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String Theory and M-Theory | Lecture 4

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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
symbol which just called don't produce a body that might be some function and maybe the or function of Q times some small parameter epsilon when you make a small transformations each queue changes by an amount that might depend on all the accused plans a small epsilon what's an example well-known example rotations space coordinates X and Y position of a particle if you rotate a little bit of coordinates each chord changes and it changes by an amount proportional small angle epsilon would be a small angle here and that case you would have dealt the Exs equal Hawaii built the wife of epsilon the minus X to have have an example conserved quantities which in that particular case would be angular momentum is constructed out of the P and the variations of cues from the right down the formula for you called the myth that charge it's also called in quite character the generator of the transformation often labeled capital Q but could therefore just to remind you owe me there and anywhere motorists Battisto assigned to some or all accord they all transformed the lake right and give the I momentum scientists shift in EI coordinator maybe we should do a example This is the Formula One blackboard This is a transformation alloy This is a symbol that represents the generator of the transit area of the operation in quantum mechanics it becomes the operated a which creates the small change the operated action on a wave function which induces dissymmetry but that's that's nurses Iran quantities conserved and it's going to come up why is come up as all sorts of symmetries and string theory but a particular 1 particular cemetery let's now forget the Mathematics emeritus pyramid compote closed strings close strings a strings it is a close friend of the cigar the blackboard is VXY axis use extra Hawaii and there's a string projected onto the ex-wife our plane now the strays from the tries by a parameter fault this strings recalled that parameter cigna and signal went from what the what do 0 At the pie about close strains were Chiron to foreclose strings at big idiot not to from 0 to prime right now so some point on the string we'll label Sigma equals 0 which point on a strained it really doesn't matter but we pick a point on the string once and for all and we label it 0 but we had a little pain and we could mark a string of we equals 0 or were they are halfway around would be sick pie a quarter of the way around would be equals pilot three-quarters of the way around would be Sidner equals 3 piles of nomads that there's a directionality along the straight directionality of increasing stigma was ahead because 80 a kind of Arrow along the strain which tells us which direction signatures increasing this way him again now astray lack of rubber band and think of it as a kind of Mark rubber band would remove those marks before finished a mock rubber band Sydney was 0 psig callers 1 degree 2 degrees 3 degrees 4 degrees always around the 360 degrees and that gives us a sense along with stretch but now that's since along string orientation of string has nothing to do with me clockwise in this Our entire clockwise news in the XY place I so that signal was increasing in a clockwise direction but now like take this strange I looked at the blackboard Ireland and put it back on the blackboard I had a physical thing I could do with a piece of rubber with a rubber band and put it back is now enable zeros over here it happened put it back the same place Sigma equals pilot till he said Nichols high over here said equals 3 piles here said increases in this direction the only point as is I don't want to get confused when I speak of orientation of the string it's not an orientation space its ascent aren't enough sense of intrinsic clueless drinking was a directionality longer and that direction our we we have keep track of OK now let's come the waves more along with Estrin can move either the right or the left but right and left are melamine in the direction of increasing signal decreasing said I do not and not complying anything about the spatial orientation I'm just imagining a little wiggle in the string the strings proceeds in direction of increasing segment I will call that a right moving ways now Turner Street and planted Back on a blackboard alright wave would be moving in the opposite direction so it's not a question of whether it's moving clockwise in a clockwise and that that's where the proceeding from smaller city with the largest right now Harry not now the figure of Yasser in particular In a rigid and the origins of string theory you break this trend the only way to break a strings to produce a quirk and mythical twice in which in squat in nearby work right now that doesn't mean that you can't invent string theories where the strings not already but they can break but there are no less so the moment pretend that the strings oriented has a sense of increases decreased by what do we say the coordinates describing the strength of the exactly the same as they were for the open string and XO and a wife of stigma and if we have more coordinates for coats and Klein told us to do we would get them in Aksu NYSE the X Y position the X and Y position of the point stigma let's go Iverson axis is the same access to to go from 0 to to party remember doing open strings we had to worry about boundary condition what goes on the ends of the string him Owens restrained but there is something that you could call a boundary conditions the boundary conditions of the relationship between the coordinates at 0 into pie what's economy x 0 0 must what equal X Of the Pied this is the same point always around so that means of what the plot Exs stigma they could be any functions as long as they overhang here's the limber as well as a common back the same value after it goes around to party so X Of 2 zinc with the X of 0 saying things With why if Tupaiidae is evil a Kauai 0 those are the 2 degrees of freedom and now I can and now we can imagine waves let's let's start with a string beans some
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
axis remember what these 2 things comes on but photons is correspond to the polarization of the book car an ex polarized photon oscillating this way now oscillating this way this is a white polarized for oscillating in the perpendicular direction I must circular polarization of 4 is a N a plus 1 plus plus or minus Ari B plus 1 1 0 0 I just being the complex number superposition our 2 polarization plane-polarized away years without I would just correspond to playing polarization of With the EIA corresponds to toe circular polarization OK so just keep that in mind dog a plus I'd be in a minus I'd be create a unit of angular momentum plus angular momentum moment OK so where are we now let's see what we can make the 1st thing we can presumably have is ground-state whatever the state aid it has some mass squared which call and North squared state has some energy and your squares what excitations command make well I can excited with 8 plus 5 more we make the lowest execs occasion we only more use 1 here is 1 he 1 he when use 1 here we are excited with 2 units of energy soda use they too would be to OK what can only make looks we can make for this state 4 distinct states after which it look-alikes so of having to follow priority sent him to photons weaker right 8 plus I'd be in a minus I'd be or 80 minus 1 and be matters what was for possibilities now for acting on the grounds that a plus 1 or minus 1 0 all B plus 1 minus 1 4 distinct states that you commit what could they correspond to what kind of structure that they have only the question is how much angular momentum may have angular momentum here are they really like alone is around him this is basically a thing them not now what's it but I don't they do it when you're playing the join you find your because you go right them as a plus I'd be in a minus I'd be 1 minus 1 so in essence it's just the doubling of what we found here which means there are 2 kinds of objects both looking like for another a bad but I will tell you right now with this it is not the right picture about the right picture but to to see that's not the right picture the thing to do is to go to the next level on No. 1 if that was the right picture then you would have to photons each behaving like a photo army is no angular momentum 0 stayed here and no angular momentum 0 about taxes is just a seasoned bees a plus a minus IBK make angular momentum 0 this combination but that means something about these hypothetical photons means the massless if you have a massive photon then it would be able it would necessarily be able perhaps angular momentum plus about the minor axis of motion angular momentum minus about that actors also angular momentum 0 we discussed that West is no candidate for that he no candidate for the angular momentum 0 which is like having 2 angular momentum 1 of forms and that doesn't correspond to any possibility of the massless particles so we might say this theory so far has produced for us a doubling of the photon spectrum but to see that that's wrong you go to the next level let's go to the next level what can you make alone Armed when you up of energy renewal but pirates armored car you with you get what you make is a bunch of garbage it doesn't look like anything sensible it doesn't fall into a rotational multiplex it has the wrong number of states to be an angular momentum tool is no angular momentum 1 piece it just the wrong is wrong combination things this is not right is not right and Italian allies right about right it's extremely subtle and is not easy it's not easy world it's very easy but Over complicated not full of things it Yukon right down correspond to legitimate states of the strength in factor constrained the rules the rules forbid certain combination and would tell you what you wanna do I think I will not explained why he can either will explain why next time us we have to occupy tonight but I want to tell you what the rulers Taylor and then will try to figure out why the mood is cold level matching and level matching said is that the right moving energy and the left moving energy we must b saying now U.S. me why why it is a toggle amount of energy going around the right half the be the same as the total amount of energy going to the left that's us that will have to come back to move your but let's not do that now let's just make it as a posturer let's make it as the assumption that they write moving energy and the left moving energy must b equal what can we say about these states while try puts you at 1 0 . 2 8 plus 1 old man has warned unit of right moving energy don't satisfied level merchant about my is sworn as 1 rule
during left moving energy doesn't satisfy level matching same thing would be either has 1 unit
of right moving energy or when unit of left more energy Chung None of these states here that we wrote Darren satisfy the conditions of the left moving energy in the right building energy sector for but the state's let's go to the next level let's see what what's at the next level at the next level with the following things so we can take 80 Matsuo we have we have I 8 plus 1 times 8 plus minus 1 this has warned you about left more formal modal you have the right moving energy and 1 unit of left moving energy Hall that satisfies the level matching condition because it has worn unit aright moving warn you of left molding and uh I appreciate this is coming out of nowhere requiring this to be true but works will come back so those a state misses OK where about 8 plus and how much energy does it had derivatives right next we have it 8 plus 0 sorry next we have beat plus of 1 B plus minus 1 saying things are really good state levels of match as much left moving images rivalry energy and we can have a plus 1 beat plus minus war young B plus minus 1 positive view of Maryland 80 right moving bit of energy and a lot of energy resource for the state and what about a plus minus Warren plus 1 also good voters their matched 1 of America's for 1 of these have 2 units of energy because they have to oscillators of the 1st frequency now was anything else with the same energy a true tried Ibarra here 8 2 plus a tool plus a minus 2 0 plus have Robbie plus what these states have 2 units of energy but that they satisfy the level match and now they have to you it's a right moving energy ought to left molding energy so these are illegitimate states gross amount was less but what's left is rewrite them a little bit different a rewrite them in terms of a plus I'd be a plus be what the operators which make circular polarization we're right exactly the same stated in a different basis in the former a plus ii 1 Warren plus plus but up times a warrant cross I have B plus 1 boy I think this makes loyal idea what did pose I'd be nickel circular polarization which meant angular momentum plus 1 what about this you will be done for the photon you did it twice 2 2 units of angular momentum going around the axis not not really mean no Oh sorry yes you're right otherwise it would be level Mac No no no Eurostar right neither right plus 1 minus 1 this is your plus 1 unit of energy this gives you might not this gives you a right moving things this gives you a left right and left his level that's the right way to think about with now same thing except with a miner's beat times Eddie minus i.e. magnesite about that would that make but this is like To right-handed photons getting up to spin is tool this is like to left-handed photons heading up to angular momentum minus to about z-axis of 1st of all we have angular momentum to him angular momentum bill for independent states year still missing something for linearly independent combinations what else can we make we can make policy we can make was just right 81 plus REB 1 and that there are put across a forget the pluses on a minus 1 minors I B minus 1 was a Class be times a Midas beat left right moving west molding and the units of angular momentum is this is Lex
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
solemn got gotten into a technical details of the spectrum of strings and so forth the 1 missing piece sources level matching where can you what level matching is now word means 1st of all and then I will have derided for you whether you will recognize the steps of variation on I don't know what is 1 formulation uses notice maybe I won't even have to use the but will say on the question is is the point Cigna equal 0 really are physically special pouring or is at all it doesn't matter which point you call certainly 0 is another way of asking the question of whether it be stay the string should be invariant under changing what you call the origin of the court Sunday's hockey cancers it could be that there could be that a special point and the strain marked by a little piece of bit a like a year and a half point can be special it could also be that there is nothing special about any point and that the theory has to be symmetrical or invariant under shifting the parameter stigma that's what it comes down and whether the states of a strain on invariant with respect to shifting that parameter that is a fundamental question could fundamental question I have stated exactly what it means yet but the stated now we could begin by thinking of string is a discrete collection of points and have instead of hiring ex of Sigma and the wife of Sigmund we were have acts of Ottey put X Y and Y was X abide by supplying extra money Weiser by just 2 positions Of the Corey find units down the strength I might run from 1 up to end but now half next was the quantum wave function of structural well if we think Oliver Reed trained as just a collection of point particles at least temporarily then the wave function of a string quantum state the quantum state electoral system what would be a function of the axes position exorcism we we got their wives X 1 X 2 X 3 but about the accident but White X 1 year when they sell elects to within cycle warrants a side of next to we got about about about 6 end ex-wife must pressure reasons why not because nothing special about the point that starts out that that's a great equals 0 no not the quantum wave function of a string which it is completely symmetric with respect to ordering the exit ordinary rhododendrons in sense of putting X 3 between X were next to us to violent video thing to do it that's a point rips string open but just in cycling 8 strain which has no preferential pouring should have the property of the wave function when expressed in terms of X 1 X 2 X 3 except kitty equal the wave function reviews substituted for X 1 X 2 X 2 X 3 x 3 x 4 1 and so forth sure also of 6 3 about about about next end X 1 X Jill Miller was away function should have a certain symmetry but the there should be invariant with respect to lumber shifting beat label on I might warned that a possible thing that we like the mayor of a string theory now where possible thing we might demand begins a really big trouble if we don't really big trouble whole works just bird falls apart OK what does that have to do what level matching nothing obvious but everything so disappear coors all blue I suppose is how I go from the discrete strain could continue string what's the corresponding thing he instead of saying that the symmetry is all of our operation of interest acts RED goes to Exide plus 1 module and cycling and what the what's the right thing to say index ii is replaced by the continuous variable said but all right I'm so it says it ripped away function which is a function of X O said no it's a functional function with a function of function called functional of the wave function is a function of X of and the same legal zeros not special that wave functions should not change if we shift as Michael said we simply shift the argument we shift the the simmer bearable the Singapore Sept 1 now Singapore subsonic was means something our view that point to Tupaiidae 1 shift it doesn't remains a rural Riordan through rotation of the year Sigma circle but see work out could this is actually just some simple of course we have to change all of the sexes we don't just change the X at 1 point said America we change the X is at all points Rosa we have some Exs along here we shift each point to the neighboring point so this actually it NATO operation which changes all of the changes Exs by
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
hands and you would be left with exactly stop the condition that the wave function or not change when you shift the Sigma axes ones at being nothing but the statement that the left moving energy and the right moving energy should be the same make curious wraps conditions if you on string or if you have a fury of strings which says that there no preferred points on know cigna axis was not have any preferred point and that the wave function a state vector of the strings doesn't change will you just arbitrarily Carmen changed the stigma parameterization new shift stigma and that becomes the condition but left moving energy in the right moving and your equal that was a level matching left moving image Eagles right will be so the content of throwing away all these states which don't match the energy going around this way than go around that way that's the content of saying that we're talking about a period of strings the which be string does not have a preferred point me segment of might have a special point calls who 0 I find this a very beautiful fact in many ways and it throws away on very large portion of the spectrum that you can make ever happens that throws away just a portion which you can't fit together until angular momentum multiple IPs of sensible particles throws away for example well throws away big chunk of the the spectrum leaves a photon photon wisely the photon because the photons are open strings and doesn't care about us Italy's the Gravatt and it is a dilettante and the axiom as the massless strings and whatever else was there sir cerebral work out for yourself will be the next level next level are you make some sense out of them anyway I think that's what I want to do today it's a lot of them are RUC areas some things you can understand but you gotta be God and somebody asked me an interesting question well 2 interesting questions from 1 of them was simply the confusion but the confusion was obviously because they are quickly I just want to remind you again begin again when he left moving right will have used to terrible are clockwise De clockwise I inadvertently used tournament to different ways our distinguished and 1 of them had to deal with In late in mind said and I had to do was waves which were moving along the plus Sigma actors the miners actors I have nothing whatever to do with orientation space the other way I used it wasn't talking about polarization states of folk time talking about circular polarization said right circular power polarization left circular but that was really in genuine space X and Y space so we must not confuse those don't even know I the same terminology for question there are as tweeting going through some of the technical details of street theory what is it what they're what kind of construction is going to it are as worked on the road at the core of philosophical questions were not sense not from suburb good philosophical questions of a definition that question about this was not a bad question was whether it 1 way or another user any either evidence of reason the believe Red In some sends strings are the most fundamental things could strain as being made of other things is I think it's kind of a question of leave mocha reductionism a some sense in which strings are the most fundamental things in the world is smaller than that of native anything and that I gave an answer which was the ultimate head Joey ultimate waffle waffle and that was we had learned that that's not a good question so it's a little bit of about How we go back to the issue of monopoles from all Member mother Paul why because someone AS the question which is more fundamental electron the Mama called is not home while fundamental Orazio electron more fundamental and I wanna raise this issue Of the of the I'm so let's suppose they really are monopolist quantum electrodynamics easier the unelected anemic so there are multiple NASA which is more fundamental element remind you look from Oracle before electric charge times the monopole charge as equal so piety In order for what in order for the Dirac strained which is the solenoid which is connected to land on a poll to be invisible this is a condition that if you have a mom poems connected to a long string strongly with make monopole mathematically charged particles which arable around the string from detect over aphasia Eappen's killers equal to that means if we elected charges very small frazzled electric charges very small we get to do quantum electrodynamics in the way that we will learn how to do it find their rooms so for fun and rooms are not very effective if electric charges large were not young because each side in diagram contains a bunch of vertices region artists each vertex hasn't square the probability 5 he is a large and it means that the finding that the values of the rooms get bigger and bigger and bigger as the size of the diagrams get bigger and bigger milk merge you can't get them up don't converge to anything so find diagrams are explicitly told the studying theories with small charges just won't work be useless for theories with big Chargers and he only have a theory which has a small electric charge from electric charge a small but if elected charges small the magnetic charges very big if we tried to move it did change electric charge in the magnetic charge we might think electric and magnetic fields this saying the same thing but they're interchangeable Maxwell's equations are the equations for electric fields and magnetic fields are completely symmetric with respect to whichever of some signs proposer Road you couldn't deal with an electric and magnetic just completely parallel with respect to each other so I suppose does
have a magnetic charges had only know which Of the 2 kinds of charge electrical magnetic is more fundamental so they might say OK let's go back and try it working with a magnetic monopoles as the fundamental charges read will find in exercise into changing electric charges a magnetic charges he could do it now it's perfectly doable but you'll find out that if you tried doing the find their rooms in terms of the magnetic monopoles cluster magnetic charges large you they wouldn't converge so it's useful to think of electric charges as the fundamental objects now another friend the magnetic charges being large that suggests that a massive amount of Paul be large wine because they have electric demand but they have field energy associated with the field energy of a magnetic charge will be much bigger than the fuel energy of an electric charge and so the big heavier because strongly interacting and that means bad a magnetic charge will be very effective at meeting of photon electric charge with amid photon about 1 out of a 130 % Percent of the time we make this charge will and Tom 137 squared times stronger so this magnetic charges do me while surrounded by an incredibly bins of photons but the font times of amid their had very strongly with pairs of magnetic charges make pairs of magnetic charges as could turn the magnetic monopole interlude very very complicated things was all kinds of internal structures designed to spread out over a larger volume it's committee heavy you it's gonna make it complex and that's gonna make it useless as a starting point for a tie does that mean that the magnetic field that the magnetic charges are any sense less fundamental well that I think is a matter of taste but he is I can tell you you could start gradually changing the parameters of the increase the electric charge Justin number in the the equations you could imagine slowly increasing at an tactical really imagine slowly increasing need the magnitude of the electric charge and at some point will become equal in Q Beyond that Q will become smaller than eat what happens the magnetic charge becomes more of the magnetic charges start to play the role of electric charges originally electric charges become complicated heavy things magnetic charges because the simple like on the question of whether something is can't positive or a fundamental is should really be asked him away in the following way should ask is useful the Think of it 1 way or the other and whether it's useful Wal-Mart may depend on the values of the parameters of the theory they depend on the back they may depend on the environment there might even be situations where you air control knobs it you couldn't turn which 1 up turning magnetic charges into something more fundamental electric charges this really can happen imagine this happening and so there's no invariant ultimate entered the question which is more fundamentally magnetic charge of electric charge it's a question which is useful remembered this this question came up bond Sollie conference Swanson Texas must 20 25 years ago I don't remember I was giving a lecture lecture was are the Higgs bones on the question was the Higgs boson on fundamental or is a composite when I was describing the theory in which the Higgs bars arms composite and Eugene Wagner famous Eugene winner raises hand vast means composite and that means things made out of a peace is so for Apple vast means composite return again is not remained composite work composite means is that it can be broken out the kid case it can't pick a thing can't dictate such fundamental means it can decay and composite means it can became break up a conformal part has somewhat Eugene you think that that means that hydrogen atoms fundamental because diagram state of a hydrogen and fundamental because can't became but you do not and we we sort talk about the product on the proton can't decay is its fundamental and the neutron not fundamental because it can begin the whole argument that really crazy nobody was making any progress in explaining why Everybody in his brother I had some opinion about what composite fundamental means and finally Gerard I was always the most sensible person these things go up and simply said I think it is fundamental when it's useful to think of this fundamental and everybody shut up because they knew it was absolutely right our that is now Australians fundamental there is 1 set of parameters in the U.S. strings are made up part of things called D brain you change the prime letters and find a D brains are made up of the string so it's a question of useful their ranges of the parameters of the theory word is useful to think of the strings of the most fundamental objects you could shift the parameters and find ranges of parameters where the things that called brains which previously had been thought of as big fat composite objects similar to morph into Allianz things and the strings themselves blow up in the big fat composites on this issue a lesson that I think physics has been driving after some kind of so of continuous mocha reduction Missouri smaller things made of smaller things made of small things and some all open sense of what the most fundamental news that does not seem to be the way things are going the way things are going is that things morph into each other when you change the parameters of the theory was fundamental to become composite laws composite comic fundamental what the ultimate lesson of the land nobody knows but our it it is a pattern of emerging West for years both in quantum field theory the era electrons and also in string theory so will hopefully get to see some of that Armed with Ernesto think we're close open strings pickled do is jump to a new subject which is string theory of course but as seen from a totally different angle the Englewood Colo In theory what he really is and then howl in theory becomes string theory or we can talk about yet is why 26 dimensions can't dimensions we can talk about that but we can talk about everything sort laboratory work in Haworth relates string theory Korea Electric was crucial for
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Formale Metadaten

Titel String Theory and M-Theory | Lecture 4
Serientitel Lecture Collection | String Theory and M-Theory
Teil 4
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15103
Herausgeber Stanford University
Erscheinungsjahr 2011
Sprache Englisch
Produktionsjahr 2010

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (October 11, 2010) Leonard Susskind gives a lecture on the string theory and particle physics. During this lecture he focuses on closed string theory as opposed to open string theory. String theory (with its close relative, M-theory) is the basis for the most ambitious theories of the physical world. It has profoundly influenced our understanding of gravity, cosmology, and particle physics. In this course we will develop the basic theoretical and mathematical ideas, including the string-theoretic origin of gravity, the theory of extra dimensions of space, the connection between strings and black holes, the "landscape" of string theory, and the holographic principle.

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