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String Theory and M-Theory | Lecture 6

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Islamic strings of been part of the subjects almost a beginning quite beginning and as said they saw off 2 important problems prefer Barrow figure firmly on us and they get rid of the backyard they leave you with penned by mentions instead of 26 so they don't solve that problem that problem turned out to be less of a problem with more of a feature feature good future people expected occupiers are mobil contoured merits a subject known as compact the occasion taking the extra dimensions and doing something with them to make them not purist or mile mark euros an interesting feature of theory or come to act on how well not to talk about a little better it it was not really historical but Of the scattering of strings a subject really began with studying scattering of particles elementary particle physics was always about scattering of particles not because it's the most interesting phenomena of it can happen if not adult you send some particles together and Barger junk comes out we're all sorts of directions but about all we can do with way Express and so we try to unravel from the scattering beta what was going on inside the collision inside the collision of course means the properties of particles so for so the natural Hong full of experiment the scattering and the natural thing that if he restored Vasquez if I have a theory of particles how do you compute the scattering amplitudes scattering amplitudes 4 was a scattering amplitude scattering amplitude you have some incoming particles part of your incoming information that I have time this way and I horizontally and vertically a wide variety of articles coming in something happens inside a black box and particles go out not necessarily the same number of particles the particles come and carry moment of course a carry other things that can be every charge they carry labels like for example of a new honors failure are or ever have Bob let's simplify story and not everything except the moment particles come in and they carry former mayor from 4 momentum means Energy and Commerce the strike Iowa for rector of energy momentum is the energy and the 3 components of momentum of authority in 26 dimensions we are 25 components he August or account for them for vector of wrote the 1st of 4 vector each particles Harris a woman the owner cause Katie Katie New cross from 1 before far from before excused from 0 to usually course 0 1 to 3 the components of moment now well you know all about the components of 4 components of momentum or particle may have something to do with the mass of a particle of course they are the energy and what's the relationship between energy and momentum match for wrote the particle remember see is 1 we will take wire he called warned what city connections between the components of energy moment the mayor Hey squared a square equals P. squared plus and squared also whatsoever following way he squared minus be squared equals M Square and just look forward to keep notation consistent where notations of physicist for many many years I'm going to write this as peace squared minus he squared equals of minors and square same formula I've just taken the negative rubber now that can also
be written in terms of the components of Katie in terms of component of Katie Earth's came not squared sorry it's came back squared minus Katie North square North standing for the time component each spatial component squared minus the prime component square is is often just called case squared just garlic a square left inside here it is a square by definition definition of a relativistic square of record it's called POP mural for cerebral Friday let's just call case squared space component squared minus harm comport square but called
case quotas and fled every particle K squared is a minus M Square you can't Mary Kay square Of course squared consist of the energy and the momentum when you say you can vary case square doesn't mean you can vary the momentum it means when you know the momentum the energy varies in a certain way and the way the various that square feet square off Cape Cod North square my case they squared equals square so it match were 1st phase of our valued by collisions is just about particles you characterize them are therefore moment 3 components of which are independent before component must be subject to this constraint now play a bunch of moment there but callers Katie 1 from fresh particle wasn't aware label them a key of Fisher Street yacht thinker called was worth OK warned us told them going on here quietly incoming moment can be unwilling to call the outgoing moment Q. What is also a former back work or to free Hugh form what very simple case in which 2 particles Goto particles Q 4 these particles from the in particles go out how do you represent momentum conservation of momentum and energy conservation momentum and energy conservation or simply bad OK want Pocekay too fast for vectors is a quota Q three-course Q 4 all compartments space components fine momentum conservation the current components of foreign energy conservation OK now because of the perversity of physicists what physicists like to do is redefined outgoing momentum and think other a common woman not crazy the outgoing member not going in common with the incoming but they don't all we really have to do to make it symmetric with respect the incoming and outgoing moment just take each Q. What queue changes signed an accord with Katie when pay it that means changing the sign of its energy needs changing sign of all of its momentum it at say trekked to be able to write this is a symmetric form instead of writing K 1 Seward here's warriors K 1 escape through minus Q 4 of 4 minus Q 3 equals now becomes KEY escaped to ask a free pass K for equals 0 you treat all particles as coming but you have to remember that the label for the outgoing particles as labeled with my miss actual moment Maria Court particle but once you do so momentum concentrations completely symmetric between the 4 between 4 articles a youthful track a useful trick that keeps labeling especially persistent Norris that when you change the sign of a momentum to redefine the momentum of a minus sign it does not change the fact that the square of the moment the music with the miners and squares so we have in this particular process we have for a moment more was all while a car although the energies Of course there may be negative for the outdoor ones each 1 of them subject to this constraint we question whether physicist were asked about this collision is what Izzy amplitude was the probability of they may have put those things square complex numbers you square defined the probability format coercion but the of functions on a number of variables a function or a probability for the coercion is a function of the moment of the incoming particles as a function momentum of the outgoing particles sort of function Ava Katie's but Scott amplitude Bingley's square as a function of all of the cases K 1 2 33 cared for butter some rebound information via bus full about because of a 2nd of all the square of each case has delayed to work minus where sows really too many variables here from many independent variables family independent variables are great 5 it before we impose any constraints each momentum has 4 variables this was for for 8 before is 12 for Morris 16 that's all are variables was going to depend on what should at least you really don't depend on that many variables let's think about physically now comedy variables a scattering amplitude or scattered across the and well the 1st thing you can do is without comment got off you can you always relativity go a frame of reference where the sent of mass where a center of mass is at rest other words with tour make to space components of a momentum are equal and opposite 2 you could always do that if the particles both moving down the z-axis well you just move fast enough to be halfway between them if they also happen to be moving in some of the direction does move in that direction you could always go toe a frame of reference where the particles are equal and opposite the space component Samoa was a special moment that next you could always we want to the system so the moment the upcoming Ian along the x-axis what's left over in the initial state was a depend on if you not at the moment are equal opposite what dissolving depend on and only the pair was initial state and on the initial state only depends on the magnitude of the moment from Apple Manitoba square of a moment I'm sorry the magnitude of the need for a vector before that passed out magnitude M Square but magnitude of the speech moment or the energy if you like what you got past the only thing left over is the total energy of the collision In center of mass frame so that's 1 thing each center of mass not surpass that are particles or all of the same kind of simplicity of particles collide what do you
say about outgoing woman from 1st figures they have to be equal and opposite white rats momentum conservation of momentum conservation they have to be quite opposite or about the energy of the straight it has to be the same as in energy of incoming state was the only thing that can differ between incoming and outgoing particles the angle it the angle of scattering I thought this valuable through all of the terms 60 variables really only depends on who would be independent variables both variables should be expressible in terms of relativistically invariant things after we could figure in the center of mass but would also be ought to be able to think about it In a frame of reference we should be able to construct a truly independent invariants relativistic invariant which are not completely characterize the scattered so let's talk about me and barriers describing a scattered across but 2 particles committed to particles go out the governor's all coming in changing momentum clearly was a scale you won key tool K 3 K 4 what can you do we were for vector what you do it for years to construct and variants is only 1 thing you really Yukos square that's about all you could do but you don't have to be talking about K 1 square or to square Jacoby talking about 1 plus Kato square right that's a good agrarian K 1 as a former rector Kaori cost caper R K 1 plus case square Merrimoor that means that means the space components of candy they want Pocekay tool which our way board arrows square might this time component which means the energies whatsoever Werner K. not King North means energy in order 1 and north to square Usama moment square you only energies square and subtract and that's called 1 plus 2 square but Celia figure out what it is but an invariant quantity lets you figure out what it is by going to the center of mass rape in this don't mess frame the momentum of this particle or to recall well K the space compartment K 1 where about the momentum of a Espada or autocracy my 51 so the toggle space component of momentum in the central Mediterranean 0 when I thought about this 1 while others from but Massey energies we didn't talk about it but what about the energies of the 2 particles in similar match Frank very Kwan opposite they have the same mass the energies are equal frame as 1 in which the particles are perceived as moving with exactly the same momentum magnitude of momentum and so little energies are the same so this just be calmed twice Cahill quantity square simple This is the center of mass energy here in the center of math frame UAE added to energies the probable energy and the squalor
of it is just the square of the center of mass energy sorry this quantity here which is called a s just call minus X is nothing but a square of the domestic energy what Garrett more the center of mass energy white it is sent don't mess frame misses you wrong and this is just a part of low-energy so my acid is just a square of to square the their strategy Squilla center less energy recalled acts now you think of any other variants that you couldn't go How about Katie 3 plus squared we took a warrant was paid to squared about 3 posted square that's a say their because K 1 takeover is might escape 3 costly for energy conservation so we'll get OK wanted cost Kato square was saying was Rico Foursquare Betsy energy minister says that the energy in the initial stages same 0 unify what about K 1 plus free square that could be something new so let's see what it was but Celia figure out what K 1 class 3 squares working in the center of mass right in the center of mass frame or other particles in an hour have the the same total energy have the same energy each part accords articles just scattered through an angle or so what happens saw the Conte and they go out with the same energy but or mangled all so let's take K 1 plus 3 square Net Irish care what force K 3 square Case 3 squared warned K 3 square How about this for the Shire reference Jason escape through a former back from Harper our a bigot what articles have Australia energy incoming territory or ignore Scarborough yet 0 0 1 0 1 twice Katie warned force theology got because K 3 and K we flew up the sign on them so and this is just K 1 plus 3 square so to with batters comes in particle morning comes in and of course also particles but particle free goes Our would part 3 1 goes in 3 goes our but if we all label these particles were Katie's Katie 1 much cause for now Q Q 3 maker outgoing particle than really what this is is it K 1 minus Q 3 square it's really the difference Of the momentum of the incident particle and the final particle is called the momentum transfer some momentum transferred if you with the thank God particle particle as same species apart then it would just be a college and some momentum transferred from 1 of the 3 being as a momentum transfer end about the momentum transferred from 1 of 3 of momentum transfer it could also be expressed in terms of the angle scattering Co you Europe 4 euros before Iraq before K want free squarish rears it's just twice energy square square mile relay would die water the energy of theology VS variable twice square square the center Matt kind 1 minus the cosine of the angle of scattered this is the interesting thing he be angle scattering angle between 1 and 3 K 1 comes eaten Q 3 goes out the particle gets deflected through an angle of that deflection angle years later the
deflection angle of a particle but between the incoming state and Short Street and that's what's here while that interest will call measured and express you put detectors in different places you scour particles and you find out of probability for them to get deflected tho angle at different values of the G this variable K cost a lot to us we do it I'm sorry we made our definition of S erase the ACS over here the definition of S was K 1 cross Kato squared how was the definition of 1 plus 2 0 coming in there was a square of the center of mass energy asked equals Energy amassed swear they want 3 square that called graduate may pick up a meeting for it but she has called pay February get trapped hot and that is some combination of energy and momentum transfer at the center of mass energy squared minus the mass squared Biggio weary or MMS a years trying 1 minus call sign Of the angle scattered I so we have K 1 plus take 3 squared How much Kato Pozsgay squared different but go back after he s K 1 plus case the regime is In this case to costly for took a 1 Kato squares same escaped recourse Foursquare fossil another combination we have a wander escape till we have K 1 escape 3 Gaylawn posted for right but some are ordered so a one-off quantity K 1 plus K 4 square what can that be well while I really do and maybe will caught my arm Cheney K 1 skate 3 square your mind to because busily T you variables about what's going on he only 2 quantities that Lee scattering depends on energy and angle of scattering we're seeing that 3 independent things only answers are not 3 independent if you use the momentum conservation and use a fact that each case squares and square what you'll find is best course keep plus you'll is equal Tom ideas forum square or something like that as a constraint among you're not free independent won only 2 days earlier so interesting quantities of food warn is dependent bomb your who did he considers that dollar the word that here just say there should be a closer sure back actor where's participate cycle With source with me there is no period that is so they are going to Eurostat if not for the fact that while got out it could be years her possible world were back you purchase it was not our world Our momentum conservation says the particles are arms Bamberger retracted I want a phased that that that's what if I said I did a lot now that frames the find of the frame with the moment door equal amount at the moment that recall opposite the energies of particles if messes are now is together but I have to go out equal and opposite for momentum conservation young so that why these variables S T N you Bureau warned which is where called Mandelstam variables and that very
symmetrically defined most Mandelstam man also varying can't Cleo active physicist Berkeley this notation columns from the early sixties vary early sixties Mandelstam variables end notice with that but they have they have this beautiful symmetric structure that's what's interesting about there's no let's not talk about finding Byram's for moment not just tell you the answers for certain farming diagrams let's suppose 2 particles collide this is a fight with Barbara create a particle of a different bastion here of course capital M Mars particles go out like what I was commended for finding diagrams coming together they cast a form 2nd article or 3rd article wherever and then that particle decays grows out of defined by agree that finding diagram has a value I'm going to tell you what it's a product of 2 cops cars as always coupling constants environment by Iran's and that is the propagator of a particle in between and that is a very simple form 1 provided by yes minus capital and squared it has also AT yacht best my best Maria's capital and that's they can restrict structure of a scattering airport to widow particles come together and merge and former apart not as it's a function on St energy possibly energy of across it doesn't depend on TU doesn't depend on the other Our who variable you won't want depends on the other variable you draw a similar diagrams accepting which instead of wanting to emerging you have draw over here wanted and free emerges but I want to prove 3 4 it looks aren't exactly the same diagram turned on its side but represents a very different kind of physical process 1 until come in exchange of particle between them and then go our and for get happy guests B and with food for this process strategy square-rigged energies of vertices exactly our team minor some squared Mayor could not identify the process which will be won over might squared would look like this are 1 Our there was a little hard to draw warned like there's 1 in and full emerge wanted for merge Luria Switzer lines like that but to ignore it it's there it's there it's important but there are a still a lot to draw a line drawn across it after then I haven't think it just a question which particles came together and I hate drawn crossed lines of a black or but there is principle Khmer 1 this 1 depends only on energy president and the angle of scattering altogether he complains the angles scary editors only a functional that means we must process happens it would just say that doesn't depend on angle scattering means every angle scattering is equal equally probable the particles growing have equal probability of getting deflected tho Haiti Engel whatever that odd but that Is the property part of the reason was simple the particles coming from some direction they form this compound state and only DKB they've forgotten which direction they came in from so can't they form the composite composite decays into case of arbitrary angle on correlated initial direction that's process here this 1 is different it depends on the energy but also on the angle of scattering this is 1 the depends on your scattering and if you work out it's easy see that it favors for awards favor small angle scattering this favors larger angles so this 1 Panzano angle and this 1 but not this past summer the 2 expressions on a real symmetry between their effective to a big change S he scattering doesn't change the property of relativistic scattering amplitudes that they have this kind of symmetry face-off get which will be a coming in which would be our outgoing particles but only if you expressed it in terms of these kind of in very good so now we have some basic idea of what is that a theoretical physicist once the computer about a model of particles that compared not because he's so interested but the thing that he can give to the experiment Boston's measure the probabilities of scattering of a function of energy so poor and use my prediction or say they that seeing you she follows this can what show we still have a lot of value shift about 50 of them Vermeer angles Gabbard right so if all it was there was yes this favors small angle scattering and RB not from didn't make it less likely will privacy White far bigger period the 3rd as there are where are well with stated was if he had a very murderer vibrant but if there was a process in which 1 4 could come together and do the same thing we would want produce squared world or normal human because we are yard they're connected Young the volume OK so what as you what's interesting you musical to ease center amassed square where squared terms 1 armed a hot 1 plus cosigned later but sure but so Miklos related if you're not s tsunami energy that but you also TEU nor the angle of scatter and this is just a function of years union angle scattered stroke Java Norway's say it is if you ass and New York cancel out the angles together look at just a function of
energy course which is query dependent on Miramax but both a separate finding diagrams and general you have your after edible but are focused on this this year the organs at this does not prescribe dot scattering of mayors arms for example very well it gives a porno description of scattering of arms and the reason is simple reason is drive there on many different particles that can be produced when tumors squad also backs them with a different matter and different angular momentum we have described the angular momentum would do we where the angular momentum with new way would be to change the dependence on the angle scattering what's giving off formulas like this are too simple described the Ruiz sticks gathering of mayors on its two-week easy exercise higher vibrations of these particles here and when the particles collide they could make some ground state 1st excited state Wrexham next excited state we can make our whole raft of different particles are gone and the people the sixties they should be getting some Byron 1965 60 667 Charlie of can't can't on mathematical import to roads within the arresting mathematical properties that would represent all over possible particles which could draw here this was trial by of trial and error woes aren't just making up formulas that try to to represent the different particles which is gone he and different particles which Gore the 1st attempt tended to be that an article that this is called S Channel process wiII because it involves 1 as miners and particles coming together killing a tool that defined this is quality teach our processed Dumont 3 coming together so they begin by saying Let's just a hair more and more stuff in Bahia S Channel but said more and more staff to be here he channel for all particles a record scattered that lasted for a certain period of time heading up the Bears particles which could go over there and we will try find more comprehensive formulas just by a young and not the right word orders some of us could could but sophisticated turned performing they a rather dramatic former was discovered which contained the physics or particles in the S Channel in awe of particles in the teacher just replaced this combination here With not just 1 particle but many of them Pawel powers where the formula which I'm going to just this is of historical interest mainly by with a formula which I will write down and you can explore its beauty is great beauty errors February sir brought his quota NXA are however were crude theirs and now he's a young Italian physicist he was younger time is not young anymore resort he just I am allowing me he just just randomly wrote down some things which have some right properties about which really did look like heading up upgrades Reyes who was a function of destined to be Everybody nor the Gamma function well the Gamma function you knowledge of discovered camera for provided began functions it is a generalization of the factorial function for the factor of culture over defined integers these gamma function is defined interpolated between integers gamma function for the integers his ego and minus sworn factorial grammar but fight among Jews to what I want as defined by the growth and continuously interpolate between germs but for over 4 0 4 but but with Wang over for this is it this is a or amplitude multiplied by the coupling constant square if you examine this template who'd you'll find our I'm not going to go into it mathematics mathematics very support the about gamma function if you if you wanna explorer Sir it's actually quite simple construction but I would tell you it looks like it looks like an amplitude but you would make by summing up large number of particles assists representing some composite particles from particle could be an of different masters the S Channel but it also looks like it or not is that it's is symmetric is symmetric with respect to S & P exactly like this is the peculiar thing about it is you could represent it as a kind of soft finding diagrams where the or particles in the S Jamil firearms like this but because its electric on the S P could also represent you don't and it is also equaled the same kind of thing going this way bottles was foot about the formula have all the important features of the scattering amplitude should her it could be analyzed as off all have larger particles were produced and indicate but could also be analyzes of a whole bunch of particles were exchanged this was something new this and seen before previous to this Everybody would have gathered contributions for rest this thing replaced with the question was what is when is is come from or what kind physics Hughes realized what kind of physics in your magic we give rise to former the answer of
course turned out b Strang theory is adventure and discovery of string theory was just looking for a physical model which would give whose answers were to answer for scattering River 2 mayors on his or were 2 particles for now to tell you what Our without a lot of drama what physical model whose rise through scattering recurred Chiluba of durable argued warning through it arms it was very hard to guess where Ms. was furious strings Murray wasn't too hard just because of her that so you she said exchange for forces visitors you could be as chairman future exterior of direct production exchange sometimes called the direct channel and cross-check PS channels called direct channel was a direct particles clean call arrest may out the question always wary of particle jumps across from 1 side or another Krause show you what her physics Our are left for you begin auto strings and nothing good calculation must show you everything away into it it's a little bit go tedious redacted nor differ nor lot about harmonic oscillators for nothin much more Soweto strings he a strength now not remember the growing Harris 8 call along it which we called it let's Strother the 2nd axis over here from stigma from Xerox here's a strengthening but the still propagates Klein let's time I was operated his and string at segment equals 0 0 right Eagles pieties all open strings now still cause moves around the space but also is located between 0 and party stop spatial position just bird parameter along restrained if the of the string of particles of may give up in here each point in the history of the string think stringers sweeping along to sweeping along from space time she called a World worldwide each point that will she was characterized by point stigma and a time which has called how power resist White Pine and where a coordinate along the stretch my real spatial coordinator star label for points along the strained Howl is this time that we used previously our in for the moment the kind of that's support point is that idea Our a world line becomes a world sheet and so being programmer tried by a single valuable how it's now Parma trumpet-wood herbal each point year as a position is based on X next meal just X already worked out what the equations of motion on X are Mayorga wave equations describing we moving are pronounced wave equation right 2nd ex by power squared minus these acts by squared equals 0 as the
wave equation that describes the oscillations of Bonner never think about where we can just imagine that this Australia is a collection our large number of particles the place Wall Street by a bunch of World War II narrowly spaced West between them was but which was going between rats a picture of the evolution of a particle knowing there was begin with 2 particles took Oracle's from repressed little tougher right next to each other but we will you know a color for a 2nd article will follow my next to each other and this necessarily but just in the parameters based here this goes from 0 parlayed we need another 1 in going from Part also from our from 0 Uzi string there's no need for of the fact that have joined the right next to each other and is drawn and their actual positions based partly might not be Jason job this shows parametritis was energizes the world sheet of left particle this have furniture is world's you right particle and they might be farm from each other busy X is over here may be very different mixes over here but with a new luck at all With some probability the end of this trend might touch the indirect however does making co-ops Basel assumptions can call write about the basic process of the theory of begin coalesced and form a single shred they really are connected if you like on Nos developed twenties touch each other a appeared connecting norwest Norwest particle of string with the 1st particle Russia an offering becomes 1 strain that conditions persist for a while 2 2 of it just for a while until quantum mechanical you happens the strings separate again randomly but you can transform renowned cleverly To make it be the same place along the lines of enough symmetry of symmetry of euro equations that you can pass put this point at the same point in horizontal rebel supporters so I after so what is the nature of this process given that it's possible to guess what the answer is probably all the important quantity in here is the amount of time that we were the strings spends call us the amount of time that the compound state saves before it breaks up into its final constituents from much of caught wipe out out Croat from this here how cruelly door with me and me and reintegrate over it there Georgia put it the jury just did it in a quantum mechanical and refer you want to start with the initial state the initial stake less start 1 part the initial state missile State Committee for love is a state of a whole bunch of little points namely the points that make up the street X 1 there are restraint X 1 per cent but 1st just a collection amassed for insects worn by about threw eggs yeah you away have functional if whose restore function of X warm through air fares vessels wave function begin with the Mostar well you'll marble the but wasn't really understand why we're functions of function of the end positioned particle press corps mechanics quite mechanics says wave functions or state vectors are functions of a position of a constituent what's a probability of find particles of position X were correct starstar Shiite reverses the wave function of starting assemblage of particles of course I'm I'm purposefully not taking the continuum limit to show you what goes on in the end you have a continuum amid reports that well you about Weller 1st thing we know about it is that this particle comes in with momentum Katie war I'll tell you exactly what that says that this wave function contains a factor he eatery K 1 time was the center of mass position this is a of away function Ito IKX is the wave function of a particle with definite woman what momentum you you lose you use a center of mass and so was a center met position and nest position was the center mass position on these points if the average position x 1 with some of the X is divided by In so have X 1 blast next tool that bad 2 X end provided by air that factor factor alone tells you that this initial particle he'll have Mormon K 1 no about the rest of the wave function which depends on the relative coordinates not some of them but the distances between the neighboring particles that some wave function which characterizes the grounds it depends on everything except the songs of but Recep some wave function of assigned nor programs they depends on exodus indexes but actually doesn't depend on some of the some of them here the differences between them he differences Araxes here they bring Exs salt pork here are some of them here in this function is computable it's just grounds describing the ground state of all a harmonic afterward is making up of strain they could really work out 2nd there will With enough room on a blackboard I could tell you exactly what was function is as a function of the correction of extra memory hard Bontcheva of a bunch of exponential workable know this is the the wave function of 1st particle What about the function of a 2nd article it's exactly the same kind of thing and accept that doesn't depend on these coordinates depends on the coordinates of be rent articles here so what's right it errors he's very R K to war now what larger right X 1 through now that original particle rose original constituents of the 1st I want to constituents of the second-string erupts right X and plus 1 X and plus tool always up text joined in the 2nd half of particles is grouped together in the second-string also provided by and also proms Aug . 1 end offices X and plus 1 20
and Leandro Ericha every red red meat or could you read it but that urea now that's initial state but this was some probability 208 points merge because I wanted to end points simply says you set X who want a sorry X is equal to X's course warned you look for that piece of the wave function with the toe and points exactly the the same pouring sorrow you begin with this next that was to say let's be if particle on the 1st James at exactly the same place as plus sorry as thanks to be of the same places irksome coursework fossil problem here X and Beaulieu I was left unchanged that's now not that's the wave function of the state right at the port after 2 particles of coal ash when Mayor coalesced point where call us to come together nothing happens for the rest of the chain but the 2 particles so this is if you like the amplitude particles cheese Koch the input to 2 particles the endpoints touched now I have a new starting point which is a function of X 1 through X 2 and it's a state of that many particles don't know that we have to eat involved we have to Ieave using the Hamiltonian Nomura Hamiltonian there's a Hamiltonian is a friend which operates you from 1 conclude next so you can't this initial state but still well-defined framed annual propagated forward in time using the Hamiltonian again Webster right rule for our bigger state from 1 from an instant will you multiply the state or something sneakily are aged eatery I did think this wave function annually Volvo you saw this floating equation for it but sources say there's more multiplying it by equally I toggle Hamiltonian times however was still Hamiltonian that I cannot only is all is just a collection of brings messed points of harmonic oscillators with just a collection of harmonic oscillators real had reduced and that gives us a state of the system at the time sheet what's the last step it is a project of final state security on 2 2 separate particles again projected onto a state with 2 separated particles of momentum K 3 and K 4 it's very very straightforward if tedious fiery PDF you take the total initial states particles well-defined ground state calls you just that the and points at the same place where constrains the wave function you let it be involved as a single strained and then you have a break up again and that's simply means multiplying it again by some final state roll out same kind of things and that gives you a transition to make a long story short you start with 2 particles you could strain that's over there the same place you Volvo and then you projected onto the final stretch of very very well-defined that gives you beat amplitude rat astray for amount power behalf you choose panel sustained in black player and all those who have them separate systems each having its own word from new more quite creep away from from Harvard Lucy a function of can't positive too articles to begin with you got set that 1st particle recalled last particle here then you look funny and airport you look for the peace of the wave functions where those 2 particles are saying place you say I know the same place fused together and as if it were a single strength a single collection of man's points and spring but not the kind power and they basically take years is a just copy Yucaipa spring in the middle and we've all that the back calculation is quite doable not even really hard it's just too much for the blackboard for a 4 1 4 1 evening value of answers but the walk arrived guarantee answer for your or that Berger murmur Is there a little will over week how do we say that we develop our aggressively and whether we integrate with power referee why we may end up being hampered truthful possible crimes this calm posit Group this is a fight of summing or breweries the only parameter here Hi I'm takes photos of the break up again and find his rollers some are all pairs which in this case just means the great over time that they spend Prodi evolving together and that gives you the amplitude for particles to particles in the initial state to become the particles finals but the result was an integral see it that way all time to spend together I'll tell you what the Indigo and after a a certain amount of calculation which is actually another hard it becomes grow our E. Howell west of Fort Collins as plus 1 is as fast is exactly this as its fleet as variable August center of mass squared of energy of the 2 particles you have a factor like that you have another factor which is 1 of my ears he Europe Powell through minus team minus 1 feed was a momentum change between initial particle here a particle here remember those Mahmoud Khalil this wave function whichever erased your coated in the wave function and kind of those exponential initial among our starters and less energy and here's the momentum transfer it appears before and they have the integrated be but that was to be another factor I think of Etowah miners tower near the Grand you're right because like this virus that Jack Clark offers 1st talent as what yes but uh yes yeah that's where get trait that plus he added but were observed by the a lot of them are and about
here on purpose on purpose began now but doesn't look particularly symmetric between town and between and yes but afterward it was computed it took about a half hour realized that you should change variables and go change variables From each hotel from some recalls but each of the power recalls Perry but Ito power recalls now like Peter towers these service becomes integral after missing some miners and Oviedo reminders reminders arose the 1st be miners here it is a rarest progress farm this saying yes becomes BSE postwar want reminders as plus 1 8th of a mile tariff marinas Martinez voluminous towers seat by definition of a change of variables while about 1 seek of a modest how that becomes a white minuses over Myers Carol as 1 also a the integral goes I think from 0 to infinity derived from 0 To infinity as because the White Tigers Ito America's cameras where was pal firms to a modest pal but just busy fish us busy Romero Fish just easy for forget that right I'm lost track of roses for from minus remember with just easy This is a former are busy in the theater logos from how it was sealed off where's Zia's Eagle go wanted Tour the miners it is my making arrest each of the mightiest hours ago was how it was affinity were you have power equals Xerox that's when it belly suddenly no Jews and then falls apart instantly the power equals infinity 1 a separation here now what happens when just tell goes to infinity is egos 0 sought to make the world from 1 0 0 0 0 4 1 from a recent signing off on integral Brookside the amazing thing about suitable shot is at the center goes completely symmetric between S & P could you see that I see the erected between us and right you just substitute was easy minus 1 you make a change of variables between scenes e minus 1 and see because Mr. de the completely symmetric although are completely on selected starting pouring between energy and momentum transfer somehow it warmed up during a completely submitted ensued between the 2 of them are why what might have realized how can he also had a lifestyle that place my were minors how shall continue make that 50 they ought not formulas news switch the order integration becomes from 0 4 1 best but it's a magic whether or not to say how much shorter of integration right and that's about the answer is symmetric between us and was it it's a process in which the particles joined a composite which would aluminum for a while and then breaks up from those analog of refined diagram in which calm positives form and then decays composite was formed but we're ones are being completely symmetric between S & P In fact this function is cold or it be a function of of function of 2 variables & P its core of data on minors and miners chief although made a function I they Fulcher mathematical physics and guess what equal to its equal yarn or exactly equal probability however crude namely this product of gamma functions gamma function bomb how did get to be that it was symmetric between this is not obvious symmetric between S channel and feature completely symmetric between yes Charyl feature on a country that next time that has to do with a fundamental extremely deep symmetry of string he recalled can't form also refers to symmetry which allows you to take these world sheets and performed them end crazy ways as if they were Turkish caffeine stretch them out different directions and for example turned this picture and draw picture which looks much more like 2 particles coming together and exchanging Inc Carol back over this should follow us back later rest for thorough yes what time it is true 0 that of them b beta function evaluated at integers God is Commodore coefficients there'd be verse overcome rhetorical which there talk about hours talking about their own and factorial factorial or and cost and factorial which occurs or replaced in commerce Forex justice just happens to be the same function no simple no simple on the possible connection with our results immersed of resort was not come coefficient were barred darn just happens it's an integral which so far has combination of Myanmar functions the bad news also below relationship it's not something comic Parkway here we see how my knowledge and certainly not a wave in the channel of founder here are what magic people find on this was if you like partly historical but so part of important logic of the theory is that number 1 you can calculate work Winter where working His harmonic oscillators anything can be done with harmonic oscillators our but a harmonic oscillators you break up process up into pieces and then you integrate over work parliament between you could calculate calculate defying an integral the end of a bite at this point by magic as a property
symmetric between S & P and somehow looks like but not some but has the features of having prophecies where particles coalesced Rivera processor draw here but somehow buried is also somehow prophecies where particles are exchanged and that was a magic coverage that was a surprise From home logic growth growing prostitutes together to make new browsers OK on any questions about Fiat above In the it structure a oscillators oscillators to where it was up front and heading women North Wednesday measures but was still has not while a staff are in Africa where this year 2 more West 1 changes from hot season EU lose to more or but now actually force 1 disappear agreed the July premier idea your yard Our Yao each other regional or about a year after her right so revenues bring reforms connects these but Yao Yao Ohio because we are big trouble you who's 1 for Murad right to go that 1 would think about it is let go of a particle by particle called and if these as Quark Siemens of strings you could think of the last 1 being Quark which annihilated with the 1st quarter of the year the other string sorry I used to attempt but OK cares about and to remember now the Bridger like you have to be careful not to lose an odd number from your toe 3rd quarter Rudy 3rd well reactor Bronaugh superstring arm of over things I will tell you strictly speaking quite push on and not through a bars are always going now I'm not going to go on to be heavy mathematics super strings what I think we're going to do next turned his are going you or are playing a little bit more about string theory about the properties of world sheets and a symmetry so stretch this thing like Turkish Taffy and make different kinds of fine and diagrams but them other move on to any another subject called and theory which is another way of deriving string theory from a totally different well I think we have a talk a little bit about compact occasion about what to do with these extra dimensions or we will maybe next time our but then move on so a totally different origin of string theory that all much much later in times of sometime around 1995 96 hour which started with a completely different picture and which a great many of the features the more complicated features a string theory are completely transparent were call them theory will will start different starting point which leads to exactly the same jerks showed as if drivers based on the last based on a yes but also the way to photons quiet period the in people who are the analog but foreclosed very similar you have to to binge you pick a particle from here I want from here and you require that will be the same poring over they do this kind of thing and then you evolved as a single string saying rules and kind of rules and that would correspond scattering who close to scatter closed strings the intermediate they look at also show your this is only slightly safer for not wrong on our moral received this this calculation originally done the bars are focused more complicated when you have to keep track of it extremism extremely slight girl different is borough not important what I should say about those is bad the scattering amplitudes we said that there was a photon in a young grab a part of System Five look like it was a speed 2 particle must been warned particle and you could force them to be mass forced a few want you and that much choice the warrant them of components to make a massive particles to do well when you take these particles and call new work of scattering scattering amplitudes very distinctive characteristic the mission of photons grab they are not just any old were scattering what they have barely very definite properties which were make them extremely special missions air emissions are absorption of photons could not be mistaken for the emission and absorption of scalar particles other things they satisfy some very very important rules those rules originate from the concentration of electric charge consumers electric charges in meeting go for pardons almost very very little ambiguity in what the emission and absorption of 4 pounds quite or were scattered 4 cars are charged particles like while even the scattering of 4 cars by 4 times there was at this point calculating these diagrams where became completely clear that these things which will calling for cars were behaving exactly like 4 kinds of things or group calling Gravitt will behave exactly like graphic satisfy all rules grab upon grab upon scattered just saying there was a particle would look like a gravel Parma was very weak things when all scattering were constructed and rigorous test of whether or was satisfying the rules for scattering of gravity bombs by massive things by Massell's and so forth perfectly exactly and people realize they really were dealing with something that looked like a scattered Gravitt columns for sold to scattering amplitude played a big role in establishing Murphy Our precision that we would dealing with objects that you behave or for Palms grant palms turned pro or for a little if a young we have talked about that but there were probably will talk about our remind me of my former lawyer
pleads visit and up steam Dr.
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Metadaten

Formale Metadaten

Titel String Theory and M-Theory | Lecture 6
Serientitel Lecture Collection | String Theory and M-Theory
Teil 6
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15131
Herausgeber Stanford University
Erscheinungsjahr 2011
Sprache Englisch
Produktionsjahr 2010

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (October 25, 2010) Leonard Susskind focuses on the different dimensions of string theory and the effect it has on the theory. String theory (with its close relative, M-theory) is the basis for the most ambitious theories of the physical world. It has profoundly influenced our understanding of gravity, cosmology, and particle physics. In this course we will develop the basic theoretical and mathematical ideas, including the string-theoretic origin of gravity, the theory of extra dimensions of space, the connection between strings and black holes, the "landscape" of string theory, and the holographic principle.

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