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# String Theory and M-Theory | Lecture 4

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if the Steiger University OK we
want to move on to closed strings but before will at the close strings it's my habit to do a little bit of mathematics 1st form a bitter reminder and thing I wanna remind you of the nite is technical it has to do With the canonical formalism of classical mechanics a reminder and reminding you about his nurses theorem where stay on but this theorem is theorem which tells you about conserved quantities and their connection with them With cemeteries the connection with cemeteries is that for every symmetry is a conserved quantity and in quantum mechanics because of quantity becomes will call the generator Of the symmetry comfort by 1 remind you of that before we before we jump into where some questions about string theory dissolute will have a blackboard but they supposedly have a look around and see amateur remember Rambo grudges and depends on a bunch of cordon itself cloak warned Q was organized in depend on those grounds in the independence on the and what else I think you got time derivatives now what's the canonical momentum conduit to killed but there may be many Q. I saw what see momentum let's call it key some of I cargo momentum related to the accused court partial L with respect to be velocity Bush for rare for a mail and I'm not going to do a writer for you know this year just going stated for you mind you I suppose you have some symmetry him not to be a summit of world guest bowl you have some symmetry which involves a transformation on Q it's a infinitesimal symmetry Regis shift things by a little bit so right it by saying the shift In Q. What under a particular symmetry operation will just call it variation accused of I wanna introduce another
restraint being flat and having a little wiggle and has a little wiggle will Komodo left or to move to the right of most of the right that's the best dissented increasing signal most of the right that's a right moving way what happens I get me in the it is really appears at the other end so it might fall off the end of it he said and you pick up over here senators would continue to circulate around around string that would be the picture of right molding labeling the right and then there are a sticker moved the left left more than ways of right waves and Barthelme could exist on the string what amount the open string with a left moving waves in right moving waves yet you could imagine a left moving waiver right away to sell left moving away but what happens when get to the Internet case figures reflected so that gives you have left going the right cargo left going the right The left Peterson the writing this it would not be considered time our kind of things which jerk conserved time of the standing waves but radio for close there running waves but last year they to arrive OK let's talk about writing me both of so it's a particular kinds of waves no particular kinds of ways we correspond the plane waves pulling up and down a string of saves right molding a moving now it's convenient to describe the waves not by signs in but by exponential R's to assist him exponential Zeus's linear X eatery I'd certainly we in our exponential sodas linear combinations of signs Inco signings In the case of open string you are either restricted through signs or cosine depending on whether it was a rich lady or not and boundary condition for the closed strings you could have both signed Sankoh signs but it could also have linear combinations of signs and co-starring so you can have exponential waves propagating who left and right the kind of way is the Duke of imagine are eaten ii In sigma which would be of course size signal plus I hate in I cosigned in Cigna plus ii sign in we can't sues to decompose things into exponential it is where free right George do that let's said what we do is exactly what we did with the open string we're going we're going the Fourier Transform XO said but now that was not among their slate just that comes back to itself after going around general form for the expansion function which is periodic with period to piety is just as a general form is that it's a summer of all integers of coefficients let's call X Internet crimes eatery I say In can be positive or negative N is positive because a right moving weight and is negative think of it as a left moving OK let's let's just checked is really have the property that it comes back to itself Wednesday those from 0 to 2 part would signal 0 0 Ito the iron stigma is worried about Mr. each of the 2 Pires what feature the items to Pine Bluff 1 I'm not eat a place to Part II 1 right so every 1 of these suppose it with the polar here instead of in there would put in over 2 payout was a comeback to itself after a cycle now so it's necessary to put integers here this is the most general form of a periodic function when it's for real expanded on a line between 0 and 2 pint assuming the all right this is right a sum of 2 while a
sum of 3 terms the 1st terrorist In for positive in but Mike somewhat end greater than 0 those correspond to the right moving way each corner right now but the left moving waves which I could write the same way exit late in negative alright derided as a solemn in greater than 0 but Our call X X my and each of the miners I Italia signals with positive coefficients here correspond right move in waves with a negative coefficient that corresponds to left moving leeway as beautiful thank you a yes Jesus in club of civil right at the time usually inherited the plus side idea of my I think it is In many cases left mowing waves and right away some of the missing here an equal 0 so it put that back in what is at a meeting you know about a dig 0 but what's a physics of what physical merely a 1st similar similar mass arrests of the of the street the actual location of the strength of our center of X 0 saves they thought why exactly the same thing for why I think I will not right But why of Sigma is equal to the same kind of some Weiser equally and Sigma wire miners and the minors injured figurehead X you could have way in which correspond to fluctuations along the x-axis ways which correspond flocculation along the y-axis the glory and the restraint and the general form what he'd do it you take low grungy and further strengthen average down but it's exactly the same grungy and as for the open string exactly the same thing is still imagine we made up of little mass points with strings of moles Uribe over Grosjean of the string exactly as before the only difference you integrate now from 0 to pipe you should well same it before X by beat Palo squared powers time minus X by Cigna squared exactly what we wrote courier we do the same thing for white plus plus white terms of exactly the same time so they have now 80 decomposition of X into discrete Fourier modes we have Lagrangian what time-dependent things in early exit in the accident absurd coordinates describing each oscillation each oscillation has a shape but also has a time dependence and the time dependences buried in the pro-independence of these X and they are the degrees of freedom what we doing this we take this expansion and plug video you get X . by differentiating gives you a stop in and you could differentiate by Cigna by differentiating use coefficient functions With a plant Dania and calculate the car for the whole system around let them do it calculate all around him and where we notice about the grungy and what what the exit of be Agonidae harmonic oscillators exactly as they were before but it will be hallmark oscillators exactly as they were before all before I do that yup before I do that let me do something else 1st 1 hurt or I harmonic oscillators look at this formula for a minute was the energy this is Grosjean energy they Lego but James Taylor plus sign visit potential energy This Is Connecticut the bank X by squared plus the expertise of a square if likewise for white like this In a funny way as D Banks buddy we
can plus the IDC said squared plus VX but trial minus V X by you stigma squared alone I think I'm off by a factor of 2 probably 1 half you see what's happening the X baby square near here they explaining how squared we double it that's why I put on him it there's also would be expired signal square and the expertise in the square from here and there is a cost the explaining how times they expect a similar comes in with a plus sign here with the minus sign here and they canceled this it is the same as you have any idea what the significance of this decomposition OK always Eve that's moving to the right is a function and out signaled plus half away that's moving to the left as I may have about I may have opposite away which is moving this way is a function of Sigma plus pound always moving in the other direction is way of moving in the opposite This is just energy stored I think this is left moving waves and this is the energy stored in right moving way right foot waves the left moving waves are transparent to each other they discover right through whichever is a linear equations equations of motion and this is just writing energy as the sum of left move in waves and write moving waves of restoring this will turn out to be of some interest to us 20 now it's come to harmonic oscillators each of these is itself a harmonic oscillator so we can they'll start over we don't need to go through the details constructing the harmonic oscillators I think we can pretty much guess is going to be a harmonic oscillator variable for each mode of oscillation Azores kind of moral oscillations you could excite sided with creation and annihilation operators the creation and annihilation 0 incidentally What is the frequency of a wave label bite in here it be the same as it was for the open string proportional in the higher in the shorter wavelength modes the higher the frequency so that will be the same OK so we have a collection of harmonic oscillators but now we have twice as many Marino have twice as many as because we have waves each integer for each frequency now labels Toll oscillation 1 would that frequency moving to left him and 1 that frequency moving to the right 1 of them is labeled with aid and the other was labeled with minors end Ms. Toma is much too much notation by can help at all these things we have to keep track of little in here represents frequency when aim as positive as in the 1st turn it represents what's waves going in 1 direction wants negative it represents waves going the other direction so let's not write down what the various harmonic oscillators our we 1st of all have created I'm not gonna bother writing unite which operates the annihilation operators adjust kind creation operators could create a X citation let's take a right no just call it 8 plus some they can also create different operator a plus and in this time creates an eye you write moving wave on the strain and the other 1 creates a left moving away from a string peso riffraff will have that decomposition what else well we have at and why I've been right everything involving why but writing widest replicates the same thing that acts what they call the creation annihilation apparatus for a while oscillators where some think it could be right now I have more we have beat plus in the and be my B plus a and B plus of minors now I could enter this if I want to write the annihilation operators form more with minus signs up here and not going below us get confused by too too many pluses and minuses a and B represent X and Y plus matches creation operators that excites the era of you keep in mind that was also miners below be using it anyway and is in and after do will 1st of all in has to do with frequency frequencies as in but plus and minus and have to do with waves moving in this direction or that's the structure of closed drink periodic that sets the whole structure of it but now we can start asking 1 of the spectrum what kind of particles kind of excited states who string half what see an alarm of the folk that we discovered in the open string your daughter's remind you about that would be open strings we didn't have his doubling of miners plus at what we Harris just creation operators 18 a plus end a B plus we started out others remind you we started with the ground state of the strain at a strained his loyalists state of oscillation doing nothing we're excited What can you excited with you can excited to get the lowest energy you want to excited with the smallest in wiII because the frequency is proportional and of course frequency and energy of the same thing so be it the lowest the next lowest energy you wanna excited with 8 plus of 1 B plus award a plus 1 and B plus 1 this is a laugh at em oscillations somehow what's associated with the struck picture he is a z-axis the X-axis in while action figures is going down the z-axis with a huge momentum but oscillating X X and the
during left moving energy doesn't satisfy level matching same thing would be either has 1 unit
soup opposing aII were right circularly polarized photon with the left circularly polarized protons to photons 0 so this has angular momentum 0 says angular momentum court and equals tool says angular momentum and Nick was minus 2 this as an equal 0 and the other possibility used to switch plus and minus 1 makers 81 miners I'd be 1 times a minus 1 plus I'd be minus 1 but for possibilities are linear combinations of beat a much angular momentum he s 0 again so we have to states with angular momentum 0 and 2 states while 1 state With angular momentum to warn stay with angular momentum minus 2 artist remember would doing with particle shooting down the z-axis whatever it is it seems the calm in a state with angular momentum to stay with angular momentum and the miners killed and maybe some pieces with angular momentum 0 what's missing how this something on the face of it seems wrong if this end to them and they must be a spin to blood and this must be representing somehow or spin to particle Hunley states has to particle has 5 weather coming in equals 1 0 minus 1 mind as to where we we have a candidate for the spin 0 we have 2 candidates for spin 0 but we don't have any candidate ever for the spend 1 minus 1 no I would go this is what will we do we sit coders busbar but to particle right right could be into particle Creal called wherever this is coca abuse and to part spin to particle has 5 states with moving down the axis deer you declare or very angular momentum about that axis with calm in a multiple of 5 possible state but we know we find the right file possible states trying for states but not the right can even now costs were missing the PM equals 1 minus 1 of those would be part of this spring to Dublin Willie conclude that I had the whole thing is a mess in the doesn't work or what the to particles what Mass West Masters because massless particles come of only in Maximo minimal angular momentum states that a general fact about all grad at times like a photon would have only a right-handed polarized and left him polarize the only difference with the photon is a right-hander polarize Gravatai 2 units of angular momentum and left him his mind to your grab at times do not calm things with angular momentum Warner angular momentum 0 so a have can only possible interpretation of this it is the makes sense reserve there's a grandma Tom here but it will leftover with 2 states with angular momentum 0 what could that be the only thing it could be used to particles which had angular momentum 0 but if the particles of angular momentum 0 1 of them 1 would be a combination of this plus this is called the time to scalar particle and they have the spectrum of string theory it has the same masses the ground mainly 0 man and the other particles called the axiom these are both familiar particles I'm very very familiar particles to learn to the number of hours tourists and they have 1 important common features they had never been discovered Our this is a plaque about In the more formal mathematical structure of the theory beginners Massell's particles interesting question is could you get rid of these somehow without getting rid of their entity that is not necessarily there is so what we found them for a for the closed string is something we found mess was spin to particles I don't think that we haven't explained his why this level matching wider we require this funny ruled that the amount of energy circulating the right should be the same as the amount of energy circulating the left made it will go through it you're Quran not complicated New supple like it right off the bat that let right were both left and right mean space means a along the and the conditions that the left more energy and red moving energy of but let's take a break for a few minutes I'll tell you what the condition is I'm not sure of the right amount of what the conditions which applies Will the writer OK with a
shifting the neighboring points OK civic right down the motorway rewrite it by by writing sigh of stigma my side of an absurd 0 2 small teams in the wave function if I make will change where every X ghostly exit neighboring position of Puerto a a number send function your send function musical 0 what's work it out we're done here is taken a function of a variable it's actually of functional but continuous the same things but less is treated as a function calculate what's going on here well we're right but this is a small change in society would you change X Point Sigma kind which injured in acts of the changes sorry when you change X plans a change elects Thomas's exchange How much as exchanging going from X at Sigma plus epsilon exit stigma partial of acts with respect the stigma and at dawn services side when you change X the change annexed simmer will you shift said and shows derivative of X with respect for times Epsilon now which said Mayor Mike talking about segment the origins of pie to know what they so as I mean he returned to his separate equation trip said not you should a them up is saying that sign leaching something a one-point plus the changes sigh when you change the next point Costa changes sigh with changes the next point what is really should be in we sit the sum of all the changes inside when you shift each extra little bit all it up 0 rats with the says we knew it epsilon now but back now what is the exploitation was sorry was designed by BX would go back quantum mechanics but go back quantum mechanics quantum mechanics Wigan have peace in but Howell is up P related to the corresponding Q inquiry mechanics Martinez on h times by be killed on onto the action of a momentum on a wave function is to differentiate respect corresponding quarter so when are you see a wave function having been differentiated with respect to a quarter let's forget it part whenever you see a wave function having been differentiated with respect to coordinate you could be right that was the action Of the corresponding momentum on society for Healy the wave function differentiated with the coordinator at points said will watch over right after it's P said what is I mean peace said Matt Save our means peace in we have a strained it has a bunch a point x said got the points have velocities the velocities of those points of a moment member we're just doing basically non relativistic physics we know nonrelativistic physics velocity His momentum so all us is now is each point on the string here each little Mass Point has its own momentum p of stigma well I find I find out that the condition that the wave function be invariant under a re Panorama traditions of the office of the Sigma acts nor adjuster shifting the sick is the condition that a certain integral he said times the FCC Zero but appears segment is nothing but Brazil is nothing but the velocity it X but of simmered but for over BXT Cowell the Indigo Alves said no on the XT can VXT said should be 0 that's a condition that don't there's no special for novices highly abstract but rather remarkable that such a condition should exist and rather remarkable Beloit says USEPA's not preferred point the same axis but now look and compare with this incidentally I wrote energy here recently integrated the left to remember what this was remedies the left energy and the rivalry marriages last moving energy and the right ring energy she With the sum of the left Mowery energy and the right moving energy is just theology What about the difference of the left moving energy in arrival in energy what's left over when you take the difference instead of the song What's that a difference the rents lot moving in a gym arrival you what's left the war is over but we that but that look they you want what you have to happen brother at City OK but I don't know if here with uh the if you want to take the difference of OF don't for whatever reason but 2nd difference somewhat moving Aegean arrived moving energy than the VAX by housewares canceled the DX by squares with canceled but the crossed arms
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