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# String Theory and M-Theory | Lecture 8

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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
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
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
us at Stamford .
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Teilchen
GALS
Andrew <Hurrikan>
Kristallgitter
Amplitude
Stunde
Eisenkern
Kaltumformen
Pulsationsveränderlicher
Elektronisches Bauelement
Gruppenlaufzeit
Omnibus
Eisenbahnbetrieb
Elektrizität
Elektronenschale
Hall-Effekt
Werkzeug
Gleichstrom
Koffer
Source <Elektronik>
Jahr
Gummifeder
Kit-Car
Elektrostatik
Lineal
Ersatzteil
PAC
Luftdruck
Marsflug
Wärmequelle
Gesteinsabbau
Messung
Drehen
Magnetisches Dipolmoment
Summer
Parallelschaltung
Pulvermetallurgie
Stringtheorie
Gesenkschmieden
Auslenkung
Energieniveau
Dreidimensionale Integration
Blei-209
Teilchen
Umlaufzeit
Prozessleittechnik
Maschine
Fuß <Maßeinheit>
Amplitude
Energielücke
Steckverbinder
Stunde
Störlichtbogen
Lichtstreuung
Kaltumformen
Potentiometer
Gruppenlaufzeit
Kiesabbau
Tag
Proof <Graphische Technik>
Elektronenschale
Übungsmunition
Gleitsichtglas
Werkzeug
Schiff
Jahr
Source <Elektronik>
Hypothetisches Teilchen
Modellbauer
Schreibware
Computeranimation