Eigenbros ep 112 - Understanding Yang Mills Theory

welcome back ladies and gentlemen welcome we'd love to see you yes folks today we have a very difficult show yeah that's to put it uh lightly i think uh this one's gonna be a real challenge yeah but before we get into it uh please like comment share subscribe if you haven't already yes yes check out the website eigenrose.com eigenbros on instagram igambrose on twitter igembrose2 on tiktok and also thank thank you to the patrons so of course uh you guys you know we couldn't do this without you we we greatly appreciate the support um and you too can become a patron at patreon.combros and there you'll get an extra 30 minute podcast every week on random crap that being one we'll uh think about during the week yeah so uh today we'll be talking about yang of bills theory yeah money no i think his name is um uh what is it uh chan chan ning or something i don't know but but yang yang mills theory it's one of these uh uh gage theories um well before we start throwing jargon all over the place yeah this is why we're this is why this could be a challenge this is like uh it's really hard to distill this concept down for some reason so me and terence we just kind of want to give you the the the punch line first sure we're going to throw up some equations up at you hopefully you're not assaulted or uh harassed and you're like i'm not gonna if they're uh if they're in physics or math they're used to being assaulted by equations mentally assaulted on the daily basis we'll throw the equations up there because they're a little bit uh they're gonna be really challenging to describe but um but there's a more compactified or compact form of the yang mills equations there are two sets of equations um yeah so we'll throw them on the board here but for the audio listeners uh we'll give you the really compact form because we're not going to go through all the subscripts here sure so the real compact form here and this is um courtesy of a great resource by michael nielsen you guys have may have seen him he's got a youtube channel he's on twitter um pretty online science dude i think he's a physicist not sure though you should check me on that but um michael nielsen has a little intro he wrote up on michaelnielson.org on yangmill's theory and i would highly suggest you check it out um maybe i can link it even in the video if i remember so his first equation is little d so that's lower case d subscript capital d f equals zero so that's the first equation and then the next one is a hodge star operator acting on lowercase d sub capital d hodge star f equals j yeah and there's a lot of uh there's there's some explanation here like the d sub d is what's called an exterior exterior covariance derivative well nielsen describes the the capital d as containing most of the information yes it's got the connection yeah so capital d is uh called the connection um and which tells us how to move stuff around from point to point on uh on a manifold and there's a bunch of other jargon that we're going to get into but i'd like to just kind of lay the foundation first on what the hell a connection is sure the manifold which you know kind of describes um it kind of has a sort of some relation to the curvature uh because they talk about the curvature and it's a compact form of a tensor and all this other stuff and so i believe the curvature is contained within the term the f term yeah in nielsen's paper yeah so really the meat of the whole thing is mostly the connection and then f contains the curvature and then in the second equation you'll see the j which has to do with current um i don't know what exactly what the current um entails but i know it entails like charges and you know the velocity of the charges and i guess a collection of charges it's like a physical it's like a current that you would think normally so it's not so bad on the surface except it took me forever to even find a resource that even had a simple version of this equation so yeah um you will notice that if you do try to look up yang mills theory there's a lot of um mystery behind what it actually is especially if you don't know topology gage theory um group theory or any of these other mathematical um frameworks yeah yeah and so the connection so i kind of wanted to add that like it the connection tells you how to move stuff around from point to point on this like manifold and typically like in physics the manifold you'll be dealing with is the mccalkey miss minkowski uh spacetime yeah um which in gr you if you've done gr you'll you'll probably know like or you'll probably be be very familiar with it right right so well and then for the layman of course it's not anything that really great it just literally means like flat space like a surface so just think of a surface that's all minkowski spaces so and i guess you're like what else is there if you want to know the antithesis there can be things like curved space so there's all kinds of different curvatures that you can allow for space time but minkowski space is the tried and true straight up flat space time yeah and then this and in this formulation the connection is or this d this capital d is what is the field that gives rise to the fundamental physics yes it's this vector field and that contains the vector vector and that's kind of the most important one because that's when we start talking about gauges right and if um you guys are familiar with e m like some of you i'm sure have taken undergraduate e m you get to um gage theory and that i think first did you get the gauge theory first and e m one whenever an undergrad what was your first class with it uh we literally just did potentials oh but you never even got to engage we got to gauge at the very end of the class no what i'm saying is that the first class that you've taken oh sorry yes the head gauge okay yeah that was the only class i took that had gauge theory i think and then and then maybe um was there anything else no yeah i think that was it so um you first get introduced to gauge theory and e m um and of course that's the classic magnetic field is equal to del cross a for the people who know the mathematics out there and that's the um the vector potential right there so a would be the vector potential yeah but we'll we'll show we'll show some uh we'll show a video but we'll also play some audio regarding this to to elaborate more yeah but this is kind of the meat and potatoes of the whole thing like the physics of it choosing a gauge yeah yeah and this kind of has to do with symmetries um and you will see hopefully it'll make more sense as we tie it all together but i wanted to play this video um of why gauges and symmetries come into play when you're talking about yang mills and this is kind of the underlying reason why and i'll go ahead we'll go ahead and play the video all right let's do it stays the same we mean the equation you need to hold it up to the yeah yeah hold on one sec yeah in the case of notice theorem when we say the environment stays the same we mean the equations that give the laws of motion for the system for example moving along a perfectly flat road the downward force of gravity stays constant we have symmetry to spatial translation and notice theorem tells us there's a corresponding conserved quantity that quantity is momentum if two cars collide on that road the sum of their combined momentum stays the same but what if the road is hilly momentum doesn't appear to be conserved it can be lost or gained to the gravitational field this is because the direction of the gravitational field changes with respect to the road it's not symmetric to translations along the road on the other hand the gravitational field across the whole stretch of road doesn't change from one point in time to the next the system is symmetric to time translations it doesn't matter when the collision happens the results are the same notice theorem reveals that this time translation symmetry gives us energy conservation and the last classic example if the so i kind of wanted to highlight here like they choose that you notice how first you had a flat space right right you had a flat vector space and it just you had the arrows pointing all in the same place so those are vectors for those of you that aren't familiar the arrows were for representing what though um just the i guess the curvature in this in this like the direction of the curvature i guess okay sure so he's just using a vector field to demonstrate how the vectors the tangent vectors can change yeah okay on this on this manifold because they had a two surface theta data surface so in this case like the vector potential um was just flat at first and then you kind of just saw how it translates symmetrically and the physics doesn't change right um and then the other one they showed that you know the curvature they change the curvature so now it's more bumpy and hilly and stuff right so now the the vector potential changes so the vector tells you where things or how things are behaving but fundamentally there were symmetries still that were sure weren't being like um how would you say conserve yeah variant yeah they were being invariant thank you so yeah when they're conserved things aren't changing and that's when they move through space that's what we typically can denote by saying or physicists denote by saying invariant right and there's actually um a very important thing called nether's theorem which pretty much encapsulates the whole relationship yeah this is symmetry and invariance yeah and this is what the video is from actually so okay yeah this video is called nether's theorem and the symmetries of reality yeah so that's a non-obvious thing especially if you're not far along in physics i guess it may feel intuitive but at the same time it's not super obvious um it's just that nurtures showed that if us if a system has a symmetry a particular symmetry along with that symmetry will come a conserved quantity yeah so that's not obvious but i guess an example of that would be like if there's um what's it uh if there's time invariance so that's when if your object or something um doesn't change over time then that means there's an energy conservation or there's a there's a conservation of energy along with that system so that's what nurtures theorem i guess basically proved and yang and that and that ties in with yang mills and gauge theory and this stuff because it basically these theories all utilize these kind of real fundamental mathematical um frameworks of symmetry to show that you can actually cons you can actually construct these whole systems just based on having these certain symmetries and then i guess a little bit of physics tied to it yeah yeah and the the takeaway here is like how we when we talk about the vector field so pay attention to that because that's in that case the gauge like you're choosing a gauge to talk about in a and and so like you're going to hear this choosing a gauge main one so choosing a gauge it's like choosing a vector potential so you choose give an example of what a vector potential would even be so vector potential is going to be i guess in in the most um colloquial way i guess if you have no reference to physics yeah you can sort of think about how um let's say you're playing in the grass sure very tall grass okay the blades of grass are these vector arrows right okay so whenever a wind blows we'll choose the wind as being the vector well the force that changes the vector um the the blades of grass right so the blades of gracious thing would represent the vectors everywhere in space yeah yeah and i guess space would be the ground yeah okay and then the wind would blow all these grass blades yeah and then all of the blades shift to a certain direction yeah okay depending on where the force is so like the wind itself is the force but in physics we like to generalize forces to potentials so we do that i mean and by definition like there's a mathematical one force is equal to the spatial derivative of the of the potential or the gradient of the potential right you remember that one f is equal to negative yeah okay sure um so we deal with this in classical mechanics but um but you can get the expression of the force from the potential yes and for for people who aren't aware of potential you can just think of as gravity so gravity would be an example of potential yeah so we're in a gravitational potential at all times when we're on earth yeah so the thing that's pulling you down is basically the force of gravity but we can we can say um go layer deep and say that there's a potential field from gravity due to the mass around us so the earth actually has gravity because it has mass so the earth is so massive that we have enough gravity to actually hold us down and we're in a potential well you can think of that as yeah and you get more information from vector fields than you do from just a force equation because like imagine if you get the yeah because if you get a layout or a view of the field and the curvature of the field then you can kind of get these your equations of motion from that yeah and we'll go more into why this is but um i just wanted to play this other video from pbs space time um called quantum and variance it kind of goes into a little bit a little bit more about gauge gauges and stuff okay here we go we introduced the idea of a gauge theory in simple terms a gauge theory is one that has mathematical parameters or degrees of freedom that can be changed without affecting the predictions of the theory an example would be a ball rolling down a hill under a constant gravitational acceleration the speed of the ball at the bottom of the hill depends on its change in altitude but it doesn't matter what we define to be altitude zero the bottom of the hill sea level even the center of the earth for the equations of motion of the ball the altitude zero point is irrelevant it's what we call a gauge freedom or a gauge symmetry and we say that the equations of motion are invariant to that parameter that's a pretty basic example but it turns out that these gauge symmetries are an important feature of most of our physical theories describing the universe newton's laws of motion and gravity maxwell's equations for electromagnetism einstein's general relativity and of course the standard model we're not quite sure why this is the case so yeah he just kind of explained a little bit more about why gage's uh gauges are are important to modern physics yeah um and it's kind of not even a thing that you would care about if you're a layman i don't think no because you kind of have to do maths even carry the gage theory is like when we would see like joe rogan uh ask lawrence krauss or eric weinstein what gage theory is i'm kind of like you care it doesn't really matter for him if you're a layman and you don't know what gage theory is it's not really that relevant to you because it's all about symmetries and things which you only care about if you're doing the math because it's like because you're trying to get things yeah it's like oh we can we can we can't solve these equations in a traditional sense so we use matrix to be able to solve this thing by by using like symmetries and arguments of arguments of symmetry to solve this equation in a way that actually gives us an answer that we can utilize um yeah and like i don't know did so this is another one of those weird things the what is it called the um like the choosing the name of like they chose the name gage theory because uh because of some railroad oh really yeah so so from what i read yeah uh was that so when they used to build railroads they used to choose so you have different size gauges yeah for the um yeah for putting down the the things i don't know things they're like so gauges are these like pins i think right yeah that makes sense um that you pinned down to the railroad so that you can just pretty much so it can hold together yeah yeah yeah so the thing is you can have different gauges for the for the railroad track that you can use different gauges and it was you could still use the railroad track it didn't matter what gauge you used you just used it and then that was fine okay you would get the same outcome you would you would still be able to use the damn railroad yeah yeah so that's kind of i i this could be one of these uh antiquated names yeah i don't like it though it sounds cool it sounds cool but i also don't know if that's actually the theory this might uh of the name or the what is it damn i'm blanking on the word there's um the uh yeah i'm coming off of a flu uh spike from my second covet shot so yeah my brain is like rattled but i'm like i'm trying to find the name it'll come to me later but anyway yeah the etymology of this yeah okay there you go yeah that's what it is so that i don't know if that's the true etymology of the or the origins of the word because i think people have just said it and i'm like this might be one of those things that people were just running with oh i see yeah yeah like they just came up with afterward yeah yeah yeah but the idea is that the idea is that you can choose any gauge and you're still going to get the same physics yeah and that's what that's what we mean by gauge and variant like right right so if you're layman don't care that much you just know what has to do with symmetry and uh yeah you you really don't aren't gonna care that much yeah hopefully i'll be able to tie it into why like we're just kind of explaining why physicists the bigger picture is is you're trying to simplify things yeah or trying to reduce it down to the to the simplest element where you can get the most information yeah the further physics goes in some sense we're getting pushed more and more to the abstract layer of mathematics exactly because mathematics really is everything at the end of the day right and as things get harder and more deep it's like we're relying on those abstractions and the mathematics to even be able to understand how the universe works because you know at those extremes and those edges intuition fails you can't really rely on intuition anymore you have to just go with the math and the way that you elucidate and yield more you know more understanding of things is to be able to have more math yeah so we're at the level of now trying to find all these real deep fundamental parts about the universe symmetries you know how things are invariant and you know all this stuff by just figuring out the math and you know the underlying kind of structure of things yeah this kind of also this episode would be good if you like are so confused about what the hell gauges are now i think about it like if you're an undergrad like i was also an undergrad like what the are you ages even after i did gage 3 i barely even knew what it was like okay um yeah so but i have this great video if you're still this kind of puts for me this put a nail in the coffin for like what what i didn't if i didn't know what gauges were this kind of gave me a good realistic picture of what it is yeah okay good good so this will be good for you folks hello everyone today i will talk about the elusive concept of gay variants we'll begin with a simple analogy of first year mechanics suppose i stand on a desk and drop a ball from some height y potential energy of the ball is given by v equals mgy with respect to the ground i can find the force in the ball which is the derivative of the potential energy and we get the following equation of motion now let's say i want to do the same experiment on top of a mountain of some height why not by the way the equation of motion is the force of gravity for those of you that don't know you start with the potential mgh um and then yeah you just end up with uh the downward force of gravity here yeah well that would mean that with respect to the ground my potential will change by some constant v naught well once again we can find the force and we get the same result in the equation for the equations of motion since v naught is just a constant and its derivative is zero so the upshot is that we get the same equations of motion even though our potentials are different this captures the idea of gauge and variant herman weil was the first to introduce the term gaining variance into english it comes from the german term eisen variance or invariance under the change of scale you can think of the quantity v naught as our gauge in this case we have the freedom of selecting the origin of our coordinate system this is known as gauge freedom the type of gate changes based on the context that you're dealing with and we see different gauges in the context of electromagnetism general relativity and field theory if a system is invariant under a particular type of gauge transformation it means that the physics of the system don't change under that transformation and the equation of motions remain the same gauge freedom often allows us to express our equations in a simpler form while still preserving the physics of the system for example we can just set v naught to zero in our example earlier and it doesn't change the dynamics of the falling ball we will now move so yeah i mean that kind of highlights nicely what we were saying yeah yeah so it's just a nice way to make the math work at the end of the day yeah that doesn't change your system at all right right and it's like um yeah the utility of it you you don't see especially whenever like we were introduced to it in e m yeah i didn't see the utility of it because they just kind of throw it at you and they're just like you know well i saw in the sense that i knew that we can solve certain equations because i don't think we could solve certain things without it um but yeah i was just like okay i didn't realize it was such a huge part of modern physics and like you know yang mills theory and stuff you know and of course we didn't know quantum field theory either so i didn't know anything about gauges in quantum field theory as well yeah so it's a big part of like the leading edge of physics like modern theories yeah so we're talking about gage theory so much because it's so important to the modern day landscape physics yeah including uh yang mills theory mm-hmm yeah so um so now that you hopefully now that you've kind of have a good conception of what the hell gauges are it's a lot more clear in like the language of lagrangians um because in lagrangians at least what i mean by that is like the energy formulation for those of you that don't know the lagrangian is this uh it's a mathematical tool that physicists use it's basically like the end-all be-all for like equations of motion equations yeah like it's like the master equation that you always want to get in terms of finding what you want in physics like everybody wants the lagrangian unless you're doing like quantum mechanics it's like the one exception but the lagrangian is everything pretty much yeah lagrangian gives you the equations of motion you know the ex you can get the velocity and the position yeah from that but that's everything right because you want to know how a system um unfolds uh unfolds over time right so the lagrangian is literally everything like the standard model can be written in terms of a lagrangian and then that lagrangian should in theory tell you everything about how the universe works so that's like the master equation that you always want pretty much in terms of any kind of physics you're trying to do and and i kind of wanted to give this little quick snippet by sean carroll um he has a good little lecture series um on this uh that we were watching the other day yeah uh we're both kind of laughing well i'm not gonna point out yeah yeah but but uh maybe you can listen to the patreon and we'll say something yeah but sean but sean's sean carroll's lecture is pretty good he uh he kind of goes over this little nugget of information here that ties into yang mills that i thought was really uh really enlightening oh okay um so here we go we'll go ahead and play but he ties it into the lagrangian formulation and tying it into sort of the yang mills kind of formulation okay the differential geometry all right cool let's see right oh you forgot to restart 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 going to 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 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 hate that 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 curfew and that is the nugget here yeah yeah that's the payoff i feel it comes from the curvature right yeah so so the kinetic energy of the gauge or the connection field comes from the curvature so this is if you've taken gr it's kind of like oh cool i see i could see that because then your manifold is uh spacetime minkowski space time okay so flat space yeah you're working in yeah and then now that you have a gauge you have some kind of uh like potential field right so if you have a gauge now you have now you have invoked some kind of curvature onto your space yeah is what you're saying exactly yeah so that's kind of and that's i guess you have a field we'll say you don't necessarily have to have courage i guess i guess you have a field you have curator then right that's what yeah that's so yeah that's right that's what that's basically the takeaway yeah yeah so i'm trying to i'm trying to translate for layman's oh my bad no true but but i will say like when you put so it's easier to see that because whenever the kinetic energy whatever or motion in a sense coming from that it like you drop a mass or something in in this curvature and you see the motion you get the kinetic energy right it's easier to see that in that picture i guess we should make the caveat that is not necessarily space that's curved i don't think it could be the like in quantum field theory they're saying like everything has its own field which i guess is its own manifold i'm not sure now i'm really reaching so put it back we'll put more in the on the uh there's another video that kind of addresses this yeah this makes sense intuitively if you think about it but the thing is we know that um gravity is the thing that we consider as bending space not like e m like lectures electromagnetic um fields are not bending space time no so you can't think of that as curved space so it's like what i think is it's saying is if you have an electromagnetic field for example the the space that's bending is the electromagnetic field um uh space yeah so in quantum field theory everything has its own field yeah so if i guess if you if you try to extrapolate the field as its own geometry then you can say that that that space for the electromagnetic field is curved which gives you a potential but you can't say that's the space time because that's that's um reserved for gr general relativity yeah yeah yeah true yeah so hopefully this kind of we have to be careful sometimes with what we're saying here so hopefully this kind of explains like it ties more into the classical language uh that most undergrads are familiar with like the lagrangian language to this sort of yang mills language of or like just talking about gauges serving as a connection and how it ties into the curvature right um but yeah this this was uh this was probably the most illuminating thing um for me at least uh this little nugget here but uh to tie it all in uh we can continue on the the origin of the standard model we're going to go back to this pbs space-time video uh that kind of ties into how we got uh the sort of standard models we we start introducing these uh phases and stuff into into uh into our picture because we're trying to get a more realistic picture like quantum mechanics is kind of this like like very at least standard quantum mechanics is like this toy model you know what i mean yeah yeah because you don't have relativity included you know and it's only red and it's only for massive fermions right that's just like protons or electrons as opposed to like light um particles photons yeah so yeah it's like it's r it's constrained to a small subset of stuff yeah and you want a more general theory that captures a lot of physics like right yeah so this is kind of what um quantum field theory is at least that's what the uh it's the evolution yeah quantum because it includes relativity now or at least special relativity right yeah so hopefully this will be more illuminating let's let's go but it seems to be a trend a theory that has these gauge symmetries is called a gauge theory today we're going to look at the simplest of the symmetries of the standard model the standard model is ultimately based on quantum field theory but we're going to use the schrodinger equation that's the most basic equation of motion of quantum mechanics it describes the evolution of the wave function which is the mathematical object that contains all the information about a particular physical system we could never see the underlying wave function of say a particle the best we can do is make a measurement of physical observables like position or momentum the wave function can represent different observables and it determines the distribution of possible results of measurement of those observables in this episode we'll be talking about the position wave function okay pay attention to this bit of math it'll be important the square of the magnitude of this wave function tells us the probability distribution of a particle's position the position that we observe when we look at the particle is picked randomly from that distribution this step of squaring the wave function is called the bourne rule and this innocuous seeming step introduces a simple symmetry that has profound implications let's see what happens when we square the wave function the wave function is an oscillation in quantum possibility moving through space and time it's no simple wave it's literally complex in the mathematical sense it has two components one real and one imaginary these components oscillate in sync with each other but they're offset shifted in phase by a constant amount phase is just the wave's current state in its up down oscillation when we apply the born rule what we're doing is squaring these two waves and adding them together but it turns out that this value doesn't depend on phase the magnitude squared of the real and imaginary components stays the same even as those components move up and down it's that magnitude squared that we can observe it determines the particle's position the phase itself is so i kind of want to highlight something like yeah because a lot of people if you're kind of like just a layman trying to understand this and you hear complex yeah yeah like what the does that mean uh it doesn't like or or like imaginary numbers like yeah yeah what the hell like i'm out what the hell what i might that's what i felt when i was a kid i was like why the hell am i learning about imaginary yeah aren't all numbers imaginary i mean yeah like aren't i imagining these horrible names god just that's my one i i guess that's kind of my my biggest gripe with math and physics they kind of pick the terminology they choose is just so head scratching it's like what who decided this anyway imaginary numbers are not um they're they're like a real thing that we use imagine it doesn't make sense though in some sense because sometimes imaginary numbers are like they don't manifest in real life right because like if you imagine if we have a wave function that has imaginary components you're never gonna like you don't know what getting an imaginary result is you're not gonna ever get imaginary results you have to you have to square it to make it into a real result so it's like there's it's almost weird like how it works out and the whole history of it is actually even interesting too because you know you know it developed from well that's what the born rule is right like you have to do that otherwise your results don't make sense you have to square yeah yeah but it's interesting that yeah anyway it is interesting it is interesting i will fundamentally yes uh the fundamental aspect of it is that basically it comes from we wanted to see what number if you squared a number would give you a negative number back which there's nothing that really does that so they created a they created a system where you can square two number or you can square a number to give you a negative number back yeah and that's where it all began yeah i guess the people in antiquity like the greeks would have like scoffed at imaginary yeah well i think people dismiss it for a long time and then they started to actually use it and noticed that it actually had some useful properties they're like square root of negative one square root you can't what's what's that square rooting a negative they probably just killed you yeah in the pythagorean in the pythagorean cult yeah they would have killed right they killed the guys for uh for finding irrational numbers and the pythagorean days are like okay he's got to die now do they really yeah some gloves like they kill people because they found an irrational number or something cause they were basically a cult back then you know pythagorean cult they had the um little like triangle whatever something inscribed in their hands they were super married to symmetry right yeah i'm talking now this i don't know exactly the history but they definitely uh some guy got marked for finding irrational numbers or finding like the square root of two was irrational or something it can so weird how yeah but but i mean like yeah so and luckily in modern days we're a little bit more open are we well well i guess we like to think we're a little maybe it's just so esoteric that people can't get raged out when they uh hear new math they're just like i'll leave the nerds there they'll figure it out yeah yeah no true um but yes so let's keep going here i just wanted to clarify that fundamentally unobservable you can shift phase by any amount and you wouldn't change the resulting position of the particle as long as you do the same shift to both the real and imaginary components in fact as long as you make the same shift across the entire wave function all observables are unchanged we call this sort of transformation a global phase shift and it's analogous to transforming our altitude zero point up or down by the same amount everywhere the equations of quantum mechanics have what we call global phase invariance global phase is a gauge symmetry of the system let's push a little further to see how far this symmetry goes this time we'll shift the phase by different amounts at different locations while still keeping the real and imaginary shifts the same at each location this position dependent phase shift is called a local phase shift instead of a global phase shift we'll try this because well we already know that the magnitude squared of the wave function should still stay the same under local phase shifts let's see what this would look like a global phase shift looks like this where all points move by the same amount however if we do a local phase shift say only this point here only that location changes as if it were part of the shifted wave making a discontinuous spike if you allow this sort of local phase shift you can change each point in a different way and really mess up the wave function that shouldn't change our probabilities for the positions of the particles but what about observables besides position according to the basic schrodinger equation we just ruined everything among other things messing with local phase really screws up our prediction for the particle's momentum see momentum is related to the average steepness of the wave function change the shape of that wave function with local phase shifts and you actually break conservation of momentum yeah and the reason why that is interesting yeah and the reason why that is is because like you know when we do derivatives we typically have smooth and continuous functions right we don't like these little shifts in the in the function yeah discontinuities equal a non-smooth function equals now you have pockets where you can't take derivatives and that's a problem yeah yeah because you want a whole function to go into derivative and you want a whole function to come out of the derivative you can't just be taking pieces of it because then it just makes your derivative or really unwieldy because then it means you have to break it up into sections or something along those lines it becomes nasty but it's probably more realistic right because in in reality we don't really have these like perfectly smooth functions yeah but of course you know every physicist just wants reality to be just completely sine waves you know these perfect geometrical structures and then you kind of just deal with real life as you have to do you know as it comes you know yeah so i mean that's what left is scratching their head like historically like they were just like well i mean the realistic pictures that we don't have perfect these functions kind of do have maybe they do have local phase shifts maybe they're there's continuous in certain parts how do we deal with that like that picture well we can use the momentum operator because well like it's getting the our back down so i imagine then one this has to do with the whole local um the whole local um what do you call it the whole local uh symmetry or whatever of um yang mills theory or gage theories i guess in general where they are they do have this local invariance as well as this global invariance i didn't quite understand what that means so this video was pretty interesting i i haven't watched this one yet but that's pretty cool um i don't know how to get away with that hmm [Laughter] but well you're in luck okay you're in luck because they're not done yet this is how physicists um so physicists saw this and they were like well what the hell what do we do uh local phase is not a gauge symmetry of the schrodinger equation how do we fix this um and i guess like if if you're kind of confused by what the hell that sentence means right local phase is not a gauge symmetry go back to what we were saying about be engaged in variant and stuff yeah think back to that and um that's where you'll find your answer but anyway we'll keep going local phase is not a gauge symmetry of the basic schrodinger equation okay that was a bust i guess we're done here all right wait just a second just for funsies maybe we can change the schrodinger equation to find a version that really is invariant to local phase shifts to do that we need to alter the part of the schrodinger equation that gives us the momentum of our particle the momentum operator after all momentum is what got screwed up it turns out that we can add a mathematical term to the momentum operator that's specially designed to undo any mess we make to the phase of the wave function if we tune this term correctly it absorbs any local changes we make to the phase and what is that extra term well it's something we call a vector potential i won't go into that right now but the important and absolutely bizarre thing about this mathematical entity is that it looks like something very familiar it looks exactly like the type of vector potential that you would have in the presence of an electromagnetic field so we've discovered that the only way for particles to have local phase invariance is for us to introduce a new fundamental field that pervades all of space and it turns out that field already exists and it's the electromagnetic field this is totally crazy we just rediscovered electromagnetism by insisting on a gauge symmetry that we had no right to expect to exist in the first place but we didn't just rediscover the e m field we learned a ton about it by discovering how it fits into the schrodinger equation we've unlocked its quantum behavior and now we know how it interacts with particles of matter to give them this symmetry we also learned about the origin of electric charge which we now see as a coupling term any particle that has this kind of charge will interact with and be affected by the electromagnetic field and be granted local phase invariance but the reverse is also true in order to have this particular type of local phase invariance particles must possess electric charge by the way applying notice theorem tells us there is a conserved quantity associated with any symmetry in this case the symmetry is local phase invariance and the conserved quantity is electric charge at this point we only need a couple of extra steps to produce the full description of electromagnetism in the quantum world quantum electrodynamics or qed first we need to upgrade the schrodinger crazy yeah did we do that in class no okay man so you can even do gauge transformations in quantum mechanics yeah who knew well this is kind of a more mod this is but this is why because it's a more um this is more quantum field theory okay yeah which i did not get to yeah we yeah i feel like i missed i'm missing a lot the modern nuggets of yeah like all of modern physics so if you guys want to get to the modern world if you're doing physics just go all the way to quantum theory and probably go to yang mills too if you can it seems that's the key to really being able to do leading edge stuff yeah cause it takes forever but i know because we're kind of stuck if you don't if you don't do quantum field theory your physics knowledge is really stuck at uh what the before times no no no it's like you're you're you're just kind of stuck if you've done condensed matter maybe your knowledge is is par to the 1960s we could get away with our knowledge doing like real life stuff because we're experimentalists right so we don't need to go super deep in the theory we're not trying to do grand unified theories and the only reason we even got into this stuff is because of the podcast right yeah but i realized that yeah if you want to really be on the leading edge you got to pretty much be in like the particle physics world of yeah true gauge theories and transformations and quantum field theory and that kind of stuff yeah condensed matter physics is kind of aren't on the leading edge i think in terms of like grand unified theories no not grand unified theories but we do need we do use gauge theories yeah yeah so it won't be entirely lost but okay if you yeah we do we do use uh sort of um what is it called group symmetry stuff like yeah algebra the algebra i imagine you could use that everywhere but i think probably the particle physicists are the most leading edge in terms of understanding the basic fundamental model of the universe yeah but um that that's basically fundamentally why you get your standard um or how you get your the most accurate picture of the standard model yeah because you have like they were saying this field that permeates right right corrects all of the local phase shifts yeah very interesting so you can just throw in that vector potential boom money and that vector potential is a it's a gauge like it's it it sort of takes care you can choose the the vector potential yeah because it's arbitrary right just choose it to fix your local faces yeah yeah interesting cool man cool but uh but hopefully that kind of ties a bigger picture um and that and that has to do with the yang mills because uh here we go hold on we have to go back to yang mills here yeah because like we were saying you have um this uh connection and then you have this uh right the curvature stuff and the connection is really important because like the math says um it is the thing i mean i'm just take the quote out from here uh the connection actually determines nearly all the other quantities in the equations um and yeah so it's literally it gives you the physical description of of uh of the the of your whatever you want to find like right yeah like it's responsible for the parallel transport which is being able to translate tangent vectors across space and that i think that connects the two different regions if you have some kind of let's say some some particle on one region of space and another particle in another region of space then you have those different local invariances in those two different places the um connection will tell you how those two are connected with one another yeah and it gives you the kinetic motion as we were thinking back to sean carroll's picture like if you want to talk about the motion the curvature right um yeah so you get a lot of information just from this gauge um the sort of what your gauge looks like right right yeah and it's interesting because um i guess the i didn't even realize that yang mills is responsible for unifying all of the forces pretty much as well besides gravity explain so yang mills theory is responsible for unifying the electro weak force you know electricity and magnetism with the weak force as well as the strong force and the electromagnetic force and that's basically the foundational theory for the standalone standard model lagrangian oh okay so they and it uses these things called special unitary groups which is that those are the symmetries that are used for those different regimes so in the electro week theory it utilizes um uh su2 which is the special unitary group two and actually that has to do with the poly matrices i didn't realize so the poly matrices are they come about from that whole yang mills um special unitary group and then um for the strong force it uses the special unitary group three which has to do with um quantum chronometer chromodynamics and you can see that two and you can remember the two and the three because you know the poly matrices they have the two the two by two matrices the special unitary group three for the strong force has a three by three matrix which has um red green and blue you know because in quantum chromodynamics there's a exchange of gluons and they call that you know they have color charge you could say which is red green and blue um don't get too hung up on that but it's it's just the strong force mediates interactions really close to the nuclei and that's what holds um holds these nucleons together basically and the weak force is responsible for beta decay which is like how particles decay into other particles um and then finally the electromagnetic force is a u1 symmetry which is a unitary group one and it's just a one you know it's just a scalar because you know charge charge uh charges always charge you know it's just it's not going to change under any kind of um translations yeah so it's interesting and that's all tied to yang mills so they and they described the standard model lagrangian as an su-3 cross su-2 cross u1 if you guys listen to eric weinstein he always talks about that that's the first time i actually heard of it and um yeah so actually this is interesting too one of the reasons we even wanted to do this yang mills is because it ties to eric's theory right you know when we when we did the last week's podcast on eric weinstein you know we didn't realize there's so much so much math you got to know about this stuff right yeah i mean because what we need to know topology we need to know differential geometry you need to know group theory you need to know um it would help to know quantum field theory yang mills theory it's like all this stuff you kind of need to really know how to tackle you know yeah these leading edge guts and toes yeah that's at least a year's worth of uh i mean for us a year it seems ambitious to me no i'm thinking i'm saying two years no a semester like i'm thinking i'm thinking of like a semester like imagine if you that's like going hard yeah i mean you'd have to i would die if i had to take all those no i mean like no cause like realistically you could take um differential geometry if you want i don't think math i don't think physicists have any business taking those math classes but well no because differential geometries related to general relativity yeah i know that's what i'm saying like you don't need to take a stand-alone uh class oh i see you can get at you you can get in jail yeah yeah you're right you're right um because you get introduced to tensors and stuff like that the language that you need to talk about uh uh calculus on on on surfaces or manifolds you get it from gr for the most part yeah i guess if you wanted to shortcut it all let's say if you wanted to do like yang mills and get to it i guess just do gauge theory because i guess gage theory ties in that group theory with um topology in some sense if you if you get it geometry i guess into geometry parts of it um and then what i guess maybe like what would you say is the most important to get a yang mills i would say probably gage theory and quantum field theory um well it builds into quantum field theory yeah that's why i think it's relevant though if you're trying to if you're trying to get to the leading edge physics of today oh well then i think you would need a theory yeah take take if you're a physics student take general relativity take um i'm assuming you already had that oh yeah oh if you're just supplementing then yeah uh then maybe you can do some reading i mean the books that were recommended um were uh i guess i shouldn't assume general relativity right uh because then i should assume quantum field theory yeah because i mean all we need to take we could jump into quantum field theory now yeah yeah like with the knowledge that we have yeah we're expected to right be able to learn but we can jump into all those we could jump into apology quantum field theory and well except yang mills we wouldn't we could do gage theory because it's purely math but yeah well because yang mills depends on all of those other ones this is why we're struggling to do the podcast on yang mills now well i'm saying like the theory the theory is built on a lot of the math that we know which one yang mills yeah because i mean they talk about like if we're gonna cut the language yeah but they also have a lot of terms the thing is like the topology aspect is very numerous michael nielsen in his introduction also addresses this like there's a ton of terms we don't know and there's operators like the hodge star operator which i'm not familiar at all um like there's there's a bunch of topology stuff in here that i don't know that you need to know you need to know like diffiomorphisms um it helps if you do it so from a from a geometry point of view this is when the fiber bundle stuff starts coming as well which we don't really know if you don't know any fiber bundles covector spaces tangent vector spaces um diffiomorphisms isomorphisms all this stuff is becomes relevant in yang mills gotcha from a geometric interpretation yeah well i mean if you really want to learn apparently this michael nielsen here he said he learned it all from this book gage fields knots and gravity by john bays and john twitter yeah and javier p munayan m-u-n-i-a-i-n and we actually have a claim to fame with john well not us but we know that um dylan berger got unfollowed from john baez because um he advertised the desk on his twitter a desk yeah john baez got all butt hurt because um dylan advertised a desk on twitter why would he get upset about that um because it wasn't a disclosed ad or something oh interesting he thought he was being a corporate shill crow you realize that dylan's a graduate student who probably makes twenty thousand dollars a year well john's okay well john thank you uh for this beautiful book i guess people love it and that's where i would suggest if you're a physics grad student watching this or if you're an undergrad who's aspiring to learn quantum field theories check out the gauge fields knots and gravity that's supposed to lead you down this rabbit hole to be able to even appreciate yang mills yeah but i would also recommend just checking out this michael nielsen intro first because it was really enlightening to me yeah simplified things in such a nice way if you don't know anything um i also posted it on my twitter if you guys want to check it out there yeah if it's not too late um yeah and i think we're done if if uh did you have any finishing touches oh i guess um well i didn't want to get into tensors but yeah we don't have time to do if you guys have any um especially people who know gage theory or any of these kind of group theory concepts or topology if you guys have any other comments to add to this video please by all means uh resources that you think are good yeah leave leave it in the comments and um yeah if you guys have any other questions too you know maybe we can answer them i mean this is this was a tough one for us to tackle we've been we've been trying to hit these really like big mon uh monster topics lately um so hopefully we did at least some justice to it but of course if you're a real specialist you may not get that much from it but this is mostly for laymen and you know undergraduates early grads so yeah please hopefully we did something for you yeah please like comment share subscribe uh check out uh our webpage at iganbros.com yes our twitter at eigenbros uh i think we're also on tick tock again rose2 yep i get bros on instagram yeah thanks again to the patrons once again thank you and if you guys want to become patreon members check us out patreon.com eigenbros support your boys three uh 30-minute podcasts every week at least on random at least all right folks and we'll see you later sayonara you

Channel: Eigenbros

Views: 5,117

Rating: 4.9272728 out of 5

Keywords: Yang-Mills Theory, eigenbros, physics, podcast, CN Yang, Robert Lawrence Mills, topology, gauge theory, symmetry, nature, noether's theorem, group theory, SU(3) x SU(2) x U(1), lagrangian, Quantum Field Theory, general relativity, magnetic vector potential, electromagnetism, electroweak, strong force, standard model, conservation laws

Id: NP0TTu-PAFg

Length: 60min 16sec (3616 seconds)

Published: Fri Apr 16 2021