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# String Theory and M-Theory | Lecture 2

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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
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
these components of momentum a finite you could always make PC much bigger
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
fines of Expo will not the acceleration but was the analog met this isn't embassy New always knew we knew won't be costly chopped this spring in the string people a little pieces the end point is very very light has amassed which is only 1 and now
we have something a little bit bizarre I'm multiplied fight and
multiply by in Webster really intermediate multiplied by we find the
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
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Stunde
Relativistische Mechanik
Kraft-Wärme-Kopplung
Waffentechnik
Windrose
Quantisierung <Nachrichtentechnik>
Eisenbahnbetrieb
Tag
Druckkraft
Band <Textilien>
Elementarteilchen
Negativer Widerstand
Nassdampfturbine
Zeitdiskretes Signal
Backenbremse
Gummifeder
Ersatzteil
Computeranimation