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String Theory and M-Theory | Lecture 8

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Steiger University let's talk
more about can't formal map and this is a very interesting in their own right there are widely applicable all kinds of mathematical physics problems the most direct is electrostatic problems electrostatic problems tool dimensions there are problems in theoretical physics which mathematically equivalent of electrostatic problem in 2 dimensions I say incidentally electrostatic problem 2 dimensions I don't mean that you have particles charges moving 2 dimensions which interact with each other with three-dimensional call forces and particles which move around electric charges which interact with each other with mathematical 2 dimensional cool forces as anybody know what the Coolong force between particles is in two-dimensional you are not 1 of our squared about the potential energy between particles a library and because the rim library has won a lot so in a world widow world where the lines of flocks key in this tape out into 3rd dimension which are really forced collide in the plane In such a world electrostatic problems were have gloom forces which 1 alarms are squared and the equations electrostatics would be towed dimensional versions of the corresponding equations 3 dimensions you allow the equations of electrostatics on let's right down the blackboard from armor Wadia described an electrostatic problem by it is god of course by any electric field but there's something simpler electric field vector fields of vector what simple Our charges but a scalar Ms. electrostatic potential electrostatic potential which is sometimes it goes under different names but let's qualified what's the connection between fine electric field the gradient of 5 is electric field gradient of Friday's your field and what's the equation fully electric field in electrostatics divergence of electric field is a facade density charge density of a right-hand side rope chemical rewrite that in terms of 580 by writing that's alright this out in detail a moment ago squared off I was wrong they that flies he about world of Bell square fire was no squared find me for a two-dimensional world related to the world let's say with coordinates little axonal that was dealt square Bell squared the 2nd derivative of why would respect X squared plus the secondary fight with respect why square and then on the right hand side is role if this charge density it's not 0 places where there is no charge West is no charge and 0 remember last time last time we talked about this equation and it's in variances conformal invariant this equation this equation is invariant or unchanged by any change in the x y coordinates any transformation and no coordinates x y coordinates such that that transformations calm formal can't formal means angle preserve another words that when you make a transformation would you be angles about ballots the society angles and no court system are perpendicular 5 and that angles In no coordinates are exactly the same angles between curbs look what the same between the your court can't formal mapping of the conformal mappings and they are the kind of mappings that cartographer is used in order to preserve the shapes and the angles on a map although they cannot in general budget of the sizes of things OK so that's an example of a place where come formal mappings occur other places would flow into a dimension His were also equations of fluid flow all kinds of problems at least approximately described by you our classes quite fast is known is not lead equations is a wave equations because they have a plus sign these are electrostatic equations and they don't describe with motion you have to and you want waves you got 18 he aware there's a reason why we call electrostatics electrostatics is nothing is moving including a later said federal intervention also is fairly dimensions to the question of which come back to it but if there is a growing Corolla 2 dimensions is if you have any field like EU which has 2 components of X and Y the curl only has 1 compartment two-dimensional world with Karole has warned that component and it is a a derivative of acts with respect Hawaii minus the derivative of you wiII with respect to x best a component of the Crow you could think of it as the component of the curl in the direction perpendicular to the blackboard but you know you could be in two-dimensional worlds where there is no direction perpendicular black storm just think of it as the girl that curled becomes all 1 component object in scalar but in Europe into village but only in 3 dimensions the curlers vector and higher by mentioned something else to form OK let's let's talk a little bit more about come formal mappings very interesting what they allow you to do incident where he is From the solution of the problem of electrostatics problem for
example arms some kind of space some cyberspace space perhaps rebounder use With some boundary conditions but you can derive an infinite number of other solutions With problems with other boundary condition just by mapping the surface and picking the solution with you and mapping the solution to solution by conformal map so was saves all sorts of applications of conformal mapping electrostatics problems we're combat collector static Snyder some point but let's talk more about come formal match formal mapping as the mapping of the plane the ex-wife pouring for the plane to itself years Cartesian Plane described by X and Y analysts is a complex coordinate Z equals X plus I wife the usual complex planes for our usual complex notation representing the point that we wanted or mapping of the pointer with South warned transformation of plane the W plane over here W cyclically plus IV you and the fee incidentally for obvious reasons X called real part of Z wise called imaginary part likewise for you and a mapping from VAX plane the W plane is just a little given any point in the X Y cycle it a unique let's assume band mappings a one-to-one Nova works for every point explained a unique point on these W. plane and likewise for each point W plain areas the unique point of export that means any pouring could thus a wrong member your America ready point W plane is unique point z planes and free point of zinc playing view of violent about this means you could take a deep polling and represent that either by coordinates x and Y all are coordinates you and and that's entity the coordinate transformation of our you'll V function of X and Y or you could say that W is a function of Ziggy so let's think governor and vise versa but for their effort today I'm going to think of W as a function of the city give it a point Zeona complex plane uniquely picks our point W is a complex functions Berkey argument the function Z and function of W. are complex numbers right now there is a calculus of complex functions like this complex variables theory and the overture spend the few moments are is both an integral and there and differential calculus but would concentrate on their differential calculus she given a function w we can try to define a derivative W will try the fine unique 3 years away we would do it start at a point Z and every point see that corresponds to some point of India W let's move the point a little bit busy plus dog but here's a Z and motor point Z plus delicacy when you do sell W 1 W passed up a W W his W. plus Elbert W what is a derivative of W with respect disease at the point z obvious it is dealt the W provided by Dell busy in the limit Belper's egos 0 usual logical derivative is the difference between the function and neighboring points provided by the difference between the coordinates of neighboring points only 1 problem something new you could approach the point z from any direction it's not all clear that in general even if the function as nice and continuous and has also took good continuity properties it's not obvious that when you calculate the ratio built the W by dope disease and then shrink the size of this thing and so will be independent of which direction you coming in from you might get 1 answer to come in From here to make another answer to come in from might get another answer you come in from here so the question then what is the condition if you wanna have a nice mathematical framework for complex functions and you want to have the notion of a unique derivative think is no good definition of the unity of the derivative but let's see what the conditions are met these derivatives the limit of dealt the W exists and is independent of which direction you comment from now show you a sufficient condition nuking along the Croton necessary conditions it is a necessary condition our because it's almost disease prove as what we will do our we've to to you from work to prove what I would do is work out the necessary conditions by asking that the race here Of the W over Delta's EU is the same if you come in from the X-axis along the x-axis or you common along the y-axis another if you've got the Z is along the x-axis or along the y-axis you get the same man it is then not hard to prove that if not satisfied that no matter what direction you coming from you get saying so how simple thing you could do slightly more complicated it's only a bit of algebra but I don't like doing too much algebra on the blackboard OK let's not do it I felt that w by Delta's 1st of all what is dealt the W W is you Class IV well it lets what's what's
not with scored W by Desi where wreckage pre-med are not allowed to use calculus every time is a derivative call it dealt W by Bell your whatever and then were not officially using calculus and that's legal but that I secretly cover he realized that the derivative Yad El Reno Burrell was afraid if I use it the code on but using calculus Inc Pres OK so the change in W. divided by the change a way of course is that lets people living there and there are a lot change the small change you plus a small Times small change in V as you go from here derivative of a differential of you plus I'd be provided by the differential of X plus might be wise it's very straightforward now that's calculated if I come only in all along the x-axis come along the x-axis BY is equal to 0 BY is equal to 0 if I along the the x-axis so this must be equal coming along the x-axis and must equal DUO might be X plus IDV by DX were keeping wife fixed another ovarian why this is the same as the partial derivative of you with respect X plus I kind of partial derivative of vie with respect to acts of sex it a clear they don't come along the y-axis come along the Y axis the X is equal to 0 why is not 0 0 and the parcel the derivatives a partial derivatives with respect the white so let's see what equal if under the presumption that W by Desi doesn't depend on which direction you commit fraud OK so what do we have to have that we have the have won over I want high comes me I'd X is good 0 is 1 of a line from the world by multiplying BY and they will have a parcel of you with respect Hawaii and then I thought I would just give you a derivative of EU with respect wife UN V are real I I neglected to say so maybe been obvious you would be a real W which is complex X and Y a real disease is complex it's at least unnecessary condition it happens to be sufficient also but at least the necessary conditions that this expression equals this expression necessary for what necessary that derivative not depend on which direction is coming in from a game now so let's take the real and imaginary part when every other equation the involves real imaginary parts really stands two-week wage equating real part and equating the imaginary part solar cell of the real part is a real plant that's got equal the EU by X must be equal DVD by wife and what about the other 1 the other 1 Sanders see that's the real party is the imaginary part of Grover here on a DVD by X equals 1 although I d you buy wife you might be by X has to be 1 of why you but you 1 that's equating the imaginary parts let's multiply both sides by and use the fact that I square is mine this war so get two-week wages 2 up we look at them 1 says deal by the exit equals Devi by Y and the other says might be y equals V by DX not mind Stevie by PX 4 of them has a minus signed the other doesn't if these equations are true and the derivative doesn't exist in a meaningful and unique way if they are both true then the function W of Z has a derivative and derivatives well-defined and the function is called analytic just name function of a complex function of a complex variable has a unique derivative from every direction called analytic at that point was Alec and analytic everywhere as it's called and Kira all also of named narrow hold more end of understood what they were meant as a non Article I was pulled out of the home or for use in the story of a fracture of fancy names for everything I've learned recently that a mapping which takes a point-to-point is called by ejection a blink back for the world Danny's coordinate transformations of the feel more physically right but only hours courtier transformations by smooth orders OK so we have now the necessary and sufficient conditions for a function to be analytic every place where these a set of those let's look at these equations more just for fun they said their coupled equations for you Camille uncouple lemon writing equations is for you and I'm an equation should be yes we can use to track armed Webb's differentiate this equation with respect acts that gives us these 2nd you'll buy squared the 2nd partial derivative with respect to X and that's equal the DVD that's equal the 2nd derivative L V with respect to X with respect to Why mixed partial derivative respect Texan white let's differentiate this 1 with respect the wife that be 2nd you'll buy Y squared that's a quote from my
b 2nd V by DX as the same as acts of from minus signed so it follows that we can eliminate it and 1 equations ears these 2nd new by B X squared minus is plus misses you govern or we just Graf just a better Ed them and memorize side will go away what familiar requires and has just look pluses equation in 2 dimensions so necessary condition or something like it fully some as a whole lot that's just what guess that's excellent this is not 1 of them was Corrowong IBM's divergent depending on which way you define the components Our now if you think of you these components that actually have think of you and Mike is a component back there what it says the curl and New year divergence of vector vanishes OK so that's that's this equation energy do exactly the same thing we differentiate this for announced that it would respect X we differentiate with respect Hawaii and differentiate this 1 with respect X will we get is exactly the same equation 3 2nd lead by DX squared plus a 2nd team IDY squared equals 0 so 1st of all and analytic function W of Z. Khan teens tool real solutions of the plus quite that's interesting amount every analytic function every function which has a derivative and incidentally most of the simple functions that you're right down or give you some examples as we go along most of the simple functions that you would be likely to write down are analytic so in this way you could generate a large family of us solutions of plus equation now not sufficient have solutions of the class equation those solutions of world-class equation just to solutions or plus equation these equations link them together so artifact our it's not every pair of what plus functions the fine and analytic function really it's just 1 of them and you could do steal the 1 earned by a trick but we we don't need to go there appears that so something about abound please Equation 17 Bentley there were 1st of my knowledge there were 1st discovered by the mathematician Koshy and course all Causey and remark by Omar who discovered the 1st coach in Nicobar pushy remind equations and get a 1st hand hot dry crochet remark now I have got somebody snooze doesn't get any credit prefer but in this case she was famous mathematician announced in Remer physicist 1st that's a promise from ADM discuss many of their Yarborough murmurs traffic promise of reels of PC it said are set me calculus but your Berggren work OK because closer equations for each local our cousin are related to a pluses equations way now I want to prove that these mappings are can't formal that was the whole point here that can't formal mappings are the seem have these analytic functions mappings work would be transformations generated by analytic function they are the same thing but let's just pull but see how we prove that yachts that's just remind ourselves 1 of 1 small point about complex numbers but suppose we have a complex number but said so accords script here it's meant to be a small complex number because doesn't need to be small with us more complex number but it does need to be small for this purpose but it can be represented as experts IYA but can also be represented as another way in polar coordinates as a radius of course rolled Times e under mangled as another form of complex number those opposing we have to complex numbers but cost dealt busy because I really do mean it to be more displacement but suppose I have another 1 but scored capital Delta z each 1 of these are intended to be small displacements but for the moment it is complex numbers and this 1 is called role prime eatery I made a prime think about the ratio of these to the ratio of these 2 got busy all over the city 1st of all has rollover of prime we could make it simple and set Romo and row primed to be equal and that's because part enough to issue interested in tables what I get for the angle a get eatery figure minus they never prime important thing here is but the angle associated with the ratio is the difference of angle by a ratio of 2 complex numbers the angular part of it is the difference of the angles keep that in mind I know you do it is our go back the
CNW planes uses plane is W. Wayne and let's take Starting in that pouring let's construct Tulu at intervals 1 of them called built Z that same summer here recently my script disease and my radio squares use of the same arable dealt busy doctors we construct tool of is an angle between them which is different angles of the 0 Our the variables themselves to little intervals themselves now I can map every point on this point 0 . on this point in particular this point he goes here b a points also get transformed so little Middlebelt busy get snapped the little build the W take joined points here and that defines a little built W same likewise with little the Surrey built a big fat but you Bell W. no I only know assuming it's an analytic napping in other words assuming that the because remind equations of satisfied assuming that the function it is hour before W is an analytic function what we know is that derivative is well-defined and doesn't matter which direction you comin along was a derivative defy common along with Delta direction if I come along the belt the direction is just dealt the W over busy right that's the derivative if I commend Delta W divided by don't Missouri but supposedly come along this directions he then what derivative big dealt the W although big Dufficy now thought W is really an analytic function it means it is Horacio doesn't depend on which direction you come in all sorts as these door equal derivatives is calculated along with belt access all the little built the axis of the Big Belt taxes should be the same analytic function we can rewrite let's multiply with some but multiply this but big dealt city and divided by big Dr. W. food out Rugova W. because like us another way it's been dealt divided by middle belt they have multiplied down wherever provide but this 1 is a quote from big Delta W or move the W through Stuart I did if but this 1 upstairs and this 1 downstairs when I was this is the race oral big Delta's Ito Little Belt Missouri is the same as the ratio of big W tho about 4 w what does that say about angles here the score an angle data and this anger over here angle associated with the ratio is a difference of angles which just means the angle the angle associated with the number Thursday Delta CEO Belper's is just a angle between the angle associated with big built the W will double using angle between here was a stellar tells us they angles are equal another word the mapping is calm formal map every angle here when you make mapping maps the same angle on he doesn't say it doesn't reorient its as the angle between curves take any curve not that Bahia angle they intersect at will be saying that's a defining property come for more rapid so the call Wanda changes that are associated with them with analytic functions are exactly the coordinate transformations which of invariants plus equation what according the transformation the boss equation of change when you do those caught that's why in string theory analytic function coordinate their conformal mappings I show port because variants Of the equations of motion almost ready OK let's let's do a couple of examples of to interesting examples what's 1st check a couple of simple functions and see if it's really true that de la state equals X Z W equals Z squared but to really easy 1 W equals busy square as Jacob it's Emma going on Zia's X square is equal to X squared minus Y squared passed twice all eyes X Y and this must be equal to you plus W is equal to the square I 1st always you you'll is equal to 0 x squared minus Y squared the the real part of this side is the same as real part of this size so that sex grammar and twice square for about a week but Mr. ex-wife the where get it right that's a direct right OK let's check whether these satisfy the glasses equation bosses equations say that the 2nd derivative of you with respect to x squared at 2nd overview with respect excoriated if is don't 2nd took 2nd delivery would respect Y squared my cell 2nd view with respect X squared per 2nd River which 1 square is 0 OK what about works on 2nd derivative review with respect to x go square
0 to differentiate once eager why you can't differentiate with hexagon again likewise 2nd derivative source also was a good solution of overpasses equation right now but throws it every function it you'd ever be likely to write down unaware it was right when that way with trial-and-error which is not however on foot but that only fails to be analytical 1 lousy point so it's not worry about arm know it was a really bad 1 1st of all W that's that's very sorry 1 what about W. equals complex conjugate of W it was complex conjured more complex transmitters you reverse the sign of the argue this would say W is Quebec X my wife are you is equal to X's and V is equal come minus why I will leave it to you to prove that occurred year Koshy Remer equations work and record direct you a wrong signed our you could see you couldn't possibly get the right signed from both plus and minus 1 of whom was a equals X. vehicle Hawaii the vehicles legal minus wife Kim both satisfy the cautionary might equations or leave it to you I will do it now give us because I can't remember the kosher Rimmer equivalent of a layoff x equals Devi baby white man a minus sign gets in the way the Soviets is not OK but analytic functions written without ever invoking the complex conjugate complex conjugate are not analytic function but W recalls ZEW square but we would kill W. equals any power there's an analytic function but still another 1 of party let's do it W. equals Largo let's let's not do work army oversee rowers Gerber yacht W. equals no z equal the W What W equals Ito at a more complicated thing let's just check it and see that it's really analytic you could also the logarithm we'll have to do both logarithm an exponential wires was in those of the other arm every comeback how we know that a functions analytic the inverse of it is and whether they will shift exactly the angles a regal going 1 way you have to be equal billing the other way R. it's also true that derivatives of inverse functions are just 1 over the derivatives of the functions themselves and so if their independent of which where you come in stopped OK but strike Our W. equal just just to learn of the experiment around on Ross to your your use of complex numbers but W then is you plus IV right where's E z z is each X plus I eater the X Placida plus I why is the same as eagerly X which is really owe it to the X Israel automatically IYT right to the I want cosigner y for starters sign of what scion warning because I sign of war the OK circuit read or fuel from this you is equal to appear you would is equal to veto the X times cosine White and V is equal to eat for the X and sign a white to reject the cocoa remote equations with Chekhov by BX what is Reube IDX what happens when you differentiating reacts with respect again so the you'll by DX saying thank you bye DX is equal to eat X cosine why what about TV by BY that where we get when we differentiate this with respect Hawaii remove sign his co-star so this is again it will be the next cosine why that of the 1st Cosira quarries in his correct and you could check the other 1 source correct answer rejected each of disease is a good analytic functions songs of analytic functions are and already Ray CEOs of analytic functions are analytic except at points with the nominated vanished points with the nominated vanishes something is infinite and that those nasty point where everything breaks down OK let's take I took 1 W equals each city let's take W. equals logarithm of Zee as the inverse function W it was logarithm of Wu and let's see if we can see how this mapping works isn't portal napping in string theory W equals logarithm of steel even works in particular I'm going to apply this mapping for the half plane they have Plane here means take they have planes which has a boundary a half-point right Rucker about market were interested in the half point for the right here is a origin and let's see if we could see when we do the mapping where the points of the half playing golf is the W right this is a seaplane and here's the W plane OK 1st of all what about the origin of Z playing what
happens the origin of the playing the yards it plane goes to the logarithm of 0 where's logarithm of 0 other 0 of course not numbers but we can take a lot of Aurora saying he hanging number and the logarithm of hanging number is a very negative number so this point that he maps someplace however the Infiniti way off the left here let's not try to figure out exactly what it goes doesn't go anywhere is really goes out to our what about logarithm of plus infinity that's way out here the best summer illuminating let's take Web tape this plane and writer and the other former Zee is equal parts are E the
I later there was this hair
flying down he yeah that's the 1 they did equals what equals minus pie over 2 years maybe equals plus 5 years are equally I was the logarithm of authority to the I made a lot of authority to the Iife there's a lot of us are London z physical logarithm of our class I say that for us I think so where is a general point go it goes to some real number plus I think what the stated goal between goes from minus pilot to propose 5 years pirate too fading equals I played a miners ii pirated 0 minus I pirate to the upper India Gross do class by Pirate too In front of a line of this incident line here it's map this infinite line this infinite head a half life and the other in hair flying its map he theology being way out here so we've done is we've taken this whole space and we've kind of been the act cities and squash the holes based the narrow rebid the ribbon between minus I piled with 2 and plus we get Johnson curves Cecelia sued going on 1st draw some straight lines like straight lines coming out of the origin here they have to go to the Orange and they also have to go out to infinity and those warnings map straight lines like jazz fans at constant angle constant dangle next were about the radio circles which required next Winnebago vertical lines why is this strip description interesting from the point of view a string theory this was the way I described world sheet of restraint in terms of the accord would make political Cigna signal which went from minus piles overtook apart over 2 hours if you like b parameter along the string the other direction was the time direction this was the world's universe trend with the a string went from minus pilot to pilot tool in would and swept out time uniformly everything was translation invariant the kind no special time and you could see from this picture here that everything has a nice symmetry along with sacked How does the symmetry along axis but say we put everything what happens here on the seaplane puts it the W plane and shift everything overlap what happened it's shrinks and shrinks by uniformed factor to push the role after certain amount that shrinks by uniform factor 1st right explained so musical dilation violations transformations which expanding contract they transformed into they can't translation curious this is 1 mapping which is of interest this means that the string will she could be described as living on a half-point like that would be incoming strain coming in at that point right there being injected Ian Wright incoming string of 12 time in the remote past be thought of as being injected on the world's right at this point let's go another map and received or more said No no no now now not all are ankles 1 is some particular wine here these lines are lines of trucks are right follicles 1 is right at the center here no I did not an the not part of the various circles here this is our equals 0 Article 1 article to articles 3 Tom sorry hours not ours mercy review was ruled out there a stays where it might not be was Yarrow it's a mapping from 1 to the other of mapping from 1 to the other than the 1 thing you can't be sure of Is that angles on here when the here across the same angle that means among other things that of a last equation X X now being X on the the export of the strength of the law plus equation satisfied for outfielder for a degree of freedom living on here it's also satisfy here about radio assists symmetry heartstrings here you couldn't stretch performed b coordinates basis 3rd fit whatever particularly reason you may want to redrawn there is another form mother mapping of behalf Blank 0 before I do that but it Natalya what happens if you will map the whole plane by the same transformation you met the whole plane from the same transit routes and transformation the angle goes from minus pilot to what there might as firewood 2 plus 2 we could also let it go it see we could also let it go from our goods because from my list pie over to miners pirate 2 plus 2 pie fires to piety and becomes an a satirist stripped of the something new what's Nova what's what's different about this strip the original when you although ways around the angular direction you come back to the same point that means Starting here when you always around you come back the same thing and other words what happens if you may have the whole place where you get what that so now Philip Torres here's this end not identified with this and the top misidentified the bottom you get a cylinder we so sorry if it's this is a space that you would use to describe open string with Dewayne about this form Close right so it's natural think of Lao open strings you could describe on they have going close brings home the whole plane with would no boundaries that area useful mathematical fact now what's good for another a stake pirouettes forget his W. plane snapped W plane we're starting again with the same place ex wife so for about this 1 W equals Z plus 1 minus 1 the Celia seaward colors who there were
a war they would say say if With or what is equals W. plus warned over but we might 1 1 0 would strive exuberance Europe Ziegler 0 equals 0 go mice work right years ago the W starter Wednesday is equal to 0 W goes to minus 1 his W. plane W. goes minus 1 about when Z is equal to 0 but woods easy go to infinity way out here is equal to infinity or another with disease is very very large was the last 1 How about way Z is pure I'm imaginary appear it's pure imaginary and other words when a line is on the boundary of the half planes where is Agarwal that means that Z is equal to some real number kind I hard times I wear W W becomes 1 plus RIA are all over 1 minus I R I think with minors about 1 plus I Oral 1 minus special about that magnitude 1 right magnitude 1 the sum of the squares of the real and imaginary parts of the city so the race of these 2 and magnitude 1 so where is this upper half flying hiya Eliza the unit circle that of lies on the unit circle because it has magnitude 1 I'm not lower half line here it also lies on the unit circle so we're X is over the entire have falling here double not Expo 1 Z is over the entire have played W. varies over unit base is called gets over the unit that but what this is useful for well practice that is useful for solving go classes equations various contexts are you put on various charges for example in various places and you want to calculate electric field if you could do it In this geometry then you could transform the same problem this geometry and solve almost problems solved 1 of these is good for mapping back-and-forth electrostatics problems is used that way you're quite frequently but say that said the 1 other thing before we finish we met before finish before we started with these circles and these wines here what they look like army unit this Kia well I look roughly like there's the radio lines start at 1 end and go out to infinity but Infinity was already at the radio lines look like they're actually circles they they actually are circles circles are differ Maradiaga end head of AT and lines of constant radius here they look like there's bunch Abbas you get closer and closer pro remember the transformation on remain here your transformations on Caribbean which moved which translated the rebound back and forth in this direction they corresponded to dilatation of violations expansions and contractions here that correspond to some complicated kind emotions here those who not circles above circles but their centers are in various places this was not a circle clearly straight line but the limit of a circle this 1 is a circle with the way down here this 1 has its centers which he a 1 hazards Morris here of course this 1 has here Kodak 1st leg will cost summer sphere of the disk now dump refusal now if he did the same thing with the plane you we get a sphere where such a merger that yes yes votes atmosphere sphere but none abused fears just intended to be a mapping From be cleaned the right so we have no another representation of the same good job who left airplane goes to be outside of the laughter at Wayne right airplane this year at the inside of the who have airplane from the outside it didn't disappear just want to be outside the East bloc a says that over the when you take a four-point the before players so so it's a little bit complicated stereo graphic is a man 3 Baxteri left the radio left spokeswoman around year they begin at this point point is that for infinity equals infinity that's W equals 1 that's over here so every 1 of these lines let's take the 1 which is 1st of all completely symmetric between wait in the lower half at except the X-axis as just this line OK the radio lines above the axis about X-axis that radio lines below the x axis I like that Frank vendors a circles on here it is a circle every 1 of these circles around point here sort of have around start or a start on the boundary around this points and they come back to the boundary so they start on the boundary of the disk you go around and they come back but as you move adult God that is did turnover but inside out as you go from here so that curvature in the office of corruption yup yup they also circles that it is a fact under these mappings well and there are under this map this kind of mapping this is called linear fractional mapping the lineage of fractions made 2 linear functions on the linear fraction map in the collection of circles and lines transformed into a collection of circles of wine was so really is a aligned that transformed to a circle Our a line that transformed to certain circles circles lines transformed the collection of Circle Laurence Tribe circles post was not under every conformal mapping on the linear fractional transformation of recalled sometimes called these transformations may have nothing to do with me be stripped from something they do SYDNEY news it OK lipstick a brick were or what they say are needed a
lot of that that is necessary life is like the 1 now that's that's some particular radius probably reviews warned as a positive point goes to the origin and now it passes through the origin of the W so where is the origin of the W planes if a thing was logarithm 0 1 year now I'm going to try to give you a Fernley complete definition of our work string theory we set up a lot of the mathematics the the problem of course is proving these things work in detail and that of course is where b real difficult stuff comes but terms but the back you may recall that we talked a little bit about what scattering amplitudes look like how you derived them in string theory which started with a simple example where 2 particles collided came in into particles went out with that OK let's just take the world's of the whole process could be any infinite strip various buttons show particles economy and let's slid down middle like that 2 particles goal out time goes from left to right to strings coming to a strings go out of here a 2 initial strings I want draw aggressively snow blocked incoming strains and propagates Susie other incoming string propagate the strings joined former single strain in here and then after a period of time how much time the answer is you integrate all over time quantum mechanics you always of all possible pads from beginning to end 1st Georgia some of the amplitudes for all possible histories and in the end because ours strings again where calculate our remind you were to calculate on well we talk about several different versions but 1 version you think of a path integral end and the girl be degrees of freedom the coordinate positions we could define a single variable and Powell variable or to coordinate on this tool coordinates on this world but the the degrees of freedom harvest based kind positions the space-time coordinates of a point on the world sheet where are they XCU where Newell goes from 1 north from 0 0 being kind but they last won 25 of 26 dimensions 9 of the 10 dimensions but wherever the collection of coordinates are described the location of a point on the strain Khaled X knew Singh manpower figment power could be took warned on world there are costs the same as diesel X someone I see the coordinates two-dimensional space and those caught a video signal power on the axes coordinates in real space time the same how's just labels which label points on the world sheet the exits are their real space time and location of I know this based on location every point on the world sheet that I know of the world sheet there's a spread out over spent 40 don't we calculate a certain path integral patentable we recalculate each from my best the action we will trek on this reminding you that the trip we will trick of letting Powell go top hideout will work to get rid of various factors of why we talk about the over again it wasn't action the action that was integral over these Cigna Andy Powell of what it was the X New Cow squared minus X new squared plus plus BX New Square odd what exactly does this mean DX niobite square means the sum of 26 the X 1 by pal Square D extol but pal squared 3 of the last 1 of these things mean this is exponential of the action where we go that we integrate at path and the world all all possible ways of showing up for surface is often written the ex- of city and power this means path integral and means integrated ensemble up of all possible configurations of the year of the world Street in space time the worlds he fills up but How does this formula he at 1st of all how does this formula know about the the moment that the incoming particles the moment that the incoming particles we have it in my hand and it's inconvenient in this picture of a say in how we do it I'm going to tell you in another picture exactly how we do do it but for the moment let's just supposed somehow we have put in the momentum of the incoming and outgoing particles somehow in this calculation going integrate all acts now there's 1 more thing was about the time interval between here and here we integrate over it I'm so we're doing in effect his summing up all possible ways all possible histories of these 2 particles oscillating forming the single particle and the forming particles again all possible histories all possible ways of the world she fluctuates at integral over here it's an aide to go up the action as a complicated thing we haven't even yet put in the year new to rule because the rule is for the external particles a moment but before we do it 1 week course we had picture what was it other sports fan base of a sports fan base right butterfly called right now we can make use of our freedom from this picture tho other sex Our coordinates for the world's Con Format can't follow mappings is a very wide variety of analytic functions and factor variety is so large that you can basically map visited this thing has 1 boundary boundary goes from here he now how did infinity here what goes on in cities subtle but Woods jumped to this boundary was always around or just like the sport beverage goes around 1 has 1 boundary apart from the fact that group pieces of the boundary or offered Infinity worry about you cannot anything with 1 boundary like that anything else we want memory by calm formal mapping in particular if you so choose just for symmetry Yukon map this whole world sheet to a circle waterway exactly the same as I showed you had a map a single strip I didn't show you have a map export Band-Aid to 280 a tour biscuits a little more complicated wants give out many years ago there were wrapping in detail the remember it but you cannot be where are we coming particles was up points on the boundary the outgoing particles are also points on
the boundary draw some wines to just indicate their look Ray the vertical lines he vertical lines here they will look like VAX noticed these vertical lines over here so around the point infinity he ever experts warn and then all of a sudden they come together like that at this point and they go like this much of mapping from year to year looks like a horizontal lines and have a go there are more complicated as works like point is you could do it exact no need to know where these lines are absolutely forget where they are playing a role the point where they Split enjoying this diagram here is determined by the position of the positions of the of of these extra points here as you move them around these points closer farther apart when you movies these points together this way is poised at very far apart when moment together horizontally these points will together so same pictures the same love physics different pictures I want you don't hear you don't exactly the same thing take the X Bybee how squared you could now call that could now call these coordinates by different names if you like you could call an X and Y or and be as matter that Coward said accord niche you calculate exactly the same quantity integrated over here but now I can tell you what to do with the external particles in a role each external particle 1 factors in the into grant thievery or factories it's the thing which injects momentum the builder function white 4 K X Our integration variables Keyes are the moment K 1 came 3 K 4 the moment the coming in and going out to put in for each external particle each particle coming in you put enough factors into the into inter-Korean here of E truly R K X at the position bus caused point z product face each external particle 1 more thing goes it's a factor of heat the RIK X wait K is the momentum of the particular particle what's coming here so I haven't eaten IKX woman on the boundary correspond particles being injected into the Byram that that rural comes from quantum mechanics comes from where we could we discusses rule another kind I want to lay out a whole set of rules you do a path integral of all possible in betting News of the world sheet into space time all possible ways at this world she could be and that space Ito action and for each external particle and the Audi Katie X X is at the point where the insertion of that particle happen on the world sheet here easy year corresponds to the coordinates of the pouring on the plane 1 less thing FPO finished you have to integrate all of the positions of these answers that's it that's the whole story that that is what string theory of this open string theory that's that's the whole set of rules peppered the through a path integral all all world sheets all the betting you could describe it honored based you could describe it Andre sport then made you could describe it however you like get out a scattering amplitude the scattering amplitude particle momentum K 1 cake to become king fall off the moment that no we did debate over the momentum we integrate over the locations on this year also those on research OK so represent the add value Bang meet place integration over right OK it does so those things I was not a reform can ease of 6 the please on the movement of the particles coming in going for it to get scattering amplitude for a particle of momentum 1 and to become a particle of I came 3 my before of the the moment our it is multiplied by its assigned missions inside that that he call that the action misses an hour earlier this notice it is a core vertex operators avert vertex operators mean spring some adjust the positive momentum in year Inter diagram they inject a floor of momentum and the diagram work that out of our currency with clashes are illegal decided we were never go inside the intervals don't side this integral known on-site that now we could write about what's right that clearly into of possible torture and the room path was just call captured the raw works not produce eagle Mama the action Ito minus the action this is action for a particular configuration and here it is the action times for each external particle a factory e for the RIT K X of Z 1 person's EIA KIT from factor for each 1 K 1 0 on the hour enough for a moment it 2 0 X Our Zito tool that got outreach external particle where's sees our positions on the desk where With the momentum is injected this is path integrated and they and integrated aboard the locations where the moment are injected into the river every this PAC IKX even tho it originally so if you have to eat the Jayhawks and I is different we should really write X Newell KNU next came you for each component of momentum and annexed Mueller at 26 Exs there 26 Katie's that's that's the whole structure of string theory will you be unit that working you that's 1 Zamudio air pressure direction the momentum is Riaz because they are the components of the moment here the components of America the X is appear get integrated over their part of the path integral here and the role bags now they could be censoring the X Baumert you could consider the be report of reaction if you like it could you could just put into the
actions of the UN them in our kayaks reach external particles this is the a place where the text is shaped the shape of the banned the town of Hangu in this drawing uses Drummond are example you Yukon Bronte any way you want right it's not real space where it's not real space to just a parameters based on right OK the amazing thing is that you could do the intervals meeting in May everything can be done but these are what are called gals integrals every quadratic in the exponent the integrals can be done and that's a good news bad news is almost always give infinity but another few in number dementia now I am not going to tell you about the wrong number of dimensions I'm going to tell you what the answer is like the right number dimensions and incredibly simple that's a solution of electrostatics problem it lowest an electrostatic problems you take every 1 of these Exs and think of it as an electrostatic potential 26 of them pain of them 11 many of them are think of them as separate electrostatic problem as a world now where there at 26 different kinds of electric charge 26 different kinds of electric field and 26 different kinds electrostatic potentials electrostatic potentials are these Exs X 1 X 2 X 3 explore what think he charges corresponding to them yes call is among 6 26 components of a moment there or like 26 kinds of charges each 1 the source of its own electrostatic field electrostatic field being called X OK yeah that here is the calculation that you do and it is really easy areas obtained by doing these integrals intervals 26 not 26 charges for 26 different kinds of charge right 26 different independent kinds of charge each creating its own electrostatic field I did do you take on this was this news think of it as a world in 2 dimensions on this this you'll have to get used to it armor the X . late Yum boundary conditions of electric the components of the electric field perpendicular to the surface of the vanished but for the boundary conditions you heard this world with these 26 different kinds of electricity on the boundary you put charges charges on the remembering each boundary point there at 26 charges to Parana the charges themselves other components I'll be electrostatic potential as a component of the momentum of this particle comes in with 26 components of Katie those 26 components of KB calm 26 charges that you put at that point 26 charges that . 26 charges at that point that would restrict drug charges this is a straightforward electrostatics problem 26 of them but it's easier it's do 26 of them as it is to do 1 of them well In principle even electrostatics problem with 1 kind of electric charge you could do the same electrostatics problem 26 times over with different values of the charge right of fixing the values of the charge and fixing these points where you calculate you calculate the electrostatic potential inside grammatical calculate electric field want the electric field you could calculate your for static energy calculate electrostatic energy of such a configuration this whole integral where electrostatic energy and then you integrated Over the points you move the pointer around In the great over them and that is the scattering amplitude 24 of the 26 dimensional particles to scatter for once said a moment that for the moment John lack the final upshot is very easy it's just an electrostatic problem with 26 different kinds of charges you calculate electrostatic energy in here what it is 26 their 26 new the mob 26 distinct 26 potential 26 electric fields he a house what it wanted 1 usual or crosses 46 waist and no cross terms between you take that electrostatic energy you exponentially that each of them minus the electrostatic energy the hall and depends on the positions of the charges and new integrated over the positions of the charges calculate the electoral 26 electrostatic energies 26 different kinds of charges that he knew that calculate electric field square the electric field why they just calculate Elektra static energies your and you exponentiation it you integrated over the positions and that gives you the scattering amplitude for particles whose components of moment on the scene as the values of the charges that you put there on these points that that's the whole upshot not a a program this can only can not now nor has more of a relationship through the applying the rules and he Barbaro varies from the U.N debris over you and debris over them and that's the string theory scattering amplitude our a fairly simple prescription to says this up for us to get a bunch of infinity from the path of the group which is away this is a final answer when it works where does it work it's just infinite of 26 and so I'm so short-circuiting it by showing you would be inserted director of a beautiful and support final went on but wants know that this is the procedure is nothing to prevent you from putting what small particles you do laughter
degrade over their locations on here but basically that 3 of the scattering of number of incoming particles amplitude within squared amplitude for any number of income particles to any number of our core particles now that we've had 3 small major small sector 4 3 long dimension we haven't talked about compact applications we have not heard our that'll be the next hour of ever be the next discussion straight connection to the because of the year Corbett said yes because recorded satisfied origin and connected to death satisfied with yes they satisfy the overpass equation not pass equation is the basic equation for electrostatic we have to call Electra static psychosis plus equation with some of the functions sources here Obama but the mathematics is the same as calculating electrostatic problem for he once it can now it's actually simpler than that you could use the cunt formal freedom freedom to recon formerly map things got arbitrarily fix the location of any 3 of them you should a degree of all of them but any given configurations you could always find the conformal mapping which will bring this point here this point here this point he said and that is only the 4th point for the case of 2 men rally it's only the location of the 4th point which to debris from here to here as a technical point place what a case I am these 2 . now be integral Albanese told a note or really only 1 integration which is the occurring between that same pot that's a good point could break apart at different points like that but you could always find the conformal mapping widow again so you have that freedom you have the freedom to pull back together a . how that ran well the answer LKB answer is that you have a of conformal In invariants freedom freedom of conformal mapping that allows it to choose any arbitrate 3 points and put them whatever you like you fix them you fix them put the first one he had saccharine he in turn around here and all the other ones get integrated over so the answer is you have in mind is and my three-year integration and my reintegration see what present here doesn't yet not rupture road hopefully corporate it was a lot of work that was off Russia make your Korea workers study it if you use a coordinates to reflect and we all have a far less cast many integration the integration you could pick 1 of these called 1 of em fixed the other 1 Arsenal all the way to Our from plus infertility might I just heard of half Abbas but there is a lot of great warrior operate let off that you do have to do that and you have to integrate all over but and the special case will you have exactly the same height theater will terminate when they enter seemed to go amazing thing is same in the world and in fact it's the same thing goes integrating all the points on the boundary of the this this is so nice uniforms symmetric picture which for particles on on the same footing and you can't call it now you can pull out push it in various ways To make it look go like different kinds of process to use by that Joe the ship Ms. is open string theory open string theory of particles Our injected on the boundaries of boundaries and these of the edges of the string and point of string the boundaries of what talk about
closed string theory Rubin next time on Jacob D mathematics of it at this level fairly simple rated fortress barracks these days this fear it is not is matter of with 4 was you can get widow I'm sorry is it's a a reasonable description of Iran's nobody has the vaguest idea what happens when you call by gravel France together so we have a good idea we have lots of ideas but the details of we're way beyond that there is a zillion versions of the bunk get this is I have described the 1 simple version of a string theory is zillion versions and the Brazilian versions connected with the way the extra dimensions of compact defied we haven't talked about that but now this summer in Rio all would expand has 26 dimensions as they compare with its permanent parable a gap that these measures he can write and you're as agrees with experiment the 1st thing you would discover if you do you work these scattering amplitudes and compared them with experiment is the number of bank mentions was radically wrong and he have the UK they get started with that we have all art work to do before we could make a sensible area how what do we want the number Bay mentions spaces 3 the 25 of them instead of riyadh but it's lot like not even integer she thought it's better in superstring theory number dimensions of mine which returned the large it like that set you better eventually will will back by saying that it is now but compact real complexity and enormous difficulty of the subject of mathematical sophistication of it is in the compact applications and it is a really hard that's where the difficulties are turned I can show you some simple examples of compact education sure you have to lead tool information about articles about gage groups and things like that but to do the real thing is far far only beyond me but far are probably on the subject There's nobody knows how put these things together make a realistic fears of particles with is a machine for making lots and lots and lots of different models the number of models or member of construction a can be made are independent of the thousands this is very much a work in progress but on the other hand there is a very very definite mathematical subject 5 of constructing such models they are highly rigorous and they produce enormous wealth of mathematical information about gravity about about mathematics but there's no question that has not succeeded in giving a theory of elementary particles or better yet succeeded in giving about can't above 500 series of elementary particles but which is maybe like that I think like that but we will find out this is I am not proselytizing for the period we tell you where they have like a gag now is a brief description must we talked about but no 1 was to preferred Oracle minors 12 member 12 times to a 24 24 what do is 26 wrapped in a flash case a lot which took a moment Wortman alive at 1 station you the only thing that 1 prize is some of the squares of the moment which is the next square of the particle more meant that is not quantized most definitely not OK but for more please visit
us at Stamford .
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Metadaten

Formale Metadaten

Titel String Theory and M-Theory | Lecture 8
Serientitel Lecture Collection | String Theory and M-Theory
Teil 8
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15104
Herausgeber Stanford University
Erscheinungsjahr 2011
Sprache Englisch
Produktionsjahr 2010

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (November 8, 2010) Professor Leonard Susskind covers the history of path/surface integrals; conformal mapping; application of conformal mapping in string scattering. 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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