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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
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
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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