The Biggest Ideas in the Universe | 15. Gauge Theory

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll and today's big idea is kind of a payoff I know that in the last couple of videos we did ideas that were heavily mathematical we did a video on geometry and topology then we did a video on symmetries which is of course based in physics but what we talked about was the mathematical language in which we describe symmetries the language of group theory so today we get the payoff today we're gonna show why we had to do those other two videos you know we love math nothing wrong with math but we're really about the ideas that describe the physical world the physics ideas and today's idea is gauge Theory gauge theory is very directly thus basically you should be able to think of it as geometry we're not gonna get too much into kepala G today but we'll get there eventually geometry wedded to symmetry that's what gauge Theory is all about you may have heard the phrase gauge Theory gauge theories are at the center of modern quantum field theory the standard model of particle physics is a gauge Theory gauge symmetries are gauge theories are theories that have a certain kind of symmetry that gives rise to fields that give rise to forces of nature that's what we're gonna go why are they called gauge theories there's this story that it has to do with railroads somehow there is this this technique back in the old days in the early 20th century that you would have a gauge a literal sort of physical measuring rod that they would use in building the railroads to make sure everything lined up correctly of course it doesn't matter how big your gauges as long as you use the correct numbers of them to get from point A to point B in other words it's just like saying it doesn't measure it doesn't matter whether you measure a distance in centimeters or inches okay as long as you get the same physical distance and so this idea was picked up by Hermann vile a mathematical physicist in the 20th century who talked about the invariance under scaling and how that could give rise to forces of nature and that that terminology was adopted into gauge theories but the gauges were talking about the gauge theories were talking about have nothing to do with scaling we're not gonna be making things bigger or smaller we're gonna be rotating things together that's the modern way that we generally have gauge theories showing up in particle physics in fact the most obvious example to look at just to fix their ideas are quarks we talked about quarks a little bit quarks are what make up the matter particles that make up protons and neutrons and heavier particles and and some lighter particles so for example the proton we said is two up quarks and a down quark the neutron is two down quarks and an up quark there are also mesons like the PI on that is a quark and an antiquark etc okay but the extra fact about quarks that really makes them different than the non quarks the leptons the electron the muon the Tau the neutrinos and so forth is that quarks come in three colors of course the idea of colors let's say red green and blue there was a brief movement movement to make it red white and blue but that was considered a little jingoistic so red green and blue was a little bit more physics oriented this is an idea from Murray Gillman the famous physicist he was one of the co-inventors of quarks and the idea is that there's not just one a pork field for example there's just one electron field basically but there are three different up quark fields a red one a green one and a blue one again not actually red green and blue if you shone light at them they would not reflect red green or blue they're just labels okay just a way of saying there's a multiplicity of three different kinds of every single quark so we could imagine drawing a little vector space it's not just that there are three quarks there's a red a green and a blue it's that every quark is some combination of red green and blue some linear combinations we can imagine here is the actual quark field it's living somewhere in this space of red green and blue that's the idea okay electrons do not have that color property but quarks do and so you will not be surprised to learn that whether you're you call any particular quark red or green or blue doesn't matter in fact it doesn't matter if you call it some linear combination of red green and blue all the matters is that the relative relationship between different quarks is going to be important but the actual value for the color of any quark doesn't matter so there is a symmetry oops I should be consistent here color wise there is a symmetry and the symmetry is rotations of these three different colors of quarks into each other it doesn't matter how we oriented the axes in these three-dimensional space also quark fields like all other matter Fermi on fields in nature whether it's electrons or neutrinos or whatever are complex-valued fields so in fact it's a complex rotation and we talked about that in the last video the symmetry here is su 3 the unitary group the special unitary group of three different complex vectors rotating into each other so there's an su 3 symmetry built into the quark Fields and in fact there's this it's a complicated history because there's an entirely different su 3 symmetry which we eventually realized was a symmetry between up quarks down quarks and strange quarks this is only an approximate symmetry it's not really as good but it was still crucially important in understanding particle physics in the 60s that's called a flavor symmetry for some reason up down and strange are flavors there you go whereas this is a color symmetry so sometimes you see it written as SU 3 color and this is the one that is going to be crucially important this is the actual one that enters the standard model of particle physics in a fundamental way this is the relationship for each quark for the up quark it can come in red green and blue the down quark comes in red green and blue the strange quark comes in red they all do okay all six of them and now when you have a symmetry like that there's a basic basic choice you need to make there's two different ways that a field theory can have a symmetry like su 3 or whatever depending on what kind of fields you have one way is to say look I can rotate in this field space in this color space in red green and blue space there's a way that I have of just rotating everywhere in the universe because remember these are fields right they live at certain locations in space no sorry let's say it the other way around every location in space there's a value of the fields so the question that we're asking now is how do you rotate in color space at different points in space at the same time okay so one option is you rotate globally that is to say you have an ocean an idea of what it means to rotate a fixed amount let's say red gets rotated into blue okay you rotate around the green axis you rotate by a fixed amount everywhere through space exactly the same way okay that is called a global symmetry and actually the flavor symmetries are like that so the flavor symmetry between up down and strange quarks is that something that just makes sense everywhere in space you know whether the quarks are up-down or strange you can change your definition globally all throughout space and that's useful that's helpful that's their you know helps you sort of classify what the field theory is doing what kinds of particles can be made what kinds of interactions there are okay but there's a much more powerful idea which is that you can rotate locally and what that means this is when you actually get what we call a gauge theory what that means is that I can separately at one point in space rotate my red green and blue axes into each other and at another space I can rotate in a completely to the point in space I can rotate in a completely different way so in other words to draw a picture here is space you know we're getting good at drawing space now and here's a point in space and at some point in space I have three little axes corresponding to red green and blue at some other point in space I have axes corresponding to red green and blue and the gauge symmetry the local symmetry says that it is an invariance of the theory that I can change red green blue I can rotate them at one point under some su3 transformation and at another point at each other point in space I can rotate them completely separately okay so I might have a quirk that I said was blue over here and over here the same quark field is green or something like right and the point is it doesn't matter it doesn't matter at every point this is a much bigger symmetry right so scientists physicists will sometimes say su3 is an eight dimensional group right because we said N squared minus one is the dimensionality of SU n and if you have a global symmetry that's exactly correct you have an eight dimensional symmetry group there's eight different things you can do when you have a local symmetry a gauge symmetry the true symmetry group is infinite dimensional because it is a separate su three rotation at every point in space and there an infinite number of points in space so the mathematicians look at the physicists and go like you clearly have an infinite dimensional group here but that's because we've suppressed the dependence on spatial location and are just thinking about the SU three transformations okay that's what a gauge Theory purports to be it's a theory where there is a symmetry where that symmetry can be separately a rotation at every point in space but like we said you might want to say okay here I have some kind of quark Q what is important is not whether that quark is red green or blue that doesn't make any difference at all that's a complete choice of labels okay it has no physical effect but I might have another quark over here I'm trying to choose colors that are not red green or blue here's another quark over here I might want to compare that quark field value to that quark field value and like they might one way to say like how are they interacting or something like that right so one way to do that would be to parallel transport the court field along some curve through space some trajectory and say like what do I get at the end of the travel transporting in other words once I have a gauge symmetry I need to be able to compare the values of quark fields at different points in space even though I can separately change the labels at those different points in space right whether a quark is red green or blue in one space is completely what point is completely up to me whether it's red green or blue at another point it's also completely up to me separately so I need some way to physically say are they the same color are they different colors or whatever and in order to do that I need to be able to parallel transport the quarks we talked about parallel transport in the geometry in topology video two ideas ago parallel transport is just keeping a vector constant parallel to what it was as you move it along some curve okay and to do that you need what you need to be able to know what it means to keep a vector constant as you move it along a curve is a connection field the connection is exactly the set of data at every point in space which answers the question what does it mean to keep a vector constant so if you want to gauge theory if you want to gauge symmetry if you want to be able to separately rotate the color fields at one point in space and another one but you also need to be able to compare them it follows logically that you need another field you need a new field you need not only the quark fields you also need a connection field that tells you how to relate Kork fields at different points in space okay so that connection field or sometimes more and more often it's going to be called a gauge field and this extra field over and above the quarks this extra field that allows you to implement the gauge symmetry is what's going to give rise to forces to forces of nature in this case in the case of su3 it's gonna give rise to the strong force to what we call in this case quantum chromo dynamics because it is the quantum theory of the dynamics of the color fields aka QCD so it's really the dynamics it's really the fact that the you need so you have these fifties axes red green and blue I'm telling you you're allowed to rotate them however you want that implies the existence of a gauge field connection that lets me parallel transport and that connection will have its own dynamics and it's the dynamics of that connection field which is going to give rise to the strong nuclear force to QCD okay in fact it turns out that this is in fact how nature works that all the forces of nature work this way one way or the other there are four forces that we like to talk about and by the way sometimes people try to get sticky about what you mean by the definition of the word force in this context the word force made perfect sense and we're doing classical mechanics and F equals MA here in the context of quantum field theory it makes much less sense there is no fundamental notion of the word force we just use the word informally to refer to these four gauge theories that we have lying around so if people say well does the Higgs count as a force this neutrino exchange counters of force does the Pauli exclusion principle count as a force who cares I don't care whether they're forces or not they exist they're real features of the world so are these that's all I care about so we know what the four forces the the standard four forces are electromagnetism these are all gauge forces in some sense electromagnetism you know we worked out electromagnetism back in the 1800s right Maxwell and their friends and in fact they invented Maxwell knew about what we would now call gauge transformations he didn't use that language obviously but now with real quantum field theory it's a much more powerful thing so let the electron field be denoted sy e - as a function of X okay so lowercase I so it's not a wave function it's a classical field that will go into making the wave function but I don't want to call it e because you know we have Euler's constant like it goes into e to the I theta that's gonna be important in a second so I don't want to call it easy so let's just call it size of U - this is the electron field and there is a u1 it's a complex valued okay it's complex so there is a u 1 symmetry of the electron field that sends sy e minus of X 2 e to the I theta and theta also depends on x times sy e - of X so in other words remember we talked about u1 back in the symmetry video last week you want assist a rotation in the complex plane so the electron field is a complex number in fact it has other indices it's more complicated than that but it's a complex field okay so I basically take that complex value of the field and I rotate it by an angle theta and I can rotate it separately by a different angle theta at every point in space so the whole theory is supposed to be symmetric invariant under these rotations by this value of theta which is completely up to me theta is itself a different value theta is not a physical field it's just the way I'm changing the what I call the real part and the imaginary part of the of the electron field at every point in space so that implies the existence of a photon field and usually in the lingo in the quantum field theory literature that's denoted a or mu the lowercase mu is an index because it's a space-time vector the photon field points in a certain direction I want to try to simplify our lives I don't know how well this is gonna work what I was gonna call it gamma of X because photons are given the letter gamma in particle physics and I'm gonna forget about the indices forget about all of that the point is this is the connection okay that's what the photon field is it's the connection in the u 1 symmetry of the complex rotations of the electron field that's what it is that's why we have electromagnetism because there is a gauge symmetry and in a word two parallel transport to the electron field from one point to another we need a connection and that connection is a separate field with its own dynamics and we call it the photon that is a way of thinking about why electromagnetism exists at all ok then that's one force of nature we have another force called the strong force strong nuclear force but you know it's inside the nucleus not between nuclei these days and that's QCD that's what we just talked about so that is an su 3 gauge symmetry a separate su 3 rotation at every point so I'm not gonna so because now there are three cork fields right for every for every kind of cork for every flavor of pork like up there's a red a green and a blue so there's again indices and you need a matrix to sort of rotate them into each other let's just not go there this is we don't need all those complications but what we need is the fact that there is an Associated connection field so the connection for QCD is called the gluon field and just like there are three different colors of quarks there are eight different gluons why are there eight different gluons can you think about it because su3 is an eighth dimensional group that's why there's eight different kinds of gluons so one gluon field for every different way that you can rotate the quark fields into each other so again this is this is a specific version of what what we call yang-mills theories the yangon mills see and yang and arm mills in the 1950s took electromagnetism which is a u 1 gauge theory they'd figure that out by I'll probably vial figure that out and Yang and mill said well look we know there are these more complicated groups these non abelian groups remember we said abelian meant that a symmetry has the property that it doesn't matter how you do one rotation versus another the order of operations does not matter if you're non abelian though and for you one that's the case if I'm just rotating around a circle it doesn't matter whether I rotate by five degrees and then ten degrees or if I rotate by ten degrees and then five degrees so can you get the same answer but if I'm in multiple dimensions I can rotate around one axis and then another and suddenly it does matter which order I do it ends you get a non abelian group so Yang and Mills were the ones who proposed this idea that there could be non abelian gauge groups which would lead give rise to forces of nature we didn't know what the groups were we didn't know what the forces were at the time so it was um Nambu and Gelman and fridge and other people who first put together the actual su3 gauge theory that is the strong interactions quantum chromodynamics in other words when we have the weak force the weak interactions are more complicated and the reason why they're more complicated is because of spontaneous symmetry breaking okay so we'll get to that but the actual way the standard model of particle physics works out is this beautiful intricate complex structure that you know you look at it and it kind of looks ugly and clutched together but then you play with it you realize in some sense it's the most general thing like given certain constraints given certain symmetries etc the standard model is what it is it couldn't have been anything else so it's kind of beautiful in that weird sense it's a beautiful thing once you learn to appreciate it so for our purposes right now all we're gonna say is that you can think of the weak interactions as rotations of the two-dimensional space that can be spanned by let's say up quarks and down quarks okay so not three different colors of quarks which are true for each up or down but rotating up and down quarks into each other that's what the SU that's what the weak interactions do so that is an su 2 gauge theory because the gauge there you can do it separately everywhere and every different point in space the up and down quarks are themselves complex so it's an su 2 gauge Theory rather than so2 or something like that and it's not just up and down it's also other what we call doublets weak doublets so the electron and the electron neutrino are another weak doublet and there's other different copies again there's complications here because of symmetry breaking but this is the basic idea here so I just need to say this out loud is always gonna bug me um these don't look like there's a symmetry between them everyone knows which is the electron which is the neutrino okay the electrons are negatively charged neutrino is uncharged it's neutral the up quark is charged plus two-thirds the down quark is charged minus one-third they have different masses so the quark colors red green and blue all three colors are really exact we the same in every way because of the symmetry relating them here this looks awkward because the up and down quarks are not the same they different masses different charges the electron and neutrino are not the same different masses different charges that's because the symmetry is broken okay I just gotta get that off my chest the symmetry is there but it's hidden from us so it's not manifest in this way of writing things down finally there's gravity now you might have been told the gravity is a theory of the geometry of space-time we talked about geometry we haven't really talked about gravity in any detail yet but we talked about geometry we talked about the fact that there's a metric that lets you measure distances along curves or areas or volumes and so forth there's also a connection in space-time that lets you move vectors parallel transport them along curves and there's a relationship you can derive the connection from the metric so up here there is just the connection this is an internal symmetry okay the symmetry is in this internal space it's not pointing in different directions in space-time at every point in space-time you can do the rotation but what you're rotating is not the directions of space-time themselves it's this internal made up fictitious space red green and blue so here there is a connection but there's no metric there's no distances there's just a way of comparing vectors by parallel transporting them here in space-time you have both you have both a metric and a connection and that's what makes it much more complicated because again everything there's this theorem that you know the world arranges itself to be as complicated as possible for graduate students in theoretical physics so there's the electromagnetic force the strong force the weak force and gravity they're all in some sense gauge theories but they're all different this is fun but also a little bit annoying okay what is the gauge group so this this idea was actually first proposed by our rukia MA in the 50s 50s something like that soon after yangon Mills proposed non abelian gauge theories Judy Amma says well let's think of gravity listening general relativity as a gauge theory and he said that it could be the gauge group was so3 comma one what is that well you know we talked about rotations right s o n is the rotations of n real vectors space time is four-dimensional so you might think that the gauge group of space-time is so4 because you can rotate things together but a little bit of reflection makes you think wait a minute I can't just rotate space and time into each other in the same way there's a slight difference there right what I can do is boost I can change the velocity at which I look at space and time I can change my reference frame so there is a slight difference in space time between the time like direction and the space like directions so this is what so3 comma 1 means it means there are three dimensions of space and one dimension of time and this is actually the group of transformations you can do at a point in space-time it turns out that for technical reasons this is not the best way of thinking about general relativity or gravity this is what is called the Lorentz group Lorentz transformations right the Ren's transformations include rotations in space as well as boosts there's also the punker a group I'm not going to get the accent in the correct direction but onry ponca Ray was a very advanced mathematician he knew about the group theory that was being developed was hot stuff back then and the Pont gray group is the Lorentz group so the rents group is rotations and boosts plus translations translations are just you move everything either in space or in time and it turns out again technical reasons but it's best to think of gravity as a gauge theory of the plunk or a group not just the Lorentz group ok it's all because space-time is special unlike all these forces unlike electromagnetism strong and weak nuclear forces all which are internal symmetries relating fields to each other at the same point in space time gravity actually is transformations of space-time itself that's why it becomes a little bit more complicated we're not gonna talk about this at all by the way we're not gonna talk about gravity as a gauge theory or anything like that I just want you to know that is something you can talk about and people have been talking about it's unclear whether you get any benefit from talking about it that way like Einstein in back in 1915 invented the theory and that's more or less a good way of thinking about it okay I just wanted to let you know that all four of these forces of nature can be thought of as gauge theories one way or the other let's now shift gears and put it to work a little bit okay let's actually make some money off of the fact that there are these symmetries so what do we know we know that we have some quantum fields we know that the quantum fields appear in different forms they're complex some of them are complex so they have you one gauge symmetries some of them are doublets so they have an S you to some of them are Triplets of color so they have an su 3 that's where the standard model gauge group comes from we know the existence of these gauge symmetries itself implies new fields the connection fields okay and the connection fields give rise to physical particles oh yes I forgot to tell you physical particles for the weak interactions they are the W and Z bosons so for electromagnetism the photon strong force the gluons the weak force the W and Z bosons are the physical excitations physical vibrations in the SU 2 connection field okay I mean think about that you totally understand that you understand that sentence perfectly well W and Z bosons are the physical excitation from vibrations in the su to gauge field but you would not have understood that sentence nearly well enough nearly as well just a few weeks ago progress is being made okay maybe none of us understand it perfectly don't don't be upset about that all right but the existence of these fields giving rise to interactions giving rise to forces of nature comes from the gauge symmetry that's the first important lesson the next important lesson is that the existence of these symmetries places suck an incredibly strong constraint on what the theory actually is remember we talked about effective field theories in the in the idea lecture on renormalization we made a big deal about the fact that we could put a cutoff on our quantum field theory we could admit that we cannot extend it to arbitrarily high jeez and then you might think that there is an infinite number of terms in the defining equations of motion for your fields right you need some equations of motion to give dynamics to the fields you're talking about and in fact the way we do that is again through an action so an action is the integral over time of the Lagrangian which is the integral over DT and space D 3x of the lagrangian s'ti the garage density is a function of fields and derivatives of fields so by derivatives the fields we mean derivatives affected time - respect to space things like that so there is a function but the hope the point is that L is a function of Phi some field some collection of fields at X and then derivatives of Phi I'll write it with the curly D because they're partial derivatives derivatives of Phi also add X so it is a function of X if you have some particular field configuration at every point there is a value of Lagrangian associated with that field configuration field configuration through both space and time okay and so as I said as I said when we talk about effective field theories there's only a finite number of terms that we will ever write down in lagrangian up to some if we only care about the relevant and marginal operators in the theory the relevant and marginal interactions in the theory so that's really very very helpful we only need to write down a few kinds of ways to multiply fields together and we have the most general effective field theory we can possibly write down and nevertheless there's a lot of fields in the standard model so you think that well ok there's still a whole bunch of interactions I can write down but you are saved by the existence of these symmetries because there's an extra requirement that every term in the Lagrangian has to be individually but that's that's that's too strong the Lagrangian as a whole has to be invariant under the symmetries that you talked about so if you have an su three symmetry you should be able to do that that symmetry transformation so let's say su three sends some quark field to some transformed quark field both these are functions of X okay and when you plug the transform quark field into the Lagrangian you better get the same answer even that is again technically a little bit too strong it's the action that needs to be invariant but usually the action being invariant means the Lagrangian of aerion usually the Lagrangian being invariant means every term is separately in Merrion that's not exactly true but that's pretty close okay so what is this how does this actually you know make us money how does how do we make some benefit off of this so think about the electron field electron field we said sigh e minus of X and there's au 1 gauge symmetry which sense I - oops - e to the I theta which is itself a function of x times sine okay you rotate in the complex plane by a little bit so so imagine contemplate that in your well sorry let me just back up a little bit let's get this exactly right this is this is tricky the Lagrangian roughly speaking is the kinetic energy minus the potential energy okay so as we go through I should actually read my notes that I have here rather than just sort of mouthing off as we go through to try to construct our Lagrangian we have to ask ourselves what different kinetic energies can we have what different potential energies can we have and the difference between them is roughly the kinetic energy is the thing that depends on the time derivative how quickly things are changing that's the kinetic energy whereas the potential energy just taint just depends on the values of the fields themselves okay there's no derivative of the fields in there roughly speaking so the kinetic energy how do you get a kinetic energy for oops how do you spell kinetic the kinetic energy for the electron field we already knew like we had that mr. Dirac told us that when he invented the Dirac equation but now we're gauging it right we're giving this extra gauge symmetry it affects the the kinetic energy of the electron field a little bit but not in a crucial way we're gonna ignore that for right now but we also have the photon field right we have a new field if if we want to have this gauge symmetry we need this connection field the photon field it needs to have its kinetic energy okay and that we just need to invent from scratch so there's a long story that I'm gonna make very very short the kinetic energy of a gauge field which is the connection right what can I say comes from is made from comes from well what could it possibly come from what do you have when you have a connection lying around if you remember the video the idea that we did on geometry connection is a way to parallel transport things and by parallel transporting something we find out how it changes around a loop if that's what we want to find and that gives us something called the curvature in space time it was called the Riemann curvature tensor so guess what if you're not a space time symmetry if you're an internal symmetry there's still a curvature you can still take that quark field in red green blue space you can try parallel transport it around a loop it will be rotated in general you can make that loop infinitesimally small and you're going to construct a curvature and the curvature unlike the connection field itself is gauge invariant it is something that can be written in a very simple way it's gauge covariant technically I am sorry I apologize for all the technical footnotes here but I want to talk to people who know this stuff already and to people who don't it transforms nicely under gauge transformation so comes from the curvature I hope that's the right spelling of curvature yes so to each connection is associated curvature we can calculate it it depends on the derivatives of the connection and it depends on it in a way that gives you a nice thing that transforms nicely under gauge transformations so you from that curvature usually by taking the curvature and squaring it you can construct the kinetic energy for the gauge field okay so for example in electromagnetism I told you the connection field is the photon field what is the curvature what is the curvature of the gauge field well it equals the electromagnetic fields the E and B fields this is the electric field this is the magnetic field again Maxwell didn't know about this Faraday didn't know about this but now you do know about it okay the electric field the good old electric field the one you can measure by having an a gal vomiter the magnetic field that you can measure with a compass these are the curvatures associated with the connection that we call the photon field it's all this beautiful underlying geometric structure okay so and the same thing is true for gravity also this is the curvature of space-time connection itself that gives you the kinetic energy for gravity for Einstein's theory of general relativity what we call the einstein hilbert action or the einstein Hilbert Lagrangian okay so that was one way in which gauge invariance helps us figure out what the Lagrangian can be it figures out what the kinetic energy of the gauge field itself is what about up here that I started this and then I got distracted here okay so let's pick this the electron field varying in that way I'm gonna copy it I'm gonna paste it I'm gonna make it bigger okay so this I don't think it's what I wanted to do I'm gonna paste it again there we go okay what I just said here was that the kinetic energy for the connection field is basically determined by the requirement of respecting the symmetry respecting gauge invariance what I'm going to tell you now is the potential energies are also highly highly constrained by the gauge invariance in this case the potential energy of the electron field itself so the electron field potential energy what does that mean well the potential energy depends on the field but not its derivatives okay so you might say you might guess that there is a term in the Lagrangian that looks like psy e - squared why not that's you know sort of like the equivalent of the simple harmonic oscillator right x squared is the potential energy in the simple harmonic oscillator maybe there's a potential energy that looks like the electron field squared but under this symmetry right cyi x goes to e to the I theta sorry X so this would go to e to the 2i theta sy e - squared so it's not invariant it changes under this gauge transformation we have to somehow do better we can do better we know how to do better I'm sort of going very slowly here the obvious thing to do is sigh a - squared with absolute value signs around it right because that is sy e - complex conjugate times ie - ok and that in fact goes - under this gauge transformation e to the minus I theta sy e - complex conjugate e to the plus I theta sy e - and then that equals oops I shouldn't write the arrow I should write equals e to the plus I theta near the minus I theta cancel so you get sy e - squared so this is invariant under gauge transformations so this beast over here turns out to not be good for kinetic for potential energies in Lagrangian this one is good and that has a dramatic consequence because again for reasons why Baraka to get into here if sigh II - is the electron then the complex conjugate sigh II - star creates positrons it's the anti particle the complex conjugate is the anti particle of the original particle that we started with I should be able to draw a better size than that I suppose I've drawn them enough in my life I should get good at it so what this means is terms like this Sai star sigh can appear in Lagrangian in fact for other complicated reasons the real well sorry so that can appear in Lagrangian so you can get sy e - squared which is sign a - star which is really the positron times ie - this gives mass to the electron x squared or Phi squared or size squared is where the is the term in the Lagrangian that corresponds to the mass of the corresponding field okay there's a longer story here with the weak interactions etc but this is the thing you can write down in quantum electrodynamics there's no problem with electrons being massive generally the rule in quantum field theory is if there's nothing if there's no symmetry or principle that prevents something from happening then it happens okay so if the electron were massless you'd expect there to be some symmetry that prevented it from getting a mass but there's not here we can write down a mass term for the electron no problem we can also write down an interaction term there's a little more subtle and again I'm glossing over some details but the interaction looks like the same thing sy star II - sy e - times gamma the photon okay so this corresponds in Flyman diagram language to that vertex that we drew where a photon comes in and how it goes an electron and positron so the point is that reason why I'm for this is a little bit of detail here I know but the reason why I wanted to go over it is you get a immediate very powerful physical implication of this gauge symmetry okay we could write down determine the Lagrangian that coupled a single photon to an electron and a positron we could not write down in a gauge invariant way a term the coupled a single photon to two electrons all by themselves two electrons all by themselves would have been this thing and that is forbidden okay so gauge invariance the demand of all the terms in your Lagrangian being gauge invariant is enforcing the conservation of electric charge gauge invariance is the thing that says that if you start with a neutral particle like the photon if it's going to turn into charged particles it will be an equal number of positively charged and negatively charged particles and that's a reflection remember of northers theorem northers theorem said that when there's a symmetry there is a conserved quantity the conserved quantity associated with u1 the gauge invariance of the vector magnetic field is charged is electric charge and you can see that at the level of the Lagrangian the level of the terms you can write down the define your effective field theory so long story short two big things just happened three big things have happened so far in this in this video one is there exists ways of having gauge Theory symmetries gauge symmetries that can separately rotate things at different points in space the price you pay or if you like the benefit you get there's a new field you need the connection and that connection gives rise to a force of nature second thing is you can calculate the curvature of that connection and use that to define the kinetic energy of the connection field so the Lagrangian the equations of motion if you like for the connection field itself is strongly constrained just by gauge invariance and you use the curvature to get there third you can also constrain the the Lagrangian associated with the matter feels with the the electrons or the equivalent things to say about quarks and so forth so the set of interactions you can get in your theory and whether or not you can get a mass for your particles is also highly constrained by gauge invariants so you can see how you know why physicists fall in love with this not only does gauge invariance give rise to the forces it's an extraordinarily powerful constraint on what can happen in physical interactions and that's exactly what physicists like you know we don't have an infinite set of possibilities to think about we have a very very constrained set of possibilities okay there's a final consequence so we talked about what do we talk about we talked about the kinetic energy of the gauge field we said the kinetic energy of the matter fields of the electrons just sort of goes along for the ride there's a little bit of a change there but it's not a big deal we talked about the potential energy of the matter fields and that is affected by gauge invariance you can give it a mass but it involves the positron as well as the electron and also the interactions another that's another kind of potential energy because it does not involve any derivatives of anything is also highly constrained by gauge invariants what are we left with well we're left with the potential energy of the gauge field right because we did the kinetic energy we did all the others right so what about the potential energy of the gauge field the first thing you might want to write down so if the if the photon field is gamma of X can you write down can you guess gamma squared okay gamma the photon field despite the fact that it is carrying it is sort of conveying information about the complex-valued electron fields gamma is help as a real field is not complex it does not get rotated under gauge transformations that's not what happens to it under gauge transformations and under u 1 gauge transformations there is something that happens to the photon fuel gamma of X oops gamma X goes to gamma of X plus derivatives I'll write a little funny D sign for derivatives of theta so theta the unk transformation itself is e to the I theta of X remember a separate little rotation at every point in space and theta itself is what enters the gauge transformation of the photon field now you might worry because up here I wrote a gamma and it looks like that would change but it's compensated by other terms in the Lagrangian that's why I didn't go into it in detail so that so forget I ever said that think about this the point is this gas is wrong right this would go under gauge transformations to gamma plus derivative of theta squared and that is not a variant that's just some messy thing it's not that messy it's a its gamma squared plus 2 D theta gamma plus D theta squared we can do that we can do that much math anyway it's not a variant changes into something else so you cannot write down a mass term for the photon there's no there's no equivalent of taking the complex conjugate to get rid of it because it transforms in a different way under the gauge transformation so that's it that's the correct result from this the answer is gauge bosons as we call them the particles that correspond to the connection field that comes from the gauge symmetry are massless that is a result of gauge invariance okay that's why the photon is massless you've been wondering since we started talking about photons why are photons massless why can't they have a mass this is why because photons are the gauge bosons of symmetry and that symmetry prohibits the term and Lagrangian that would give them a mass so this is a feature of gauge symmetries that the bosons associated with them are just massless this is enormous ly useful this has giant implications the massless nests of the photon to me it's intrinsically interesting right the photons are massless I already told you the theorem that says that things need to be as complicated as possible so this is not the end of the story but it is a very important part of the story the examples where the photon the the gauge bosons are obviously massless are photons and also gravitons people sometimes wonder like you know you never seen a graviton etc if you if you've been following this far all the ideas videos we haven't gone into general relativity and gravity just yet but general tivity is a field theory and quantum mechanics is out there so it's going to be a quantum field theory at some point there is already an effective field here you can write down for general relativity and that just predicts the existence of gravitons we haven't observed them directly but if you don't believe in gravitons you have to undo something absolutely fundamental about the nature of quantum mechanics itself you can try to do that good luck with that but the rest of us sorry is gonna believe that gravitons exists even if we haven't observed them yet so photons and gravitons because of this gauge invariance that we cannot write down a mass term for the Lagrangian and therefore they have zero mass and this gives rise to Coulomb's law for photons and Newton's law for gravity that is to say when you have some let's say you have an electric charge okay the fact that photons are massless that this is not very respectable what about to say but it gets you the right answer and it's close enough to being respectable very roughly speaking the fact that photons are massless says we can make them meet implies that we can make them without any cost okay you can make an arbitrarily low energy photon the energy of a photon can be as low as you want energy goes like you know e equals H F the frequency can be as small as you want giving rise to a low energy photon as low energy as you want so from an electron there will be lines of electromagnetic force coming out okay in all three dimensions and if you draw a sphere of radius R around electron these lines of force go out they don't end they just go out forever because it doesn't cost any energy to make these photons and therefore the density of lines of force goes down as the area of the sphere this sphere here the sphere has an area that is proportional to R squared and therefore the density of lines of force goes down as 1 over R squared and therefore the force is proportional to 1 over R squared that's true both for electromagnetism and for gravity and it's true for the same reason in both cases in both cases these are forces that are mediated by massless bosons and massless means the lines of force never end because you can just make a particle of arbitrarily low energy you can stretch out there so massless gauge bosons seem to give rise to force laws that goes 1 over R squared it seems to be just a geometric fact right so on the one hand this is a great triumph for this whole story I've been telling you because we've explained we say now that we've explained both why electromagnetism and why gravity obey more or less 1 over R squared force laws we know that in general 2v that's not quite right because general relativity is a non abelian gauge theory so there are couplings between gravitons but still in the solar system this is extremely close to a very good approximation the problem with this is that it doesn't seem to hold true for the weak and strong nuclear forces the nuclear forces are short-range they are not proportional to 1 over R squared there's no Coulomb law for the strong force or for the weak force and in the 1950s everyone knew this stuff like this is the story I've just told you was know you know when yang-mills proposed yang-mills theories this we thought we understood magnetism in the 1950s QED right quantum electrodynamics we thought we understood gravity at least classically general relativity the strong and weak nuclear forces were still mysteries so in yang and Mills proposed that you could make gauge theories with non abelian gauge fields with rotations that did not commute with each other everyone could instantly say well that would give rise to massless bosons and we haven't observed those that would give rise to long-range forces and the strong weak nuclear forces are not long-range what is going on well something is going on in both the strong nuclear force and the weak nuclear force and again because of the theorem that says things need to be as complicated as possible what's going on in those two cases is completely different so we have to examine in different ways the strong nuclear force and the weak nuclear force okay so let's think about the strong force the strong force QCD right quantum chromodynamics is an su3 gauge theory we're the three is the three colors right su three color red green and blue rotating the quarks together so this is non abelian and what that means is that the rotations in color space do not commute with each other rotating around green and then red is not the same as we're taking about red and then green what that means for reasons I'm not gonna get into right now oh yeah you haven't yet to trust me sometimes I'm doing my best to give you the concepts but things get messy very quickly what that means is that unlike photons photons are not themselves electrically charged okay photons carry the electromagnetic force between charged particles but they're not themselves electrically charged so photons this is a little parentheses here photons can do this you can have a photon come in and give you an electron and a positron but not a photon coming in and giving you two photons okay I need to sort of X that out because this is not allowed photon is not coupled to itself because the photons not itself electrically charged but quarks or sorry gluons oops that's not what I wanted gluons do change non-trivial D under su3 there's eight gluons and they all very very roughly speaking you can think of the gluon as a combination of a color in an anti color like red and anti blue or something like that doesn't quite work because if you count that way you get nine different gluons and there are only eight but roughly speaking that works but the point is gluons are them sell themselves colored in a very real way so gluons can do this where here's a gluon that's a photon this is a gluon a gluon can make a quark and an antiquark this is Q anti quark okay if the gluon itself were red anti blue one of the quarks might be red the other cork the anti quark might be anti blue for example and you can have a gluon come in and coupled to two other gluons just like that that is something that can happen in this non abelian gauge theory that doesn't happen in ordinary electromagnetism that's one feature it's non abelian so it can couple to itself non abelian means couple sorry should be tell you what couples gauge fields couple to themselves gauge fields in this case the gluon couple to themselves unlike in you one electromagnetism the other feature is something called asymptotic freedom this is really I mean this is really the payoff lecture here in many ways the payoff video because we're bringing in all these ideas from different other idea videos that we've done remember we talked about in the video about renormalization we could put a cut-off on our theory to get an effective field theory we admit that we don't know what goes on above the cutoff the price you pay for that is that the coupling constants in your low energy effective field theory depend on the value of the cuddles okay so you can ask as you change the cutoff how does the effective value of the coupling constants change one example would be in QED this is not an example of asymptotic freedom yet but this is good old ordinary quantum electromagnetism as you have the energy going higher and higher what happens to alpha the fine-structure constant the answer is that it goes up like that to get stronger and stronger we even do the little picture of the electron being renormalized by a cloud of virtual particles right I give some physical understanding of what's going on why the effective coupling electromagnetism is stronger at shorter distances therefore higher energies so it turns out there's a calculation you can do in the strong interactions in QCD call it alpha strong the strong interaction version of the fine-structure constant you can ask how does it change with energy and the answer is it does this it goes down okay this is the phenomenon known as asymptotic freedom it's not an easy calculation to do david gross frank will check david Politzer did it in the early nineteen seventies they won the Nobel Prize for doing it it's really that's it that's why they won the Nobel Prize the curve goes down stead of going up a - line instead of a plus sign but it's not obvious it depends on all the fields all their interactions you have to actually work it out okay and what that means is high energy means a weak coupling so that's why it's called asymptotic freedom freedom means not interacting at all free don't interact okay so this is saying that if you really push the particles to very very high energies the interaction stops being strong that's it's weakly coupled you can use Fineman diagrams everything is great but as you let the energies go lower and lower as you settle down rather than smashing particles together in high energies you just let them sit there the coupling becomes stronger and stronger if you have particles moving and hitting each other at very low energies so there is a value here an energy below which the coupling is about one remember the whole point the wonderful thing about the fine-structure constant even though I drew it changing by a lot here it's always pretty small 1 over 137 had very very low energies and not that much different in high energies so in the strong interactions below a certain energy and that energy works out to be about 0.3 GeV okay but a third of a GeV remember a GeV is the mass of the proton or the mass of a neutron below that energy the strong interactions are so strong when you can't even really use Fineman diagrams anymore you can't use perturbation theory the theory is just SuperDuper strongly coupled these quarks are really interacting with each other and the gluons are interacting with each other a lot that by the way is the reason why the proton is about 1 GeV because there are 3 quarks and we remember I told you in the sorry if you watched the video on scale but then also watched the Q&A video in the scale video I talked about the mass of the proton being 1 GeV only in the Q&A video for scale that I revealed that the quarks that make up the proton are much less than a third of a GeV there are only a few times more massive than the electron ok the up and down quarks it's the energy of the interactions of the quantum chromodynamics interactions that give rise to the mass of the proton and that's this energy times 3 1/3 of a GeV times 3 is 1 GeV that's the mass of the proton the reason why the proton is a is about 1 GeV and mass is because there are three quarks in it and each quark is surrounded by this energy from gluons up to about point three GeV and there are three of them that's where you get that mass has nothing to do with the mass of the individual quarks themselves and what this means is as synthetic freedom means as you get to higher energies the interaction goes away you get the lower energies the interaction becomes stronger and stronger and what that means is confinement so quarks if you have two quarks if you just simplify your life and just imagine there are two quarks interacting with each other so here you have a quark here you have another quark okay and there there's some gluons between them so they're interacting with gluons and there's a lot of them because they're strongly interacting so a lot of them and you try to pull them apart okay so you try to pull apart the quarks what's gonna happen is not that they just get further and further apart like an electron and proton you just pull apart no big deal what's happening is these gluons become more and more energetic to the point where they there's so much energy in the gluons then you can just create more quarks so what happens is actually you just create here's a quark you create an anti quark oops you create a quark antiquark pair in the middle that here that you're the two quark you started with here's the quark here's the anti quark and now you have a quark talking to that quark and this anti quark talking to the other quark that you started with so when you try to pull apart a quark two quarks to get individual quarks out there all by themselves it will never happen literally never happen it's not that you haven't tried hard enough you pull them apart it's like pulling a rubber band apart you never get only one ended rubber band you just split it in the middle and you get two new ends it's much like the magnetic monopole store you cut a magnet with the North and South Pole you don't get a North Pole all by itself you get a north and a South Pole on both of them so confinement is and this is because as you stretch things out remember longer distances is lower energies lower energies the coupling is stronger and stronger so you never get a quark all by itself and what that means is you know instead of this nice Coulomb force with lines of force going out you might think well I have a quark this is what you might think you'd be wrong and I have lines of force from gluons going in all directions right so I should get a 1 over R squared inverse square law force but in fact the gluons themselves are interacting with each other all over the place right I can't even draw it nearly realistically enough it's a mess and the whole thing can fines together into a ball and no gluons escape to the outside world that is confinement and the size is around this QCD scale so that's why in the case of the strong interactions the forces are short ranged because rather than lines a force going out infinitely far the lines of force interact with each other and they bundle up into a ball of yarn that forms a proton or a neutron or a Payan or something like that all those are massive particles and there's no massless particles that you can actually observe in QCD so gluons and QCD are massless because we showed the gauge theory implies that bosons are massless right that the gauge bosons are massless but they are confined so you never see them so there's no long-range 1 over R squared force that's how the strong interactions work still got the weak interactions is the final bit of the puzzle the weak interactions ok this is a longer story not going to get to it in any detail but the problem with the weak interactions not the problem the feature is that there's an su 2 gauge symmetry but it is broken spontaneously broken by a little field we call the Higgs the Higgs boson field and let me just give you the briefest of glossary of what that actually means so there's an su 2 symmetry okay that means two dimensional space two complex dimensions so really four dimensions so it's kind of hard to draw those really four dimensions but two complex dimensions okay so let's let's stretch our imaginations a little bit and draw the to demand the two directions in higgs space but the call it you know h1 and h2 but keeping in mind that both h1 and h2 the two directions of Higgs field space are both complex okay and so what happens is this is the there this sort of for a four dimensional space of Higgs bosons but what happens is in the early universe the Higgs boson settles down and gets a nonzero vac value in the vacuum in empty space so you might think that if you have some fields there minimum energy state is near zero value for the field so let's let's back up a little bit let's uh draw in a slightly different way if we draw the potential energy here okay so now we're gonna draw the the potential energy of the Higgs field which is a function of hey h1 and h2 two complex valued fields okay there's h1 there's h2 you might think one possible way could have been was that you get a potential that looks more or less like this our famous simple harmonic oscillator potential okay and then what that means is that the Higgs would just sit there at the bottom and everything would be great the symmetry would be respected by which we mean you could rotate h1 and h2 into each other su 2 rotations and that field value would be unchanged it would not do anything by doing that however that's not how nature works that ain't it that's not what's actually happening so in fact let me erase this thing which is fine but I can do better here's what what actually happens you again are gonna do field space oops that's not right so you're gonna do potential energy here is one part of the Higgs field here's the other part of the Higgs field h2 is the potential energy as a function of h1 and h2 and this is just a fact about how nature works you know the potential energy for the Higgs field doesn't look like this drawing on the Left what it looks like is what we call a Mexican hat potential I do not know why they don't just call it a sombrero potential they never asked me for some reason particle physicists like to call this the Mexican hat potential okay it's symmetric around rotations with respect to rotations of h1 and h2 that's it needs to be symmetric this this rotation in this direction is the SU 2 symmetry of the weak interaction so it needs to be symmetric but the lowest energy state is not at h1 equals 82 equals 0 the lowest energy state is the brim of the Hat at some non-zero value so the Higgs field in the early universe would have started out here okay it would started out near zero but then it would have fallen into the brim of the Hat as the universe expanded and cooled down the Higgs field goes down to the bottom where you know where along the brim of the Hat does it live doesn't matter completely symmetric right that's the whole point in fact there's literally no difference between it going to h1 or h2 or anywhere in between you can always do a rotation so it goes wherever you want the point is it goes somewhere oops the point is it goes somewhere and that breaks the symmetry the symmetry is still there since symmetry is still underlying the dynamics of everything but now you can ask yourself the question am i pointing in the direction of the actual direction the Higgs field is pointing or am i perpendicular to the Higgs field so even though this theory itself has a symmetry the low-energy manifestation of the theory does not because low energies means small tiny oscillations and you're either in the direction of the Higgs field or you're not ok so there's something that breaks the set the tree between all these different values so in in yet other words you can you can think about it this way if here is space and you have remember let's say the up door up quark and down quark a little doublet su 2 doublet an su 2 symmetry that rotates these two guys into each other so that means that if it weren't for this Higgs getting in the way there'd be no difference between the up quark and the down quark but now at every point in space there is a Higgs field pointing in some direction in SU 2 space not in physical space not an up-down left-right space but in SU 2 space okay and that's true everywhere so now you can ask are you the quark that is pointing parallel to the Higgs or perpendicular to the things okay and that breaks the symmetry that is why they are different and also long story short you can have an interaction just like you had this interaction way back up here where was it that yes you can have an interaction between the photon the electron and the positron likewise very similarly even though you could not have an interaction between the gauge bosons and themselves to give a mass you can have an action interaction between the gauge balls let's call let's say the Z boson Z complex conjugate times Z times the Higgs this isn't allowed interaction and the Higgs field becomes just a constant value this little number here for some reason that value of the Higgs field is called little V so this turns into little V times Z Z star and that is a mass term for Z boson there's a very similar thing going on for the W boson so the mass of the W and Z bosons is around 80 or 90 GeV 80 or 90 times the mass of the proton so they're heavy that's one of the reasons why the weak interactions are hard to notice it's hard to make a z boson and the other thing is that rather than the force law for photons where is it blue photons the force law 1 is 1 over R squared there's really it's hard to imagine what else it could have gone as there's there's arguments ok but but that's a very natural thing for it to do the force law for the weak interactions for the W or Z bosons is proportional to e to the minus the distance over the Compton wavelength of the W or Z bosons remember the Compton wavelength is just 1 over the mass so this is e to the minus the mass of the W or Z bosons times the distance so the force law rather than going is 1 over R squared it goes as e to the minus arm which is very very quickly going to 0 so that's why the weak interactions are short range this is part of the scale discussion that didn't quite have time to do this is another way that masses get related to distances because the mass is just 1 over the Compton wavelength and basically if you imagine you're a little quark you're an up quark okay and you're trying to give off W bosons in some different directions to create force law it takes energy to make those W bosons they're not massless there's a lot of wolfs that are required so roughly speaking you can push them out lambda W which is 1 over the mass of the W which is not very big at all I mean sorry M W is big so lambda W the Compton wavelength of the W is very very short so the weak interactions are not only weak they're also very very short-range this is why it's hard to notice them here is the big picture okay I know this was a lot it's a lot to do on the one hand as a payoff lecture but on the other hand is a lot of concepts going on I get that so the point is that gauge theories have phases completely different use of the word phase than eally i theta phase that we talked about before here I'm talking about phase in the same sense as we talked about water being liquid or gas or solid ice right different phases different ways that the physics the physical properties of the substance can manifest themselves even though the same underlying dynamics are going on the a gauge Theory can be in the Coulomb phase and there that's when you have free massless well I shouldn't say free free implies non interacting this is this is free free in a different sense of the word free Friesen allowed to propagate all by themselves not confined okay free massless gauge bosons and that leads to a force law F proportional to one over R squared okay this is for gravity this is for electromagnetism that's one phase that a gauge theory can be in the confined phase so here you have confined massless gauge bosons and their gluons QCD would be an example the force law is a little bit trickier it's again an exponential fall off but the point is it's not the gluons themselves that are escaping out and creating the forces forces between protons and neutrons can be modeled as exchanging pi ons back and forth between them but even that's not a perfect model it's really that if you get a proton a neutron together they form a six quark bound state they're no longer the identity of the proton the neutron by themselves is not separately conserved but it's all blob together is a bunch of quarks and so to talk about the range and the force law with respect to the strong interactions it's a tricky thing to do most people just don't even try finally there's the Higgs fades and that's where you get broken symmetry and massive it's not very elegant massive gauge bosons so on the one hand yeah life is complicated we have su3 the strong interactions QCD we have su 2 and we have u 1 also gravity they're all there in all the different phases are actually realized there's there's other phases you can imagine talking about but these are the main ones for gauge theories so Nature has chosen to give us a bounty of different possible physical situations going on that's why physics is hard but that's also why physics is fun I'm hoping I'm not sure this is true but the hope at the end of this long disquisition is that the underlying ideas remain clear there's a lot of details here and because the world is full of details in the details matter but the underlying ideas are actually very simple and beautiful the existence of a symmetry that is separately realizable at different points in space implies the existence of a new field that tells you how to relate what's going on at these different points in space that field the connection field gives rise to gauge bosons these Keys BH bosons can appear in different phases and that explains everything we know about all the forces of nature in any experiment we've ever done so far there's no force of nature we don't know that we know about that cannot be explained by one of these different realizations of the gauge symmetry underlying things so hopefully you get why the combination of geometry and symmetry is so important it explains an enormous amount about how we understand the real world

Channel: Sean Carroll

Views: 110,624

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Id: AuqKsBQnE2A

Length: 77min 57sec (4677 seconds)

Published: Tue Jun 30 2020