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Supersymmetry & Grand Unification: Lecture 6

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Lance Steiger University yup within
a combative the supersymmetry business boy this is very abstract stuff and I really am surprised anybody returned after a time when the keep going we will see what who had was sealed gives up 1st OK now I guess they'll get fair last time drive to lure order Kazimir affected as well it's an effect of quantum field theory we describe the number number a number of ways but certainly take show reflecting surfaces called them a certain distance apart our then there are effects of quantum field theory roughly speaking roughly speaking you could say that will let that's not have the conducting plates there then there's a vacuum energy in the vacuum energy compete for love coming from my virtual our production of pairs of particles in the vacuum that's 1 way to think about it could also think of it as a 0 . energy our oscillators harmonic oscillators Her associated with quantum fields saying then end on an empty space purely empty space there's a certain amount of energy that present as a consequence of these virtual process to use the prices is also take place in the presence of these conducting plates but the conducting plates kind of interrupt the process for example roughly speaking virtual loops which would have been present in the vicinity of these plates have the plates not present are going to be affected by this reflecting waits for example with this as a reflecting pair plywood plates and blue pure of photons the font tons roughly speaking will allow will reflect off plight of virtual ritual process will be kind of interrupted by the presence of these plates in ways that would not have been
there if the plates that in there that's all it was in effect of the plates on the virtual processors use virtual photons in the back you get reflected off a coffee rip off the reflecting surfaces and so forth and that modifies the value of energy density the vacuum energy density between the plates also somewhat on the outside of the quake our if you take the plates a far their leaders far apart than true that each played by itself still does some interrupting Of vacuum diagrams and modifies the energy but modifies e energy in a way which depends on the distance between the place so far away you 1 as a bringing together more complicated things can happen photons condemned by virtual photons condemn back and forth between them and the result was vacuum energy or be 0 . 7 g becomes a function of the distance between reflecting plates when energy as a function of this they say that it is any indication that was a force you could think of energy routers a function of the distance between the plates and even think of as potential energy derivative that potential energy with respect to dissidents it it is is a form of Porsche so there's a quantum field theoretic origin of force between reflecting plates which disappears rather rapidly as you separator plates because as 1 over the distance of the 6 however remember what our 4th so much faster than the force because of what color the energy I can remember we we yet you cannot you could look at just them by Mitchell grounds in fact are but I won't do that right now and that creates AT a force between the plates which is typically attract so that's a detail after work out forces attractive inside of these firmly on loops here before so probably have the opposite sign an hour so this is a and this force to be measured it actually is measured difficult measure at a very small forest and falls off rapidly with separation column but often take in Kobe confirmation of the idea of vacuum energy is a summer extent but none of them other here and you could also just think of it as a force between between the plates were McGee close together anyway USB about the chasm in their quest is a part of it is there that he does not let on where all kinds of processes in our armed detailed high-precision behaviors avenues for comic prospectus for which a sensitive to the use of vacuum Oops now sobering director of measurement direct measure was what we really need our vacuum energy what you mean by Energy Energy is Max E equals M C squared mass is the source of gravity the basic things our that mass and energy do walk out very fundamental level as they gravitate so the question is would this energy that's stored between East the plates year modified both the man of a grant these plates and would it affect the gravitation of some other objects out here and that of course is much too hard to measures way way beyond the ability to measure but the vacuum energy that present just an empty space gravitates and the effect of that gravitation is the accelerated expansion of the universe so really want people talk about vacuum energy and we'll be alarmist evidence for it the only real evidence for it is accelerating expansion of the universe that I would I would put like this other ways think about the force between these plates you and thinking about vacuum energy but just think about the force between place half for death off for the 3rd called fled Guardado depends on the kind of particles that of think believe that if there were firm eons which were mediating the force that it would be a positive and negative I think that's true I have but a will let part of that all there is a saying that explains Auburn Oregon all quantum field theories predict the existence of acumen Is that mysterious well OK it's something that is very very well established in quantum field theory that there is a 0 . energy firm require the field called back in managing its really there is a measurement of it and that's and that's not mysterious 2 mathematical consequent of quantum field theory that really mysterious thing is not that there is vacuum energy dark energy aware your Call it the really mysterious thing is that there's so of it that's the really mysterious thing why is we're so little of it the I say it seriously nobody knows what it will assess assured that the right that the writers does not have lessened the mysterious thing is absolutely not president and and as a look at what we call it is such a feeble recourse missiles could source more from many
many years physicist including Einstein's thought it must be exactly 0 they convinced themselves that there must be an explanation of the fact that it's exactly 0 there was no argument there was never an argument about why shouldn't be vacuum energy that gravitates but since it seemed to be absent the composure seemed to be absent a high-precision a psychology was well it was really not be here at all since it's so much smaller than is expected on the basis of the quantum field theory plus a little bit of dimensional analysis said that hundreds of orders of magnitude smaller than what you expect there must be some principles of physics which says that it's not there the right theory of nature as we all know it is strength therefore there must be a principle of string theory which says that there is no vacuum energy and that must be the explanation now that we have an explanation is is extremely mysterious fact that when the next round of experiments were done to probe the existence of vacuum energy a little more precisely it turned out that it was there RSO was entirely of psychological in mysterious ways we convinced ourselves that there must be an explanation of its absence and then when it wasn't Babson but all our old my good said mysterious mysterious but what's mysterious it is the fact that it sought actually sold Braun for what it is the Shirley some form of back you do not get for those it is not so they have their own set of those states it by Noonan and not now not for the symmetry had broken that's exactly this fact but symmetry transformation is generated by despite these operators killed there analogous some extent like other generators of the symmetry generators angular momentum but there's 1 thing aware they do an angular momentum generated just the rotates by infinitesimal angle column was supersymmetry generated does is it doesn't turn in a system of arms in the air and a system of firmly on it removes 1 Boaz Island and replaces it by firmly on job I had a thousand bows on the supersymmetry operation the super generator generator of the cemetery would take 1 bows on and replaced by firmly on all 1 for me on and replaced by bows on it wouldn't take Old Firm eons and replace them by all bowlers arms or vice-versa now say awake but why can't had multiple times with this Q and take large numbers of Fermi arms and replace them by bows on the answer is accused Queiroz equal 0 that's the character of these fermionic dog these grants many and variables that they have the property that when your back twice with them you just get nothing sold on compared with the generators of rotation you make a small rotation of space and that changes everything a little bit but then Yukon compound that rotation many times by rotating a little bit rotating a little bit rotating little bit rotating a little bit and turn everything around you can't do that would be used fermionic generated you operate with Cuba you replace a affirmed on Bibles on your operate again with Q and you get nothing sorrow the way they think about it is White Cube does is not to replace all from eons by on but remove 1 Firmiana replaced rebels or vice-versa so it changes you buy when you knew the size of this case Year's that will bet young writer that's so it's a talk about that now what there are 3 things that supersymmetry does for you well the which sounds quite significant and may be quite important or are they might just not be and the answer is at the moment we not by supersymmetry ending supersymmetry measurable within the next decade be machines that are now being built the LHC particular on the 1st thing they do for you it is to control enormously big numbers which occurred and which force you into thinking about extreme fine-tuning the variables in particular the Higgs mass the Higgs masses we talked about our would naturally if it was no funny thing going on but would naturally get shifted up some very large Maris cricket 6 unit and the only way to control at Cook is I either yes I agree on it is I extreme fine-tuning of the parameters or by the symmetry which replaces premier arms embargoes rebels on so that's a 1st for a fact that controls beat let's just calm divergent South energies energy of our South Our Lady of the Hicks and prevents the
entire remember all the masses of particles are responses to this pigs phenomena that the Higgs particle got dragged up to some very very high energy babies divergent finding diagrams Everything we get cold weather and all of the masses of all the ordinary particles would be your way way up at some enormously energy if could presumably some more parts Carol a 1st cricket high equates the cancelations progress would talk about that 2nd it gives you candidates for a stable particles which may be in for a stable new particles gives you candidates for new kinds of particles which cord plenty various roles in physics in particular dark matter but matters not dark energy dark matter is believed to be particles and is believed to be particles which have not been detected yet saying We're not been detected yet probably means they're very heavy released by the scale or current experiments on supersymmetric particles when they decay will always dictate their particles the orderly particles and there are their partners particles and partners of a particle afford ordinary particles of fermionic partners of bows on an ordinary particles of bars on the partner in the firm on so actually separator world into 2 kinds of particles those which you ordinarily think of and those which are their Super partners no whenever a super partner decay is a superb partner an Eno and you know Higgs Eno gloomy nor of faulty 1 of the supersymmetric particles which have not been detected decays it gives off and ordinary particle plots another supersymmetric pot supersymmetric partners don't doorway that means there's always going to be if you start with the support of 2 start with a partner you'll wind-up apart you can get rid of them became get rid of them you can is always although 1 of minimum energy which is stable the 1 of minimum energy which is stable is usually call the LS people like this super partner a K easily can't became principal perhaps a convict Katie but that's the story supergravity also complexity but the simple a simple story is the lightest super partner will not the case now whatever dark matter particles are believed to be red ahead maybe a TV something like that why aren't they the ordinary particles something must prevent them from the became the ordinary particles had only know they don't decay because they've been around for about 10 billion years that's how we that doesn't mean they'll never decayed but sure means that they don't a year at a rate that you might have expected art from a from ordinary particle physics they seemed to be exceedingly stable and supersymmetry provides you with another class of particles the lightest of which is likely to be stable are from experiment we know when we expect that the life of the foreigners away something like about 1 TV and so provide a candidate for dark matter whether it's the right candidate but we don't know and the orders of magnitude our of ministers of these particles that properties and so forth seemed to be good respectable candidates about matters that 2nd fact no later say it is look we already know that there are particles but had not yet discovered in NATO base discovered I mean discovered in a laboratory or even discovered in cosmic rays are anything like that where Gravitt 1 of them but that a remote thinking about that is a dark matter particles so whenever the dark matter particles are they doing choir another we use the word sector another another sector are of particle physics the very weakly coupled to are kind of matter we know that otherwise if they were strongly couple what happens if when they banged into nucleus they would be slowed down in a body of a that's supposedly electrically charged electrically charged and not tremendously heavy what would happen to them in the dark matter in the galaxy the interiors it would scatter off the ordinary matter and scattering off the ordinary matter be enormously large Hey law which is much bigger than the Galaxy a large halo of dark matter we get scattered and when it gets gathered it would lose energy and bowling and basically coincide in there be a safe place the the ordinary matter so areas as dark matter it whizzes back and forth through the ordinary matter who play the Galaxy toward ordinary stuff without interacting with the March supersymmetry provide good candidates for this kind of thing so that's another reason not to believe it but but the thing go this is interesting directions to explore the last reason is extremely interesting with talked about a little bit we talked about renormalization of coupling constants I told you for example that electric charge of an electron at small distances is larger than the electric charge at large distances This is this phenomenon of of screening and it's also the phenomenon renormalization renormalization changes the input parameters for the output parameters the input parameters generally governed physics and very small distances or equivalently very high energy and how the parameters governed physics at lower more accessible energies are not the same so in particular there are the coupling constants such as electric charge the fine structure constant are loan which is just a square of the electric charge of electrons some units they mention was quantity and if you want the plot be elected charge which called How the fine structure constant it would have the property that as a function of energy all are 1 over of Waverly smaller and smaller distances while larger and larger energy then the fine structure constant would increase somehow fine structure Council would increase on its usual to plot In the 1st week in inverse remind us warned because in various approximations inverse the nice straight
line and it would decrease the inverse coupling constant would decrease energy that is this exactly this phenomena that point lecturer and surrounds itself with opposite charge and that opposite charge a serve lowers the charge at large distances and or for right there on the coupling constants 2 other important coupling constants of the coupling constants of the electroweak theory me out here it is these coupling constant that's associated with the emission of photons that's electric charge for Carter out right there is the admission for example of a W bows on the W is governed by another coupling constant which is the SUV tool coupling constant of the electroweak theory and if you calculate it if you calculated using Inc the rules of quantum field theory No. 1 and No. 2 0 some experience and experimental data tells you what the value of these constants at low energy some value low-energy our tendency EEV some appropriate energy these constants are determined by experiment but we have electroweak coupling constant is a little bit larger that means it and verses the smaller and it also has a kind of screen phenomena associated with it also varies moralist universal also varies moralist linearly and they happened the cross a certain point or say when they cross nobody has ever measured physics at such a high energy these trajectories the trajectories these curves are very very shallow they vary appreciably only over very very large scales energy the coupling constant only very logarithmically they're very very slowly and so a few at the plot known electroweak coupling constant except follow its renormalization a functional smaller and smaller distances a higher and higher energies you would find that they crossed and when they cross the across but some very high energy roughly somewhere in the neighborhood of about 10 of the 15 pence that was 16 Gigi right that kind of 15 per cent of the 16 times a mass of protons way way way beyond those Our they expression and is likely the happened next million years with lot of the items 1 while coupling constant which is interesting and it's the coupling constant training a glow on electroweak which coupling constant just Nordic course I'll electoral is 1 more coupling constant that the coupling constant for a quarter to a good that's a much bigger coupling constant which means it in Mercer smaller but it has on property because of me properties of quantum would not only knew properties of quantum chromodynamics that this screening phenomena happens in the opposite direction instead of roughly speaking this is this is a metaphor for the real thing a roughly speaking a bit of color color charge creates a ended up color charge of the scene and that increases each charge at large distances decreases its charge at small distances that called asymptotic freedom Back at small distances be quantum chromodynamics coupling constant gets smaller not bigger as the opposite behavior from from electrical forces and that means that this coupling constant inverse coupling constant increases that we haven't talked very much maybe will get to a little bit about unified unified theories a attempt for port In 2 0 0 1 basic framework all 3 of these interactions to think of the photon the W bars on both arms and the glue arms as warming belonging to a single family of objects which have a symmetry among them symmetry is obviously spontaneously broken if there is any such cemetery because otherwise otherwise you wouldn't be able telephoto on from the double euro gone so there are different but but 1 your it is the symmetry between these things is spontaneously broken and spontaneous breaking please 3 coupling constant have followed the strong coupling constant it also attracts any crosses the other 2 someplace if you can't just the Standard Model fund modified standard marble and you take these 3 coupling constants they have a tendency more or less cross each other at more or less the same point but I think a less is a better description the more they miss each other a pretty significant margin even tendency is for them to so appoint a tempting way and can't wait point for the same value will he solve that leads to the speculation that really there's is 1 prime ordeal coupling constant spontaneous symmetry breaking B which is the cemetery which relates these 3 phenomena spontaneously broken causes them display but some very high energies perhaps they come together and form a single coherent unified theory that the whole that's an idea that's tempting idea there is some substance but the problem is what you really like it is these 3 coupling constants to join a point you like at some very high energy these 3 coupling constants to compete gathered and to tell you that there really is a single phenomena are extremely small distances which is unadulterated by this spontaneous symmetry breaking spontaneous symmetry breaking always has the effect of influencing physics more at low energies of high energy that's it that's a common denominator for spontaneous symmetry breaking it tends to affect physics at low energies more than high introduced so the hope was that these coupling constant would come together at some point but they don't win supersymmetry is a of liberal then we find in diagrams which the flying the way that these Constance varies is somewhat modified was a modified by its Mr. your kind of finding diagram that among other things effects electric charge the electric charges for example affected by electron-positron loops like Mac electron-positron heaps inside of time proton is a photon 5 this
could be a photon exchange between 2 charged particles this could be a proton and another prime the repulsive force between 2 protons if you bring the protons very close together protons another good example because protons have I have a have a sizable so let's take a very small Our electron pair of electrons and you look at the force between electrons if the distance between electrons is large enough To allow this kind of look to be president then it affects the electric charge makes smaller sorry young mix of smaller than it would be Our if you Bruk electrons very close together when electrons are very close together because between them for this kind of ritual could take place and just basically the single find Bagram by itself as solve the source our variation in these couplings variations couplings is called running the couplings are running coupling constant the source of the variation this finding diagrams were you a hand superb partners in liberal you generate no find diagrams for example the fine and I diagram that I just rule which has an electron positron could be replaced by a similar finding by a which the super partner of electrons goes around the world and that the upshot is that the way these curves vary is affected by supersymmetry a mail apes a striking fact that if you would take every ordinary particles and is had to it super partner with the same properties except opposite from bows on this of apart from yes White then 3 careers of crib cross-over point and they cross at point 2 of very good accuracy of about 1 per cent no less say cost point said nobody has ever measured anything up here what you do is you measure down here in Japan he emerges here you take the fury whatever the periods you start calculating finding that rooms and you extrapolate the extrapolation of the coupling constants after is within the accuracy of the calculations to about 1 per cent the 3 coupling constants cross report is no trick in making the took coupling constant across reported that's not attract the fact that that the fact that the energy is way up high that's interesting but but to make 3 coupling constant of 3 things cost a point that either requires lock in this case I would say bad lockup it's just lock because we don't want be misled but by lock so either requires an accident Beliveau of about 1 part the tool although really represents the addition of more kinds of particles in these loops and if you actually really just take the supersymmetric partners in they really do crossed to within 1 person at a single point pocket are got his 1st audits of the energy group don't Norwich blew up our business said nobody's ever measured themselves to say what they do or do we don't do is a matter of theoretical arm argument but the theoretical argument would be that want to get the high enough energy that the effect of the spontaneous symmetry breaking news gone then believe is just joined single public-private not for a Seth 1st not not not is not right OK Ireland me at the end of the question the far right if the Super partners was blade the words Super partners were exceedingly heavy sell heavy that their contribution to these loops would not become important I'm very high energy than they would miss each other you have to assume the supervision of the supersymmetric particles are not too heavy for this to be the right picture how not to heavy not too far from the experimental range now this all sounds like a marvelous accent the sort of things people cooked up borrowed to to get funding for big accelerators but In this case these calculations are quite serious quite honors and it really does look like with the Super partners is a chance this is called unification of his coupling constants come together this is the 3rd day of power that's call higher precision precision but precision like that higher precision unification I say that's the other things state one-liners point is that if you will also the plant the gravitational coupling constant the gravitational coupling constant targets Armed with a gravitational coupling can't work the gravitational coupling constant which corresponds to the gravitational force but say between a pair of electrons at low energies that coupling constant is incredibly small might be gravitational force between 2 electrons and distance of a comic diameter or even nuclear diameter of even a earned our any reasonable range that gravitational forces forests the electromagnetic force that means of plot the gravitational constant on here we way up here somewhere and low-energy but the gravitational interactions increased but rather rapidly with energy to know why the gravity of increases with energy had more energy you have them 0 gravity you have that that there is no tendency for electromagnetism to do if you have more energy that doesn't mean you have more charge the charge of electron might this effect here might be considered a little bit of a variation of the charged with energy of distance but just the ordinary sends electric charges not energy but gravitational charge is energy therefore the effect of gravity the analogous effect of gravity on particles would be something which gets stronger with energy and therefore they in Of the gravity of the gravitational coupling constant would fall and fact it could also plot where it would cross these 3 lines and that's not terribly far away actually it that actually crosses fairly close to this point over here our so it's another indication of a common scaled that might be governing the various forces 6 more or less
common scale but these 3 really do cross and they're very similar to whichever gravitational force be different you don't quite know how compare it don't quite know how compare gravity with forces Soto argue 1 way or another whether crosses at the same point that dicey but the 3 key directions three-year our leaders we we are just not there far what a fall guy have the same interactions Pamela have electric charges sorrow part of electrons called electron as a scalar stands scalar it has exactly the same electric charges electronic different mass a similar charge our partners water squat recalled quirks however the same charges as weak work a corresponding quarter also the same color quantum numbers so squark interact with the same kind of color forces quirks and supple down our melodies these Theo do not define new interaction so it's part of that and Armenia the restrictions of supersymmetry of very very tight they're very very tight they tell you detail the relationships between our interaction strengths of supersymmetry gives you are a lot of detail in the particle spectrum and the particle interactions and those details it would be a pretty clear the only issue is whether the supersymmetry supersymmetric particles whether their masses are low enough to be detectable or the LHC if you make until having this picture doesn't work fine but you might be out of luck could be it could be that the supersymmetry really is they're just by bad luck it happens to sit in the particles are a factor of 2 too heavy to be detected a QHC Elleby awful we we would know what to say so on the other hand if you make a particles but the superpower chore heavy In this control of the divergent self energies doesn't work very well and the Higgs particle gets pulled up the higher energy you need the cancelation and have a cancelation you must have not too big a discrepancy between the masses of particles in the Super partners so what if you trying to say that the super particles of their but there at a hundred times mess that could be detected the LHC a number of these things would just go wrong with the dark matter particles would be the wrong energy wrong the higher precision unification will work out nearly as well and this control of the divergent South energy of the Higgs will fail it won't be as bad as it would be if there were no super partners but will still fail solve if supersymmetry is gonna do something good for us in these 3 domains experience in these 3 domains here it had better be not too far from the experimentally accessible range that's what has that's what has had Armed no ideas come and gone usually don't last very long when they don't do something really good this idea of supersymmetry has been around for a long time ago hasn't gone away that's no proof that correct but it did some evidence that the something rather attractive and compelling a compelling not in the sense that has to be true but has to be passed it has to be tested that sort of thing was a further sign later while I'm Richards acrid you're asking the existence of body has a lot that it's a ratio of Dili is the put a play with a problem and the problem is 6 without a suspicious more or less specific period dark matter art all you really know is a density of the dark matter came as density of the dark matter and that mass density per pubic something or other could be real for a heavy particles what painted the 15th very light particles and you can't help yeah it could be it could be all could be it could be a boson condensate of extremely light particles In fact that's 1 fury of it call the axiom theory of dark matter and it could be just a very very large number for you to volume of very very light particles they better be both arms otherwise you can squeeze that that many of them into a small volume and or they could be heavier particles so without some kind of theoretical prejudices our you can't tell what you can't pilgrimage about an hour on that they have to be here in order to control motion of galaxies missile 4th rotation curves of galaxies before other than gravitational effect we don't have a completely compelling a set of arguments about the properties nevertheless with some assumption was some assumptions earlier saw robust assumptions there the class of theories falls into 2 interesting possibilities 1 is particles of a massive about TV and the other is extremely light particles of former boss of condensate and all them could work of ideas a feasible and furthermore anything in between will also work is his no fury of stuff in bequeathed the ear a 1st for half an hour you have to have some reason to believe that I will at the very beginning wars but there it was a point in time when the universe was a whole lot harder than in fact in which thermal energy scales were let's say way up here somewhere in that situation it would be totally equilibrium which would dictate how many of each kind of particles it would be and more or less you would find approximate equality between the various kinds of particles in fact just about all the particles would pretty much have the same abundant HMOs arms firm eons more or less all about scene and so at some time in the past particles in that particles both guns and firmly on all more or less equally abundant and pretty much determined by the thermal equilibrium that takes place if they would to many
ordinary particles are not enough super particles in the collisions of ordinary particles would produce super particles produced too many Super particles of the abundance favorite Super particles in collisions between support particles would create the ordinary particles and thermal equilibrium there would be some kind of balance between our so the expectation was what was going on at the very very beginning but if we traced back as far as we can it looks more like ours equal abundance these various kinds of things which work part of that the young yet very likely a neutral almost certainly neutral particle White well for 1 reason is that charged particles likely to be similar or colored particle is likely to be somewhat heavier than the neutral neutral version of a white because of electrostatic or R Of course energy densities so the expectation would be that it would be a neutral particle but in various versions of the theory it could be different things could be a part of photon could be the partner of the Higgs boson I could be the partner of grab which is called the Graphic Kino our and the various different possibilities I don't think there's any definitive idea of work over a report soup partners but he did say that the hour is different arm it is different the vacuum energy divergences just the energy of empty space and in terms of finding diagrams From the finding diagrams it's just 5 grams or virtual particles which saw populated space find complicated also of surviving environs convict place but there is no other White completely empty space a contribute to vacuum South energy of a particle in particular the Higgs particle is a class of diagrams in which the president Of the basically say persuade the presence of the Higgs particle or any other particle provides a kind Our impurities in in the backyard it's good that's for particle visits an impurity in the back of the impurity has In effect on these vacuum diagrams its president modifies them in a way which is not 2 different then the plates of the Kazimir effect but you could summarise it by saying important find diagrams renormalize single shifting the massive particles diagrams which are connected to that article that so a fighting in diagrams which are sensitive to the presence of the Higgs particle give a mail or shift the mass of the Higgs particle and that it is a mistake in the same kind of phenomena are except it is what we say modified by the presence of the Higgs particle itself and therefore shifts the mass of the Higgs particle I love whereas the vacuum diagrams are just shifting Over energy density of the of empty space it is 1 thing is an interesting fact Twitter we haven't explored cosmology at least for the number of quarters but there is 0 1 feature of this vacuum energy which is what distinguishes it it makes it rather special but stopped 1st about their ordinary energy stored in the form of particles what part they could be dark matter particles it could be any kind of particles on eventually at some point is the universe expands is particle separator enough that they had not been violent interaction with each other they're just marvelous sitting back and as the universe expands they separated and as a seperate energy density decreases energy density decreases because the volume was space increases the number of particles doesn't increase and the result is that the energy density of the vacuum obvious are the energy density of space looked ordinary particles armed decreases as the universe expands its it works differently for photons particles with may ask the reason as the following the particles man totally expansion of the universe eventually calmed arrest relative to each other now they can't arrest relative to each could adjust participate in the overall expansion are but think of them as roughly being at rest the local frame over here their energies just M C squared all their masses just messages their mass and the way you find the energy density of a massive certain region is just a couple of particles and give each 1 a mail that's the way you would calculate the energy of just ordinary particles you just take all particles of a certain volume and divided by that volume the energy density and that is the volume expands the particles just move out with the expanding volume adjust the participate expansion the energy density decreases but the total amount of energy In a volume like effects but another example it would be volume a space photons in it photons our and you can't think qualitatively of just having a cavity with photons because in reality the photons pass out through the walls of this region freely there are no real walls but a photon the passes out is also likely to be 1 comedian you could imagine a EU regional space expanding regional expanding regional space where the number of photons effects but each photon each even he gets stretched by the expansion into Waverly gets stretched and so as is Waverly gets stretched the corresponding energy of each book on decreases so that means relative of massive particles quotas man nonrelativistic massive particles the energy density store radiation will decrease faster are as the universe expands the energy density stored ordinary particles this issue white candidate the energy density in radiation blackbody radiation is vastly smaller than the energy density and protons a nuclei electrons what because he expansion has stretched the photons but hasn't changed a mess of particles here if you go back into the past at 1 point the energy densities a represent Mercury acquired this fact that I have basically to look for all of us had a and half for let's let's
not tried there were given the video subtleties of mechanics and let's just say that we're talking about more less classical radiation and are treated with classical radiation stretch the energy density decreases in OK now what about a vacuum energy vacuum energy is just be energy of empty space it has the property that as the universe expands the vacuum energy density simply doesn't change it just days there stays the same energy density it doesn't die loot as the universe expands our solar system displaced how good I look empty spaces and be space well that's a that's a caricature of an argument about the point is that as special property of vacuum energy that it doesn't die loot universe expect that may be that at some time in the the early universe and it was true it is true that a nearly universal vacuum energy density was much much more the energy density in radiation or particles was much much more than I was not very important but as the universe expanded be ordinary sources of energy by but Divac listeners promoted this being radiation became smaller than this being more ordinary nonrelativistic particles but the vacuum energy did not and that means that's somewhat point to the vacuum energy density will become more important larger than either of these 2 sources of energy that is already happened Ray happened that we energy density of the vacuum energy density is about 3 times larger yeah it's about 70 per cent of EU energy density of the universe today so at some time in the past maybe our roughly a a billion years ago amateur exactly where work to calculation 7 6 billion years ago it was a crossover where other forms of energy earlier on more important the density was larger they control the behavior of the universe and as it expanded loaded and unloaded until about some X number of billion years ago the vacuum energy became more important and today it's more important by about a factor of 3 0 4 or something like that 70 % terms of 30 % versus 70 % our 70 % being back to review how we did all the talking about the various find diagrams forever of vacuum energy and the Higgs phenomena are we go off Our kindred never crossed his wrote for personal injury write anything down Seth what was it that if that is OK you I think you're asking me about the role of energy conservation Cosmo could be inserted is bizarre answer the answers energy is exactly conservatives always exactly equal 0 there of various forms of energy 1 of the forms this form of energy that former of energy vacuum energy but there's also another kind of energy and it's kind of the kinetic energy of expansion its Justin energy that's associated with the expansion of space the energy density but what are associated with the expansion space according to the general theory of relativity happens to be negative that happens to be if we the finding the radius of the radius of the universe and given time quality a call the scale factor there is an energy density store just in the fact the universe is expanding it a kind of kinetic energy a . time to reverberate as kind of kinetic energy is proportional to square all video of the expansion velocity happens to be negative as 1 source of energy and then all the other terms and energy there's this fat something called curvature energy and they all have it up 0 passed a rule of general ability general relativity insist that total energy Our cosmological universe show all Ayad up 0 current sold the the answer is that only other forms of energy Naylor may not decrease if there was no vacuum energy these energy densities would decrease but would be compensated for by on negative increased the of a over a square forever so energy conservation general relativity is very peculiar it's not simple conservation or few tried of force To be simple conservation than answer the energy is always 0 being compensated for by some comedic energy expansion or is not what must be 0 it's a consistency condition of road it's 1 of the equations of motion general held up that's a different issue at 2 different at your whether it be opened Close story on the boundary between them there are various forms of energy including ICR yet you don't know your question a fact so what other forms of really want is not happy to do it but focus used by young visitors Einstein's prime time component of the Einstein equations I was right on the left-hand side is geometry to terms associated with the geometry of space geometry of spacetime 1 involves the kind derivative of expansion factor was just a few like to sink above the radius of an expanding universe a . Over 80 square why it's a dollar a year of square the key map of any particular significance seek multiply the whole equation by any power you like it traditionally expressed in terms of quantity that's 1 terminal left-hand side the mm remind you of the Einstein equations on the form something geometric equals or something that has to do with what you would ordinarily call energy and momentum energy or mass already work or on the right-hand side this it's not meant to be taken literally just our left inside you have some our something that involves the geometry of space on the right he inside you have sources and those sources are the form the energy momentum tensor right side are translated into the equations all 1 of the equation has a form of something involving
8 . 0 a squared is another term involving geometry depends on whether the universe open closed or flat open closed of flat and it has a form of but wealth depending on close to plus or minus 0 0 with a flat and something involving world a square that's the left inside it depends on a rate of expansion of the geometry and it depends on the nature of our space whether it be positively curve negatively curve of flat so that's what's on the left-hand side on the right hand side is the energy density of ordinary energy density it is call that role the ordinary energy density means it contains particles nonrelativistic particles usually called matter it contains radiation and it contains vacuum energy and the matter emerging as ordinary matter and dark matter that's a form of the cosmological equations motion but another way of writing there is just put the this on 1 side of the equation is the Magic equal sign equals 0 put everything on what sorry but everything on 1 side of the equation and now reads that something is equal to 0 was is that's something that's something is actually the couple Hamiltonian of general road really busy energy In this sense all of canonical Hamiltonian be strict definition of energy from the disappearance from theorems are connecting symmetries and conserved quantities yes there's a conserved quantity we call it energy but it contains this expansion stuff it contains stuff about the the curvature space and all things instead of 0 but rules the rules of the game other than that there is no notion of energy conservation and her personal no other notion of energy conservation don't following a volume of space missed the road this part of the energy Community decreased Greece with buyers will be compensated for by his side when get into supersymmetry but circuit with up as a breath test up for us but rather a death while they're still up in 2 waves Bristol early on HBO's on on-field their values the values that they take on an ordinary there the bows on ordinary numbers electric and magnetic fields are are ordinary numbers firmly on the field of grass they enter commute among themselves so 1st of all they show up and uniform of the family on fears of a show up and another way which is what I was going to get that made me we may decide the moved the postponed till the next time looks like it's getting late but hey the other way they show up it was either really D worry trivial are hard pesewa tracking on it appears that in supersymmetric theory that there are additional dimensions of space those additional dimensions of space time acts not only have X 1 2 3 and those of the ordinary dimensions space all areas but in supersymmetric theories there are additional directions a space lab space time space and those are the directions on grass and variables many of them are there will get the bat basically tool complex for a real grass and variables call offered to think of it as complex but complex simply means to real 4 of them altogether it is as supersymmetric theories are a series which if you like lives In a space which is composed of both ordinary coordinates the grass court of this is extremely odd very very abstract but at least 1 thing you could say the grass coordinates are small they don't correspond to big extended directions a space you can move around in those ARMY do I say this small this small because they the squared is equal to 0 so like any really really small number it's square To be taken to be 0 breast coordinates are small but nevertheless they are important to the structure of the theory and the way a supersymmetric furious set up is by defining the the Lagrangian density and will talk about this a little bit I was going to go into will bit tonight but it's going through it but it's over grant density which 1st of all not worked before we get to Lagrange density let's talk about field field ordinarily are functions of the axes but fight X but normally the way we think about fields but a supersymmetric theory the fields are a function of X and the failures so it's as if indeed they were additional dimensions of space and yields could depend on both axes and the now the favored the image is very limited wise eliminated expanding let's suppose this fire was an ordinary number as opposed to a grass number then Hua right at this although nearly number fire X and later but did make this a capital fiery called super field scoreless Super field because the market cost but the call was super field and it depends on positions ordinary positions space and time and his grasp of variables right now you can take any function of grass variables and expanded powers of the breast bearable let's suppose for example there are 4 distinct stated this is correct when ordinary supersymmetric theory was will understand why but not right now there are 4 what favors a 1 2 3 in for bacon be combined to complex paid but puts just taken for then why have action commutes landed fight but got little fine now X which is independent of data at a power series expansion the figures plus let's call it Beta Alpha Psi house 1st is just some index which labels of various our grass when
members of the problem for them in this case then there might be something proportional Beta Alpha Beta Beta plans I'm not give me a name of another field Our sorry z z Alpha Beta this is also a function of X also a function of X and so it goes until you get the full powers of data all what traders had better be different but also have to be different here because any favors square 0 because they don't want stated tool but you can have a warm square Our eventually get 1 fated to within 3 feet of former times some of the things I we need to give it a name Our X and then there is no more than that there is no more than that another way of saying it is that the soup field is composed of several ordinary fields on HBO's on-field dialects are firmly on field society YYC firmly field well because it has to be aggressive in number grasp a number transgress the numbers an ordinary number and then another bows on field and then so a car was a finite number of them are finally multiple at finite multiple put a fields associated with different particles and so you can either think of it as a collection of ordinary fields or you can group them together and do the bookkeeping by saying the initial family of grass court and the field could be expanded norms grasp court with the kind of bookkeeping 1 still had the idea of a super four-year-old Ucon write a Lagrange a Lagrangian density which depends on excesses and the favorites this bill up out of these super fields products of them functions of the various kinds of derivatives are them I won't bother writing down a specific around but it's built out of the Super fields and that Monday who runs in density depends on Exon later the important question is always when you have a Lagrangian would do with around him you'll integrated before action and then a classical physics at see the principle of least action or use the Lagrangian and findings path integral formulation of quantum mechanics the important thing is the action the action ordinarily isn't in the goal of the Lagrange density all over the former dimensions of space but now it also becomes in integration over the various favors but that is the structure of supersymmetric theories that everything is exactly as if there was some additional directions of space and the action of the theory becomes an integral not only although ordinary space but also for all that they know that raises a question what the devil do we mean by the integral over grasp of variables what do we mean by taking a function of grass and variables and integrating it over fate that's obviously but as yet undefined concert I think we did fine arithmetic of rum of grass variables and IndyCar multiplied in the ruler of usual arithmetical operations Boatwright dividing by dipped by sea Our but ailing in multiplying Zurich a thing to do you can do the differential calculus the differential calculus is very simple if you want to differentiate with respect to the maybe you don't follow us the obvious thing if you were a different share with respect they ones you just find out whether they don't want appears in here editors could appear here and you just differentiating with respect to it just differentiates exactly the way you would ordinarily differentiate you don't have to worry about higher powers of data given faded only appeared linear order and no higher so the table of derivatives is very very simple the derivative of a constant 0 something proportional Bader just a proportionality factor and that's a very simple table over derivatives the question is what do we need body be integral or favors are grasp variables against going to tell you there would be the justification for the rules of integration a useful a useful in various contexts it's completely abstract an integral is not in any sense a sampling intervals although grass variables are analogous to definite integrals may not indefinite integrals thought others definite integrals so when you integrate all the data here you doing it definitely the grow over the entire status space I won't bother writing from minus and Finley infinity because they do the kind variable which very very much but Our it really is no snow analog of a indefinite integral is only me and what I say it's a very much you're getting away from their peers as it turns out integral the derivative are exactly the same thing if he were a lot of faith that it is also a definite integral yes yes yes Saunders don't tell you for the purposes of knowing exactly what written here I'm going to tell you what the table of integrals for a year progress integration endeavor then we could start to talk a little bit about supersymmetric theories how you construct highly construct the means point of supersymmetric theories is that they whatever they do what they preserve this symmetry between bows arms and Premier they ensure the delicate cancelations if you construct properly they can be complicated interacting field theories but they will always reserve in the same way he did if you construct on an ordinary field theories carefully they will preserve Lorentz invariance they will preserve rotation variants they will preserve the various and variances that you built Lagrange in the same way by being careful To with every step preserve the supersymmetry you can be sure that at the end of the day day that these delicate cancelation of find diagrams and things like that are
preserved and don't get ruined body of work by interactions by all various complexities in theory that's the the point of point of it is to set up a kind of calculus of how the construct supersymmetric here battle would do more to tempting to build the calculus which is analogous today tensor calculus Our relativity which insurers that every stage we preserve the invariants is on the rotation of the Lorentz transformation of what we're talking now about preserving the invariant was on supersymmetry OK so they have to know how to do grass intervals then and that going to spend a lot of tournament is going to tell you are what the rules are rules are mostly justified by the usefulness about tell you some of the community the rule pretty unique pretty unique follows from a cop or a small number of assumptions the 1st assumption is arity let's just talk about the into global 1 quart posing as only 1 grasp quarter I want to consider off an integral of some function of data these data the 1st rule is linear Lenny says if you have a In a beacon B ordinary numbers 8 times what court half of data sorry a efforts say that closed P G of data he made a very simply be answered and a hazardous ordinary numbers proper that's just a quarter a transient aglow played a data past transient integral due later G. of data as simple enough just usually EU rules are referred calculus the only other ruled that you really need is that they I let's go back but would functions for a minute what is it integral with respect to an ordinary court may put call VX follows a derivative of a function with respect to acts let's make assumptions about assuming that the function f uh Inter will definitely from my innocence Vinny that's a had more assumptions that the functions that will be interesting to us all ones which go to 0 very very far away after most purposes in their own space time we're interested in concentrated regions of energy and momentum and so forth so for most purposes British functions which are vanish far away large large positive and negative acts or sausage and the there there are right just 2 of the rule for integration by parts of your life so the interval of a derivative a definite integrals spaces you go 0 that's a rather important Formula 3 cars or were not over a game we use it left and right all over the place but ensure the correct for around of grass and functions derivative ID data favre effort is equal to 0 Decatur sorry sequels show that pretty much determines OVER may not be table of integrals for example supposing we take 4 and of data Lara some constantly have data equal any can't to speak they don't want to speak later but ever faded be equal to the data whats DES by fate 1 just differentiating after favor with respect gives 1 so long as follows In the glow water which is sworn data it is equal to 0 so the definite integral of data on all figure whatever that means is by definition now equal to 0 are only the next thing Indian the next day we can have his efforts they there could be proportional 3rd arm sorry car yet the next into growth we could study is the integral role here In the role played a deep Bader what about Vader Square D data they squared is 0 so we don't have to worry about squared QB his only other thing we have to worry about what is In the goal of data we favor this is not grass number because it's got to grasp the numbers they beat data was campers aggressor number This is an ordinary member in fact you need a you will hear the rule doesn't matter it's a convention to convention but it's a convention that this is equal to 1 and that's it that's the whole table of intervals of functions of 1 Grasston variable right now we can write we it will be data on any function of data in any function of figure Khmer written as some 80 Crosby data and that's all there is nothing beyond that but being integral our fate at times a the interval of faded is equal to 0 so that gives you 0 and the integral our faded faded that's 1 I should put the over here the interval of faded data is 1 by assumptions and so the answer is just beat now interesting not no fears happily square right None of the square right so we can't if there was if there was such a function we would immediately conclude this has to be 0 right this is part of the story to can't abide by faded but her right so here's an example of the interval be data a function figure is the last tournament expansion in this case it determined the expansion with 1 data that seemed to be known as it's the same as the derivatives the same as the derivative but that's the interval although 1 data out of an arbitrary function of data an arbitrary function of data has to terms and just the coefficient of West the definition of these will be played at a pretty simple it's amazing that has any nukes where there were 2 skaters let's suppose there are 2 faded well I'll play what the rules are you could work amount by the same kind of
argument exactly the same argument effective PT argument pretty much exactly who it it is but a we had to favors or 8 complex data which has a real imaginary part puts you would do it that way they never had a fatal bar made a bar means the complex conjugate you could also think of it as a warm and they did to it doesn't matter on the general function of faith and bar consists of several terms consistent Haiti which is independent of data it consists of works called our data bar society where sigh is grass number but suppose a he's not a grasp the number that was our grasp number and then there's another turn which is some sidebar later a lures things Lanier's in 2 stages and then the last term but I'm aortic be bath the quadratic if there are only 2 ladies and there's nothing beyond the quadratic squared is equal to 0 bar squared is equal to 0 paid month later is not equal to 0 but later but they the bottoms figures you could 0 so that's it that's that's the most general function of tool grasp variables as I said you could ride in terms of they want to it as matter there are altogether 4 terms OK the integrally as a table of integrals be in Togo he faded bar definite integral although both favors Article 2 0 the just like the interval dictator data is equal to 0 3 in the growth of of them he David played bar Kitty guesswork batters 0 why because it contains an indigo faded later just to get this for a moment and concentrate on the 1st factor here they favor also recalled 0 likewise be in the vote paid a bot Decatur data also equals 0 only none 0 in the disease they don't bar a bar deflator Miller has arrived with a certificate dictated edible I remember when you switch sign as we use which orders things young I do look at this and say look 1st of all I see the interval data that was 1 nite so we do that and to go 1st Oracle at 1 and then left with begin to fade a body they by analogous reasoning it also warned of this is equal to 1 that's a whole table of integrals hour intervals of arbitrary functions of 2 status I'm not supposedly integrate functional state and we integrate not theirs but just some general function therefore trader at a bar a reading of the Arabs what is that they
said city integrated what will reviewed Wan right now there's a reason why was it gave me a lot lot R yes if it does backward and Obama's Duran contributed backward right yes but to you right life so what's what's the upshot of an integral like this being of this former while the only termed the contributes is the term of the fate of Arcadia 0 so the answer is just the integral part a function of grass and variables is all the ways the coefficient of the land turn that's it that's the whole that's the whole theory of grass integration you simply find the coefficient of western the expansion acts grow its sound very very trivial really does center tolerance enormous will will do some examples now it changes yes you have to be careful about being the rules for example of our data bar later times D data bar data a is legal Tom minus 1 white or say that because if we interchange fees bar that of course I would say it was plus 1 and now we look at it and in a sandwich here between the fate of Barnaby data bar we see something which we already said was a good 1 the poll our and then we have made a body here by radical toward so you have to be careful look this whole subject of supersymmetry divided physics in 2 groups of people who were those who were able to follow it those who warned they the ones who are not able to follow it were probably the most distinguished physicists in the world Richard find be Coleman I would include myself among that camp not because I was distinguishes because I was unable already and the people of followed people who good at keeping track of signs that I I go berserk tried to keep track of the signs supersymmetry and I have no wish for another imminent deletion I have no it's just pure torture it's a form of torture OK never mind yacht so huh it is a we build supersymmetric O'Brien Adam super fields commercial according to some rules I haven't told you the rules had but according to some rooms which preserve the supersymmetry Winnie that concept of road derivatives and so forth which is a useful concept of derivatives of doing these things and then we he did great reaction all over X and all with but integrating all but they don't always means just pulling out the last turn the expansion of which seems public trivial and yet it has this enormous How would you want to be able to control and keep both from our exact relationships between particle masses exact relationship between coupling constants and the avoidance of all of these divergent quantities which occurred prior I think I think I had intended to go through a couple of real examples of constructing supersymmetric theories and show you how they work starting with the very simplest armed in just 1 dimensional space name one-dimensional kind but I think I'll say that for the next time around Mazur said he is subject to is very very abstract is not the abstraction is not would bothered people or herself who was a simultaneous crack it is also the concrete this of needing to keep track of yet the head that was just too much anyway calm by Mel Edwards the subject pretty well but it took about 30 years Bureau set yes yes yes we celebrate Memorial Day by having a Europe said lecture left what is morally well now we will we will continue on to I will not be away anymore we bosses of the program says I can't go away so I will be here I removed I have a laugh I had a does anybody know as their courses now this a lecture 6 so a former lectures press a above all but the idea of the boss for me that I have pronounced in my while this is a draft a 3rd of this is that preceded old and I believe 19th century Russian was a 19th century mathematician ought to do a differential forms her and his wife the well let me out of there will be less day because I will not be here on June 12 OK as everybody following the supersymmetry is very happy with your abstractions as long as you are there for more please
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Titel Supersymmetry & Grand Unification: Lecture 6
Serientitel Particle Physics 3: Supersymmetry & Grand Unification
Teil 6
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/14975
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

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