New Revolutions in Particle Physics: Standard Model | Lecture 7

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New Revolutions in Particle Physics: Standard Model | Lecture 7
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(February 22, 2010) Professor Leonard Susskind discusses spontaneous symmetry breaking and gauge invariance.
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Steiger University so the goal
of today he was to show you how hot particle like the photon particles like the photon King Gina maps can get a flank something called spontaneous symmetry breaking Our frolicking from something called the Higgs unarmed tool important ingredients that I haven't emphasized tonight north of and technical but also 1 of the more technical and the other 1 of them called spontaneous symmetry breaking and the other called gage invariance we have to go through and if we really want to seriously understand what's going on mathematical ideas restore we can't avoid mathematics as usual last time I think I discussed a little bit an example of spontaneous symmetry breaking namely are very simple model of magnets with magnets magnets individual magnets could point up or down the energy is such that energy is Lolande that parallel use whatever sway and larger peril there is no fundamental asymmetry between up and down through neighboring up little magnets have the same energy is 2 neighboring down 1 up and 1 down the same images 1 Barham or not was no fundamental difference between pop up and down but nevertheless because the spins aura ended because it's energetically favorable for them to be parallel then all have to do is go very far away with supports imagine that were talking about the ground state state of lowest energy magnet has been frozen all its energies fatah which state is at hand given think of the possibilities occur not the Kobe down 1 up and warned that as will cost more energy but which is it all up borrowing all that well the answer is determined by one's been very very far away very far away and you freeze 1 of them just once been very very far away the effect will be that its neighbors In order to save energy for parallel and so it's 4 of the 6 neighbors around it and dimensions based and 3 dimensions 0 6 neighbors on square Latimer cubicle artists sold and well tell its neighbors looked you wanna save energy point the same when I point but then the 6 neighbors have a budget neighbors connected to them and so forth and so on and the effect will be heard that the entire lattice if we really want to be in the ground state will all point in 1 direction and half sold there is a symmetry the symmetry is spontaneously broken let's let's just distinguish that and compare it with while call explicit symmetry breaking explicit symmetry breaking the different imagined instead a slightly different systems processing system but 1 ingredient is a magnetic field and the magnetic field points but In that case we just an individual Spain 1 spend that time forget its neighbors will look prefer to point along with direction of the magnetic field prefer a wasn't amazing energy will be lower when point along the magnetic field this is an explicit breaking there's something in the system which is singled them up as different and down we expression for the energy is no longer symmetric if all I had was once in the energy would be different if for up and down and that case it really doesn't matter what you do very very far away doesn't make much difference on is enormous savings of energy if all of this beans appointing along the magnetic field and you try to take a substantial number of them and flip them in the wrong direction the energy cost would be enormous so be proportional a poll numbers spins it you try to change so been far away holding it in some direction is not going to have much influence far away white because they have an influence far away he and the lines things up with it cost and energy you would be enormous so that's explicit symmetry breaking up the bible of Jackson gave up we can has a magnetic maiming big moments at no charge magnetic moments In but elected neutron which honors a magnetic moment but no less all the things that I would've thought they had a good fortune and in the real world with proper get up here now if you were dead because if you had doors fans lined up in 1 way no raw and in some way and the UK spontaneous case New Canaan new football the spins and held down yes away he would go out there what's that church new lifestyle that now that brought their freedom you have all the way all of this would have thought about waves or moved waves Earl of
Rio's Union waves of reorientation yellow talked extensively about waves a real orientation off but OK so question how would you know you find 1 of these magnetic systems that real magnet it's just a mathematical model was spins which could Libya but you find the sample and all pointing out that you will find out whether it's spontaneous symmetry breaking or what's Co exports of record-breaking what's a difference what the other phenomena which are characteristic of 1 versus the other yes in the case of the simple magnets which need to be up there special phenomena associated with spontaneous symmetry breaking his existence of domain walls was a domain wall here had a big sample is a big step for him end you'll have the site of the experiment is a finite size sample but very big city of Goma the boundaries of the left memory and you make all the spins are you just freeze them to hold on to them perhaps by putting me the spins on the boundary over here this is not time to space directions just for simplicity you freeze although the spend upward Over here what you expect a happened well of course the system will save energy by filling itself up will go up spends payrolls not because they think around here which is favored but just because each neighbor influences it's all its neighbors and says the save energy of Iraq body of color difference between that and explicit symmetry breaking which is which is the presence of a magnetic field in the bulk of the sample here orienting the spends up well 1 experiment you could do is illegal the side over here and place of the spins now then what do you expect in 2 cases took a city expect quite different behavior if there really was a preference explicit symmetry breaking said you save energy by being on the effect of what's going on here would not be very important to them my propagating in a little wages but it certainly would not flip a large macroscopic percentage of the spins because that just cost energy to turn them relative to the explicit magnetic field so frozen magnetic field pointing out the effect of turning man he only property a little ways and would be too costly and energy to turn over large macroscopic our fraction of them on the other hand it up is just as good as boundaries tours if there is nothing external to system saying up as preferred and effect is quite different instead they favorable energy configuration will be 1 in which have the a sample on 1 side is bound and half a sample on the other side is up now the Split might not be at the center that's not so important but this would be a perfectly good configurations where half of them would be up half of them will be done you cannot find the state lawyer energy there is no state of lower energy and have been sharpened have down but this will be called the domain wore separation between the 2 types of behaviors will recall the domain of this simple example the phenomena that would characterize spontaneous symmetry breaking would be the existence of domain walls were on either side of the domain wall there would be basically identical things except reflected buyer 180 degrees from walls would be the existence of domain walls would be the signal the signature of a spontaneous symmetry account of this case now on weekend think of our field theory which has very similar behaviors it a field theory we have any feel whatsis qualified as green OK was great worker the Saigon but black is better and strike with the black areas I feel fine the simplicity now find is a real value field only 1 component fury can take on any value any real value from plus to minus infinity and this AD a dynamics this field which is controlled by a on let's ride Lagrangian for relativistic field and the usual expression for the Connecticut energy of a relativistic field is differentiate Friday with respect to the direction New square that value square with summer signings plus signs and minus signs of a characteristic of relativity and that we describe it putting up law indices but basically this is just the sums actually differences of squares of the time derivative squared the difference between the time derivative
squared and the space derivatives square missing here is 1 half 5 . squared minus the derivative with respect that X of 5 squared same thing for y so for me and just
represent that I did you find me from getting represent the most simply but represented by means of simple notation let's just call it 1 half and the 1 here it is completely on conventional meaning to say it's a convention that you could change it would make any difference of physics on simple they should just be right defied square and means by square my serve radio the Fire scored a layup command another terms which is a potential energy a potential energy are minus V
of 5 minus a potential which depends on fire potential energy it doesn't depend on the gradient of body the kind the river a space to religious depends on by itself and imagined that theophylline look something like that completely symmetric exactly symmetric with respect to changing plaster minus acts of you would be fine equals 0 right away he said but scoreless . fi equals and here Friday equals minus plus 7 miners and plaster the Minnesota completely synaptic this function assume it is completely symmetric with respect of flipping about these vertical axis and under that case it will once again beat tool ground states field through states of lowest energy incidentally the energy has a plus sign of 5 Lagrangian as a minus the energy has plus signed it will be 2 states in 1 configuration the field will sit at this point he everywhere space so everywhere in our sample warned basis flight could be minus F no I wine kit that make it John but obviously the images lowest ever fight is at this value of this value but White jump from here he air without a big cost in but now but there's a lot of positives opposing the field just went of the sharp boundary here fight equals f far-right equals minors everywhere in sample vii is equal to ease the plus a minus F and so the potential energy here is minimized but what about the derivative energies here what about the terms of the energy proportional go but save a derivative of FIA with respect that acts squared misdirection could be ex there is a huge jump or a sudden jump in fire going from here to here a sudden jump means large derivatives in fact if the jump was sudden and I mean really concentrated the derivative of fire would be very big square root would be even bigger and you would pay a large price an energy for a sudden jump on the other hand you pay no price of 5 is either equal to ask or minus F everywhere this is very much like the magnets lined up by that firm and again is a case of spontaneous symmetry breaking they have different possibility where they would be spontaneous symmetry breaking would be if the potential energy instead look like this then is ground state would just be fined equals 0 while Flagler 0 it is the same as fight was minus 0 and there would be no bias Our you would not observe anything in here it is fight with 0 everywhere and there would be no breaking of the symmetry of left to right of plus far-right combined this flight OK so that that would be the case other a spontaneous this would be the spontaneously broken case this would be the unbroken case in 1 case the ground-state simply has the field equal to 0 Amelia case they wrote to ground states identical ground war were physically identical ground states and again In this case here if you insisted that the field was f on this morning F on the sent by going into the system and holding it down then there would be a transition somewhere in in the middle from minus after class that it would be costly in energy because of these gradient terms but still you have no choice to said plus on this and minus understand and there would be a domain wall down them down center Yao a little influence on 2nd engineers thing only news that's not but what they're looking this when area up enough there because you at secondary nearest neighbor but they will be seen 1 the voters a 2nd nearest neighbor interactions also preferred power it wouldn't make much difference if you have a situation where the 1st nearest neighbors preferred Harrow and the 2nd nearest neighbors and that terror you might get into some kind of competition not that for what that the war no it refers to the idea that there's really no fundamental asymmetry there was a question of chants which way it goes or not chance that you can you could set off the different by a very very kind perturbations very tiny perturbations just 1 spin all day it is the kind of instability it's a kind of instability that wasn't stability instability means that a very very small perturbation on the system will change a lot so small in this case may and may need it is 1 been far away is being tampered with you are only think that's not much of a change in the system you go far you have a system which is millions of miles by millions of miles and that's loaded up with a huge number these spins and somebody off on how the jury turns 1 of them all what's the effect the effect because of a propagating waves which would come in and have a fixed time of course but which were cause that stopped all the energy out of the system which caused the
ground state to be realigned he is they need both at the depends on the particular of physical situation with a little bit more about it at higher temperatures where there's lots of random fluctuations in May Be in typically would be be that the random fluctuation would be such a bad the average there would be equal amounts of up-and-down random fluctuation may start the cooler you call it and take energy and take energy out of it which weighs in Senegal wealthy answer is advantage so a random thing they catch my form plus over here minus over here another patchwork might start performed the law the energy with Parra and then the patches would connect themselves together in the patches would try to go like each other and then until it's a matter of random chance which 1 will take over the whole system OK so and that's that's what's called spontaneous symmetry breaking now in this case here you could ask what would correspond to explicit symmetry breaking but yes they know it hurt mother 1 lawyer got back there but if the potential hand in the balance so that 1 was lower than that then this would be an explicit symmetry breaking and be ground state would sit here explicit symmetry making that means it's something asymmetric it if you like Norma gene itself or the potential energy just an explicit terms in the fundamental set up to begin with which favors 1 side of favors 1 configurations that of the other at explicit symmetry breaking in that case but they are very very big sample it will not be domain walls it would be very energetically unfavorable tab domain walls of domain wall was simply 1 minimize the number of wrongheaded wrong directed spin so wrong directed fields and simply move over and try to wipe out the wrong spin my happy completely do it but warned the domain wall will not be in the center because that will require too many spins to be in long state for many degrees of freedom to point space to be wrongly align OK so I think we have the basic idea more spontaneous symmetry breaking is but now we're and in this case the cases are discussed the symmetry which is broken is a discrete symmetry and spin case it was been up goes to spin down in this case left Costa right it's not continuous symmetry a continuous symmetry is 1 way you can interpolate between Neukom interpolate continuously in between our whatever configurations they are you want is a continuous symmetry will discuss it where a moment of discuss right now as you tool as you 3 whatever they're all continuous cemeteries where you couldn't Tupper Lake configurations by using the continuous generators of a group to go from 1 configuration on the magnetic Kate Real magnets real magnets often in the real world have continuous symmetry is the continuous symmetry would be not just new flat spare but you can't rotate so for example some examples there are magnets real magnets have continuous symmetry where the spins preferred to be aligned all right but it's not just up and down but anyways in between in that case the symmetry group is not just reflection but rotation into continuous group so we will wanted since were interested in things like this you to arrest you create a new 1 we want to understand what it would mean to spontaneously break a continuous symmetry under the duo 1 example tonight of both the notion of continuous symmetry breaking what the idea of Goldstone boson which replaces domain walls and how this can radically change its nature when the symmetry has to do with gage both arms has to do with gage bosons and morphs from spontaneous symmetry breaking into the Hanks phenomena in particular how figure takes phenomena gives particles like busy bows on NASA's our provides a mechanism for the masses but I would try to go through that mathematics 1st of all we're talking as I said about continuous cemeteries and I don't wanna do the general case some interesting for us at this point to do general case through a very specific case the specific case involved field which qualified again but is no more complex field instead of just existing on the real axis instead of having a value on the real axis has a complex value or another way of saying it is a can take on a value in the plane the two-dimensional plane was a plane the plane is the plane of the real value of fine and the imaginary value of 5 and complex field his fiery own class why can't find imaginary I want fight it's a field depends on position but knowledge of that field involves knowledge of the 2 independent components of the components are not components space this is just a field which is composed of 2 fields and grouped together the
form something called the complex field and the cost is also a canonical conjugated canonical conjures a complex conjugate 5 star which is just tickled fire Rio minus I fire imaginary IS a foggy is a thing which lives and the plane I need to to say the values that takes on are in the place and let's suppose let me in that case we were right the comedic energy here and not in this form but as the product of the derivative of the kind you 5 times for itself this is equivalent to the summer so turns 1 involving fire real the other involving fire magic the rewrite this if you like you could rewrite it as being you'll find a real squared plus these new fly imaginary square as is using the idea that a square of a complex elevate complex conjugate and complex numbers Condit song complex conjugate is just a product of Rio components of square the real components plus square imaginary components that's so it's right this way and back the half is normally not put for complex field now this little June has a cemetery and the cemetery is the U 1 symmetry what's right and you replaced everywhere fired by 5 and 5 equal e problem minors I made a times another words we multiplied the field at every point prime does not stand derivative here just redefinition of the field will we multiplied by the guy not that means on the complex plane here now we rotated by angle theta now it's this would here does however that the 3 were at the same time by Prime Star is equal to eat them minus I later times vise prime direct SR fiery stock scale there is a constant a moment but it is a constant constant with respect the space it could be any phase but was not allowed to vary from point-to-point or current time at the moment OK white why it is invariant under that because we eatery I will cancel out the derivative what's right down the derivative Of fight crime and just not right debugging NewWave destroyed derivative abstract derivative of fight crime is just eaten EIA played at times a derivative of 5 wiII because they constant by assumption likewise derivative of 5 start trying by Prime Star is Ito miners data derivative of 5 star that means the product of a D 5 times would be staffed by Prime is
exactly the same Peter I think cancel out and Sevilla Granzien is unchanged by rotating the fueled by a fate that call the symmetry of 1st symmetry means operations that you could do which don't change Lagrangian got a ride on the low grungy and which don't respect the cemetery but what else can we a head of the Lagrangian let's get rid these news just caught it a derivative of 5 times the rhythm of 5 star you could remember that some signs of their armed ukulele potential Seretse had a potential Ramsey and that always means subtracting 305 but I want to be a fight have the same symmetry in other words I wanted to maintain symmetry under these days rotation had I do that the answer is simple he requires that the potential as a function only of 5 star 5 awards only a function of fire real squared plus 5 imaginary squared said yet was it was er question job now wants clear that this will run Have the symmetry even with the potential I could break the symmetry also 2 ways of breaking the symmetry if I made the potential of function or some other combination with adjust fire real by itself without financial legendary that would break the symmetry OK what is this symmetry from the point of view of the picture over here it is the symmetry I rotating the fight playing at every point just take whatever the field is at any point in space and rotated here rotate Oldfield simultaneously and that is the symmetry that with talking about OK let's now focus on 2 different kinds of potential energy is 1 of them spontaneously breaks the symmetry of the dozen the situation is very very similar to Our food previous carries Russell as a simple example let's take a potential well that's draw was right but just take the potential for star 5 by stuff I wear I minimized 0 it's always positive except . 0 and so the potential is minimum right but point is obviously a symmetric point with respect to rotating the plane and saw potential like this where child draw offer you a minute of Orchard Road draw a potential like that is symmetric and there is no either explicit or spontaneous symmetry breaking associated with it the grants it is just fight equals 0 completely symmetric but now let's write another potential our astride a minus any times this plus P times five-star square this potential has the property that nearly origin incidentally near the origin is bigger than next revised small fight before goods like fight before by 2 a 4th as much smaller 5 square so near the origin of the potential energy it is not a minimum of 0 but maximum at 0 at that the deal doesn't want bid 0 field 1 0 because cost more energy for would the be 0 than to move away from 0 potential energy because of the minus sign he favors having find move away from 0 really it doesn't want moved too far from 0 if these positive for supposed these positive B is positive for large flight is bigger than this bigger you don't want to have watch field because this will get big and overwhelmed all art competition between these 2 will take place and the other will be of favorable value of staff I some bad we can drop picture struck picture there's a picture of a potential here a fight plane make fighter plane horizontal I'm vertically is potential energy In the 1st case the 1st case was plus and I didn't even write this year but we could was a potential look like it's a function only of five-star 5 that means it's Of function only above the distance from the origin and the potential would look like something like this it would be a paraboloid would be a paraboloid which would have complete rotational symmetry about 5 0 0 GMT energy is minimum here the potential energy is minimum fight was 0 and if we move away from Fico 0 there asymmetry which make sure that the had to energy doesn't depend on angle that's character of the 1st case where we just had a plus a five-star fine and the in fact we could put it the another term as a matter of knowing specific ingredient here Is the potential is upward and continues to grow upward so that the minimum of energy is ratified or 0 under those circumstances would say there's no symmetry breaking no spontaneous symmetry breaking Iranian let's go back to the case with minors he then the potential looks like start origin and goes negative untilled recorded turn becomes as large as a quadratic terror and that it turned up it's still symmetric about rotation it's a bit hard to draw it's a surface of revolution meaning to say it's symmetric with respect the rotation about the axis here and the minimum is not point but the minimum lies on a circle and lies on circle in the complex plane from this point of view Olivia minimum energy steady somewheres along a circle with a radius which is determined by the parameters here is simply minimizes functions as a fortune a five-star fine and you find out where that point let's call the radius F. that's a standard notation for instrument the minimum of the potential is not ready point along the track so along a track like theirs but track of radius death right now what Is the ground state of the system there many ground and all of the same energy they correspond to the field being at any point along the circle here everywhere was awake think about this it is to us imagine that the fuel a bunch of little arrow was not arrows in space arrows in this abstract if they want to be all lined along the same direction at I know that because it his variation in the field from 1 point to another in this tournament has a lot energy and replace the gradient turn the field once the state parallel In this internal abstract space everywhere is a space the field will want a light at the same angle but however is an ambiguity which direction the energy is the same along any direction as long as the field has magnet half words as long as stuff in the trachea or equivalently in this track over here then energies minimum could you say anything beyond that now that's it energy will be minimal somewhere everywhere so but it will cost energy cause the vary from point-to-point even if it stays in the trachea Libya cost and energy of having a change from point-to-point starts a datagram state is any 1 of these possible positions for the field but always still locked together again spontaneous symmetry breaking with this time the symmetry is a signatory of rotation of our here I'm now are domain more interesting questions this case are they domain walls so let's go back to the rebound an experiment that we might imagine to check whether of domain walls take big
say amp book of end 1 place in the sample weak but the field at this point already at promises call at any of the other and we put it be along here the field lies in a direction along here headlines indeed direction what you think the ground-state state of lowest energy subject to this constraint is that the field is you will not until it gets some point and then makes a rapid transition to the other value is not that's not preferred energy configuration the reason is that cost a good deal of energy because making the transition from a suddenly has the effect of making the gradient terms in the large a much cheaper thing to do is have the field gradually vary from year to year this was not an option in the case where the symmetry was just a reflection on symmetry but now have the option of going smoothly from beat so the field at 1 end of space is at a and gradually as we move across space varies staying in the trachea staying in the track so that it never cost so much potential energy I'm a slot that should possible ways to go there to possible so let's resolve that by not putting this point be put over here just to resolve that ambiguity but you're right there is an ambiguity about which way you go from 1 point to another if if this pointers over here not the extreme diametrically opposite point it will pay go this way and not go around the long way because the field gradient will be less yup was or he could get out of the way but given pair points a and B unless you have very singular situation where exactly diametrically opposed there will be a particular best state but it will be a state with smooth variation from 1 end to the other no words effectively Miss a year of field in this system for which it does not cost much energy the very smoothly from 1 place to another hobby energy costs and having a gradient that's all there is only the Energy Cost of having a gradient and nothing else a field with that character ratify itself but will discuss what that is a field of that character that there work is only only gradients energy Is basalis it corresponds was part we are now has a lengthy but a little bit of later the jurors plus the case so it is a 0 this this you can prove that the way to go from a to in the minimum energy is have this angle vary linearly from 1 side to another part young the and not going away from the ground just change the angle linearly now that a that's a steer you could prove vital heart so Batiste about her NDI April approved by year using go somewhere calculus of variations pipe minimization prove answers but if this is he always does have not killed that the potential Yukon unimagined an explicit symmetry breaking which would correspond to coping with potential little bit if you tilt did it 1 point would be preferred road trip to the other and it would be the ground OK let's so let's discuss From her point of view of wave equations it just wanna remind you he wanted wave equations Our a field with only in terms of its energy is massless wasn't mean that matchless it means that the quota associated with but also means if you imagined always here all wave of wave number can words away which looked likely be X or from 1 point to another just a single field now forget let's forget the complex from just a few which goes like people IKX odd then we would say that it has a momentum case has a momentum k k is related to a wave number as a momentum Kerry and if we energy Mansfield boast 0 sk goes to 0 that means that the quanta of that field have 0 energy when the momentum goes to 0 cases like momentum if the energy stored in the field with equals your oral vanishes and that's says there's no energy for a particle of very very low wavelength a very very long wavelength Excuse me very low wave number 0 5 and that would correspond the saying a particle at rest would have no energy particle arrest has no energy it's called a massless particle so if we energy level waved his Lauren lawyer as the way gets longer and longer wavelength yours varies more and more smoothly then call that a massless particle that's the case the mess was particle on Keyser let's let's look at various potentials let's start 1st of all potential which like that and squared over fighting squared off I start vise president on its clear that if you created a wave vary from place to place no matter how long that wavelength wars as long as it was variations field away from 5 equals 0 there's gonna be an appreciable cost of energy white because any place we feel that is not 0 particularly away like this is gonna be a lot of places where it's not Zero and those places of cost energy that is the case of a massive particle of case particle which energy no matter how long the wavelength of the particle is going on this is absent in a very very long wavelength has very small the only turns in the energy comes from the gradient terms and very very long wavelengths have small gradients the gradient a very very small when the Waverly small and that's the case of a massless particle so a term like and Square square is called amassed represents massive apart we discussed as before store were in detail again religious show how you could think about decomposing in this case here with field like stole lying on a circle here let's go through a little bit of mathematics and see if we can identify wiII there's a massless particle and what their missiles particle missiles particle incidentally just corresponds total way you where the fields In the lowest energy stays in the lowest energy then the long wavelength expectations that just means a few varies gradually from place to place only an angle and never creeps out of this track Bell have lawyers very low-energy will correspond to something less but he received some mathematics for on so let's take the potential to be a derivative of but sickly and Avila Granzien could be derivative of 5 times the derivative of 5 star miners leader of five-star 5 and that lets
represent the field in a way not by the real and legendary part of the field but what's right is equal to roll current Ito alpha offer is angle on the complex plane role it is e radial value of a field lowers the magnitude of the field and offers angle Roland enough hour also feels Rowan Alpha could vary from place to place and let's rewrite the Lagrangian in terms of role for the notice I'll from notice moving around on the trachea corresponds to keeping role constant and varying Alpha they could think of how fat as a field which whose value is on circle here if you like but it has to do with an angled role is a field which when you change it changes the potential energy and pushes you away from me here from the art from the minimum energy OK let's let's try right there were poor grungy in terms of Roh and offer very easy the derivative of vying with respect to any direction space space time derivative of 580 is equal to the derivative of Roma fans eatery I differentiating this factor leading this for a long and then other terms is lost
rubble times in the river the I offer What's that now I'm vast ownership role here careers later derivatives however roh thank you through a remote turns into the Arthur was rolled times a derivative of the eye offer and that's I rolled 5 times a derivative of Alpha all usually I offer let's factor Audi I often outside so that's the drew fire about derivative of five-star start multiply sort that out the river of 5 at exactly the same thing except with a minus sign over here the river Arroyo minors ii derivative of Alpha times people like I offer now I have is only multiplied these to multiply these through the cross turn goes away and we just get 2 terms 1 of them is just a derivative of royal squared Peter I offer cancels eaten minus I offer here so that it just dispense with that from the beginning because the I offer hours away and we get we get derivative of role squared and then we get a plus Rose squared times the derivative all squared and then finally V of was fi stuff Friday in terms of raw roh squared notice that the potential only depends on a roll and not Alpha if cemetery butternut say that about that very very low energy excitation very very low energy excitations very very warm Waverly configurations with a few very slowly from point of forward stays down the trachea No. 1 and No. 2 words very slowly we only have we don't have much energy just trying to make a very low energy expectations it's not gonna pay us throughout the change role very much will want to keep role right on the track he sold in some approximation of where we don't ever excite me most in up and away from the track here we just think of role was a constant Rosen vary from point to point and therefore we could say that rose to political role we because it an other rower actor minimum f so we could is by squared derivative of alpha squares and then the rest of it has only to do with role has to do with variations of roll from place to place climbing up and down as well for example and also potential energies that have to do with the value of role the potential energy is a safe paid don't get softens point that but still it's there this serve was required veterans just looks like the grungy and of a field With no potential of all were about to step toward we do this at numbers just number fixed doesn't vary from place to place we can absorb it into the definition redefinition of Alpha because they as s terms alpha-beta no Field Woodward called for a and this would just become a derivative of beta square that would be characteristic of a few would no maps so the F here that's that's unimportant as a cynical absorbed in here and just change the definition from Africa at times out for a constant and this is becomes they will grungy and from massless field with no potential energy at all was meaning that the meaning of that is if all you're allowed to do is very angle here then excitations of that type behave like massless fields behave like field whose quanta have no masses not really what our role role is an independent feels 2 independent feel that has a Lagrangian derivative of Rose squared an eternity of suppose we choose the ground state to be over here at this point which to choose
grants they've been in a direction that we can move until direction so we can move around the circle that's just exciting out there we already taking care of that that's just a massless excitations that costs no energy except for the gradient terms but we could also move radially outward what would happen if we started the fuel over here radially outward and there is a saying about orders that India any rotation here just oscillate about this point were oscillate about this point is very much the same way as the field oscillate about the origins of it was a five-star that corresponds to a Mets oscillations if you take the field every winners and displaced just oscillate back-and-forth that corresponds to a man who was real role here which could contain for example rose squared are contained a year return which are so it's about here that those correspond there were more romance it is it's an energy that's associated with Zero moment 0 momentum means that the field is homogeneous and doesn't vary from point to point 0 momentum means no variation all sources energy stored in the a way and now awaits hardly away energy stored configuration you more everybody simultaneously van it would urge rocks back and for that frequency the frequency of its motion is the Mets if you took the field everywhere displaced up on the side of the potential would stop oscillate everywhere would correspond tone on the other hand the if you displaced along the trachea nothing much happens it displays the everywhere along track you would do we know we story force the restoring force which gives a periodic oscillations motion that is what a massive fueled by oscillate so if you excited homogeneously and I'll take whatever you'll have a asymmetry and the symmetry are already on a continuous retreat in this case the continuous symmetry is a rapacious returned here and that's symmetry is spontaneously broken there is always the possibility our changing orientation of things very gradually from place to place if you have a continuous please and that case you have massless expectation mast was particles of particles called Goldstone but Goldstone Boaz Island is 80 particle or feel the Goldstone field is a field which is massless which has no restoring force by virtue of a symmetry by virtue of the fact that you moving the view of the direction the cost no energy and so when you have a spontaneously broken symmetry hero Goldstone was hero Goldstone boson Her Goldstone north of the river back in this case the Goldstone boson is Arvie quanta on the field Alpha which is just a motion this trachea that's a very important concept and physics a term it's very important to me in particle physics gives rise to a massless particles with which will see you are really on 1 of the assailants ended so also a very important concept and condensed matter physics and condensed matter physics for example there are magnets reflect a lineup can be if magnet where are where Yukon orient the magnets direction but symmetry on the rotation there or spin waves spin waves are gradually variations of the direction of spin from point to point gradual and for the same reason that I've described here very gradual variation of directions from place to place is not cost in energy cost almost no energy and no spin waves behave like massless particles server for minority idea of Goldstone most on is really important concept and few appearing condensed matter physics shop where else was appear kind prices history physics which is indeed a few Peria condensed matter physics In general relativity also appears there reward tonight
about along elastic strain is an exam uh with emotions the waves along a long string was just taken my mother string string theory just a year or longer rubber string from 1 place the next longer that's an example of Goldstone was approached on-field what would be the symmetry associated with against Iraq also set up a set up an example for you forget gravity grantees for your exposes poll over here another pall over here arrange around the with a string connected to it and centering around here the reason I've set up this way is Whittaker move the whole thing up and down right OK we we know that if we make is very very very long and there are nasalis waves which propagate up-and-down masses means the cost very little energy only gradient energy could you guess what the symmetry in this case the translation up and up-and-down right tires as imagined no gravitational field it costs no energy Comeaux this whole thing up or down the symmetry its translations of the translations symmetry up-and-down so 1st of all as a cemetery next you can imagine changing the orientation and orientation position gradually as you go from point to point is Goldstone boson idea where you'd do we imagined a little symmetry transformation but a gradual change from 1 point to another for that right now is a sort of like a very slow variation of the year of the vertical position so on overall motion up and down cause no energy very small variations across here will also caused very little energy and the waves go long wavelengths which propagate up-and-down a very very long string also Goldstone on the Goldstone arms of the symmetry of translation spin waves are Goldstone boson on rotational symmetry and this feels is Goldstone bows on of a U 1 symmetry awards the tree now polls show that true it was the thing was standing still there's no comment from but whatever volume on the York give some oscillation energy and I be careful be careful what you mean about momentum you speaking about vertical momentum now on vertical now could certainly make it have 0 vertical moment but what about horizontal momentum young well not as his waves as wages with weight numbers sorts actually if you have a standing wave would be a linear superposition a postponement of miners moment from Seoul why would be momentum in this case would not be the vertical momentum but the momentum along the way along a year along with VAXes the vertical direction here just a stand-in for some internal Our symmetry armed dead another example another example slinky pin them slinky a torrent of slick Mininera circuits Eds the you could move the whole slinky howler horizontally doesn't
cost any energy so the symmetry of horizontal just think of it as being infinitely long for all practical purposes but if you move it horizontally cost no energy services symmetry in this case horizontal more than you could imagine slowly varying from place to place how much you translated it that case would make Tiger over here style here but you do it slowly enough then the than they have in the way might cost you much energy slowly enough from ParcPlace to place a of the gradient of course too much energy so in this case the symmetry would be translation horizontally and her waves of striking also sound waves are also Goldstone bows arms they also correspond to translation of all the molecules and you have sample system with infant system of water molecules and you translate all the molecules it cost you know energy but if you translate them little differently over here that that only he that creates a very long wavelength pressure wave density where was also called stumbles on and they all have the character cost no energy when the wavelength goes to infinity when the momentum goes to 0 we call them matter source OK so we seen some connections Meryl between symmetries and mass was HMOs Army Corps there are also members on Goldstone I'm sorry it's apologizing to my friend Goldstone who was a friend of mine recalled engagement mistake sorry Jeffrey Geller anybody instantly fails a scores of what now stock about a totally different subject also connected with U 1 symmetry like there so it's not totally a disconnected let's talk about a gage invariant so I'm not sure we'll put the 2 concepts together tonight it may take until next week concept called
gage invariant also associated with this U 1 symmetry remember that you want symmetry symmetry under this kind of operation is what we identified earlier past lectures with electric charge conservation it doesn't have to be electric charge but let's think about electric charge with a wind-up with gage bosons with masks while the real photon doesn't have a mast but we could be thinking about some other gage bows on like bows on which does have amassed was so for the moment orders called the charge electric charge but it might not be electric charge it might be something of a different character and with thinking about something which is not really electric charge but we did identify the symmetry of this height With we conservation of the electric charge carried by 1 of the 5 field charges of various kinds are always carried by fields which are complex and the Conservation Law is associated with the symmetry symmetry is associated with you want transformation namely multiplication by a phase of it we talk about this a grain that previous time so going toward again but let's our butts a race squiggles here Eliezer South user began Ender Is it the symmetry that were interested in fire goes fight crime question must wait lastly required in order for it to be a cemetery must we require that leader is constant over space all we allowed to make transformations with made it can vary from place to place not baby always allowed to imagine transformations where are symmetry parameter they later it is not field now just symmetry parameter where is allowed to vary from place to place another lawyers will you rotate the phase of the feel differently in some places and others is that a symmetry so let's find out let's find out by doing such an operation and see whether changes will Lagrangian not that's a test using something might be a symmetry tryout by applying it to the fields and then sticker back into a grungy grand you change OK so his or transformation find time is equal the eagerly cleanup acts could be space time and space and part general space turn right running out of ink again Astra but good OK so here symmetry operation lets blips calculate the Lagrange in our calculate rabbits 1st calculate the more until around here were presented above the grungy squares of some derivatives and maybe some potential energy so let's start with derivatives derivative are arbitrary direction does not direction picket directions they kind and calculate derivative of fight crime His 5 prime well 1st of all a low contained eaten EIA played a kind a derivative of fire him on modified before the transformation takes place and don't contain a 2nd term 2nd will be far I 5 a derivative of fado with respect to acts times each of the iPad and fireworks factory I made Our overall multiplicative I notice 0 to it terms 1 of them is correctly figured Young 1 of them doesn't care 1 way or the other whether they'd varied with respect Exxon mark that was history here the other time actually involves the variation from point to point the OK now let's right will grungy and as we might think of it in terms of 5 private contains we know will fight crime were interested in the river 5 prying eyes derivative Of 5 star although it's right that what's right to the revival of the conjugate derivative Our 5 Prime Star but that's just River firestorm minors I Tri-Star derivatives trader with respect to x times each the minus side know well bill is multiplied and see if it's the same as is multiplying the original derivatives of fire five-star while easily answers no because there's symmetric terms he ability with they are just check 1st all the eatery I figures will cancel out when I multiply the river of 5 prime time's a derivative of 5 Prime Star at least these factors will cancel out another words the factors will cancel out if 5 beta doesn't depend on position if they did independent position easily cancel out so let's just erase them erase them I don't want to right a wrong equations solely mock boredom multiplied these together I'm going to get but this is all I was hoping to get there was hoping to get the original Lagrangian back and show that the Lagrangian is invariant but it's not theirs a couple of extra chairs 1st of all was turned on let's Stewart areas I find star minus 5 star Tiphani misses is not 0 5 did the river of five-star this suffice derivative fight is not 0 times favor body X that no I don't want that there and then finally visitors square term which is five-star five-star trying times derivative data with respect banks square it was a disaster area that's a disaster if we were hoping that this would be a cemetery it's not a cemetery be Newell Granzien when expressed in terms of the Lagrangian when expressed in terms of 5 prime is not the the same as the Lagrangian were expressed by what about the potential energy if the potential energy it is a function of only if I start side if I start fight will not change when we do worry symmetry transformations even if it depends on space so the potential energy Batson that but they can they make energy derivative terms in the energy and which I sit down we what the problem is the problem is simply associated With the fact that the derivative of 5 prime is not the the same as the river of fighting for the upper equation even if we ignore the seat of the data that does cancel even if we did not back the criminal fight prime contains this extra Turner's so we just don't have a cemetery well I can only do what I need to restore the symmetries of anything we can do Lagrangian well yes of course we just throw with these terms and Lagrangian not be very interesting over grungy Inwood potential has no interest in dynamics Whatever the answer is yes but to do so you have to introduce more fields and the additional fields are we gage bows on fields in this case who really talking about electromagnetism it would be the electromagnetic field let's introduced electromagnetic field incidentally a transformation like this which depends on position the a symmetry of a hoped-for symmetry transformation which depends on position is called a gage symmetry I do actually I do know where it came from birth a historical of glitch it doesn't have to
do with a tire gages about to start port where it came from the terms but that's called a gage symmetry gage symmetry implies that the symmetry program can be chosen vary from place to place their number they did not suffer field just a generalized symmetry parameter which is now allowed to vary from place to place but secular happening here is only make the phase of the field vary from place to place would picking up a little bit O'Grady and energy the gradients our own changing cars and picking up radiant energy that wasn't there before or OK let's grabs I'm not that like we had similar cast them be now is just make a configuration with the field where the phase of the field varies from place to place it costs energy that talked him that's all but what's happening yet but I've been very explicit about it and I've identified this as a villain the problem here this year right so the question is Kalie find some clever way to cancel out why should we want to do that love ultimately it's an experimental factor that that all of the interactions of nature are associated with gage symmetries like it or not only interactions but all the gage symmetries Polegate interaction photons busy bows arms W. bows on all are associated with symmetries not just of the kind we could rigidly everywhere do the symmetry transformation but symmetries will you graduate work in the manner a gradual arbitrary variation of the symmetry parameters from place to place OK so let's so let's see what the ingredients that you have to add you have that and collection a field in this case just 1 no field I should say for new fields 1 for each direction space you must enough for vector field and therefore for there is if this war electrodynamics would just be the vector potential the vector potential but for vector potential Out of which you build the electromagnetic field Rogers remind you quickly with the connection is our the electric field well 1st of all the kind component of a year's electrostatic potential its gradient space Is electric field the space components of former freeze and curl of that 3 vector Is the magnetic field from borrowing apparent standard that the potential out of which you build electromagnetic field tents composed of electric and magnetic fields will come back progress but let a in the symmetry average we now elsewhere in field and you're interested in the cemetery you'll always Our address have to address the question how this is no field transform when you do the symmetry operations another words if we don't symmetry operation that takes us from fight the fight crime is also something that we have to do with could take us from a they prize and the answer is yes but cost a justification for this it is the result in the end that you can keep gage invariance but what you do OK is it to it the derivative knew of data the transformation property of a is not some multiplication by a phase of vary but just having a gradient of is a former vector derivative of fate is also for vector that's the operation that that a brawl allow us to have a cemetery next we have to change the definition of low grungy and definition of Wenger is no definition of grungy and will be to replace ordinary derivatives by a new kind of derivative called the co-vary derivatives it's not that it's not the cold derivative general relativity it's a core wearing through gage theory the define it for real defined don't Bond fight call variant derivative of a charged scalar fear over for at 1st the call interim is defined the big the ordinary derivative we can take away these music we want wouldn't the that's right reverse plus disarm straight plus vii a new war crimes trial definition that's a definition of a coal variant derivative if you like it and just think of it as a replacement of derivative by derivative plus my hand any that may not be really doesn't mean anything by itself it's something which acts on a charge field this is definition a co-vary derivative same being analogous things would be new five-star it's a goal the being new five-star might lips minus I knew five-star given mind this wouldn't make any sense if I was a real valued field when not because of fire was a real valued field we wouldn't want to get something imaginary defines itself a complex field and adding something additional complex that's fine no problem OK this is the definition Of the colder in and they can make fans in Lagrangian forget that forget the potential terms they Connecticut terms of Lagrangian are just replaced by the square or you and these new five-star star Michelle you why that Skagens fly that now has the variants that was lost over here I was extra little trick of inventing an extra field giving it a transformation property no no thanks a Lagrangian which did not have this gage invariant and makes we don't have to work out a Lagrangian it's enough just to look at co-vary derivatives let's approved let's find the connection years body in 5 years for his forays by Prime but With let's calculate coal variant derivative of 5 what's calculator co-vary the river of 5 from what we want a shot Is that the co-vary derivative of 5 prime is essentially the same as physical variant derivative a fight that would then established that the the new look around which involves square views would really invariants alleged check out calculate 3 calculate the coal variant delivery 5 prime that's equal to the rest of our 1st areas just ordinary derivative 5 prime and there it is up the of fighting plus on fly a derivative of later With respect X I wonder if I ever signed roared from Greece arms may have assigned wrong here on to quick and fix signed by a change in the sign away I think over here condemn role change of attitude on OK now that's the 1st in the call variant derivatives or fight time just the derivative of 5 prime but then we have and plus
ii something times 5 prime for October here ate a prime is a prime occurs and calculating the call variants derivatives Of the prime field for while the fear a prime source put a prime in what is a prime time it is a new money this hour you say that do think at times sorry this is 5 prime a new minus being used all kinds 5 prime is a right leaders right up 5 such which right but at the and I must of left them out overall phase from yacht both of them have a factor eatery RIA data but but they look at healers this extra nasty turn which destroyed the invariants Of this expression but here it is exactly the opposite to you exactly the same term away sorry what turned coming from the transformation of the derivatives of failure and the other turned coming from the transformation of a not exactly the same light flight here we have high trying Italy data indeed of IDX brought their two-year 3 days later skip rid of all news I saw this Turner and Mr. and rural Turner Mr. our exactly the same cancel succeeded in canceling out of I'm playing off a transformation properties of the derivative with the transformation properties of AT and causing me extra turned to cancel so early fine we find that derivative called river a fight crime is exactly the same as the call variant derivative of fine except for this factory to realize here we have to put that back what about the cold area derivative of the complex conjugate fired Prime Star that's equal to the recall variant derivative 5 minus are very rare guess what when we multiplied by this cancel out and genes is gage invariant so this a field here was an invention of the Vice art soft forcibly make the Lagrangian gage invariant but we had to invent a transformation property of the gage field itself of the vector potential itself but next we're missing something in or around June were missing Oh yes of course we are missing the potential but said that in Britain Regan multiplied by this yet yes Friday were granted as being find a new five-star minus V of five-star this was gage invariant begin with freedom after modify that polio missing something rather important gage theory and the the this would be a very interesting gage theory without some dynamics for the vector potential itself perfected the venture was really the thing which out of which you create b electric and magnetic fields of can anybody know what Lagrangian for the electromagnetic field is annuity energy for the electromagnetic field these Clippers be square about a Lagrangian look at eastward miners be squared is good manners is good but it can also be represented in terms of the field and so tried in terms of the field concerned the field cancer as close as you know I just remind you of this new new isn't an isometric object it's derivative and newest direction of 18 known minus the derivative In the newest direction of a new so to kind of four-dimensional curled its it had a diagonal components of 0 0 0 0 0 1 actors 0 8 0 times 0 0 9 0 8 0 0 all of the diagonal elements of 0 it's Dyson the new no interchange and idiots components will 1 space in 1 time to worry the component of new no warm space at 1 time electric field it 1 space higher component and that determines the direction of electromagnetic field what about the terms with them our space-based components the magnetic field for example F X Y and Z would be Z B is the magnetic field so what cycle so this is the electromagnetic field concerned and the scanner dynamics of Maxwell orderly Maxwell by Memorex is determined by a look around you and mg is just as is just as New Moon New raising the indices is just a couple assigned changes here and there around he squared minus be squared the is square be square but the important thing is that the Lagrangian for the electromagnetic field is involves nothing but as you know what's also have you is s new gage invariant which jacket to find out we have calculated f minnow prime well edges tickled to Denio a prime no miners Dino a prime meal at the ballot to remember race with a prime minister we did a prime just quarter a figure was minus the derivative our faith right so it's a targeted this is going to be derivative knew of known minus derivative knew of a derivative of of data has just been fur a pioneer aII and then derivative respect Miller said what's gonna happen tonight but Mr. subtract derivative norm of a new home and there's a plus Deno new welfare OK but did you fate is the same as being newbie new faded doesn't matter which or you differentiated misses just the fact that there are the order of differentiation partial differentiation material was what we find we find it as prime is the same exact is at so the electromagnetic field cancer is gage invariant that means that we can construct a completely gage invariant dynamics are of electromagnetic field coupled of field whose quanta are charged particles is is this is a field whose quanta are charged particles What is it into deal if ropes is Mr. Kiepper Rice's theme your only these covariance derivatives not ordinary lives a potential Arab plus and squared of square meaning if you knew the appropriate thing he squared square Thyssen numerical factors which a conventional is usually a quarter and front of this but that that's not port the important thing is what
terms here are gage invariant so we've discovered the fact that we can promote symmetry like you 1 year tho a local cemetery a local symmetry means a weakened locally vary the phase weakened locally do the operation whatever it is from point to point and still have the symmetry but it's always at the cost of an extra field an extra vector potential field and that vector potential field can also be assembled into a Maxwell Our Lagrangian that would determine Maxwell's equations Maxwell's equations coupled to a charge carrying that with this OK what can we not bad really tell you if we can get and still be gage invariant 1st of all we can and potentials which are functions other than functions of five-star fight if they were functions of just real part of 5 that would kill the symmetry but as far as modifying electromagnetic field while we can in the might but scored and squared over tow times U.N remember the man's turned from HBO's on is typically an end squared terms a square of the field a man's turned a photon would be a termite this if we entered this Sharon the Maxwell theory here will be aiding amassed 4 times however it's forbidden if we want a theory to be gage invariant This is not gage invariant this is not the same not equal fans square or till 8 prime new Prime this is forbidden by gage invariant is no way you I can give the photon a match that took his me means photon is necessarily restless if you believe in gage invariant and every physicist bullies engage variants for a wide variety of reasons including mathematical consistency with theory really breaks down badly we try to do it that's a photon can't have mail or can Ted characters in the 0 device year by which taking their Lagrangian and just as it is is a any trick that we can do to give the photon wraps McCurry right now it visible it next week if fully spontaneously break U 1 symmetry by making the potential here have 1 of these were shaped by mayors field Lexus over here that will mean that you wonder symmetry is spontaneously broken by explicitly spontaneous symmetry breaking that took is this the Granzien is gage invariant spontaneous symmetry breaking will have the effect of giving the photon and it will also miraculously removed Goldstone bills on so the effect of spontaneous symmetry breaking is twofold removes the Goldstone bows on that was associated with slow variations of far-right from 1 point faces a fine of 1 hand and the same kind gives a man's of potential figures a amassed of photon photon real photon doesn't have a mail blows on that this is called wouldn't work out next worker worker to next time and just see how spontaneous symmetry breaking gives rise to a mess for the gage Mostar and that's called the Higgs phenomena and this would be an example of how the Higgs phenomena gives may ask for particle which mathematically you're not allowed to just put in a mess here that's a that's this signature of basic idea that big phenomena that may be deep mathematical reasons why you cannot just by he and give a particle or manners but spontaneous symmetry breaking stuff but all I they say it but I us now there is also career while we said that only what they what I'm telling you it is that if you have some free 1st state you can create a Lagrangian witches agent 2nd statement coupling between charge carrying fields of electromagnetic potential would become inconsistent or engage around like Cartwright demonstrate that really does become mathematically inconsistent gage invariant bond so why should you believe what was it was a question the question is not really answer but you can construct a gage invariant Lagrangian result is a nice consistent mathematical structure each if you try to do anything else you you'll wind-up with theory which quantum mechanics which will have some terrible things will have negative probabilities in it even worse things and are a barometer go through that gage invariant is extremely fundamental principles in theoretical physics will be a like amount and drop purposes if we just have to accept that we can also accept that as AP empirical facts that all of the particles namely photons e bows on W. Parsons reminders bows on our nature seems to work chosen there were granted seems to be exactly gay that's a good thing because we don't want negative probabilities we are not that passengers division got plan is that we had the animal field in order to implement the Serb right now we're not chief it will rock yes we're verse right arm what we had to get field To do it without the field we could not keep that is this is much deeper mathematics you haven't the fiber bundles all kinds of all kinds of tourist reprimand 18 is technically order mathematician court connections as all sorts of interesting which were you and after giving a hockey club if if if if a week before the start of the 3rd Court storm Boswell was a feature of the theory where Al vector potentials and without gage invariant or the Goldstone boson was the energy of the Goldstone bombs was exactly the energy of making these various variable phases but now we find that a way of making the variable phases with a cost of energy White because the Lagrangian is invariant undergoes the gradual Chase changes of was I mean that means there's no cost energy in this gradual change of fair use no Goldstone but will do that next time will do next time will see that this Lagrangian does not have Goldstone boson islands and it does have an e-mail yeah is that what this book I'm a little made him if he has a list of a this I don't know our but I would not be a surprised that somebody has found a gage theory rivers I don't know of any connection our I do know that there are connections with the emotions says not at all waters there is a lot would be polled were surprised rewards correct remember our world for more
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