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New Revolutions in Particle Physics: Standard Model | Lecture 8

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to now as a matter of fact I do not want this or talk about tonight and it's called the phenomenon was eggs eggs started last week and I didn't really he's talking about a man but if we don't actually massed on the way to look at the track Russia a real photon these symmetry group associated with it is not spontaneously broken and I'm going to talk about Howell if the symmetry groups associated with the photons were broken spontaneously how it would give the photon maps no reason we're doing that His because exactly that phenomena happens with busy bows on W. burst on the photon the photon was a little bit simpler indeed it might have had amassed the fact that it doesn't have a mast is perhaps an accident of nature our absence L an appropriate Higgs field where talk about what the Higgs field to get it maps so that's what we want about like how gage bows on like the photon get a naps when spontaneous symmetry breaking was the symmetry incidentally was associated with photon is a conserved quantity which is a charge conserved quantities always go with symmetries what's the cemetery the connected with BT concentration of electric charge you want it's the thing which multiplies the charge field by a phase it is a kind of rotation in the complex plane complex field solve Araujo explained to you tonight her spontaneous symmetry breaking that symmetry induces a mail from afar now there is a situation in nature where the photon does get announced and the symmetry of U 1 symmetry is spontaneously broken has been ordered what is Connecticut is condensed matter OK the interiors in the superconductor and a superconductor photons propagate With a cast ballots in the superconductor when they leave the superconductor their photons again when they're in a superconductor they behave Adams if they have met a source that's an example of the Higgs phenomena are we can talk about it stop that that superconductivity only happens when the full Thomas traveling to something other than the no no superconductivity or the photons No no no photons is not propagate with demands the preserve its velocity is slower than the speed of light but it's still true 0 momentum infinite wavelength its frequencies 0 let's talk about the different sets of interesting different rumors will talk about arms the relevant issue is shape of a certain current is he there energy vs. momentum or are for waves were corresponds some momentum for waves either In 1st wavelength or wave number K veteran a K K is basically the inverse of a wavelength and vertically were have instead of energy Dr. is likely vertical here was energy but the frequency frequency omega member equals H bomb maker a P E egos each bar OK is wave number the number of waves that fit into a given us into a unit that now a distant acts of this these 2 planes of the same except for a factory I was Omega vs. Katie look-alike for an electromagnetic waves and empty space but what's the connection between Omega omega equals see Times k that's it make legal now relationship incidentally is true of sound waves waves our on our long strings and so forth approximately the same relationship the difference of course they have different velocities of propagation but notice that all major zeroing Tadeas 0 that means on this side only here that the energy is a quantum of always the energy 0 when the momentum 0 recall particle when its energy is 0 1 and energy 0 0 if it's messes with call it would call for contest protocol nasalis I'm so any
wave which has the property that Omega goes smoothly de 0 risk a goes 0 we call and typically video the shape of the curve looks like that have taken a positive and negative correspondent away going to go left right but forget going to OF be frequency goes to 0 when the wavelength becomes infinite that's nasalis and all that happens in a prison Is that see changes see changes now won't vote on what about a particle of maps for particle of mass its energy is equal to the square root Of Peace squared plus M Square now subsea is there because seat of the bought an nor C squared see square in them that our grew out of a C squared he squared plus and squared see what kind we said Siegel want square the peace quotas on square was that look like better curve which it's a hyperbola effectively square 380 square but said squared warned he squared minus P. squared equals M Square that's a hyperbola is seeded different case want P goes to 0 the energy is not 0 that's the rest match was just a massive the rest mass is the value of the energy when the momentum goes 0 I'm so I love through different situation 1 of the velocity of light may be different in a prism that it is an empty space but that just changes the slope of the current does not change the fact that the 0 at the origin here the other situation is that the energy is not 0 at the origin and has this hyperbolic shit that's called having a most of the frequency same thing here that a wave with infinite wavelength has as a finite frequency wasn't at a wavelength mean it means you disturb the field every place the same you rigidly you make the field absolutely homogeneous homogeneous means it everywhere is the same you shift the field everywhere and you let it go where happens is everywhere start Stossel simultaneously that's a phenomenal he infinite wavelength 0 wave vector the whole field oscillate also corresponds to particles arrest oscillating because they had met press that's what important thing to keep in mind is that oscillations fuel where mere homogeneous where infinite WaveLAN rows of things McCormick residents question about 3 4 ways ages what with self far ahead a red is less that that's right if you if you have a way you would given frequency signs a piece of glass frequency doesn't change when it goes into the interior of the material but as I figure for to different materials might have a different slope so the same value of frequency you would jump from 1 in could jump from 1 way or another the fact that this situation our ships would that be shut your rural will make his 1st class and we stand at the same time says that as far as our full here which instantaneously by Utah viewership back no no bomb entanglement situation where you have to particles 2 different states which are entangled with each other a over here be over here or a mail here be over here that they have to be last week to different states lost been up and down the man's been up after a classical field with all of the particles missing state look for the and that it was aware that represent that this school the relationship between Omega and Kate between frequency and wave number is called the dispersion relation is saying is no dispersion when is a linear function of K or make it as a linear incidentally the slope of the curve Is the velocity of the omega decay is a yard is the velocity of the way so if the slope of this is constant everywhere as always move with the same velocity private life always move the same same velocity if these slow varies from place to place for example as it would if it was a embarrassed different wavelengths propagate with different velocities In that case the way you will disperse so call the dispersion relation with name and now Omega Omega as a function of Katie that relationship is called the dispersion relation of the way so the words go bad I'm massless certification is 1 with a linear dispersion relation a massive particle is 1 with a dispersion relation that does not go to 0 as scapegoats goes to 0 but has a curvature and shape like that and the man is nothing but a energy of the quantum when the momentum a 0 and energy of quantum quantum as a function momentum when the momentum grows 0 is amended to that that's the notion of mayors and say it is connected with the curvature of the potential energy of the field at the of the curvature of the potential energy at the equilibrium point wherever the equilibrium point happens to be white collars if you shift the field away from the equilibrium point that's it everywhere is homogeneously everywhere is the same what happens stocks oscillate about polls Lorena G overshoot goes up and goes back and that corresponds to homemaker not being 0 At Cape equals 0 cable 0 means to shift a few everywhere simultaneously homogeneously and so we take the field shifted everywhere simultaneously and the frequency at that point as the Mets the frequency 0 and the only way the frequency will be 0 is the potential energy is flat as a function of the field of potential energy happens to be flat and you shift the field away from the minimum here it's not a minimum anymore it's just a flat directed flat erection of all things it called a flat direction 5 if you shift field along on a flat direction it will respond to the set so that corresponds to the homogeneous field cable 0 having no energy no frequency frequency 0 as use spot a mess was object when there's a direction ofthe lacked direction of the potential he ever flat erection of the potential is going around the rim on the other hand if you
exercise the field perpendicular to that oscillate so this can this kind of field has 2 distinct kinds of quanta associated with it 1 man's list when rested with semester 1 corps in the context of the standard model massive figs goes on solve oscillating this way recall the Higgs boson Island has amassed what about oscillating this way that's the Goldstone on the man's list but there is no mess was called stumbles on nature so something happened to it in the parlance of our over high energy physics the Goldstone boson on got each by a gage bowlers on resulting in giving the Higgs bows on the Mets so that's the phenomena that we wanted to mathematically tonight the Goldstone on each by free gage bows on resulting in a mass of the Higgs boson member of Goldstone on is a 1 associated with emotion around he and the Higgs boson as the 1 associated with or obstructs growth it was mathematics service of I could do without mathematics I would but I can't so we could do with mathematics already set up the ingredients rise the basic ingredients for it have those combined together work are also how Earth so are the is only 1 spontaneous so directly said that as part of its not a simple U 1 symmetry as you to 3 of the weak interactions that take up Stu bounds and neutrinos to work for lecture Soto more intricate than just the you 1 I certify Waters said the case was has concluded itself young yes well enough known only yesterday not just OK case so let's make tell you what the point there are a collection of particles in the Standard Model which form 1 mathematical reason or another came not have any mass on unless the symmetries spontaneously broken no illegal through the trees where go through them from various particles and see what the reasons are white a mail is disallow by the standard model without spontaneous symmetry breaking Bedouin spontaneous symmetry breaking happens all of those particles get masks and they become the masses Of the stand of the Standard Model particles Jimenez wine 1 about particles that they require a spontaneous symmetry breaking again and are there any particles that don't require a spontaneous symmetry breaking togetherness in theaters yes it can be the loss articles on all the particles that we don't see in a laboratory wiU be that we would see those which symmetry breaking give the maths and we don't see those that symmetry breaking doesn't give them any India's speculation the why that might be yeah at can the macho mass might be way up at the plant scale natural masses of particles may be very much larger leaving over a bunch of particles which don't have masses what your calling natural reasons and those particles only get from spontaneous symmetry breaking those particles of alike what was the ones we see in laboratory our we have every reason to believe that in fact I think we practically no by now that there are additional particles in nature for example the dark matter particles we don't know with absolute certainty what dark brown particles ah but it seems very likely that the dark particle dark matter particles of particles a thousand times heavier than proton for example why is it but those particles have large Miers and the other ones don't Annie and so is the mathematics of the theory goes particles which have large mayors don't require From our or spontaneous symmetry breaking together so we'll go through that that's my goal for tonight and if we don't make it through it entirely next time the Higgs phenomena and how and what the words mean the Higgs particle gives such a such unrest by the 1st class of particles which cannot have a man on spontaneous symmetry breaking money analogs of the photons the gage bows on matter which I assure you however for conduits mayors from spontaneous symmetry brick let's begin with a bows on-field fight we did this last time but the subtle and difficult concept but do it again and fight it's a complex that means if I start to go along with find can right fly into weight is my real plus I 5 imaginary was a correspond corresponds program drew before or fire real fire imaginary what we can write it as 5 equals role Ito of RIA Alpha where row distance here real distances basic distance and fields and now offers the angle to different ways polar coordinates a Cartesian Courtnage field that our incidentally 5 star is of course fiery only minor sci-fi imaginary it's also equal as here so he rolled the mightiest eye out for functions of position this is a field is a function of position and as you could say in either case there are 2 fields real feel the imaginary field on the radio field and the angular field might begin to guess about the connection between role Alpha and took kind of emotions that can take place here Alpha corresponds to exciting angular HMOs if you very Alpha from place to place like varying around the circle if you very role like moving radially back and forth which 1 do you think gets a mess Red which 1 billion amassed offer which was innings HMOs rout which 1 is Goldstar miles an hour you gotta give that's that I have hardly article for the mathematics not really that sort of mathematics is now believed to be the trekking going through the mathematics is to write the Lagrangian field river of five-star times the derivative of flying this of course means time derivative squared space through square Brodeur's right away the derivative of fire with respect to space and time as for vectors bothered into each other bats gradient terms in the energy and you can work them out in terms of clients livestock for you this becomes derivative of fire real squared plus derivative why imagine square another words it just behaves as if they were tool independent fields
real unimagined are all are quick and rioted in terms of role an alpha and this would become all I've done for you work at yourself derivatives of Rose squared in the end plus rose squared times the derivative of alpha squared role is the magnitude of the field Alfred angle but each of these off fields and this is what Will Rogers figure in 1 way or the other problem now wants in a potential energy the potential energy I'm going to assume is symmetric symmetric with respect to rotations in this plane it a really does have a U 1 symmetry potential energy can 1 symmetry it means it only a function of rolled equivalently it's only a function of 5 star 5 it does not depend on angle so it's right that class Our of roles but but that also corresponds be right let's take actually potential always comes in with miners grungy and so were minds a simple case will be rose squared in energy which increases quadratic Lee away from the origin that's a possibility let's think about that possibility Roe squared it's not really well and it's a possible expression on Willard that correspond to intensify real and fine imaginary does square fire real squared plus firemen jitters squared that's the Pythagorean rose squared is fire real square passed by imaginary square if the potential was very simple form of quadratic through it it he finally real squared minus subtract report by real squared plus fire managers square aid notice of a would actually be the sum of 2 turns 1 4 5 3 0 1 4 5 imaginary and both of them would
have amassed turned it could be a coefficient here and squared over to it Hey then you have this situation which I drew only here where both directions of oscillation corresponds to a mess and they're quite independent of each other don't coupled with don't talk to each other just 2 separate fields 1 that I label fire really the women label fire magic that I could be more complicated things he which we couple together but this is the basic pictures of the potential looks like it's 1 of the potential looks like that's also a function of role but it would be inconvenient to represent it in terms of fire real environment unary especially since the minimum is stuck out here away from the origin they Cohen this firing and fine imaginary not obviously the best coordinates when you start here that accord Mich are the angular coordinate and the radial quarter white because the correspond to a different kinds of oscillations with 2 different frequency another words just think of this as a function of Roh and Alpha know this potential energy doesn't depend on Alpha and all but does depend on wrote it depends on enrolled in a way which is minimized when Roh is equal to half and in fact it takes a good deal of energy to displace it away from sell it's natural to write that rolled natural right that role is equal to half plots a bit of flocculation a minimum energy is when role is stuck at any lies along here Rose stuck it out but you can this place a little bit and let's represent that displacement away from half by a letter of the alphabet and I'm going to Chile the next letter of the alphabet if the path which is age Free H for what each Higgs and a figs so thinks field is just the displacement of the radio direction real part of the field away from its equilibrium but now it caused a good deal of energy to excite is Higgs field White because it's got potential potential might be rather sharp fact it is rather a sharp sold on the some approximation is another good approximations teammates say that a energy behavior is as if rolled off rose in value F now this is a very common thing in physics to say drive on low-energy expectations low-energy oscillations of system low-frequency oscillations and high-frequency oscillations and ignore the high-frequency oscillations that would be equivalent to ignoring each year just saying role is equal to half plus something that takes too much energy for us to worry about today will worry about day when our accelerators have more energy while would this look like and what role is stuck at at matters will ignore derivative and furthermore about V of Rome Lee of roses VOF so V is stuck at a minimum it is stuck at a minimum it doesn't doesn't vary from place to place this and they are irrelevant to the dynamics Of these angular field furthermore since it cost so much energy to make a bit of a Each we just approximately say that rule is equal to half and we find that the lower energies of a small energy expectations of our grungy square number times the derivative I love how fresh square there is no potential energy associated with Alpha lies in no potential edgy associated with offer for the obvious reason that we don't need any potential energy wanna go around the angular direction on weak this is a little bit less familiar because it has this effort but we can't we define field we could redefine the field by by sending no fee of code data base is a quota s firms How about a sister Reese dealing of the field and is a redefinition and then we would rewrite this just the Alpha debater squared but no Norman no master because it doesn't cost any energy to displace the field homogeneously everywhere no and sell or beta depending on which you want to write is a massless was for you use the Goldstone goes on and as I said what its energy corresponds to its slow variations of refuse to vary the field from place to place slowly you get a little bit of energy to various slowly enough to get a little bit of energy from the derivative terms here but no potential energy wraps the hallmark of a massless particle Goldstone boss Goldstone bows on however massless particles which consequences Our of be spontaneous symmetry breaking of 8 continuous symmetry the buzzwords that that have a story about a spontaneous symmetry breaking him together with Goldstone bows on phenomena and the other half of the story which also has to do with varying the phase of a field from point point is the gage invariant Sarwar spend a bit of time give a break for a few minutes they will review again told last time Dejan variants that take us another 15 20 minutes 10 minutes mature and then put the 2 2 of them together and see what comes out and what will come out will be the Goldstone boson was eaten by the gage buzz on giving the Higgs boson that's not giving giving it bars on their source of that but I think he's further previous complex players feel that had reduced across the field it is a point in the complex plane value that a field takes on the point in the complex plane at every point in space the field has a value which is the of a complex point plane no words at 1 point in space the fielders 3 D plus I times 6 and another point in the plane that supply plants my someplace else a video macaroni number plus kind is a more orbits number like that it says that what you I would With that the value of the field point is not radially symmetric value the fuel at a point as the value of the fuel the point is that radio cement tickets appoint complex plane what really symmetric is the potential energy the French or energy is only a function of the squared plus the squares rain 6 feel at some point has a value over there at some other point in space the field on value over here at some other point takes on the value of the here potential energy is only it has a a a a a a a a a a a a a a exports I wipe the value of field is next wine and not positions base is
not the field how the point Texan in wife businesses the field whose value would is a point in the complex but abstract idea fields of fields commit complex Cygne real be multicomponent you can assemble them together if they have took components where until a complex variable the field itself is a complex variable at a complex variable ways you move around the space field takes on different values on the complex point it used that idea fields don't just have to be numbers it be complex numbers Vermont policy but it's not were still not there so would be the vessel if you get a bit of the yacht that no part of it all man well the problem is that amount of energy that would be involved for Higgs is way beyond the energy our that even a big accelerator can now so at the ballpark from you LHC but yes In principle you could give them give the feel-good give big enough not knock over the top 5 everywhere simultaneously space that would be enormous amount review if that there are a lot of germ likely gallon maker bit would make a monstrous splash of some kind Otsego at stake a deliberate let's go let's go through it again at stake what we mean by a field as a function of space Command what this idea Our a field which is a complex variable is just to make it clear once more with space x y and z right now I can have fields in space that's a scalar fields because of a scalar field and space followed which now what characterizes it as being a it means it doesn't change when you rotate coordinates that means it that now when you rotate the call and it's talking about here when you can't rotate the coordinates of Rio space doesn't change that's for a scalar field they have several scale of heroes let's say particularly opener scalar fields let's give the names let's call them find on and fired on 2 scalar fields our Ucon then a symbol and a few light this is cult only a matter of convention is partly a choice a useful choice it's 1 of these things in mathematics which sometimes are really just useful conventions this is 1 of them and the whole idea of complex numbers is a useful convention it's useful convention from representing pairs of numbers song and the pairs of numbers recall the real imaginary parts of the members X not ruled that column X X stands for space but just Real and imaginary the Our components are just ways of speaking about Paris of real numbers that what is a little more than pairs of real numbers and more structure that goes into year and of the complex numbers but no fundamentally it's a way of talking about pairs of numbers you comply pairs of numbers on Nunavut Cartesian plane the Cartesian plane is not real space it's just the Cartesian plane which of plotted the real and imaginary parts of complex numbers if I had 12 fields 2 scalar fields 1 on which I qualify fiery on the other the name fire real and I imagine I could assemble them together in complex field not complex field and at each point of space will be of fire real and a fire imaginary each point of space I could go and plot the value of the field on the complex point keep the complex plane is not space is just an auxiliary construction an auxiliary construction which is useful in replacing to real fields by 1 complex symbol Close to trek for representing cutting down the number of
equations secure right cutting in half basically by writing in his equations for complex variables OK itself in that language you would say that the scalar field complex scalar field phi is that just that it's a complex scalar field and at every point in in space real space it has a value added value is not a single number it's a pair of numbers like isotope airfields to begin work that paired numbers plowed disappointed at this point the space of a year the field was over at that point the space fields over here at that point it's over so that's the idea of a complex gala feels sometimes the real and imaginary parts are called the real and imaginary components of world but don't get those confused With the components of the vector field I can't reveal the components of the gravitational field those components have to do with projections on real space time coordinates this has to do with the projections onto this fictitious too dimensional plane the teams Our writing prayers numbers that once you realize that you could write off the field as a complex variable and plotted on the two-dimensional plane like that then it becomes a possible symmetry of nature of possible symmetry of mathematics to rotate the feels the mother was there might be a cemetery which says that you could really label or redefined fly audience by all our by rotation and that happens waited for years when now we have a right like this which don't change when you do what when you multiply fired by eat and of is a class her grand which don't change would you multiple when you rotate the fields simultaneously everywhere in space when you rotated in this fictitious plane words take the field at every point the space and you rotate its value you given no value to the field which is related to the old body changing the angle here rigidly everywhere and space than this Lagrangian does not change we went through that last time and that's the basic idea of a complex field with asymmetry the symmetry of called U 1 symmetry it is a global so that's called a global symmetry global symmetries 1 way and not allowed to change the phase angles In space in space you just rotated everywhere so we conceded symmetry just by looking at where dead a function of road well you could look at Member waves you could see that this has real square plus imaginary square things when everything is a function of real square that if does work any of now Brent services a fortune role so you could see it here incidentally what does the symmetry operations duel on Rowan Alpha but Rose said you shift but constant namely the angle rotate midfield wireless shifting think Alpha by constant doesn't change derivatives tension that that's correct right so I'm Malek agree on the basic idea now let's go through again I know yes works go through the idea of a gage transformation we went a last time some scary mind you Americans and a lot of time on the resources we can but please raise questions it is not asymmetry of this program here to multiply the field 5 by an arbitrary function of position by arbitrary fees the score the phase we are like this and 5 of them next just is not a symmetry will enter this last time we knew we wrote battlegrounds in in terms of fight crime and found out that it didn't have the same form another words from the low grungy in written up their particular turned derivative of 5 stark terms of derivative of flying I don't after putting me indices mule you know about that that is
not the the same matters the derivative of 5 trying star derivative of 5 prime not the same therefore is not asymmetry is not retreat wine that's because derivatives act back not only on fine but they act on he found favor favors variable if they had the invariant space then this would be a cemetery but they is based not to work that out in detail less time and I'll show you the moment we found but to make it a symmetry you have and another feel the kind of compensating field a compensating feel that will compensate for changes in the law grungy and when you shift the phase of the field keep in mind it's useful to remember that shifting the phase of the field is just 8 heading function for shifting a phase of the field is a heading of function fire that later vexed for changes phase angle everywhere by amount later if is a constant symmetry fate is not constant not surgery in order to make cemetery half-day another field and other fields called the vector potential for vector hazard meal it's also function a position out of it you build electromagnetic field he and be you're electric and magnetic fields the transformation property when you do it gage transformation like this is called a gage transformation when you do don't gage transformations have to do something that 80 at the same time factor to pick a prime of X say people X plus or minus there was confuse bird signs here and it's plus the a derivative a new of that another words you you'll end AD the gradient space time gradient of data if you do these thought voracious together and you construct the right kind of Lagrangian it will be invariant under suffered for it it's a very it's a cemetery so let me remind you how works all incidentally that's good enough the for less than a month ago a but a mail and he is aid goes a crime scene you obvious saying right with arrow location of the transformation from starting with a is what they might be right transformation is a goes as a Class D you a better writer OK sisters but this is a transformation that you make these are the prime variables visibly and front and I may have written that the opposite way less time member but I will compensate for any mistakes not only mistakes but any change of notation by changing it consistently everywhere from 1 week to another day I don't remember that precise picture but also your you could do would eat away if down 1 plus IAEA a in a story and they may have bird we change notation but you do it consistently everywhere far so let's now RWE seemed that ordinarily Riviera of fight crime is not simply related to the ordinary derivative of fireworks right up the relationship Bush but figure derivative of fight crime threaded with respect any access or stop writing news in those derivative of 5 prime is 1st of all it equal to eat ARE data which is a function of X is a derivative of fire if that's all they wives and we did multiplied derivative part of the complex conjugate of 580 but a derivative of finally Vitaly I. figures would cancel out right facts of that's that was we be in fine shape we would say is a symmetry but there's another Turner and the other turned is fighting times the derivative times a derivative of Salida also trying we also need to be out who is this Sharon here which is a nuisance although laws was something which carried and if they not only through each I they hear it would cancel out multiplied by the complex conjugate but it comes into its derivative over here that's a problem right to fix that problem why should you want that works affected nature that there is a gage that there are gage symmetries source or an interesting fact of mathematics of such symmetries of possible to make it possible you replaced ordinary derivative by what's called a call them in derivatives misses a mathematical concept that comes from the theory of fiber bundles but goes I don't know whether the fury of fiber bundles was invented before or after the notion of gage invariant physics and Ophelia was invented afterward but some of the safe area by fraud that's right I think so we are very proud of our but the mathematicians and they did it completely independently of but I really don't know any case some of it is both a mathematical physical concept and in order to make the Lagrange engage invariant you invent a new kind of derivative call the call area derivative With respect again
with respect each their acts in the space of and your right this is equal to be ordinary derivatives plus the rest of confuse try plus those work with take plus on a and 80 times 5 there is an implicit index here once moral right a user vector to reveal plus multiplying by a sometimes this is indicated by saying the derivative is replaced by derivative plus my only makes sense if you apply it to a complex field like 5 so this is the definition of a cold variant derivative and if I were to write the call variant derivative of 5 it is by definition just a complex conjugate Siegel theme you find star might next i.e. genial five-star now I assert easy check is check that the relationship between the call variant derivative of 5 chemical area derivative of fight crime is very simple all you have to do is to evaluate the co-vary there was a cold this is a common area derivative 580 let's write down the call manager of 5 primary had well that's exactly the same except declines everywhere being you fight crime is just a deem you'll find a plus on May 8 for time plans 5 fight crime thank you bye primarily by Prime Time OK if sit fight crime over here you're going up with this extra terms over here a fire ratify evaluate Dean you fight trying to write a new 5 prime areas but Caesars right now but I do here this was a new 5 prime right With firefighter that former memo abysmal differed from capital not can refer to a Pfizer actually work and when you use the bar we are but if he you don't have that I on the top as the former board if am Estonia is no distinction in my notation between uppercase and lowercase sometimes I like them proposal things when I'm feeling in a good mood these things on when I'm not feeling in a good mood of the amount so a In most of the book spent yes but then works what he did None Mariah Carey fighters fighters fired and I have not I did not need to distinguish uppercase lowercase the vote Ray have about this November consistent here what but he Ms. went on now was uh uh for me a yacht yup what's really meant by being neophytes being you know him we are OK so let's and you have a fortnight the arbitrary for vector being you you can multiply by Jean you symmetric and yet we knew we know the meaning of this symbol for ordinary special relativity is that the time components of these 2 the Siemens signed and the space components operas are so high right the meaning of this as it should be take it to beat the opted for the lawyer took their culinary vector times her right but I don't wanna right that every time over oversaw just write an abstract symbol derivative derivatives now I'm yeah that Florida has slowed down right sorry the but you're right but we've already gone through special relativity a number of times and so I don't feel the need to be careful about relativity indices shots only 1 Bala writing them for the most part I keep saying derided the keep writing them OK 5 that now here idea of fight slug theme you fight prior I will get his unwanted nasty piece over here but if I also plug-in a prime times Friday wears a prime ready prime over here a prime equals AT to a plus or minus new data then they will cancel they will cancel it will be a turn DRUG youth later times you know I mean right so fight the tool be new status will cancel each other 1 from calculating derivative here and the other from the explicit difference between a prime in a Seoul
I out exactly what fine I will find it the you fight I'm this 1 over here who's equal but not quite just you fight or missing but the way I played an overall lead the eye so you say Well didn't anywhere knew of fluff fight crime is not the same as being new funny it's not Dejan very true enough but only differs from being gage invariant by an outside overall factory the data which will disappear when you multiply this by conjugate notice that In this formula he had the derivative of faded does not appear to have recalled variant fight prime is equal to the call very derivative of fired up a fool this overall factor Vitaly I. figure which will cancel when you multiply this by being knew of 5 stop crime that will be equal to each of them minors I figure times D knew of stock selling you multiply stood together the unwanted unpleasant things that had the rivers of favor in them disappear was the cost the cost is you have introduced no field which transformed in this way would share in this way over here but once you allow itself that freedom to introduce a new field of vector potential this slogans you over here becomes invariant and the gage and was later on will rise about in some more detail but 1 more point before we jump to the conclusion and that's electromagnetic part of the Lagrangian this field Is a charge carrying field it's not electron because electrons Nova bows out but it could be some charged bows arm of some sort some charged bows out of some sort yeah there are charged both arms helium atom was helium nucleus is charged Vos armed so far I represent some charge bows on May represents a vector potential and electromagnetic field has also governed by Lagrangian Lagrangian is defined as you know people being you pay no mind is being a new space time the next space time component as electric field the space-based component of magnetic fields and the squalor f now that there are some signs that go into it and knew knew what have you know that East square miners be squared that's the Lagrangian New York from magnetic field the important thing for us is that this is gage invariant does not change when you make a gage transformation what happens that a a changes body the gradient of data so what happens to this crime becomes ever plus being later minus being these 2 terms canceled so if it is itself a gage invariant when you of squared it still gage invariant so they hold up shop is only 1 more thing 1 more thing you could hear the words Dejan there and that's a potential which depends only on fight a potential which depend only on fire staff so full of Iran's Ian a simple gage theory involving the interaction of electromagnetism with a charged scalar field would look like have been you vii being you purchase I defy star Villa have V. of Tri-Star fight which is just role if your number
and then plus squared plus square miles be squared and I fully gage invariant no change in it when you make huge transformation that's the simplest gage theories nutshell but but but but bicycle Aguirre that's basically restructure all beach theories they have structures markers nuts asked how what happens when the symmetry of rotation of fine spontaneously broken windows that happen it happens when the potential hazardous upside down and back some Our upside down sombrero where has that shape so that the ground state of fight breaks the symmetry what happened what new thing happened new thing happened all before we do that lets which just remember that this potential was such that angular direction is a Goldstone balls on known as the radial direction which we call the Higgs boson has a mayors and furthermore be full time does not have a mail how I the photon doesn't have a mail the reason is simple B is only a function of the derivatives of the vector potential that means if you were to shift the vector potential everywhere simultaneously there would be no change in energy shift the path and their vector potential everywhere is together no change in energy back means a match was particle so the two-man Toll apparent massless particles floating around in this theory 1 is really Goldstone boson and the other is it is a photon and now but watch what happens let's take a while Wildwood a minister for the photon look like if you had worked well and this term is always something which is quadratic in the field for example the mass of fine if it existed would be phys a five-star firefight squared fired by next Friday real square toes by imagine square or the photon this be it would be something proportional 80 square the vector potential square if you allowed yourself but the vector potential squared until Lagrangian you would have amassed for the 4th time that would not be be gage invariant look at years a square which means the a new way to put that in the Lagrangian while what happened when you make a gage transformation which shift 80 plus the gradient of data square right but that's not the same as a square a squared it is not the same as a poster of square so a square just a square a squared would not be you will not be gage invariant so messed term is not allowed by Dejan very gage invariant is a symmetry which prohibits the photon from having a mail would does well let's look at the wanted a little more closely at a particular let's look at this time over here but In fact let's concentrate on the Goldstone bows on a piece of it we can do that riding verified is f turns eatery on at no it's not works let's just let's go for it is is that and Lagrangian so what's India d fight derivative of far-right class as a class REA times but today right wherever he I star is derivative of funny minors I five-star right now let's suppose that we're looking at the ground state when the field is right over here when flight is equal to half but say just a real value the real value of fires equal efforts simplicity let's say imaginary part of 5 Zero so a few sitting right over here but so the various turned 1st of all Our well the most important Turner is this 1 here 8 times flight times five-star but can't find star is stuck at a minimum he stuck at the minimum because it takes a lot of energy to shift fine away from the minimum so as long as as we don't excite a lot of energy is an effective terminal run June which is just a squared times five-star Friday which is just equal to half squared pounds a square somebody said that was because you immediately see that this is behaving like a minister for the photon is behaving exactly like a mystery for What's the matter lesser basically just f actually twice FOL square 2 times f but this factor numerical factor here is playing the role of forget there's playing the role of a square times Square which would be a mass photon will be an energy that you get when you shift the field homogeneously so where we learned we've learned that when you spontaneously break the symmetry when you study the theory in the neighborhood of the minimum over here the effect of 1 of the terms Lagrangian is just and make amends for the time net interesting when the photon propagates in this world where the symmetry is spontaneously broken it behaves as if it had a man that doesn't rest for all practical purposes so it got Paramount the statement that says photons can have met no real photons of cost on grass but this photons in this world with the spontaneously broken symmetry do have passed the 2nd thing is the Goldstone bows on disappears What is the Goldstone bows on disappear well let's remember what the Goldstone on is it this angle alpha the angle of the field on the complex plane what does indeed strands of 8 well 1st of all what Is the Goldstone on the Goldstone boson is simply a slowly varying varying value of alpha offers a field can vary in space could in space it would be interesting African variance space but you could make up for another way you could make it very by doing a gage transformation is getting a little bit of later on sex so you could make Alpha vary from place to place you might think that that makes a Goldstone goes on but it doesn't be because this as a symmetry symmetry means that you don't change the energy it means that the system stays In the ground-state this is not a real change it to gage transformation so the stumbles on completely disappear completely disappears it just becomes a Our gage transformation no real energy associated with it at the same time the vector potential gains a mail sold us so a zero-sum game you get rid of 1 man or you don't have a zero-sum game you remove the missiles Goldstone was armed and remove the missiles photon replacing the missiles photon man massive photon and at the same time the Higgs blows on now has a mail are so it's about here that's the that's these things but if there are by plugging in basically just by plugging in the value of the Higgs field the value of body at the minimum here we generated a mystery for the photon away they come from came from the score variant derivative so is king conical variant half years later it was as if we were like those get cash and if you can you could rewrite of the grand jury in those terms you could do a
gage transformation and basically completely remove this angular degree of freedom completely remove the angle from the Osseo symbol baby version of that if your completely forgot the radial degree of freedom or not this completely froze it just completely froze you would find that the Lagrangian Our young coal variant derivatives on Friday would be common death think times a derivative of Al plus the vector potential itself may be kind eatery on our our foray is just a quote that at times each of AI Alpha and of the derivative of flying is f on May Alpha F quite a derivative of Alpha frenzy to a guy out there would be a need to for multiplying the whole thing this is just setting the magnitude the review rude rolled equal to F King of big Africa money outside here would if would come earlier outside this would be the core derivative of fire and it would contain the derivative of the angular part that's important contains derivative and you apart what about the derivative of fire started we will do the same things through Alpha a minus Ito the miners offer after from getting tired of sorts getting a bit late on bets that's what do we get when you multiply and together we get exactly what to expect OK now you also have ever knew New will you have this time as the snow run gin and you have a new New now let's make a gage transformation researchers a clever gage transformation which is designed to get rid of this term altogether How could you make a gage transformation the gets rid of this just said later Eagle to minus Alpha remember what happened to the vector potential when you make a gage transformation adjust picks up a derivative of data if you make a gage transformation using for the gage function later How the itself is just goes away completely orders are those Angola completely our it just becomes and this terminal a grungy just becomes a square by designing you can construct engage transformation which just removes there's this time this gets rid of the offer dependence altogether and you just get better squared times a square all you get is a minister or what's left then the cost you still have square but this is gage invariant so it doesn't change when you delete gage transformations that result after the angular degree of freedom just disappears a magic happening angular degree of freedom disappeared the Goldstone Bozo was gone and photon has meanwhile the Higgs boson on which is as radial oscillation that's still there so this in the magic of the Hague phenomena and it's the phenomenon of giving a mail the gage bows arms buying spontaneous Is it there is that start salacious are distinguishable slipped from defeat at the that's fair to say it's only fair now I will tell you right now that this is also the physics of superconductors superconductors are charged bows on the charms balls on our bound pairs of electrons called Cooper pairs Copa pairs are bound are loosely bound pairs of electrons but they are bows on so there's a charge bows on-field system and the charge bows on fuel has shifted away from the origin by some complicated dynamics and superconductor creates a condensate of those charged pairs and its equivalent of exactly this kind of spontaneous symmetry breaking where flight would be replaced the field describing the corporate parent so this is also the physics of a superconductor and the superconductive the photon also behaves Arcadia like a massive particle 1 less then how can you lose a degree of freedom however if degree of freedom disappear angular disappear degree of freedom it didn't disagree disappear what has happened a man's plus photon has polarization states member polarization is a little arrow Perez perpendicular to of the vote on its if you like there be aware of the direction of the electric field oscillations so every photon carries with it a a polarization tool orthogonal directions for polarization 4 times going that way the polarization that later 2 polarization you could also think of them as state of circular polarization of the 4 there is no state in which the polarization vector of the photon points along the direction of motion Maxwell theory doesn't allow that why is that consisted of a photon is described as a vector how could it be that only has components and not 3rd the answer is very simple the fault time was the speed of light it can never be brought to rest suppose it could be brought to arrest supposing the photon could be brought to restore could be brought arrest and you brought a terrorist was going down that way and you brought it arrested a a polarization along the x-axis and you brought correct it would be a particle the rest the polarization along the x-axis now you could accelerated along the axis and all of a sudden it would become a particle whose polarization was along its direction of motion so the answer is up if a particle like the full-time work that may ask it would have to have to read actions of polarization it could where have you could bring up the arrest and wants its arrest is no distinction between the 3 directions nor have 3 directions of polarization and then if you spend up along some Max's it would have 3 directions of polarization of massless particle could never be brought arrest perfectly consistent I only have 2 directions of polarization namely circular this way or circular this way no other way of thinking about it is if you take a particle circularly right-hand polarized and you brought arrests and then accelerated a off the opposite direction it would all of a sudden have left-handed colors it would have left him left him right that's also would affect about me point is that you have not lost degree of freedom the Goldstone boson has really become br others missing near polarization before other saying it is you started with 3 did Google 4 degrees of freedom the took on a took component for each Rick Fox took components of the charged scalar and 2 components of polarization of the photons Indian you left again with 4 degrees of freedom 3 polarizations of the photon and the Higgs balls on so you really haven't lost any degrees of freedom by various turned the Goldstone bows on into the what's called a longitudinal degree of freedom of the photo we want in which the vector potential or fielder's pointing things electric fields pointing along the the direction motion and that's what's happened but they all have never next time I was sure you the physics in a simplified context the same context of 180 on electrons and these the quarks get amassed from the pigs so far we haven't talked about the quirks of Elektra they also have amassed what is the matter what does the MAS has to do with the 6 phenomenon that will be the next time we finish that we will have a set up in which all particles including the Higgs boson albeit demands from the same place namely the spontaneous symmetry breaking then we can start to talk about this very interesting situation that all of the particles in nature of the we know Nunavut with measures are the ones which would become massless if there was no spontaneous symmetry breaking the amount of enough formal
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Metadaten

Formale Metadaten

Titel New Revolutions in Particle Physics: Standard Model | Lecture 8
Serientitel Lecture Collection | Particle Physics: Standard Model
Teil 8
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15080
Herausgeber Stanford University
Erscheinungsjahr 2010
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (March 30, 2009) Leonard Susskind explains the Higgs phenomena by discussing how spontaneous symmetry breaking induces a mass for the photon.

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