String Theory and M-Theory | Lecture 1


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String Theory and M-Theory | Lecture 1
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Susskind, Leonard
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(September 20, 2010) Leonard Susskind gives a lecture on the string theory and particle physics. He is a world renown theoretical physicist and uses graphs to help demonstrate the theories he is presenting. String theory (with its close relative, M-theory) is the basis for the most ambitious theories of the physical world. It has profoundly influenced our understanding of gravity, cosmology, and particle physics. In this course we will develop the basic theoretical and mathematical ideas, including the string-theoretic origin of gravity, the theory of extra dimensions of space, the connection between strings and black holes, the "landscape" of string theory, and the holographic principle.
Weapon RSD-10 Pioneer Trajectory Neutron Magnetic core Visible spectrum Supernova Global warming Relative articulation Group delay and phase delay Pattern (sewing) Gear Roll forming String theory Barrel Spare part Elementary particle Quark Magnetic moment Stagecoach Digital electronics Space probe Kilogram Cocktail party effect Map FACTS (newspaper) Cogeneration Excited state Spin (physics) Computer animation Standard cell Autumn Mass Measurement Universe Restkern Last
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Theory of relativity Analog signal Effects unit Noise (electronics) Scale (map) Sintering Continuous track Spring (device) Limiter Tool Elementary particle Magnetic moment Energy conservation Photonic crystal Source (album) FACTS (newspaper) Shoelaces Excited state Direct current Particle physics Cartridge (firearms) Mass Mode of transport Plane (tool) Ruler Hot working Speed of light Day Year Summer (George Winston album) Force Access network Rotating radio transient Roots-type supercharger Mask Band gap Lecture/Conference Ship class Automobile String theory Steckverbinder Electric charge Rail transport operations Eraser Fundamental frequency Regentropfen Ballpoint pen Field strength Electronic component Inertia Hochsetzsteller White Relativistic quantum chemistry Classical mechanics Kopfstütze Spant Focal length Permittivity Hour Radioactive decay Wire Expansionsturbine Restkern
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Ruler Hot working Effects unit Transmission line Scale (map) Year Order and disorder (physics) Force Cosmic distance ladder Ring strain Bird vocalization Roll forming Brickyard Lecture/Conference String theory Condenser (microscope) Cardinal direction Spare part Elementary particle Kardieren Tape recorder Electric charge Quark Quantum Magnetism Magnetic moment Collision Measuring cup Field strength Gate (airport) Photon Hochsetzsteller Superconductivity Vakuumphysik Electronics Tuesday Drehmasse FACTS (newspaper) Excited state Fox Density Direct current Sizing Strangeness Mass Measurement Sheet metal
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just tell you a little bit about the origins of the arms is a string theory really were in Haiti they did not have to do with quantum gravity they had to do with protons neutrons on was particularly as on the fact that theory woes put forward at a time when it really wasn't even all that assured was suspected by there was known for sure Mayor protons and neutrons man's arms and so forth at a quark content through them diller suspected that was a pair of quarks the idea of global ones did not yet exist have gone well and that it actually did are about where nobody paid too much attention to what number will have postulated something right but saw forms were not part of the standard discussion of hate crimes and Alexei around 1969 68 70 what was part of a standard Hey John physics was 1 and interesting fact that the number of particles states was launched there was the proton and neutron of course are not interested in the difference between proton neutron just think of them as 1 thing the nuclei and then there was another article which is very similar to the broke neutron and a little more speed the heavier maps and then there was another 1 above that with a little bit heavier mass wounded largest people group pictures 5 grams called true Frank G. Flower flower but that tops plots and they were diagrams were it that indicated the spectrum of objects like proton for example and a plot our horizontally here to reply plot vertically angular momentum and horizontally the square of the Mets who decided to put the square of the Mets there instead of mass nobody that diagrams have nicer looked at them Clark may have squared and what was discovered experimental use of experimental fly half back Pisa might have screwed up experimental facts but the spectrum of particles OK let's start with Riyadh but stole the probe that has had the universe spin romance Of 1 in certain units namely the unit in which the masses of the proton 1 at approximately 1 GTV we're back time GEV didn't exist it was the B E V 4 billion barrels gear and solve the proton and the neutron with were already a mess of 1 mess square warned and an angular momentum over here this is another particle with angular momentum 3 these were freely on this and angular momentum was quantized in half integers those another 1 appeal bigger mass of 0 1 0 1 another 1 and readily remarkably these particles all formed a straight line a straight line India and the plot hello and I tell you when children Frau chief 1st but this fall logic of drawing a straight line you only 2 points on us Clark and they thought was a furor that so any talk points you could draw a straight line will visit Europe but these guys were not the smartest while they were pretty smart but they and so often they said all of his 2 points last rostrate like through them and miraculously as experimental and on the additional particles Orlando on the same what square justly actually said square plot the function and scored 2 points for North part of a stage where workers had eyes we're they were why with a white advance that what know what you're lucky also have some were value it what's on Friday wasn't entirely market was entirely and was stupid their guest it was an interesting reasons for it but the same pattern held truly Bullhead grant released for major and studied in detail for example the has on the pioneers also exists are trajectory is on Omar's Reggie trajectories are GGT of Italian physicist who originally 5 core razor trajectories and if you plot me matters on spectrum the mayor's on spectrum as on verbose on so the angular momenta way integers for example applying resigned as almost 0 maps that mess squares smaller than the Mets in units of upon as I was almost massless with almost zeroing you a moment and then the next 1 up I forget what's called armed medal in the of next one's quarries Nova remembered more Max went up and that's fine art and 5
6 7 articles along a trajectory like that the
Roma as which was another matters on which stocks with angular momentum 1 also pattern and what small these trajectories power which your parallel to each other which said it would end a square meters it takes exactly the same energy while my exactly approximately the same energy to increase and square by 1 you love to increase and squared when you increase the angular momentum over the spectral most quantize of course Of course I do particles of choir mechanics spectrum was quantized but in each case the sale relationship between L & M Square and the same quantum jump in and squared with new increased L by when you when you now in units of parked cars of course solve this something going on but was giving rise to large numbers of particles of higher and higher angular momentum higher regular moment was not that on common you take a basketball River arrest that basketball has a certain energy and therefore certain masks and Yuka plots on worries are you spend the ambassador Formula 1 unit of plants angular momentum Mets uneasy due late but when some angular momentum will be rotating it will have some rotational energy so if you increase its angular momentum by 1 unit you will have to increase its energy by a little bit kind kind of pro basketball and you can keep increasing the angular momentum of the basketball as you do so the energy will increase and factor will not look like a straight line will look like curve what and all Aidan somewhere why does the curve In the current ends simply be course at some angular velocity and tropical waters of that PM basketball would be torn apart so a someplace at some high angular momentum which represents Of the strength of materials from our strongest whatever basketball's a made out of so trajectories like that were not unusual you comply than that Adams Adams also have the property that as you increase during your moment you increase man where energy mass messed things they are but again for a as part of his or only so much mess you could get it before you wineries Oh well was unusual particularly unusual again soon we Adams it would not be straight lines pose unusual here was the simplicity of the formula 0 simplicity of the rather straight lines renewal straight and power of the rich ended on Tuesday there if I mean that the plot them as art or other Barry R. or the proton or neutron climbers on or the Romans armed or its excited states would form the the same 1 member would you take a 7 hour family of particles since well since locker was called the Universal Reggie slow strain both arms and firmly on stained with different families of our of both arms and now this is strictly for those objects which I will pay drawn those things made up Out of quirks and warm which we now they recognize is being made up of quirks and want are fellows observation occasion of this observation was fairly clear you know it was misinterpreted time many many quarters wireless family clear it said that Haiti Jones will come that you could spin them off this is not something you could do with an electron there is no excited state of electron with higher angular momentum at least not a not mayor current expandable pounds oral electrons or elect points you can't you can spend point of us spinning a point of mean anything turning point you Karen screamed a lot so somehow these objects were not a simple point part of that was the message that should have been banned in many quarters was taken from us M like they had a stretch ability that were from this picture you could be Goose if you want we will we were you could reduce the fact that they perform as they spare you wouldn't do it's obvious from here but you can the or something else too was another observation which was a very bizarre Pizarro observation but describe it to you in terms of mayors on Amazon's scattering let's say Ms. honors on hand for a particular what they claim on partners on scattered a priority coming here primers on party was a matter quite fast Our miners have scatters or from other partners are really is 8 particle called the matters on which while not can't pass it on to as arms to part as I can come together at verdict find diagrams and make aromas are what make the the Roma's on my back I would have talk lily may or may not talk more about these matters on the importance the ideas of important particularly names of important Roma's and they backed by could Roma's arm could materializes pair of pie Amazon's again this thing or finding diagram conventional farming diagram showing govern the properties of party on hard
scattering probabilities OK but was 1st now use draw here then he quantum field theory where immediately tell you if you have this diagram here with plans come in this way and make a wrong as on him in office to pile it will be another diagram was like this is just a saying diagram turned on its side where a world is exchanged between 2 partners are in this case a Roma as on drums from here yet and then were rapper pylons ever waiting but this is just the same diagram aside 1 exist the other has to exist that a consequence ah principles of quantum field theory salt that's something of a regrettable but then once it was recognized that this will roll man's was not Our unique creature but came along with his whole family whole Reggie trajectory other excited states it became credit was no reason not only Harold aromas are here you could end together all the various you could add pie Amazon's committee formed not a wrong on the next excited state of aromas on the next excited state of the rumors are actually would you draw the diagram you really committed to it heading up with contribution Of all the arms along here likewise here if you could exchange aromas on you could also exchanged all the excited well it was something very very suspicious vary very peculiar when people numerically when do they actually a lot from my own experimental data the contributions of the Roma's on a rope Prime is on the road double prime on and so forth and so on we do back and they did the same thing for the exchange of robbers and they found something rather remarkable they found some approximation the sum of all the rumors on going this way give rise go about the same thing as the sum of all the rumors on the was going be opposite me other channel was called lower as it appeared to be Over account thank all you needed to represent the data in all the needed to represent the physics of scattering was summoned all rogue Rob prime bubble pride in this annihilation process violation re-creation crosses and that you didn't need dead this In because this already seen that contained no matter where you are on the other hand it was also true that you could ignore this altogether and they all all this up again get the the right answer it's something which will pretty much like experimental Bay that I was worried peculiar any quantum field theories will look at this as a complete how rubbish you keep you if you have this you have a head back and you don't you don't get say this all of it is to get the sale and yes going but peculiarly when you ended all contributions that's simply gave you for free these effects of a Member for that reason I another other people began to draw diagrams which look not somewhat like find diagrams will which look more like this with oil knowing the sank lock something's going on here 1st of all we have the idea that quarks quirks were not on your idea was ill must be going on there must be a picture which looks something like this replacing the primary rival this would be a quarter this would be an end by quarter this will be of court going this way and or go this way likewise over here we did Ted Williams like back our took some time for us to think about filling in what goes on in here we do drew pictures like us with a look best if you will think about somehow what's going on has an apology score apology which will there's the luck if you cut it this way if your majesty slicing in an instant time time is running off of course role draw but pictures for by prime were allowed to run upward of his slices right through the middle then you see something that looks like a fine and diagram which took particles come together and joint and make another bout particle if we think of particles of pairs of quarks and this figure here Cabrera Rosegg I press are the other hand it we take that same venture and slice it this way it looks like a picture in which something's jumping across Mike but of the other way it looks like these 2 particles produced thing would you jump from the site of this size sold list was kind of the origins of pictures 1 or ingredient was it was just for fun this but just curious curious question if it while these quirks what's holding those quirks together or maybe it's something he he is a space time diagrams of course maybe something it is bridging between the quarks and this sort then when you slice through his diagrams and what you would say he is 2 quarterbacks with something bridging between here and that's something of course would have the structure of something
one-dimensional connecting quarter nearby quarter strained a pair of quarks Jacoby the other way he that called Original Sheet hero could be sliced picture where a string was exchanged for more sorry this was a very crude origins of the idea of string theory factor exactly where it came from but the core of bid some very different people for different ways gone so if more what's more worth more once a page Ryan is a straight with quarks connected it Ucon spent something you can't do Electra you can spend you can try to calculate when some assumption about the the nature of the material forming the strengthened its ask the city of various properties you can start asking how they energy of it whatever energy is the parent employee restores kinetic energy the 2nd more was stretching Energy Solar some kind of assumptions about the nature of these things or of the material here you could start asking questions about how the energy increases as a function of the angular momentum surprisingly a relatively simple assumption that where we make it today we were today half will see that these pictures are not as arbitrary as they might see no vs. and squared a linear function of elders and Square is exactly what we need now I got it he shares if it is a shame for us about how this is not a path change this is something sitting up all points of between New York between the court Europe a slice it remains birds take a look at it as an instant here it is because the stock poor and backward and forward to stuff between that aren't just often between might have been a collection of articles were forming a strength of of former string like they it might were nor were not prejudice about whether it was truly continuous or whether it was just something which was approximating something continuous between but they will want things that make you think there might be something more fundamental about it then just a shoe and that was that as far as we could tell these trajectories didn't seem you take a shoelace and prepare a golf balls at the opposite end of it and spin around eventually Yukon reporter were breaks end as far as could be called these rigid trajectory not right so it seems maybe that but it was something not for their activities there are only extend but the standard Model gives rise tourists dreamlike behavior that can prom yup standard model can describe a great deal about arms depending on the energy at very low energies the standard model has a very good description of car and Driver his the energy goes but where you start getting into the issues of these particles being change back and forth the standard model begins to get a little bit more difficult to deal with a string white pictures becalmed Her because more will not be don't the gets a lot and they are now but signed giving out are so what that made with the period existing could be stretched more or less indefinitely without much happening to it will welcome back I was pouring well in fact not entirely different people on some sense they may be collections of glue warns the fact they have to be collections will use the present understanding of it today is something like this wrapped around the globe field was like the Maxwell field off the quirks U.S. about bar magnets are case of quarks alike polls North Pole South the are a bar magnet North for South for you can't have Mama polls the only way you could have a North Pole are softball was found was the end of a bar magnet or use a different way you could have bike calls you you could have magnetic fields with a wider Gutierrez here it looks like the magnetic monopole lines of flock spreading out here it looks like they're for spreading out there might be some reason why the wines of flocks former Romero built between the current understanding our connection between global arms and strings goes something like this quantum mechanically fields could be described as the particles 4 fields let's take the field description in the field descriptions field if we took quarks more quarter demand by 4 it would be all like the field configuration between a particle between a charge and composed truck to look like the rest of us the deal between a positive charge and negative charge looks on the energy as a seperate them busy energy stored the field between now fraud no electrodynamics of field lines spread out because they spread out the field diminishes in between them as you of separately fewer hours you seperate part of a charge feline spread out the field diminishes in between and but the usual pattern or understanding today is that the nonlinear errors we talk about this before adjustment again with the normally use In quantum Cromwell dynamics have the effect of causing these field wants to attract a certain way and the effect of it years that the field lines form stringing the Uruguay fat as you pulleys apart a string Gaza spread this way it just gets longer and longer among us that's
our current understanding and if you like it's permissible think of the string is being made up about as you pull apart it's not like us shoelace or rubber band as you stretch a rubber band the number of molecules in the doesn't change molecules major stretch eventually because number molecules doesn't change eventually they get too far from each other and back robber barons Brecht's bar Mantilla rubber band which as you stretch every time there was a gap opening up between albums on Nowak was between that case you could imagine that you could stretch this the ad infinitum forever without breaking it 3rd that's the nature of the glow field between quark back worked as a stretcher energy of stretching goes until creating more in between you in districts the however Her reps the more got ideas says that alligators say that they say how those kimono were 1st or we don't Norwood streamed watch on really Norway now all of this was taking place leaks scale both the size of a proton this is an enormously scale by comparison with the scales on Our of quantum gravity which take place Clark warned it some more or less perhaps accidental back that that the mathematics of string theory has described both thing quite different car a completely different length scales but ruled these considerations on other people began to work How the mathematics of the interacting string can you really quantify this can you really make her a theory of interacting strings which will give you all of the physics of the interacting Hager on BSE at the time it is you look a look promising kept it looks very promising a packet promising and promising and promising likes being theory about huge promise but he never quite get it right for reasons that in hindsight are fairly clear the precise mathematics so we were using was not quite the right mathematics froze a it was the right mathematics for studying quantum gravity and so we kept getting removal as much as we didn't want we kept getting particles in the theory was you all mass since being told what is that we a bad at times but we want better grab Thomas was loosens call a grab and we could make it go away nothing we could do could make it go away eventually some smart trying to John Schwartz Joel Sharp said when it when they were baby be gone maybe this is a theory of quantum had about you're and every string theory of Higgins final has been put together with proper Maddox does work was a little bit different OK let's talk about well before talking a bar army brings the mathematics of strings let's talk about relativity versus nonrelativistic kinematics for serve just a bars were this food dimensional structure in here which was its replacing the idea of a worldwide a world line is being replaced by a two-dimensional she such as she the term actually go back for me was called a world today standard terminology Our so strings all worlds you look back on this thing In the same since the particles are world wines look back and so that's euros jargon world sheets worldwide but that there a is a stresses that the lawyer now so I get well relying evokes cannot ones are going the other side your from North Pole go south for magnet not many a magnet has to polls is always proposal opposite signed no nomads has to North no might have to South for every nite has North for one-stop north and south what past war for what black wall Ramon bar forwards despite the fact remember fewest walk professor of physics akashi Kyoto this is a team love the late place in any urinary relieved the subtleties of quantum mechanics is electromagnetic field continuum on some ways yes electromagnetic field made of Greek want yes somewhere are here for mechanics tells you that trapped distinction between continuum the screen supple 1 Our her will try to answer it over right now I think both are true continuum of discrete depending on where you think about it for life next question on not nonrelativistic vs. relativistic kinematics kinematics or simple ideas about the particles energy momentum the cemetery is were of motion of on the face of a relativistic non relativistic physics look very very different of course we know that connect to each other but let's but Our quantifier abroad distrust different yes your Michael you could always as requested eat young a former could bring recalled saying an offer a parliamentary seat has not changed and that passed
a bill that Iran's transformation of review would be brought talking of Lawrence Korb who rents transformation of molding stirring full-strength so right what they consider a fight in which simultaneously was this line here you would still be seeing a string but you would be seeing a string emotion as opposed to standing still are my wife is standing still incidentally from massive standings things we go a lot go up and Germany bribery or cerebral stands still similar masters let's come out too the issue of relativity vs. How would you describe a relativistic stress laughs awfully complicated describing anything relativistically is complicated for example just 3 let's begin with the energy of a particle the energy of a particle relativistically point particle of peace squared off over cruel and momentum squared a R depending on the number of dimensions we will have the end up the various components of momentum provided by twice the assets of my simple algebraic quantity square functions easy compute so are you like it progress a constant and the constant you would think of as a binding energy or just below energy of the particle be courses vary so might put something else there of course be the energy picks to assemble a particle whatever it is a characteristic of it Is that does not depend on the state of motion doesn't depend on key ally in in a relativistic theory was a natural candidate for what is Bedias energy of the things went P is equal to 0 alright relatively Nora Over the years the energy of a particle arrest warrant was formerly special theory wrote what would prepare and squared off quite so the natural thing but there which would be MC Square Baluchistan covered as a of constant and constant only in so far as was not the kind on P. vary from a different kind of particle of different kind of particle it could be the binding energy holding together an ad could be whatever but doesn't depend on euro overall motion and for many purposes you could just drop this because always her their energy differences don't depend on base Cuero and these quota where most terribly easy to manipulate things that you were just quadratic allow ahead of course if you have many particles in the system and would you do if you end up energy and then I'll interact you also an upbeat binding energy in town where you could cause wear them all up again this doesn't matter because energy differences don't there are insensitive to it now get this from relativity would remind ourselves what a formula for energy of a particle is the energy of a particle we are rocket it's equal the square root of Peace squared off and squared at a cost of on the correct estimated of course equal the sum of all particles just right EU equals the sum of all the articles in the system are P square % square where peace square PX was PY square was busy square however the many dimensions we wish to take into account a white pickup can our dimensions are appropriate for problem of would be PIA an MIT for the ice particles and also X warrior z had a legal from here to Hilo 1st of all I left something out and left about the of off that's what power of the speed of light and Square and square he was so as I finish this 1 room demonstration and nor is seeing a lot more about the lack of your right sorry and Square 4th very Seurat P square C square crazy before that for our central OK the nonrelativistic limit is appropriate for problems that particles moving very slowly which means its momentum is very small and particularly in which peace glitzy squared is much smaller than em square feet of 4 under those circumstances you could take scored appease Quebecers squared write it former and squares seek a 4th plans 1 last piece square Over and square feet 4 press you go back there out of the square root beer and squared of fought and that gives you look and see square on the outside with sign Bull with but with correction and the correction is a he where you do and then you expand at the square root to use a formula the square root of 1 plus a small quantity is 1 plus a small quantity divided by the square root of 1 costs more quantity physical could warn plus small quantity over to assert approximation of course back but as a small quantity gets more smaller it becomes better and better so you get you get M C squared plus these squared C squared off and square and Seiko for us divided by tool all kinds and MC Square here source you would care for M C squared less familiar about the relativistic rest energy but this year has for what powers of sea innumerate fall powers of C in the see goes away it has won power of them enumerated to powers of cancel right and you get good old nonrelativistic formula but some approximation of approximation then is good it's good when old particles moving forward Our 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 Pua nonrelativistic physics for all of these particles because they have relative motions which erupted near the speed of light source speaking not alchemistical limit is a good thing to do what all of a product moving story and it is an approximation now with another sense in which nonrelativistic physics is an exact description of relativistic effects from this this is something that goes back a long ways and particle physics car or work on a 19 sixties he said some Parma call the infinite moment frame what's called White Kong trying to look corrupt like called for him to see these things described OK but said that it it's easy it's easy if I don't want to do it in great enormous generalities it's easy and he is what the trick there instead of thinking of system and its rest frame when we said foreign here of arrest frame modes a frame in which every moment the slot where do a different track we all look at it from the point of your frame where everything the entire system has boosted up perhaps you would normally the along 1 axis roads both survey moving down the z-axis 4th pick up the abuse z-axis so that has modestly large moment more easy access to mull tumult generality there we can take any system end boosted that along with the actors and then rewrite what former looks like appraisal I'm Loomis said equal a from Barber keeping city the energy is the sum of all the articles of again square peace Quebecers Square which is equal to square wrote of HP's yeast square course P X squared plus PY square square but now boasting the hell out of the system along the z-axis until every single particle as a huge momentum along the z-axis every single 1 of of them if there's any particle which is going backward the z-axis you just haven't boosted enough just goes that of those going forward with a large momentum and that case all of the PC's ovary wha what happens the PX PYT and when your boss on nothing that's the rest mass we don't even speak about moving necessary more arrest and the composer momentum perpendicular to boost don't change when you go some focuses not big quantities PC's and P XPY and are kept effects and are much smaller and easier so the appropriate thing to do here In taking no limit is next spring and then a large pieces EU expanded this being small away the that right this and the former square wrote of 1 plus a just caught peace squad IPX right about its Web of PY squared plus M Square provided by PC squared all use P Zia me outside right PCI outside of her brother PC inside the square root it would have to be squared it would be easy square and there would cancel was easy here OK what's the next step expand use a binomial expansion binomial approximation do exactly the same thing we go here this is now the small for the so this becomes P easy times 1 Class B X square possibly y squared pass and square all were twice he disease twice PC square Excuse me please squared Mora to summarize all the energy some or all of particles of key easy on the eyes particle plus summer of coed P E R P numerous stand for PX PYA but to loopy will be stands for a PEX little please squared off over twice a hazy boss M squared over twice big PC now now what it was a PC up here and easy squared he said brackets OK 1st observation if we believe in momentum conservation which we do rule in this class of free buoy the momentum conservation then 1st of all this is just a focal momentum of the system 1st here it is the focal component of momentum going down was the actress its huge Merrill large but it's a constant it's a constant has various things gone system the problem momentum never changes if you have a constant energy which doesn't change in any way during the course of a year constant edit thing any of the energy or subtracting from the energy doesn't do anything for example if you edit electric charge the energy since electric charges conserve and the only thing that's ever reported in physics is differences of energy you could drop it will keep it doesn't matter the same is true here you have a combo conserved quantity which is closer other reasons than energy conservation energy of course is also concerned but PC is considered father reasons here something which never changes you could just drop it if you want to think of the energy is being the Hamiltonian system it would make no difference whether you want you drop a ball drop at a cost considerate quantity would never changes he can drop will make no difference the rest of the energy here this thing here not 1st of all known the PC is in the denominator was I why is there why is the energy so small in particular energy differences for example differences depending on the state of
motions in the XY plane they will be kind he why they kind anybody know why quietly energy Keller for this it's useful to remember a bit of quantum mechanics even always don't need to be doing quite 6 was the meaning of the energy the energy of course inquiry Canucks is the same as the Hamiltonian and is also a classical mechanics but what's the meeting are the Hamiltonian in quantum mechanics remember your quantum mechanics off it is operator it's demolition operator but also the opera as my energy but also associated with something else it's also associated with Carnival I remember this is the same as theme by P. T. they really are H. bar probably Dubai this was actually cause them operator on state back where means the state energies are very small is a systems are changing very slowly as auto joint currently in classical mechanics just 0 . 0 struck hard these energies of the system of very very small means changes take place very very slowly the smaller the energy of the energy scales are over for more Europe PC here it means a lot to that please years flora things take place what's going on here very simple it's time dilation the more you booster system up behind higher moment your reference frame slower things take place OK that's interesting but of course we have all the time in the world are system will movie weakened wait as long as we like to see things take place if we're trying to make a theory of radioactive decay jurors boosting up will make the radioactive decay go slower but no we can rescale Africa say instead of working or scale microseconds work scale of millions of years also see will also see these are nucleus decay every just has to be rescheduled so this 1 up easy there the cult offices will 1 of be this is it incidentally from are particle noise and a more sober fact a PC's get who origins of Ibrahim's fixed proportion the Mayorga a larger proportion that said the energy got that's a completely expected from apart from that if we re scale all we PC's nor the fact they get a gorgeous Reese jail the evolution of from this Hamiltonian all this expression for energy really does look like the the nonrelativistic nonrelativistic expression with respect the motion in the X White Plains for most of the x plane the energy is proportional to the square of the X warned moment just as it is for hour wrote article but that the role of the man's article in his nonrelativistic analogy is not the rest Mets it's the momentum along the z-axis what this me was masses on right that got to do with the difficulty of deflecting something are this is saying is that the momentum along the z-axis is functioning as a kind of inertia with respect the forces in the perpendicular direction and the whole thing is looking very very much like our it we think of PC is a constant and always uses the nonrelativistic formula for the energy of our tool the eventual particle notice we now have only dimensions emotion I was aware about this term over here which allow which we interpret that again remember that we think of Z as being independent of the state of motion at least two-dimensional motion so with respect the fundamental analogy analogy between relativistic and nonrelativistic physics the analogy between relativistic physics and cool them motion here which easy plays the role of the masks and have about the subject over here are the plays the role of a binding energy have a right properties to be a binding energy deal only thing about a binding energy is that it should be independent of the state of motion that should not depend on this was not end on two-dimensional motion so this is kind of interesting Bali interesting it's incredible useful studying particle by Memorex absolutely central who are discovering strings is back in a very precise exact way the motion of a relativistic system boasts of up to enormously large moment behaves completely numb relativistically with respect to the motion in the plane perpendicular for the year boost it for this reason that string theory is also often described in terms of mathematics which is the mathematics of a nonrelativistic string nonrelativistic strength from nonrelativistic stringers a collection of point particles by some limit in which a point particles get more and more continuous 0 moving relativistically wire by what they could by the way you do we use nonrelativistic physics to describe anything as complicated as a relativistic God strained while the answer is In the infinite momentum frame which these days is callable icon frame mostly because nothing to do with Collins nothing whatever the from his are of career Mozart court conference not port Indians the momentum frame motions number or mistake and you have a chance that perhaps the most similar strained when the boost up may be described by by Paramount wrote Mr. Kwan this seems to be borne out now seems to be this has been the techniques have been used for since the very beginning of string theory can analyze relativistic strings on let me show you the simplest back here are nor permission you or me very simple connection that rat follow from thinking this way let's no hypothesize posture that we can think Our particles as strings using it to using Twila dimensional analogy with non relativistic physics to explore those strings as if they were conventional nonrelativistic must shoelaces but something close to the rubber band stretch of all five-acre move because slapped ignore things a rubber band and ideal rubber band and about what the Oh what a mathematical description of a tool damage on rubber band which is moving around in 2 dimensions let's take our rubber bands to be open rubber band that somebody purposes Ricardo and there are opened up 4th begin with open strings open strings mean strings with may or may not be something interesting attached to the end but were interested more in strain on but but let's right physics of a year of a string was worth energy was energy stored in a string but we can't think as a collection of points point particles which later on we will pick
limits 1 of the things doing Whitacre limit his will let the the massive each point guard 0 that's because would go ahead with the whole string the finite mass where I think of it as being a collection of a virtual infinity of point masses had better be vetted taking a limit we let the massive each point 0 I bought what's the energy of this energy is gonna be proportional the kinetic energy to be the sum of all the points of X on dog squared he's a two-dimensional now so we could write this says X plus white . squared ICI pouring provided by tool and we might put here in a massively particle which later on we it goes Zero but so it's not be too were just tell you have to do the continuum I'll show you have direct just carrier I worry missing out of this interaction young are these points are attracting each other if they were attracting each other they would the acquire by performing a strain they are in addition to the points we have to put in a little sprained and connect them think of it as 18 are little balls move brings you see the springs will balls and little springs from but just cause extra by square XO by Square now stands for X X squared Exxon square was white square OK was the potential energy between reports potential energy is a summer also overall neighboring pair so there's another storm of idea was a spring constant Court K. all messed points the same Mass as a spring constant and what is the potential energy between a pair of points it'll be proportional the distance between them XII minus X I'm plus 1 square probably a true there but this is talks a lot with folks Laura the energy stored overstretched sprained his proportional from a distance of stretching squared off the hook slaughter Our formula now what happens when you go to a continuum limit you points get enter and denser and then more and more of them you have to do 2 things you afterward a mass of each 1 goat 0 and you have also at the spring constant waiters who want spring Council to get because more big big cleaner you know I suppose take a rubber band figure rubber band grow a big long Presa band stretches Desi stretch not take 2 points very close to each other driver stretch of same distance much harder so the spring constant gets big and mascot small but only a look you get the figure for me would you get is of course the role of represent our are an integral replacing the somber the Inter over up parameter along the straight after Richard introduced a mathematical parameter along the straight ring call out parameter were given a stigma goes from 1 end of the string where we can arbitrarily say 0 the stigma 0 at this and that deal and we cannot arbitrarily say signatures equal pie I could think of 1 that could taken B 7 as a matter of our it will be useful to think of her court pie the reason is later on a study closed strings which go way ways around the globe but it's nice to say they go from 0 to at all about your from Xerox piety so this somewhat the Mass points is going to be in if the growth From 0 the party these stigma a walk they were on have comedic energy Our will element of strain at point continuous train now we will element of points Sigma Whittaker's velocity squares of I'd like to know about this firm Aurelia I II I have chosen the master Gore with the appropriate way of dropping the past year by the time you finished army you can absorb may have something else my to backstop square clearly kinetic energy I worry about this term he was 2nd or quite about XI my Cyprus warned Witchard ways that way where they have the wrong this is like a derivative of X with respect the stigma but what squared Salt the other terror here will be derivative of X with respect for Sigma misses derivative of X with respect kind this is derivative of X with respect the stigma square but this is the energy of the string if of I wanted to write or grounds in your moreover grungy is kinetic energy plus potential energy Lagrangian kinetic energy over what the Grand Union but it would be best if I Waveney energy it would be with plus star hearty energy Oradea Hamiltonian class let's focus for a little bit we must offer a few minutes and progressed but let's focus I will on a strained which happens I have no overall center of mass motion in the true dimensions of the x y plane coming back every year for all I could do is use a model for relativistic string where is simply based on the kind of phenomena thinking but in which there are only 2 0 X 2 X is moving unity in the direction perpendicular to the motion service to be called X and Y above the X parte square really consists of ex parte square cause why that squared the system the expertise was square forced Wiberg District square is clear yogurt at all I just absorb it I just shows all starters tourism . 5 shows Cheney in such a way to make sure that when I got a final continuum limit call coefficient was warned remembered something has has various you vary the spacing and can be chosen such a way as to make this and this is a conventional energy are vibrating strings has terms kinetic and potential potential proportional stretching the BXT stigma is stretching of the string work at what point out warned each arresting fact this Hamiltonian here for this expression for energy it is a generalization of this expression here for a system of particles which also has a interactions between but the whole thing the whole object it may be vibrating doing things for the whole object is an object call it a particle boosters say it's our particles are Our protons and neutrons have spin they wrote papers all sorts of internal motions particles win over most of
articles most Adams and her emotions of quarks inside of protons neutrons the best bet would be there all sorts of internal motions of every particle so this stringy vibrations internal motions or for that perhaps perhaps but warned he are fully the internal motion particle all the internal energy particle internal energy would be contributions the energy from the potential stretching and from the relative motion of the different parts we although will separate not be overall motion of the string the center of mass air that would just be treated as what is the position of the particle but the relative stretching and the role of the vibration that's true orgy Saleh when we calculate the internal energy of this particle what should be related to we should related not for the Mets but the vast square in this correspondent stunned the allergies to exact statement about the properties of relativity reserve a very precise mathematical statement which I won't make now workers of reserve exact in which fast-moving systems are completely wrote nonrelativistic in two-dimensional stands what would be internal energy Doris filled would correspond the mess of a particle but mass square so for a while for a string of arrests think of a spring which has an almost India and the X-Y plane was doing was vibrating has internal energy that internal energy has to be identified with the squalor Of the mask the entire assembly of constituents of the strength of the constituents of the strained are gaining for something that we want the core particles the article has amassed squared which is the the sum of all of the internal energies in start particle this is an interesting fact we get Mass school where 3rd for the energy of a particle immerse from that the there are some seas around which I have tried to keep track of their season Merry right are yah haven't tried to keep track of them but this connection was something I knew that nobody else knew with that kind I worked on both these figures are the soul it is an interesting is exciting another fact another factor is a strained is not so different that spring if you look at a spectrum of energies of string its quantized and pretty much the same way will come with quantization are carefully but each basic fact about quantization Ave Is 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 an integer multiple of some something each time you increase the energy of a spring or a restrained you buy 1 unit if create increases the squalor by you Chris is mess square foot yacht a duties yes His work the focus is evident 3 young men all of us they Al over there there were no news what I OK but ballet lab OK let's let's look at the formula will be careful ring now and is he was my fault that I put it back a credit card or the U.S. last visit by U.S. News Of the remained in a frame of reference work Bob zipping along but we're not alone but not for long a load of tractors for from mass but just how sponsor and equal 0 on the right side the U.S. warehouse this thing be 0 will go from bad is a very significant and arresting . com or some other that's just man on the highest-ranking entire including Aerts Internal Energy Inc. stretching energy all the the energies that you would normally set up to find and see square find the rest mass but publicity was just over this just classical mechanics relative we had deal on but if we didn't use corn carrots we would know this string would be quite where have quantized energy levels and therefore quantized Mesquite In fact if we increase angular momentum by 1 quantum then for quantized energy the quantum of energy that would be introduced would be a quantum of em square rock quantum of him this was an immediate piece of evidence that moreover got this was this workers were making a woman with another interesting fact here on supposing you took restraint which was not moving forward you stretch out which is stretched out to research Norway are much energy Where Have All in all of its energy would be a potential energy kinetic energy but calculate what it would be our biggest TEXT stigma our she stretched out uniform Lurie then the change in X along a wave of it would just be the length of the string allow just be were stretching out with physical distance L stretch after a physical distance L although a distance from 0 5 right I the river X with respect the signal would be something like elder by the by I don't care about pies right now they're not what's what's interesting BX signal would just be proportional to the length of the string stretch dapper distance L provided by the range of signal from 0 applied that would give you the expert said the story concerning the expo is proportional to the length of the string and the expertise in the square would just be the squalor of the of strength this is folklore this is hoax Lofaro stress to stretch out the distance L B energy-storing nonrelativistic Lee will be all square foot blast last week would have squared but now we know something interesting about how about the energetics of this string if we were to study it the arrest In the rest of the string energy of the string is a man with without from estimate square boosting friend from want that now waited we always equals M C squared the rest what Is the rest mass of restraint and the answer is that arrest it is proportional to lose a proportionality factor proportional to its like another words this strained has the property that these energy if you think about it and rest frame if you stretch at our
distance L it will have to energy which will grow old With a L and be proportional arrest frame it doesn't look like a hoax lost going all we ever looks like a different kind of strings whose energy is proportional late last very interesting because if it another picture fits with the picture which I described before blinds of flocks connecting quirks by 4 life is a flocks would produce a slice of the electorate they would produce a patch of electric or magnetic field Europe doesn't matter whether electrical magnetic why a of toolbar like this would produce a magnetic or Nordic field an electric fields of energy and the energy density and depends on the field the energy density all all this long would be uniforms have number of flux lines passing through this slow areas as saying the number passing through this the Solari and so forth the field strength would be uniform along with toward both Loxia this is in factory property of tool magnetic flux in superconductors so so fuckin ah superconductors you don't have of course are for superconductors but you can have a longer lines of magnetic flux live the property of the magnetic field as uniform along and therefore the energy densities uniform along and that means that the energy is proportional prevailing this is a common thing in there are in field theory condensed matter physics a variety of different context with we're field energy in field forming a long strings is proportional from going from strengthen our point square a different kind of stress another way to think about it is drive the string is made up of our little particles but as you pull are and separated distances you know particles formed between so as to keep the number of particles per you only fix another way to think about these long flux lines that uniform along railways as you pull them apart what particles form the fill the gaps in that situation would also be true the energy per unit length would be fixed you would be proportional GREGORY this is right now actually experimental fact about about grants that you can spend them are stretch them and they have a property that the energy per unit length is fixed they have a distress call the string pension string tensions are constant but would not be the case in ordinary folks who are restricted but bad but this is a show in the rest frame and the rest frame the energy of the strings proportional explained in these instead of momentum frame with a physics is all nonrelativistic be energy is proportional to to the square going quite coleslaw so they still kind of strings talks a lot and there are folks who related to whichever some sense that justice saying object being described in 2 different reference 1 arrest and warmer if you are a role that violate that will be put what barons of bribery raw speed and because we all know that now are here with we've we've made an indirect induction remain indirect induction 1st first-ever reduction was the incident momentum framed everything nonrelativistic at least in 2 dimensions we use that to discover the fact that we stretched energy of restraint which is L squared because describe nonnotable Stickley like oak floor Michael Oakes last spring is all square fast be related in square it indirectly from Matt weakened quotas that of rural the rest frame the energy of the string would be proportional was Lane and that's interesting because as a wide variety of interesting string like objects current field theory our money that Adams may of fuel the field configurations which have exactly the same property on Our so it's from back in a few minutes I think God figure
probably exhaust to tensions today or Lumley summarized this summer of experimental properties Our vote of hate crimes indicated this kind of our expectations along a line Ben a lot of time will probably will realize that this pattern yeah is the appropriate pattern for strings whose potential energy is is proportional to their way of something we actually could reduce directly from here a fact missing we could I that there are over the energy growers the link of such a strained that was a consequence of a of this relationship here that was warm next fact not wrote that our relativistic physics in a frame which everything is moving fast is same as non relativistic physics except the 1 less than an hour in 1 West dimension and so we can try to build a simple theory of relativistic strings by going to such a frame and just using nonrelativistic but 1 must eventually here it is nonrelativistic strained in two-dimensional space knowing we have to remember Is that wherever we saw energy we have figure reporting however G particular interlude you should be really identify with the square of America's not rapper fury member came from the food different expansions if you like 1 of them was an expansion look yeah which this was the big and then the whole thing is approximately of war MC Square Square Root thing here the other expansion was a terminal which this was big and then the excess energy was proportional and square and also bill that you go through this little exercise your conclusion is that Seahawks lawyer energy of these effect nonrelativistic string should be identified the as square which indirectly tells you that cover the rest of the strings proportional right and finally there are lots and lots of field theory in condensed matter systems which have the same property sold about something encouraging OK let's take a rest From back arrest questions I don't think over Strawser quantization stress quantization is easy we've done most of things were necessary for a figure that is the collection of harmonic oscillators are bar RO for Greek APEC arrest 1st without a clear about what they saw that gave you use is not eager busy your Kansas you get that with which is years so you don't a 4th that had a lengthy report last In both cases In both cases the massive proportional lengthen the mask were washed away square in both cases at but in 1 case you call the energy the mass of the other case you Collier energy square but go through that tokamak question as broke through again because it was some sleight of hand there was some tricky because their wrote that energy is he called the square root of peace aquarium across square let's set the moment of forget motion and the ex-wife plane just concentrate on busy direction and higher energy everywhere the please this chilly sake expand 1 of them is good when he is small and and was large other words when I'm enough and when the particle is moving slowly in my frame of reference in that case what's he rebuild but we write physical peak times the square root of 1 class and Square square right sorry I said I wanna start them the situation with peace more than large God is smaller members large so I right to stand as and squared times 1 plus piece squared or M squares P a small yeah as large soapy square of M Square is a small quantity of March there are
PM could come out and this becomes and Times Square Root of 1 P square was square about approximately 1 course P. squared over where progress toward which is a am which really means M C squared plus peace Cuero widow and sewer in that context the internal energy when the particle is at rest in in space is proportional to the Mets that Grove eagles and squares or case and that tells you that for a particle at rest its inertia particle at rest nurture Nova words its usual resistance to work acceleration are is same a mass and energy messed me inertia and the Jammu and proportional guage of OK now what are the other expansion the other expansion we most like held a moment was very large we expand the other way OTP is very large now so this is the square root of p squared times 1 class and squared off a peaceful way all where is he class right guess PE classes and Square a to pay right now this is the moment the mulled direction that we did the boost tour the directions of momentum and Rick preparing here by drought notice that energy apart from this fact Peace which is just total momentum which we could drop because structure forward interesting things is proportional the squalor of the when this form the energy is proportional to the square of the what that's areas is out of a particle or system system of particles of moving down the axis with enormous moment them Miller measure of relative this direction he followed his money go back go back when they put in here the other terms past PX square Kauai square they just want together with him square this was PC squared off Sam square past PX square foot PY square with thinking is to be small and easy to arch of 1 of the things that the says Is that me and I shot was now not mail of particle it's the momentum along the z-axis matches slots and it's not true number or reverse quickly but relativistically it is true that a given force perpendicular to the direction of motion will produce a small or acceleration who want to of a lot momentum so that's the 1st thing this P and and B squared is playing the role of an interim manager and squared these going role of Turner due return low-energy becomes mass square in the frame and just becomes high P easily reports awkward worker primer somebody asked me about the connection between these kind of strings which have this property us having a image agreement late and superconductors are think I mentioned as we were talking our will spell out superconductors have the property of repelling the magnetic field the repelled don't want to accept magnetic field Preparatory to learn naturally with polar magnetic field ah because they repelled a magnetic field is kind of pressure that was pushing magnetic field out of the way but a few somehow pushing magnetic field of Korea do that if you push a magnetic field in the our into the superconductor researcher with the lines of force are passing from 1 side of the conductor put big pieces of lines of force are passing throw it back what it would do would be to squeeze will cost of lines of force away push them until a year until an narrow string like things like back but a cough look story Goff look sorry sorry superconducting flocks line superconducting superconducting books like magnetic flux now electric Fox magnetic flocks and how could you make 1 principle and they the superconductor here's would you might I doubt very much wealth was probably believes that the real real but uh you take appeared is superconductor you drill incredibly narrow hole through it is a good bargain experiment was not something that I want to go away with practical expanded you drill so it is incredibly small arms organ but see get this right off before you put before you call them a superconductor before you call them the soup you pick along a bar magnet and you stick it through the halls like a flocks lines com out this way from a bar magnet and this way weather now you call it becomes a superconductor and if you wrong How the magnet these lines of flocks hero again pulled into a magnet even AFP of pour out magnet for warns of flocks will go through the of the form of faith tool through that nite the feels along with tool would be uniform and because the field is uniform energy per unit length effects of came out let's go before ever imagine that we have magnetic monopoles we get actually simulate magnetic monopoles but let's suppose we really did have magnetic monopoles really were magnetic monopoles world there may well be but we have discovered let's suppose we discover them from manipulate them OK there we could take a month poll and the entire mama Paul but they plant properly each other it annihilate the apart take them and put them into the superconductor Monaco both and and Pamala for market you which ones which image could enjoy a total value of course the matter for me and my mom polls have Samoa flux lines between them now called on superconductor take up for me and porn starter seperate what happens exactly the same thing we think happens between a pair of quarks except that this is just an ordinary magnetic field between Mamma Paul my so inside a superconductor on my pole and for would have energy which would be proportional distance between why distance between them because the strain between her has radio energy proportional goods Wayne Mark Foley Obama Paul could not separated from each other it would be confined because as you start a seperate them the energy goes up with going a rare actually that's a character of what happened Tour are too a for caught a drive separate America he goes up so OK so this is a system for the energy is proportional and our work has the characteristics of the same kind of thing that requested me whoever away because Yukon if the US was left was 1st pseudo Asia would if that is the bridge or Mom appalled at prefer to be virtual mom from our view from mom calls or chrome or by Member Mylapore walls are yeah
the superconductor here the condensate superconducting condensate is made up of electric charge and causes confinement of magnetic charge in quantum Cuomo by Memorex quarks are confined and they have kind of electric charges of quantum chromodynamics the things which Our magnetic charges our sir where are those magnetic charges why don't we see them because they're always convinced him the vacuum sold purposes this is beyond birth 1st were intended to talk about it OK what other questions the command Court From Roseburg right now will be fair he or she is the rule for higher well a superconductor a highly wrote it it's that the but normal but string life so the gate of regular folks Wall Street had this inside the mosque about half of the year after the start of the year but nor would do and the details not progress something special about grows as I said what it really means in particle language is that as you separator the constituents over strength as you pull apart With the but it these energy of calling card instead of just separating making large distances makes more explore articles so fears of density of them along with string always that's a character of restructuring and their way to think about it is their strings made up of field where a cost field lines are not allowed to leave few lines are not allowed as long as the field was spread out this way has remain uniformity of the direction of sorts a characteristic of flocks flying fox lines which are complying with any kind of situation where electric field welcome find prevented from spreading out from the perpendicular direction the energy per unit length of would be can't above all a sheet on Tuesday I are a good question young remarkably away string theory works is new are not independent degrees of freedom for emotional along the other direction this is a remarkable strange fact that in string theory you do not include directly degrees of freedom for the motion of a stranglehold on the direction of the year the boost it is thought that that connected with something called a holographic principal really are gravitating system you need 1 less direct mention described form very remarkable features of string theory but you don't describe the strain in what say in three-dimensional space you and limit you describe it only by the two-dimensional motion and yet as rules say that's consistent with our were low and there's no way I made a last May I don't know what the maximum number lose our of such good start and worse but doesn't give any parents talk I'll I'll talk about it from a time but we do have a bath remember that would not really interested in Haiti around so I was giving you a historical or prospective aware came from Hong amount we want to study this mathematical theory and not insist that looks like a Johns but just ask what it does look like they will find out as looks more like gravity Over the Hedge Run along with a letter he did this what did BA are part our world order does well or carry butts what Brickyard sort of question is can be expressed in 2 different ways let's say are it can be expressed departure could be expressed in terms of the smallness of the particle warrant terms Of these energy that it takes to increase the particle by warned you about so much your number the formula for the energy of a rotating system as a function of angular momentum l squared where that's the angular momentum provided by two-time something removed Mama owner said I was a moment of inertia related through the best the and peace Kuwait on the size of the system by MR Square Times Square Square and are square but the point is that for a given the moment of inertia gets more and more of the system gets stuck yes from entrepreneurship give very small when the system was very small now what is El Al angular momentum comes in quanta was a force in square age 4 square from that menu could ask how much energy does it take go from an equal 0 equals 1 an equal 0 the energy there might be a little bit of energy but the energy piece of energies 0 the 1st excited states would have energy bars square still times 1 provided by the moment of inertia and if the object is very very small the moment of inertia is very large not 1 a moment of inertia is very large so the XI pension Energy Energy thanks to increased speed angular momentum by would you becomes very very big if the object small while hey don't up big objects they have small moments of numerous charm Garner on scale of were quite rare has on just north of now big blobs they have large moments of inertia the energy taken to excite 1 is not very big electrons unknown to be much smaller the expectation is that exceedingly small maybe 10 of the 16 times smaller you squalor their radius to get the moment of inertia so I could mean the 32 times smaller moment of inertia and that means the energy of expectation would be kind of a 32 times bigger well maybe that's that's that's probably too much but there so if electrons had excited states be be courses of more of their rotational expectations those rotational expectations I think I lost my true fractured plot by Mass necessary to increase the angular momentum Irwin unit would be way off not bad and of a plot just too much it just takes too much energy spent about I it to much I mean too much compared to the energies that are available on particle collisions and so the energy needed a true exciter Haiti drawn by 1 unit here's West news roughly of ordering GUV call light particles of GUT whatever it is you see these expectations the energy needed to expose spin-off up an electron is much higher yields than about the energy needed the spend up area of a basketball but when you angular momentum Mary small very small effect you can even see votes corn prices over there will be a look whatever it is 1st record your or cup of electron volts because the mother member when you're Cobaw work charm bolts of Beirut God for more please
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