying to make a radius of compact applications more and more you eventually get to the point where when you cross it you can't I never think of it as saying sizes smaller than 1 or are you could interchange winding and momentum and say it's bigger than 1 is no cents and and a certain sense there it's not possible to think about string theory where aura is smaller than a set aside Oh you could think that if you want all size what is rearranged itself so that it looks like string theory on the larger space this is a lesson 1 of our readers there's a lot of every hour from eons bows on survey all our mailed said Yahoo bars every compact direction not only indicated 1 compact correction circle we can apply the problems with several compact directions but this is a kind of freak a surprising result that characteristic really of string theory is not characteristic of a point particles moving no no enclosed spaces like it's interchange of winding with momentum hardly seems an obvious thing to do remarkably the entire series scattering amplitude spectrum of particles of old works is symmetric under this the change None of the Above close close string forestry open strings it doesn't make any sense to wander around this morning not now they're core both closed strings BOA II shows said that the theory of closed strings this interchange open strings more complicated now talk about these black hole where a lot of 4 whether this is it saying that there's a certain
distance where when you try to think of string theory on a space smaller than that of coolant string theory trying to shrink the diameter of a compact of occasions arbitrarily small distances you get frustrated it just really Rangers itself so that it looks like the did that helped give the self energy well the best In some senses a smaller share of small size here during her arm keeps things are being infinite yes INDUSTRIES TO life number analysts also say that there is a of visitors yeah or model or 0 . 2 the state as room get out of here is your have compact Berman Chen's and it's possible was on a U.S. gained maybe you asking how you make them if you don't have the sea floor FDA off if a winding number or particle with a warning you could make them pusher Makeham how we butter start was something to restart work particle which has now winding number through spring that looks like that going now we take this and we pull around stretch about still down a winding number it's not worth around and now we stretch it always earned he's comes back their backside comes up over here 0 . it still has no winding number winding number of this piece is canceled Labrador so you take your arm you take the rubber band and you pull around him and take care of take care of it to pieces and that's what you left but still has no warning member remember now that string theory the basic process of interaction is for things like this to become things like this rearrangement so if you sit there for a while the string will rearrange In format now lawyers have it has particles with opposite winding number we still have a net 0 winding No. 1 was one-on-one weight waters where the other way but now they are disconnected and their free to seperate throw that waterway just ejected out of the system you it's some a velocity and when it gets off Balfour century forget about it and you're left there were particles sold where whether you like him are not you can't forbid them some things you could try the debate even safe and that they had to begin with whom everybody back our possibly some things you can't forbid because you can imagine that the processing of the theory of self important crosses the theory can force them are new and used a force Of course they always give forced in pairs but you can bet that jurors to throw away half of them and just the gruesome experiments with remaining is increasing its this this flows close flows so large that EI does it get some energy from not only from momentum but from being stretched takes energy do it 5 but those are the sort of thing which would happen if you call lighted to particles With no winding number just to collude to Klein particles collide hard every possible combination of stuff happens and among the things which happened these winding and and that winding things go flying off later in the day end the logic is the same as the logic of electric charge suppose world started with no electric charge that mean they would be no electrons not any kind of collision that would take place are we create electrons and positrons in and the positron to go flying off the summer place a left for Electra logic is really no different than electric charge you can't not have electric charge because you could always make it in Paris the net charge in the world might be 0 Frohling harm to the net winding number might be 0 but who cares we can do experiments on her your arm region which contains a laser warned so 0 young so this is a name incident be equivalence between theories which look different usually called a duality this is called key do our capital he nor our and that he in this case against the Taurus peace stands Taurus cars are key out already has a history of the rally which relates compact occasions very small geometries compact applications are big another startling thing there was 1st discovered that surprised everybody would means is that geometry defined in terms of how strings geometry can also often be recovered op partly recovered from the spectrum of mode Armenia the geometry of a drumhead the shape of a drummer a famous mathematical question condition 8 over drumhead be predicted from the sound that makes from the year from the spectrum of vibration is is not quite but you can't predict a lot of the shape of the drum here same question applies please compact vacation from the spectrum of particles a spectrum of vibrational energies vibrational and other kinds of energies could you predict the shape and road the size of the compact direction and the answer yes to a large extent on but there are some ambiguities and some door allergies that it can't tell if it's a tiny geometry or it's a big geometry because these 2 little into each other that would not be the case with drummer history you would not mistake a small drum for big drug OK for more please
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