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New Revolutions in Particle Physics: Basic Concepts | Lecture 6

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on to the Dirac equation or let's move backed of Dirac equation I gave you the last time the simplest example of the Dirac equation and how and Dirac equation is always in equation prefer Barbosa was not satisfied Dirac equation acetify something else will come to call decline Gordon equation for Maxwell's equations after this simple equation now describes particles which move outlined always tell the right now that the left and in fact went push will discover all we had discovered that describes both particles and in particles we could think of them as electrons but only moving to a rights only go over again briefly before I just write a function black warned but all the Ito IEK X minus some but it yes KB can only go up positive well with such a repository look to be anything a moment Garth remember the velocity of the strictly speaking media the phase velocity of aware of the Phaser loss of away is just Katie overall maker if or maybe it is a linear function of K that it's the same as the good work the ratio of capable may get is universal Katie then phase in group velocity the same but many cases this is a group velocity list just call regroup and when you're right there so you may think I mean the
absolute value of
k ee taken be positive cake to be negative Omega could be positive or negative negative but I really needed K overall may go With it's appropriate sorry if is positive and a case study of is negative and Omega posit this describes an object whose velocity is total left negative if both are positive and it's a velocity moving to the right again k omega have seems signed it's a wave moving to the right of Kano may have the opposite starring in his ratios negative find no problem with it it just means away most of the left the Loretta Ronk I'm sorry it's so they write so may go case a good enough of of assigned the same urban scale maker of lawmaker overtake OK now let's try and a very very simple equation for the motion of a particle on line Bolinas ex did we have time then it is these signed by X plus Deeside ID equals 0 this describes waves the fact describes waves of this form here could you tell whether left right you on the right hand trying to direct your right OK so let's let's check designed by the X will just pulled down a factor of K this for the form of this will form of side we can use this equation could tell us what the relation between Katie make it signed by DX taking derivative respect X pulls them Katie taking in the river respected pulls down From my I O Omega or electors multiplies whether the original function he and if we set equal does it just says that I can is equal maker and that's obviously a wave that most of the right were set at 5 most of the ride but in general there are 2 possibilities for a given ground Our there are 2 possibilities we can plot this relationship Omega vs. K acts a straight line I'm actually left out the speed of light the speed of light going this equation that goal hero of those ago here now pharmacy hero of seek is like X will be 1 was CDs Tobias here now will succeed equal Glamorgan convenience than Omega is proportional to carry and you can see and of course Omega Boras also recalled a 0 . 1 0 at a virus eagled the 1 that make it is also the energy of individual quantum self our here and this branch OK we have positive energy humanist French we have negative energy negative energy electron not a good thing because any electron which has positive energy could proceed down this line here bomb bomb bomb bomb bomb for them the stairs by continuously emitting photons of positive energy our and there are no electrons would be stable in fact the world would not be stable it would afford fill up with more and more and more negative energy electron but certainly not electron were boosted by sewer we don't we fill up the Dirac sea they're actually use the fact that you can put electrons in the state with stabilize everything against this but from falling down stairs process by just saying all negative energy electron states are filled filled to begin where that is the lowest energy state if you want the lower the energy step the energy of the system maximally put as many negative energy objects into it is you combine just keep piling in negative energy objects in that the urging so filling this up completely gives you the Dirac city and it gives you where ground state which will call back it it got close distorted and it yet the moment the with discrete it would be like a staircase as continuous just like them continuously emitting photons you girls but once you fill up the Dirac see the negative energy then a positive energy electrons has no place to go well before then is but Kinkel indefinitely the fall down perhaps in some photons our but can't go far are OK South that's incidentally you could see why it's important that equation like this that it had better be an equation for me OK now we removal particle from the negative energy but struts over here a year all the energy levels for making equals 0 0 energy positive Omega negative lawmaker with fellow these are so these are all filled out can't put any more in there it is a and
B. That's a vacuum and now want to put an electronic put a positive energy Electronic that's why we can put a negative image electronic bowl we can do is move an electron from negative energy city and wouldn't the positive energy see doesn't have to be symmetrically located we can take electron from here and anywhere is over here and that creates 2 objects 1 of them is a hole in the Dirac a hole in the Dirac has now the Dirac sea is always associated with these particles of negative Katie and negative or major what happens if I removal of particle of negative Katie Moors negative momentum to create a hole was a whole have but will forget put charge yard a positive charge or suburb in terms of momentum we removed a particle of negative momentum so we have created an object with an extra excess relative to the vacuum relative to the vacuum it has 1 positive unit of momentum relative to the vacuum when we remote particles from here and put here we create something positive energy and positive momentum in fact only take up articles from here and here we create 2 things we create or hold with positive momentum and positive energy and electron with parts of our momentum parser energy so boss the whole and the particles move to the right motor right they both have positive momentum and positive energy and that's 0 OK that's the fact that this equation here positive charges fully in that particles negative charges is just a definition of course negative charges for the particles positive charges for the and that particles everybody has positive momentum and everybody has positive energy all real objects in the theory OK that the simple Dirac equation sigh is built up our of creation and annihilation operators the standard already what had planned to reply something then get Long Point heads came as you can see I'm not actually side of what it does is afforded by the river of giggles Katie a reggae post her record Real think probably won't confused confused by the fact that three-act sees except that didn't Tilak season occurs OK now there's a little bit weird half particles which can only go to the right by really consistent but has another weirdness also vacuum as a mob property that full of momentum which negative momentum the whole thing must have an enormous momentum shooting off the left and don't cares if that's what's called a vacuum which just find it have 0 momentum but still it is a little bit on particles can only go to the right and that particles can only go the right nobody knows laugh but nevertheless the vacuum is filled with these negative momentum is memory symmetric between left and right for thing Yuka may ask can you what can you do the out the words to make it more symmetric was left to right the answer is you'll have to introduce 2 kinds of electrons right movers and left movers these right movers so that indicated by playing are he yeah right off right and let's In another field describing the left moving particles less lonely particles will describe antiquities it'll be described by an equation which is similar except that will want Omega to be mined this case only have to do is she the sign of 1 of these 2 turn was which 1 so if we rewrite Equation let let Lemieux I write this here sigh right sigh arrived by sea it is equal minors sorry right by he acts as a writer moving waved left molding waved on another field completely separate field different creation operators different annihilation operators describes a different kind of particle namely a left more over physical class desire right by the X. if we work out the relation between Omega Katie in this case that some may go because minus 1 Ms. case was Omega overcame people's plus 1 thought left Is that right overseas a lot more the only difference is the sign of this term here now they have a bunch of left moving particles with positive energy other was another branchia it is our right more is our left movers right Morris and left more than another branch with positive energy but also another Brentwood negative energy what do we do it to stabilize the last move electrons so that they don't fall down the stairs we fill the seat of left moving electron negative energy sea of a more electron now have as many left moving as right moving particles in the city and solid balanced it's a symmetric with respect left and right vacuum warranty space what we think of this empty space no longer has a huge momentum in 1 direction has been canceled out and we can now have left moving alright moving particles or left moving and write moving holes the White House Council of our on white wine is yet there was is some of which the just so-called filled states a you Coffill negative energy states and these states of Poles say a filet negative energy states is an empty and that particles state
leaders real label renamed and now we say this is a theory Our particles and that particles both of which could move both left and right and we never talk about negative energy again but said but there I find it when I 1st learned think that's where the because we for 1st learned it I found there ever comforting to realize that yes this whole thing it makes sense and you could prove to yourself that it makes sense just by saying that the kind of like a metal you fill up all the states and holes and particles and from metal was consistent while this is as consistent as that storm on what the massive these particles the man's source why they mess less because they move with the speed of light rain or come back to the moment and try and make massive refer make a mess of a moment but just as a matter of notation let me teach you a notation which is important in subject let's take sigh right and so I laughed and put them together and make a column that they're out of that now this is purely notation no physics and what it's our occasional trek which will use over and over again and let's make a column vector Alabama and then rewrite this equation in the following way I well . means time derivative but tough resorts call size is now just side but it's a column records neither left nor right but contains both of them now let's take these 2 equations and combined them together into a single 8 equation for citing arrived for the column vector site I'll write down what it is no just look at what it says Is that TV our society is equal the something that recall Alpha games decided by DX firm minus shine in there by definition of Robert definition was Alpha a the that good work the thought said sigh by Alpha decide by DX what is out of is a matrix is a matrix or lose value what is 1 for right and minus 1 for lack case or what matrix as it offers the Matrix 1 minus 1 0 0 another words write it I love to write it in not combat the notations and matrix notation it just the but this it Alpha which is 1 minus 1 0 0 these side right by X. decide left by acts but as a minus sign should be a minus sign in here like that that I'm so this is just a fancy trek for writing to separate linear equations like that and putting them together into a year matrix notation that's very convenient but hasn't done any no physics now so far the 2 particles are both the left movers in the right moves on vessel if they move along Rosiak area taxes with From the speed of light make the massive can we enter these particles are his equations something which makes the particles have met about important thing after electrons the mess they do move with speed of light so that we have to remember what the connection between energy and momentum is for massive particles From massless particle energy is just proportional to momentum Omega equals case about fur a massive particles Iran away he young what's the relationship for a particle are finite positive you're pregnant mass Omega vs. case omega squared equals Romero not only that that's nonrelativistic we want relativistic now really doing it's obvious we must be doing relativistic physics because we're doing something which is approaching the speed of light nonrelativistic is a special case of relativistic when things movers slowly back bullets formula for a relativistic particle Maksim square root our case squared for Simms squared mega Eagle Square root of K squared plus and Square weenie equation everyone described massive electrons where the relation between frequency and wave number is going to be that's when it is equal to 0 0 it just says Omega is Katie lawmakers might escape on makers Kaori makers might we have amassed that's the case but now it won't the case with the Mets how we do that not not obvious it's not obvious but the howitzer with play with Bobette but you're right this equation 1st in a year and need forum which will be convenient for for us when you take a of an you hit it would be by deep-sea what it does is it just give your minus I'll make and illegal but Ciara here we like that of the I remember this just means Debye IDT outside gives you a factor minus 0 I'll make it that according to this equation is minus Alpha now wears a scare DXK minus IKB canceling out be Andres and minus signs it says is aren't equation that all make it is equal to alpha case now this is too short and for this relationship now there is not and the reason is because Deeb IDT gives you might sign plus I Omega Debye DXQ humor in this case so that cancels emeritus this is just a short he and trick furry rewriting Rios says when you apply these objects the plane wave you'll get you get the relationship that all make OK all my skate as the South but is it's just the instruction buddy upper components here it's 1 for the lower components of minus 1 at
that that's a brief shorthand for for these 2 equations that made it is equal pay for all right movers on May gives you a minus Khalifa left over a belief that it for flee this is only if we ask how fast a particle goes with this relationship here what we want is computer you'll maybe decay that no that's the group velocity and that's something like carrying over square king square % square and that thousands of velocity of a particle or wave laws proposal puts momentum divided by the case could force square was built in to how fast the waves moved OK let's try a let's just be a little were clever end trying to correct this equation Omega is equal outlet Katie a much but yeah now that the no get out make it is really a toe by 2 matrix strictly speaking through it to buy the matrix of Seoul and uh and Keyes a toe by 2 matrix proportional to the EU but Tom but as I said were really stands for resist that's a trick from rewriting this in an efficient form I don't have rewrite is derivatives of return Merry work with the thinkable may get as ID by P and is minus ID by BX unimagined the whole thing is operating on the wave of services was shorthand ability and something to this now something with a mask all right now man's plant something plans while I either know maybe another the egg Beta Beta but in order made his point really know what it is at this point presumably it's also a matrix and making Eagles Alpha Cape was M Percent beta and let's required of whatever alpha and beta or are let's requires that all major automakers squared His eagle took square Plesant squares that place that as a requirement on the matrices Alpha and Beta gave away what they is that face a lot square that's Omega square is equal but my of Beta and plans Alfred Casey Close made em where we get we get alpha squared case squared at good we won a case quarterback we want plus data squared and squared at everyone and squared but now we get some garbage we get Alpha times Bader Kate turns is how further Beta Kate plans and and beta offer Kate funds M. images numbers of course they just the adjuster momentum and a massive apart from sewing get Alpha Beta was made offer our fans K and this is good we want that tells us immediately whether house that is whether whatever else they can mathematical object it is alpha squared needs to be 1 we also want em square there so beta square root of the beta it needs to be worn but we don't want this Mr. has no place in this formula store we want we want whatever alpha and beta are that they have the algebraic properties that out Beta opposed Alpha is equal to 0 this would be impossible without mandate is being numbers you could certainly satisfied itself was quoted as equal one-dimensional problem beta squared equal 1 that Japan also solved but how to solve this together with this if they would just numbers it would require you abated to be 0 but this is the neither from elevator recalled 0 million if matrices quite possible so let's begin without squared equals 1 we've already specified what our borders the square of this matrix is indeed 1 the square of this matrix is just 1 1 whether baby is it also had better be a matrix our Carlos Queiroz 1 but it can't be the same as our 1st was the same as our than Alpha Beta plus Beta offer would just be twice square and elegiac be tool so it can be data can be out it also has to have a square which is equal to 1 but there are more matrices around here is an example Beta news equal rub beta squared the fate of data is equal 2 0 1 0 1 0 let's check back at square at every square exports Davis Square is equal to 0 0 1 1 0 pain in the upper left-hand corner we have 0 0 plus 1 times 1 good that's good and the lower it it's trivial Yukon work that out yourself the square of this matrix is indeed 1 good but more and more important but equally important is Alpha Beta times data Alfred C. quotas your own words of Alpha Beta His Beta Alpha let's jacket wants just do these things wants in on a blackboard and then after that you believe me OK beta is 1 0 0 minus 1 that Alpha Beta 0 1 1 0 this is Alpha terms beta offer data when multiplied matrices wrote column I get 0 is when times as well as 1 when that he attends is is minus 1 this time there's 0 that's how 4 times beta about data pans out forward stood me up a Sevilla 0 1 1 0 that's beta after is 1 0 0 minus 1 line that could the corner elements for left corner element a bag over the 2 0 0 0 times 1 plus 1 0 good that's 0 but let's look at the upper right hand corner of 0 times 0 1 NYTimes minus 1 so whatever it is we get a minus 1 here 0 0 end 81 the whole warned them here so Alpha Beta it is negative of Beta Alpha misses the negative events so through this matrix here square His 1 ended in that I commute without proper Bader postpaid offers equal 0 it's another way of saying that as use these things and Beckham commute
there are other possible choices Oconto another 1 Wyo OK so we've actually found the solution of of we chose the Alpha and Beta the 2 matrices the written here and then re right the Dirac equation what's right about specifically are in terms of its components right I was really quite Asians I really don't mean real as opposed to our imaginary just explicitly very very explicitly writing down the equations to see what they mean when chewing ignore Bader Canal Plus OK so let's so here we are in a column notation says that sigh right sigh left here omega equals Alpha case omega is always the same as I transit time derivatives so I kind of time derivative it is equal to allow minus I had believed times the X derivative of sorry for her 1st ride only X derivative sigh left column not missing a minus sign here right where that minus I've come from it's Alpha MOL favors the matrix warned minus 1 0 0 was minus 1 here it gives you a negative sign Darren here but now it's rightly other terms plus bait at times sigh itself at times sigh right sigh 1 as they now that it is the matrix 0 1 1 0 what does is matrix dual where over if flaps em so let's do it sigh left right now we see what the Koreas equations are eyeing sire right Dr. Zeigler minors Oh I D X sire right clusters of mass M plus and so I left it throws sigh left into the equation was right likewise heist I left God is equal plus ii VX so I left plus and sigh right that explicitly what the equations Sary pariah any number of former across arrived at a body so I got is equal to heII minus Hi how far he signed by VAX plus an Bader society in this former must remember that these columns and days of maker seized in this former it's just absolutely explicit our what the equations are and Bader interchanges baby interchanges sigh left and right so these equations become coupled equations the left and the right move was become couples that's earned very strange in the nobody would've expected this before Dirac that that somehow where amassed from electronic is visited turned the equation for sigh right which is proportional force I left and termed Versailles which is proportional Messiah right off by but more than that on this these equations automatically concern momentum because the translation their right but the particle now has a match this equation electoral now has and messages and so that's a remarkably beautiful and simple equation that is a beautiful and simple equation but he lets not brought to it is a beautiful and simple equation all of that Is it Lorenzen Marino's solution should not purists invariant solutions should be learnt area the meaning of that as a given solution for example might describe an electron moving with a certain momentum transformer at the moving with a different momentum so it should not be in where warrants transformation should do is a ship carrying a wave functions describing a particle of a given momentum until the warrants transform momentum and yes that's since reserve when I drove it but it's satisfied the 1 key test filmmaker square a square cent squared and that is a Lorentz invariant relations Omega square case square Square and yes is these equations are Laurent OK let's let's go to the limit just the the look at a little more clearly Let's go to the limit narrow which we could not do in this case without the minister let's go to limit would elect from a standing still you can't make a massless particle standstill it always move the speed of light for once a particle has amassed you can allow Newcomb study words at racks arrest means its momentum is equal to 0 9 2 at the moment the Beagle 2 0 all we can just a where is our equations are equation was all maker was equal to look out for I was a minus after classmate am Westchester are manipulated at this level 1st and then will write down what the equation is what equation motion as the cassette Katie equaled says set electron Toby arrest that's saying the setting its momentum to be equal to 0 momentum equal to 0 0 1st of all says that the wave function has no space variation all K is equal 0 there's no space variation it's just constant with respect to space if the wave function is constant with respect to space meant that only control this term away in any case were acting on a wave functions which have 0 momentum more 0 wave number then Omega is just Eagle debate at times for for particle at wretch its energy this just eagled data planned and all our weekend was at me a means armed that's right here in this form I side by that's a you get all major by hitting it would I be by C. that's ego of
data of for particle rest Wickham throw away the space variation the endorsed Dirac equation just becomes I decide by deeply physical debated In time site or the right away sigh right thought the authority is equal and not left an IAEA sire left thought is equal to em sigh right did that normal might whether as a minor yard Beta 1 as a minus sign below normal
moments beta has no minus signs in state is 1 1 off the bag so this is a simplified Dirac equation for particles arrest were you're being at rest notice that this still coupled together in this left to right away but could be coupled defined ordinary equations which are not coupled in this way all you have to do it is a add and subtract equations let's get them where we get there we get I sorry that's plus side plus means I Write plus so I left we had then I saw right plus I left is equal To End tie-ins sigh left plus I write which is just plus we now have a equation for an object which is deep coupled with just 1 equation not coupled to anything else and what about the other possibility let subtract when we subtract them we get our sigh minus 5 signed minus aside I write Marine this I left normally on the right-hand side Is it an inside minds Martinez end sigh my interest so by aiding them and subtracting them another which taking linear did did different linear combinations we find 2 objects 1 which satisfies an equation with a plus sign and 1 was satisfies an equation with modest which if we just look at every just imagine that sigh has a frequency and this is also the same as or maybe equals and not for plus sign and all maybe equals minors and minds all may get equals plus a for this equation and make it equal Tom miners and this equation so notice what's happening when the particle rest it also has positive and negative frequency when it was moving with the speed of light so I left inside a right had pop positive and negative frequencies about 4 per hour but when the particle was at rest it's the linear combinations pluses I'm writing this which have definite frequencies plus and money so sigh plus is the field operator describing positive energy particles arrests and sigh is the field operator describing negative energy particles we when negative energy particles arrests the bad things we don't want them what would do we just feel that their acts to you and we are not so what we have left although there is B plus electrons here of electrons which are linear combinations linear superposition of electrons left Moving in Right willing electrons linear superposition electrons left them right makes an electron arrest off with a frequency or an energy proportional romance with different gives you an electron with negative energy you just fill up the Dirac Stewart soros Mel your particles of man's both positive and negative energy and factor every momentum there are positive and negative energy you fill up all the negative energies end you leave the positive energy called a positive energy electrons and positive energy holes they both follows and particles now have met our negative energy electrons when you remove 1 has positive energy and so will all hold has positive mass just as an electron has positive the original particles that positive male so you see that it's so that a very odd thing is said nobody had any idea told that a better way to make a massive particles in quantum mechanics was to take a left movers right movers and have the left movers was coupled with the right moves in this on way now commonplace the idea that coupling left movers to write movers gives you mess massive of articles but still it's surprising Intourist back you're torn stocks my proof I just in case you don't know what constants are minus our plus is equal to desire right plus I'm my left and my mind is equal to right Minor sigh left wears it means get and wave function or did feel operators sway Yukon what happens if sigh right plus I laughed acts on the vacuum What does it get busy give you 2 particles 1 left and right side lost sight of that kind right would be to particles so left contains creation apparatus I write contains creation operated as you multiply the together and it gives you products which create 2 particles where last time the really you 1st of all I want is a moved Holloway they want v that's a particle with half a unit of probability of being left over and have been the probability of being right it's a coherent linear Quantum superposition so these plus and minus operators off-field operators which create particles which are linear coherent positions of other words Quantum superposition of left nor right
moves which means they have equal probability for either equal probability for Ivory saw particle at rest is linear coherent superposition are left more right over a massive particle and general contains both left moving components might arise emphasize does not mean to particles it means 1 particle with probability of a half bit in this case the bill it a toe left or right over I need to say that if you didn't have this term in the equation that it would be moving a little left of the right its components you that's its matrix components in your original equation describe left more was right in the new equation with the end turn the describe something it rest our yard you shouldn't think of it there's a coherent superposition something moving to arrive something moving that left both at rest on the whole thing is but look at his if that Minister more not clear end I applied sigh right plus I left to evacuate end it would make a coherent superposition of a particle flying off to the left of particle former right the man's term modifies whole thing significantly and it simply makes a waiver which just sits there it's not going to let not the right it just sits there Our but it's made up of these scenes objects the opera component and lower component so that's how you make a massive firmly on all massive from eons in particle physics are built have left movers right movers it doesn't matter because so if we are to you could erode the prescription pollination phase actually matter that the number of equation basic saying they were just ignore it is for the 2 of we're all by saying that additional side by side minus how have basically same side right there your it or you're asking about the relative size they have for particle at rest for particle erect new phase in the field patients there don't but we do and that's a consequence of beta having been chosen to be a real matrix there are other choices of data that 1 could chosen for the only rule Badeau workers that was an object which cut and that commuted with without and the square was 1 is not unique but when she chose a teachers choose stick with it but and rule equivalent the roll equivalent to each other but as a consequence of the particular choice of data the coefficient here's 1 I chosen are the choices there might be some phases in interest How does Over the past but have he's half its that things which will labeled a which when the mayor's wasn't there really corresponded full of Moria right moving away no wants the ministers they'll we might we might give up a fee on the idea that's composed of left moving arrival in waves that were due to it is misleading to think of it as leftover rival waves his shift away that's not moving all but it's built our on the same mathematical objects which were left moves right movers when there was met with a mouthful of the best thing to do was play with it and perhaps once the semester give up foreign L meaning right or left to mean something else by them foreign the opera component of law laws so that they can move a one-dimensional particle from regular movement the mucous angular momentum was mixture only 1 image so we have not gotten that angular momentum we're trying with him we were here where we were last week signed by America but the point is with this kind of equations specifically this kind of equation you have no choice but quantizer with the rules of Fermi arms because otherwise you have this terrible problem over unstable vacuum we will we have that we have not done wrote this to on need to historically of course to direct quite well no much which showed decline Gordon equation I think before a direct equations with young you may reference Over the years last year right to the end of the will Honda words Is the law and if you think about which were not according the combined group this over for years but that
Fox argued that year the last funnier author of 1 to go over the last year need I know which may mean supposedly wrote them the equations for a nun 0 momentum yes and solve it they could you get the equation with 0 momentum by Laurence Tribe yes but I would have to tell you how Lorentz transformations act on site which I'm not worth it has lots of love gave if you and Lorentz transformations as right say on this is the theory of electrons arrest and now what will rinse transform them into yes that's correct but but you would have to know how to transform young that's correct you're so this such a step that somebody not that that would he's also as the late there were value yeah a on a bit of money just thievery brief about this point How what's the connection between this formulation of and these ideas that Higgs field give particles premier are you on the equations of motion Florida the firmly on fields and the Higgs field the Higgs field as another field qualify score of 5 x there's a scalar field as a scalar field where this Pyrex the go equations of motion 1st of all contained equation motion flight of some worry about what it is but the Dirac equation becomes a nonlinear equations and becomes a nonlinear equations in which there's a coupling constant here and flight coupling constant and a fight no man's a Dirac electron has no man's land if the vacuum was such that the field 5 or equal to 0 as you normally would think novacula empty space y must be 0 right OK if that were the case and looking at these electrons in a vacuum in a empty space where fire was equal to 0 0 this wouldn't be there and you just have a massless electronic but for some reason for some reason we energetics on the Higgs field themes were the lowest energy having the nonzero value of 5 but say constant nonzero value of 5 5 x equaling a constant which is codified as that the vacuum effects would be on empty space will light was filled with far-right then fight here would just be a constant value and all of this would play the role of the man John equation the masses of electron is equal coupling constant times the magnitude of times the value of the Higgs field in the vacuum that city of us a connection and we wish that that a collection so any place this can come from it will provide a mail In particular 1 place it could come from is non-linear equations including couplings between bows ions and Fermi eons and for some reason the Bulls ionic field having a value in the vacuum and empty space 1 of the things that I would to say more about how they expect as well young OK rude we could talk more about right now from what I was going to going to 3 plus one-dimensional Dirac equation since US the stocks that b of the Higgs field a Ozark field bows on Uckfield's qualified they have potential energy they 1st of all have energy which depends on the gradient of the field are both space gradient of climb gradients Empress I worry about that they also have energy which just depends on the value of the field itself the Maxwell field for example has energy which is e squared was be squared electric field square past magnetic field squared now both electric field and the magnetic field are proportional to derivatives grow vector potential climbed the rivers in space derivatives so that case all the energy is associated with gradient of fields after the Higgs field there are 1st scalar field you could have the energy which is a sort of called potential energy of the field of fire it does not depend on the space derivatives of time derivative energy density and energy density in the field and then principal unless you have some deep principles to guide you this I could be anything any reasonable functional 5 now there are some symmetries for the case of the Higgs field for example symmetry is that fire positive and negative 5 are identical to each other as a symmetry of becomes minus 5 symmetric and then easily if I should be a symmetric function YTY I there's no important right now is just means it should be a symmetric function on the left and right as a bulge over here bowled over here now for whatever reason the reasons are still not completely settled completely via fight as a function which this why that is as I said nobody really knows what sorrow but to pick a function 1 functions as good as another it has to be symmetric but that's all and from what we know about the laws of a particle physics it appears that the potential energy of
the Higgs boson on looks like he used his fine is fight 0 and the minimum of the potential visit some nonzero values of 5 either over here for over here you get the pick 1 here hours if he OK so if the potential look like that you would say the minimum was at the origin if that were the case the vacuum value of 5 state of lowest energy the state laws potential energy where 5 equal to 0 antielectron would have no man s is for some reason the field potential looks like this the state of lowest energy will either be over here or will be 2 configurations which would otherwise be identical 1 with positive value of the field 1 with negative value of the four-year-old either of them give rise to a messy either of them give rise to amass a V G times and it are you could start asking why did nature chose still have a feel potential which looks like this not completely clear at the moment OK but say our real vacuum over here it's assumed that the field Kabila here and the vacuum but cannot also be perturbed it can oscillate about here just like a particle on a potentially vibrate back for the Higgs field can vibrate it can however frequency what corresponds to a vibrating Higgs field right the frequency of the Higgs field is related to the mass of the Higgs particle and the XI pensions of the Higgs field on in which its oscillating are like any other oscillation common quanta those quanta are the Higgs particles of its particles correspond oscillations in here but if the Higgs particle is very massive means it takes a lot of energy to get this field vibrate in the vacuum adjusts its electron as a man sorts vibrate that's a huge particle now Higgs field is coupled in an interesting dynamical way proved electron field then by the laws of action and reaction which I'm not going to be terribly specific about how the Higgs field will react the collisions premier collision affirming Young's will stop Higgs field vibrating stop still bribe vibrating and create Higgs particles in his oscillation how much energy does it take it depends on the mass of the Higgs particle of banks particles very massive it means it takes an enormous amount of energy to excite 1 Quantum's worth of vibration and here topics particles massive it means that collide electrons were a lot of energy get vibrated wanted vibrating those vibrations are the quantity of the Higgs field so content the Higgs field is itself a legitimate quantum oscillating object which is described by Quantum of quanta recalled eggs particle men there to be electron in another fermionic fields quark fields and so forth in such a way that a collision off To firmly on fields stop the Higgs field vibrating actually half of the year with a path to get that sealed that if you get the Higgs field to move an appreciable amount for example if you could somehow get the Higgs field they get in balance up here and hold it there electron would have right now the stakes are huge amounts of energy you could To create a region of space and the all that With the Higgs field LaPierre would require an enormous amount of energy so much energy try to make that region big enough to do an experiment and would create a black hole so a very difficult to arrange for a regional space had a Higgs field are sufficiently displaced so that you could see an appreciable change in the massive electrons but yes the answer's yes electronic workers ripping out the answer is yes that the displacement of the Higgs field would correspond to effectively a shift in the mess we work a lot easier you know but we don't
think there's only 1 nurses Alaska in the Standard
Model particle physics is when he exerted does the work for war are articles elect From Zurich worked so In a supersymmetric version of the corrupt cool 1 of them Schroeder up quarrel is too but no more than 2 and it's not that 1 is for electrons awards for quarks it's a little more complicated than that in some ways I'm more complicated and some ways support so our supersymmetric theory there 2 Higgs bows on regular ordinary period is only 1 turned things a little more complicated than I didn't give it its full of glory but I give you the basic idea the Higgs particle was an oscillation of the bottom of this potential well the shift Of the Higgs field in the vacuum is the origin of the mass of electrons and the quarks and the real question is why do you have to go to such lengths why don't you just give the electrons the equations mess in the whole of the Higgs field that has lock the do with a weak interactions and birth unification of the case a no no no no really on it just as 1 in which gives me 2 0 up quarks and the other 1 which is bank works electrons let's will find it it's a technical feature of supersymmetric theories which itself is not terribly interesting you were not terribly conceptually interesting it's just to keep all symmetries and everything a force of throwing talking spots on the on my mind where the ugly features a supersymmetric theories but it may be true but we are not on it said she if you the getting in over our heads now because much of this now and we want we want to take it easy winners in pigs mess come from nobody at the moment the real question is not why the Higgs particle has amassed a real question honestly is why the Higgs particle masses so small look at it was symmetries symbolic missiles stories symmetries are very important to the story but suppose we had a year on a reason to believe that it was a symmetry armed with see what workers but she said let's With Commack this is an interesting model so explained to understand the relation between symmetries and particle masses but come back to it but I will tell you the basic symmetry I'm thinking about is it's left right symmetric symmetry here if the vacuum was left-right symmetric it would mean left right now doesn't mean space means and field based here that it would mean the vacuum value of the fuel have to be 0 and the mass of the electron would have to be 0 so it's the breaking of cemetery where she is so what provides a massive electoral now many of the particles the basic structure of the furious such that their symmetries which would tell you that the vacuum a symmetric those particles would have to be massless and they are only yet amassed by virtue of the vacuum being asymmetric like that is all of the particles that we know if if if all the particles knowledge of the exceptional warned namely the photons get their maps for would-be man's tourists would not have mass of the Higgs field was at the center of the boat is exceptional because it does not mean that is not an exception it has no amassed . particles have met have man's be cause the Higgs field is now center arms that's where the I'm sorry there is 1 exception apart from a photo on the Higgs particle itself the Higgs particle itself could have a amassed even if it didn't have an offset like this so the Higgs particle gets its name from Pope lead different mechanisms then electrons quarks easy both arms W. bows also the particles have massive because of a mechanism like this only the boson on gets messed from some other mechanism but we don't we haven't gone far enough yet to work to discuss that will come where the normal with the big my work full-time staff would not default on has not but there were no Bush 1 of these known the objects in nature the only things which move the speed of light are for grab part of known things at 1 time was thought that neutrinos amassed now know that neutrinos have played a little bit of gas which means it left right coupling between them I'm very small my wife's was more of a question try to get these things but sure we talk
about for the members of the Big 3 them 3 plus one-dimensional Dirac equation give apparent apparent what's want to understand this was very similar again particles of described the oral fields waves of described by a frequency and wave vector wave number becomes a wave vector which is proportional to the momentum of a particle wave vector has components KXK wines too easy and the Dirac began by writing down a generalization actually reckon right the same equation major equals underwear was Alpha Times k was remind us that Ogata minors there minors health care and Barak said Let's generalize this plus pay their past and again standing for a particular equation of particular differential equations kind derivatives space derivatives and just match that I leave at minus source where is on not sure I've lost track of workers should be minus 0 maybe not not no minus back your martyrs back murmur OK or make giggles Alpha Katie was dated him now they're acts Red look let's try this she and three-dimensional space I have 3 components of momentum closed momentum and therefore I have to have 3 components of Alpha it better be of the form of out forewarned K 1 plus Alfred too Kate tool plus Africa 3 case 3 components of K K I want to win 319 X warning are for K 1 offered to cake to post up 3 K 3 is a natural generalization of the plus Bay then started out this was the most general linear thing I can write down on its nice is simple let's saying if we can do business with an equation like this with alpha and beta is being matrices Eames such a way that Omega squared is equal to K square for some square we will require Omega squared equals K X squared K 1 squared plus 2 squared escape 3 square Square that's relativity now courses water relativity and justice equation this time without strike back Our civic and do business with that and get some missing some like anyone else look at us square lawmakers Guerrero together what's more fight for K 1 suffered took a 2 0 South 3 catering was beta why he chose this particular form is a historical factor was simple was nice a simple and let's try it was in that spirit of strife as you will we get of 1st of all we get things like How 1 square K 1 square have similar things about 2 square cake to square 3 square 3 squared it's pretty obvious we had better choose all of the B the square of each Alpha to be equal to 1 OK so we can write that and alphasub squared equals 1 for 1 . 4 5 equals 1 2 and 3 square of each Alfa Matrix each 1 of them has to be 1 of the 1st condition but LaPierre half 1 square was offered to squared equals offered 3 squared equals 1 of 3 of them now where about data but we multiplied beta squared were also going to want the match our readers to you yard where 1 elected this so also beta squared better be acquittal 1 where we get some other terms really terms for example which have K 1 skate to tho come from took places where K 1 Kato from these 2 terms but will also get terms from these 2 terms saying K 1 K too so utterly half a K 1 will have K 1 cake to 0 have How former off tool plus South to allow for 1 or more to do with that 0 0 0 and more generally be intact commutator of 2 offers had better be 0 if they're not the same if the out of were of the same so we have out Friday Alfred J. Ross Alfred J. off I equals 0 0 I not equal to jail 5 equal to JD has to be 1 c could write this in the form of like a patient 5 Alfred J. plus Alfred J. equals way saying 0 490 medical the J 1 what
about the terms which have products for example Kate kind and we're Times MK one-time K 1 times from here and K 1 doctor at the lake 81 times em from here and K warned plans and from here I saw also have the have each 1 of the Alpha plus plus K 1 and How foreign Bader plus made out for 1 raft around the after all in their commute with Bader it's a of that for every Alpha Phi Beta plus Bader I thought I must equal 0 if we can satisfy these relations for matrices altogether 3 ounces and 1 made a very symmetric every 1 of them didn't die commutes with any other won the ball for me and the systematic our before matrices press a lawyer have square 1 they all have to enter commute with each other and that's it basically fined for such matrices then we can find a wave equation of this seemed column matrix type which would give us in 4 dimensions 3 plus 1 dimensions will give us Omega squared equals case squared per cent square so Dirac set himself the task of finding a collection of 4 matrices which satisfied through his 1st observation must through a bit of a disappointment is that you can do it With 2 by 2 matrices you could find 3 matrices of the right former which our square 1 in which mutual we can't wary you freely and 3 Polly matrices but there is no 4th 1 hour and sold 1st attempt failure animal Directorate 10 probably know this so intuitively didn't bother her 1st case where you could have matrices which satisfies for not for them into the space time that's not . 4 by 4 matrices 4 by 4 matrices 0 4 by 4 matrices which satisfy all of these conditions there are no 3 by 3 and Noto by 2 matrices just 1st cases for about 4 makers so direct figured out 4 by 4 matrices which are which satisfy these rules and were given to you pay what they are used in the study itself some of the firm the details for by 4 matrices which satisfy these they cold of course the Barak matrices and like that raises a puzzle from the beginning the chill by 2 matrices corresponded had something to do with moving left and moving right where this fall but for matrices were 4 possibilities associated with before matrices Our that's something will come to next time I will do that tonight but the new ingredient is spin that OK learn a little bit about angular momentum before Wittenberg about their OK so tell you what these minimal set of matrices not unique and not unique Gupta some simple transformations but there equal Our that unique but equivalent in the same sense but can pick of I'm interested in mutually orthogonal taxis space mutually orthogonal axes or unique among us mutually Nevada sees them unique but they are equivalent in that the the laws of physics in the eyes of axes of the same thing you could make transformations warned the other in the same sense the choice of Dirac matrices is not by equivalent and he has a particular solution OK so Bader His equal to 1 0 0 0 0 0 0 1 0 0 0 0 0 0 minus 1 0 0 0 0 0 minus 1 as data that before already others I want to simplify well maybe I think I'll write them without support for rotation OK as Bader for 1 jar job to go home and check these algebraic relations 0 0 0 0 0 are 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 after 2 equals 0 0 0 0 rackets start saving myself work by just providing these things into tool by 2 structures in coal in each 1 of these give each 1 of these toe 2 matrices a name but I'll do that 5 here goes to win comes next minors party party 0 very nearly 0 minors ART I 0 and then 0 0 0 0 0 0 0 after 3 Kirk Shirley you could yes somebody wanted a Stabile not 0 0 1 0 0 0 0 0 0 1 minus 1 0 0 1 2 0 0 0 0 minus 1 0 0 0 0 0 more nifty notation is the divide things in a box of show by 2 matrices each 1 block of Tilburg 2 major issues and then write them and the former Bader is equal to the Europe of hot identity or unit matrix he said unit to buy 2 6 0 4 0 0 0 0 0 minus I went each entry is now it's too late to make tricks Alpha is equal to 0 0 Sigma Sigma 0 With these matrices are the 3 polling matrices 3 Polly matrices if you don't know what they are here they are signal 1 who said tools and to refrain from back use matrices the report create these objects satisfied the Dirac In fact commutation relations and if you write a wave equation what would the wave equation b B equivocal Asian would be on works 1st of all maker for component sigh warned that 2 centuries for a corresponding to the 4 by 4 metric centuries and you write that derivative of society with respect see firms but it is equal to minus all eyes were on revenue of siding with respect to x Heidi X
ii means X Y and Z this means out 4 1 times the river of respect XL offered through the river respect to worldwide and so forth some plus beta em start at the famous Dirac equation stands for 4 equations for equations in which the components get a good deal more mixed up than the 1 in the year and the easier case of the one-dimensional BMO mixed up because there's a lot of off diagonal matrix elements here means off bad news of literature on the skipped mix-up do you have different components thoroughly intricate way but still it's coupled set of linear differential equations for components the matrices entangle or entangled with technical term Russian news it where a young women matrices coupled the various components together called direct question our room we will come back to it in the next term will be discussed where's comes from West Baden comes from Izzy extra doubling if you like are fine the functions as that an his world and has over here worry you go from actually tried it it is a yeah it's not enough wood the spinners the for that therefore component Object but they're not for vectors before you is not the same as saying the space has for dimensions Our happened in the previous case the space time has 2 dimensions and matrices widow by however that was just an accident and in this case also or Max as you go between different dimensions the matrices get big fast they're not there not the same dimensionality always even dimensional and they're always of dimension to To the end in any dimension In any dimension a space the size of matrices full of an inside a lot of money and where best where some sort of momentum money but where we matrices mixed up components and some lively intricate worry that's again interesting to study the case with Norman From particle arrest and that just has this and you notice Bader has to positive entries into negative entries it means that there are positive energy states and negative energy states saying deal send go up the may do you when equation out I still they are correct program I love for cattle were 0 0 till you're right you are right there is To hear Thank you y 0 to avail of a powerful 1 square should be 1 but this is obviously if both I 1 this is Opel 1 squared plus sample 1 squared which is to strike the a residual of 4 if you have absolutely yes sir I mean until Barak wrote down this equation and that particles never been heard our spinner had been been heard of but nobody knew where it came from what Barak postulated this equation and track and its consequences in 1 of the consequences that particles you have the consequence was Spain like was there Storer yes but this is not the more all the Orioles Norway over fears that there was much later and it keeps getting in the car as with all theorems there are assumptions you begin where and sometimes beginning the assumptions were let's just take 3 feels Richmond fields which didn't interact rubber feels completely linear equations and see what possible ally was approved that case man people study coupled fields and earned and that more more elaborate urine by now I don't think anybody doubts that the only possibilities of ozone incidentally This is a statement about three-dimensional space one-dimensional time relatively in 3 dimensions a space when the mention of time in 2 2 dimensions of space in 1 dimension of time it's different there are things which interpolate between those on the formula Our and their interest in blood there are 4 dimensions that propose arms and premiere of but as such the 1 sure our what's recall the index will be the label Paul name it means that I want use I because use I at least twice in this equation at least twice in this equation we could we could fix that square reminds 1 right arm others are high but I would not want at this are you labels What roses space 1 x y and z but now there's also a label here's a rhetorical out court Oh Boy River South Emma the president as the budget I we think this is paid the matrices themselves Have indices P Q 0 and this would be killed and this would also be PQ and this you and this case P Q would run from 1 for days if you want to know where all the hidden in the city's this is magic later right this equation so that it looks more relativistic and majority of them next time our basically is more symmetric way derided but this does the trick any case that's suspects the Dirac equation its simplest a incarnation only next time well I say we would do next time Spain Spain and the way fields for relativistic bows doing that we will be ready to start to talk about real particle physics we need to talk about what we theory of HBO's Oz light about half the particles nature of HBO's on and Dirac equation is is not the right prescription for them and we need to talk about that a little bit and then we move on serve corpse lap times Zeeb ozone expose on normal expose on the wide bows on his state's bows on photons there and the radical order of the group symmetries reactor use more for
more please visit us at stanford .
Schaft <Waffe>
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Formale Metadaten

Titel New Revolutions in Particle Physics: Basic Concepts | Lecture 6
Serientitel Lecture Collection | Particle Physics: Basic Concepts
Teil 6
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15069
Herausgeber Stanford University
Erscheinungsjahr 2010
Sprache Englisch
Produktionsjahr 2009

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (November 9, 2009) Leonard Susskind gives the sixth lecture of a three-quarter sequence of courses that will explore the new revolutions in particle physics. In this lecture he continues on the subject of quantum field theory, including, the diary equation and Higgs Particles.

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