The Biggest Ideas in the Universe | Q&A 15 - Gauge Theory

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll this is the Q&A video for idea number 15 I think was 15 which was a gauge theory as we said in the video itself gauge Theory was sort of a culmination of a summing up a high point in which were able to bring together the payoff of talking about quantum field theories and geometry and topology and scale and a whole bunch of other things in an interesting way so the QA video is going to be actually the things gonna be an important and a good one I mean the questions and on the blog post and and the YouTube videos we're extremely on point them let me put it that way they were very good questions and in part I think that's a reflection of the fact that this is a very rich and rewarding topic where in the original video I'm not able to cover everything interesting that there was to say about it now there would be an obvious strategy to do here which would be to have a separate video on the standard model of particle physics because the standard model is the best theory we have for how the world really works the quantum field theory world anyway apart from gravity but I'm not gonna do that I decided not to do that because the standard model by itself is not one of the biggest ideas in the universe it's on the one hand really really beautiful and certainly extremely effective in fitting the data on the other hand it's a concatenation of a lot of different ideas from a lot of different angles so basically for these videos what I'm interested in is giving you the basic insights on the ideas themselves the major concepts that go into building the standard model so things like renormalization and symmetry and so forth but for the actual standard model many of the details are going to be left aside they're interesting they're they're well worth studying and I encourage you to go do that they're just not what we're about here in this series so what that means is that I had the opportunity in this Q&A video to fill in some of the gaps between the gauge theory video and the actual details of the standard model so let's get to doing that one thing is this issue of mass in quantum field theory I said something the mass of a particle the mass of a field so we've said before that quantum fields when you quantize them they look like a collection of particles so even though the mass of something is something we attribute to an object something that can have a rest frame and you measure its mass in the rest frame we often very casually talk about the mass of a field by which we mean the parameter which when you quantize it and when you get a particle out will be the mass of that particle so but how do you know if you look at a quantum field theory what the mass of the particles is so I said that the mass term comes from if you have some field I said I think I did in terms of gamma the photon field but let's just simplify our lives and consider a single scalar field okay a scalar field with just a single real degree of freedom as we say Phi of X and T so at every point in space-time there's a value for Phi it's a real number the simplest possible field we can invent so we're so the punchline is the mass term is the term in the Lagrangian City for this field which looks like some parameter times Phi squared okay and that's what we won't try to explain a little bit I'm not gonna be able to explain it fully because you have to go through the details of quantizing it and you know showing that there's a limit in which it looks like particles again and so we've done that part so I'm gonna sort of hope that you just trust me in some of those details and instead say you know why is it fighted cubed or something like that what's so special about Phi squared so we know that we have the action this is how we think about the equations of motion the dynamics of this field classically and then we'll quantize it etc and the action is we've done this million times I'm going to keep doing it is worth doing the integral of Lagrangian okay DT and the Lagrangian is the integral over space of the Lagrange density and the garag density is a function of Phi and derivatives of Phi so of Phi and so therefore as the way we usually think about it is the integral d4 so the integral over all space-time were over some volume of space-time of this Lagrangian city and what is the Lagrangian City L well it is the kinetic energy the kinetic energy density but I'm not going to bother with those niceties now usually when we just did point particles we said cadet ik energy minus potential energy for a field there's other kinds of energy that can come in so we have minus the gradient energy this is how the field changes from point to point in space whereas the kinetic energy is how it changes from moment to moment in time and then then you have minus the potential energy and there's another term which we could have had just for the point particle which is fine it's the interaction energy so the field is going to interact with other fields and that's going to contribute to its energy and that's going to contribute to its equations of motion okay but let's ignore that for right now let's just think about fields all by themselves just a scalar field all by itself well let's see now look at some of the details here the kinetic energy as it turns out is one-half Phi dot squared Phi dot in fact you know if I dot is might be notation I may or may not have used I forget it's the time derivative of Phi okay D Phi DT squared so literally how fast the fuel is changing in time that squared times 1/2 I'm not going to justify this but it doesn't seem surprising the kinetic energy for a particle is one-half MV squared V is the velocity so it's one-half M DX DT squared for a field replacing DX DT with D Phi DT does not seem like a big stretch where did the M go if you want to know while kind of absorbed into the definition of v when we were you late phi 2 particles gradient energy so it turns out this is actually a very nice thing we didn't go into details here but we know exactly what the gradient energy is going to be even before we do anything because of relativity because of special relativity special relativity says space and time are on an equal footing so if this is what the kinetic energy is we instantly know what the gradient energy is it's minus 1/2 derivative of Phi with respect to space squared now I'm just writing d Phi DX squared it's really D Phi D X and then squared plus D Phi D Y squared plus D by D Z squared if you're in three dimensions but you get the point it's the rate at which the field is changing over space squared so what about the potential will just write V of Phi okay so this is what we have to work with so when you write a scalar field theory the kinetic energy and the gradient energy basically are fixed those are universal those are what we have for every scalar field that's not always true cosmologists like to mess with the kinetic term in interesting ways but we're not gonna do that usually what you get to play with are the interactions and the potential those are those are the things that come into your definition of what you mean by this scalar field theory okay so what is the potential going to be since we're ignoring the interactions what is V if I going to be well what you can do is remember the whole thing that we are the whole philosophy we're taking towards these quantum fields is perturbative that is to say we're imagining you start with the vacuum state with a state with no particles the field just sitting there very very quietly and then you perturb it a little bit okay you let the field vibrate very very gently you don't have to do that you can imagine you know the feel fluctuating very very wildly but here on earth you know even in a big particle accelerator most of space is just empty there's a few particles most of the modes of any particular quantum field theory are in their vacuum state so this idea that you start with the vacuum state and then add a few vibrations is a very good model to what we actually have in the world okay so therefore when we do this V if I thing we can imagine that Phi is small right if we imagine that Phi equals zero is the minimum of the potential is the vacuum State is that is the classical vacuum state then we can look at small variations of Phi so you might say that V of Phi well it's gonna look like I'm gonna draw a squiggle there so we don't need to worry about the parameter the specific numerical coefficients or anything like that you might say it'll be you know Phi to the first power plus Phi to the second power plus Phi to the third plus Phi to the fourth etc okay that kind of expansion and these terms should become less and less interesting less and less important as time goes on and the first thing to notice is that in fact you're not going to have term proportional to Phi by itself linear in Phi because if you did that would mean that the potential via Phi as a function of Phi what does this look like in general well that Phi term means that the the potential is a nonzero slope at Phi equals zero and then you know goes on to do whatever it does depending on what the other terms are but the point is if that were your potential the field would just roll down to the minimum okay so you're not actually at the minimum if you were at the minimum you can't have a term proportional to Phi there for if you're slightly more happy with calculus the way of saying this is at the minimum of the potential the derivative the potential has to vanish so the coefficient of Phi has to vanish so in fact let's get rid of all this there you go so we have a term proportional to Phi squared that's the very first thing that appears and then if I cubed and 5/4 and so forth now if I cube to 5/4 are different in us in an important way from Phi squared in a few different ways for one thing you notice that if you look at the kinetic term it is Phi and it's squared right it's their time derivative Phi squared but it's still there's one appearance of Phi to the second power so there's basically two appearances of Phi in the kinetic term likewise there are two appearances of Phi in the gradient term right D Phi DX squared so this first term Phi squared in the potential kind of goes along with those it's the same order of Phi in the equations and that actually has an important consequence so let's imagine that we only have five squared let's imagine for the moment that let let V of Phi be something which I'm going to pre cognitively I'm gonna predict the future I'm gonna say let's let this be one-half M squared Phi squared okay let it just be that okay nothing else then the whole Lagrangian City is looks very sort of nice and similar right 1/2 D Phi D P squared plus 1/2 D Phi DX squared plus 1/2 oops - I need some minus signs here minus 1/2 D Phi DX minus 1/2 M squared Phi squared so if I appears twice in every single one of these and remember how we got the equations of motion the equations of motion come from minimizing the action you knew that was gonna happen talking and writing at the same time it's not my strong suit so by the way let me let me say this Q a video is going to get a little more technical than the regular videos I know you might think of the regular videos are already technical but I'm gonna use this Q&A video and we can let her hair down as an excuse to get a little bit more into the weeds here that it's just inevitable so I'm gonna be writing even more equations than I usually do so the equations of motion come from varying or differentiating sorry the action and setting that equal to 0 so you're taking the derivative and that's the derivative with respect to Phi right that's that's the idea you change the value of Phi the trajectory Phi is on the field configuration of Phi and you ask for what is the field configuration of Phi that minimizes or extreme eise's the action as a function so therefore and that's gonna lead to equations that you get is there's some details I'm skipping here but you get that equation from the grunge density and so if the Lagrangian satine which is what goes into what gets integrated give the action is of order Phi squared right if it has Phi squared in the kinetic term the gradient term and the potential term and that's all it ever has then the equations of motion will be of order Phi they will all be linear in 5 because you're gonna get them by differentiating I'm not gonna write down what they are now why we could write down what they are the equations of motion for this particular Lagrangian density work out to be what are they gonna work out to be when you take a derivative of something squared you bring down a two and it's gonna cancel the one half that's why all these one halves are there because you're gonna eventually take the derivative of them but then you get a second derivative instead so we're gonna get d squared by DT squared minus d squared by DX squared minus M squared Phi equals zero okay this is the second derivative this notation d squared Phi DT is just that's just the derivative with respect to time of the derivative the spektr time of Phi okay so this is the equation of motion for that Lagrangian City I'm not gonna tell you why but you could look it up and you notice that every term has one appearance of Phi in there these squared terms here you know this this d squared that's not Phi squared that's the derivative that you're taking the derivative twice but Phi itself in every single term only appears once okay why why do we care about that because that means we can solve it that means this equation is exactly solvable and in fact we sort of secretly did solvent by solvable you know many of the differential equations in theoretical physics are complicated Einstein's equations are horrendously complicated typical equations for an interacting quantum field theory horrendously complicated that's why we invent Fineman diagrams as a simple calculation 'el tool that lets us approximate the answer we're very very happy if we ever find equations of motion we can solve exactly right this you can solve exactly so by solvable I mean we can get exactly what the answer is and it turns out the answer can be written you know you can look at modes of the its a this is this is a free quantum field if you look at this equation there's no interactions in it except for this Phi squared and what I'm saying what I'm telling you here is that Phi squared so V of Phi equals one-half M squared Phi squared is not an interaction strictly speaking oops it is it's part of the free theory so he made a big deal when we did our original discussions of quantum field theory about starting with free fields and then adding interactions to them and the free fields the reason why we could do that is because the free fields had equations of motion that we're solvable here it is this is the equation that is exactly solvable and gets us a free field and basically the solutions are I'm not going to write them down because the chances that I get all the I's correct etcetera are very small but they're plane waves they're plane waves with a certain frequency a certain momentum moving in some direction okay they're the modes of the field those are the exact solutions to this equation and they have a relationship that the energy really the energy density but you know what I mean squared equals M squared plus the momentum squared where M really is this okay so this looks just like the energy equation for a particle and that's not a that's not a surprise that's not a coincidence that's actually really what is going on so that's why I know I'm skipping steps and asking you to believe me a lot but that's why Phi squared or field squared is what gives you the mass of a field because field squared in the Lagrangian City corresponds to equations of motion that are still exactly solvable so even if you have the kinetic energy gradient energy plus Phi squared you can solve them that's what gives you the equations of motion for the free field and then you add on to V if I you know Lambda Phi cubed + ADA v to the fourth + etc and these will be interactions once you have those in your theory you can no longer solve the equations of motion exactly instead you use Fineman diagrams and guess what Phi cube will give rise to fireman diagrams of like that five five five five fourth will give rise to five in diagram to look like this Phi Phi Phi Phi etc that's why that is hopefully convincing you enough of why it's Phi squared in particular that gives you the mass term it's part of the linearized equations it gives you this relationship between energy momentum that we're very familiar with from particle mechanics and it does not count as an interaction all by itself so we can think of free fields as either being massless if there is no Phi squared term in a potential maybe there's a higher-order terms but that's okay or they're massive if they have this term so you see that this structure here for the Lagrangian is very very general because even if even though there will be in general many other terms in the Lagrangian you will always start with these okay you'll start with a kinetic energy and remember the time energy time derivative and space derivatives are related by special relativity so that's really sort of one thing that's going on let me always have a Phi squared term you never have a fight erm because otherwise you wouldn't be at the bottom of the potential and these are the two things you have you have the kinetic energy and you have the Max those are the two parts of defining your free theory okay and that's gonna be true for everything so not just for scalar fields when you get to gauge fields when you get to fermions like the electron or the quarks it's always going to be field squared in the Lagrangian that corresponds to the mass and this parameter M really is the mass of the particle okay of the particle once you know to that particle description now one of the reasons why I wanted to talk about this well I should have talked about it before I mean it is part of understanding what's going on but also you've heard that in the standard model of particle physics masses come from the Higgs boson and I wanted to explain a little bit about what that about is about because number one it's not all obvious why that is true number two it's not even true all right I think we already said that in protons and neutrons most to the masses come from QCD come from the strong interactions coming from the gluon mediated interactions inside the tones and neutrons but some of the mass is the mass of the electron and the masses of the individual quarks for example do come from the Higgs boson I want to explain next I didn't really get to do it so think about spontaneous symmetry breaking again oh I guess one question Trafford I'm sure it was asked but I forgot to write it down was why spontaneous symmetry breaking like what is what counts as non spontaneous symmetry breaking the answer is that non spontaneous symmetry breaking would be explicit symmetry breaking so if you simply wrote down a theory in which most of the terms in the Lagrangian had a certain symmetry but one didn't so the theory as a whole just doesn't have that symmetry that's an explicit symmetry breaking spontaneous symmetry breaking is when the theory has the perfect symmetry but the state the theory is in typically the vacuum state the theory is in violates the symmetry okay so like here on earth this is an analogy this is not exactly the same thing but here's the analogy here on earth if I drop the pen it goes down right there is a spontaneous breaking of rotation symmetry if I'm out there in space the earth is not there if I'm closer to the true vacuum then every direction seems exactly the same but here on earth the existence of this planet below me spontaneously breaks rotational invariance it picks out a preferred direction called down okay so spontaneous symmetry breaking happens when the configuration of the fields breaks the symmetry even though the theory itself in this equations of motion is completely invariant so that's what we have in the standard model of particle physics and let's pick a very simple example of an so2 symmetry remember so2 is just a circle right it's just one thing it's just rotations and let's say Phi 1 Phi 2 space okay I'm throwing it this way because then I'm gonna draw the potential V of Phi V of Phi 1 Phi 2 and so2 is just rotations in Phi 1 Phi 2 space it's topologically and mathematically equivalent to u1 so if we called if we defined Phi capital Phi to be a complex valued field 5 1 and Phi 2 or real valued fields but we could define capital Phi equals Phi 1 plus I Phi 2 right and then there's be u 1 symmetry e to the I theta phi you just multiply that's rotating that fine but we're not going to do that right now anyway even though that is what we get in the standard model but we're not gonna do it for our purposes right now okay so now we're imagining that we have a potential V of Phi 1 Phi 2 that implements spontaneous symmetry breaking so what that means is that the quadratic term the Phi squared term will be negative so what we want is the Mexican hat potential ok the famous Mexican hat looks something like this and there is a brim of the Hat at the bottom here which is the vacuum manifold the place where the field will live with zero energy and to get that long story short you write something like minus 1/2 mu squared Phi squared plus lambda phibes I should be more explicit here because we have two fields so this is Phi 1 squared plus Phi 2 squared then I got this wrong in the in the actual video last time but plus lambda and let's get it right Phi 1 squared plus Phi 2 squared squared ok so morally this is five to the fourth but this is written in such a way that it is actually invariant under the symmetry okay and so I wrote mu squared rather than M squared because this would be the mass term for these two fields if the field were living up there right at the top of its potential where Phi 1 equals 0 and Phi 2 equals 0 but if it were there it wouldn't just live there it would roll down to some point down here on the brim of the Hat and the vacuum manifold so mu squared here is a parameter of the potential but it's not the mass what do you have to do is expand the field around the place it's actually living okay and in fact let me move that dot let's that's put the dot over here okay let's imagine that the field is at 5 1 equals 0 fine 2 equals something okay so what you should get from our previous discussion is that I don't know if I if I drew the potential here maybe I didn't even draw the potential here but after saying that V squared V is one-half M Squared 5 squared what does that look like it looks like just a parabola right V and Phi and so even if there are higher order terms Phi cube by 4 etc the first term you're going to notice looking at tiny deviations from zero is the Phi squared term okay so that works even when the potential is more complicated but you're not at Phi equals 0 you can re-expand the potential here near the bottom into Phi squared plus Phi Q plus 5/4 etc you can rewrite it by shifting and changing variables it's a very easy thing to do so that will be so the mass of the actual field is not mu in any sense it's some combination of mu and lambda that you get by changing variables okay you can do that but that's one thing you can do the other thing actually draw it this way so there's basically this thing that the field can do it can oscillate back and forth perpendicular to the direct to the radial direction sorry along the radial direction is what I should say so away from Phi 1 equals Phi 2 equals 0 that kind of oscillation would be a massive gauge sorry I almost a gauge boson a massive boson there's no gauge that you're here yet we're just doing a global symmetry okay there's a massive boson corresponding to oscillations that go up and down in this room of the Mexican hat by going radially either inward or outward from the origin okay it looks locally like Phi squared is not 0 and in fact it is if you read if you change variables but there's another direction you can go you can also look at oscillations around the brim of the Hat right hard to draw here in a very convincing way but you can look at oscillations that stay in the vacuum and fold the these are they're sort of the two perpendicular directions starting from this place where you're in the vacuum manifold that's the value the fieldhands you can sort of oscillate in two different directions what you notice is along the brim of the Hat it will always be true that the mass is zero because the mass is the coefficient of Phi squared and there's no potential along that brim of the Hat the potential in that direction and only in that direction is zero it's flat so this is a massless boson so if you had just had both the coefficient of Phi squared and the coefficient of Phi to the fourth be positive then you would have the minimum would be at the center of the potential because everything will be going up okay the minimum be at the bottom both oscillations would be a positive mass because you'd have curvature of the potential that was positive in both directions but because of this spontaneous symmetry breaking phenomenon you have one of the directions in the potential gives you a mass but the other one is massless and it is always true because of the symmetry there's always a direction in field space where the symmetry relates what happens to what what happens at one point to what happens to another point you notice that the brim of the Hat every point is related to every other point by doing an so2 transformation by rotating around in field space in that way and because it is a symmetry nothing can change from one point to another and therefore the potential cannot change from one point to another and therefore the potential is flat and therefore the mass is zero so that's a theorem that is called Goldstone's theorem although Goldstone definitely came up with the theorem that the idea predates them a little bit too Yoshihiro Nambu so these are often called Nambu Goldstone bosons these bosons that are in the flat Direction Goldstone's theorem says that spontaneous symmetry breaking which I will now spontaneously breaking for global symmetries so this is not yet a gauge symmetry this is just two fields which I universally rotate everywhere always leads to massless bosons this was a big worry when people invented this this is like the sixties and people are thinking that Nambu and and Goldstone and others because they liked the idea of symmetries they had noticed Nambu especially was a pioneer for understanding symmetry breaking spontaneous symmetry breaking but then there's a theorem that comes along and says when there's symmetry breaking there is a massless particle I think about massless particles is they're easy to make right there they don't cost any energy to make them that was another question that was asked you know why doesn't it cost energy to make massless particles when I say it doesn't cost any energy what I means is what I mean is it takes arbitrarily small amounts of energy if you have a mass there's a minimum energy that you need to make equals mc-squared to create that particle if you have a mass less particle for any fixed momentum of a wave or a particle in that field still has energy but since the momentum can go to zero the energy it takes also goes to zero so there's no lower bound on what the energy is so we don't seagulls don't bosons in the real world we see approximate Goldstone boson there's a whole discourse about pseudo Nambu Goldstone bosons PNG bees for symmetries that are broken explicitly as well as spontaneously but this this was sort of this line of reasoning was thought to be bad news for fans of spontaneous symmetry breaking since we don't see any Coldstone bosons okay well it turns out I've been emphasizing that this is for global symmetries because it turns out the story is very different for gauge symmetries and I kind of hopefully this discussion for global symmetries made sense and you understood it you understand that one direction in field space is flat there for massless bosons the perpendicular direction is curved there for massive bosons once you get to the gauge symmetries it's much harder to actually through the details in this way so I'm not what I'm gonna do is tell you the answer okay so for gauge symmetries well you start with some massless bosons right you start with not scalar bosons but vector bosons the spin one particles that are actually carrying the gauge symmetry right the force bosons the photons the gluons the gravitons or what have you okay so you start with so pre spontaneous symmetry breaking you might have massless spin-1 bosons and maybe you also have some massive scalars okay so if we had a theory like this but it was all pluses in the potential so there was no spontaneous symmetry breaking the minimum was just at zero you could have some massive scalar fields that will give you massive scalar bosons and they could be coupled to some gauge potentials right so to some so in this case in this theory that we have right now it's an Esso to symmetry so that's the same as au 1 symmetry so we'd have the equivalent of photons this would be a kind of photon like theory and that that those would be your massless spin-1 gauge bosons and then you have some massive scalar bosons that's let's let's say this is without so no spontaneous symmetry breaking then you have with spontaneous symmetry breaking what happens is the colorful language that we use is that the gauge bosons eat the Goldstone bosons and get fat they get massive okay so you get massive spin one bosons so in this case by the way let's like stick with this theory without spontaneous symmetry breaking what we mean is the potential looks like this you did not have spontaneous symmetry breaking so here's V of Phi Phi 1 Phi 2 and we're imagining that it's plus Phi squared plus 5 to the fourth okay so you have guess what - can I move this yeah oops no always forget what I can do oh yeah like that I did it two massive scalars one massless spin-1 boson if you have an su s o 2 gauge Theory the photon basically with spontaneous symmetry breaking what you have is a massive spin one boson and one massive scalar so this is the case where the potential looks like the Mexican hat very sloppily drawn Mexican hat so the direction that is along the flat part the vacuum manifold is now eaten by the gauge boson and it becomes massive so the point is another way of saying this I I know I'm skipping over a lot of things there's just too much stuff going on here a massless gauge boson well a massive mess let's spin one boson that's saying I'm glossing over one of the facts is that except for gravity which is weird all the gauge bosons the force carrying particles have spin 1 we haven't even talked about spin yet but you're just gonna believe me for right now the massless spin-1 boson has two degrees of freedom and what does that mean well we talked about degrees of freedom of degrees of a degree of freedom is you know how much information you need to tell me to fully specify what's going on so if I have one single scalar field filling the universe I say one degree of freedom if it's a real scalar field if it's a complex scalar field or it's too real scalar fields I have two degrees of freedom a massless spin-1 boson is 2 degrees of freedom and basically the way we think about that is if you're moving so if this is the momentum of the boson then it has spin and it can either be spin this way okay spinning right handed compared to where you're moving or it can be spinning this way those are the two choices so it can be spinning right handed or left handed right handed or left handed circular polarization for you classical electromagnetic fans out there so the spin can be either along the axis or opposed to the axis and those are the only possibilities because when the boson is massless it has to move at the speed of light for a massive spin one boson it doesn't have to move at the speed of light so there are three degrees of freedom and that's because there is also a the particle is spinning perpendicularly to the direction of motion or I should say the direction of motion is not even defined so if if I go to the rest frame I could imagine it's spinning this way spinning this way or spinning this way you might say why can't it spin that way well because this is supposed to be symmetric around the circle so spinning perpendicularly that way is this the same no matter how I oriented I know that's not completely convincing us cuz we haven't talked about spin yet but this is the true thing the point is that with spontaneous symmetry breaking in a gauge Theory you don't get Goldstone bosons ever what you get are massive gauge bosons so you know or you you've sort of been glancingly referred to this in the case of the standard model right what do we have in the standard model the standard model of particle physics we know that it's based on a symmetry su3 cross as to cross you 1 and that symmetry is spontaneously broken via the Higgs mechanism down to su 3 cross u 1 now you might think that what that is saying is that su 2 is broken and the rest are left unbroken it's not quite that simple the su 3 that you get the strong interactions su 3 is just instill unbroken okay it's just left unaffected by the Higgs boson but the Higgs boson affects both su 2 and u 1 and it breaks them both but it leaves this u 1 invariant okay so it leaves are you one behind morally it's equivalent to just breaking one su 2's worth of symmetry but this is what actually happens in the real world I want to be I want to be telling you what actually happened so we're not going to go through the details but we're going to count degrees of freedom that's that's that's something we can do it's those numbers are small they do not require a lot of arithmetic Talent so you're breaking su 2 cross u 1 down to a single u 1 so it's kind of like the U 1 just goes along for the ride it really doesn't it's more complicated let's pretend that it does okay so it's like you're breaking su 2 to nothing it's like you start with an su 2 symmetry gauge symmetry and you completely break it okay so what is su 2 su 2 loops su 2 is almost again sort of morally it is almost so3 it's a three dimensional group okay and what that means is therefore there are three gauge bosons okay this is before symmetry breaking happens so let's just say there are three gauge bosons that's all we will do but then you also have the Higgs which is a complex doublet of scalars this makes sense cuz it's su 2 right so the Higgs let's call it H can be written H 1 H 2 where both H 1 and H 2 are complex valued fields so this is one two three four degrees of freedom because these are complex valued fields every complex scalar field has two degrees of freedom and these are two complex valued fields okay so four degrees of freedom overall so it's again I can't really draw four-dimensional space but it's kind of like I can draw a three dimensional space so if I had if I had a three dimensional set of fields Phi 1 Phi 2 Phi 3 I don't I have four-dimensional set of fields but I could imagine that the potential would be minimized on a two-dimensional sphere right that would be the equivalent this would be three dimensional Mexican hat and so the vacuum manifold would be these points on this sphere and I would have a number of degrees of freedom they'll be along the sphere okay and those would all be Goldstone bosons in the global symmetry breaking case and this is not a very representation so I'm only able to draw three of them but the point is these four Higgs degrees of freedom DF degrees of freedom after spontaneous symmetry breaking go to one massive scalar boson plus three massive gauge bosons so if the symmetry were global rather than a gauge symmetry there would be three Goldstone bosons from the spontaneous breaking of the Higgs by the Higgs of this symmetry there'd be three directions you could move the field that would keep it in its vacuum manifold this by the way is why there are no so I don't know where to where to draw this let me just say it in words this is why there are no cosmic strings or mono poles or domain walls in the standard model of particle physics because the vacuum manifold is a three sphere a three-dimensional sphere that's why you get three massive gauge bosons three bosons three what we say would be Goldstone bosons if they hadn't been eaten by the gauge bosons and a three sphere has no PI 1 or PI 2 or PI 0 no non-trivial first second or zeroth fundamental group so no strings mono poles or domain walls there you go sorry about that but anyway this is what you actually have in the standard model of particle physics these should be familiar one massive scalar boson we call it the Higgs boson that is what we discovered at the Large Hadron Collider in 2012 three massive gauge bosons which we call the plus W particle the minus W particle and the neutral Z particle the Z naught ok these are three gauge bosons that were massless before symmetry breaking they ate up the Higgs boson and they got backed and this is why the weak interactions are short-range because this these fields that carry the weak interactions are massive if you go way back up to this equation of motion that we had right here for massive scalars you could say well let's change the right-hand side by adding a source okay by adding some particle that creates that the couples to this v field and what you find is this sum I got wrong last time what you find is a u Kawa potential so the potential for this I'll just write it in a different color so the the the field value what what could what can we say here if we if we plot as a function of distance from some source the value of Phi I'm using the word potential in two different senses here that's why I'm hesitating but it goes down exponentially okay it goes e to the minus M times R with this value of M right there okay I said that the force went like e to minus M R but that's not right the force is the derivative of this field value so it actually goes as e to the minus M R over R just to be very very careful there you go the point is that this fades away very very quickly as R gets large compared to the inverse mask Compton wavelength of the particle so because the Compton wavelength is 1 over the mass of a particle so when you have these massive gauge bosons with these the masses are very roughly 100 GeV it's more like 85 and 90 or something like that but still over the order 100 GeV that's a very big number the Compton wavelength is a very small number the range over which the weak nuclear force is noticeable is very very tiny that's one of the reasons why neutrinos can just pass through your body all the time the tree nose can pass through the earth all the time because neutrinos don't feel the strong nuclear force they don't feel the electromagnetic force they're neutral they do feel the weak force but the weak force is really weak yet we get really really close to something before you feel the weakened Equipe force and this is why okay so I'm not sure how convincing that whole discussion was but the point is this is actually so I'd need I haven't even gotten to how electrons get mass yet right a lot of people you'll hear people say that the Higgs mechanism was invented to give masses to the electrons with the quarks or whatever which is completely false no one knew certain yo Higgs and brown on glare and the others who were thinking about these ideas in the early 1960s they had no idea that you could use the Higgs to give masses to fermions electrons and quarks and things like that they thought that you could spun they what they cared about which was true is that you could spontaneously break a gauge theory and make it go from being long-range to being short-range okay and the reason why they cared about that is because they weren't even thinking about the weak interactions their thing about the strong interactions they didn't know about QCD they didn't know about asymptotic freedom or confinement or quarks or any of those things what they knew was it was it seemed like an attractive idea to make the strong interactions a gauge theory but gauge theories gave one over R squared forces and the strong interactions were a short-range that's why they invented the Higgs mechanism they thought that the strong interactions would be governed by a theory like this and that's why it was short-range didn't work out that way because nature is very complicated and tricky but anyway this is the explanation for why the gauge bosons in the standard model have a mass it's because the Higgs mechanism gives it to them they've eaten up the Goldstone boson so this is supposed to explain both why there are no Goldstone bosons and why there are massive vector spin 1 bosons in the standard model ok so what about the electrons why do they get their electrons etc in the standard model what does the Higgs boson have to do with them getting mass again I'm gonna be a little bit technical which means in this case sketchy so usually people say they're being technical they're gonna flood you with details what I mean is I'm hiding the details from you but just saying words that are true and you'll have to go fill them in so this is you know again Higgs and brown on glare who should the Nobel Prize for this also Guralnik Hagen and kibble wrote important papers about this Phil Anderson wrote important papers about this Nambu had thoughts about it at tuff Ted thought about it mostly in the 60s they were thing about the strong interactions it was Steven Weinberg in the late 60s who came up who wrote this charming paper very short paper and the title is just a model of leptons and he was trying to explain different features of the leptons and one of the features of the leptons like electrons cuz they violate parity ok they act differently there are right-handed leptons and left-handed leptons and they seem to behave differently very roughly speaking the weak interactions tend to interact with the left-handed leptons but not the right-handed leptons they act with left-handed particles and right-handed the anti particles okay but right-handed particles and left-handed anti particles are just not doing anything in the weak interactions and so Weinberg was trying to understand this and what you would guess what if you were clever enough to know the underlying particle physics would you would guess is left-handed electrons have a gauge symmetry of some sort right as we now know it's an su 2 gauge symmetry and right-handed electrons don't so you can just say that that would explained how electrons could have this interaction with the weak interactions but the problem is the mass term for an ordinary electron if you're Paul Dirac and you're writing down the Dirac equation and you think about the mass term for the electron I said that you know all mass terms are field squared right Phi squared or gamma squared for the photon or whatever but for fermions it's a little bit trickier it's still field squared but it's actually the right-handed field times the left-handed field very roughly speaking okay so the mass term actually looks like psy bar sigh if psy is the field let's call this cyi so this is the electron field or this is what you would guess it looks like mass times that it's actually only mass to the first power not the second power because fermions are weird and different it's but different in the scalar fields but okay here you go and it turns out when you this bar is like a complex conjugation kind of operation but it also flips right-handed with left-handed so if you have a symmetry that has an effect on the left-handed field but not the right-handed field so this is again morally it's not exactly right this is kind of like psy right x side left okay not exactly right but okay close enough if psy left has a gauge symmetry and psy right doesn't which is what the data seems to imply in the weak interactions then you can't write down this term so this was a known problem like if you have a symmetry that it effects left-handed particles differently than right-handed particles then you're not going to be able to have a mass for any of those particles not just the gauge bosons but even the fermions that are related to those particles so what Weinberg realized was that what you could do is with the Higgs boson remember we have this doublet here right capital H h1 h2 so what you could do is you could write down the right-handed electron so psy electron right which is just a single thing all by its you could multiply it by H which is this doublet of Higgs fields and then you have a doublet of electron and psy II know and the neutrino psy knew L so in other words what Weinberg says is maybe deep down in the bowels of the Standard Model of particle physics there is a symmetry relating electrons to neutrinos they're a doublet under an su 2 symmetry the right the left-handed ones the right-handed ones don't have that symmetry they're just single all by themselves and but what you can do is you can make a interaction term not a mass term but an interaction term with the Higgs boson and then what happens is for spontaneous symmetry breaking let's write it this way under spontaneous symmetry breaking the Higgs field which is a dynamical feel - can fluctuate gets the form some fixed value V and then 0 plus fluctuations so this is the sort of equation way of saying that you know the field in its Mexican hat sits there at some value V ok so the first component of the higgs field takes the value v the second component is just zero and what that means is that this term here starts looking like sy e r v x sy er x sy e L which is just the mass term for the electron and there is no corresponding term for the neutrino and that's why neutrinos are massless in the simplest versions of the standard model because there's no right-handed neutrino to do anything quarks are more complicated but anyway everything is more complicated this is why I'm not doing a whole video on the standard model but the point is Weinberg realized that the Higgs mechanism which was invented to solve the strong interactions could be applied to the weak interactions and would explain not only why the weak interactions were short-range but also how you could have a gauge theory that affected left-handed particles but not right-handed articles and yet still had those particles be massive through the genius of spontaneous symmetry breaking so this was not the original reason reason why the Higgs boson was invented but roughly speaking this existence of the Higgs vacuum expectation value the Higgs field has a nonzero value out there in space the electrons are able to react against that value of the higgs boson in empty space and thereby get a mass it's a completely separate mechanism from how the gauge bosons get a mass but it is nevertheless true at the end of the day that the Higgs mechanism gives a mass to both the gauge bosons W plus W minus and Z of the weak interactions and the fermions of the weak interactions the electrons the muons a Tau and all the quarks okay I understand that there's too many details being glossed over to really get that but I wanted to say a bunch of true things that by following all these videos you are in a position to at least kind of understand in a way that you know I wrote a whole book about the Higgs boson and the standard model particle at the end of the universe and I couldn't quite explain things in this much detail there so you get more than they would get okay anyway lesson how mass comes in the standard model from the Higgs boson and as we said before protons and neutrons get mass from a completely different mechanism but the quarks inside them do get mass from the Higgs mechanism exactly like this the other thing I wanted to say about the standard model is people you know I don't know what street corners you people are spending time on but I made a plot something like this with energy and I plotted the running of the coupling constants right the you know as you go to higher and higher energies the strong interactions get weaker and weaker that's called asymptotic freedom asymptotically as you go to high energies the particles just become free they are not interacting though interactions get weak so whereas for electromagnetism the interactions actually get stronger so in the standard model s oops su3 cross su 2 cross 1 in the standard model there are 3 different coupling constants one for you 1 you're tempted to say it's electromagnetism but it's not remember in the standard model the actual u1 that you get for electromagnetism is not the u1 you started with when you say yes you to question because if there's a admixture of s u2 and u1 so this is actually you want hyper-charged I'm writing that very tiny so no one can read it because I'm not telling what the details are but just trying to get it right just trying to say true things so this is the u1 hyper-charged and it has a coupling constant alpha 1 and there's a coupling constant for the SU 3 bit alpha 3 and there's also a coupling constant for the SU 2 bit alpha 2 okay and what you see is if alpha 2 and alpha 3 are going down and alpha 1 is going up maybe they all meet at some high energy scale right maybe they're converging on each other and they can somehow meet each other this is not an original idea this idea actually Stephen Weinberg and others talked about this in the early 70s and this was put to good use by Howard George I and Sheldon Glashow in the mid 70s when they invented grand unification so this is the idea that at this energy scale near where they meet grand unification occurs this idea from the seventies it may or may not be true but if you have all the gauge groups of the standard model su 3 su 2 cross u 1 as subgroups of some bigger group you might imagine that bigger group has a single coupling constant so it only is one value but then it's spontaneously is breaking the symmetry there's some fields they're called cleverly x and y bosons that break the symmetry of some high energy so this energy is about 10 to the very roughly 16 GeV remember the Higgs boson is down at 10 to the 2 GeV and electrons and quarks or even lighter than that so there's a very high energy scale the Planck scale is 10 to the 18 GeV so it's pretty close to the playing scale and the scene in the seventies eighties like a good idea right like you know the coupling constants would run they would unify and not only will they meet but we will literally unify them by saying that s u3 s e2 costs you one are all subgroups of some bigger group like SU five or s o ten or six there's different famous groups that people tried to use so without really trying to but you know without not trying to either the way that I drew it here they don't quite meet and it turns out that as we got better and better data you know these these curves the way that alpha 1 alpha 2 alpha 3 change with energy depends on not only their values at low energies and not only the gauge groups therein but also things like what particles they couple to the masses of those particles and things like that you know the interaction strains with the Higgs stuff like that so there's a bunch of parameters that you measure to calculate these curves and in the 70s and 80s we had measured them not so well so to the error bars that we knew then these curves met at some high energy once the 90s rolled around we had measured these curves well enough that they no longer met like the precision became good and we realized they're not actually intersecting so this was this you know there but turns out there's two attitudes you can take it with that one is okay grand unification is wrong by the way grainy invocation also predicts proton decay and you haven't measured that so maybe granny vacation is just wrong but the other possibility is that there are more particles in between you know here's what we actually observe down here like Large Hadron Collider okay there's a lot of room up here in between the Large Hadron Collider and the grand unification scale by the way also a grand unification scale that might be the scale at which something like inflation happens in cosmology so you know there's a lot of good happy feelings warm fuzzy feelings about grand unification but it didn't it didn't work in the most naive way and so maybe one idea is that there are new particles in between the energies we've seen and the granny's vacation scale that affect how these different curves meet so one simple idea is supersymmetry so what you can do is if you put supersymmetry at some intermediate energy scale then you can make the curves tilt a little bit so you get something like this and this this and now they actually do all meet or they would meet if I was a more accurate crafts person so there was this big belief in the 90s and you know some people still believe it today that if supersymmetry exists at energies slightly higher than what we were able to observe it can salvage the dream of grand unification and without it or without something equivalent to mess with these curves it's harder to see that happening it's not impossible nothing's ever impossible particle physicists are very clever and inventing models you notice some of the romance goes out of it though because if you have three lines in a plane and they happen to meet at one point that is a non-generic thing and that's kind of impressive like two lines will almost always meet somewhere but three lines intersecting at the same point is a special thing here you have three lines but also a free parameter you can move where you want the supersymmetry scale to be so it becomes much less impressive that there is unification at all and also we haven't found any supersymmetric particles maybe we'll find them tomorrow we haven't found them yet so honestly people don't know what to think about this people don't know is their grand unification is there supersymmetry any of these things these were very theoretically attractive ideas from the 70s and 80s there's no empirical evidence for them yet and and there's been anti empirical evidence there you know there's things we could have found easily so as good Bayesian z-- our idea our our credence that these ideas were correct is lower now than it was 20 years ago and by the way I'm this is all quantum field theory we haven't mentioned a super string theory yet but there was in super string theory string theory this is old-fashioned string theory so old-fashioned string theory there were five string theories and there was one that was especially popular to be the real world and that was the e8 cross e eight heterotic string and e 8 cross e eight were the gauge groups so we've talked about s o n gauge groups as you and gauge groups there are other ones or others other continuous groups lis groups that would be candidates for being gauge groups of something or another and there's a set of exceptional gauge groups special ones that sort of don't fit in into that other classifications and ei tis one of the best ones it's one of the you know most impressive gauge groups and the string theorists didn't put it in the string theorists went through and said what could possibly be the gauge groups of string theory and they literally only found five answers okay and E 8 cross e eight was one of them and one of the nice things about that one of the reasons why that was thought to be very promising is that there is a very natural symmetry breaking pattern where you go from e 8 cross e 8 and you break one of them entirely so you just get EI t' and that breaks naturally down to what is called e6 I'm not explaining these at all I'm telling you this is true an e6 breaks down to su 5 which can break very naturally down to su 3 cross su 2 cross u 1 and sometimes instead of su 5 is that so 10 or something like that but this is so it was all so even though not necessary for grand unification it was possible to fit grand unification into the scheme of super string theory very very easily easily I said old-fashioned string theory because in the early I guess the mid-90s following Jo polchinski ed Witten and others showed that in fact there's only one string theory in some sense because what we felt were five different string theories and also there's another one 11 dimensional supergravity these are all just different limiting cases of the single underlying theory that is now called m-theory but even though that's an advanced now we know that these are all related to each other we know very little about what M theory is and you've heard the story probably about how there's differ ways to compactify m-theory to get the world and there might be 10 to the 500 different ways and so rather than having five different candidates we have 10 to the 500 different candidates for constructing the real world out of string theory and that's a lot harder task so we don't know quite what to do with that all right we've already gone on an hour here and I discussed the most important things okay I did discuss those but let me let me just do a couple of other things very quickly before going away so electromagnetic gauge invariance I want it so we talked about sorry should not talk and right same time magnetic gauge invariance so I said that this is a u 1 gauge Theory electromagnetism is U 1 gauge theory and in particular you have something like the electron field sigh e goes to e to the I theta which is a function of X because it's a gauge Theory cyi we're ignoring spin and su 2 doublets and all that complicated stuff but this is the basic thing that happens this is the gauge invariance of electromagnetism at every point you take the electromagnetic field and you rotate it by a complex phase that can be different at every point in space and so there were two very very good questions relating this thing that I told you two things you learned somewhere else okay so what about quantum mechanics you know in quantum mechanics you have a wave function and that's also capital psy just to distinguish it from the electron field capital psy is a complex number and I can imagine taking capital psy and multiplying it by some phase call it e to the I alpha may be alpha is not good now health is fine it's not the fine-structure constant just some number alpha times sign and we know the probability is size squared and that just goes to size squared so this is an invariance that is the probability of getting a certain outcome in quantum mechanics is unchanged if I multiply the wavefunction by e to the I alpha right because it doesn't change the square so doesn't that is that related is that the same thing is it similar and you know the short answer is no they're not the same thing they're not related but there's a longer answer he knew there was going to be right so the quantum mechanical wave function remember is a vector in Hilbert space if you have two particles for example well let me say it this way if you have a single particle in good old nonrelativistic Schrodinger style quantum mechanics then the wave function sy is a function of position and time also but it's a function of the position of that particle okay but if you have two particles then the wave function is a function of two copies of space x1 and x2 right and if you have 10 to the 80th particles like you have in the universe then the wave function is a function of 3 times 10 to the 80th different coordinates three dimensions of space 10 to the 80th particles okay whereas this sigh E is a function of X that's it because it's a field it's a classical field it is not a wave function so this thing when I say you know you multiply for the u 1 gauge Theory then you multiply by e to the I theta X that's the same X as where psy is for every value of x psy has a value and the gauge parameter has a value okay here in the quantum mechanical wave function there's no equivalent of that there's no when you say sigh at what point it's you know every different particle this is describing has different locations in space so there's just no map onto that so there's no parameter this this alpha here is not a function of space in any simple way it's just once and for all you can just multiply the whole wave function by a complex phase and everything is invariant so there's a similar structure but it's not the same thing okay the invariance of the quantum mechanical wave function under a single phase rotation is not the origin of the gauge invariance of electromagnetism however having said that there is a relationship I'm not going to go into it in detail but if you go through the details of starting with the quantum field theory wavefunction and then you reduce it to the situation where you just have a single particle electron and then you look at the equations of motion for that particle you can turn it into a Schrodinger like equation for the wave function that looks like the wave equation for the electron field it's you know the Dirac equation gets involved and it's very complicated so I'm just letting you know again if you if you dig into this very deeply you will see there is a relationship but conceptually they're very different the relationship is not central to at what we're talking about here the phase invariants of the quantum mechanical wave function is a different thing than gauge invariants the other question very closely related is what about classical what do you might call Maxwell electromagnetism so if you've taken an electromagnetism class classical electromagnetism there is something called gauge invariance but it doesn't look anything at all like this it looks very different so in what you have in Maxwell electromagnetism is you know you have the electric field E of X and you have the magnetic field B of X and these are the things you know the fundamental dynamical variables Maxwell's equations tell you how these couple two electric charges and magnetic fields are created by moving charges in that whole bit but what do you figure out what what someone tells you or you figure out is that you can have a vector field a called the vector potential so I should write this as a function of X just like a and B and a scalar field capital Phi of X called the scalar potential and from these four numbers so a is a vector so there's three numbers right a X a Y Z Phi is a single number single function so there's one number there from these four vectors you can derive what E and B are and since I don't prepare for these videos I'm not going to get the formula is exactly right I know formulas that are correct but they're like fancy four-dimensional formulas these the old-fashioned formulas I forget but basically E is like - the the derivatives with respect to space of capital Phi the X plus the derivative expected time of a so there's three numbers on each side good that matches and B is what we call I'm just gonna have to use some vector notation here vector calculus here the curl of a which is a fancy way of saying that the x equals the derivative of a Y with respect to Z minus the derivative of a Z with respect to Y and other permutations okay so you don't have to remember any of these I'm not trying to explain these level of detail that you would actually understand I'm just trying to tell you that they're there but the point is there's gauge invariance here so I could always in Maxwell's theory I could just work with E and B I can just work with the electric and magnetic fields but it was even Maxwell knew that you could also rewrite them in a somewhat simpler way in terms of a and Phi so a and B you might think there's three components of e three components of B so there's six numbers six functions right but this little bit of prestidigitation shows you there's really only four functions that you need to derive what both a and B are doing so they're not independent from each other and there is gauge invariance here so the gauge invariance in this case roughly looks like you take a and you send it to a plus the derivative I don't know what the best notation use here is the notation everyone actually uses is this grad lambda of X so grad lambda this is a funny notation with the upside-down triangle which means a vector and the components of the vector are d lambda d xD lambda oops D lambda T Y and D lambda D Z so the derivatives they should actually be partial derivatives if you're being careful but the derivatives of this function lambda in the different directions and there's another thing for Phi it also changes so this is gauge invariance this is a change in the vector potential at every point due to this function lambda that leaves all the equations in Merrion you can go through you can check that E and B don't change when you do this particular transformation so this is what we call gauge invariance in classical electromagnetism does it have to do with the what I'm telling you is do you wanna gauge transformation yeah what it has to do is lambda is the same as theta that is to say ie when you send a goes to a plus the gradient as its called gradient of lambda you send sigh e to e to the I lambda so I eat that's what it is so again in an ordinary non field theory Maxwell electromagnetism you don't need to worry about this because just have particles with charge okay and the particles with charge don't change under gauge invariants at all but once you go to field theory and you want a couple everything together in a simple way this is what you have to do you have to actually let both the gauge field and the matter field in this case the electron change under a gauge transformation but all I'm telling you here is that they really are the same gauge transformation the gauge transformations I'm telling you about are the same ones that we told you about that you were told about in your electromagnetism course and there's more to do with this okay so from all this knowledge you can figure out you know how you can take the fields and combine them together to get things that are invariant under gauge transformations and that's what you need in the Lagrangian the Lagrangian s'ti that defines the theory so I mean maybe one more thing maybe one more thing I gotta tell you okay maybe two more things well one is very quick let me just say it in words people people wanted a little bit more clarity on this claim that if you have a local symmetry so let's say you have quarks red green and blue and you want to compare them at two different points in space so you should know how to transport and the claim was if you wanted to say there was a symmetry that let you rotate the quarks independently at every point in space then in order to compare them you need a connection field and that's literally what a is here this is this is why the photon field is usually not called gamma in particle physics it's usually called a because of Maxwell's electromagnetism this is the field that you quantize in field theory for electromagnetism it's a in fact what you do is sorry I tell you more than you want to know the vector the four vector potential because of special relativity is a mu which is what we called Phi and then what we called a so a four dimensional vector because it's relativity so space-time is four dimensional this is the thing that gives you the photon when you quantize it and so people said well you know what do you mean that this demand of a symmetry causes you to have a connection field in some way and you know so let me just confess that's sloppy language that's a language that we always use when we talk about these things the the more strictly correct thing to say is you have a connection field the world has one okay it's not that anything forces you to do it the way to think about it is if you wanted in order to have a gauge symmetry a local symmetry symmetry that is allowing you to do symmetry transformations independently at different points in space then you need a connection you need a connection field otherwise you have no way of comparing what's going on at different points in space now you don't need to say that connection field is dynamical you could imagine a theory in which the gauge field is just given by God it's just fixed here it is it's not fluctuating it's not changing overtime it's not responding to what's going on elsewhere but it's still there and in case this sounds crazy to you there's a theory just like that it's called special relativity okay in general relativity where you have gravity is dynamical and fluctuating the connection field changes from place to place it has dynamics Einstein's equation is what gives it to it but in special relativity there's still space-time there's still a way to compare vectors at different points in space-time in other words there is a connection but the connection is just given it's just fixed and it's flat the curvature of it is zero so there's no you know there's no necessity that comes along with having these fields you know you you do experiments ultimately when you want to say like people ask well why is it su three across su 2 cross u one do you do the experiments you do particle physics and you say that's what's needed to fit the data right and they say like well if you have three quarks how do you know there's three different if there's three quarks and a proton how do you know three different colors well as we will see there's something called the Pauli exclusion principle two fermions like quarks cannot be in exactly the same state but it turns out that the two up quarks let's say in a proton basically are in the same state how is that possible well one of them is right of one of them is blue or anyway they're two different colors so they're not really in the same state so we know that we need these different colors in order to make sense of what's goes on a proton so all this is driven by data all of this is driven by trying to understand the actual data that's the only real reason we know that these fields exist because you need them to fit the data and the final thing is let me say again a little bit more detail than you want but one more little connection between gauge fields and curvature remember so we said that if you have a gauge field you can parallel transport comparable transport vectors in some generalized sense like a quark with its colors I'll take a little internal vector and parallel transport lets you define the curvature of that gauge field and we talked about the Riemann tensor and depending on whether you watch the Q&A video one way of writing the Riemann tensor is are Rho lambda mu nu but it's a map from three vectors V 1 V 2 V 3 to a fourth vector V 4 and the map is this little thing I'm not going to go through all the details but V 1 V 2 backwards minus V 1 minus V 2 and then if you have V 3 it gets changed by an amount before okay so that's what the map is and so you might say so there's two questions here can you have there's a lot of questions here sorry can you have zero curvature can you have a connection without curvature can you set the connection which is flat the answer is absolutely yes and that would be special relativity for example in space-time so the connection this is the Riemann tensor what I didn't tell you is that there's something called the connection I told you there was something called the connection but I didn't tell you the formula or the symbols the connection looks like this gamma lambda mu nu okay and this is roughly what tells you so the real intention tensor tells you how does a single vector rotate a little bit when you go around a loop defined by two vectors roughly the connection is saying if I just have a single direction that I'm moving in okay the one and then I have a vector that I am moving the two and I move it a little bit it sort of changed infinitesimally okay compared to where it was and that's v3 so that's why there are three indices on the connection and the way so this connection tells you how to transport in one direction but that just depends a lot near the details the actual equations depend a lot on what your coordinates are and what direction you're moving in etc whereas the Riemann tensor is what we would call gauge invariant it is independent of your coordinates or anything like that it's a much better defined thing and to get to the question that was being asked there are exactly equivalent structures in the gauge theories for internal symmetries like su 3 costs su 2 cross u 1 so for su 3 instead of the connection sadly we still use the letter A whether it's su 3 or su 2 or anything we would write a mu a B why do we use different letters why is you know the connect so for the Riemann tensor and the space-time connection the reason why gravity and space-time are harder or more interesting is because it's only space-time there's only vectors that literally point in directions so the question what about the vector you're moving and the question what direction are you moving it in our questions being asked relative to the same space namely space-time here when you have an su 3 connection then you're still taking a direction in space-time V 1 okay but you're saying I'm moving a quark field so let me draw in a different color Q and that gets moved to here under parallel transport so it changed a little bit and I when asked by how much does the quark field change so these indices a and B are quark indices or really s u3 indices and so now unlike in the Riemann tensor and space-time case where all indices are in the same direction or in the same space up down left right forward backward past future here there's indices in color space which are a and B and an index in space-time which is the direction you're parallel transporting it and there is curvature su3 curvature and roughly speaking we call that capital G I shouldn't call it capital G let me call it capital F mu nu a B and yeah it's complicated sorry but I got to tell you what it is I started so I shouldn't stop it is the derivative in the mewed erection of a new a B minus the derivative in the new direction of a mu a B minus or plus on them you're going to get it right a I don't having a this right a mu a see a new C B and call that plus and then minus a new C who something like that this is a very close approximation of the true thing anyway the point is there's an absolute parallelism from the connection in space-time you can construct the Riemann tensor from the connection in the gauge Theory su3 construct this curvature tensor and from this curvature tensor you're going to make the equations of motion for the gauge field so for electromagnetism for example I forget I think I might have said this already for electromagnetism F mu nu doesn't need to have any indices because it's just a single complex number that is being moved around there's no indices it's not like su 2 or su 3 where there's two or three different directions there's one direction so FME doesn't need any indices for E&M just a u 1 gauge field and it all circles back this is there are zeros it's a four by four matrix zeros on the diagonal how do you know that because if you look closely this formula I wrote down its anti-symmetric in mu and nu so whatever the value of f mu nu is for one value of mu and nu when I switch mu and nu it better be minus that so this is F 0 0 F 1 1 etc so if I switch 0 and 0 I get 0 and 0 so the number that is equal to minus itself is 0 so all these diagonal elements are exactly 0 okay and the top line here I never get the minus signs here right but this is the electric field e x ey ez minus ax - ey - easy and these other three empty slots the magnetic field B Z minus B X B Y minus B nope I didn't get that right did I I mean how I had a good chance of getting it right just didn't quite see that's B Y this is B X I'm not getting the signs right but you want to get the indices right this is minus B X this is B Y this is minus BC anyway way more detail than you need way more detailed you can possibly follow all this is to say that the good old fashioned way of doing electromagnetism that Maxwell would have recognized is still here it's still there in this fancy Dancy language of connections and gauge fields and things like that they're all related to each other and you know I'm not giving you enough information to understand all these indices and but it's actually pretty understandable like it's way easier than quantum mechanics to be honest so if you can find a good book on classical gauge Theory I'm not sure if there is one but I think there Valarie Rubik off wrote a book it's a big hard book but it's called classical gauge Theory that's one but anyway Gabe's theories are worth learning about they're really interesting there's a lot going on I am many many more things that I could say but you know ARS longa vita brevis we have other things to do so thanks for sitting around with this one this was a long complicated Q&A video but it's worth it this is important stuff this is how the real world works

Channel: Sean Carroll

Views: 40,866

Rating: 4.9361701 out of 5

Keywords:

Id: KIs5mYNZ15o

Length: 87min 53sec (5273 seconds)

Published: Sun Jul 05 2020

Reddit Comments

[u/fubarbazqux]

I feel like this episode was impenetrable if you don’t already have some background in particle physics. He’s just spitting new unexplained things left and right.

How am I supposed to understand what spins different gauge bosons should have? Even in most general terms, there is no explanation where spin comes from. Is a scalar just a spin-0 particle here?

Why do left-handed and right-handed electrons have not the same kind of wave function for Higgs coupling? I’d of course naively expect symmetry there.