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# String Theory and M-Theory | Lecture 3

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Erkannte Entitäten

Sprachtranskript

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Stecher University I have

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got pattern unlike the pattern of starting lecture with a couple for Brewer Mary's sometimes reminders sometimes new things by now I've taught things summit current that of I may remember that there are reminders you may think their new things about our lesser puzzle over things will go all really quickly really really quickly is a hallmark oscillator that exceedingly fast but the other thing I wanted to go over before we begin is being of massless particles the spin on massless particles is a little bit different in some ways the spin on master of particles the basic difference it is the massless particles can never be brought to rest removal was pure white long boundaries do more with the speed of light long more store over the speed of light whereas particles were massed you can truly down bring arrest and that makes a difference not certain properties were banned our illegal 1st spins or but to harmonic oscillator 1st together are very quickly you not all arms is going to adjust the set the euro the notation mostly to get the notation consistent of before we're getting our energy oil grungy animal harmonic oscillator is ext . squared and then for reasons that is if you go back in you know about the harmonic oscillators the describe the oscillations of strings 1st of all they have little index ended which resume which had to do with the number local science coastline and stigma it was the and that their co-starring comedy oscillations company oscillations over the interval from 0 party a string makes so there are those acts of man's Avi Cohen describing various crimes of oscillations that was there and for reasons that I have no special rules that prevent it turned out that was of foreign and the nominated here that was kinetic energy and then the potential energy are energy at plus sign near the potential energy was proportional back square saying X sub and Square looks like we were going to do with the next summer and squared provided also by 4 and they are sitting in front is of the square on the frequency of the oscillator and Alice actually just in the square and that's what we got 4 energy of these in the mode of oscillation X Fox William Osama kinetic energy pose potential energy Russell grant Tamil energy quirky we will grant GM is X parte and square enormous operating in but remember I'm talking about acts of an ex parte square miners and squared x squared for water around him from this you go work out bronzes equations of motion and you can discover the oscillate oscillate with frequency In our next goes current wanted the choir mechanics awaited got acquired mechanics is the 1st construct the Hamiltonian then do some tricks were lecturers construct the Hamiltonian remind you hopes Gulf Richard corrected me will fail text of grow 4 OK so what do you do the 1st thing you do as 1st step in and constructing Hamiltonian is the 1st step is to replace the X Xbox by the moment moment a peak of remember the river grungy and with respect to to the velocity is called a momentum so this faced with this particular factor of 4 here the momentum would be ex parte provided by 2 why because why different you with respect textile to it cancels nominate through a bus 1st next that is you rewrite the energy In terms of instead of X not you use so we just have song for ex parte Florian wires and give us that gives us 40 x 5 square before b squared the right of our form so this will be P square with Norway for revenge the defense origin of change plus and X square 4 where we are Call the Hamiltonian 8 Kwan mechanical Hamiltonian the except that it becomes an operator and operated in the face of State our track is always the same for harmonic oscillators that's just erase all of this is covered our starting point right here you look at this sum of squares a square Was square you're right when a square Crosby square a square miners be square white a b cleansing miners about a square is quick past be squared that's a plus I'd be times a day I work about may square side terms plus side is class you must be squared across rows away so you look at this sum of squares and you realize it's a sum of squares In what will term square on this square of an X over to write 5 an ex all tool Plus RIP times in X or to minor side as some next step you give these 2 things names but not quite pushed wet think about commutation relations between these tools of course 1 of these is essentially the creation operator or erasing operated the other 1 was essentially the Loring operator I have to figure out which is which and the numerical coefficients nor make them creation of Irish shoppers burst over here your remind you connotation elections between annihilation operators of combing my nest and creation operated as a plus 8 plus 1 Max raises the energy by 1 unit off a miners who loathes the energy by when you're at this commutation relations just 1 so that's how we identify creation and annihilation operators that operators which are complex conduits of each other these 2 of trumpets 104 Java commutation relations worn so let's work out the commutation relations between these and just check if freer warned we will get what I what the commutation relations between X and PD remember complicated between extra pay I'm setting but 1 Wells precise age are quite go back and a remember like natural XP miners PX His are gonna healing have X ray of of in all the tool a minor side arms the commutator of excellent piece elects 1st all 1st all what about the curry favor between isn't it but well because everything anything commutes with likewise the only turns on a cross arms X with P and Q X fights

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over just look 1st term here or crushed here that will give us an auto career or Simitis on from here but then become fear of X with he is plus I worked for apply but plus side altogether makes it just in order to so far as commutator of this guy without 1 on other sort of us once again I see out I see him and all to Iseyin or ii but now I have the keys and the actors in the office of water the result exactly the same thing does not 4 0 restaurant and warned do the same thing his doubles of this just gives us and now I wanted to get 1 but I got it well I don't think I just don't violate both equations both of these operators by the square root of in other words provide each of these by the square root of end likewise over here 1 over square Romanian met eats up In here and that's all I remembered if you're desert island in you can't remember what combination of taxes and fees are creation and annihilation operators compared with he had offices and was wondering if I had my creation and annihilation operators interchanged I would give my 1 so Oh we can identify the annihilation operate a minus was won over the square and that means square of and X over tool plus I over the square root and he and the creation operator is saying the same except for my son but with I just like to remind you the creation and annihilation operators the creation operators increase the energy by warned unit and when I say 1 unit I know me when you hear the frequency of this offer waiter years wrote in Europe the ruling Uri which is what comes which XM and talking about we can now Greek amount would back guardians as a creation operator and annihilation operator for each oscillation for each possible oscillation of the strain and the frequency is an the frequency of course is the amount of energy that edited by citing the afterward awards if we can work out our what the energies are working energy spectrum is and do require the character of the oscillators and that's basically the spectrum of these strings will come back did the other thing to remember that I've talked about last time waters if we're talking about strained in realistic number of dimensions of space and time namely 3 dimensions of space one-dimensional harm ordinary based end we think about these things moving with a great deal of momentum them z-axis than they are really only told dimensions of oscillation could take into account another time maybe even Leydecker they'll talk about the directional oscillation how would determined by the others but these independent degrees of freedom away oscillations in the plane of somebody or watching a string going down the axes and I'm oscillations altogether well oscillations along the X or with the X-axis that with he found that direction I told you was acts for the ceiling of white off the oscillations Kerr currently X direction they commit carnal y direction so that means it is another term I really don't right before Hamiltonian here I should have written plus another term where X is the change were exactly that means quit each frequency will be colossal rigors the oscillator oscillations along the x-axis in oscillations along the line but given names but call and 8 for the x axis and B for the Y axis say equations we might be plus equals 1 end same equations here except for substitute they will would be X I will put wiII were up you about peace sorry for a repeat the others these Aleksandr piece why the momentum and the exploration Malaga 1 while already back from this equation here Luxembourg last Koreans and symbol in here also rose here wet solve this equation X and used jaw equations for lesser governors to equations for 2 1 known X and please How can I saw mystifying network exit In terms of the creation annihilation operas is something will use later not must show you tonight or not but what I want to write is the core warmly position of restrained interns Of the creation of annihilation operators that will turn out to be quite useful so I know it is if I want the isolated I want to get rid of pig that's easy just these 2 equations high Mr. will cancel solo yet we get a course course being minor so we should index here for and oscillations that's equal to produce just an Mosquero and sorry and are square root of end Trans XAN Square a man forums X game what so I was refined wheat and corn it is just the sum of creation and annihilation operators provided by release Justice glutaminic you I remember now let's before we leave Marque are sober but just remember with his oscillators came from a cave from writing X of stigmata remember what at all was a member 1 similar to it why cement musical from B sub and plus plus piece of and my of a square Rivera and basic Bentley as I'm sure most of you have heard rumors string theory doesn't work very well wrongly 3 dimensions of space oracle confirm a rumor recovery we need more anymore and I mentioned we just dead more quarter to forge UK bearing coordinator for you run out of court sorts that's the reason Samarra brute strength theory is in 26 dimensions address

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shot I actually actually knocking Mayreef but remote Exocet XOC MY of certain right mirror string more around the X-Y plane will rebound reactors and and there along with spring reinvent a parameter called sued not sir oil Ross 0 fighting for and it would take any point on the string characterized by a stigma that signify point if you like and has a particular value of X and Y VXY taxis each point the string has therefore vocation Whitcomb Texas wives when we use our freedom who'll Fourier expand X and Y refineries right reverses the and of course are in the same way kind of caught X sub was Axel X him it is simply the Fourier coefficients on and cosine and the Fourier expansion of extra likewise white why do I use coastlines enough signs of the boundary conditions are such that the derivatives next Obama should be importing that's the conditions ripe for philosophy and not we so strong enough to cause infinite acceleration but this is the right way To Expand BX likewise with the Warriors some additions and why some and cosine incident where and that all weaker alight these coordinates in terms of creation and annihilation operators right there some already and Our a Class a and Class minus over and always square root and branch cosine stigma and the same thing down he'll of bees civilian Bob it it was an interesting well the best way to manipulate the quantum mechanics of harmonic oscillators through the creation and annihilation operators if you want a find out where particular operated postal quantum state your apply the operator who applied in a form of creation and annihilation operators you immediately see work hours increases the energy decreases the energy and that of course will be very useful before moment adjust equations writing down black our because we use for Kellogg offer these again so if there are questions about the right now neither would do USB philosophical questions about the meaning of life also not no extra men's arbor Fourier series coefficient all of city's independence is here this is what the only extends the I'm part of this kind this is momentum frame which we gave me to we called Powell when I write Box box always refer to a time in the nonrelativistic analogy which I did call feed because they didn't want to confuse it always time and the rest store called course former leader called it now that means every point on the string is not just a function of Sigma is also a function of how it actions which are dependent on how why they depend on how because they're oscillators they are sort of library reactions vibrate with oscillations they depend on time read call signs of fingers those are the ones which depend on other any other questions for both really struck off they were those years prior to last now use you areas you with a plus must and was somewhat more Warner here I think we could trigger check it out whitey a figure subtract energy why what makes you think what you they not for what course roughly what companies and how could you see what happened you look at them and see which way you may be right but how could you look at and I don't think you're right check check and say he a at Cameroon Mama complex summer storms CIA could look bad easily and see which 1 raises the which went with him he light on Hey this where side I don't area you have to go through require mechanics have to check whether the commutation relations want or minus do so but this might sign doesn't career covers something Brooker Bob next From assets should be better way of absolute revered not may all OK so what how much energy well I occurred because of what I have to say next question next preliminary the spate of mass and master of particles I movie components for a does a system of spin J have 1st thought was 0 they have a right to jerk performed I saw spin J has to J. postwar history it's been 0 romp has 1 state and 1 of here or there the case 0 1 state space and 1 3 states spin through 0 5 states and so forth however metals particles are special don't fall into this category matchless 0 0 state Omri has warned state biomass for spin 1 state has 2 states are free on vessels 2 Space system has 2 states are fired arrest was 3 Tuesday which is exceptional on an hour to the euro's 0 was only 1 state all have to for metal from how we understand that how can that be of questions how could be consistent broker because I will tell you what the states are really KCS particle a Maginot moving along a stretch Max for the

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moment it could be a mess or master unless opposes moving along certain taxes the makers of the component of spin along that was the maximum that let's forget 0 0 here where only has warned state perplexed go was and wondered what the maximum speed along any given access while answers the spins along any axis could be 1 0 and minus 1 shot off a sprint through rule 1 0 minus foreign miners in each case there was always a maximum spend in this case a maximum and minimum me because they are our maximum positive and a minimum which would be negative buzz can be thought of as a states with this spin is maximally all along the direction emotions world maximally of opposed the direction of motion not as far as whether this is along the direction of motion or against the direction of motion that's really on or arbitrary choice of right here and all versus left him rule spin it has to do with rotational momentum is a vector along an axis my right hand my fingers Kroll right-handed with my left hand across left him a spin that way is considered cars available along the actress and spend nite away is considered a negative along the same access or simple reflection symmetry have you if you have a lot you better have the other the have right hand and were sold with reflection symmetry you will have a West here was arriving in the particle would spend that way the OBO left him away and the other way so if you'll have a particle moving down the axis with the maximum component of spin there must be on another possibility of particle must also be capable of having the mainly women from there was any reason why have have been in between so let's take In 1 case the spin 1 case the 3 states along axis have spend 1 0 and minus 1 goes over 3 states we see because of reflection variants that has been long reactors must also have Spoon minus 1 must also have a spin 0 well read against component of spare V component of spin-off products and we have resisted depends on whether the article is massive all missionary wife deposit particle as massive to begin with and supposing it has when argued that the must be free states it was moving down the z-axis and there has Spoon not all only was exorcist rotating away no magic bringing that particle arrest or better yet not only in such a way you keep up with the particle is now moves Pidilite gum s you keep up with you look at the particle particle rash of but it was arrest so now your discovered Represa particle arrest but moved downward president change it's been something this way going on that access you keep probably will still be rotating worry so it still has been 1 of the longest active spin 1 along with access but now that you brought up the rest you could a bit and then boosted back up in that case you've created a particle which is 9 billion our which is neither has mine will spin in that direction on the new direction but which is now has Spain's pointing out on the blackboard even the particles moving found erection and you did that by 1st of all bringing particle arrest this saying if you're living in a world with rotation variants of the particle arrest who's been pointing along Baxter's must also be possible however pointing a Womack relative think about it is you could imagine if brought arrests boosting along the perpendicular axis of evil lady you wind-up with particle who's been this perpendicular to the axis of motion that must be the case because you can't bring a particle Koresh and an accelerated in the perpendicular direction without changing its OK so our conclusion Our if you have a massive particle it must have all 3 states spin-off of Axis but along bad actors and that slalom off you have to be able to make up spin along the perpendicular axis million if the particles may there is nothing you can do to bring it to arrest any Lorentz transformation will leave moving with the speed of light you cannot bring it rest and then rotate out from under the spin and and boosted the direction the implications is that of particles have 0 man perfectly consistent all they have is maximal spring and minimal spirit and nothing in between the equivocation Are you know all this and another the photon what guys the photon who the polarization of light the polarization of light can be described in a number of ways to commute described as linear polarization circular polarization of described it in terms of circular polarization circular polarization and we call sponsor of times depending on the circular polarization rising in the left-handed corresponds to what assemblage of photons which alive that right hand spinning or left hands and arms there are no which are not polarized little polarization is always perpendicular to the direction of motion it's you I the left in a one-hour linear polarization was linear polarization armored carrier now what Lee of polarization is a linear polarization if you have William correlation it is not as you or angular momentum along the direction of motion it's a quantum superposition of the 2 possibilities circular polarization right and circular polarization from laugh what's the difference between saying it's a it's a linear superposition of polarization and saying it has no angular momentum rural has equal probability from being spilling right and spilling left so does not mean that it has on average knows yes it does mean that on the average most perfume measure the angular momentum Rafael Tom what I will be right or left him you will never measure of photon which has low angular momentum or erection so that only states of the cars and 1 other

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point let's go back to rule are Leopold variation of photon moving down actors can be polarized here's a plane perpendicular promotion if you face and own plane polarizer it can be polarized but very along the Y axis polarized along the x-axis Apollo's issues of vectors as a vector which is always perpendicular to the direction of motion before it's always perpendicular the words that the polarization as always transverse there's not polarization of the photon with polarization with electric field barrel points a longer direction before prices a FOR by Maxwell's equations Maxwell's equations Terry of electric and magnetic field and therefore the polarization of cars are perpendicular direction more solid you could have 2 states of 2 2 different polarizations you recall the X polarization state of the of photon or recorder the white polarization for our neither of them is circular polarization What not circular polarization circular polarization is a linear superposition of quantum mechanical linear superposition right-handed Colorado hand polarization which means right here got sick or perverse state X plus Kansas State wiII I have no I won't say they left there is X minus Iowa is State of the Art of America Europe Roberts Bristow circular oppose and the circular polarization purer circular polarization circular polarization is a quantum leap superb reach far on a circularly polarized photons is a quantum superposition a X polarized proton award for 4 years supposed put the why are supposing just considered X plus white what kind of photon that that would be less Ely polarized but a 45 degree angle to a actually sold for you are allowed to do was put real numbers here you would make linear polarization which could point in any direction perpendicular to the direction of motion for but having I is available gives you something new gives you circular polarization if you or you lose as your basis state circularly polarized state instead of the linearly polarized there are only 2 of them so the angular momentum of a as you go along the axis of along we are composed of reactors and there is no followed which has 0 angular momentum along the same is true of gravel path a slightly different way grad icons osce Pinto object if there were a mess they would have fired states program can't only have to state they are roughly speaking and this is how would you want to keep mind too good you will draw a graph icon moving down the axes with Maximo angular momentum tool units some angular momentum moving along the z-axis is mathematically the seen as process columns each with euro with 1 Murat from earned your momentum from taking photons sticking them together but move down the axis of the angular momentum of the 2 of them in the bakery those possibilities they could both be circulating In the right-handed cents and the angular momentum would be 2 years so they could be both circulating for the left the angular momentum would be miners to your would you don't have as a grandma time which corresponds to a on right polarization and left grammar tons have only 2 states maximal rotation when their action axonal patient over their actions and mathematically you can play golf just having pay Harare 4 cards of right-hander multiply this by a mother for part of the same kind and think of it as a gravel path that will be useful for use later think about spin tool by combined who's been wonders what's that gave out here because he was so if if not well nor to of Barry which studies find diagrams from gravity times by thinking of members roughly speaking squares refined by Iran's 1st warned particles but mostly I think it's trickery there's no real sense in which a graphic who from Christopher combining this spring to 40th think of any of us which particularly useful that require royal number vote on house or don't any elsewhere momentum momentum as energy and spent I'll does not charge remember remember of merit reveals that I had a job that EU foreign bond has he said before autonomy given moment only has warned that arguably and I would say he is that what they saw are you continuous spectrum arrest Mercer or if you light a continuous spectrum for a given moment Her popular replied off particles with removing inside Iraq copper American painter of some sort it doesn't really have continuous spectrum has extremely thin spectrum for what I mean by that is if the entire partners at rest there a lot amount of energy levels said only we can generate a continuous spectrum just by moving it but I was speaking about the pot at rest now folk dance troupe can be addressed Secale quite user definition use a different definition of forgiven momentum where terror now we know something about harmonic oscillators some us the contour is a spectrum of strings the low-lying spectra which going most lying spectrum meets for the 1st few very show just remember energy and former analogy between nonrelativistic physics in relativistic physics energy becomes not indices quiet not the mayor of the square of a mail so why we speak about energy of a

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vibrating strings and we em all opt for find total energy of a vibrating strings they corresponds to the square of America's Malcolm self just Bergeron OK what ought be discreet energy levels of restrained talking about straight-ahead arrest me to say it's not moving in the perpendicular direction rural Robbie energies while we just a look is just a collection of harmonic oscillators let's see what we can make the 1st state in the lawless study is state which is annihilated at the ground-state all the annihilation operators shut the state the ground-state all bees and always or annihilation operators Kazuo 0 the state in which neither of on only is oscillators of all excited what's the energy of our state well if we would only ordinary quantum chemical Hallmark oscillators we might say it's a half each borrower may right around but let's let's leave it open from let's not prejudice the answered Our Lives leave it flexible for a moment and call man's square this is or was therefore does not stand for the vacuum it is not empty space is on excited afterward as a particle particle moving down reactors but it's got it's in the ground state of war with relations say it has no oscillations because every harmonic oscillator 0 . oscillation because of the uncertainty principle with some fluctuations but everyone the oscillators as as quiet as can be consistent with our mechanics so this is a particle and I'm not going to begin by prejudice thing about what the massive of a particle is I'm just going to court and Nora square whatever happens to be the energy on the ground state for the square of the restaurant we'll know what it is we don't have a cool yet for finding out what is or whatever it is when we have only 1 we're excited we will increase by integer amounts of energy selected how we can excited what the lowest energy excitation was the cheapest thing in energy weakened due to it and the cheap this thing in energy that we can do it to it is thought apply the creation operators erasing operators but the lost 1 a wonders remember each oscillator increases the energy by it in units if we want to preserve lowest energy we want to hit it would go with oscillator or of smallest frequency of the oscillator of smallest frequency corresponds in equals worn out we had with a someone the fish what energy Ovapa straight all incidentally you draw 0 0 at when increased by 1 unit was 1 here indicates that the frequency of appendicular oscillators warn you have the energy that you wear is 1 unit misses increase the energy of the Rams by warn your neck so the energy of state is whatever and editors plus 1 nobody's asking me about know how could I be talking about energy 1 you want when you warned her Jules 1 horse Hyundai force Powell horse powers parlor horse power 2nd hour with the British euro British thermal unit yacht with from British thermal unit this injury unlikely governors so as is 1 when we're talking Aurora warned will come U.S. suddenly Beckham back to Rebecca back units but we combat units but is sung since some appropriate units units work after the 1st excited state which is a dagger 1 gives us 1 extra unit of energy now that's not the only state that has 1 extra use remote or make me think of be warned for so remedy what about any linear combination of them did you take 2 0 eigenvectors Our Lady of the energy and you superpose them gives you back from our Rogge America with the same back a linear combination of these has the same energy now 1 water Albert properties of these object and be in particular I mean 1 of their transformation properties when you rotate the the space we have an actress of prefer exercise the Finder the preferred access to boost the system amount of moving like hell along with the actress but I still have a rotational symmetry of the entire thing about the axis boosted how ill 80 be transform undergo Will you ask where came from they came not from Ex-Im white we wrote that X Y some some involving bees and not starting a use Hawaii was some involving was and XIY are components of lack of so that the president of the components of a vector transformed the components X and Y transform like a component of a 2 dimensional record what about the AU these are obviously are also transformers vector a and B if you worked around take the the coordinates which transformed into some 80 co-starring phase of course signed later little rotate like a two-dimensional vector so this is and reuse of vectors that means that you can consider these 2 things out of character this 1 it is point the ball is in the plane perpendicular could marching always new plane perpendicular promotion which is large this 1 has some kind of actual real property in 1 direction this 1 has pictorial property in the little direction so 2 states transform into each other's records they have the same properties as the plane of polarization of photon photon polarization is also a vector mainly because once the direction of the electric field of summer summer a vector points in the direction perpendicular to emotion over photon and so it's looking as if these 2 things must correspond

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somehow through a vector character of a particle how can we turned it into an 8 circular polarization yes we can turn to for a circular polarization and instead taking a 1 or be 1 we take 81 plus I knew want and that's a circularly polarized something it's a stream has the properties of a circularly polarized particle and of course to compromise a mathematics is identical problem mathematics polarization states a particle moving down there be transformed into each other and rotations we could linearly policies with real coefficients and then we would just be be rotating the direction of our of oscillation or we can what I with tournament to build upon this is none uneasy visualize front-runner visualizer the transformation properties which count most but there is not a Florida State Route energy there is no 3rd oscillator it was only the X Wasola why afterward there is no 3rd state so there is no 3rd polarizations there is only 1 conclusion which is consistent with Lorentz invariant relativity with other properties of these things this object of these 2 objects must be Aracelis they must be be like the amassed if they're only 2 polarization space that would not be consistent with a massive particle because a massive particle Yukon bring arrest and turn direction emotion and create the 3rd polarization of these 2 things these 2 states behave exactly like the 2 polarizations states of the photon moving now axis website Yemeni emotions perpendicular reactors nor overall motion moving down the actors and are 2 polarization states they do have the properties of the polarization states of photons of vector character primarily in the perpendicular direction Eliza transversely polarized they identified as photon like objects string theory off and they must be another was something deeply wrong Lorenzen variants cannot be maintained and soul become you must realize with contour disaster from Brewer vary on fortunate the clash were at this point we should fully string theory White because in North square course 1 has to be 0 mess of this object which is not the ground state and excited by 1 unit resist if makes sense as a relativistic object it has to be massless means of North's squared negative OK Biggio Piazza Vermont squares negative problem with man His M. North squared was energy Auburn Mass their squares of a ground state without these words the stream which was not a vibrating along the X or Y axis ground-state that's intolerable have particle would make it mess squares crazy square erred is positive Alcan square big negative if a lot so remember how Roger Maris record OK that now wrote what buying a new degrees and what maybe the mess quit is a bad thing said Mega Man square imagine addresses a bad or make that war Hawaiian Abed training so what you've discovered something a certain parameters mega wasn't me I'm that would make it a mayor squares a name a tacky arms trafficking your Frankie comes from the same it's not the khaki particle were not reported this techie her what's so what's the thing in your car which was enough rest was going Tech younger pack on procure path from start khaki what the Agassi I guess it's a Greek word Tekamah packet packet refers to speed speed I think it I think it's a word for speed and was Nauru think I don't think so I think it could be linear free I don't know I don't know my Greek immigrant over the Greek word baby to Yiddish word but but it does it does it does but think I figured I think prefix tacky reforms speed in general I don't always sure but I do not know whether the why they call it Pak backyard attack young men removes fast backs off as was bathrooms of light now that's clearly some back case lines thanked almost a remote figure from fast so we seem to encountered something which is a packed with negative squared let me tell maybe I should stop pouring and tell you a little bit about factor that Montague their intolerable and if every string theory had attacked you would not have heard of string theory order abandoned sometime around 1969 and you would never heard about so of course answer is that there are many string theories many good string theories but do not have the backyard there are no string theories that don't have this photon like object is a good thing because the world has photon ripples in the backyard of just tell you about earns a little bit how physicist use terminology today although a crackpot strength the move faster than the speed of light look at me so don't fall into a trap at 4 although they could somehow sends signals the speed of light that's not the issue of ownership Hawaii there were some ideas that they exceeded the speed of light the important quantity of interest for the moment now is frequency of a weighty youth Vs it's a wave number what is the frequency of a young Granada area which is equivalent of course course equivalent to be energy of an object apart from a factor of age bar is the same energy as a function of a momentum suffer less let's write that down from ordinary nonrelativistic particles E R P P squared with 2 wear right I'll just point something out to you we could we could go across from the about but the derivative of with respect the PUY was MAS varsity that's a general rule is actually a property waves if instead of the EU put Omega and set of PU per carry various a property of waves group velocity is given by

1:01:44

derivative Of the frequency with respect for the wave number the and due respect for the moment let's try it out for something else about relativity what EU vs. P but our full time as equals Pete fancier Eagles P. piracy right now he is able to pee currency was a derivative only with respect to pay which is of course the velocity of authority her 1st trial more generally equals the square root of p squared cause and squares OK now I get a little hard derivative of EU with respect but gives us 1 divided by twice the square root of p square pass and and their Mortimer enumerated to K. P Over Be Square Square P.O. be Square Boston scored OK I'm not gonna try to convince you that this is the velocity of a relativistic particle but does have 1 property noticed less than 1 square feet square Boston squared is bigger than he square feet square profound squared is bigger than a square just peace squared soapy Nevada by this is always less than 1 that's another way of seeing a relativistic particles always move with speed less than what are the speed of light whether the miners here that goes Athens pure white that's what people call these things pack where somebody a rubber and I have no idea what was 1st decided that it might be interesting to think about particles of fast speed of worried that postulated negative square questioners was really wrong with that other vendors seems to contradict something that so I started for us what does a pack yarn really mean and the answer is it doesn't mean something moving or extends transmits signals faster speed of light it means that the vacuum is unstable something can happen rifkind destabilized the system that's the usual sort of khaki answering give your younger do example were 1st West think about wave equations stowaways quick these equations are also ways of thinking about wave equation is aware of course but with but with key what wave equation 4 0 the function of sorrow hero 3 sway it what kind of waiver Zappa Zamora was more with speed of light we by he squared stems from Agace square with bones before those many partners if you have a wave of time derivative each time derivative gives you a factor of Omega what about each space derivative Katie right based the review your case so this equation says Omega squiggles case square or maybe equals case I was about the speed of light at the speed of white recalled 1 source of Austria persuaded the omega decay 1 how I turned into something more was slower than the speed of light or air for a while a N mashed carriers we have yacht which side that's why not writers Meyers Maria's and scored twice and we have is currently is the remember the respect that all may grow Wyoming I am a arise rare some vehicles as equally RIO McGerty in different because scenario make so this this is really a novel way of talking about minus Omega square is minus case squared and miners M squared which is the same as Omega squiggles K square Prosser squares from here that is what to do to arm was cast AT&T energy Delaware corresponds to the energy of the wave the energy of a way of Harris P 5 ID key squares where it has plus B 5 IBX square owed him Ms. Gillis plus and squared squared but then I remember from last quarter wave equations amassed terminal how you put an arm for a particle in terms of the way the equations Klein Gordon equations with mass this was the rules no notice ready emerging whether this is a crime potential energy from the field was positive if positive thimerosal part that meetings wrapped if you increase fine the energy increases the result of that if you increase 5 it will oscillate back it's just like sort of positive energy when you displaced to feel a little bit because you gain

1:08:51

some energy class summarily because it costs of potential energy increased fighting will swing back and the way stories while I have had a negative territory here so the potential energy of firehouse look like that for you fall off a cliff if you fall off cookware but at Chang so what's in this stability and instability on the vacuum drive to thought vacuum was just 5 equals 0 2 gave it a little push they we go it is a kind of interesting question supposing you really did start With very carefully Bowers at the top of a potential everywhere everywhere based fire was absolutely Bowers top Willis it that way forever for absolutely balance it would be like I your shop cancel perfectly balanced give opposed the price pushed Norfolk goes but supposing you pushed it only a 1 . 81 a model for this clinical model for Yakima command it is a model for it think about you have a bunch of political paid arms and that coupled together by Springs area gravitational field if you set the Norwalk perfectly vertically they will stay that way if you only have a single penny on and you give a little push or what happened tip In fact it wouldn't it wouldn't fly off to to infinity boat would make a big deviations from work so the starting point for the stable what happens if you push 4 of them simultaneously you apposite or acquiring or the car but he is an interesting question supposing you put 1 of them only 1 of them this 1 right over here how fast with the armed with signal propagate outward you push 1 of them it passes an ex it pushes the next 1 and sure enough if you wait flopping over but how fast best signal spread out well what thinking about like being Girl br velocity where more grow faster than the speed of light the way packing on suggests it might it doesn't matter falling all over domino effect proceeds where Be a jury the boundaries are involved of the arms of a domino effect was spread out no faster than the speed of light so the concluded his attack dealers I'm not to be thought of as things go faster speed of light strained theory has not provided a way of exceeding the speed of light was but as give us a theory with them on a stable ground unstable vacuum a little crying need cost will destabilize the whole world that's not a good thing that's something that we need To avoid and that's where superstring theory comes as where we needed for us so we have to do better than just as very string River prescribed OK we found if we ignore the fact are just say somehow magically we get rid of it we found the theory which has a photon like object and amassed 1st book on like object with 2 polarization state and it's a candidate for something like the fault or I said to you there are tricks and ways of getting rid of the pack yeah you might ask me on their tricks and ways of getting rid of the photon like object romance was wildly as nobody of within the within a month of the discovery of string theory people have figured out how to get rid of the Acura never got rid of the 4 complex object wine might have liked to get rid the 4 calm like object massless spin warned particle Winehouse OYO of stars good thing it could be the folk art the reason was at the time string theory was not supposed to be the theory of 4 times it was supposed to be a theory of a dry as armed Barry Bonds and there is no match for us as on Ronald resort service was thought to be what got a feature at the car and people want to be get rid of not only the backyard but also the massless like object all these years people have been doing string theory that pack Young was very easy to get rid of a nobody ever 1 object for part of the note is not matter so far is the faux pas what a look at the quality of our would be of little bit misleading ah there are many sources photon White object string theory which 1 is the real photon and what we do with all the others those very model the pain goes depend on details to say that this is the follow would be misleading ah is a faux pas Mark object in the gage rose news a thing would behave or for where you are and were me to come back brewery gestured Lepista over assured us right right now it is thought to have those if last year had not asked love laugh From our we're not yet what I won say murder From a strictly theoretical point of view of quantum mechanics were upside down chose perfectly good is something wrong with it just another ground state the man's back yard it doesn't look like the world no cost are good in this case here we know what we're happen varies guys there were just turnover among warned of oscillating about Obama here who nor was it was a bomb forget the Gravatt we have something new something closed strings soft font these are open string open strings means they have to win fans of career taught them a stream of from Serb borders at getaway Wednesday and Buddhist prep Hargeisa resemble Rice's graft it seems that your new properties of both our organization forms of the use of force that it was a matter of not borrow properties but stress also if on the now the interesting question would have to do with the radiation of 4 cars downtown radiation of arms you have the have objects which carry charge we haven't yet discussed what type of object might correspond with things which carry charge once if we found strained if we found String objects which played the role of charges we could ask if we the rules talk about the rules we could ask about the rules of of 4 times

1:17:49

by accelerated charges

1:17:51

the answer is yes we would find out if we behave

1:17:54

exactly the same way that emissions are for condoms from

1:18:00

charged particles work that will Beverly Ruelas after a probability distributions definite radiation patterns God knows patterns would be reproduced quite the quiet 3rd precisely Joao young crew OK let's let's more mentioned was Westheimer so you really the basic process of interaction it takes place or follow with talking about free spring free Springs me isolated strains moving by itself on accompanied by the armor and doing nothing except the propagating go along and Boeing is only turn away goes persuade goes manifestations of wheels are simply color at discrete collection of energy levels and also angular momentum for our strength move in Chungli in many ways for example 1 of kimono remember these strings of the mathematics of them is a mathematics of nonrelativistic shrink was 1 of the things of strength and do it it can rotate among Mexico's wrote letter number about Mexico's B send difficult for us being the thing which stretches about so this not be something that is relativistic strings control this means that they can have all sorts of angular momenta not just 1 unit of angular momentum of 2 units of angular momentum but you could spend them out but when you do because we're doing Kwan McCarrick each unit of angular momentum cost you an additional unit of energy a all no way at all they what it will not window trade about this their man right are any system where drug rotating Co will rotate about Turner so Questrom where's the tremor maps and other some ice straight string like this rotating about an axis of course the match would be at center because of you have something that anchor the end of the string Stringham rose around the back of banker the heart anchors strength theory but but rash energy level that to is what I call it a lot said if all our missing whether electron toaster during her am nucleus electron circulator nucleus reason the reason that electron circulates around nucleus with nucleus standing still is because nucleus was very heavy of the center really strictly speaking they both rotate about comments and your Mac but because the nucleus is a couple of thousand times heavier and electron it looks like a nucleus of rivalry with you quote pitcher Paul Byrd you always better road thrown out by Scottish what led Fogus tracks your that I thought with Jews of regular Favre Russians yes OK the basic process of interaction or allow strings the talk to each other to interact with each other do things are discussed was less time of player is remarkably do everything mathematical with tonight tried prescribed Rubin mathematically rare bird to squat with things that happened a particularly to these open strings is very simple it's really are only process everything is built out of it and there are fighting even a simple description but at the moment the 1st of all we only thing perhaps apart from these strings wiggling in doing their own thing really saying that allows them to talk to each other is that when endpoints find themselves coincident points in space in other they however there is a probability pour scorn and for them to join called joining interaction so some probability they don't inevitably do it there is a constant periods call the screamed coupling constant and what that does it takes a pair of strings as a pair of strings normally space time off this time of trend it comes most ruined their endpoints meet at this Point may join together performers shingle that we cannot string theory olive what in quantum field theory is the vertex with 2 particles come together and join and make a 3rd particle as could be of electronic an electron and photon or it could be any of the other interactions are studied 2 particles come together drawing and make a 3rd particle 2 strings come together join and make a third-string but the process is very simple wood doesn't just happen any old way it happens by the end points coming together punching and combining perform a single thing not a thing that can happen in quantum mechanics kill Anaheim need save quantum mechanics Missouri reversible theory anything can happen kind in in 1 direction can happened the opposite way in so that means that there is also an amplitude for a single string the break before if that probability while Lord was strewn with brutal integrity we constantly be breaking up if that probability is very small and string might last a long time and her before it breaks up like that sold our warmly assumptions in string theories of probability is small the probability is called the string coupling constant pressure that analog of the constant nature of which would be what would be for the electron electron for cars it would be a be electric charge will be the fine structure constant right the square of electric charge that cover your problem loser squares of Emperor our so be probability for 2 strings join in we like the electric charge both the seems quite depending on which kinds of strings Our joining sense that they could be the excited strings a could be the rotating strings that could be vibrating strings some probability went to Toll in point touch each other with a few and the same problem Drury bears a string of Norway along minding its own businesses that that what's the episode of Fisher now if you was elected fish right McKinney was awarded car and those are fights this is basic structure but 1 of the things that can happen is just like a snake which encounters tell stupid to know that found own pale might find its own tell the same thing can happen with strains that also spoke opened and if the the string come around like that in the course of its evolution and Mike was due

1:27:15

back when it does the Hague detects a mail just what rappers they don't so it inevitable there was no way around it because you have a fury of open strings and those open strings interact and words have just joined us living process happening then it is quite inevitable orchestrating a rehash have closed string of closed strings Our unavoidable he'll have a theory was closed to within a work out the mathematics of closed strings we were removed from right would cover the mathematics of Closter users very similar opens things will work out Western values stiff vision grass he said that yes the does yes absolutely goes far above wrap up this is now reservoir the motivated for you why closed strings are inevitable straight or a fury of interactive now what about a theory of closed strings that Avenue stocks with closed hearings is inevitable that there will always be open Strang known North End the Nixon said they would have me with a new kinetic fiber and held dual closed strings interact with each other they don't interact by splitting touching strings in the way they interact is different interact used to close the and they come together and then few was the former another close friend and no point today of a formal open string became beyond this touched a former number closed so this is it is what quite possible have theories of Purim close friends with no open strings but it was not possible to have a theory of open strings when our cost at adjusting it means that a prediction of string theory is that there are closed and there's not prediction of string theory that it will exist opened it is a prediction of string theory theory that must exist closed just because Vickery closed because we open string theories Baker because string theories without opened people what I was saying Cloer 30 or open string be interesting go expectation of open of photon we think interesting expectations lowest expectations of a close friend possible the graphic ran so toward of this thing where things a string theory literally predicts because you cannot have it is gravity remain any saying theories that have things like electromagnetism but some of off but there was not string theory because paragraph gravity every string theory necessarily have our gravity just because it will have close I never heard string theory have close got some kind still have our story from why was the open string necessarily but times well we did some manipulation and found that it was the only interpretation of the same thing exactly the same thing with close exactly among time tonight a they were close to end course In some a string theory is the involved the splitting enjoining Prada in those string theories coral strengthen become problems further also string theory is that don't have this particular process of breaking my career he that list true if did get a half-full wherever they can still be rattled such numbers among Norway user closed curriculum directly problems rapist's imports was forward when up whoa springs in fact what would you have this is even implicit in open string theory you could have the basic thing that can happen this tool in his Kentucky enjoying but the something else second-half building it can happen to strings could touch point which is not a name and then they can rearrange sorrow still we started with we started with 2 strings the restored it will warn we start it was 2 0 this way and this became again the same process work through things Carmen coach In rearrange as a process that can happen in string theory or I will have to prove to you it could happen I'm telling you now OK so I'll see what happens if you have AT closed and open strain of your close an orchestra and close friend is a closed during the simplest thing that can happen is the closest friend common it can't see opens couldn't rearrange and now was a formed 1 friend OK so let's think about what that means that means and open strings more belonged to begin with it was drawn use Our opens closest friend comes along and closed there was left over a period Opa bladder processed will turn into these absorption of gravel upon my Prague particle which is not so something we found that closed strings can absorb grab times were open strings control grab on track everything can absorb upon that in string theory was worried everything gravity In essence as a simple-minded way the fact that orchestra strain or close string can always absorber grab part of this process is we all more hours 1 of the ways of thinking about the universe shower gravitation are set up to both Syrian all yellow Parker we should talk about that arm in weakly coupled string theory would be interactions weak the strings a transparent to each other but they can do this but were small probability was small probably really come a along and they do that but they can also pass right to reach over some these screens obstacles like shoelaces became have richer but they can't have rocked path which are our public knew was superstring next Parma you how her her way to remind summary chart America burners to push for new mails from what we hat nominally fleeing the city yacht is that there will not be that it has told us of choice there is the norm the basic

1:36:32

probability per unit kind if it touching to interact with the same for all strings but keep in mind is moving fast they spend less time Merrin near each other so the probability per unit time His saying if they're moving past each other fast there will be less of a problem of joined the trouble occurred basically a probability per unit kind of Asia follow a off their shares up like that edges said that less focus of 1st thing we should take up next time you and I think that's addressed Is there a unit right member must be you because I've done all sorts of equations which didn't really make dimensional stand was was Eric Exs always right so Muslim youths of about creation annihilation operate a well have commutation relations daggers to 1 so what's the beverage now neither stop NXP proportional the some missing units in the story are frequencies was sealant frequencies In the 1st time but my frequencies are in the jurors integers don't have any some missing Europe but I have improved in here and I think the 1st thing we would talk about next time you hear what units another way to ask the question news I tell you over a heading in expectation of energy increases the mass of square by a certain amount in what he wanted madam routes saying forum what units kilograms and there would be no electron volts and slugs what unit was over as 1 unit of expectations on March the increase the Pike what enter would you guessed it a fundamental theory of gravity you might think it's upon graphs you might death it is to revise the rules more upon of her something next time we will take up explicitly issue of units which is clearly important that import not flee was Healy is not the same as it is yes joy in the simplest thing theory just a single can't turn back and generally strangle can't I not a fact really strictly speaking there are no constant strength theory but they will look at that before we all right Close the possibility of her burned breaking not our wine and what would not here has been annoyed me why no no matters and that not over for the U.S. composed with him right they stood at figure asking whether you could hear more complicated objects like there's now this doesn't correspond for finding by REMIC corresponds a 3 strings joined a vertex at the center as I what you're asking yard about drawstring series to have that possibility is it does it end not a variety of different theories the question is whether they all have in common US was a generic prediction of string theory that you really asking what they all have in common and the only thing I really think of the pit wall having commerce is a all have closed strings and they all have grab some versions have open strings other versions have records of ball Boas Argentinian swings around tangles of tangles as feed falls over enforce space Our yard in fact in Hagiographa 6 lender Hey don't physics is a version of string theory when we talk about units next of Co you difference between string theory has a new theory of Haiti drowned develop this kind of thing with me vs. this kind of thing work for mystery quarks Mariano the Sevier caught and they're caught string theories can have these kind of the 1 thing they always have the coach cover change Our mission of interaction you're natural questions the Corcoran got 1 we shall 0 while also not shot also should yes yes are there is a dimensional like French forming the number of dimensions of space there is a story about the values of the year coupling constant as a function of dimension of courage but pick Basra come to it 1 more or humor are 1 were workers young I a letter movie 8 a day that I will not lower on the moon ring north from myriad is a graduate makes removed not with to makes the bull was storm Warburg it was Was Aristotle said the has to be a cause for motion was noticed that nice things to 7 ramosa will keep moving strings of the same it's nice gravity which makes things more Oh what does gravity do gravity causes things to accelerate shall we haven't gotten the gravity earthenware gravity comes from are so far this is a theory which we have no particular reason up till now have any gravity what makes things Norman General workers also service and there to keep more what's starts moving the exchange of all those who voted for half of the time they get to bowling something more will ring now you look at it from frame of reference this morning so what made a move we that give you have system you have better retrieval system which led what but I won't tell you a bit of a thing is not moving in 1 frame of reference is moving another friend 4 0 you Delaware 1 of the best Galileo knew enough not to answer the question what causes motion but there's question what causes changes of motion laughs forces forces core course changes emotion no flower a classical mechanics forces course change of motion what causes the most in the 1st place well initial conditions and show condition why causing the 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1:45:42

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### Metadaten

#### Formale Metadaten

Titel | String Theory and M-Theory | Lecture 3 |

Serientitel | Lecture Collection | String Theory and M-Theory |

Teil | 3 |

Anzahl der Teile | 10 |

Autor | Susskind, Leonard |

Lizenz |
CC-Namensnennung 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. |

DOI | 10.5446/15129 |

Herausgeber | Stanford University |

Erscheinungsjahr | 2011 |

Sprache | Englisch |

Produktionsjahr | 2010 |

#### Inhaltliche Metadaten

Fachgebiet | Physik |

Abstract | (October 4, 2010) Professor Leonard Susskind reviews harmonic oscillators, the spin of massless particles (photons and gravitons), the low lying spectrum of strings, the tachyon problem, and the basics of string interactions. 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. |