Lecture 1 | String Theory and M-Theory

Stanford University all right so let me just tell you a little bit about the origins of string theory the origins of string theory really were in hadron physics they did not have to do with quantum gravity they had to do with protons neutrons mesons particularly massan's in fact the theory was put forward at a time when it really wasn't even known that for sure was suspected but it was known for sure that the protons and neutrons massan's and so forth had a quark content to them it was suspected that Amazon was a pair of quarks the idea of gluons did not yet exist the idea of gluons well in fact it actually did but nobody paid too much attention to it Nambu had postulated something like it but gluons were not part of the standard discussion of hadron x' and let's say around 1969 1968-70 what was part of standard hadron physics was one an interesting fact that the number of particle States was large there was the proton and the neutron of course I'm not interested in the difference between proton and neutron just think of them as one thing the nucleon and then there was another particle which was very similar to the proton in the turn had a little more spin and a little bit heavier mass and then there was another one above that with a little bit heavier mass and a little bit larger spin people drew pictures diagrams they were called Chu Frau floods flops not Platz Platz Platz Platz and they were diagrams which that which indicated the spectrum of objects like a proton for example and they plotted horizontally here let's say they plotted vertically angular momentum and horizontally the square of the mass who decided to put the square of the mass there instead of the mass nobody that they the diagrams had a nicer look to them if you plotted mass squared and what was discovered experimentally this is an experimental flat fact getting my keys in my F screwed up today an experimental fact that the spectrum of particles okay let's start with the let's start with the proton that has half a unit of spin and a mass of one in certain units namely the unit in which the mass of the proton is one that's approximately 1 GeV incidentally at that time the GeV didn't exist it was the B ev4 billion now its Giga okay and so the proton and the neutron would are over here a mass of one mass square root of one and an angular momentum of 1/2 then there's another particle with angular momentum three-halves these were fermions and so their angular momentum is quantized in half integers and so there's another one up here with a little bit bigger mass another one another one and another one and rather remarkably these particles all formed a straight line a straight line in the in the plot of L versus M now I should tell you when qu and Frau G first put this forward the logic of drawing a straight line there were only two points on this plot and they thought it was a theorem that through any two points you could draw a straight line well it is a theorem but these guys were not the smartest well they were pretty smart but they they and so they said oh there's two points let's draw a straight line through them and miraculously as experiment went on the additional particles all landed on the mine what an M square yes they actually said that M square they plotted it as a function of M Squared and album and said two points let's draw a straight line and that works out just fine but I mean that explains why M squared is the axis there because they're hypothesizing L is proportional and squared so now they were why would they why did they postulate that instead of L proportional to M they were lucky they were lucky or else they had some deep no it wasn't it and it wasn't entirely luck and it wasn't entirely and it wasn't a stupid guess either there was some interesting reasons for it but the same pattern held true for all hydrants or at least for all hydrants that have been studied in detail for example the pine is on the PI meson also exists on a trajectory these are known as reg a trajectories are eg GE for the for the Italian physicist Tulio Reggie they're called Reggie trajectories and if you plotted the Mazon spectrum the Mazon spectrum neurons are bosons so the angular momenta were integers for example the PI meson has almost zero mass its mass squared is even smaller than its mass it's true in units of GeV all right so the PI meson was almost massless with almost zero angular momentum and then there's a next one up I forget what it's called I don't remember what the next one is called I used to know but I don't remember anymore the next one up and the next one up and five six seven particles along a trajectory like that the Roma's on which was another mess on which starts with angular momentum one also same pattern and what's more all of these trajectories were parallel to each other they were parallel to each other which said whatever this M squared thing is it takes exact Klee the same energy well not exactly but approximately the same energy to increase the N squared by one you to increase M squared when you increase the angular momentum towards the spectrum was quantized of course it was quantized these were particles and this is quantum mechanics the spectrum was quantized but in each case the same relationship between L and M squared and the same quantum jump in M squared when you increased L by one unit one unit now means in units of Planck's constant of course so there's something going on that was giving rise to large numbers of particles of higher and higher angular momentum higher angular momentum is not that uncommon you take a basketball you leave it at rest that basketball has a certain energy and therefore a certain mass and you could plot at some place and now you spin the basketball give it one unit of plunks angular momentum that's not easy to do incidentally but when it has some angular momentum it will be rotating it will have some rotational energy so if you increase its angular momentum by one unit you will have to increase its energy by a little bit tiny tiny bit for a basketball and you can keep increasing the angular momentum of the basketball as you do so the energy will increase in fact it will not look like a straight line it will look like a curve and it will in somewheres why does the curve end the curve ends simply because at some angular velocity the centrifugal forces are so large that the AB basketball will just be torn apart right so in some place at some high angular momentum which represents you know the strength of materials how strong is a whatever basketballs are made out of so trajectories like that were not unusual you can plot them for atoms atoms also have the property that as you increase their angular momentum you increase their energy the mass energy and mass being the same thing but again for an atom there's only so much mass you can give it before you ionize the atom so what was unusual particularly unusual and again incidentally for atoms they would not be straight lines well as unusual here was the simplicity of the formula or the simplicity of the observation rather straight lines I mean they were all straight and parallel to each other and when you say how do you mean I mean that if you were to plot the Mazon or the baryon or the proton or the neutron or the PI meson or the Roma's on or it's excited states would form the same the same line in other words you take a set of part of a family of particles same slope same slope and it was called the universal reg a slope same for bosons and fermions same for different families of of bosons and fermions now this is strictly for those objects which are hydrants those things made up out of quarks and gluons which we now today recognize is being made up of quarks and gluons that was one observation the implication of this observation was fairly clear even though it was misinterpreted at the time in many many quarters it was fairly clear it said that hey drones will composit that you could spin them up this is not something you can do with an electron there is no excited state of an electron with higher angular momentum at least not not though within current experimental bounds so electrons are like points you can't see you can't spin a point spinning a point doesn't mean anything turning a point you can spin a lump so somehow these objects were not simple point particles that was the message that should have been and then in many quarters was taken from this and in fact that they had a stretch ability that from this picture you could deduce if you wanted and we will we will you could deduce the fact that they deform as they spin you wouldn't it's not obvious from here but you can all right but there was something else there was another observation which was a very bizarre up to observation let me describe it to you in terms of mess on mess on scattering let's take mezzo mezz on and then for particular let's take pine there's on primers on scattering here's a PI on coming in applying as on let's call pi doesn't matter whether it's pi plus pi - doesn't matter and it scatters off another prime is on there is a particle called the Roma's on which while it's not a composite of two pi mesons two parmesans can come together at a vertex and a Fineman diagram and make it Roma's on let's make the Roma's on like that we're going to talk when we may or may not talk more about these massan's it's not important that the idea is important but the particular names are not important Roma's on and then that PI meson could then that Roma's on could materialize as a pair of primers ons again just being a Fineman diagram conventional Fineman diagram and it would govern the properties of pyon pyon scattering probabilities okay that was the first thing now these are all pylons here any quantum field theories would immediately tell you if you have this diagram here where the tip ions come in this way and make a Roma's on and then go off as to PI ons there will be another diagram which looks like this it's just the same diagram turned on its side where a Roma's on is exchanged between two - ons in this case a Roma's on jumps from here to here and then without the peons ever annihilating but this is just the same diagram turned on its side and if one exists the other has to exist that's a consequence of principles of quantum field theory so that's something that everybody believed but then once it was recognized that this Roma's on was not a unique creature but came along with this whole family this whole Reggie trajectory of excited states it became clear that there was no reason to only have a Roma's on in here you could add together all the various you could add pi mesons come in form not the Roma's on but the next excited state of the Roma's on or the next excited state of the Roma's on and so actually when you draw this diagram you're really committed to adding up the contribution of all the mesons along here likewise here if you can exchange a Roma's on you can also exchange all the excited States well there was something very very suspicious very very peculiar when people numerically went to do this they actually added up from known experimental data the contributions of the Roma's on row Primus on the row double prime measure on and so forth and so on they did that and they did the same thing for the exchange of the Roma's on and they found something rather remarkable they found that to some approximation the sum over all the Romans going this way gave rise to about the same thing as the sum over all the Roma's ons and its partners going in the opposite there any and the other channel it was called in other words it appeared to be over counting all you needed to represent the data and all that you needed to represent the the physics of Pi scattering was summing over Rho Rho prime Rho double Prime in this annihilation process annihilation and recreation process and in fact you didn't need to add this in because this already seemed to contain it numerically numerically on the other hand it was also true that you could ignore this altogether and they are all all this up and again get the right answer get something which looked pretty much like experimental data I was very peculiar any quantum field theories would look at this and say complete how rubbish you give you if you have this you have to add that you have to add it you don't the you don't get to say it's this or this you get to say is this and this okay but peculiarly when you add it up all these contributions it simply gave you for free the effects of another diagram here for that reason I and other people began to draw diagrams which look not so much like Fineman diagrams which look more like this we didn't know what we were doing just saying look something's going on here first of all we have the idea that they were quarks quarks were not a new idea we said what must be going on is there must be a picture which looks something like this replacing the Fineman diagram this would be a quark this would be an anti quark this would be a quark going this way an anti quark going this way and likewise over here we do diagrams like that now it took some time for us to think about filling in what goes on in here we just drew pictures like this and we said look at this if you think about somehow what's going on has a topology let's call it a pathology which looks like this then look if you cut it this way if you imagine slicing it in an instant of time time is running upward of course and all you know pictures finding diagrams time we're allowing to run upward if you slice this right through the middle then you see something that looks like a Fineman diagram in which two particles come together and join and make another dial particle if we think of particles as pairs of quarks then this figure here can represent a picture like this on the other hand if we take that same picture and slice it this way it looks like a picture in which something is jumping across slice it the other way it looks like these two particles produced a thing which jumped from the side to the side so this was kind of the origin of pictures one more ingredient was added it was just added for fun and just just a curia a curious question if there are these quarks what's holding those quarks together well maybe it's something in here these are space-time diagrams of course maybe something is bridging between the quarks and if so then when you slice through these diagrams then what you would see is two quarks with something bridging between them and that's something of course would have the structure of something one-dimensional connecting a quark and an antiquark a string and a pair of quarks if you cut it the other way that same two-dimensional sheet here could be sliced into a picture where a string was exchanged from one side to another this was the very crude origins of the idea of string theory in fact it isn't exactly where it came from but they could have it did to some extent different people thought about different ways um so what's more what's more once a hadron is a string with two quirks connected to it you can spin it something you can't do to an electron you can spin it you can try to calculate with some assumptions about the nature of the material forming the string its elasticity its various properties you can start asking how the energy of it what kind of energy is there incidentally first of all is kinetic energy and second of all is stretching energy so with some kind of assumptions about the nature of these strings or this material in here you can start asking questions about how the energy increases as a function of the angular momentum surprisingly with a relatively simple assumption that we're going to do when we may get to it today I hope we get it to Attila to tour today we'll see that these pictures are not as arbitrary as they might seem l vs. M squared a linear function of L versus M squared is exactly what you need now I yeah why did you suggest that it was something like a string something you normal particle exchange through the two forks well this is not a this is not a particle exchange this is something that's sitting there all the points of it between the between the quarks this is if you were to slice it at an instant let's take a look at it at an instant here it is at an instant that consists of a quark and antiquark and a whole bunch of stuff in between now that bunch of stuff in between might have been a collection of particles that are forming a string you know that are forming a string like thing they might no no prejudice about whether it was truly continuous or whether it was just something which was approximating something continuous in between but there was one thing that made you think there might be something more fundamental about it than just a shoelace and that was that as far as we could tell these trajectories didn't end if you take a shoelace and put a pair of golf balls at the opposite ends of it and spin it around eventually you come to the point where brakes and as far as could be told these rigid trajectories did not break so it seemed maybe that there was something new going on finally to the extent that the standard model gives rise to a string like behavior it can yeah the standard model can describe a great deal about Pi on scattering these days yeah depending on the energy at very low energies the standard model has a very good description of Pi on scattering as the energy goes up where you start getting into the issues of these particles being exchanged back and forth the standard model begins to get a little bit more difficult to deal with and a string like picture becomes becomes more useful did they know that the trajectory does not pass no no no no but didn't give any sign of giving out and yeah so what that meant will it appeared is this thing could be stretched more or less indefinitely without much happening to it will we'll come back to this point well in fact they're not entirely different just glue arms in some sense they may be collections of gluons in fact they have to be collections of gluons the present understanding of it today is something like this that that the gluon field is like the maxwell field the quarks you asked me about bar magnets okay so the quarks are like poles of the North Pole and South Pole of a bar magnet North Pole South Pole you can't have mama poles the only way you can have a north pole or South Pole is to have it as the end of a bar magnet okay or you can say it a different way you can have dipoles you can have magnetic fields which look like this here it looks like is a magnetic monopole lines of flux spreading out here it looks like there are lines of flux spreading out and there might be some reason why the lines of flux form a narrow tube in between the current understanding of the connection between gluons and these strings goes something like this quantum mechanically fields can be described as either particles or fields let's take the field description in the field description the gluon field between two quarks nor quark and an antiquark would be like the field configuration between a particle between a charge and an opposite charge it would look like this that's what the field between a positive charge and a negative charge looks like the energy as you separate them is the energy stored in the field in between now for ordinary electrodynamics those field lines spread out and because they spread out the field diminishes in between them as you separate they feel as you separate the part of the charges the field lines spread out that the field diminishes in between and that's the usual pattern our understanding today is that the nonlinearities we've talked about this before but I'll just mention it again that the nonlinearities in quantum chromodynamics have the effect of causing these field lines to attract in a certain way and the effect of it is that the field lines form strings that look like that as you pull these apart the string doesn't spread this way it just gets longer and longer and longer that's our current understanding and if you like it's permissible to think of the string as being made up out of gluons as you pull it apart it's not like a shoelace or a rubber band as you stretch a rubber band the number of molecules in it doesn't change the number of molecules in it then they just stretch it eventually because the number of molecules doesn't change eventually they get too far from each other and bang the rubber band breaks but imagine a rubber band in which as you stretched it every time there was a gap opening up between atoms a new atom was inserted in between in that case you could imagine that you could stretch this ad infinitum forever without breaking it and that's the nature of the gluon field between a quark and an antiquark as you stretch it the energy of stretching goes into creating more gluons and B Queen and you can just stretch the hell out of that that system and won't work yeah can you extend the analogy to say that the strain that makes up saying electron is its component well first of all we don't know that string that electrons really are made of strings no no this all of this physics was taking place on a length scale of the size of a proton this is an enormous length scale by comparison with the scales of quantum gravity which take place at the Planck length it's a more or less perhaps accidental fact that that the mathematics of the string theory has described both things they're quite different they occur at completely different length scales but through these considerations I other people began to work out the mathematics of interacting strings can you really quantify this can you really make a theory of interacting strings which will give you all of the physics of the interacting hydrants the answer at the time is you've looked it looked promising it kept it look very promising in fact it kept promising and promising and promising like string theory today a huge promise but never quite did it right okay for reasons that in hindsight are fairly clear the precise mathematics that we were using was not quite the right mathematics for studying hagans it was the right mathematics for studying quantum gravity and so we kept getting the blue as much as we didn't want it we kept getting particles in the theory with zero mass and spin - what is that a graviton nobody wanted a graviton this was a nuisance go away graviton we couldn't make it go away nothing we could do could make it go away and eventually some smart guy named John Schwarz and Joel shirt and said wait a minute wait a minute me well maybe we were being dumb maybe this is a theory of quantum gravity and not a not a theory of hadrons incidentally string theory of hadrons final has been put together with a proper mathematics it does work but it's a little bit different okay let's talk about well okay before talking about strings the mathematics of strings let's talk about relativity versus non relativistic kinematics Oh incidentally just a buzz word this two dimensional structure in here which is what is it it's replacing the idea of a world line a world line is being replaced by a two dimensional sheet such a sheet I think the term actually goes back to me was called a world sheet today it's the standard terminology so strings are world sheets looked at at an instant in the same sense that particles are world lines looked at at an instant so that's the that's the jargon world sheets and world lines sir that diagram okay feed it there's string in there as a stretching that put the gluons in tension no and so how do you get from plus to minus well the lines of flux come out one side and go in the other side same way you go from the North Pole to the South Pole in a magnet I don't do any magnet any magnet has two poles there's always two poles of opposite sign no magnet has two north poles no magnet has two south poles every magnet has one north pole in one South Pole North and South are like plus and minus four what a black wall I don't know what a block wall is despite the fact that I'm the felix bloch professor of physics I don't know yeah - the string be considered either continuous or discrete or does it make any difference well you see now you know you're running into the subtleties of quantum mechanics is the electromagnetic field a continuum well in some ways yes is the electromagnetic field made of discrete quanta yes in some way yes and quantum mechanics tells you that that that distinction between continuum and discrete is a very subtle water and I won't try to answer it right now I think both are true its continuum and it's discrete depending on the way you think about it all right next question arm none of nonrelativistic versus relativistic kinematics or kinematics or simple ideas about the particles energy momentum the symmetries of motion of on the face of it relativistic and non relativistic physics look very very different of course we know that they connected to each other but let's let's quantify or discuss that difference yes yes Michel you can always ask you a question as long as you keep bringing me cookies screaming a point in time that we hit the water no change the angle of that cross-section of a fine you still describe whether something like a spring yes yes the Lorentz transformation of us what you're talking about is Lorentz transformation of course the Lorentz transformation is a moving string but it's still a string absolutely right so if you were to consider a frame in which simultaneity was this line here you would still be seeing a string but you would be seeing a string in motion as opposed to standing still right when I say standing still incidentally I mean the center of mass of it's standing still strings wiggle a lot they've got a lot of tension and they vibrate a lot so they don't stand still but the center of mass can stand still all right let's come now to the issue of relativity versus nominal conductivity how do you describe a relativistic string well that's awfully complicated describing anything relativistically is complicated for example just the let's begin with the energy of a particle the energy of a particle non relativistically point particle of is P squared over 2m momentum squared of the depending on the number of dimensions we would have to add up the various components of momentum divided by twice the mass that's a nice simple algebraic quantity the square of a function is easy to compute and so forth you might add to this a constant and the constant you would think of as the binding energy or just the energy of the particle because it's there so you might put something else there let's call it B the energy that it takes to assemble a particle whatever it is the characteristic of it is that it does not depend on the state of motion it doesn't depend on P in a relativistic the there's a natural candidate for what this bee is it's the energy of the thing when P is equal to zero right I mean with relativity or not relativity it's the energy of the particle at rest what is what are we in the special theory of relativity what do we put there MC squared of course so the natural thing to put there which would be MC squared but let's just think of it as a additive constant and it's constant only in so far is it does not depend on P and I vary from different kind of particle the different kind of particle it could be the binding energy holding together an atom it could be whatever but it doesn't depend on the overall motion and for many purposes you could just drop this because it's always there and doesn't the energy differences don't depend on it so P squared over 2m and P squared over 2m is terribly easy to manipulate it's just the thing that you just quadratic of course if you have many particles in a system then what you do is you add up the energy if they're not interacting and you also add up the internal binding energy internal energy you could call it you weigh them all up again this doesn't matter because energy differences don't are insensitive to it now how do we get this from relativity let's remind ourselves what the formula for the energy of a particle is the energy of a particle this is e I in relativity it's equal to the square root of P squared plus M Squared it's of course this I'm going to correct this in a minute but it's of course equal to the sum of all the particles let's just write it as e equals the sum over all the particles in the system of P squared plus M Squared where P squared is px square plus py squared plus PZ squared or however the many dimensions we wish to take into account but why take into account however many dimensions are appropriate to the problem so this would be P I and M I for the ayth particle and also XY and Z and whatever else okay how do we go from here to here well first of all I left something out I left out the C to the fourth that's the fourth power of the speed of light and M square C squared here as soon as I finish this one little demonstration I'm going to set C equal to one going to us which do I haven't run you're right sorry MC squared yeah C to the fourth very good C squared P squared C squared M Squared C to the fourth good well we're having trouble okay the nonrelativistic limit is appropriate for problems where a particle is moving very slowly which means its momentum is very small and in particular in which P squared C squared is much smaller than M Squared C to the fourth under those circumstances you can take the square root of P squared plus M Squared and write it first in the form M Squared C to the fourth times 1 plus P squared C squared over m squared C to the fourth you can factor out of the square root the M Squared C to the fourth and that gives you MC squared on the outside that's good sign but with a correction and the correction is this over here what do you do with it you expand out the square root do you use the formula that the square root of 1 plus a small quantity is 1 plus the small quantity divided by 2 square root of 1 plus a small quantity is equal to the 1 plus the small quantity over 2 that's an approximation of course it's not exact but as the small quantity gets smaller and smaller it becomes better and better so what do you get you get MC squared plus P squared C squared over MC squared MC to the fourth divided by two all times MC MC squared here okay so let's see what cancel MC squared that's familiar that's the relativistic rest energy but this here has four powers of C in the numerator four powers of C in the denominator C goes away it has one power of M in the numerator two powers of m in the denominator cancel them and you get the good old nonrelativistic formula but it's an approximation it's an approximation and when is it good it's good when all particles are moving slowly it's not just the whole system which has to be moving slowly to use nonrelativistic physics you might for example have a box of particles and the box may be moving slowly but inside the box the particles may be moving with close to the speed of light you cannot use pure nonrelativistic physics for all of these particles because they have relative motions which are up near the speed of light so strictly speaking the nonrelativistic limit is a good thing to do when all of the particles are moving slowly and it is an approximation now there's another sense in which nonrelativistic physics is an exact description of relativistic physics so I'm going to show you this this is something that goes back a long ways in particle physics when I worked on it in 1968 or 67 or sometime it was called the infinite momentum frame now what's called a light-cone frame so if you look up a light cone frame you will see these things described okay but it's easy it's easy if I don't want to do it in great and enormous generality it's easy and here's what the trick is instead of thinking of a system in its rest frame when we said or near the rest frame in other words a frame in which every momentum is slow we're going to do a different trick we're going to look at it from the point of view or a frame where everything the entire system has been boosted up to have huge momentum along one axis in other words boost it up so that it's moving down the z axis let's take that to be the z axis so that it has a mungus lee large momentum along the z axis there's no loss of generality there we can take any system and just boost it so that it's along the z axis and then rewrite what this formula looks like ok so I'm going to you know I'm going to set C equal to 1 now I'm not going to bother keeping C the energy is the sum of all the particles of again square root of P squared plus M squared which is equal to square root of PZ squared plus px squared plus py squared plus M squared but now we're boosting the hell out of the system along the z axis until every single particle has a huge momentum along the z axis every single one of them if there's any particle which is going backward on the z axis you just haven't boosted it enough just boost it more until it's going forward with a large momentum in that case all of the PZ s are very large what happens to px py and M when you boost something nothing that's the rest mass we don't even speak about moving mass anymore the rest mass and the components of momentum perpendicular to the boost don't change when you boost something ok so now the big quantity is PZ and P X py and M are kept fixed and much smaller than P Z so the appropriate thing to do here in taking the limit is expand it for large P Z expand it for this being small a way to do that is to write this in the form square root of 1 plus let's just call it P Square P X let's write it out P X square plus py squared plus M squared divided by P Z squared all times PZ on the outside right PZ on the outside if I brought the PZ inside the square root it would have to be squared it would be PZ squared and then it would cancel this PZ squared here okay what's the next step expand use the binomial expansion and binomial approximation to do exactly the same thing we did over here this is now the small quantity and so this becomes P Z times 1 plus the X square plus py squared plus M Squared over twice P Z twice P Z squared excuse me P Z squared or to summarize it all the energy is the sum of all the particles of PZ of the iPart achill plus the sum of let's call it P P will now stand for px and py let's use little P little P stands for px and py little P squared over twice big PZ plus M Squared over twice big PZ no no those are PZ up here and a PZ squared down here okay good I don't need to put brackets in okay first observation if we believe in momentum conservation which we do in this class if we believe in momentum conservation then first of all this is just the total momentum of the system the first term here is the total Z component of momentum going down the z axis it's huge very large but it's a constant it's a constant that as various things go on in this system the total momentum never changes if you have a constant term in the energy which doesn't change in any way during the course of a a constant additive thing adding it to the energy or subtracting it from the energy doesn't do anything for example if you added the electric charge to the energy since electric charge is conserved and the only thing that's ever important in physics is differences of energy you could just drop it or keep it that doesn't matter the same is true here you have a conserved quantity which is conserved for other reasons than energy conservation energy of course is also conserved but P Z is conserved for other reasons here's something which never changes you can just drop it if you were to think of the energy as being the Hamiltonian of a system it would make no difference whether you whether you drop it or don't drop it because it's a conserved quantity which never changes so you can drop this will make no difference the rest of the energy here is this thing here now first of all notice that PZ is in the denominator what does that mean why is there why is the energy so small in particular energy differences for example differences depending on the state of motion in the XY plane they will be tiny why are they tiny anybody know why are the energy difficult I will tell you for this it's useful to remember a bit of quantum mechanics even though we don't need to be doing quantum mechanics what is the meaning of the energy the energy of course in quantum mechanics is the same as the Hamiltonian it is also in classical mechanics but what's the meaning of the Hamiltonian in quantum mechanics do you remember your quantum mechanics it is an operator it's a hermitian operator but it's also the Opera mm-hmm it's an eigenvalue of the energy but it's also associated with something else it's also associated with time evolution all right remember that this is the same as I D by DT namely IH bar probably D by DT this is its action as an operator on a state that what it means to say energies are very small is that systems are changing very slowly this is also true incidentally in classical mechanics or just to point this out there quantum mechanics if the energies of a system are very very small it means changes take place very very slowly the smaller the energy if the energy scales with someone over P Z here it means that the larger the P Z is the slower things take place in the system what's going on here very simple it's time dilation the more you boost the system up to higher and higher momentum in your reference frame the slower things take place okay that's interesting but of course we have all the time in the world are a system moving we can wait as long as we like to see things take place if we're trying to make a theory of radioactive decay sure boosting it up will make the radioactive decay go slower but we can rescale that out we can say instead of working on a scale of microseconds we'll work on a scale of millions of years and we'll also see the we'll also see the the nucleus decay everything just has to be rescaled so this one over P Z there the total this is the one over P Z this is incidentally for the ithe article and we add them all up so the fact that all the PCs get large incidentally in fixed proportion they all get large and fixed proportion that said the energy got smaller that's a completely expected phenomenon apart from that if we rescale only pzs ignore the fact that they get big or just rescale the evolution of the system this Hamiltonian or this expression for energy really does look like the nonrelativistic nonrelativistic expression with respect to the motion in the XY plane for the motion in the XY plane the energy is proportional to the square of the X Y momentum just as it is for the non relativistic particle but notice that the role of the mass of the particle in this nonrelativistic analogy is not the rest mass it's the momentum along the z-axis what this means what is mass mass is inertia right that got to do with the difficulty of deflecting something well this is saying is that the momentum along the z axis is functioning as a kind of inertia with respect to forces in the perpendicular direction and the whole thing is looking very very much like if we think of PZ is a constant then all this is is the nonrelativistic formula for the energy of a two dimensional particle now notice we now have only two dimensions of motion and what is it what about this term over here well how will should we interpret that again remember that we think of PZ as being independent of the state of motion at least a two dimensional motion so with respect to this two dimensional analogy analogy between relativistic and nonrelativistic physics it's an analogy between relativistic physics and two dimensional motion in which PZ plays the role of the mass and how about this object over here it plays the role of a binding energy does it have the right properties to be a binding energy the only thing about a binding energy is that it should be independent of the state of motion it should not depend on this does not depend on the two dimensional motion so this is kind of interesting and it's not only interesting it's incredibly useful in studying particle dynamics and absolutely central to studying strings is that in a very precise and exact way the motion of a relativistic system when it's boosted up to enormous ly large momentum behaves completely non relativistically with respect to the motion in the plane perpendicular to the to the boost okay it's for this reason that string theory is also often described in terms of mathematics which is the mathematics of a nonrelativistic string a non relativistic string a non relativistic string is a collection of point particles in some limit than which to let the point particles get more and more continuous all moving non relativistically why by what the hoods bar do we do we use non relativistic physics to describe anything is complicated as a relativistic string well the answer is that in the infinite momentum frame which these days is called the light-cone frame mostly because there's nothing to do with cones nothing whatever to do with cones I'll tell you another time where it's called a light cone frame not important but in the infinite momentum frame motion is nonrelativistic and you have a chance that perhaps the motion of a string when it's boosted up may be described by not by a kind of nonrelativistic quantum mechanics and this seems to be borne out not seems to be this has been the techniques that have been used for since the very beginning of string theory to analyze relativistic strings let me show you the simplest fact yet ugh no okay let me show you one of the very simple connections that that follow from thinking this way let's now hypothesize or postulate that we can think of particles as strings using the two dimensions using the two dimensional analogy with nonrelativistic physics to explore those strings as if they were conventional nonrelativistic math shoelaces but something closer to rubber bands stretchable they can move they can flap they can do all the things that a rubber band an ideal rubber band can do what what's the mathematical description of a two-dimensional rubberband which is moving around in two dimensions let's take our rubber band to be an open rubberband that means somebody took a scissor cut it and opened it up let's begin with open strings open strings mean strings with two ends that may or may not be something interesting attached to the ends but we're interested more in the strings let's write the physics of a of a string what is the where z energy what is the energy stored in a string we can think of the string as a collection of points point particles which later on we will take limits one of the things we will do when we take a limit as well let the mass of each point go to zero that's because we're going to have an the whole string has a finite mass we're going to think of it as being a collection of a virtual infinity of point masses it had better be that in taking the limit we let the mass of each point go to zero all right but what's our what's the energy of this the energy is going to be proportional the kinetic energy it'll be the sum of all the points of X I dot squared these are two dimensional now so we can write this as X plus y dot squared that's the ithe point divided by two and we might put here in a mass of the I part achill which later on we're going to let go to zero but let's say let's not there be two I'll just tell you how to do to contain the continuum limit I'll show you how to do it I'll just tell you how to do it all right what are we missing out of this formula interactions yeah are these points are attracting each other if they weren't attracting each other they would just fly apart they're forming a string they are in addition to the points we have to put in the little Springs that connect them so think of it as a chain of little balls and little Springs can you see the Springs little balls and little Springs let's just call this X sub I squared X sub I squared now stands for X x squared X dot square plus y dot squared okay what is the potential energy between the points the potential energy is a sum also over all neighboring pairs so there's another sum of I here there's a spring constant let's just call it K all the mass points have the same mass there's a spring constant there and what is the potential energy between a pair of points it'll be proportional to the distance between them X I minus X I'm plus 1 squared probably a 2 there this is Hookes law this is Hookes law the energy stored in a stretched spring is proportional to the distance of stretching squared that's the Hookes law formula for they now what happens when you go to the continuum limit in other words you let the points get denser and denser and denser more and more of them you have to do two things you have to let the mass of each one go to 0 and you'll have to also let the spring constant what it is you want the spring constant to get bigger small big big can you earth you you know why supposing you take a rubber band and you take a rubber band a big long piece of rubber band and you stretch it it's easy to stretch now take two points very close to each other and try to stretch them that same distance much harder okay so the spring constant gets big and the mass gets small but at the end what you get just take it from me what you get is of course an integral represent an integral replacing the sum the integral is over a parameter along the string you have to introduce a mathematical parameter along the string we can call that parameter we'll give it a name Sigma Sigma goes from one end of the string where we can arbitrarily say it's zero so Sigma is zero at this end and at the other end we can arbitrarily say Sigma is equal to PI I could have taken it to be one I could have taken it to be seven that doesn't matter it will be useful to think that to call it pi the reason is later on we're going to study closed strings which go all the ways around in a loop and it's nice to say they go from zero to two pi that's all but they go from zero to pi so this sum over the mass points is going to be an integral from zero to PI D Sigma this is adding them all up and we're going to have the kinetic energy of the little element of string at point Sigma now continue a string now we take a little element a point Sigma we take this velocity squared and divide by two and what about this term over here I've chosen the mass to go in the appropriate way of dropping the mass here by the time you finished you can absorb the mass into something else doesn't matter what it's just X dot square it's clearly kinetic energy what about this term here what's that going to look like how about X I minus X I plus one which you replace that with derivative derivative this is like the derivative of X with respect to Sigma squared so the other term here will be derivative of X with respect to Sigma this is derivative of X with respect to time this is derivative of X with respect to Sigma squared this is the energy of the string if I wanted to write the Lagrangian you all remember what a Lagrangian is energy is kinetic energy plus potential energy Lagrangian is kinetic energy minus so if I wanted the Lagrangian it would be this if I wanted the energy it would be with a plus sign okay I'll write the energy will write the energy Hamiltonian plus let's focus for a little bit I'm going to stop at a few minutes and we'll take a rest but let's focus a little bit on a string which happens to have no overall center of mass motion in the two dimensions in the XY plane we're coming back now to here what we're going to do is use a model for a relativistic string which is simply based on this kind of infinite momentum thinking but in which there are only two X's the two X is moving in the in the direction perpendicular to the motion so this could be called X&Y but I'll just call it X dot squared it really consists of X dot square plus y dot squared this one consists of the X by the Sigma squared plus DX plus dy by D Sigma squared is that clear yeah okay oh I just absorbed it I just chose oh sorry there isn't two two is important I chose K in such a way to make sure that when I got to the final continuum limit the coefficient was 1 remember it's something that has to that has to vary as you vary the spacing and it can be chosen in such a way as to make this and this is the conventional energy of a vibrating string it has two terms kinetic and potential potential proportional to the stretching this is this DX T Sigma is the stretching of the string okay I want to point out one interesting fact this Hamiltonian here or this expression for energy is the generalization of this expression here for a system of particles which also has a interaction between them but the whole thing the whole object it may be vibrating and doing things but the whole object is an object we can call it a particle who's to say it's not a particle protons and neutrons have spin they rotate there's all sorts of internal motions in particles we know there are internal motions of particles internal motions of atoms internal motions of quarks inside protons and neutrons the best bet would be there are all sorts of internal motions in every particle so this stringy vibrations and internal motions and so forth that perhaps not perhaps but would add up to all the internal motion in the particle all the internal energy in the particle the internal energy would be the contributions to the energy from the potential stretching and from the relative motion of the different parts the overall motion will separate that out soon enough the overall motion of the string the center of mass of it that would just be treated as the ball as the position of the particle but the relative stretching and the relative vibration that's internal energy so when we calculate the internal energy of this particle what should we relate it to we should relate it not to the mass but to the mass squared in this correspondence it's not an analogy it's an exact statement about the properties of relativity is a very precise mathematical statement which I won't make now but there is a there was an exact sense in which fast moving systems are completely relative and non relativistic in the two dimensional sense what would the internal energy now respond to it would correspond not to the mass of the particle but to the mass squared so for a string at rest think of a string which has no motion in the in the XY plane all is doing is vibrating and it has internal energy that internal energy has to be identified with the square of the mass of the entire assembly of constituents of the string if the constituents of the string are adding up to something that we want to call a particle then that particle has a mass squared which is the sum of all of the internal energies inside the particle now this is an interesting fact we get mass squared for the energy of a particle in this framework hmm there are some seas around which I haven't tried to keep track of yeah there's seas in the yellow season right yeah I haven't tried to keep track of them but this connection was something I knew that nobody else knew at the time I'd worked on both these things and so this is interesting this is exciting another fact another fact is that a string is not so different than a spring if you look at the spectrum of energies of a string it's quantized in pretty much the same way we'll come we're going to do the quantization of it carefully but the basic fact about the quantization of it is that the string is a collection of Springs and springs have quantized energy and what's the formula for the energy of a quantum mechanical oscillator and integer multiple of something each time you increase the energy of a spring or a string the internal energy by one unit it could increases the mass squared by one unit increases the mass squared by one yeah else energy zem trip at the end is this work the simple confusion is M the mass of the strength yeah it's a mess of the whole screen the whole script and the energy is socially doubled with the size of mastering four times because M squared would be is that correct yeah yeah you've stolen my thunder okay but now look well okay let's let's look at this formula a little bit carefully now and see if I see anything interesting you like sir you have them squared there that would kind of kind of Connaught the rest mass of the system today it is in fact the rest mass of the string in a frame of reference where it's not zipping along but where it was really stationary by the equation seems to be simply not oh but I'm describing it put on it will that's right because the photon is massless well that just corresponds to M equals zero on the right hand side and you're going to ask me how can this thing be zero well we're going to come to that that is a very significant and interesting point we're going to come to it for the moment it's just the mass of the entire string the entire including its internal energy including its stretching energy all of the energies that you would normally add up to find MC squared to find the rest mass that's what this is here okay it's just a this is just a classical mechanics and relative relativity applied so far we haven't been it any other we haven't introduced London currency and things like that okay but if we did introduce quantum mechanics we would know that this string would be could have quantized energy levels and therefore quantized math squares in fact if we increase the angular momentum by one quantum then the quantized energy the quantum of energy that would be introduced would be a quantum of squared not a quantum of in this was an immediate piece of evidence that that more over yeah okay this was this was one of the hints that one of the hints there's another interesting fact here um supposing you took a string which was not moving but what you stretched out but you stretched out to a certain length okay how much energy would have have well in all of its energy would be potential energy not kinetic energy let's calculate what it would be how big is the xt sigma well if you stretch that out uniformly then the change in x along the length of it would just be the length of the string l would just be we're stretching it out to a physical distance l we stretch it out to a physical distance L over a distance from 0 to PI right all right so the derivative of X with respect to signal would be something like L divided by PI I don't care about the PI's right now they're not what CL what's interesting the X by D Sigma would just be proportional to the length of the string if you stretched it out by distance L and divided it by the range of Sigma from 0 to PI that would give you the x by d sigma and so we can say that the X by D Sigma is proportional to the length of the string and the X by D Sigma squared would just be the square of the length of the string right this is Hookes law this is Hookes law for a string if you stretch it out the distance L the energy stored in it none relativistically will be l squared but that's what has to equal the mass squared now we know something interesting about how about the energetics of the string if we were to study it in the rest frame in the rest frame of the string the energy of the string is the mass we got from mass is squared by boosting the string but if we went back now and we said look wait a minute we know a equals MC squared that's rest mass what is the rest mass of the string and the answer is that the rest mass is proportional to is a proportionality factor proportional to its length in other words this string has the property that the energy if you think about it in its rest frame if you stretch it out the distance L it will have an energy which will grow with L and be proportional to L in the rest frame it doesn't look like a Hookes law string at all it looks like a different kind of string whose energy is proportional to its length well that's very interesting because it fits with another picture it fits with the picture which I described before of lines of flux connecting quarks and antiquarks lines of flux would produce I slice something like this they would produce a patch of electric or magnetic field here it doesn't matter whether it's electric or magnetic lines of flux in a tube like this would produce a magnetic or an electric field in here electric fields have energy and the energy density and them depends on the field the energy density along this long tube of flux would be uniform if the number of flux lines passing through this little area is the same as number passing through this little area and so forth the field strength would be uniform along this tube of flux here this is in fact a property of tubes of magnetic flux and superconductors and so forth and superconductors are superconductors you don't have of course monopoles and superconductors but you can have long lines of magnetic flux and they have the property that the magnetic field is uniform along them and therefore the energy density is uniform along them that means that the energy is proportional to their length this is a common thing in in field theory and condensed matter physics in a variety of different context with with field energy in a field forming a long string is proportional to the length of the string not the length squares a different kind of strings another way to think about it is that the string is made up of a lot of little particles but as you pull on it and separate the distances here new particles form in between so as to keep the number of particles per unit length fixed that's another way to think about these long flux lines that they're uniform along their length and as you pull them apart more particles form to fill the gaps then in that situation it would also be true that the energy per unit length would be fixed and the energy would be proportional to the length this is by now actually an experimental fact about about hydrants that you can spin them up you can stretch them and they have the property that the energy per unit length is fixed they have a stretch called the string tension the string tension is a constant that would not be the case of an ordinary Hookes law if you strict them but that's but this is the picture in the rest frame and the rest frame the energy of the string is proportional to its length in the infinite momentum frame where the physics is all nonrelativistic the energy is proportional to the square of the length like Hookes law so these two kinds of strings Hookes law and flux tube are related to each other some sense they're just the same object being described in two different reference frames one at rest and one now father if you had an ordinary world rubber band and I was vibrating it with that most relativistic speeds in one there no way order your rubber bands if they vibrate with relativistic speed we haven't got the vaguest idea how to describe them we don't want to do that no uh we've we've made an indirect deduction we made the indirect deduction first but first half of the deduction was in the infinite momentum frame everything is nonrelativistic at least in two dimensions we use that to discover the fact that the stretched energy of a string which is l squared because it's described non relativistically like a hawk's law like a Hookes law spring is l squared that is to be related to M Squared indirectly from that we conclude that if we were in the rest frame the energy of the string would be proportional to its length and that's interesting because there's a wide variety of interesting string like objects that occur in field theory not made out of atoms but made out of field field configurations which have exactly the same property all right let's say let's come back in a few minutes I think I think I've probably exhausted your attention for today but let me well let me summarize let me summarize let me summarize experimental properties of of hadron indicated this kind of excitations along a line had we've been smart at the time we probably would have realized that this pattern here is the appropriate pattern for Strings whose potential energy is proportional to their length that's something we actually could have deduced directly from here but in fact why am I saying we could have I did that that the energy grew as the length of such a string that was the consequence of this relationship here that was one fact next fact non-relative are relativistic physics in a frame in which everything is moving fast is the same as nonrelativistic physics except in one less dimension or in one less dimension and so we can try to build a simple theory of relativistic strings by going to such a frame and just using non relativistic physics but one less dimension here it is here's the non relativistic string in two-dimensional space the only thing we have to remember is that wherever we saw our energy we have to think of it or internal energy in particular internal energy should be really identified with the square of the mass not the mass okay that if you remember came from the two different expansions if you like one of them was an expansion in which this was the big term and then the whole thing is approximately of water MC squared square root of this thing in here the other expansion was determined which this was big and then the excess energy was proportional to M Squared not M all right so when you do that and you go through this little exercise your conclusion is that the Hookes law energy of the effective nonrelativistic string should be identified with a mass squared which indirectly tells you that that the rest mass of the string is proportional to its length and finally there are lots and lots of field theory in condensed matter systems which have the same property so that was something encouraging if you like ok let's take a rest and then come back and either ask some questions or I don't think I'll discuss the quantization of the string today the quantization is easy we've done most of the things that are necessary to figure it out it's just a collection of harmonic oscillators um but I think we can take a rest first to help my engineer it seems you start with a classic strings dan you you lose the hell out of it without to the z-direction Jenny basically do a Lorentz transform houses Janu you've got the same physics when you go back should you still you get back to the physics that you started with you do with me but you do it's just easier this way no but you don't have the same physics yeah I first started with a spring that had a length proportional to its mass square in both cases in both cases the mass is proportional to the length and the mass squared is proportional to the length squared in both cases but in one case you call the energy the mass in the other case you call the energy the mass squared let's go through that low I'll come back to your question in a minute let me just go through it again because there was some slight of hand there was some tricky business there we wrote that energy is equal to square root of P squared plus M Squared okay let's so for the moment forget the motion in the XY plane it's just concentrate on the Z Direction in the time direction energy is related to time P is the related to space there's two ways I could expand this one of them is good when P is small and M is large all right in other words when I'm enough and when the particle is moving slowly in my frame of reference in that case let's see what we do then we write that this is equal to P times the square root of 1 plus M Squared over P squared right sorry I'm sorry I want to start them the situation where P is small and M is large good P is small and M is large so I write this then as M squared times 1 plus P squared over m squared P is small and M is large so P squared over m squared is small quantity okay not good okay the M can come out and this becomes M times the square root of 1 plus P squared plus M Squared but that's approximately 1 plus P squared over 2m ok squared which is equal to M which really means MC squared plus P squared over 2m so in that context the internal energy when the particle is at rest in space is proportional to the mass that's your good ol D equals MC squared ok and that tells you that for a particle at rest its inertia a particle at rest its inertia in other words its usual resistance to to acceleration is the same as its mass and its energy that's mass meaning inertia and the general energy are proportional to each other ok now let's do the other expansion the other expansion we boost like hell so that the momentum is very large okay and then we expand it and the other way let's see P is very large now so this is the square root of P squared times 1 plus M squared over P squared which is p+ did I do that right yeah P plus M squared I think over to P all right now this is the momentum along the direction that we did the boost there are two other directions of momentum and we can put them in here but notice that the energy apart from this fact this piece which is just total momentum which we can drop because it drops out of all interesting things is proportional to the square of the mass so in this form the energy is proportional to the square of the mass and what that says is that if a particle or a system system of particles is moving down the axis with an enormous momentum that its inertia that its inertia relative to this direction here well let me let me go back let me go back let me put in here the other terms plus px squared plus py squared they just went together with M Squared this was PZ squared plus M squared plus px square plus py squared we're taking this to be small and PZ to be large okay one of the things that this says is that the inertia is now not the mass of the particle it's the momentum along the z axis and that actually makes a lot of sense though it's not true none relativistically but relativistically it is true that a given force perpendicular to the direction of motion will produce a smaller acceleration the larger the larger the momentum so this that's the first thing this P here is the inertia and the M Squared is playing the role of an internal energy or M squared over 2 P is playing the role of an internal energy so internal energy becomes mass squared in this frame and inertia just becomes PZ oops that doesn't look good but all right now somebody asked me about the connection between these kind of strings which have this property of having a energy per unit length and superconductors I think I mentioned it as we were talking I will spell it out superconductors have the property of repelling a magnetic field they repel they don't want to accept magnetic field penetrating through them they actually repel the magnetic field ah because they repelled a magnetic field it's kind of a pressure that was pushing magnetic field out of the way but if you somehow push a magnetic field I'm going to tell you how to do that minute if you push a magnetic field into the into the superconductor in such a way that the lines of force are passing from one side of the conducts super here's a big piece of superconductor lines of force are passing through it like that what it will do would be to squeeze will push those lines of force out of the way and push them into a into a narrow string like thing like that that's a call to flux I'd it's called a flux ID or a superconducting flux line superconducting yeah superconducting flux line magnetic flux not electric flux magnetic flux and how can you make one in principle in a in a in a superconductor here's what you might do I doubt very much well this is probably not the way it's done in the way in really but the you take a peep a piece of superconductor you drill an incredibly narrow hole through it this is a Gedanken experiment this is not something that I want you to go away with as a practical experiment you drill through it an incredibly small let's see my going to get this right ah before you pull it before you cool down the superconductor before you cool down the soup winter conductor you take a long bar magnet and you stick it through the hole like that flux lines come out this way from the bar magnet and this way now you cool it down so that becomes a superconductor and you draw out the magnet these lines of flux here will get pulled into the magnet even after you've pulled out the magnet lines of flux will go through that magnet and they will form a thin tube through that magnet the field along the tube will be uniform and because the field is uniform that means the energy per unit length is fixed okay now let's go a little bit further imagine that we have magnetic monopoles we can actually simulate magnetic monopoles but let's suppose we really did have magnetic monopoles that they really were magnetic monopoles in the world and they may well be but we haven't discovered them yet but let's suppose we have discovered them and we can manipulate them okay then we could take a monopole and an anti monopole put them right on top of each other don't let them annihilate keep them a little bit apart take them and put them into the superconductor monopole an anti monopole I'm not going to tell you which one is which I'm just going to draw two of them and of course the monopole and the anti monopole have some flux lines between them now cool down the superconductor and take the monopole and the anti monopole and start to separate them what happens exactly the same thing that we think happens between a pair of quarks except that this is just an ordinary magnetic field between the monopole and my anti monopole so inside a superconductor a monopole and anti monopole would have an energy which would be proportional to the distance between them while the distance between them because the string between them has a an energy proportional to its length okay the monopole and the anti monopole could not separate from each other they would be confined because as you start to separate them the energy goes up with the length and and that's the that's the character of what happens to two hydrants or quarks and a hadron you separate them and their energy goes up so okay so this is a system for the energy is proportional to the length and it has the characteristics of the same kind of string does that answer your question whoever asked me and who asked me how well I guess he's gone the question was what's what's taking a role of a superfluid for the hey groans what is this super superconducting tool virtual monopoles that's what it's thought to be virtual monopoles but not these with different monopoles or chromodynamic mana poles in the superconductor here the condensate the superconducting condensate is made out of electric charge and it causes confinement of magnetic charge in quantum chromodynamics quarks are confined and they are kind of the electric charges of of quantum chromodynamics the things which condense our magnetic charges you say where are those magnetic charges why don't we see them because they're always condensed in the vacuum so but this is this is beyond the what I had intended to talk about today okay what other questions come up oh it's almost nine o'clock okay yeah so is the rule or highly relativistic sir well a superconductor is not a highly relativist it's it's a it's not no no but it's string like so if you took a a regular Hookes law stream made of Springs and had a going relativistically inside the circuit market so assuming you could you have to make up some theory every after this in theory but that that construction would then have an energy to work into its life an order would do depends in the details no there's something special about energy that there grows at length as I said what it really means in particle language is that as you separate the constituents of the string as you pull it apart the energy of pulling it apart instead of just separating and making larger distances makes more make small particles so that the so that the density of them along the string always stays the same that's the character of a these kind of strings and the other way to think about it is they're strings made up at a field where because field lines are not allowed to end field lines are not allowed to end as long as the field doesn't spread out this way it has to remain uniform in the other direction so it's a characteristic of flux lines flux lines which are confined to a tube if you had any kind of situation where an electric field will confined and prevent it from spreading out in the perpendicular direction the energy per unit length of it would be constant along with yeah that's a very good question yeah remarkably the way string theory works is there are not independent degrees of freedom for motion along the other direction this is a remarkable and strange fact that in string theory you do not include directly degrees of freedom for the motion of the string along the direction of the the boost here it is thought that that's connected with something called the holographic principle it really in a gravitating system you need one less direct dimension to describe it but it's one of the very remarkable features of strength eery that you don't describe the string in let's say in three-dimensional space you in this limit you describe it only by the two-dimensional motion and yet as we will see it's consistent with with Lorentz invariants yeah how many how many data points do you have on the Reggie plots in five or ten or I don't know what the maximum number is now uh seven something like that so it's not endless but it doesn't give any evidence of giving out I'll call I'll talk about it another time but there the the evidence for that but remember that we're not really interested in hadron so I was just giving you a historical perspective of where the whole thing came from but now we want to study this mathematical theory and not insist that it looks like hadrian's but just ask what it does look like and what we'll find out is it looks more like gravity than it does hydrants well the idea that we had had sometimes spatial element of the part of the composite Ness of the particle for a particle it doesn't seem to be composit well okay let's let's be get so the question is can be expressed in two different ways let's say um it can be expressed though the answer to your question can be expressed in terms of the smallness of the particle or in terms of the energy that it takes to increase the particle by one unit all right so let's let's remember the formula for the energy of a rotating system as a function of its angular momentum l squared where that's the angular momentum divided by two times something number one of this moment of inertia and what's the moment of inertia related to the mass the mass and the square of the size of the system right the sky mr squared not M squared R squared M R squared but the point is that for a given mass the moment of inertia gets smaller and smaller as assistant gets sorry yes the moment of inertia gets very small when the system is very small now what is L L is angular momentum and it comes in quanta so this is proportional to N squared H bar squared or something like that and now you can ask how much energy does it take to go from N equals zero to N equals one N equals zero the energy it might be a little bit of energy but the energy this piece of the energy is zero the first excited state will have an energy H bar squared over 2 times 1 divided by the moment of inertia and if the object is very very small the moment of inertia is very large no 1 over the moment of inertia is very large so the excitation energy the energy that it takes to increase the angular momentum by one unit becomes very very big if the object is small well hadrian's are big objects they have small moments of inertia on the gun on the scale of quantum gravity hey Jones are just enormous I mean they're they're you know they're big blobs they have large moments of inertia the energy taken to excite one is not very big electrons are known to be much smaller the expectation is that they're exceedingly small maybe 10 to the 16th times smaller than a proton you square their radius to get the moment of inertia so that could mean 10 to the 32 times a smaller moment of inertia and that means that the energy of excitation would be 10 to the 32 times bigger well maybe that's an that's that's probably too much but they're so if electrons have excited states because they're so small if they have rotational excitations those rotational excitations well I think I lost my true for our cheap lot but the mass necessary to increase the angular momentum by one unit would be way off the know that end of the of the plot just too much it just takes too much energy to spin it up when I say too much I mean too much compared to the energies that are available in particle collisions okay so the energy needed to excite a hadron by one unit is less is roughly of order a GeV you collide particles at a GeV or whatever it is you see these excitations the energy needed to looks to spin up an electron is very much higher you don't see them how about the energy needed to spin up a a basketball by one unit of angular momentum very small very small in fact you can't even see that it's quantized so atom would be okay whatever it is first there are a couple of electron volts need to spin up an atom by one you a couple of electron volts good okay for more please visit us at stanford.edu

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Keywords: physics, science, atoms, energy, string theory, einstein, electrons, protons, neutrons, graphs, regge slope, particle physics, theoretical, black hole, spin, scattering

Id: 25haxRuZQUk

Length: 106min 55sec (6415 seconds)

Published: Wed Mar 30 2011