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String Theory and M-Theory | Lecture 2

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Automatisierte Medienanalyse

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slalom Stecher University where begin
with a little mathematical preliminary of she'll mathematical breweries which if I didn't know the man off in advance I would have to do with them during the course of showing you some things about the period and that will be on the other hand is nothing but thick you don't know well most of world fresh thing just a little bit of cow robe of calculus formulas couple calculus formulas the useful have a blackboard off if we have a function let's call the function acts that's the dependent variable X is a function here was a function of its gotta be a function of variable called stigma why am I using XYM raising stigma because it turns out that these are standard member patients used string theory asserted options but they could be any functions Kobe Y is a function of X to Kobe acts function Signora Toby was are on G were an unwilling to all our for the purpose of this discussion now segment the variable which runs from Zero coors half cycle around a circle so segment is an angular variable except the quite regular variable Tarango bearable alleles halfway around the circle that I want approximating continuous function by a discrete function and eventually make my description better and better and better by filling in the by filling in the axis with more more points effort calculus does we replaced x 0 stigma by X which caught the well I runs from 1 of the later on Monday Lavinia Neuberger bag let's golf with French between an X at all an ex and I plus 1 we call I don't exit we like but felt the X between or and plus 1 or might warmer than maybe it's just a quarter acts of acts of alright minus 1 I could chosen it I plus 1 sign backstop X and it's well approximated women who we put many many points in that assuming the function as move the modes assuming that limiting functioned as a small differential will function as well approximated by derivative of X with respect said 9 times Delta Sigma and how they has dealt the stigma Delta said felt the same segment is the stigma interval between 2 neighbouring values so probably do that that's the Holland double plight provided by Jack up into little segments another 4 world make yourself and finally of Formula misses derivatives formula for growth but stressed the magic adding up war with exit some of all the extra by adding them all out what's the approximation for back well if you multiply it by Delta Sigma no it's just Delta Sigma which is and and multiply that by some of the stigma is best becalmed was in no way be integral In this case from Xerox the piety of Texas said we simply replace acts by Texas did not and dealt the Sigma becomes these certain right that basically it's a definition of an integral but I want to have these equations on the blackboard because we use them our card OK that's the 1st mathematical program Mary straw wine underneath it but you try to mathematical preliminaries blackwater over here and the use of more their pocket next surpassingly have a functional now or continuous function here factory Bosnian really have to be continuous but over the functioned figure to be contiguous the finally interval from 0 and such functions were cavalier a rural describe a merit any such functions
can be Fourier analyzed another way of saying it could be written as a sum of signing the call signs are signs in call signs are right out and give you examples of a moment but before you do so I should say signs of old coast off before you do Sul before you do do it you have the will establish start features about functions which put them in a clash which is I called Buechele my mom rebuttal of the Barbara conditions boundary conditions the behavior at both functions at in the pouring the boundaries of the I sold for example this functions could stand for the displacement of violent history from the horizontal brought horror draw horizontal not far from probe violin strings your whole in the areas of current firmly in place and FOR the value of X Games 0 just because costly ends are being held next are called jurors batters called dear boundary conditions our right at once Videsh parish where was the richly Irish here and there's like the of course French mathematician and studied waves moving things like strings and classified 1 of the boundary conditions life and meaning of back their sway boundary conditions means exit the end points that means x 0 0 equal 0 and X of piety equal 0 now another class of functions which are so the opposite mystery stands our capital but just the class of functions are described by something called no I'm I'm boundary condition armada of course was was a German mathematician and iii you and AT and end no I'm I'm boundary conditions and also would ever find those are the appropriate conditions for discussing restraining emotional strain now tell you where come from as we go along but the moment I'm not doing physics and doing just those some mathematical fact not Mandalay conditions are the statement that the derivative of the fortune is 0 at the end points so let's 1st Roderich like function of the Ridgeway functioned might look like this it's came down the end points so XVM points 0 4 North on functions the derivative of X with respect the stigma but it is equal to 0 and all functions which oak like that put a vertical axis script to radical access they don't necessarily go through 0 appearance points but their derivatives go through 0 that means and not delivered at the ends here flat at and so suddenly among functions or but earlier yet but the volume on boundary conditions now why would you would pose 1 versus the other will discuss that this is clearly you would want to impose this if you have string of which was held back when would you use this theaters where you have a string musicians are much better off there are but but both kinds of functions can be Fourier can be written as it was by far the best word to the forest borders are 80 on a play where you would use nor among under Organ Pipe and if you close up the end of the Organ Pipe Band sound can the sound waves have through a B 0 displacement if you open up the Organ Pipe and so nor among boundary conditions so odd yacht shrink was brutally songwriter where they are we're not talking about strings moving gravitational field vertically and horizontally of research that is a global overtly has roughly vertically and serves as although you mean if it has
a vertical jump a yacht on
qwest physicist of the rule don't like such vertical jump this carbon or use Commerce discontinuities usually mean infinite energy with a plebiscite let that the show John yah yah but really sharp jump blues even that some Ripper Yarnall you allow the end and your allow that have sharp jumps in these functions but they have to be piecewise continuous piecewise continuous music and jump your job here but always 4 are purposes and particular frustrated by restrained perhaps jump and it would mean a string was broken now with Norbert district pocket next Fourier decomposition aureus yes job urgent might look like that what's that what I'm satisfied by what's more likely dissatisfied this 1 this 1 however after think about it and our of our from up there right now it it satisfies anything it's gotta be this 1 but Rogue River with a back foot yacht as he has because you're probably right back and I don't think about our normally get my hero offer I'll tell you I'll give you an example of an idealized string system which does satisfy your probably don't pay and another poll over here according to the ground but you have a light almost massless ring which goes around these polls and they're connected by prosperous again X corresponds from a height of the string works above the surface this kind of system of these if these rings were very very white would have the Ridgeway boundary condition instead of holding the end of the street now the end of the string is free-floating OK twat be as a string free-floating like this that noise on conditions Our appropriate got some obvious that's obvious we're going to explain the physics about why the ends of strings freely floating Stringer satisfy this but for the moment we just dealing with mathematics right now are getting comeback through Fourier decomposition of such functions for decomposition very variability essential role when so many things but a particular string theory functions which satisfy Dirichlet boundary conditions both can be caught FOX however XA Exocet stigmata but can be written actually signed from end equals 1 until infinity off call fisheries Exxon may pose a set of coefficients times signed an X and stigma why sign because sign other functions would various IBM pouring sign of an extra any he and his euro and lexical 0 at Mexico's pie the most general function that satisfies rich Laker boundary conditions and which is continuous and a proper Pasillas continuous differentia good things with general function which satisfies can be written as a song of raw our oscillating function like this is Dorisa function whose derivatives 0 0 0 call sign co-starring next cost signed annex any and is flat at end points and they not call signs Corsa X Frank electrosurgery X was cosigned Zero X and was signed Buffalo's begin was signed 0 ex-wife and started an equal 0 assigned a 0 X Amin was assigned as you're alright which is 0 so serves no point in putting a term for our which is 0 for about call sign 0 0 acts 1 and if the function happens staff have them average of its averages Zero overly in the world it will start out with a constant terms which is just flat no it's a flat function is a good nor among function sought in this case XO Signora are an extra where Astana from indie equals 0 because euros now worry over an interesting case X abandoned cosigner stigma what about the derivatives Our acts but stake could be richly
case was a derivative of excerpts stigma with respect the stigma well if you differentiate us I know what you get we a call sign so if they function is to assure its derivatives my mind a function was going on it's 2 removes Dirichlet butterfat shot but did a good number West mathematical flak from flak less mathematical fact from growing on the blackboard is a fact about integrals integrals off signs and co-stars most of you know anybody knows everything about Fourier analysis knows if not used a fact surpassing a tag Let's work with co-starring supposing I eat the great from 0 to piety call sign of ends my kind course are love and said my 0 I get 1st of I get is not equal pay him it is 0 if it is not equal to air start with 1 cosigned here and this article and means euros no multiplying 1 cosigned by another car starring and his most of you know or suspect if you do that integrated average a product of them you get 0 so this is your if and is not a equal quota and where that is oil and that is just a right of course is just a co-starring squared are answered so I a figure that out of so I think will co-starring squared and stigma cult was cosigned squared vary between varies between 0 and 1 In fact the square of the cosine if the the call sign if some cosigned function looks like this some cost on looks like that the square of it would of course were Iris warned us stay 1 bowlers dips below the axis years the square would be above the reactors squaring looks like the square before that Wednesday average of a course 0 where the average of a cosigner square at half time some particular if you integrate cosigned square you just the average of cosine square was a half with integrating it be seen between 0 and Pa so this function on the averages just a half born you integrate between 0 piety you get away from the interval with me in the world and this is just high over to pile 4 In an hurry for any in equal him order right now it is equal dealt and that's a symbol which is 0 0 if and is not equal to and and 1 if any is equal to at times higher no I made a mistake for 1 special case rebound or special cases equal 0 so it's taken equal 0 a double 0 wrote the party Of course I do what's cosigned 0 for a cosigner zeros 1 of us is worn a stigma and that's equal the pipe sober 1 except showcase is when a kid in a on both equal to 0 OK assist rose a vicious trophy and I'll I bother writing it ahead of itself is not save on 1 of roughly have 0 was about 0 now OK and not exceptional way except for any glows I am equals 0 and then we get at just it is a mathematical fact that we will meet but the other thing that I had planned to ride out as a branch of mathematical fact graphical hold off for Oakham baccarat direct properties of harmonic oscillators yes but we just write down 1 fact about harmonic oscillators ought to have a harmonic oscillator described by a quarter X the harmonic oscillator or hoax loss are Florida the young lady a quarter of the oscillator you could write as X whether or not sustain Exs hero back from Omar watches call coordinator of the oscillator the displacement of the oscillator tax was a kinetic energy was the kinetic energy of it was a kinetic energy of . 1 FM backs square right 1 half an hour transit time direct squared but you can always working units in which the masses equaled the 1 you could always use rescale things you could risk Gaelic warned that saw and ex squared is just take take me and you take the square root of it to absorb it in here and rescale X Ukiah although it chose units so the kinetic energy X stopped squared hand over to its conventional put to pass a kinetic energy are a point maps point what about the potential energy of a hoax or so his energy scorer energy but plus one-half time just Britain constitute court camper x squared what's the
frequency of this soft or anybody out there with the frequency of squared of Kayla em but and 1 I've chosen and B 1
working in a formulation way a musical Kauai what is the frequency of reporters to squared of Kang on Europe's Katie is the frequency square it was just put Omega square this is the
formula from across choices visit energy Connecticut energy potential energy if we want to work with were granted we were right kinetic energy squared minus potential Orange a square peg wraps that's the basic formula that will need for harmonic oscillators a quantum oscillator we'll come back through the quantum oscillator of course quantized with energies in units of
Omega so on the energy
levels of the harmonic oscillator any integer
Transamerica under each bar such electric or on the same page on next question I want to address Sousa perhaps a philosophical questions but quit after winter it what is a particle were leading by a particle Ruby talking about strangers string particles strings are assemblages of large numbers of particles were Springs between our so in what sense do we want me Litalien always say a certain particles a string that raises the question What do we need in general by a particle Michael what properties the particles have never location but are 80 no particle is known could be a point of fact no particle that is known it is a point Courier not particle what we know of is a point particle even electron was not a point particles it has a real clout of photons around it before concert virtual parish borrowed the electron if you could look at powerful microscope would have some flowers around it and that hours would be virtual photons of virtual Paris electron positrons so much on pouring so a proton or neutron was far from a pouring those are big gigantic objects that there are why are so big the use of wrote turn and for this class protons are very very big so a particle was not a which is a point particle it to become a composite committee made of things away with you Tankleff stake a claim which I think most of us would not call a particle of Mary ordinarily let's take 80 box filled with articles a box were just ordinary box can can filled with a larger gas on a large a large number of articles it has amassed it has positioned news center mass position we speak about position we usually mean the center of mass position John we apparatus that user thing about it is that a particle of course starts a cup of coffee on has a position that has amassed wine we call a particle apart from the fact that it's big and sincerity said league is a relative terms our souls from Why is this not a particle water other ingredients the week ahead when we speak of particles Obama physicist speaker particles our original BSR hotel you can't be In the visibility in not Proton was certainly not indivisible as well as shell vision is that if they want to fall again loses supposedly will not try it is particle happens off-court ratified a ripple bigger OK I will I will tell you what a ingredient How are poor or important distinction between things which we usually think of particles and things which we usually think of as highly composite it has to do with their energy spectrum remember that energy is equal to maps equals M C squared and it has to do with their mass spectra an electron has gained more or less what has a unique maps of the electron you can it of a mass of electron packet the masses to shake it shake it stopped shaking your why have commensurate don't that put some energy into it that's probably started up and a mail to its energy and energy is Ismail electrons there is nothing you can do to increase the mass of an electron I should say nothing nothing we can do in a laboratory a president can exercise the electronic state of higher met its discreet it's it's of all by itself energy sector with the plot vertically the energy it would be the electron the photon would be down at the bottom with 0 so they it would not be all whole bunch of excited states of electron just above the prop does have excited states but the excited states are pretty dish Greek league different than the proton self it takes a couple of hundred 80 million electron for the spin the proton what the cause it are story so proton also has to some extent this discreet and the expectations of the oral well above works it's got a friend of rock isolation in energy or maps worry about their cup of coffee a cup of coffee has amassed some fracture kilograms but what I started up a little bit Soto he appeared kilograms or circular what about the 1st excited state of coffee protocol coffee down to 0 temperature nor have some maps How much energy can I get to where I can and an incredibly tiny amount of energy to it why while 1 weighs just go Pope 1 molecule it but even less energy by making sound waves through a four-month through it incredibly tiny amount of energy so the neighboring state right there which is so close that you really can't distinguish it as a separate interviews individual quantum state and there are zillions of them will stay the course bike practically forming a continuum of energy levels that is how that's how we distinguish particles From our from Morse much can be
excited by I need a little bit of energy particles it takes a significant discreet amount of energy are of identifiable discrete amount of energy worksite that's really the only difference is a strained quantum strained a particles or not and that depends on the excitation spectrum of energy levels above the ground state if they're well separated for some reason the answer is it will behave like a particle their extremely close together so close together that the experiment whatever happens to be can distinguish them but it won't be like particles are matched characteristic difference between Bush's the particles the energy that it takes perturbed a proton is a couple of hundred all summer 15 15 per cent of its own dash to get excited state of a more 20 all about 50 % of expressed 40 % of smashed to get approval next energy level so that but a significant war about a strained of quantum strained while the strains that we are going to to talk about particular ones were going to talk about the beach separation between the 1st race its excited state is more like a plant a huge mass of experimental point of view so huge that we have no hope of in a laboratory exciting expectations so far that's why we called strings particles Young is usually was his you see the Gaddy off your of the energy levels of a energy rebels if you don't think you might be able see that was made up of pieces but nevertheless you would discover that even those made out of pieces to excite them with cost of very large amount of energy on the scale of particle physics energy levels and other words if you were to try and Electra which was truly a string and it's a mass is what happened media or something like that and the excitation energy rotated Book of vibrator vote to do anything else to it Is the Plonka maps which is how many times bigger than the electron intent 23 and 24 times bigger so it was a big gap specter existence of gaps in Spector was what points will make particles different America have violence strength for the those very little energy by comparison with actual maps for violence trained is made up of a cat guard would require Mr. Mike Catt for kill caps acquire yes there are no more or less of a gram what for a purpose significant graphs and the energy that would take a worksite 1 quiet the worth of energy a violent string when translated into is on measurably small sold down a lot of quantum state I knew and the of Ilan string is not something we would call for particle OK so now we have to begin to explore the mathematics of mathematics and Black Forest suppose we should say we're exploring physics but the physics it's fairly mathematical a is appoint Potter co-op us some Cuesta reported that particle there's nor Samoa particle with Willamette costly formalized a goal gave me a hard is but it is Our alone is not enough photon is not a point current and associated to an electronic broke on a positron Papa who Archana positron so few with the look of proton carefully you will find that also offers structure of electron-positron pairs of more virtual photons are set free dance more bar there are not in which more and it's even measurably it has measurable you can actually measure the of the fact that they electron for example is not pouring user resource to among other things were anomalous magnetic moment for a child's life and no no not too much lot me correct definition of don't twayblade stars but not physical folk now we talk the last about describing systems at very very large momentum this is important part of the logic of war or reproach to string theory would or could take call like all frame momentum frame now just remind you for the moment we week we could come back to it cars but at the moment we beat me fact that we established order we talked about last is that if you take a system which could be highly relativistic where means to me highly relativistic now I mean better has part pieces which move relative to each other with close to the speed of light a up the time twirled around so fast the ends of the Parnham moving with cost the speed of light that's a very relativistic system and the question is can you in any sense that tore describe it using something like relativistic physics if the baton is moving slowly for all parts of the movie with only a small fraction of the speed of light of course to use nonrelativistic physics describe it but it's an approximation they really an S. spins faster and faster as parts of the top wrote boss to use relic of quick cost the speed of light that approximation break spent a good approximation Soto not sent start even if
the but time at the center of mass of a arrest issue of using nonrelativistic physics to describe the relative motions the pieces are it is not question a the sooner massive Ezra arrest it's a question of the relative motion there's no good approximations no good nonrelativistic approximation to but pound which is spinning fast enough parts are wrote mystic motion were richer on the other hand the trek but I showed you last exact its exact following sent you could take the baton twirling around an actress going in this direction or and boosted all you'll have to do in order to bring it into the nonrelativistic formed talk about last time was just boasted To win moment where the momentum of the whole thing is much larger than the momentum going around in these planes or equivalently much more relativistic along the axis boasted that it is the other direction then the formulas showed you last time become exact they say they say that with respect to the told dimensional motion two-dimensional motion In the plane perpendicular to the direction that you Bush to throughout the description is completely an exact wrote respect show but was the trick that we used over were going to use the describe the properties of strings Rossak trick which makes it which gives me the courage to explain it this class because I would try to explain it before we wrote to perform shot young last reverse he commented that so far I am a man of letters life is like half cut off for a B 1 of that again just briefly in like a of fully a day but are you take you take limits you define things which have went so even show you sort of thing you could define which was as long as you just Beckham actors any actors any access no matter how the system of moving pick any accident boast along axis it was a long way from where rotating a story on the that way because he was as bad as it was moving this way it still has a nonrelativistic prescription where would look like none brokers trickery from pointers you is a romp going projected onto a plane projection onto the plane was world there's a curious thing about strings as it's completely enough to to know that projection on the plane know everything about this is a bizarre an interesting fact about things that are very do not have any independent quarter too few Bush more relaxed which they have no independent degrees of freedom along the axis or with degrees of freedom of perpendicular our it's a rather remarkable fact about an analyst said the Union for any coverage drilling then they can you know they can't arbitrarily will directions emotional directions completely determined by emotionally yet so that it is it like it is an by our trappers downward erection but more than that not just a protracted share would appear motion well 1st of all oil Iran's contract but in a way off you find that firing of examined within that Lawrence contraction yes they would be moving but the emotions completely and entirely dictated by the motion until the election of a curious fact is a source of the where our was 1 of the things there was 1 of the things that were speculation began young that is closely connected with R and I be we're may try to get it right now for sure instead which is going to lurid let me just remind you about that like story will spend 2 minutes added you take any system of collection of particles moving doing what I'm doing it as a center of mass enough frame of reference to which the center of mass is at rest it has an energy that energy is called a thrashed energy and that's the thing which went abided by C squared you call the maps multiply provided that the energy arrest means the energy when its momentum is your and a frame of reference which momentum was 0 that's cold the rest maps that doesn't have to refer to a single particle can refer to any system whatever In a frame of reference which accused has 0 moment the abroad the frame of reference which arrest in that frame of reference its energies court arrest energy or rest occurs when a man that man is in general composed not only of the masses of the constituents but the relativistic motion the comedic energy and even a relativistic kinetic energy whatever it is it all Arabs out for them now what moving those molding it has an energy which is not just and it's a square root off the momentum squared plus the mass square I've set sequel toward the speed of light 1 fassel formula for the energy of it right back a former square root our Peters each squared that's the direction wrong class right PEX square plus BY square but I'm not just gonna write key square P well what's escorted by PX square at White PX were forced to watch square Glass M squared when you boast the system along an axis sorcery z-axis the other components of movement known change so as long as
these components of momentum a finite you could always make PC much bigger
than anything else problem figure PX PYE and much for years and is not limit this becomes equal to 0 PC plus P X squared plus Y squared provided by twice PC plus and squared the derided by twice PC where done his expand this quantity here I'm thinking of easy as the big day in and the rest of it as being much more and there's a lot you get frustrated as will this does not look like it has a good limit as it stands with PC gets a large press of Otis logic it since but was a since February at best affinity was a Norris but this is the really does not depend on the X momentum wire momentum or even a wholly internal motions which are making up the Mets it's just a constant which doesn't depend on the internal structure of what's going on inside the object it's just moment Yukon subtracted offices a rule that conserved a quantity which is conserved and richer for reduced Yukon subtracted from the energy because in the end of the day the only things you're interested in our energy difference was generally speaking I at least before you worry about gravity the only things we worry about about energy energy differences between systems are so you could subtract off or bring it to left hand side the override eat minus PC and not worry about there were about a PC there because it simply never contributes to differences of energy of different for example if this what have and Adam Mike case where man those with decays is related to the energy difference between the states which became just as long as Adams momentum along the z-axis doesn't change during the decay this term he doesn't go around and not about the PC in the denominator why that keep in mind that energy is related to cotton that energy is Hamiltonian Hamiltonian in quantum mechanics has to do with the rate at which the wave function change energy is Hamiltonian and Hamiltonian is RID by GTE time derivative on away from Why is the right hand side going 0 for a right-hand side is going to 0 the left Paris I'd be going 0 another words the wave function must be changing very very slowly for a particle moving with a large moment the internal away fortune away function having to do with its internal motion why is it that the internal motions look like they're going slot on Malaysian which has so it would you are magic is you want to rescale the kind of fast other system growers found Rosiak the slaughter its internal motion but we'll want the internal motions you wanna keep track of them we want or understand them so we be scaled the time of we gotta what kind variable which as we booster systems we go toward car and variable which is suitably redefining so the eternal most levels a year Was Adam we want to reach gala time so that the have more of their rate of firm of revolution of electrons around the island would not to 0 and that simply means falling away the PC it is the time dilation fact we don't care about it and we want speed up everything by going to a new time variable all we have to do is always easy and marmalade I
suppose that's a quota multiplying easy and then you say we have an expression on the right-hand side which was not witty or art which does have a limit as pleasing goes to infinity it's a Hamiltonian for the energy function which keeps track not only of the internal motions by the motions along the X and Y directions emotions along the X and Y directions woes internal motions are kept track of by Tamil career over here and it has a nice limit as PC ghost be No. 1 number told it looks very nonrelativistic songs of the squares of a momentum components of momentum of a square the momentum provided by tool that's the royal yacht and Will you be Hilliard easy on the individuals involved not what we have to worry about the relative peace use to some extent on if the system is composed a rancher apart so we are the core of the degrees of freedom there we have to keep track of it is the ratio of momentum along the c-axis of the different parts but in string theory you don't have to worry about this is quite a marvelous of string theory you don't have to worry about it constrained Her but that is so it is with regret young it's not missing year because you're thinking about Poeppel system of the system is made up of parts each part would have we give a contribution of looks like there's will less he lives on he arrived Brittany expression for the entire compost system of the compound system of made apart this would be the center of mass motion this would on toast match motion the center mass motion two-dimensional plane that the system is made of a larger parts so that a peach Here are some of momentum of particles EADS in principle should the worry about the relative components accessed our energy really have to worry about them it be very complicated string theory of magical reasons there only partially understood but there are spread around but so the magical be relative motion of the apart along the z-axis is completely constraining completely determined by their emotions you don't have to think about Schroeder away more about but it should it's not intuitively obvious but stroke OK the main message here is that the physics of Erie fast-moving system as it moves but we are in the perpendicular plane has the form as exact amount approximation exactly has a form of two-dimensional wrote script for 6 but would you have to keep track of it is that they these portion of the energy which is independent of the state of motion thing you would None relativistically think golf as beyond the bounds binding energy of the system just the energy arrest is not proportional per match this formulation was proportional per square sir of that was a review for 1 hour review our try not to make 1 or reviews for everything we do that well over the bar I thought party it was worth the effort with crayfish finish what we said last time lapse Oh of gone let's go want this trend is now the right model full we're talking about now is strings moving in tow dimension by plane perpendicular to the direction of large moment don't let the let the dimension of the large momentum longer perpendicular the blackboard Then we want to make a model of relativistic strings which is illegal in around moving stretching going all the things were strings do Memorial only in couldn't fight to do that again I think I'm still reviewing but year's work with failure strain is a collection has . org begin my thinking of string a collection amassed points story back to this approximation here capital in the West for its capital and last points but but In minus 1 Springs between and think of a string of a collection amassed points with between them and minus Warner Springs and an energy forthright energy is narrow nonrelativistic physics each particle as a kinetic energy some of all the particles the kinetic energy is X start squared all twice its mass times twice massed not messed over to forget arrival choice mess over to the city of Iife spring X I need your to they had every place right X and is also whitey X Mexico's Hawaii baritone directions that everything can our call Aleksandr ii it does not stand for effects of what it stands for which particle here we're talking about change saying and White Dock Square over to force a kinetic energy and what about the potential energy the potential energy something like Kappa
times X all right oracle Court dope Baxter record barracks before God help OPEC's which caught dope Texas Delp X I square a day or 2 where Cappaert is spring constant between the springs between neighboring grow between neighboring mass point that sealed you up around here that's the difference is Connecticut energy potential energy runs use kinetic energy miners potential energy so we might want to write that down some point exactly the same crime expression except where X everywhere is replaced by white that's a store trainer of . matters now I want to take the limit in which angles at launch so 1st of all we need a parameter the label of spherical label positions local labor leader of the mass points back and there will take the called stigma echoes from 0 pipe arbitrary Coleco from 0 to a 4 1 necromantical from 0 to Itel article from 0 to tow it's just the labeling of bodies we label going from 0 to 0 Papa how the reason for going from 0 hi instead of from 0 to party is because we going from 0 to party for closed string rubber bands they got restrained as a rubber band the car opened and has to wear rubber band which is close together or our could go from 0 to 2 parties so it's just arbitrary reminder that is not a closed cyclic strain will go close a cyclic strings pocket about the Masters has only let me we want the mass of a whole string to remain fixed at each mask Point waiter grammar and I let them number mess points Goto affinity what happened the string we get incredibly massive that's with we wanna keep mass of a whole string fixed by subdivide who was smaller pieces so that means let the Mets shingle was 108 data track I think we can just let it be worn off and we don't let it be something else could Let it be toolbar and this is actually a choice of units an arbitrary choice of units after a fire or your choices and make now could be absorbed into Europe's so we let the man's gotta was 1 in a while a total mess of string may not all want sorry just a moment when we go we go back this mess here is not the Truro Mass of mass it's the nonrelativistic NOR maps I shouldn't caught and because then you identify with the relativistic MC Square let's just call it more mule the euro stands for just a Pereira and analog mash of Rio Sol LeWitt newbie 1 what would the tall Kalama Wyaga mayors of the system be 1 so this is a weather analog mass war hero but With the analog worn what about the spring constant what happens if you'll have a bunch of very very stiff stray spring Nigeria bunch of very stiff springs and because he can't stretcher-bearer March now we combine in a series of large number of these but in what's the spring constant overhaul thing is it easier to stretch a harder at stretch as much easier the spring constant Oliver Kahn past spraying of the compound or are overcome part of spring is like 1 already in 1 over divided by the number of hours the meaning of that is if we want to keep the the can't pass it spring affects everyone a keeper composite spring tricks and we have put the spring constant of the individual little Springs he get larger small large large if we have a robber baron we stretch it it might have a hoax constant of normal number these you stretch but you try to take toward the neighboring molecules a structure might guarantee you we were at the program a lot more for a stretch of between 2 neighbouring molecules sold the spring constant K E I think I will take be works In this is arbitrary is convention Katie cause and although pi squared the past where's put them only for the purpose of minimizing the number of pie formulas don't want a worry about a pies just ignore it radio every frame nor central role in anything I put them in In or appeared get rid of pies somewhere the states to about set while Ryan I said nothing illegal Doric suborn sometimes I said some things like free equaled told but warned not more right now I checked through normalization warm spring from Robert you could ignore that by let me right these things integral us using their exactly the formulas we have are OK all we need 1 more formula Dr. stigma it is equal pi divided by him I think I actually have set up bit your yacht segment pi divided by taking the pie interval and breaking it up and in the pieces OK let's take the kinetic energy 1st the kinetic energy is going to be 1 of end find the sound of X I got square kinetic energy in the mouth for Europeans to restore the potential energy I now want him but see warm and is dealt a certain the pot I just right this equation by apartheid so let's say so this is equal to Sharma Delta said 1 although party firings X I got square in fact I worked backside to act as we to out here but now we use a connection between intervals and a some times dealt the stigma is the same as an integral vistas becomes if the girl a two-party integral part of derivative of X with respect the stigma square with respect to prime square remove X with respect for scored squares powers time powers analog or abused period we like cone nor infinite moment when he was backstop means DXT T BXT town our somewhat of a stigma times a function is related this the integral interval golf from made the rule goes From 0 that's a Connecticut Energy where about the potential energy for Italian our Portugal a court about yourself exactly the same things the new thing that we need we have built by next year and next is related to a derivative of X with respect the stigma stock we have a camera we get here across a review of our X with respect to said squared all Cosby said Have you seen major energy function like that anywhere as before mom is therefore instead of calling X X I call it off foggy of calling stigma
said Mark a recorder acts durative I thought our stigma as a warning in space in excess of field this would be the energy of a simple matter were simple wave field a simple way field odd and it would satisfy a wave equation was physics is exactly the same physics as always like physics waves run up and down this should make trouble by the only thing that you have to keep track of is what happens 4th support we have a wave but she was job suppose and we have a wave to more so after a right where Abbas away when it gets it doesn't keep going it bounces off the boundary condition either do rich late war going on boundary conditions will cause the wave the bounce off you know what happens so the wave when you have Nollamara boundary conditions yacht gives footballer it gets flipped upside down in order to keep a few feel at home among Member condition but it is thought that just reflects those same America our just reflects left to right but it does reflect the bounce back so that's that's the basic wave equation weaker right down the wave equation that corresponds to this how would you write down a wave equation while you're right over grungy and were granted it is kinetic energy potential energy and then you would work out what granted equations for grudges equations would just be a simple wave equation and a writer because swimmer Rivera needed but it was would just describe waves moving up and down reflecting off the interest and was Palestine OK I'll tell you exactly what towers remember that we had to rescale with time because of the small band of Clark's on how was really proper part of its time the prime clock that's moving with on infidel menagerie was a clock that we boost together with the rest with a system that Clark slows down so we're not measuring real time in in the rest frame bold measuring and that the system of interest moving next of a clock and so we can compare the internal motions were clerk of Africa's tell us of this style so it's time except slowed down so we keep track of the you are the kind violated internal motion propre gripped by bus with powers at proper time but you guerrillas kind from the point of view of the nonrelativistic go picture price about energy Ma restrict energy if this strain is at rash and trend this plane and Marija bearing that motion of and the total energy will be the square of the merits of stress eventually uses for over a year for identify our this energy with square maps from America new is the shit amassed only also point of reform is not nearly as his yacht air at it's a complete and all our object that borough furry even need to specify we could just just wages right arrow was formula he will write down formula is that what has made human disease Aegean these cargo icon energy reserve our call icon who has nothing to do with cones Our firmer smirk more suited major we're road ADDS if you get the same thing what is none earned a point is cold call Collins is reasonable like coal crime where are the handles all those are cause the wages coalesced around right I find that shift is your schedule he has none on servicer a its from the right word would be white front yes some rump terminology it is not right but it too like clean so world brought about purposefully did you want to do for hake is not pointing giving you wrong terminology explaining what's wrong climatology might figures of 4 P E is gaining steel that is not what this is this is not right for Marie Ricard our preferred a was a misnomer White from toward a bigger better word in your head he broke bricks Russia will not be easy where he did have like Concorde but his white Concorde notes were white called court she also talked about something which I recall white frontcourt two-year reserves so what's not artworks not the Loyborg OK 1 nite last pouring before we well I think what I think all of this we we win over West partners more completely now now I think I was something within engulf the good anyway do we talk about the boundary conditions at the end the strain why they're going on and now they're showing but ropes to act now this up from a blackboard here but now we want to come the boundary conditions from exactly this issue incidentally let me me reiterate again as another terms and the other 10 is of the same form except with wiII replacing X. Marshall string involves knowing book X and Y are functional should point on Australia's labeled orders and excimer wife so we are both Rivera must forget about their about 2 quarts isn't real string theory orphans makes use of anymore accord them many more coordinates we're just going show X wiII Mr. letter spoke Tuesday because we're ready Uzi W V resource for further Jack Harloff far-right let's talk about boundary conditions on me and those things the boundary conditions are actually determined by nothing more complicated than the law they say why we are to of what we're talking about relativity works because of the two-dimensional analogy two-dimensional motion strained his laws Our correct now Newton's laws apply the white no user laws applied for the Mass Point no laws apply to as boring tell us how for each accelerate given the four-star what's the forest are 80 mass for now tentacle Mass point is being cold from the left as being pulled from the right so we forces they know they won't necessarily balance but it will be getting a 4th from left on force from right there 2 special points which are only getting forces from 1 side the other boroughs of the mass point end of the Strip so let's concentrate on the Mass Point in the Australian and then he was a West Point on the strength of the ultimate point restrained in India's pen ultimate point and minus 1 and that collected by spring now want the use of force the x component of forests on the end point
of the spring ox lock box lawyer tells you that before Sunday in the string is the displacement or the distance between the end point me and my 1st point when other words before us on the strength before Sonny and point here is proportional dealt x 4 West I was just caught dealt X but means the separation between the end point to me and my his 1st point that's the worst on all sorry about the spring constant spring comes Kate measure member Katie is very large it grows like me and I won't bother putting in the party's for and the end of the last molecule from is dealt scales or gulp X times and now was dope ex Delta X is approximately and the limit of a large number of points it becomes exactly the derivatives of X with respect for Signora lawyers doctors said 9 dealt the send felt the same Gore's like 1 over and solve cancel I don't care about the commerical constant factor of us on the end of the strained there's going to be proportional but just for proportional but the a derivative of X with respect as well as interim Alexis Specter said that some might be amount but the string is stretched nearly stretching crack community if the signal 0 0 would mean the end point to me and my 1st point arrive top of each other 1st separated than those of the expertise there was a happy equal by by Newton's equations by nuns equations Alaska equal the analog amassed the acceleration and
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we have something a little bit bizarre I'm multiplied fight and
multiply by in Webster really intermediate multiplied by we find the
acceleration will go off to infinity and gets larger and larger acceleration should not me that doesn't make sense but the acceleration of Ian point of string uses of wildly wildly violent the answer that the right boundary condition at the end of the string is my mom numbers so not that prevent acceleration quite physical an author prevent the string from having an infinite acceleration of the end point you imposed 1 on boundary conditions VX by a stigma 0 at the end Port cash but but but this was in fact Bugel argument about a reverse merger was replaced level crushed OK so now I have a fairly complete system a system of Lagrangian well-defined members should question which we will take up after about 5 minutes is how we do the quantum mechanics of bachelor after where after the quantum mechanical energy levels worker computer corner chemical energy levels and having computed the declined chemical energy levels will or something about the masses of these objects that ball so we mount we give Everything else off a blackboard and say Here is 8 system classical system more probably make quantum mechanics and the however we refined its energy levels now what do you study this spring and particularly the study of quantum mechanically while a fresh thing you do is you write acts as a sum over co-star from Alexis axes functional said white X function as did my wife is a function of Sigma stigma rose from 0 to party what can we do 0 2 maker Our concrete investigation of this weekend Fourier analyzed acts you're virusoid right nor on boundary conditions mean call signs and so we write EXAFS of segment in this former also why absurd and then we take these 2 expressions but concentrate on X and we club limit is to go around or energy of our republic mentally energy allure grandeur of the kinetic energy in the potential energy and we described as a storm in terms of the comedic and potential energy I was set no degrees of freedom the new degrees of freedom are just these actions on white so straightforward West let's begin with the kinetic energy to read a great deal of about sold for family and start with a kinetic energy at work out for you and grand opera all men who potential and what I have that will have something a Wickham were free throw more easily than those for here OK so we take X and we write this 1 he what text and not about me kind derivative sorry about what Teixeira about what X . X about certain X also ignored escaped by differentiating now the cosigner and send doesn't depend on sorry time at all which cosigner inside Iraq has not time-dependent have time-dependent yacht something is staff time the strings know West time-dependent is the Fourier coefficients the singer dependents it is coated in these co-star the Colleen dependents extras also a function of time otherwise Wentling strength a wiggling straining the position of every point on is time-dependent Austria where roles where we put time-dependent before what would prevent so think on the X Games as being object which have a time-dependent Cabot X and ah sorry Texas 1 . woes that that's going to be some money and we're just going to kind differentiate were a different shape this was the kind of person us as an equal 0 . 70 of alright it has X and . time derivative wrecks and plans call science Malek striker kinetic energy right kinetic energy we 1st have to write the square of the exodus is saying is BXT cow act pointed however the square it the square that will give us a double start to stop the X square will be a bubble son started from In equals 0 and am equals 0 . very a M Nichols 0 . 3 x stopped and co-starring not just rewritten this multiplied by a multiplier joined 2 steps no OK X and co-starring and said Price sought access of sorry X and staff which group of together X area X M . co signed and stigma cosine and said but were also instructed to eat degraded the integrated said the be stigma bill is saying dependences in these costs so it can interval of Acadia he said no and the last step was to provide by 1 or 2 Price won over Tobias a convention to convenient convention How about this integral With the interval has 2 kinds of have terms with In the end end not equal to 0 and other terms what I'm not firm would end in an equal to 0 we have to be a little bit careful about them off 1st of all after we've integrated over Segner there will be no alternatives would end not equal pay him a orange we're going to get a single song not a bubbles audibly in the single sum 1 to party not a first-term come from an equal 0 1 M equals and that is just x
0 . squared provided by true now no no to short all not even any part yet to win on Friday where that comes from a came from a piney canceling a pie downstairs at 1 is this was a 6 0 incidentally you have any idea what I 0 stand for if the average position right but that Mexico City Spurs center amassed pollution which orchestra restrained in addition to everything else envisioned all of its wiggled it has a center of mass motion center Mass which is a position this is just the kinetic energy of the Riemann everything else is associated with these cattle vibrations and promotion this is the peace that has to do with your overall Moshe over strength so x on unwise euro on nothing but the general brokerage puts suppressed why just what Quebec and no 1 on the other the other firms are a son from equals 1 February but 0 have plots for auto price X and about squared a remote terms with end not equal to an and in what state integral are cosigned squared orange I want to write a pilot so the pines cancel an organ extra factor to downstairs which Omega overall a quarter Hey OK sir Mrs. Jenner may at Shea start and this year has to to do with the internal relative motion are of these little constituent elements of strength not that turned minus STX by said let's work that worn out I think I can erase so its guerrillas from the what's work at our about BX by said direct stop it's just the expertise and we start with this formula here bats X and now we want to differentiate it would restrict the stigma what happens when you differentiate a cosigner stigma with respect the stigma are my nest sign becomes signed but was also a factor right shows Kim he s co-starring becomes strike so that's a derivative of exposition sector said not not us X involves call signs derivative of X with respect to stigma involves shot if 1 is a rich lady of North Miami OK so now right down to the square of the expertise will be a adding man class a song is always in an inorganic because we square just over square at the minus sign goes away hired in plans and arms X M X signed a asserting cited here say were signed him so well and sketch integrated for a missing anything yacht for price OK what see the of the integral of sign in the stigma prime starring and said a stigma from 0 this was a 0 SC 0 0 equals area and the chemicals and far richer wary of ship poverty OK so again this or a heads-up where constrained at any cost and my be integral into rule be 0 on equals area so it just and squared and squared and pilot to erect trial 2 a careful of apartheid and put before downstairs so it looks like this same factor of a quarter a process of not squared it has in squared injectant squared it would owe it all graduates writers or grant the minors for the energy of course so if free terms or together the 1st time mosaics nor the 2nd turned and also Pinter serves at Connecticut Energy this is a courses also kinetic energy comedic energy of a whole string and end the something which involves Exc would let's go back to here this is the energy of a harmonic oscillator a manager of a harmonic oscillator extort squared MMX square evidently each end her the low grungy grungy and Mama Cass choice programs harmonic are sorted out Free chair what frequencies of oscillators that this system as it as a collection of infinite collection of infinite number of harmonic oscillator serves goal you'll have a discreet accountable infinity of offspring with volleys of course is just a harmonic Serbia restrict your harmonics over noise what we've done is to rewrite the law grungy and in terms Of the coefficients of the individual and not away find 1st of all and that coupled to each other at the end harmonic not coupled to to the end collection of on none interacting spray interacting harmonic oscillator West the the frequency of each 1 of results what's the frequency of B and passed away yes but you could read that offer France here I wouldn't change if you Pro 40 at the frequency with stood up in both places or with people in both places Omega Is the frequency and Omega squared is the coefficient of X squared is the coefficient of X squared can right was a collection of oscillators each 1 with a frequency or may get an equal pay what about Omega 0 what happened
to it you realize of course is the restoring force of the afterward the restoring the hoax law stretching energy of the oscillator restored for this not only starring boss of the center of massive clusters of restored under arrest on matter of free the Away is nothing holding onto a center of mass so there's no restoring force here just text or about squared and the internal energy This is the vibrational internal Remigio strength is this which has to be identified Wurzer it we lost it with the Mets squared the true man square of a relativistic string internal energy is the internal energy debate the 5 the squalor of the OK with Milo derivatives just as we Rick its stitch trivialis just the overall server mayors motion this is the interesting part he was a collection of harmonic oscillators now I've written only half the other half Our why so another identical double E infinite collection of harmonic oscillators each 1 with frequency and so a has frequency 1 too much low-frequency twice the frequency of rectal has 3 times frequency and of course simply a different different harmonics of restraint a whirling kilogram we quantise with archives that we subject end of the process of quantization which is very easy from across Florida in fact we don't even really have to go through the whole exercise with before we just have to remember that the energy of harmonic oscillators are quantized search for our 1st ever against you repeat what the energy levels are going to be as good a lot of energy levels and the number of course but we we can make some allergies NATO with a start organizing I think tonight won't write down creation a annihilation operators and let's for the moment suppressed White will combat Hawaii was just concentrate on X on that will include the White of the White each and there are 2 kinds of oscillators oscillator along the x-axis off are short of water so let's see what they and they have the same frequencies BXR afterward BXT extends don't have frequency same frequent the axes and the wife of prickly a replica of exactly the same thing and got on the weather harmonic vibrating along the x-axis or wire by combined together you could make oscillations along different factories so let's see what kind of energy levels system will have on the bare ground state where none of the oscillators or excited all also in their ground state let's call at old ground forces not correspond to the back doesn't correspond in a space corresponds to a single strained with no energy expectation photo stringer world form of particle experimented you've got a moment and ended has a gap in the spectrum there's a gap mixed energy level because monoculture leaders have gaps villages quantum Wal-Mart picks song energy sunk by Nativity excited saw as a part of our West 1st excited state the 1st excited state you can create by creating 1 unit X energy I promises that correspond to it corresponds to the requester remember energy levels I should write out a formula the energy of a harmonic oscillator it goes in cage borrow maker Omega different and a different number of quanta what short Q. What number of course a number of integer a different integer music these integers labeled different ocelot but each oscillator itself can be excited I'm so remote we take energy are a hostel use Q. be integer cue cards a frequency would go step plants constant 1 works forget form part of the war and I a guy is just a sofa and oscillator it's just Q would kind and and or the energy of and offshore a so what's the 1st XI patient you can excite you could site I X all-white the lowest possible and equals 1 a waiter could excited 1 time you could do it 1 quart of better How much energy doesn't have Q was 1 in his 1 it has 1 unit of energy in some units we have specified unity and solve whatever the ground-state years bidders to state amount to states a barber whatever the energy of the ground state would not specify the ground-state energy was 1 it above it the Brown state above a bar to state waterways to states BOX you couldn't get site along the x-axis over wire exodus corresponds to a string vibrating or reveal the are sort of I bring along the x-axis or Y axis of course linear superposition of Labrador so a 1st warned you
about 2 state a multiplicity of 2 our operate more a multiplicity of this 1 is unique a multiplicity of this 1 is to possible farmer from possible states what's what comes next well you can go on a number of things you can excite the always fossil later quite you correct excite the lowest XI story priced hooker excite the lowest a White House story twice you correct site the lowest X oscillator and the lowest why our Sauredo each 1 boils could you do excite the 2nd afterward world that will work this out more carefully next time when we introduce creation of annihilation operators bought you can excite either the X. over why after Fostoria warrants that will give you 2 units of energy appear or you can excite the X lost afterward a once in a while I was passed away the ones that will also give you 2 units of energy but you could also cite the 2nd oscillations wants you XOY I will give it to more so altogether there are 5 states here 5 states X excited place white excited twice X warns why want or the 2nd oscillator reviewed BX white white white for if if 1 OK camp with we can take these works tens of creation rich and operators 84 X B for white typical lost oscillator a plus warned 1 hour twice squared but of but at bay Rugby 1 plus squared off all this is exciting X twice a wire acquire a plus warning b coughs warranted always free and Marrakech 8 2 plus 4 beat 2 plus 5 state checked by state press half exercise count the next 0 next to levels country states are the role Europe bearer exercise for the next but the point is the most important point is that there are gaps between significant high significant gaps between the states so discrete particle like arm will work out a little more detail spectral and then having done all of back was tried a white identify summaries particles some of them are bad guys we want to get rid of others are good guys the road but are we recognize we say those particles aware wife Harry that will want to close British study right now our next 5 minutes 2 minutes what the relationship is called open strength Europa strings cause Fairbairn they have it imports the basic process of interaction of string theory is stringy and coming together and joining to make longer spring brings go together if you like in the context of hate drama as on the strings have quirks of Ian's quirks court at a quarter not that's not true flows from string theories Olympiad thrust and bought for ordinary drums quarter hit record whatever happened was a quarter-million back work come together and I await and create a long string of basic interaction process which we have taken up yet of discussed is the joint and splitting of joining can happen splitting can happen at work joining a splitting of strangers now here's a strength or by a self and as it happens the string executes of fluctuations which happens to bring the other around close to original set strain or sees another and the string only here doesn't know it's part of the same strain all endorses is found the mother strains are another and another aide to annihilate were so wants to introduce string interactions your committed to something new committed closed strings every theory of strings always has closed string Yukon right series of course strings it don't have open strings but you can't write theories of open strings that don't have cost so something more general quorum of forwards general something that you have to make a prediction about string theory it would not be that open string but it would be or strings is nowhere close strings which in theory there are ways around the open so the next day we have to do it after we were out and figure out what open strings like we work our close I work over the answer right now now open strains often behave like for top closed strings behave like grab there's no theory on strings which doesn't have a graphic on the off theories of strings which don't have but for
more please visit us at Stanford . EDU
Schaft <Waffe>
Magnetisches Dipolmoment
Diwan <Möbel>
Satz <Drucktechnik>
Konzentrator <Nachrichtentechnik>
Nebel <Astronomie>
Angeregtes Atom
Knüppel <Halbzeug>
Masse <Physik>
Dreidimensionale Integration
Relativistische Mechanik
Ausschuss <Technik>
Negativer Widerstand
Concorde <Flugzeug>
Gebr. Frank KG
Öffentliches Verkehrsmittel
Virtuelles Photon
Band <Textilien>
Source <Elektronik>
Closed Loop Identification
A6M Zero-Sen
GIRL <Weltraumteleskop>
Elektronisches Bauelement
Römischer Kalender
Walken <Textilveredelung>
Holländer <Papierherstellung>
Elektrisches Signal
Magic <Funkaufklärung>
Maßstab <Messtechnik>
Optischer Schalter
Zeitdiskretes Signal
Stoff <Textilien>
Elastische Spannung
Schwimmer <Technik>
Optisches Spektrum
Radioaktiver Zerfall
Schwache Lokalisation
Maske <Halbleitertechnologie>
Fuchs <Panzer>
Elektronische Medien
Direkte Messung
Avro Arrow
Echo <Ballonsatellit>
Spiel <Technik>
Fuß <Maßeinheit>
Mark <Maßeinheit>
Quantisierung <Nachrichtentechnik>


Formale Metadaten

Titel String Theory and M-Theory | Lecture 2
Serientitel Lecture Collection | String Theory and M-Theory
Teil 2
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15122
Herausgeber Stanford University
Erscheinungsjahr 2011
Sprache Englisch
Produktionsjahr 2010

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (September 27, 2010) Professor Leonard Susskind discusses how the forces that act upon strings can affect the quantum mechanics. He also reviews many of the theories of relativity that contributed to string theory today. 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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