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String Theory and M-Theory | Lecture 5

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discussed some units called and let's let's discuss unit to bear In Haiti drawn DFARS irks wave the discussed the idea wept a Pedroza the and there can be excited that can be excited by sitting in the rotation are setting in the vibration our my ex citing the harmonic oscillator that make up the stringy character of the year cover proton or what ever happened to be the 7 energy scale a certain amount of energy that each excitation will give your In particular the Energy jumped from a ground state for the 1st excited state amateurs it a depend on well In a strength theory really only 1 parameter we talked about a little bit we discussed the idea that if you stretches returned from it is go back our way is if you stretch Australian from 1 point to another and it pretty much like us bring it does behave like a spring it has an energy if this is nonrelativistic strength just ordinary nonrelativistic sprang on rubber band foreign idealized abandoned its energy is potential energy court where's folks the energy some Katie which is some spring constant current 3 separate times a square of the distance between me and pouring missing something factor 2 yup that's a potential energy that strengthened our and we've identified energy this nonrelativistic energy we've identified it not with the romance of the strain but with the square of the massive because with thinking about this and frame of reference where a string is moving very very fast and when nonrelativistic physics actually works for the motion of the strain perpendicular to the direction of motion so let's keep that in mind what has been going down the z-axis like there's a it's been stretched to a distant X works collect for distance L is becomes L square and we've identified the energy with the square of the man's L. squared there is some spring constant at spring constant borders core carry it says that a man's of the spring which in the rest frame in the frame of rest at which the string is at rest that man is the usual energy that's going to be the square root that spring constant Times l perhaps divided by the square root of 2 and not interested square of to now so the man's overstretched training is proportional to Woodslane kind of some
parameter which I call the square root of cake the square a has another Maine called the pension Cripps it tension the strain energy per unit Lake amassed a string arrests were not looking at it In train which it moving fast it's energy is man we stretch it which gets out they the energy of it grows proportional to the length but it's like surface tension except with Liddy retention and of the energy proportional Lane the coefficient is called Pension Pension and the spring and so those attention which is just the square root which is just a square of William courts where a we just call it attention on now how much energy who we get each oscillation for each oscillation all now what it where units of a square cage let's work in our favor units in which the speed of light and plunked constant quarter award in those units energy has units of worn over Wayne solar energy has units of 1 Mobile Lane what's the units of capped a year of square root of capped by the string tension yards is squared over links squared energy Square if you like so this subject over here with quality Seifert attention that tension in the shrimp is being the which depth the fundamental scale Our and sets the scale that everything is to get a sets the scale of units of energy squared and play what it does it tells you that each excited state each time you excite going by when you it a adds to its mass squared essentially attention Ed so they would test units thing which determines the units of theory string penchant the baby and the string tension the bigger the jump in mass were new excite something of the same Tobin ordinary sprang into with ordinary springs you have a lot of friends let's so let's take a mass of the spring to be 1 of the spring constant Casey the frequency is just about the square root of Taylor and the frequency is proportional to the tension and of course energy that you bump up the strip the sprint by every time you excited is also proportional so the energy giant energy jump between the ground state the 1st excited state next excited states soulful is controlled unit is controlled by this string OK you could ask What is spring and string tension is force per unit of Surrey is energy per unit length named for energy pre Lane energy per unit length force or poised times Lane his energy force can't work force times distance is work right all this attention years is if you poll this subpart bets on which forces pulling you back at summit forces pulling you back the force within the string another way to say it is if you don't want a strings of the surface of the Earth a at some place and suspended away from it it
would be the way that the spring but the strength could support a heavier weights were just shrink work for them however small weight would be pulled out and press the character of a strings had 1 of the laboratory the force be wait that support is independent of the point incidentally you see that from this formula you see that from a formula that may ask our energy saving energy is equal to spring car or tension at arm's length forests is energy per unit length energy per unit length is forests and so the force that this strengthen support is independent of the way that's the character of the East that most OK so the new guy just how much wait at the surface of the earth How much weight accorded a young pay drama Amazon do that somehow Ankara 1 of the quarks and Amazon for some support over here stealing a much weight because they can't support their suit is about truck about I can remember it's an 18 wheel truck 16 will try for maybe some maybe just a half panel truck a member but it's that order of magnitude that's what 8 could support With so that pretty strong microscopic little things but they're pretty darn strong and the straw that are the higher the frequency of the oscillations so the larger the gap between the lowest energy state the highest energy state had allegedly nuggets that anybody ever support truck with them would major cost not only actually know is the amount of energy that it takes to excite strength from that we deduce what the engineers from attention back and tell you with your support these are not the strings that strings the U.S. unimagined are associated with really fundamental particles like Gravitt times photons so for strings are much much stronger very much stronger attention and vastly larger this again you took the Earth's and you supported him awake namely the weight of the hold galaxy the galaxy now of course the nonsense it's not galaxy the removed your goodwill but you could concentrate and you could concentrate a mess of the galaxy small volume and make the earth heavy enough maize ever make your heavier increase gravitational field went not necessarily sell we can make the Earth much bigger but keep its gravitational field was saying it would somehow keep the the gravitational acceleration the acceleration of gravity the same than the way that you could support would be about with Galaxy so the strings it would talking about in fundamental string theory of vastly higher intention much higher meaning of that is that beat energy in expectation that excite the is much much larger is a much stiffer springs they have much much largest bring calm and therefore much larger frequencies of oscillation if larger frequencies of of oscillation than larger energies In exciting How much energy while the thinking goes that probably somewhere near park energy of talking about Gravitt times are gravitational scale is a really know what the funk energy use and I know you've heard of the United Chile would be the 1st of all the number 1st of all in the correctly and 2nd of all what actually where comes from now can't see squared but that's that's a fairly big energy it's enough to but a car or a little bit so we irradiated side Graham and the minus 5 grams and and then you take that square and multiply by we're we're right right but when I asked you you know what funk energies on Park Lane plant whatever it is I was asking theoretical questions I was asking you know what it is In terms of the of the of the constant nature maybe we should go through that users were also please do not or could not out of the question that the question this stiffer they are the smaller where are now cuisine stretch the mentally Lane is no limit to Whittaker stretch them but if you ask how much India ground-state remember things in their ground-state oscillate how much delay oscillate in the ground state in what we know what the meaning fluctuations in the size of these strings vibrations of and a lodger the spring constant the smaller they are a much stiffer much harder to pull apart pulling this thing apart by 1 centimeter maybe cause you where work cause allowed the effect of this smaller the bigger the spring constant how big the people think they are full somewheres of water the plant cooling plant commands Park Lane How fast as 1 vibration plant kind of picked to let's talk about his plant here that this some reason to believe it's a little bit more of that may be affected can may be a factor a hundred smaller but that's that's fine pores so let's talk about what the Parkway I didn't prepare so we have to work it out in real time what is the Plonka for great Web plump units Furcal where 80 Europe's and physics we need have 3 units massed Lane and kind and I usual choice of units is deep usual units so we use a laboratory are not determined by any fundamental a 6 meter is convened unit so far around the measuring role OK as working for measuring role looked like cloth or averages are so no doubt it originated from a link of human arm probably a foot the course is also a similar kind of unit soon came from the Kings for it Our so what's the real physics and was the real science of what a meter is the real science comedy How does it take to make a link the law armory useful arm has nothing whatever to do with any fundamental physics as to do a biology politics maker are not a very deep unit may messing things kilogram as a counterweight Paducah manipulate the laboratory and 2nd a 2nd was about the time that you could measure with a pendulum that's it that's where they came from and they have no deeper meaning that we would like really fundamental reasons in physics to choose 2 units which have some very fundamental meaning now for example and if we do then many of our equations will be much simpler for example reaffirms the size of a proton comes in the various equations in nuclear physics and the size of a proton enter of
minus 13 centimeters it we work centimeters or tendinitis 50 meters is gonna be can't minus 15th solid such as candle minus 50 1 . 7 3 4 4 1 9 8 7 and the physics messy and that they had nothing to prevent us from a single lets you lose the US which the radius of the proton is exactly what that nuclear physics will turn out to look a little bit simpler an nuclear physics will turn out to be a little bit simpler but atomic physics won't look at how a lot simpler Our physics Our sub nuclear particles sub hadronic particles warm-up look a lot said Brewer there's nothing universal about broke broke others just some article why is proton instead of some of particle no good reason says nothing really universal about broke down on the end of the rules of physics will not in any way be especially simple we use the kind same thing so we would like to greedy units which have some deep fundamental significant it's a Qantas saying we would like to choose 3 constants of nature and set them equal toward it's completely equivalent choosing 3 units the Schillaci 3 constants of nature and set them equal the 1 constant nature 19 dimensional can't be size of the protons want but that's not a good 1 you will use things which are really which really very universal 3 there are 3 constants in nature which truly universal what they are the 1st constant is the speed of light YYC its universal because as a rule rulers nothing can move faster than the speed of light nothing everything's cool is that every velocity of every material object is bounded by the speed of light the use of the word everyday tells you something fundamental about something where protons are electrons or photons or anything else all are bounded by the speed of light so the something universal about speed of light the can't succeed this is like constancy of it what yes yes sir the fact that it's sustained in every reference frame and the fact that it's the same for all particles North Central particles they waited elated protons pure white the limiting velocity proton is speed of light sold in that sentence universal so that's the 1st thing we want is that sea he Gold Award from a good thing to do here we do battle a kind of course the next thing which is very universal is plants can't be given example plants constant is not just any old constant like a radius of of a proton but connected with the uncertainty principle Burke every object in nature the uncertainty principle between the momentum and position is greater than a recalled 8 far and maybe some forest To win their number but it doesn't matter what I object to talking about talking about electronica because about broke ties you could be talking about a bowling ball all constrained by the uncertainty principle and so in that sense H. bar and it doesn't matter what use aged borrow or the owner or each other 1 without to that's not point here like this that plants constant applies that everything in the world its universal soul the next thing you might want to do with my life but we will is to set H Oracle warm so what's still less really Universal is there another fully Universal constant the gravitational constant remember according to every object in the universe matter what made out of the matter what it is composed of every object in the universe has a gravitational force which is the product of the masses provided by distant squared kind is universal constant G so again something which applies every are they units are they units in which C H borrow GER equaled war that's same as asking the question Can Yukon body C H bar and G until into a combination that has units of mass could you combine them until combination that has units of Lane and could you combine them into a combination that has units of time and the answer is yes our let's let's see let's do late they would even better but still late square receiver complying a combination of G each bombs city which has a unit of length squared Catania we don't just some simple dimensional analysis rebutting the dimensional analysis OK orders simple the Mitchell analysis we want the a combination Jeter some power cord P H by some other power and see some other power P Q Bar which has units are laid I when link squared say has units of Lane squared now this is a it's equal going square means it has the same numerics sometimes people are sometimes cruel bracket around more you have to indicate that units units Our decoder PATH the accused we are to be linked Square on Tuesday killing squared-toe avoid a square root the final form of a visit there is another reason it is widely believed that b fundamental unit is a unit of area volume of Lane but will come to that some of it but let's see if we cannot see we find PQN or OK 1st question is what that now we have the figure out what the units the conventional units of G H Bomb CAR units of city that's easy units of the at Slate over time right where about units of plants can't see this as best as I can play fly as is a man moment Momentum NYTimes distance right backs Dr. Pete Bell backstop Peter Elaine and remedy momentum is a Mass fans of all our city and velocity is over time so as length squared off over time playing square mass over time that wants constant and the last 1 the 1 that I can never really remember these 2 I remember when I can never remember his G figure out would feel and Jr if can get you we we use 1 of the equations a G appears in my before as an equation that our work good will which was your equation by acceleration equals yeah I will get better it but that's equal to 0 rest G R square the squared acceleration is white late per times squared again equal means I was mean equal sought l killed over he squared provided by him that looks like right OK ahead
but snobbery get his always you last year you're right you're right whose work OK so now we have a left side length square now go Jeter the power PowerPC want to G. again at a Q of all over 3 squared times and the but Price L tell 3 Peter the took pity and pity right you that right as Filipino about 8 Portola killed 8 part of a Q is squared length 2 Q. end the 2 Q no end to the queue but and fifa killed figure barite think of the last 1 is the speed of light the art power which is linked to the ah time we are now we have to do it it was fined 3 constants so that everything on the right-hand side cancels out except to powers of light OK so the 1st thing you can see is may only appears in 2 places master McEwen peace outside Mississippi better Beagle 2 killed West just sent people to kill we now know that end we can cancel out the masses have done now well story have we have to get rid of all the kind that looks like rights a harmony times do we have we have droopy Topete plus Q was the same as payee 1st 3 Perry plus are people 0 3 people us boras equal you're wrong so that tells us that aura is able to minors 3 bay but I do that right what's just check Europe to P 3 P Plus oracle 0 articles ministry a payout from Evans right minus How repay at 3 I make a mistake gave a Pete River a figure that she was so if I'm not mistaken I think we can get rid of all peace he is able peculiar so what's your we have reappears Eagle peculiar that looks like 5 p void why not recognize at 5 p the Tokyo that's taking care of and now Boras what Oryza Cooper minus 3 N they to flood yet the fight has I think I had to pay what they have to pay the and then that tells us what players physical don't want so let's see what we have our is equal to 1 that tells us that the plant area this is the plant area plunk area squared is Jean H Morris looks like all CQ killed CQ this is usually expressed by saying plant square of the square root of it thank you love letter should put some numbers my problem is I can never remember plants constant but no points constant and and my 34 yet he is 6 times the amount of 11 which is about 10th of a can the order of magnitude the minus 10 was age Barton my 34 the numerous 34 In the usual MKS years extend the rumors 34 and CQ is 3 times and we ate 3 times in the 7th Armored drive take the minus a bigger tendinitis 24 hour faced more like the minus 25 rumors prior so a square meters users up as baseball but a small gap some some some playing a tiny number on the plant life is very small if we had that right we would've gotten 10th of our money is just 35 needed the tender I should depend minus 70th square centimeters I'm not sure if right print that but said they are right sorrow bats small that's more than anything that but the OK now punk time a plant point is the square root of What about the plant time gotta plunk Lane is being a sizable new thing he had his thing the size of a park where the plant kind at the time for a life greater cross that Wells could be right kind like raked across several days so that means that means we have to wear divided by the speed of light now we have provided by the speed of light so plant Elaine L. plant you can't remind us 35 meters CD-ROM Clark kind is end cover minus 43 of 42 some sort 42 seconds 44 43 seconds America brought about the plant met we haven't figured out what the plant messes but we could do exactly the same calculation work out work plant masses and from to borrow from 8 4 and the plot commands would be about 10 to a minus at 8 kilograms so this big that's a big maps and ominous 8 kilograms observable thing too little dust grains tour bus grain these are impossibly small small current smokers OK this it is our hope plant Units things measured in these plant units are measured in some very very fundamental players some things the universe has a radius is known observable universe the horizon size is about 10 to the 60th portly and the 60th complaints age of the universe is also about can the 60th plant times our parents good serve a gallon Dr sorry a tank of gasoline is about b and energy content a tank of gasoline is
about the point rest so that means you could drive across the country where can masses of energy bread that's what but these are the units that ring theory is In the size of vibrating strings fluctuations in it good a quantum uncertainty order of magnitude Parkway the frequencies of oscillation kind of frequency but period of oscillation public and the 43 centimeters sorry sorry demise 43 seconds right now the amount of Mets that would be involved in exciting a fundamental particles searches a Tom electron which electrons if the electronic is a string Eucharist from much energy does it take to excited the units are expected to be somewhere in this range 10 to the minus a kilogram now that's a Hulu each number Hymenaea mini is Hugo electron volts 1 other useful No. 2 Member of her remember is that 1 plonk mass is about came the 19th use which means kantor 19 Proton 1 plant mass is about to enter the 19th Proton can the 23rd Proton masses is the program of water or something and the 19th which is for orders of magnitude smaller is a trainee clean you drop with visible quite visible about so so exciting electrons by a particle physics collision is it well we'll middle particle physics collision will just exploded tank of gasoline in the vicinity of electrons stop electron vibrating E. M. I a system that are hard to get the energy to be concentrated in such a small loan such a small this sorrow said these are the reasons that but that that yes Roberts cram state at you not what remember it may Where's which common integer multiples Ms. squares which common integer multiples so few want to know what the unit is square the plunk maps and that's the amount that takes occur bump you up but the goal it but the Gulf from the ground state of the ground is almost massless then it's about 1 plunking of energy what it right right right is just much too much energy the company such a small warriors there and in all its for this reason that a such string theory is the way way direct direct the detection how would you detect thing fingers a strength the same way you detected a proton string you hit it you sediment the vibration and you discover that was a bunch of excited states with the mess squared is proportional to the angular momentum at the kind of thing you'd like to do with them lecture on the kind of thing you'd like to do a bear with Merkel of busses kind of thing you like to do with the time but decrease in masses we would be ready structure a now with the a straight as a whole have to be somewhat dated this they now what is this letter a topic here's what makes basis state that we get that erupted last year said too would like yes yes based on a number of things no no it's not just a normal known it's not just numerology armed the Kabila wrong it could be a thousand times less nobody thinks that the string skill could more of our worry about that look quite impossible but that string scale could be larger than the energy scale lower that as possible How much lower well I think from the things it we know about particle physics all we can say with any precision of the current time really all but from me precision with which particle physics standard model so forth in the work and think there's plenty of evidence that the energy scale is very very hot weather it's plant scale 100 upon scale thousands of the plant him nor are there is another scale in physics another scale physics which keeps coming up over and over again we talk about it's unification scale if you remembered all from where squatter we talk about the the scale at which the various coupling constants seemed to come together the scale which electroweak forces in the QCD forces seemed to emerge until common structure experimentally in there there there are experiment not experiments to get energy but experiments to extrapolate the extrapolation of what we know about physics seems to say that ordinary quantum field theory probably holds the something like a distance scale no as of roughly a thousand times a larger than the plant scale so we still have evidence that the ordinary quantum field theory without any string be scrapped until 4 or equivalently the electrons fundamental the scales perhaps a thousand times more power and water park scale but that's that's still pretty small and you wouldn't be able suspended galaxy maybe you could only cover German I don't know globular clusters from the 5th that it seems that a letter it is but that we also learned what thought thoroughly not not for plant believed to be the mass of the lightest possible black hole 1 Park area 1 unit of entropy and how long would it take for operate 1 plant kind here today gone I think all of the above both this is fine back at least 4 rout that he said it was a test of our priority is evident are from explain in detail this is just to remind you Egypt plot the coupling constant all of which coupling constants the ones we measure in laboratory beyond our the electric watches called electric coupling constant power he he squared really weak interaction sorry electrode electroweak Mario Rivero 3 of them if you 1 S U 2 and S Ukraine 3 of them this is energy this way in energy is the
same as in numerous and restoring energy gets launch Waverly short energy Holly Jimmy's more pockets we measure these coupling constants at some energy scales remember coupling constants of interchange remembered your Cheney reduced scale we measure them somewhere where in laboratories which means 50 GTV hundreds most mesoderm here we get 3 numbers we also measured slightly indirectly derivatives worries that the fact we weekend computer derivatives are very weak computer derivatives of them Chile theoretically but was good confidence because we know that you hear very well what we find is that a few extrapolated to use all crossed about common points that point and that pouring is way way out here an energy which is not the plants get but about a hundred or maybe a thousand times somewhere between 100 thousand times smaller scale maybe more like a hundred dollars could be an implant roughly divided by a hundred somewhere in between and the plant scale would be over here all this is a logarithmic scale on a logarithmic scale it looks like tore a logarithmic scale are giving gap convene a very very big changes in looks like this now cost we don't know with certainty that is not all kinds of things going on at higher energy here which would not this up badly we don't all we do know is that that if there is nothing in here among things up badly in the some reasonable you back that these coupling constant were just come together scale meaning of that is a quantum field theory conventional quantum field theory seems to be working to about here as other evidence for the scale coming from neutrino masses and a few other places our but it's far from height it will become much tighter the LHC is lucky and makes the right discoveries will become much tighter and will have a much better knowledge about this extrapolations of is 1 of the things that the learned city is how reliable the extrapolation areas but Clark as being out here and this is solved doesn't tell us with string when here what it tells us is that the stringy character of particles does not become important uncle energies above so somewheres between hearing he if string theories all right somewhere you may be a cut further slug that now should be easy and follow was quoted a yacht done if you were to plot the gravitational coupling constant behaves differently in different character but let's see it started very very small it goes up somewhere right there's a vendor would cross it doesn't cross at the same point across somewhere nearby but basically the place with this 1 gets to be about rural 1 that's the plant scale so really it's really exactly the same statement to say that gravitational constant a running coupling constant crosses at roughly the same place as to save his unification scale is not too far on Apple yes that each but they are not Capital G. does not change the capital G not exactly the right measure of its capital GE 5 let's such stood at your question let's look at 4 slower between 2 objects in electromagnetism its East glared provided by our squares and best of forests and that's the electromagnetic coupling constant here in gravitation into GE and squared large square so the thing which is analogous to ease squared is times and squared for what it assiduously well the energy use there it is associated with a particular scale a physics that you studying I'm vacancy then that this quantity here increases like M Square now so this this is the real thing here that mentor most men will get a on that dealers where they say will accelerate because of that start using a new yet the store Bill yards total now Mr. waste to produce black all 1 way or another you've got a whole lot of pressure for squeeze things was more Formula One way is Connecticut Energy small objects from lots of kinetic energy blast them together and you have a chance of making a black hole the other way is that Connecticut Najib gravitation gravitational potential energy put things together that takes a star make a black hole using gravity you need something as heavy as a star maker Blackall by the collision of particles you need energies some of order the Plunkett due to think about that for a minute accelerator when I was a fund that's explore the following question how big what kind of accelerator would take Lula said do experiments of Clark Air do interesting experiments of Parker G. well basically the energy of an accelerator on a linear accelerator scales linearly would be the size of the accelerator but you get a certain number of Of GUV per unit slack is a good example it takes approximately 2 miles to accelerate thing to worm 2 are 100 Jedi 2 miles from the G maybe tho miles 100 but the plant energy is about 10 of the 17 times bigger than the extent of the 19th GUB so it can 17 times larger so we just did just that but the parameters others like accelerator Ukiah multiply this by roughly 17 so what they can to the 17 miles high biggest into the 17 miles from and other 7 I can't say I think can the 13 miles is about like year of the 30 kilometers from remark that member and the because of and and the Ford light years galaxy size galaxy size to work to accelerate something of what I could do do it at least at least not tomorrow but suppose it could build such a accelerator next question what kind of luminosity dear after have an accelerator hostile luminosity on the chances what we wanted do now is we want to call light particles head on make a black hole but remember the radius of that black holes is itself a plant the plant radius so that means it that even these particles the call I with In a plant distant so well that accelerators that that's not the way work the way accelerators workers procedures get other particles going left
water particles going right and you get enough of them so that some will collide at the distance scales if you're interested in need bigger luminosity what kind of luminosity would you need all animal let's say was a slap luminosity can 13th particles per 2nd or something you know I'm not the well kind 13 particles but that's just but you know the distance scales a reprieve emerged better you have to do I I don't know what they can the 20th particles enter the 20th particles colliding each with a that's 20th Hanks of gasoline per 2nd had not yet had the 17th barrels of gas per 2nd fuel it as big as the Galaxy leading couldn't do it that's so we would need to explore small black holes and something pretty close to it to explore string theory directly really direct the question is are they indirect I think is the indirect test will pretty much be varied theoretical man on less somebody constructs a very very convincing string theory that gives rise of the Standard Model exactly you Sabri computable way CEO we hope of direct test figures vary wrote or that but that's will face in October up against OK Obama did you get were really have intended to Gulf renewed touch Emery the next thing that pops up on me maybe I would beat yes yes it is believed again without really terrific every but again it's the same kind of evidence of where where is our plot coupling calm if they want extra dimension and the extra dimensions have any sizable distance if they were all sizable than ordinary quantum field theory would break down when particles begin exploring the extra dimension the conventional use Of course the fuel theory as we use it would break down at points where are we energy became large enough to explore these extra 20 dimension if that's true and if we continue to get evidence the quantum field theory theory makes sense as it is How to these distances here little nor the extra dimensions all must lie somewhere between the plant calling maybe a thousand times bigger than the plant growing so exploring extra dimensions will also be something that d said left orders of magnitude or that's another question that's the question the Yuri really doesn't there were on your arm this but this way if you try to make them smaller than the plant pouring you wind-up making them larger than the plant graphs making small things with big man's means you make AT black hole Black hole so if you try to think of strings which were much smaller than the plant and the MAS would accordingly be larger than the plant met you would be talking about black holes he wouldn't be talking about things that problem would collapse to grab the gravity demands and their small distance would cause gravitation basically tournament bicycles so that's the reason that the 1st thing years do not ever consider the possibility that a string scales more what remains of it is thing that there's a problem only gave us what T then you you do exactly the same thing you say Let's make unmasked squared however these parameters here and I forget exactly what it is we could think about was that the others Alain Juppe as you wrote would if the tribe that yet but that's the body of a rock and what's met as follows from what we well know it doesn't follow by itself from but if you would have on top of that he had get asked for for 8 years ago take to particles whose distance a part of the plant place now because it distances so small the uncertainty principle says that they have a lot of energy right the uncertainty principle says if things are very well localized than they have a very large momentum fluctuations of the momentum very large take 2 particles whose distance it is known to be within the plant distant as says something about the momentum their average momentum was extremely large from Vatican compute what their average kinetic energy of these 2 particles are genetic energy will be plant of 2 particles to try take 2 particles of localized them to a distance comparable to the plant distance the answer will be that that would be energy that you
have to put in to do that will be department for just uncertainty principle with a work out per nite look at what cigarettes struck I will tell you next about some mathematics and tell you about the mathematics of conformal transformation White because that's the basic mathematical tools of string theory we have a choice I can't even tell you why there are 24 26 dimensions in string theory Arkansas constructive about conformal transformations will get the both of them at some point but Our Moses on Thanksgiving Thanksgiving Thursday or think the answer is yes OK we have 45 minutes try and 45 minutes to give you the easy version of why there have to be 20 feet 6 inches of Ozark strength theory be easy version is not easy and but it uh doesn't a really get to the root of it but the hard version is very hard but that doesn't mean that we can't fixture we can talk about a qualitative but I thought I would do we what I call the easy version which is not easy and not satisfied but nevertheless is correct lets as when people see this they walk away in a society where armed that's a sort of magic button the bad but it is correct and it is sort of a way that be a we need for extra dimensions was discovered it has a dual if you remember In order to get the photon to come out right we had to do something on we had to take the man's squared of the ground state of the oscillating strain let's take the open strident pick in the case of the open string mess square had to be my this warned you hear that was so that when we applied a creation operated to create a photon polarization it came out mask works like this 1 when you're on top of that is something that we haven't talked about how we get rid of this particle of negative mass where I assure you that we got rid by armed let's don't worry about that agreed mass squared of the ground state minus 1 Why is minus 1 how we we get minus 1 well harmonic oscillators have 0 point energy so the ground-state energy I love this vibrating strings should be nothing but the sum total of all our these 0 . energies Of the vibrating stress 0 . energies or positive but but and in fact they depend on the frequency anybody remember what 0 . energy harmonic oscillator a half age for may was forget Elmer 8 bar cyclical 1 1 a of Omega and you remember the frequency of and harmonic oscillator was is in in each 1 each 1 of these oscillating modes of spring restrained each while the oscillating mode and oscillating mode has frequency and of the ground-state energy of these and oscillator just in older tool in some units in over to their this doesn't sound good we have a lot of oscillators 1 for each integer wrap ALR poorer reduce troops mud quite manager and how we can in the up and get minus 1 cell or something strange going on and there but actually we don't have to get my we have to get some little bit different let's go back to the energy are very very fast moving systems the energy of a very very fast-moving system 1st of all it has overall momentum going down z-axis it has the overall moment and then on top of the overall and that conserved quantity but we really don't care about their marched into never changes during the process sees the place particle physics and then for a particle which is not moving from center of mass is now moving German the plane what's the works the rest of the member the mayor square provided I think twice the moment and that's why we began to identify energy nonrelativistic energy with em squared revenue and that the energy and to make precise this idea of very rapid systems which become nonrelativistic due that let P become extremely large basically you have to take the limits of an infinitely large momentum in the limit of infinitely large woman Our This is the a form of energy makes silhouette think a formal logic and thing goes 0 but but you really do is you take minus P. you subtract the
momentum from the energy but some writers is a perfectly conserved quantity never changes would never changes don't care ever much about it and then multiply the whole thing by Sato peas and that gives you an square this to pee here the reason you multiply by 2 P is that's got to do with time duration of fest over things move them axes slower its internal motions go slower the internal motions that means the smaller the internal energy for example if you haven't Adam what's the ionization energy of a hydrogen atoms 13 . 5 electron volts or something I take that as a new speeded up a new standard allure the energy difference between around state-of-the-art mustn't think creatively ionization energy BLG difference between the ground state of a hydrogen atom and the 1st excited state what well How threefold both something like that for fear for electron now what happens as you boost out what happens to the overall energy increases of costs right as the tho I have the energy difference between media between the 1st excited state between ground state in the sector and the 1st excited state is good smaller smaller get Smallwood was 1 of a P fact the energy of that Adam differences of energy between Adams a proportional to the square of the massively provided by 2 P so that the formula is well we don't want that what we don't care about this piece that just has to do with overall motion c-axis just subtract off doesn't have to do with the internal motions and the appeal of the later yeah that's just associated with slowing down of the internal motions so let's just multiplied through by then this is the way we really think of energy this quantity is the energy and proportional square OK the point is now the actual energy Of the ground-state state has it it is a piece of peace with that's because peace backwards put let's not subtract exports they spectrum side plus Suarez plus peace squared opposed to peace we diskette Frank P. frenzy of St. peak times is and squared plus 2 P square I notice 1st of all interesting thing which is the mass squared which would interested but there's also this PC this infinite peace so it's not really quite true that a be energy has stayed up from minus warned it would be OK if it is something like minus 1 plus an infinite can't stand that was similar to pee square so much on how that works Cheyenne works how you could and up 1 was to oppose 3 postwar fibers or only internal energy and get something which looks like a constant mainly minus 1 which is what we want to get plastered in affiliate piece that can be absorbed into something that's already there visit trick of physics takes place all the time you will it's in it happened all the time all kinds of context we you get some infinite you don't know what to do with me the answer but you realize that it was all of that they think which is becoming infants already has a constant peace but you just a head the extra infinity the constant peaks in example the vacuum the energy euro over Mr. mass of our mess particle where self energy is also took things like that where you can hide infinite constant the things which are already there nothing a satisfactory state of affairs it's just something that we do all the time and energy in this room has 0 . oscillations from all over Fort Collins and everything else laughter 4 pounds over here but from Diego oscillations of electromagnetic field how bigger those oscillations the energy industry where we do with it we just subtracted off by saying we could they at constant to the energy of the room wouldn't make any difference served as director of the same thing he'll begin constant piece which is just it but constant walls its constant never effect plaster terms which looks like this with radio additional Mayor square on when they show you the mathematics we condemn discuss some of the Parma physics mathematics is tricky but but easy In mathematics adjuster question having not 1 or 2 1st report for just some 1 post to put 3 where the all factor half outside that comes from the half From the half age for major but the basic calculations up all of the integers obviously fair and let I can extract that event infinite peace in infinity plus something like this is always a quantum field theory and her relativistic quantum mechanics work OK so 1 post Opel staples for this is not something we can easily a opposite so let's do something and then take over what they would do something to it so makes stands and then take over and let's multiply each 1 of these but I introduce a small constant epsilon each other miners epsilon number less than 1 but when epsilon is very very small this number close to worry right OK let's enter weight loss tool each of minus to Epsilon this is
just a trick for making the song courage each we're gonna put in something which gets smaller and smaller but when epsilon goes Zero it give us back the original site so epsilon of 0 infected or yard With 1 Pastora Cimex warned 3 8 7 minus 3 epsilon post minus for Epsilon but mineral if epsilon is not Zero the sequence of terms the that formerly of my tulips ministry epsilon Dugat small small epsilon small it takes a long long time for them to shrink we were just a plot the Etowah Midas touch minors so epsilon the event function the integers it would fall vary slowly if epsilon is a small number but eventually he come a minus in Epsilon will wind and it will be make beach terms successively smaller and smaller be able remark so ability he had these up and then dating up you let epsilon go 0 it's a mathematical tricks works see what they're up to what we get thanks a bunch of steps and I'll show you the steps are the steps are elementary OK on this can be written as this summer from any equals want to infinity on and each of them minors an epsilon minus 1 epsilon 21 miners go epsilon times to so for themselves right of exercise now the trick is to get rid of us in press triggers a garrison in the waning it with us and it is just different the take somewhat out the Internet and differentiate with
respect to epsilon what happens with different with respect that song what does the derivative respect epsilon near that brings them miners and so this is a quote from minus hours rural Europe minus here this is able to the original summer interest and each of the miners and as a 1st step next step is to observe that we can end this series up course series is a geometric series starts out purchase you would've hasn't eaten a minus Absolon in each other miners to epsilon course and Ito minus 3 Epsilon which also can be written in another form of each other miners epsilon time 1 plus each minus epsilon reminders 2 Epsilon and so forth I pulled out a factor of e-commerce research on new notices exactly this is a geometric series 1 force a number square the number succuba but the number and so we can in although the Saab where we get each other miners Epsilon divided by 1 minus Eco reminders that's the thing that we want to differentiate with respect to Epsilon best compromise resigned from this is something our our your window at home or work of mystery roughly what happened but they were interested in small epsilon registered small epsilon so on let's To a that yacht but expanded the smaller but 1st expanded for small epsilon and Bender the operational epsilon grew 0 interested small epsilon 1 the minus 1 where expanded in terms of Epsilon 1 minus epsilon class epsilon squared over to going to need things the power epsilon squared we won't need things beyond that of course a lot more but their rural we know about the denominator Watson the Obama later 1 miners case let's right out a minus Epsilon Murray warning plus epsilon minus epsilon squared what so would go next whatever they would me the next 1 of you advance word to write for sex workers it is checked assigned 1 minus 1 plus epsilon Cuero next want plus epsilon fuel or what but Bob website fuel or 6 now we have trouble trouble is ones cancel the denominator it is proportional to epsilon what's factor out 1 power of Epsilon right derivative the object was derivative would take but factor out a factor of epsilon dominate everything vomit as epsilon factor about 1 epsilon tenderness becomes 1 minus epsilon plus epsilon squared plus higher turns pose a higher terms disappeared when we when we go to war small epsilon there will be report but it was a minor Cyrus what but that lot I thought I've got it right to Europe 1 minor brackets 1 minus epsilon plus epsilon square over 2 0 minus epsilon Q all 6 at everybody happy with that No. 1 dilemma cancel the times I might will make plus Miners plus I have no idea if that's what I had written them before that would pull out a factor of Epsilon and that this until 1 minus epsilon over 2 plus epsilon square Over 6 right now the trick is we're expanding everything and powers of epsilon role here we have a thing in the denominator I want this and the denominator I want in the numerator somebody a deal with the thing in the armory use the formula that 1 minus a small number 2 scorer is Global One Plus SE process squared past rescue so for your ex both something not think right I think did everything right there so we do know is we take what's the denominator he ever we expanded forget fully epsilon dependence enumerated over to we get thank God that I get 1 plus epsilon ought to minors epsilon square oldest but then visitors term coming from squaring the small quantity here if I keep things the epsilon square wanna get but think plus epsilon squared for Brett you could expand about yourself bond or something like that plus higher order I never but now I figure was epsilon square just epsilon scored for you now they would chose epsilon cube in the denominator he became epsilon squared pulled out of him and I was purposefully keeping everything powers or Epsilon square I know that the next orders for OK so let's see if we can combine this together I think the chances of it right negligible Heisserer get minus D by the Epsilon 1 over Epsilon Our Times k 1 times 1 further everything get from 1 warned plus epsilon or tool minus epsilon scorer over 6 while African combined the together what's epsilon square former intercept Court of epsilon squared over 12 he at 4 4 but of when you do that so who did it U.N
400 but the me about 30 minutes to get that square bucket that's we do what he does I know I know I'm 1 plus epsilon might and when plus the salon square over 12 right that is this business and then we get Martinez epsilon my Epsilon 1 and then our minors epsilon square Over show but this time he only get something of what epsilon demand that interested keeping anything epsilon due to higher order care about 2 1 moment where we have plus epsilon square work to rule and nothing pass that because the next 1 would be epsilon for what I think this is everything short epsilon square should be plus now Mr. B plus right now right but 1 kind of Sean square with world locate now epsilon minus epsilon that's my intercepts alarm to the best and these 2 cancel the or formula right there as Aussie anything of higher order is not going to be in look at 4 0 that's 1st drove 1st focus on the warrant it's easiest the easiest turned epsilon over to it gets multiplied by 1 of epsilon that means is just 1 half after what happens when you differentiate 1 half with respect Epsilon 0 so this is we'll have to write it's not that OK now have a term which is 1 of epsilon as bad thesis this is something fitted epsilon equals 0 hours removing a good epsilon equals 0 I what's what's the derivative of 1 Epsilon Minor Suaava epsilon square right might this for Repsol squared and is another might signed plus 1 epsilon squared thing yes and they end while the Epsilon Epsilon 1 of the epsilon canceled continued differentiate with respect epsilon when you get might miss 12th this is a famous formula which is usually stated half-jokingly they want us to oppose reposeful 5 equal combined 12th the forget about the storm Maliki here and the reason is request Turner combined well this generally is an infinite constant which really doesn't I would have to convince you that way then what happens it gets it absorbed into the Peace squared with the peace squared attorney energy infinite turned the energy and careful Ramallah really careful analysis every armed allows you 2 0 absorb In the sum indoor and taunt edit of constant energy that doesn't do for the moment you have to accept that are the real answer really right answer really right and is in good string theory properly defined string series this is but I'm not sure how he warned Michael tell you why now understood I this In Philippe Kahn is not important you know the famous story of Dirac and power you Her that Barak calculated the vacuum energy found out that it was in for that and there are around said I don't care about an throwaway away Polly but diagram just because at does a mean of 0 right well in this particular case rack winds Barak wins but this is something we will have to come back through and
explain why this 1 over epsilon square remember epsilon voters euros is revered in for the constant and in this particular case the infinite 1st really is not port where Mecca pursuant right now I'm going to tell you the answer is my support might support for now this is not yet a particularly good answer is not particularly good answer yet remember what we wanted to get was altogether minus 1 we want to get this mind is 1 which 1 we excited by 1 unit of energy will give us the mass particles In fact the miners 12 was not acquire right and the reason is because of this I have from the half a bottle maker so it's actually won over 24 hour he worked for you for that latest 24 it is the energy of the ground state including 0 . oscillations but throwing away a certain infinite which 11 have to explain away later Our but OK we get however thinking I left out 1 thing remember each mode their eyes not 1 isolated but 2 oscillator ease in the B remember the eighties and the bees X sorcery was in the White House orders so actually it's twice twice this 1 for the X also is 1 for White House it is it's still it's only 2 0 minus 24 we really want minus 1 that's how they get minus 1 who say they want to would that mentions of oscillations what went for a but 24 hour directions of oscillation plus the direction going down z-axis plus kind make 26 that's with 26 right now this is a crazy story at the time nobody believed it this is a little bit too silly to be trove but In fact the mathematics did together this is the simplest way this disk which is fairly complicated already is the simplest way see is historically the way came about that that requiring a cigarette beat them mentions of space perpendicular to the direction of motion and requiring mystical reminds 1 just so that when you excited you can get the full time that that was the original argument about calm about D having to be 24 or the total space dimension having to be 26 hours the arguments this was hardly a convincing argument that kind of virtue is the easy presented road he shows what is to close the idea this is a rare come it legislative taking all kind 1969 novice goes back 1st 1969 In 1969 a return maybe early 70 but I think it still took a 9 the yes as well this point people would just exploring the mathematics of string theory and they have and are but that but the well experimentally rather rigid trajectories but at some point when the mathematical structure was put in place people began explore different on site and it was realized rather quickly that there was some kind of funny thing going on that it was a spin 1 particle when you lit up when you look up me but it didn't have all the states that it needed it had to polarization not 3 polarizations and it was realized rather quickly that this could only be the full-time or something like a far apart but that then left the question what was this 0 point energy that had to be minus 1 and was realized that quickly that required 25 24 dimensions of oscillation now as I said this was by no means a convincing argument other much much much more convincing arguments came about highly mathematical really highly mathematical and I'll tell you about them in words but not now we need 1 more month the trick a a a a a guest at summer trip to get aid but there before I tell you that when they go back the close strains
remember called strain your hat excited Whites Letter gets to Europe hit it with 2 oscillators that meant that the ground state have that might list tool units of energy but now we have a problem b over 24 hour these 24 is only minus 1 would see for a while they must be able Whittier I know but you only have 24 directs New Kids a number of direction and space when you go from close close but None of the same hearing the same theory has both closed it is that you you can't change the huh right to give the now we will agree with theory has 24 dimensions now what happens to the cause follower of similarly that remembered you have twice as many oscillators the closed proved that the ones a corner left and you gotta once a corner of right everything gets double once again not only in oscillation for every directions based procure have warned for right going modes of mournful who have gone so in fact it's true that you get twice as many which means that the ground state energy minus through which is just exactly the right thing so the 2 units of oscillation will bring you up mast What's that began personnel better we're just for well better but they are far from our far from convincing ordered the real thing has to do with something called for more variants of take up from formal invariant next time would come conform with very means if you want to prepare yourself learn a little bit about complex functions feeling very much with the door to a bet but what the cost you remind equations are not there will explain it I'll our early will calm toward Hawaii string theory is a theory of Turkish Taffy your request that you put us where you put that way that way you stick together somehow stays together From formal transformations anyway I think were for tonight where Causey Remer equation ships culture fortune just let Alberta Barbara for more please
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Formale Metadaten

Titel String Theory and M-Theory | Lecture 5
Serientitel Lecture Collection | String Theory and M-Theory
Teil 5
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15130
Herausgeber Stanford University
Erscheinungsjahr 2011
Sprache Englisch
Produktionsjahr 2010

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (October 18, 2010) Professor Leonard Susskind delivers a lecture concerning plonck variables and how they relate to string theory in the context of modern physics. 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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