Superconductivity And The Higgs Field

hey everyone today I want to talk to you about super conductivity and the higs field now you might be thinking super conductivity the higs field huh what what do those things possibly have in common great question you’ve come to the right place but before we get into it we’ve got a bit of housekeeping to do so first of all here are the timestamps for this video now I’m very aware that it’s kind of unusual OPP post a 5-hour physics video but think of it like an audio book now of course in physics you can’t just have audio because you got equations and figures and animations so it’s like an audio video book so even though it’s long there’s a lot of good stuff and it’s all in service of this profound vision of the facts that we basically live in a superconductor as we’ll talk about so I know the length of the video is a bit unconventional but I really am enamored by the online video format and I think it has a lot of potential ultimately it’s up to you to decide whether this is something that’s worthwhile but uh I put it to you for your consideration I recommend watching the video in sequential order because the ideas build on themselves but it’s up to you you can watch it however you want to watch it maybe there’s a topic that you already know so you want to skip ahead maybe you’re watching this in multiple sittings and you want to pick up where you left off well in any case do as you wish in this video we’re going to be drawing ideas and equations from three books first up we have introduction to Elementary particles by David Griffiths and this is like the sacred text this is the most amazing introductory text to particle physics that has ever been written I cannot recommend it highly enough it covers everything from the basics of quantum mechanics to relativity to gauge Theory to all kinds of stuff so definitely check out that book in addition to that we also have introduction to Super conductivity by Michael Tinkham don’t let the name fool you okay this book is quite intense it’s very rigorous and mathematical but that’s a good thing think of it like a cookbook with a bunch of delicious recipes in this book you can find a very very comprehensive overview of superc conductivity and finally we also have gauge theories of the strong weak and electromagnetic interactions by Chris quig this is a fantastic book and I highly recommend for anyone who’s interested in a gauge theoretic perspective and also a nice comprehensive overview of the standard model in addition to these books we’re also going to be referencing a variety of papers throughout the video and you can find links to those in the description below and last but not least I will also be posting video note PDFs as well as animation python codes to my patreon and how that works is if you sign up on there for five bucks a month then you’re in the club and then you can access all of my posts whether it’s the video notes or the python codes or anything else I don’t charge per post it’s just if you’re in the club you’re in the club so thank you to everyone who has signed up on there and if you’re considering signing up please do please support my channel no these videos are a labor of love I truly enjoy this and I hope that passion comes through it is also a ton of work and if you want to support my channel that really means a lot so thank you for your consideration and with that let’s get into the video hey let me ask you a question are you in space right now well no you’re in air there’s air all around you but there is space between those air molecules and in fact that space plays an essential role in allowing air to do what air does whether that’s creating air resistance or blowing around in the wind or just bumping into stuff you know air as we know it depends on having some space in which to exist and I know that’s pretty obvious but hang on to that sense of obviousness because things are about to get weird Okay so the properties of air like for example air resistance this is not a fundamental property of the universe when you think about space and you think about how most of the universe is just this empty frictionless vacuum you realize that air resistance is actually a particular phenomenon that happens down here on Earth with our atmosphere or I guess on another planet with an atmosphere but it’s not the norm and it’s actually not as fundamental as our everyday experience makes it seem to be so then suppose we want to discover the ultimate truths of reality the Bedrock truth set the foundation of physics so not the things like air resistance that are particular to a particular environment or particular material but I mean what are the fundamental truths of nature you know and to understand that we should think about things in the vacuum of space where everything is really isolated and clean and pure to get at the most basic most essential pattern of nature if we can understand the bare minimum aspects of reality and empty space then we can better understand what happens when that space is filled with Arrangements of stuff but here’s the problem empty space itself is not empty and I don’t just mean that space isn’t empty because there’s light and gravitational waves and all that no no no I’m making a far more provocative claim I’m saying that the thing that we consider to be the empty vacuum of space is not some empty platonically ideal notion of mathematical extensivity but rather that all of that supposedly empty space including all the tiny space between and within atoms is filled with a kind of substance although a substance is not exactly the right word a material Essence might be a better way of describing it though that’s kind of vague and confusing sometimes people describe it in terms of jello or molasses but it’s not edible and besides it’s not viscous either if we’re being technical we can call it the higs field and that’s all fine and good and that’s the proper terminology to which all kinds of formal details have been attached but higs field is not a very inviting term it doesn’t give the intuition anything to grab on to and that’s why so many people even today have no idea what the higs field is they’ve just heard that there’s some field or whatever but they don’t really have a picture of what it is and why it matters so our goal today is simple to develop a remarkable astonishing awe inspiring mental image of what the higs field is and how it radically Alters the reality that we perceive and our approach to developing that image is based on a peculiar Insight that in all the English language the single best word for describing what the higs field is is superconductor so that’ll be the central theme of the video we’re going to look at the higs field through the lens of superc conductivity now to be clear I’m not exactly saying that the Hig field is a superconductor in Theo technical sense of the word there are some nuances in caveats and we’ll get to those but I will say that the higs field is a superconductor like a goose is a duck that’s about how similar these things are as we’ll see so on the one hand I don’t want to make a too low resolution of a statement and just say that the higs field equals superconductor because that could be misleadingly simple but on the other other hand I really cannot overemphasize the similarities between the higs field and super conductivity and so I think there is actually some utility in just imagining that these two are the same thing and then carving in the nuances and caveats as you go along like that’s actually not a bad strategy for learning about the hfield and don’t just take my word for it okay this video was heavily inspired by the brilliant paper from superc conductors to Super colliders by Lance Dixon which is one of my favorite papers of all time Link in the description you should read it it’s great and it lends credibility to the bizarre ideas we’ll be exploring today okay so you know the observable universe like all the things imagine as a thought experiment that there’s like an atmosphere everywhere throughout all of that hypothetically like all of space you could breathe if you were on the moon or way out in space there’s air right so imagine and feel what that would be like okay but now instead of air imagine that it’s something like a superconductor condensate which you don’t have to know about yet we’ll get into it but for now just know that while this condensate doesn’t have air resistance it does have some other properties which pertain to it rather than to the underlying reality of things and so this substan likee thing that’s spread out all throughout our universe is actually contaminating our laws of physics even in the vacuum of space where you expect things to be clean and pure okay right away you see an apparent problem here if If there really is something like a superconducting condensate spread all throughout space here and there and all over the place then it should be radically modifying the laws of physics our universe should be all messed up but when you look around it seems like the laws of physics are totally fine nature seems completely normal there’s mountains and trees and sunshine and all of that it’s all fine so what do you mean there’s a superconducting condensate everywhere if we don’t see the effects of it then obviously it’s not actually there and this whole idea is preposterous and absurd but here’s the thing to understand the higs field we have to peel back a layer of the onion of nature this requires a kind of radical open-mindedness actually it requires an epiphany what we consider the normal natural regular old world around us actually has a severely messed up laws of physics so the World As We Know It with all its mountains and trees and sunshine and all the patterns of nature which are mostly electromagnetic and all that you touch and all that you see and all that you taste and all you feel and all of that is immersed in the higs field this strange superconductor likee field that has somehow been spilled throughout our universe and the result of the higs field is that it turns off the thing we call the weak nuclear force and also makes it seem like electromagnetism is this simple little little U1 symmetry thing when in reality that U1 symmetry is just a fragment of the world of weak isospin and hypercharge which pertains to the su2 Cross U1 symmetry of the electro weak interaction if not for the higs field if the electro weak interaction were able to manifest without being suppressed by that higs field then the whole fabric of reality would have a completely different character symbolized here by that exotic rainbow pattern which of course is not to be taken l Lally but is to be taken seriously as a representation of the unimaginable tapestry of cause and effect which the comfy blanket of the higs field protects us from this seems unusual and surreal because we’re talking about a layer of reality which is deeper than the one that our perceptions are based on it’s a bit like trying to understand the physics of space if you don’t have a rocket and so you’re stuck here on Earth and you just have to imagine what it would be like if not for all the air molecules well you could do it but you’d have to mentally immerse yourself in a hypothetical but it’s more than just a hypothetical because it is all around you if not for that other thing that’s all around you you know what I mean it’s like man you have to read between the lines of reality again we are in space right now space is all around us but there’s air here too and so we have air resistance and all that and we are in this deeper mind-blowing reality of Electro weak unification it’s all around us but there’s a higs field here too and so the weak Force has turned off and electromagnetism manifests as a simple phase Symmetry and we have all the patterns of the world in which we grew up in which our Common Sense was developed but Common Sense is a small subset of Truth and in physics we go beyond and although it seems very unusual and crazy at first eventually it clicks and then you’ll find yourself closer to truth with a deeper understanding of the world in which we live okay so before we get into the details here is an overview of what we’ll be talking about today first we’re going to have like a crash course on superconductivity and we’re going to take a historical approach looking at some of the main discoveries and theoretical developments in chronological order so that we can see not just the equations but also where they come from and why and that’ll give us a good conceptual framework for understanding and exploring superconductivity so then when we say that the higs field is like a superconductor will have a good idea of what it means for something to be like a superconductor then when we get to the early 1960s the story is going to naturally segue into the Anderson higs mechanism which is the way in which otherwise massless gauge bons acquire Mass the photons in the case of superconductors and the W’s and Z bons in the case of the higs field and we’re going to linger on the Anderson higs mechanism for a while because because that’s really the core concept at the heart of everything we’ll be talking about today towards the end of the video we’ll also briefly talk about Electro week unification at the highest level just the big picture of the basic idea and how the higs field fits into the picture so that you can start to see and feel the deeper reality underneath electromagnetism and the weak nuclear force but we’re not going to get too technical on Electro week unification today in the interest of time that’s a topic I would very much like to come back to someday but it deserves its own video or maybe a series of videos let me know if you’d be interested in that well anyway that’s what this video is all about so without further Ado let’s get into it super conductivity came into the world as a total surprise without explanation so I’m not going to try to Define it as if it’s a normal Phenomenon with some boring explanation no it’s a mystery so I’ll tell you the story of the mystery okay so first of all something that was known way back in the 1800s and that you can measure at home is that the electrical resistance of a thing depends on its temperature so if you take something and it can be just about anything and measure its resistance with an OHM meter while heating it and cooling it you’ll find a trend like so and this trend has a good Common Sense physical explanation when you heat up a thing the atoms start vibrating more aggressively and so they’re going to scatter the electrons that are trying to flow through so it takes more effort to push electricity through a thing when it’s hot and conversely if you cool the thing down then the atoms relax and the electrons can go through more easily without being disturbed as much along the way one way to imagine this is to picture yourself in a crowded room but the people are all standing around or walking slowly well you can walk through that room without much effort at all just like excuse me coming through on the other hand if you’re trying to cross a mosh pit at a heavy metal concert it’ll take a lot more effort people are crazy in there they’ll push you around you might catch an elbow maybe fall down you might not even make it to the other side you just have to chug a Mountain Dew and hope for the best hope that fate is on your side so that’s why resistance goes up with temperature and we can explain this graph in a way that makes perfect sense and even though this story about electrons bouncing around is a simplification cuz in reality of course the details are more detailed nonetheless the story is directionally true and in a meaningful way it is true that electrons scatter more in a hot material and so the resistance goes up well so anyway that tells us something about the material essence of electrons and their behavior in a material but then one question that comes up is what would happen at super super Cold’s temperatures where that thermal scattering effect goes away would the resistance approach zero at zero temperature or would it ASM toote to some non-zero value or what well that’s a good question and it’s worth exploring because that could tell us something about the nature of electrons in matter and the nature of things you know which is what we want to know so it would be interesting to measure the electrical resistance of materials at super cold temperatures and that brings us to the story of high cameling onas ones was a professor of experimental physics and the director of the cryogenic lab at the University of Leiden in the Netherlands he was also a Pioneer in the field of low temperature physics and in 1908 he became the first person to ever liquefy helium at a temperature of just 4.2 Kelvin in other words 4.2 de C above absolute zero and that was awesome because once you have liquid helium you have a way of putting that very cold temperature into a material form into a substance and this facilitates all kinds of very low temperature experiments so owns did a lot of experiments on what happens to materials at very cold temperatures for example measuring their electrical resistance as a function of temperature and in 1911 while measuring the resistance of ultra cold Mercury he inadvertently opened the Pandora’s box of superconductivity let’s take a look at some actual data from one of his 1911 Mercury experiments we’ll put temperature in Kelvin on the x- axis and on the Y AIS we’ll put resistance in ohms let’s look at the data points in order of decreasing temperature first at a temperature of 4.37 Kelvin we have a resistance of 0.13 ohms then a bit colder at 4.33 Kelvin we have 0.127 ohms and then at 4.23 Kelvin we have 0 114 ohms and so far it looks like a linear Trend which is not so surprising you know but something really weird happens when we go down to 4.2 Kelvin now the resistance is only like 0.01 ohms and even crazier if we go down just a little bit colder to 4.19 Kelvin we get 0.0000001 ohms which is within the sensitivity of the experiment so for all we know that may as well be precisely Z ohms what and if you go a bit colder it’s the same thing the electrical resistance has just somehow magically turned off which does not make any sense whatsoever it’s crazy by the way you can find the original 1911 drawing of this plot and own’s paper on the sudden change in the rate at which the resistance of mercury disappears Link in the description below now at the time of ‘s Discovery nobody had any idea what was going on here it was a crazy new phenomenon but it was clear that there was some kind of phase transition something new and exciting and mysterious and weird was happening and because at first all anyone knew about this phenomenon was that electrical resistance disappeared naturally they called it super conductivity but the word superconductivity is actually kind of a misnomer because there’s so much more to this phenomenon than just the lack of resistivity as we’ll see in a moment well okay so after this discovery one thing you want to know right away is are there any materials other than Mercury that are able to become superconducting so onas kept studying various materials and in 1913 he found that tin and lead both go superconducting at temperatures of approximately 4 and 7 Kelvin respectively and later on in the 1920s and 30s superc conductivity was observed in F ium niobium tantalum and vadium in all cases with a transition temperature of less than 10 kelv so as more superconducting materials were discovered that started to add some context to superc conductivity it appeared to be a phenomenon that happened at very very cold temperatures just single digit Kelvin above absolute zero but aside from that nobody really knew anything about what superc conductivity was or why it even existed in the first place because it seemed to defy all reasoning and common sense you know it made no sense whatsoever and then the plot thickens because in 1933 the German scientists Walter Meisner and Robert asfeld discovered something new and profound about superconductors superconductors expel magnetic fields and what it means for a superconductor to expel a magnetic field is that uh suppose you’re looking at it above the critical temperature so before it goes Super super conducting and in that situation if you put the material in a magnetic field the magnetic field can just poke through the material but when you cool it down below its critical temperature so that it becomes superconducting then a magical thing happens and the magnetic field just gets expelled from the superconductor it goes like boing like I don’t know it just gets pushed out of the superconductor which is super weird like what is going on here and even though this effect is super mysterious it’s it’s actually pretty easy to imagine and the way to imagine it is to pretend that the superconductor is a water balloon you know with a water balloon you have the water in the balloon but then you also have the thin layer of balloon and so what happens in a superconductor is the balloon layer the rubber of the balloon in that layer you have these super currents that are whipped up by the magnetic field that swirl around in a way that kind of mirrors and counteracts the magnetic field so that it can’t poke into the the balloon and inside the water of the balloon there is no magnetic field and so this is really weird like other materials do not exibit this Behavior because I mean if you think about it in a static magnetic field that is a magnetic field that’s not changing in time and just has some constant strength and constant Direction what a superconductor does when it’s swirling up these surface currents is counteracting the magnetic field but the supercurrents are able to persist indefinitely without requiring any kind of external source power to maintain so let’s say for example if you had some ordinary matter and you wanted to make it do this thing well it couldn’t because the surface currents would get tired they would dissipate just like electrical current does it dissipates in the form of heat unless you’re constantly putting power into it and so we find when reflecting on the nature of the Meisner effect that this effect is contingent upon the material having precisely zero resistance and this is actually a very nicely measurable thing because if there were any resistance at all even a tiny amount the current would dissipate pretty quickly and the meiser effect would get tired and the magnetic field would be able to slip into the material so if you do an experiment where you show that a superconductor is able to expel a static magnetic field and you check on it some time later and it’s still doing that it’s like okay well we can put a very very very low upper limit on any resistance that exists in this material and so yeah for all intents and purposes if you have have a true pure meiser effect that material genuinely does have precisely zero electrical resistance and which is extraordinarily weird I can’t overemphasize like you know electrons are matter right they bump into things like you see that in the way that resistance goes up with temperature right so how can you have a flow of matter in a material that’s made of matter with precisely zero resistance that defies every Common Sense notion of what matter is the profundity of this mystery cannot be overstated now I should emphasize that just because the material has zero resistance doesn’t necessarily imply that it is going to exhibit the Meisner effect so for example in principle you could have a material hypothetically that didn’t have electrical resistance but that just didn’t do this thing about expelling the magnetic field so at the time that the mner effect was discovered this genuinely was a new aspect of superconductivity in addition to having zero resistance superconductors also expel magnetic fields and so if we want to understand what a superconductor is we have to take that into account and we have to be able to explain that now even though the discovery of the Meisner effect deepened the mystery of superconductivity and made it more puzzling it also gave us another piece of the puzzle it added some depth and some nuances to the mystery but that also gave people more to work with another handle to grab on to to try to figure out what was going on here and sure enough just two years after the Meisner effect was discovered the famous brother scientists Fritz and Hines London came up with a pair of equations that modeled superconductors pretty well and was a significant step forward in terms of our understanding of how to model superc conductivity and the flow of supercurrents these equations relate the supercurrent density to the super electron number density as well as the elementary charge the electron Mass the electric field and the magnetic field the supercurrent density is the same as a current density it’s just that it has to do specifically with these super electrons which the London equations don’t tell us anything about as far as like why they exist or how it can be that they have no resistance that’s still very mysterious but you know Suppose there are some super electrons that can slip through a material without any resistance well then the supercurrent density is going to be the current it of those super electrons oh you know one of the great things about the London equations is that it deals with the elementary charge squared I don’t know about you but whenever I see an equation in electromagnetism where there’s the E I’m always like wait a minute is this the positive absolute value of the elementary charge is this the electron’s charge which is negative I make so many sign errors because of that but uh fortunately when it’s squared you know it’s positive okay so if we look at the first London equation this one tells us that the way the supercurrent density field changes in time is proportional to the electric field in the superconductor times a proportionality factor that depends on the super electron number density the electron’s charge squared and the electron mass and if we look at the second London equation this one tells us that the curl of the supercurrent density field is equal to the negative of the magnetic field times the same proportionality con Conant and so what that equation tells us is that if you try to put a magnetic field into a superconductor it’s going to swirl up some swirly supercurrents and the more magnetic field you try to put in the swir those currents are going to be and that we can use to understand the Meisner effect but before we get into the second London equation first things first let’s take a closer look at the first London equation so the first London equation tells us that the time derivative of the super current oh by the way now I’m calling it J we don’t have to say J sebs all the time it’s okay in a superconductor when you’re talking about current you’re talking about super current okay so the time derivative of the supercurrent is directly proportional to the electric field so the first thing you want to do when you see an equation like this is be like okay well suppose we put some electric field in a superconductor what happens well if the electric field is constant then the supercurrent is going to be constantly increasing and increasing and increasing and increasing and oh my gosh this is too much supercurrent what is happening how could this possibly be this is crazy it’s just going to go to Infinity so what happened here is the London equation wrong no we just used it wrong we made a false assumption that you can have a constant electric field in a superconductor you can’t what happens if you try to put an electric field in a superconductor is at first the super current is going to increase and increase but it can’t do that forever eventually it’s going to max out and actually what’s going to happen is the super current is going to reach some steady situation at which point it’s no longer changing in time and so by the first London equation if the supercurrent is no longer changing in time then the electric field must be zero and in this way a supercurrent is able to Schoop away the electric field so the electric field starts off non zero it whips up a supercurrent the supercurrent reaches a steady state the super current’s time derivative then is zero the electric field therefore is zero it’s been shooped away so that is a very interesting equation but it makes sense when you think about it because if a superconductor can’t sustain an electric field then it can’t sustain a voltage drop and so in a quasa equilibrium steady state situation everywhere in a superconductor is going to be the same voltage with a superconductor even when the electric field is zero that just tells you that the super current isn’t changing in time so you can have a super current that persists and keeps on doing whatever it was doing even in the absence of an electric field field so in the absence of a driving impetus to keep the current going it just keeps going all on its own and so yeah in the first London equation we see that a superconductor has zero electrical resistance all right well now let’s take a look at the second London equation this one says that the curl of the supercurrent density is equal to the magnetic field up to a negative proportionality factor in other words if you try put a magnetic field into a superconductor that’s going to swirl up some supercurrents which is a part of the whole thing about the Meisner effects you know how a swirly currents in the water balloon mirror and expel an applied magnetic field but to really see how the London equations imply the Meer effect we have to pull in another principle from electromagnetism which is ampere’s law in the last video a couple hours in we derived ampere’s law by relaxing the QED Lan with respect to variations in the photon field and its SpaceTime derivatives and we did that using gaussian units which is the easiest system of units for seeing how electromagnetism blossoms from local phase symmetry but anyway the London equations are often written in SI units so let’s go ahead and convert from gaussian to SI and actually while we’re doing that let’s also notice that in a quasa equilibrium situation where we let the superconductor relax for more than a few NS the electric field becomes Zero by the first London equation right it get shooped away and so the electric field term vanishes from our equation and then we can go ahead and solve for the supercurrent density Vector J while on the right hand side what would have been a factor of C over4 pi in gaussian units becomes a 1 over mu in SI units where m is the magnetic constant also known as the permeability of free space now what is this equation telling us well never mind the m not Factor that’s just a number and all that number does is it stretches the vectors but it doesn’t change their Direction so the main Insight in this equation is that the supercurrent density J gives rise to a curly magnetic field B and you learn about this in electromagnetism where if you point your right thumb in the direction of the current density Vector then your fingers curl around like the magnetic field so let’s consider these two equations at the same time do you see the beautiful interplay between these equations forget about the constants those are just numbers the equation on the left says that the curl of J is proportional to B up to a minus sign and the equation on the right says that the curl of B is proportional to J so which field is really the curl of which other field yes they both are these two equations exist in a seemingly paradoxical tension and so it’s not too surprising that they’ll have the effect of turning off the magnetic field within a superconductor without doing any math that’s one of the things we might expect to come out of these equations that the magnetic field should be turned off but let’s convince ourselves of that well first things first we can plug the equation on the right into the equation on the left by substituting that 1 over mu curl of B in for J and now we see quite a bizarre equation again ignoring the constants it’s basically saying that the curl of the Curve Cur of B is proportional to Nega B huh what how can the curl of a curl of a thing be proportional to the thing what a strange field and let’s not forget that b is really just a notational or linguistic shorthand for the curl of the magnetic Vector potential a so really this equation has a triple curl setion on the left and a hidden curl on the right what a dizzying equation if you feel dizzy right now good that’s a good thing that means you are properly appreciated the swirlin of this equation so where do we go from here well fortunately we can invoke a very convenient Vector calculus identity which says that the curl of the curl of a vector field is equal to the negative llan of the vector field plus the gradient of the Divergence of that field and that’s not physics that’s pure math it’s one of those things that you just have to look up in a table or maybe memorize if you use it a lot or if it’s a rainy day and you’re looking for a miserable activity you can prove it by writing everything out in terms of derivatives and doing some pattern matching all right so if we look at this equation the term on the right involves the Divergence of the magnetic field and being good electrom magnetizations we know that the Divergence of the magnetic field is zero because B is secretly nothing more than the curl of the vector potential a and the curl of a vector Fields doesn’t diverge so that’s gauss’s law for magnetism and it lets us delete that diver term from the equation so now we can replace the curl of the curl of B with simply the negative Lan of B when we do that we find that the llan of B is proportional to B up to some constants and we can clean up those constants by moving everything on over to the right side of the equation and actually we can go one step further with our housekeeping by defining this parameter Lambda which is the square root of m/ mu NS e^2 and then we can interpret this equation as saying that b is the kind of field where when you take its llan that has the same effect as simply multiplying the field by 1 over Lambda squ by the way if you’re an igen person you can imagine that b is an igen function of the Lan operator with igen value 1 over Lambda squ well anyway the Lan operator is the kind of vectorial generalization of the second derivative so if you just plot things out along one dimension and imagine theion as a second derivative of some component of the magnetic field along that Dimension then you find that the magnetic field drops off as e tox over Lambda and you can prove that by taking the second derivative of that exponential so this Lambda parameter is the length scale over which some magnetic field outside the superconductor will drop off within the superconductor in other words Lambda is the thickness of the water ball material of course it’s not a sharp boundary you know even at just half a Lambda away from the surface the field is already noticeably attenuated and if you go a few Lambda into the superconductor well you’ll still find a little bit of field in there but not much Lambda is the order of magnitude length scale in which the magnetic field is being expelled by all of those swirly surface currents so yeah the thickness of the water balloon by the way I remember this because Lambda kind of looks like an upside down slingshot and when I was a kid we used to slingshot water balloons at each other so this Lambda thing is called the penetration depth of the superconductor and it’s a material property that can depend on the conditions of the situation typically Lambda is between 50 and 500 NM so pretty small about 100 to 1,000 times thinner than a human hair okay so the reason we just did this analysis is to see how the London equations give us an analytical handle on the mystery of super current and the Meisner effect even though these equations are just phenomenological and they raise as many questions as they answer at least they’re a good starting point and they give us this concept of penetration depth Lambda which is still a useful concept even today okay so you might be wondering where do the London equations come from and why are they what they are well back in 1935 the answer was basically we don’t exactly know but the equation seemed to work and so they’re phenomenological equations however they’re not completely arbitrary so they can actually sort of be derived from deeper principles although if we’re being technical this is actually a pseudo derivation because we have to kind of make some illegitimate moves here and there and put in a little bit of mystery and so anyway it’s worthwhile to see this pseudo derivation so we can see what the London equations are really telling us okay so to start off we can write write out the supercurrent density in terms of the number density of super electrons the electron charge and the super electron velocity vector and this is just the idea that the current has to do with how many electrons are there and how fast they’re flowing but then we can write out the velocity Vector using the velocity operator which is basically just the idea that momentum is mass time velocity solved for velocity but then also instead of just the kinetic momentum p which is mass time velocity we also want to add to it the product of the electron’s charge and the Magnetic Vector potential and this is the way in which the vector potential augments the momentum of an electron so this is known as the principle of minimal coupling and as we saw in the last video the fact that a augments the momentum of charged matter is really a core part of what a is and why it even exists okay so this quantity P plus EA is known as the canonical momentum and so far we haven’t had to make any weird sketchy assumptions but let’s go ahead and make a weird sketchy assumption we’re going to assume that for some reason the kinetic momentum of the super electrons is zero that’s very strange because you would think that the kinetic momentum should be non zero but okay whatever suppose that it’s zero and we’re just making this up we’ll leave this as an unexplained mystery for now now we have to be careful because when we impose by Fiat that P should equal zero you never know what this might mess up with the other variables in our equations right so another thing we’re going to do is the vector potential a we’re going to write this now as a Tilda and the Tilda is just there to signify that hey we have to be careful with this Vector potential a because we don’t know exactly why the kinetic momentum should be zero and we don’t know exactly what that’s going to do regarding a and the behavior of a and the constraints on a so imagine that the Tilda just represents like a curiously raised eyebrow it’s like okay what is with this weird Vector potential now if we look at these two equations we can smos them together and come up with an expression for the supercurrent density in terms of this weird Vector potential a Tilda times some constants and this right here is actually the two London equations in one equation we can see this by writing out the definitions of the electric and magnetic fields in terms of voltage and the vector potential as we saw in the last video right remember the six ways so when we do that the first thing we’ll do is say you know what we’re in a superconductor so in quasa equilibrium we’re not going to have voltage gradients that’s going to get shooped away and the Expressions shown here for e and B are in gausian units but to convert to SI we can just delete that factor of 1 / C so then from this one equation involving J and a tlda we can recover the first London equation simply by taking a Time derivative of both sides and we can recover the second London equation simply by taking the curl of both sides so what this really tells us is that the London equations in essence are just the statement that in a superconductor for some reason the supercurrent is directly proportional to this magnetic Vector potential and that is very very strange because usually the magnetic Vector potential is not a directly observable quantity but supercurrent and current in general is an observable quantity so if these things are directly related up to some proportionality Factor then that means that this weird Vector potential a Tilda is actually directly observable so that’s kind of strange and you can imagine that in order to get this math to work consistently we have to put some guard rails on it you know we have to impose some structure to prevent these equations from breaking down and giving us nonsensical results and as we’ll see later we can derive these guard rails from deeper principles but for now just think of them as a result of having to keep the math consistent after imposing this constraint that the kinetic momentum has to be zero well anyway the first condition is that the Divergence of a Tilda should be zero this is the coolum gauge condition but in this context it has extra meaning because it also implies that the Divergence of the supercurrent is going going to be zero and that can actually be regarded as a continuity equation on the supercurrent now also we have the constraint that a Tilda should equal zero in the bulk of the superconductor away from the surface and this constraint is compatible with the misner effect so this has to do with the expulsion of magnetic fields in the interior of a superconductor and finally we have the constraint that AA should be tangent to the surface of the superconductor another way of saying that is that the dot product between a Tilda and the surface normal Vector should equal zero so if a Tilda is perpendicular to the surface normal then it’s necessarily tangent to the surface and what that implies is that currents are going to be swirling around the surface but are not going to be like accumulating onto the surface now I want to reflect on this for just a moment before we move on so the London equations are telling us that the supercurrent J is proportional to this mysteriously gauge invariant Vector potential and that this fact is a direct result of the super electrons somehow having no kinetic momentum and that’s very strange and it makes us Wonder like how can that be are the super electrons somehow all locked together in a rigid Quantum state with no kinetic momentum I mean what how can this possibly be you know so the reason I bring this up is to show you that the mystery at the core of the London equations really all boils down to like why is it that super electrons seem to have no con momentum or at least in my opinion that is the most intriguing question that we get out of the London equations given the knowledge known back in 1935 and also if we think about this from a more modern gauge theoretic perspective the London equations are based on the idea of the magnetic Vector potential being gaug in variant meaning directly observable meaning like there’s extra reality to it than we normally expect from a vector potential and that necessarily means that there’s something going on in the superconductor that is modifying the way that we normally think think about phase Symmetry and electromagnetism and that’s going to be a theme that we’re going to explore in depth later on in the video especially as it pertains to the Anderson higs mechanism okay so where are we now well we have seen how the London equations give us a pretty decent description of super conductivity but guess what in 1935 the same year that the London equations were published there’s also another dramatic experimental plot twist and that is the discovery of a whole new type of superconductors which exhibit behavior that the London equations do not explain and that brings us to the story of Lev shubnikov Lev shubnikov was known as the founding father of Soviet low temperature physics and he actually worked in ‘s lab in lien back in 1926 to 1930 this was actually just after ‘s time there when he was there s had already retired and unfortunately had passed away but uh during that time in On’s lab shubnikov and another scientist uh Vander Johannes deas I I hope I’m pronouncing that right discovered the shubnikov deas effect which provided direct evidence for something known as landow quanz which we’re not going to get into today but point is it was a pretty cool Discovery but uh anyway so after that in the mid-30s shubnikov went on over to Ukraine and started his own low temperature Lab at the Ukrainian physics and Technology Institute upti and there in 1935 he discovered a whole new type of super conductivity in single Crystal lead thalium and in 1937 he discovered another example of this new type of superconductivity this time in lead indiia Alloys so the discovery of type 2 superc conductivity is a pretty legendary thing you know it’s a big deal and shubnikov is rightly known as one of the heroes of super conductivity but shubnikov was a really accomplished scientist even Beyond his work on superconductivity for example in 1936 he along with his wife Olga th noova and the legendary Lev landow discovered antier magnetism oh and yeah so um we’re not going to get into the story today but uh you know this was just one of the terrible tragedies of uh of history and and physics you know 1937 you had Stalin’s terror and if you want to look into something called The upti Affair look into the uh ket’s landow leaflet you can read about what happened there you know I really don’t know what I can say about that today or you know this is not really the Right video for telling that story but uh we can only imagine what other discoveries shubnikov might have made butth you know what to honor his memory let’s go ahead and talk about type two super conductivity okay because uh the people that took away everything from him they didn’t take away his legacy okay so to understand what a type 2 superconductor is first let’s talk about type one so a type one super superconductor is just a regular old superconductor like the kind of superconductor we’ve been talking about so far where it only has two phases either it’s superconducting or it’s not and you can draw a phase diagram if you put a temperature on the x-axis and magnetic field on the Y AIS then there’s going to be a curve in that space that separates the superc conducting phase and the normal phase because superc conductivity can only occur when the material is sufficiently cold and also in a sufficiently gentle magnetic environment but if you overwhelm the superc conductivity either with temperature or with an extremely strong magnetic field then the superc conductivity is going to break down and the material becomes a normal material but the problem is we don’t just have type one we also have type two so what’s the deal with the type two well it’s very similar to type one except in addition to the superconducting phase and the normal phase now you also have this mixed phase and what the mixed phase is is the material still has zero electrical resistance so it’s still superc conducting in that sense but it no longer exhibits a complete misner effect instead what happens is actually some of the magnetic field is able to poke through the superconductor and at first glance you might think okay big whoop what’s the big deal I mean it doesn’t look that surprising or astonishing but the thing to appreciate about this is it’s not just that type two superconductors kind of fade in between superconducting and normal like a nice smooth transition no it is two transitions there is a phase that’s purely superconducting and purely exhibiting the Meer effect but then if you apply a strong enough magnetic field all of a sudden some of the magnetic field is able to start to poke through the material not all but some and the onset of that effect is sharp enough to constitute its own phase transition okay so mixed is not just like a combination of superc conducting and normal no no it’s really a different kind of thing and then if you go up to an even higher magnetic field even the mixed superconductivity breaks down and you notice that as a sudden onet of electrical resistance and so when you think about the two Transitions and the material’s ability to let some of the magnetic field poke through and then you think about everything we’ve talked about so far about the London equations and the m near effect and the water balloon it’s like okay hold on now no this is different something weird is going on here this is like a whole another chapter in the story of superc conductivity it’s like what the heck now we have type one and type two like oh my gosh as if this phenomenon couldn’t get any more mysterious and just to emphasize you know the London equations do not explain the existence of the mixed phase also known as the shubnikov phase in honor of shubnikov because you know in the London equations I mean there’s some fancy equations but they’re not that fancy see really it’s just the idea that the supercurrent is proportional to this mysteriously gauge and variant Vector potential and that’s all fine and good but it doesn’t have the necessary structural richness to it to explain what the heck is going on in the mixed phase so now we’re about to dive into the theory around what’s happening there and how we can model it and how we can understand both type one but also type two superc conductivity and the mixed phase and all of that but before we do that we have to jump ahead into the future because I I want to show you some experimental data or really just a picture that’ll help us make sense of what a type 2 superconductor is and then that’s just going to make it way easier to get into the theory because we’ll have a picture in mind that’s going to help us out okay so if we skip ahead to the modern era we can take a look at a type 2 superconductor in a magnetic field what the picture shows is a very thin type 2 superconductor that we’re looking down on from above so let’s say the plane of your screen right now let’s call that the XY plane and we’re looking at that from the Z Direction and what we’re looking out with all those little dots is that each one is actually a Quantum Vortex of supercurrent so imagine that uh well you know what it’s like this imagine you’re in a hot air balloon above Oklahoma and you look down and you’re like oh no it’s a tornado like one tornado right well the thought that’s going to cross your mind is man I picked the wrong day to go hot air balloon but you’re not going to have like an existential crisis in terms of like the tornado breaking your sense of reality no it’s just going to invoke a sense of danger but now imagine instead if you look down on the beautiful fields of Oklahoma and you see a bunch of little tiny tornadoes and then you take out your binoculars and because of course you brought binoculars and you look at them closely and you see that each one of these little tornadoes has the same exact amount of swirlin it’s as if they’ve been copied and pasted all over the beautiful fields of Oklahoma now that’s a different kind of predicament because maybe it’s not so dangerous to be flying above a field of little tornadoes I don’t know but one thing I do know for sure is that’s going to make you question your sense of reality because you’re going to be thinking hold on now how is it that there’s a whole bunch of these little tornadoes and they’re all swirling by the same amount now obviously this scenario I’m describing with the beautiful fields of Oklahoma filled with these Quantum tornadoes such a thing is preposterous and absurd and so you don’t have to contend with it in a very serious way however uh this thing with the superconductors is actually real so yeah it’s smaller and it’s less perilous but like look at this picture okay like look at it these are quantum vortices of supercurrent each one contains a single Quantum of magnetic flux and you know the flux Quantum this quantity has been measured to 10 parts per billion and for whatever reason when you look at a type two superconductor in a magnetic field in the mixed state when the magnetic field is able to start poking through it does so via these flux vortices that’s how the magnetic field is able to poke through because each flux Vortex carries one flux Quantum of magnetic flux and so it’s the number of vertices you have times the flux Quantum is the total amount of magnetic flux that’s going through the material but in between that you still have the superconducting condensate doing its thing so you can still pass a current along this material so yeah that is the mixed state of a type 2 superconductor and I just bring this up to give us some context right so the superconducting state there’s no vortices the mixed state you have some vortices and then the normal State the superc conductivity just breaks down there’s no supercurrent there’s no vortices and the thing that we now have to explain is how can we have such a Preposterous thing of quantum flux vortices that’s question number one question number two is why do some superconductors have them and some don’t but then also how can we explain this in a way that builds upon and expands the London equations cuz we know the London equations work pretty well for type 1 superconductors so whatever our explanation is for type two it has to be compatible it has to be reducible to the London equations in the appropriate scenario right so okay so that gives us some sense of what we’re trying to do so let’s do it we’re going to travel back in time now not all the way to 1935 but to 1950 that is when there was a big theoretical breakthrough in superconductivity so in 1950 the legendary physicist Lev landow along with another great physicist fatali Ginsburg came up with the Ginsburg landow model of super conductivity and to this day the Ginsburg landow model is what you use to understand the flow of supercurrents and their behavior in an electromagnetic field so the equation shown here is like the heart of their model this is the free density of a Ginsburg landow superconducting condensate I know it looks a little complicated but we’re going to break it down bit by bit in a moment and you’ll see that it’s actually not so bad and actually the Ginsburg Lando model is really based on one idea which is that inside a superconductor there’s going to be a schinger like wave function and it’s going to obey pretty much the same laws as the Schrodinger equation but it also kind of wants to repel itself so we’re going to model the state of the superconducting condensate with a complex valued scalar field now before we go further we have to think wait a minute what are we really saying here are we saying that all of the super electrons are in the same Quantum State yeah that’s what we’re saying and okay that should be a deal breaker right from the start you should hear that and be like no you can’t do that you can’t do that at all because why because electrons are Fons firion are half integer spin particles according to the spin statistics theorem if a particle has half integer spin in then it’s anti- symmetric under particle exchange and therefore it’s going to obey Fido statistics and therefore you have the poly Exclusion Principle and that’s the principle at the core of the stability of chemistry and matter as we know it this is not some flimsy principle you can’t just do away with the poly Exclusion Principle and yet at first glance it seems that the Ginsburg landow model does exactly that all right so why would we even entertain such an outrageous idea in the first place well because it turns out that the Ginsburg landal model and the equations involved are able to derive the London equations from deeper principles and explain type one and type two super conductivity and explain flux quantization in superconductors and basically it’s right like it’s the right answer okay so it works and it explains experiments so it is actually a really effective model at explaining superc conductivity but you do have to wonder how can it be that you have all these electrons in the same Quantum state but we’ll come back to that question later first let’s explore the Ginsburg landal model all right so let us derive the laws of physics for our superconducting condensate our derivation is going to be based on the idea that the laws which govern the superconductor can be derived from the energy contained in our wave function S as well as the electromagnetic field and also the interaction of our wave function with the electromagnetic field so the condensate and the electromagnetic field are going to do whatever they they have to to relax their energy to the minimum possible value now for everything that follows we’re going to assume that the superconductor has just a little bit of time to relax into a quasi equilibrium situation and so all we mean by quasa equilibrium is that we’re not going to worry about super high frequency transient situations in all of this analysis we’re going to assume that the condensate has a few NCS to relax into its low energy configuration okay so what we’re looking for is an equation that gives us the free energy density of the system which is the local free energy per unit volume as a function of something that depends on the wave function S as well as some electromagnetic stuff okay but then the question becomes what should that equation be well first of all there’s going to be some energy associated with the condensate density that is the number of Super electrons per unit volume so that if you have more or less of the condensate this energy term in the equation is going to be bigger or smaller and we can write this as Alpha times the amplitude of s squar where Alpha is some material property that by definition gives us the amount of energy contained in the condensate density and the absolute value of s squ is by definition the condensate density that is number of Super electrons per unit volume and then of course there’s some other stuff that we’re going to include as well but for now let’s focus on this term well if you make a plot where the absolute value of s is along the x- axis with zero on the left and on the vertical axis you plot Alpha time the absolute value of s squared well if Alpha is a positive number then you’re going to end up with the green curve shown here and as the free energy minimizes that’s going to push the absolute value of s to zero and so if Alpha is a positive number s doesn’t want to exist however if Alpha is negative then it’s a different situation see because then you end up with with this purple curve where now SII wants to be as big as it can be because a negative Alpha means that the more SII you have the lower and lower the free energy is going to be well of course there has to be some kind of limiting principle that’s going to stop Sai from running away to Infinity right well yeah so as it turns out there’s also going to be an energy due to the condensates self interaction and that is going to provide the limiting principle that prevents the amplitude of s from running away to Infinity when Alpha is negative now the way to think about the self- interaction term is that it depends on how much condensate density you have somewhere so that’s s squar but then it also depends on how much condensate density you have there for the condensate density there to interact with the condensate density that is there so you have another s squar and you end up with a s to the 4th and so what this is is an energy that has to do with sigh not wanting to exist around itself so we can put a number on this beta and actually we’ll go ahead and use beta/ two so that we end up with a cleaner equation later on doesn’t really matter but anyway this factor is a material property that sets the strength of the self interaction that is the energy per condensate density squared now when Alpha is positive even with this beta term we’re still not going to have any super conductivity but now for the case that Alpha is negative this beta term comes in as a limiting principle to prevent Sai from running away to to infinity and so therefore in the case of negative Alpha the free energy density is going to be minimized at some optimal value of the absolute value of s and that is what determines the density of a superconducting condensate and that can be calculated with freshman calculus because all you have to do is think about this equation in terms of the condensate density row where then the free energy is Alpha time row plus beta2 Row 2 well then the minimum Point occurs where the slope is zero and so all you have to do is take a derivative of the free energy density f with respect to the condensate density row set that equal to zero to find the minimum and when you do that you find that row is going to be the absolute value of alpha over Beta And so therefore the absolute value of s is going to be the square root of the absolute value of alpha over beta now I use absolute value here because bear in mind Alpha is going to be a negative number for a superconductor and beta is always going to be positive so by saying absolute value of alpha over beta what that really is is the ratio of how much the superconductor wants to exist divided by how much it doesn’t want to exist with itself and so it makes sense that the condensate density would be that ratio so you see now why we started with beta/ 2 it’s so that we don’t end up with the two in this equation you can think of beta as basically being a constant positive number but Alpha changes as a function of temperature in a really significant way so if you look at the plot on the left you can see that Alpha becomes a lower and lower number as the temperature decreases you go to a high temperature you end up with positive Alpha you go to a low temperature you end up with negative Alpha now the curve shown here is something of an idealization in reality it’s kind of a little messier than this but this curve applies pretty well near the transition point where Alpha equals 0 and that transition point where Alpha equals 0 that’s the critical point where if you go up to a higher temperature then Alpha becomes positive and now it costs energy to have a superconducting condensate and so the condensate doesn’t form and you end up with a normal material not a superconductor but if you go below the critical point then Alpha becomes a negative number and so therefore there’s negative energy associated with the formation of the condensate and so the condensate is going to spontaneously form and so the material will be superconducting if you subcool the material below its transition temperature then your Alpha value is going to be more of a negative number and so there’s going to be more of an energetic incentive for the condensate to form so that’s why if you subcool a superconductor well below its transition temperature you’ll end up with a denser condensate and remember as we saw just a moment ago the optimal value of sigh is going to be a tradeoff between s wanting to exist right there’s that Alpha term that makes s want to be nonzero but if s is too big then the beta term becomes dominant and you would have to put in more energy to bring in more s because it doesn’t want to exist with itself so the actual value of s that the superconductor is going to take on will be that optimal point that minimizes the free energy density at a value of the square root of the absolute value of alpha over Beta And so in this way you can see how the condensate density becomes non zero when Alpha goes negative but remember s is a complex valued field so in addition to its absolute value it also has a phase and the question is what is the phase of sigh well if you look at the alpha and beta terms these only have to do with the absolute value of SII so they don’t care about the phase at all and as we’ll see later on the rest of the Ginsburg landow free energy density equation it does care about phase gradients but it doesn’t care about the absolute phase but because SII is going to have a nonzero absolute value its phase has to be be something and the way to see this is suppose we put the complex plane horizontal and on the vertical axis we’ll put the free energy density pertaining to those Alpha and beta terms and you can see that if we bring the superconductor down below its critical temperature so that the alpha term goes negative we end up with a kind of Sombrero shaped function in the complex plane and while we can predict the absolute value of s by minimizing the energy the phase is going to be just randomly selected to be just some random phase and so even though the equations are symmetric with respect to the phase degree of Freedom the fact that Sai needs to take on some non-zero amplitude is going to force it to pick out a particular phase and this is the essence of spontaneous symmetry breaking and there’s a subtlety here which is that this spontaneous symmetry breaking one of the things that this relies on is the idea that phase gradients are going to cost energy and so the phase of the condensate is going to want to align coherently from one place to another and so in this animation you see that dot what that dot represents is the phase angle picked out by the condensate as the material goes superconducting and the fact that there would be an energy cost associated with phase gradients is what keeps the phase at one place in the condensate the same as at another place okay but now there’s a very subtle principle that we have to keep in mind which is global phase symmetry so when we’re doing quantum physics and we’re using complex numbers we’re really just using complex numbers as a kind of number that can roll around and so the absolute phase doesn’t really matter so much is Phase gradients right phase gradients are what encode momentum and energy and so we’re always free to Define what we mean by a phase angle of zero and one way to imagine this is uh if you want to use a complex number to track the phase of the moon you can use e to the I Omega T where Omega is the angular frequency of the Moon and that complex number is going to evolve at the same rate as the phase of the moon evolves but it’s up to you what you want the phase angle of zero to correspond to you can say Okay phase angle zero is going to be a new moon or a half moon or what you know whatever it is and so as part of setting our coordinate system we’re always allowed to say what we mean by a phase angle of zero and so when you’re dealing with a superconductor it does spontaneously pick out a particular phase but as a matter of convenience we can set that phase angle equal to zero and that doesn’t diminish the physical implications of spontaneous symmetry breaking but it’s weird right because it’s like okay hold on we got to parse two different things here like there’s the facts that the material is actually picking out a particular phase combined with the facts that we have the freedom to decide what we mean by zero phase okay so like what is the interplay between these Concepts and so I was trying to think of a way of explaining this and I was in the shower and I was looking at the water and I’m like okay water flows and super current flows and soap is clean and math is clean and then I’m like no it’s just it’s not working out the wisdom of the shower has let me down but then I turned and I looked at the shower wall with the tiles and I had this epiphany of like oh yeah that’s the thing right there that’s how it is because imagine if you had a bunch of tiles that were superconducting and you cool them down and you let them all condense and each one is going to pick out a particular phase and when you consider multiple different superconductors that’s when you really see the essence of spontaneous symmetry breaking and when you see this and then you think about the fact that we have the freedom to choose what we mean by zero phase you realize it’s like oh okay yeah I mean we can rotate the definition of zero phase and that sort of uniformly would rotate all these different phases but that’s kind of not super important I mean it’s sort of like if you have a crystal with multiple different Crystal grains and you rotate your head and you’re deciding what you want to call your X and Y axis you know what I mean it’s like that doesn’t actually change the relative orientations of the crystal grains relative to one another and likewise a global U1 redefinition of what we mean by the phase is not going to change any of the phase gradients in our superconductors and so all of these different options for how we want to globally Define our phase angle are going to lead to exactly the same physical description of the system oh let me mention a concept in passing and I’m not going to linger on this it’s a big topic but while we’re here imagine two tiles that are next to each other and they have a different phase just to give you a feel for what that phase difference means well if the two tiles are totally insulated and not connected to each other and spaced apart a little bit then each one is just going to have whatever phase it has but if youing bring them together close enough and maybe you put some material in between eventually what happens if they’re very very close together is that the wave functions of the condensate are going to overlap a little bit and you’re going to get a kind of quantum tunneling effect where the two different superconductors are going to feel each other and that’s effectively a phase gradient from one place to another and so that’s going to drive a current across that Gap across that Junction and what you find because you can do this by the way you can you know you can make superconductors and put them real close and what you find is that there is a current that goes across from one superconductor to another which is proportional to the sign of the phase difference across the two superconductors so if you make a plot of the current versus the phase angle difference what you find is that if there’s no phase angle difference from one superconductor to another then there’s no current because there’s nothing to drive that current but then if you get a phase angle difference of say 90° then then you’re going to get current flowing from one superconductor to the other now if you have a phase difference of 180° the current doesn’t know which way to go so you get zero current again if you have a 270° phase difference then that’s just like 90° but actually now it goes the other way so you have current flowing in the opposite direction and then if you have a 360° phase difference well that’s really not a phase difference at all so then you have no current oh whoops I almost forgot to turn on the bubbles so yeah you see that there’s really an observable effect that has to do with these phase differences and that tells us that this thing about spontaneous symmetry breaking is actually real and does actually have some physical consequences by the way this is known as the DC Josephson effect but let’s talk about that more some other day I only bring it up in passing as an example of the reality of superconductors picking out different phases but uh for the rest of the video we can actually imagine dealing with only one superconductor okay so we don’t have to worry about the phase difference from one superconductor to another but one superconductor can have phase gradients within itself and that brings us to the kinetic part of the Ginsburg landow free energy density okay so gradients in the phase of sigh and in fact gradients in sigh more generally have something to do with current and momentum and kinetic energy and we can understand all of those Dynamics by precisely defining how our wave function s and our Vector potential a contain kinetic energy so we’re going to start off with the principle from Newtonian physics that kinetic energy is momentum squar / twice the mass and that’s the same principle that we used in hydrogen part one to construct the Schrodinger equation just as in that video we’re going to bring in the momentum operator P hat acting on Sai which by definition is negative I bar time the gradient of s but here we’re also going to pull in the vector potential and we can go ahead and slap that right onto the momentum operator with negative e over C * the vector potential a * s that is known as the principle of minimal coupling and as we saw in the last video the vector potential A’s ability to affect momentum in this way is Central to its definition as the gauge field that sets the kind of local phase Transformations that we can do with our wave function sigh and we’ll talk more about that later on in this video oh and in this equation E star is the effective charge of a super electron and as we’ll see later on that’s actually going to be twice the electric charge of an electron because super electrons are really secretly electron pairs but we’ll talk more about that later okay so then to assemble the kinetic energy density what we’re going to do is we’re just going to plug in the definition of the momentum operator into the equation about kinetic energy and when we do this we find that the kinetic energy density is 1 2m star * the magnitude squar of I gradient of s minus e/ C * the vector potential Time s and so all that is is the combination of those two principles notice here we’re using M Star instead of M and so M Star is the effective mass of a super electron and that might take on different values because bear in mind we’re not dealing with a free electron in empty space we’re dealing with super electrons inside of a material but in your mind you can imagine mstar as playing exactly the same role as M from the shinger equation and so you don’t really have to worry about its numerical value today it’s more about the concept and the role that it plays in the equation all right so then if we write out what we have so far for the Ginsburg landow free energy density we have this term with the Alpha and the beta and now we also have this term which encodes the kinetic energy density of the condensate in the magnetic Vector potential a now then we have one other kind of energy that we have to account for and that is the energy contained purely within the magnetic Vector potential so to do this we’re going to borrow an idea from the last video where we saw that the electromagnetic field strength tensor contributes to the quantum electrodynamics LR as f muu f muu over 16 pi and you find when you pick out a particular reference frame that you can write this term as 1 8 Pi time the difference in the magnitude squared of the electric field and the magnetic field now in a superconductor the electric field is zero in quasa equilibrium because it gets a shooped away and if we translate from a lonian framework to a free energy framework we’re going to drop the minus sign on the magnetic field term and then remembering that the magnetic field B is just the curl of a what we find is that the energy density contained in the vector potential is simply 1 8 Pi time the magnitude squar of the curl of the vector potential a and that right there is the Ginsburg landow free energy density so the condensate is going to do whatever it has to do in order to minimize this equation and so that minimization principle is at the core of the landow model and if we can understand what has to be the case about s and a in order to minimize the free energy density F or technically in order to minimize the free energy which is f integrated over space then we can derive some governing equations for S and a now that derivation is very long and complicated and it would take us too far a field for this video so we’re not going to get into the details there but in essence this is just like your standard Oiler lrange situation where where you have some functional that you’re trying to minimize with respect to different field configurations and when you do that you can convert an equation involving in this case the integral of f over space into a set of differential equations in s and a that give you the conditions that s and a must satisfy in order to minimize the free energy density F and these differential equations are called the Ginsburg landow equations they come directly out of the definition of the G ber GL free energy density so these are not additional assumptions or additional axioms no they’re a direct mathematical consequence of the definition of the free energy density F now the first Ginsburg low equation is very very similar to Schrodinger’s equation for a particle with mass m star charge e and energy igen value negative Alpha but in this equation we also have an additional term which is nonlinear and that’s that term beta * the amplitude squ s * s so that’s a number that is basically s cubed but with the same complex phase as s and that nonlinear term causes so many problems because when you’re actually dealing with these equations very often you have to either make an approximation or take a numerical approach but uh anyway in spirit the first Ginsburg low equation is very much like the Schrodinger equation and you can interpret it in much the same way and that beta term remember beta is a number that has to do with how much s doesn’t want to exist with itself and so what that term is going to do in the context of the first Ginsburg L equation is the beta term is going to want s to spread out away from itself okay now the second Ginsburg Lando equation is the usual equation that we have for current in quantum mechanics as a function of s and a so this equation is all about the way supercurrent flows and also within this equation we find the London equations if we make the Curious assumption that the mysterious a Tilda of the London equations is the vector potential a minus H C over e time the gradient of f where f is the phase angle of s now at this moment we haven’t yet seen why that assumption makes sense as we get into the Anderson higs mechanism we’ll recognize hey that’s the vector potential in the unitary gauge and it contains special physical significance because it has conjoined into itself the nambu Goldstone modes in our wave function Sai but let’s not get ahead of ourselves for now we’ll just take it as an assumption that the London equations a Tilda is related to the vector potential a in this way to see that the second Ginsburg landow equation gives us the London equations all we have to do is plug in that definition of a Tilda into the second London equation and when we do that a bunch of terms cancel so the five terms cancel and we’re left with this expression involving e star and M star and s^ squ and negative e star over C and A Tilda and if we convert from gaussian units to SI so the 1 over C on the vector potential goes away because of the different definitions of vector potential in si versus gaussian and if we also recognize that the amplitude of size squared is the number density of super electrons NS which is sort of a core Assumption of the Ginsburg Lal model that s^ squ is the super electron density then we can rewrite this equation as J = NS e over M time the mysterious Vector potential a Tilda and as we saw earlier this equation is the two London equations in one if you take a Time derivative you get the first London equation and if you take the curl then you get the second London equation so the London equations are nested within the Ginsburg landow model which is super cool but there’s also all this other quantum richness in the equations which presumably are going to tell us something about the difference between type 1 and type two superconductors and just in general this gives us a much richer and a much more interesting mathematical landscape to explore okay so let’s get a feel for some of this Quantum math and what it can tell us about superconductivity so imagine we have an interface between a normal material and a superconducting material well our wave function Sai is not going to be able to transition infinitely sharply between Z and its bulk value of root of Alpha Beta but instead there’s going to be a gradual transition that happens over some characteristic length scale Cai now that length scale is called the coherence length and it’s defined as approximately the distance that it takes for the condensate sigh to return to its bulk value away from a disturbance or from the edge and this is one of those order of magnitude things s of like Lambda you know it’s not exactly a precise Edge width but it’s sort of a characteristic length scale so this is like classic quantum physics you know when you have a wave function there’s never a sharp boundary there’s always some blur because if you had infinite sharpness that would require an infinite gradient in your wave function but then you would need infinite kinetic energy density and that’s not a thing so in the quantum World things are always a little bit fuzzy all right so how do we model this mathematically well imagine a superconductor in no magnetic field and we can say that the vector potential is zero just don’t worry about it this is not a magnetic situation and without loss of generality we can also say that our wave function s is going to be real valued because we have the freedom to Define our phase angle such that the phase angle of s is zero all right so now let’s apply the first Ginsburg low equation now the first thing we’ll do is because we’re saying the vector potential is zero don’t worry about it we’re going to go ahead and delete that term and so the equation is going to simplify like so next up to make our lives easier we’re going to define a normalized wave function f which is just s divided by its bulk value < TK of alpha over Beta And so f is going to transition from zero outside the superconductor to one inside the superconductor and to simplify the math a bit more we’re also going to consider the equation along with just one dimension so that ation term becomes a second derivative in X and when you make these simplifications and you substitute in F into the equation and you simplify you end up with the equation shown here now an equation like this has the form of a constant times the second derivative plus the function minus the function Cub equals 0 and if you spend a lot of time with differential equations then you’ll be tempted to identify that constant on the second derivative as the square of a characteristic length scale and I’ll show a brief argument for why that’s the case in just a minute but for now we’ll just suppose that’s the case and we can use that then to solve for that length scale by simply taking the square root of the number that appears in front of our second derivative when we do that we find that the coherence length is h / the square root of the absolute value of 2 mstar * Alpha and bear in mind Alpha is that negative number that gets lower and lower as we cool down the superconductor and the condensate wants to exist more and more so we see that there’s an inverse relationship between the coherence length the blurriness of the edge of the superconductor and the amount that the condensate wants to exist so if you super cool your superconductor so that the condensate really wants to form and you have a lot of super electron density then you’re also going to have a Sharper Edge of the superconductor conversely if you warm up your superconductor and you get a close to that transition temperature then the coherence length is going to increase and your condensate is going to have a blurrier edge now the coherence length of a superconductor it varies from one material to another and also from one temperature to another but typical values are going to be sort of on the order of like 100 nanm could be more could be less you know some are you know 10 nanm couple nanometers and other superconductors are microns of coherence length this is something that really varies quite a lot from one superconductor to another but even still it’s very very small so it actually is a pretty sharp edge it’s just when you’re really really zoom in that technically there is a little bit of quantum blurriness there but that actually does have physical significance as we’re about to see okay now I know if I don’t address this I’ll get comments about hey why didn’t you address that so let me take a few minutes to give an argument for why it is that when we saw that differential equation we interpret Cai as a characteristic length scale so first let’s imagine that we’re inside the superconductor so our normalized wave function f is equal to 1 so if we give f a small disturbance G so now f as a function of space is going to be one plus this little disturbance G is a function of space where G is a number whose absolute value is much less than one and we want to know across what length scale is that disturbance going to vary in space well all you have to do is substitute 1 plus G in for f in our differential equation and you find the equation like so when you evaluate the cube and the second derivative of one is zero so you end up with the second derivative of G well anyway you end up collecting terms and a couple of things cancel and you can absorb the G terms together and we can also use a small G approximation to uh discard the G2 and G cubed terms and when we rewrite the equation and divide by x i^ 2 we see that taking the second derivative of G has the same effect on G as multiplying it by 2^ 2 and so therefore G is going to be something like an exponential whose argument is < tk2 X overi you can see then thatai appears as a sort of order of magnitude length scale over which our disturbance is going to vary in space okay but that was a bit of a mathematical tangent let’s get back to something really deep and profound how about the flux Quantum let us talk about about Quantum flux vortices man what a weird concept it’s actually kind of frightening we’re not in Kansas anymore we have now drifted over the beautiful fields of Oklahoma remember from earlier if you think about a flux Quantum it’s like this is the most surreal unusual thing ever how can this ever make sense how can we possibly understand it what a bizarre thing but check this out what if we had a superconductor and we poked a hole in it so that there was a point where our wave function s had a value of zero so no super electrons at that point but in the bulk of the superconductor away from that point we have the usual bulk value for the condensate sigh well now as we know the condensate is going to rise from zero at the defect up to the constant value in the bulk over some length scale which is approximately the coherence length Cai and that makes sense based on everything we’ve talked about now check this out with one move we’re going to make a flux Vortex okay you ready for this that’s I guess that’s the sound effect for putting a Twist in the phas okay so now we have a very similar situation but you’ll notice that the phase angle fi wraps around the defect and this is a flux Quantum it’s a Vortex of supercurrent and the reason it’s quantized is because the phase wraps one time around around the vortex core and in this two-dimensional picture the vortex core is a point defect but in reality a superconductor is a three-dimensional thing and so it’s actually a line like a tornado so we’re looking at a cross-section of that and in real in three dimensions the phase winds around that tornado and so you end up with the topologically non-trivial but topologically stabilized field configuration where you have a phase gradient that’s going to persistently swirl around and around and around and because phase gradients encode currents when you have a situation like this you’re going to end up with a persistent tornado of supercurrents okay so there’s a couple of questions that come up right away when you see something like this the first is why does there have to be one phase winding why can’t we have two well mathematically you could imagine the phase winding two or three or n times around the vortex core but in practice it’s more energetically favorable for a superconductor to have n vortices each with one phase winding than to have one Vortex with n phase windings so when we have these vortices in a superconductor the most energetically favorable thing is to have very many of them each containing one phase winding and that is where the flux Quantum comes from oh by the way the first person to think that superconductors should have quantized flux was Fritz London back in 1940 8 but it wasn’t until 1950 that Ginsburg and landow came up with the model that really provided a solid mathematical foundation for exploring the nature of flux quantization and it actually wasn’t until the late 1950s that the scientist Alex abazov studied these vortices in a lot of depth abazov was really the first person to intensely study these vortices and to look at how they pertain to the phenomenon of type 2 superconductivity and so in honor of his work these are often called aazav vortices but you can also call it a flux Vortex or a fluxon now for a Vortex like this to be energetically favorable the supercurrent needs to be able to swirl far enough away from the vortex core that you have a good density of condensate to swirl up so as you move far away from the vortex core and the absolute value of sigh Rises to its bulk value you want to have currents being able to swirl around out there in in order for a Vortex like this to form in a superconductor and remember what is it that determines how far into a superconductor you can have swirly currents well that’s going to be the magnetic penetration depth Lambda remember way back towards the beginning of the video we talked about Lambda as being the thickness of the water balloon material wherein you have these swirly currents well in this context you can think about Lambda is setting the length scale away from the core that you can still have a good amount of swirly currents and because it is the coherence length Kai that determines how fast the superconducting condensate restores its density away from the vortex core what matters as far as whether or not we’re going to have these vortices is whether or not the penetration depth Lambda is Big compared to the coherence length Cai if the penetration depth Lambda is very short compared to Cai like let’s say Lambda is a 10th of Kai then you’re not going to have these vortices on the other hand if the magnetic penetration depth is many times the coherence length then you will have these vortices and the material will be type two and it turns out that the crossover point between type 1 and type 2 superc conductivity occurs when the penetration depth is about 70% of the coherence length and so 7 is the magic number where if Lambda is greater than 7i you’re dealing with a type 2 superconductor and we’ll talk a little bit more about that in just a moment but first I want to say something about flux quantization in general not just dealing with the case of one supercurrent Vortex but broadening that up to the general situation involving flux quantization in a superconducting material so magnetic flux quantization is the idea that if you try to poke a magnetic field through a superconductor the flux is going to have to be quantized meaning that you’re going to end up with an integer multiple of f KN the flux Quantum as you poke that magnetic field through the material and this fact arises directly from the topology of our condensate sigh so if we suppose that we have a condensate with a phase angle of fi and you imagine a closed loop in this condensate so a loop where the place you start and the place you end are the same place then if you walk around that Loop and you add up the change in the phase of sigh at each tiny step DL that’s going to be the dot product between the gradient of sigh and your tiny step DL well okay so you walk along and you’re adding this up like an odometer that can increase and decrease and you’ll find that by the time you complete the loop because you’re going to be back at the place you started at the total sum of all the phase changes along your journey is going to have to be some integer number of complete rotations because s is single valued and so if you do a full complete loop around s the phase you end up with is going to be the phase you started out with and you can write this in terms of an integral around the path of the tiny change at each step along the path gradient of. DL and by the single valness of s that has to equal 2 pi n where n is an integer now if you’re no fan of phase gradients you can gauge them away using the local U1 symmetry of electromagnetism and that lets you exchange a gradient in the phase of sigh for some Vector potential a scaled by this quantity negative V Star over H C and so this comes from the principle of local phase symmetry now there are some nuances here that I’m going to gloss over involving Branch cuts and Global topological structure as it pertains to phase Transformations never mind that okay it doesn’t really matter for now just if you focus on each tiny step and you exchange at each step some gradient in fi for some Vector potential then you can rewrite our integral in terms of an integral of a around the path and you can pull that e star H bar and C onto the right side of the equation and what this statement means is is that one way to think about going around this path and calculating our phase odometer instead we can think about that as a circulation integral in the vector potential and we can see that that is going to be quantized by implicit virtue of the topology of Sai and that right there is the principle of flux quantization but normally this idea is dressed up in terms of a flux integral of a magnetic field and to get that all we have to do is apply stokes’s theorem which tells us that the circulation integral of a vector field around a close path is going to be the flux integral of the curl of that Vector field through the area enclosed by that path and that’s pure math that’s not even physics that’s pure math and so then if we take our integral and we cast it in terms of a flux integral of the curl of the vector potential a and remembering that the curl of a is the magnetic field B then now we end up with an equation that says that the flux integral of the magnetic field through some arbitrary cross-section of a superconductor is going to be quantized and is going to be some integer multiple of HC over e now here we’ve absorbed the 2 pi and the H bar into H because remember H bar is defined as H over 2 pi so then what is the smallest unit of magnetic flux that can poke through a superconductor well that corresponds to the situation when n is one or equivalently negative 1 the minus sign here just means it winds the other way but in any case what that shows us is that the smallest amount of magnetic flux you can poke through a superconductor is going to be Plank’s constant H time the speed of light C divided by the effective charge of a super electron and remember we’re using gaussian units here so this formula is in gaussian units and you know what’s really cool about this is H and C are very well-known quantities and the flux Quantum F not that can be measured and so we can use this equation to measure the charge of a super electron and when you do this you find that the charge of a super electron is equal to precisely twice the charge of an electron so by examining flux quantization in a superconductor we have direct experimental evidence that super electrons are actually electron pairs simply by examining the topology of a condensate sigh we’re able to relate the flux Quantum to the effective charge of a super electron thereby experimentally determining that a super electron is actually an electron pair I think that’s just the coolest thing ever it’s like a super wonderful result all right so before moving on from the topic of flux vortices and type 1 and type two let me say a few words about flux penetration which sounds kind of weird but it is what it is okay so remember earlier in the video we were looking at the Meisner effect and we talked about how if you have a normal material non-s superconducting non-magnetic then if you put it in a magnetic field the magnetic field is just going to poke through it’s going to penetrate through the material so one way to think about that is if you make a plot where along the X AIS you plot the magnetic field that’s being applied to the material and we can call that H so by H we mean the magnetic field that we’re trying to poke through the material now on the Y AIS we’ll put the magnetic field B that actually pokes through the material and so first of all if we consider the case of a normal non-superconducting non-magnetic material then whatever field we put the material in whatever H we apply that’s going to be the magnetic field that goes through and so we’re going to have the equation b equal H and it’s just going to be a line that goes straight up and is very boring and there’s nothing exciting about it but now consider the case of a type one superconductor when you start to apply a magnetic field to a type 1 superconductor as you know there’s the Meer effect and so you put in a little bit of H and it just gets expelled from the superconductor the superconductor does not allow ow any of that magnetic field to poke through and so the flux density B is zero you can try to apply a field H but the superconductor just doesn’t care it gives absolutely zero flux however what if you keep turning up the strength of H well you know if you look at this picture you can kind of get a feel for the fact that it costs energy to pull the magnetic field apart like this it’s almost like stretching a bunch of Springs you know they want to be straight they don’t want to be curved and so if H is too high then eventually that magnetic energy is going to really want to poke through and the superconductor is just not going to be strong enough to keep it expelled so there’s going to be some critical applied magnetic field we’ll call it h subc for the critical field where beyond that point the field is just too strong and it overwhelms the condensat ability to stay condensed into a condensate you know what I mean so like thermodynamically there is going to be this limit to where the low energy configuration is the one that just completely suppresses the super conductivity and allows the magnetic field to poke through and so if you take a type 1 superconductor and you apply a magnetic field stronger than HC well then you turn off the super conductivity and from that point onwards if you apply a stronger and stronger magnetic field the material is just going to behave like a normal non-s superconducting material okay so now what happens if we have a type 2 superconductor and assume for the sake of argument that we would expect it to have a similar HC as the type one except for the fact that the type 2 superconductor can host flux vortices well what’ll happen is at first if you apply a relatively weak magnetic field the type 2 superc conductor is just going to exhibit a complete misner effect you know it’s going to be like hey okay A little bit of fields no problem let’s go ahead and expel that but if you go beyond a certain point less than HC eventually the type two super conductor is going to be like hey wait a minute okay this field is getting a little bit strong and I’m a type 2 superconductor so I can have vortices so rather than completely trying to keep this field pushed apart why don’t I make a couple of vortices and we’ll let some of the field poke through and as the field is able to partially poke through that’s going to relax the magnetic Field’s energy a little bit you know because now it’s not completely expelled it’s not like pulling a bunch of Springs apart no it’s like pulling some of the Springs apart and some of them are able to just poke right through and so it’s kind of a more relaxed situation and because the field is able to partially relax by poking through these flux vortices what that means is that the material can stay superconducting up to a field that exceeds the thermodynamic critical field HC because the vortices take off some of the magnetic pressure that would otherwise want to suppress the superconducting condensate however this effect can’t last forever at some point you’re going to end up with so many vortices that you just can’t have more vortices okay like everything has its limits and eventually the material if you go to a very strong field is going to break down and the field at which it breaks down for real like completely breaks down we call that hc2 now in physics as in life that which bends is harder to break and so for a type 2 superconductor hc2 is actually greater than HC by the way this is why people use type 2 superconductors for very high magnetic field applications now when you do that you have to be mindful to pin your vortices on some nanoscale defects but that’s a topic for another day well anyway the point is if we say that the defining attribute of a type 2 superconductor is that it has this critical field hc2 that is greater than the critical field HC that we would expect from thermodynamics and if we say that that is the defining attribute of what makes a type 2 superconductor then it can be shown in an analysis that we’re not going to do today that hc2 is equal to the < TK of 2times the magnetic penetration depth Lambda divided by the coherence length c times the critical field HC and so if we think about that as the essential characteristic of a type 2 superconductor is that hc2 has to be greater than HC and we Analyze This inequality divide by HC move some stuff around we find that the constraint is like like we saw earlier that Lambda has to be greater than CI over the < TK of 2 or about 7i so once again that is the magic number between type 1 and type two superconductors also on this topic there is a dimensionless parameter that we can make simply by dividing Lambda by C so in other words what that is is the magnetic penetration depth expressed as number of coherence lengths now even though the magnetic penetration depth and the coherence length both vary as a function of temperature getting smaller as you cool down your superconductor their ratio Kappa is actually pretty constant and so when you think about the Ginsburg landow model and these two length scales that naturally emerge Lambda andai you can see that their ratio is a very defining aspect of a superconductor and if you want to know about how a superconductor behaves one of the most important questions to ask is what is its Ginsburg landow parameter and is it type one or is it type two or if it is type two like how much type two is it right how big is Kappa anyway so that’s the Ginsburg landow parameter all right now I want to reflect for a moment on the explanatory power of the Ginsburg landow model because this model is really the gold standard of how physics should be it’s a beautiful thing cuz we get so much more out than we put in cuz think about it what do we put in well all we’re really saying is that that we’re going to be able to model our superc conducting condensate with a complex scalar field with the super electron density proportional to S squ so that’s kind of you know basic quantum mechanics and we’re also going to bring in the magnetic Vector potential a which is very much standard quantum physics although of course it’s weird why are all these electrons in the same state but other than that it’s pretty straightforward and then we also make some reasonable statements about how s and a contain energy in quasa equilibrium by defining that free energy equation and all of this is constructed using very reasonable principles and evaluated using the standard methods of physics so it’s not a crazy model even though it makes us wonder how can we have so many super electrons in the same Quantum State beyond that it’s not actually that bizarre it’s pretty straightforward now what we get out of the Ginsburg low model is we get the Ginsburg low equations when we vary the free energy with respect to S A and as we saw that’s a generalization of the London equations so that’s going to model the way super currents flow and from that we also get the magnetic penetration depth Lambda from the quantum aspect of the Ginsburg glow equations we get the coherence length s and we also get the misner effect and we see that the topologically non-trivial solutions perfectly explained Quantum flux vortices and flux quantization we find also critical magnetic fields hc1 HC hc2 and the model naturally splits into type one and type two super conductivity depending on the ratio of the magnet itic penetration depth and the coherence length and the model also tells us a lot of things about the surface energy of the superconductor and we can also do all kinds of analysis on Vortex matter and flux pinning and a whole lot more okay so like pretty much all of the Salient aspects of super conductivity are encoded in this relatively parsimonious set of assumptions and presuppositions it’s like a beautiful thing so the math works but as beautiful as all this math is we do need an explanation of how it is that a bunch of electrons can be in the same Quantum State all right you know what it’s time to address the elephant in the room which is why is it that the Ginsburg L model is so successful in explaining superconductivity if it’s based on an apparent violation of the poly Exclusion Principle and to find the answer to that question we have to turn to BCS Theory which was published in 1957 BCS theory is named after John Bine Leon Cooper and Robert shrier the basic idea of BCS theory is that some electrons can actually be attracted to each other in a material via their Mutual interactions with vibrations in the atomic lattice also known as phonons and you know this is often pictured as imagine you have an electron traveling through a material and as it goes along it kind of draws in the surrounding atoms a little bit right because the nuclei are going to be kind of positive ly charged and so as an electron moves through it kind of pulls them together but then another electron somewhere else now sees that density and that concentration of positive charge like a little cloud of positive charge and is drawn in towards that and so in this way these electrons can pair up that’s a gross oversimplification because in reality the details of this are very uh you know quite complicated and if you want to see those you can look at the BCS paper from 1957 Link in the description below but for today we’re not going to go into details on how exactly BCS works the main thing I just want you to know is that these guys figured out that superc conductivity has to do with electron Pairs and the way that they pair has to do with how they interact mutually with the crystal lattice by the way you can prove that by making superconductors with varying amounts of different isotopes because that’s going to sort of modify your phoneon landscape modify the acoustic properties of the material and you can see that that has an effect on the super conductivity so why does this matter this thing about pairing who cares well see what happens is when the electrons pair their spins add up and so as a pair they behave as a bon which has integer spin now in the classic BCS model a spin up electron pairs with a spin down electron for a Net Spin of zero for the pair but in some more exotic superconductors it is technically possible that you could have like two spin UPS or two spin downs and then you get a spin of one but that’s still a bon and once the pair is collectively a bosonic entity then by the Spin statistics theorem now that they have integer valued spin that pair is going to be symmetric under particle exchange and so they are liberated from the poly Exclusion Principle and that allows them to form into a condensate and so that is the microscopic story of how the macroscopic condensate emerges anyway again there are a ton of details in BCS Theory it’s a very complicated Theory but check out the BCS paper Link in the description um actually the first couple pages are really nice and prosaic and then the math gets pretty intense but you might be interested in at least reading the first couple pages also if you want to wield super currents if you want to use this for engineering applications you don’t really need to know BCS you can just use Ginsburg Lando BCS is really like um you know what it’s like it’s like trying to understand Water by using molecular Dynamics it’s like actually if you’re doing something in fluid dynamics you want to use the your Stokes equations to model the flow and the pressure and all that but philosophically if you want to know what water is you also have to know about H2O so BCS theory is like understanding water in terms of H2O and Ginsburg low is like understanding water in terms of the Navia Stokes equations and the way that it flows oh while we’re on this topic superc conducting electron pairs are often called Cooper Pairs and fun fact about that they’re called Cooper pairs not because they make barrels but because like Sheldon their name after Leon Cooper and so you often hear Cooper pairs or super electron pairs or you can just call them super electrons that’s okay as well but just keep in mind that actually the unit of superc conductivity is a pair of electrons so when we talk about super electrons we’re implicitly talking about electron pairs okay so BCS theory was a fantastic Triumph in understanding the nature of superc conductivity at a microscopic level but in 1957 the question still remained are we sure that the Ginsburg landow model emerges from BCS in the appropriate limit and I mean one would assume so based on the experimental success of Ginsburg landow but technically if we’re being philosophical we want a mathematical rigorous derivation of that fact and that derivation came along in 1959 thanks to the brilliant scientist Lev gorov in 1959 he published a paper which demonstrates that yes indeed BCS Theory does logically imply the Ginsburg landow model in the appropriate macroscopic limit and there’s a couple of other assumptions in there as well like uh Ginsburg landow is a good approximation especially near phase transitions where the absolute value of s is relatively low so you know there’s some nuances in caveats and you can see his paper if you want to read about those but the big picture is now we have a microscopic description of how super conductivity can come to be in terms of electrons interacting and pairing with each other and we also have our macroscopic Ginsburg landow level of description and now we have a logical Bridge connecting the two so this is a really enlightened comprehensive picture of superc conductivity and as of 1959 Humanity had really kind of started to figure it out in a serious way but the plot thickens because in 1963 the American scientist Philip Anderson taught us another way to think about superconductors by emphasizing the importance of symmetry and symmetry breaking Anderson realized that much of the behavior of a superconductor can be understood as a result of the phase rigidity of the condensate and the way in which that phase rigidity Alters how we usually think about electromagnetism to be clear Anderson didn’t modify the Ginsburg landow model he just found within it a tremendously insightful way of looking at things a mechanism Within the model that accounts for much of the bizarre properties of superconductivity in terms of photons acquiring an effective Mass inside a superconductor okay so this mechanism should be called the Anderson mechanism right well actually Peter higs came along in 1964 and in a beautiful Nobel prizewinning paper of just a page and a half he took Anderson’s idea and ran with it generalizing it not just to superconductors but to space itself which as it turns out is apparently a kind of superconductor as we’ll explore at length in the latter half of this video and so you often hear it called the Anderson higs mechanism not because they came up with it together but because Anderson came up with the basic idea and then higs saw in that idea a cosmic Epiphany which forever changed our understanding of reality itself so I think they both deserve a little credit all right to understand the Anderson mechanism we have to bring together a handful of Fairly technical Concepts this is graduate level physics so don’t be demotivated if it takes some time and effort to learn that’s just how it is so first let’s go ahead and summarize the Anderson higs mechanism just the big picture and then we’ll go through each of the component Concepts one at a time and then we’ll assemble those Concepts into the Anderson higs mechanism in a more technical way but hopefully in a way that provides memorable and visual Insight okay so in a superconductor a condensate forms and it spontaneously picks out a particular phase the condensate also has some phase rigidity in that there is energy associated with phase gradients so if you try to locally twist the phase of a superconductor that would take some energy so our condensate lives in the context of a global phase symmetry but it picks out a particular phase and then it enforces that phase via its phase rigidity fine okay but then we would expect to be able to excite the condensate by creating waves in its phase and letting those phase waves Ripple around because by global phase symmetry they should be allowed to Drift But by the phase rigidity they’re going to be brought back in line so the phase at one point of the condensate is going to be brought back into alignment by the pure pressure of the phase at neighboring points and so it’s going to be an interplay between a symmetry and the pure pressure that breaks the Symmetry and these kinds of phase waves would be an example of nambu gold stone modes which are waves that arise when a global symmetry is broken like phonons or magnons or apparently like phase waves in a superconducting condensate however the physical nature of a phase wave in s is a really complicated thing because the local phase symmetry of electromagnet antism means that we can simply gauge away those phase Waves by applying a phase transformation which by definition takes out the phase angle of the condensate everywhere and that puts us into something called the unitary gauge in the unitary Gauge by definition there can be no phase waves in sigh but everything has a cost and when we gauge away our phase waves we end up with an irrotational contribution to the vector potential a and that energetic contribution to a will give us a term in our free energy equation which depends on the magnitude squar of a well such terms are forbidden for a massless vector field and so we come to find that in the unitary gauge it is as if the vector potential a has acquired mass and that seems like we’re violating the laws of electromagnetism but bear in mind that behind the scenes it’s the phase rigidity of sigh that props up the ability for a to have mass this is often described as the nambu gold stone modes have been eaten by the vector potential thereby giving the vector potential Mass okay so there’s a number of technical Concepts in V Anderson higs mechanism so let’s take a few minutes to review each of those Concepts and then we’ll bring it all together into a beautiful picture of the Anderson higs mechanism all right let’s talk about nambo Goldstone modes nambo Goldstone modes are waves that are born of a broken symmetry so in the laws of physics there are often symmetries where a symmetry is a variable that doesn’t really matter and a good example of that is spatial translational symmetry which says that the laws of physics in one place are the same as in another place and we know this because our planet is hurdling through the void at very high speed and yet we don’t have like seasonal changes and the fine structure constant or that sort of thing like no actually space over here is pretty much the same thing as space over there and so imagine that you’re a free atom just drifting through space well as you drift along the space that you travel through at one point is going to be the same as the space that you travel through at another point but when you have a crystal and a bunch of atoms get together and decide to do a periodic Arrangement well if you’re an atom in the crystal then now all of a sudden it kind of effectively does matter where you are because if you start to drift away from the you’re supposed to be at the neighboring atoms are going to be like hey get back into that position we’re kind of doing a periodic thing right now you know so and so a crystal effectively breaks spatial translational symmetry now that’s kind of a dramatic way of describing what’s happening and even though now if you’re an atom in the crystal it’s as if you don’t have spatial translational symmetry but the principle of spatial translational symmetry is still an aspect of the laws of physics and so a dynamic interplay emerges between the Symmetry and its breaking now similarly if you have a magnetic material the magnetism arises from the alignment of the spins of the electrons in the material and so what happens is as their spins align it’s going to spontaneously break rotational symmetry because there’s not any cosmically preferred spin axis but because of the magnetic ordering of the material the spin in one place is going to want to be aligned with the spin in another place and so once again we have a nambu gold stone situation where you can get these spin waves going through the material because rotational symmetry allows the spins to kind of rotate their axis but the magnetic ordering is going to pull the spins back into alignment so these waves are called magnons and much like phonons they are nambu Goldstone modes and so then likewise in our condensate it exists in the context of a global phase symmetry so that’s going to let the phase angle Drift But then there’s also some phase rigidity which is going to apply a restoring force that brings the phase back into alignment and from that we should expect some wavy waves to emerge now one of these things is not like the others the phase waves in a superconductor have a particular kind of existence because they’re exactly the kind of thing that can be gauged away and so this reasoning about expecting superconductors to host nambu Goldstone modes it is well motivated and correct but when you factor in what those waves are made out of you realize that there’s actually a really subtle Dynamic going on here where to understand the N Goldstone modes you have to look at it in the context of the vector potential and local phase symmetry now to set our expectations there’s something you should know about nug Goldstone modes which is that typically we would expect these waves to be massless so what is a massless wave well consider the photon now now a photon is not in U Goldstone mode but it is the quintessential massless wave and so it’ll work just fine for thinking about what a massless wave is so when you have a photon there is a direct relationship between the energy and the momentum so if you double the photon’s momentum you get twice the energy now as we’ve seen in quantum physics energy has to do with how much the wave is flopping around in time and momentum has to do with how much the wave is crunchy in space so when you look at a photon you see that the more scrunchy it is the more floppy it is and so if you imagine a very stretched out wave like for example a red Photon or even infrared or radio or something longer then as the wavelength gets longer and longer and longer the momentum Trends towards zero and so does the energy and that is the essence of masslessness so a massless wave when you imagine stretching it all the way into the long wavelength limit with wavelength approaching Infinity the energy should approach zero and even though photons are not nugold Stone modes that same principle applies when we talk about nugold Stone modes being massless if you imagine for example a phonon of very long wavelength as the wavelength increases and increases the waves get slower and slower and slower and the energy Trends towards zero and there’s actually a theorem gold Stone’s theorem which says that nambu gold stone modes are massless and when you have a system that hosts nambu Goldstone modes you can always put in a very small amount of energy by exciting a very long range nug Goldstone node but there’s a very important exception to Gold Stone’s theorem and that is if your nambu gold stone modes are the kind of thing that can be gauged away well then this thing about masslessness goes away because when you factor in the gauge Fields then you see that it’s more of a complicated situation as we’re about to see okay so I just bring up photons as an example of a massless wave but now in a superconductor are nambu Goldstone modes these phase waves they actually do acquire mass and the way to think about that is they get absorbed into the vector potential a and then the resulting a which has been enhanced by the nug Goldstone modes inai well that upgraded Vector potential a is going to have mass and when you try to excite the massive Vector potential you’ll find that even for an excitation with zero momentum that is a super long wavelength the energy of that excitation is not going to approach zero but instead we’ll approach the mass energy corresponding to a quantized excitation of the upgraded Vector potential so now we’re moving in the direction of seeing how that happens and we’re building up a picture of the Anderson higs mechanism but as we’re doing that there are a couple of vocabulary words that we’re going to be using and these are solenoidal and irrotational and in order to see what those words mean and how they’re relevant to the Anderson mechanism let’s take a moment to talk about the concept of Helm Holts decomposition so let’s say we have some smooth Vector field in three dimensions and suppose that it’s well behaved so for example if you go out to Infinity the vector field vanishes out there well interestingly you can decompose that arbitrary Vector field into a part that doesn’t diverge plus a part that doesn’t curl so the field that doesn’t Di Verge that’s just going to swirl around right that’s all swirly so that’s called the solenoidal field think of like a solenoid you know but on the other hand the fields that doesn’t curl that only diverges it only pokes out and pokes in and so that’s called the IR rotational field so one way to think about a vector field is the sum of a solenoidal field that swirls plus an irrotational field that pokes out and pokes in and diverges and today we don’t have to go into the details of how to do a helmoltz decomposition but I just bring it up so that for example we can talk about the solenoidal part of the vector potential or the irrotational part of the vector potential because as we’re about to see these are useful Concepts when thinking about the masslessness or massiveness of photons in the context of the Anderson higs mechanism okay okay so I have this animation here and I hope it helps to kind of illustrate the concept it is a bit cluttered I I don’t know I mean this was my best attempt at this but um apologies if this looks cluttered oh and by the way this animation Loops so don’t worry you don’t have to pause it we can just let it go and let it do its thing so anyway what this demonstrates is that if you have some arbitrary Vector field so that’s the gray arrows then you can write that as the sum of the solenoidal field so that’s those blue arrows and you can see how they swirl around and the solenoidal field is all about curl and then the green arrows you can see that this field doesn’t curl it only diverges it only pokes out and pokes in and the sum of the blue and the green that is the sum of the solenoidal and the irrotational field add up to the vector field for whatever physically realistic arbitrary Vector field that is okay so that’s the concept of hemholtz decomposition and I only bring it up so that we can use the concept of the solenoidal part of a field and the irrotational part of a field okay so if we want to learn about the Anderson higs mechanism one of the Core Concepts is the local U1 transformation also known as a local phase transformation now as we saw in the last video the local phase transformation is at the core of electromagnetism and in fact this is electromagnetism in one principle and I’m not going to rehash all of that today cuz we’ll be here for an extra 3 hours and 12 minutes but let’s just take a look at the definition of the transformation because this is going to be helpful for us as we explore the Anderson higs mechanism so the basic idea is that we’re doing some quantum physics and we have a wave function Sai but the phase of Sai isn’t really a physically observable quantity on its own and specifically what we mean by that is if we transform our wave function s into some wave function S Prime by multiplying s by this e to the I Theta term which is a unit length complex phase field so you’re changing the phase of s arbitrarily in space and time then the resulting transformed wave function S Prime should still be just as good of a description of the situation but with the important caveat you also have to apply a compensating transformation in the electromagnetic four potential a and that transformation is going to turn a into a prime by subtracting this quantity H Bar C over Q times the four gradient of that phase field Theta and if you want to refresher on how that comes to be just go ahead and watch the last video and you can see where that comes from but for now we’re just going to take this as a starting point and we’re going to translate it from relativistic language into a particular reference frame namely the reference frame of the superconductor cuz when you’re dealing with a superconductor you don’t actually have to use relativity you can say we’re going to work within the inertial frame of the material itself so in order to do this translation let’s go ahead and write out the definitions of the four potential and the four gradients in covariant form with the downstairs mu the four potential has voltage for the Tim like part and the negative of the magnetic Vector potential as the space-like part and the covariant 4 gradient has a 1 over C time derivative for the time like parts and the gradient for the space likee part so now let’s write the rules for a local phase transformation in terms of just the spatial Parts because that’s what’s relevant in a superconductor we’re dealing with the magnetic Vector potential and we’re not so interested in voltage so looking at the spatial part we can see that the transformation rule can be written as the transformed wave function S Prime is once again s time the exponential of I Theta where Theta is an arbitrary phase field this context it’s just a function of space and doesn’t depend on time because we’re in quasi equilibrium and then the vector potential transforms by the addition of HR C over the charge of a super electron times the spatial gradient of theta so if you look at these two transformation rules the spatial part only as well as the relativistic rule you see they’re basically the same thing for a there’s just a difference in sign that has to do with the mostly minus metric and also instead of Q for the charge we might as well write it e as we’ve been using for the Ginsburg Lando model for the effective charge of a super electron and this brings us to a very important idea which is the concept of choosing a gauge so when dealing with electromagnetism we can transform Sai and we can transform a in a complimentary way in accordance with the local Yuan transformation Rule and that doesn’t change anything observable whatsoever right so that is the principle of local phase symmetry and that principle blossoms into electromagnetism and this principle also means that we have infinitely many equivalent ways of describing the same physical system in terms of different sigh and different a that are all related to each other by local phase Transformations and this is possible because s and a are not themselves directly observable the quantities that are observable that are sort of encoded in San a are encoded in such a way that they’re insensitive to this kind of a transformation so then to choose a gauge is to narrow down that infinite set of options with some convenient gauge conditions and this reduces the redundancy in s and a without affecting any observable quantities like for example we can impose by Fiat that the Divergence of the vector potential should be zero this is known as the coolum gauge condition and this condition can be imposed because you can always add the gradient of some phase field that will ensure that a is diverg less or alternatively we can also say you know what let’s make s a real valued field because we can always apply a local phase transformation that takes away the complex phase at each point in space and then whatever gradients were in s they can just pop up in a as some addition of the gradient of that Theta field that we took out so when SII is a real valued field that’s called the unitary gauge and that is a very important gauge as we’re about to see so yeah that’s the the concept of choosing a gauge and it’s very similar to setting your coordinate system like when you choose where you want to put your XYZ axis and how you want it to be oriented and all that sort of thing you have the freedom to do that and you can still describe the same physical situation and likewise we have the freedom to apply some gauge conditions that are compatible with these local phase Transformations and that’s a part of defining how we’re talking about the physical system so now there’s another very important concept that we’ll have to know about if we want to understand the and higs mechanism and that is the gauge covariant derivative so the purpose of the gauge covariant derivative is to give us a way of defining the derivative operator that is essentially unaffected by our choice of gauge right because for example if you just apply a regular derivative or a gradient to SAI and then you transform Sai by a local phase transformation you’re going to mess up its gradient so what the gauge covariant derivative is is a kind of derivative where when you you apply a local phase transformation to S and you apply the corresponding transformation to a the gauge covariant derivative at least the magnitude of the gauge covariant derivative is not going to change so that’s going to be a quantity that does not depend on the gauge that we’re in and as you might expect then that quantity is going to be closely related to physically observable things like momentum all right so the gauge covariant derivative is defined as the gradient of s minus 1 / e h C time the vector potential Time s and by looking at it you can kind of Imagine okay so we slap on a phase transformation to S we transform a in the corresponding way and this quantity should remain pretty much unchanged maybe it picks up a phase Factor but its amplitude should remain unchanged let’s go through this real quick just to see that that is indeed the case so if we write out the definition of a local phase transformation where we slap on a phase Factor e to the I Theta to S and that becomes S Prime and then likewise we add the term proportional to the gradient of theta to the vector potential a and all we have to do to see how the covariant derivative transforms is to plug in S Prime and a prime for S and a in the definition of the covariant derivative when we do that we find the expression the gradient of s e to the I Theta minus i e h c * a plus h Bar C over e gradient of theta Time s * e to the I Theta so it’s kind of an algebraic Mass but we can simplify it first thing we’ll do is we’ll apply the product rule to the term s e to the I Theta and first taking the derivative of s we get the gradient of s * e to the I Theta then you take the derivative of the E to I Theta term and you end up with I * s * e to I Theta * the gradient of theta okay now likewise if we look at that second term let’s go ahead and multiply through by that factor i e star over H Bar C so we pull that in we distribute it we cancel a couple of terms on the Theta term and we end up with the expression shown here now this equation simplifies quite nicely because you can see that these green terms cancel each other out and then it gets even better because if you look at this equation with the gradient of s * e to the I Theta minus i e h bar c a e to the I Theta man these equations are mouthful anyway you recognize the blue terms here as the definition of the covariant derivative D acting on S and then you see that there’s an e to the I Theta term and so you can go ahead and fact that out and what we endend up with is the conclusion that the covariant derivative acting on the transformed wave function S Prime is equal to the covariant derivative acting on the untransformed wave function times a phase Factor e to the I Theta now e to the I Theta has an amplitude of one and so therefore the amplitude of the covariant derivative is the same before and after applying a local phase transformation and so what this demonstrates is that no matter what gauge we’re working in maybe you’re working in the Kum gauge and I’m working in the unitary gauge we can both agree on the amplitude of the covariant derivative and this is a useful concept for simplifying a lot of calculations and also connecting different concepts from one gauge to another which as we’ll see is like essential for understanding the Anderson higs mechanism oh and you know what another cool thing about the gauge covariant derivative is that it makes the Ginsburg landow free energy density equation a lot simpler because check it out okay so first let’s write out the equation for f as we’ve seen now if you look at the definition of the gauge covariant derivative you realize that this middle term in F the complicated one with all the H bars and I and gradients and E you know that whole term you can see in that the pieces of the gauge covariant derivative and you can kind of assemble it together and do some pattern matching and you end up finding out that that whole term can be replaced simply with h^ s over 2m star time the amplitude squar of the co derivative acting on S so that is really cool I mean look at how much simpler it is and not just simpler in terms of number of characters but also simpler in that now we have this quantity which manifestly does not care about which gauge we’re in and that makes the equation a lot easier to parse philosophically if you want to color code the different parts of this equation we see that there’s the free energy density associated with the condensates existence as well as energy density associated with the kinetic energy of the condensate that has to do with that gauge covariant derivative of s and then we also have the energy density of the magnetic field now when you look at this equation and you consider the various things that it has to encode in order to give us the full description of a superconducting condensate you actually see that it’s about as simple as it can be I mean I know the first time you see this equation for the Ginsburg land free energy density it’s like oh my gosh what is all this Quantum stuff this is crazy but when you really dive into it you like oh okay actually this is a pretty elegant model you know it’s not very contrived it actually makes a lot of sense and while we’re here let me just make one more point about the Ginsburg landow free energy density which is that the whole equation the whole F equation does not depend on our choice of gauge which makes sense because energy is a real observable quantity so by definition it cannot depend on our choice of gauge whatever gauge we’re in all the observables should be the same and you can see this by imagining that we apply some arbitrary local phase transformation and then look at whether or not the individual parts of f are going to be affected by that well first of all the energy due to the condensate density and density squared that doesn’t change at all because this phase Factor e to the I Theta that we slap onto s does not change the amplitude of s and now if we look at the kinetic energy density of the condensate we find that it’s a constant times the amplitude squar of the gauge covariant derivative and we know that the gauge covariant derivative its amplitude doesn’t change when we go from one gauge to another so that part stays constant as well and then last but not least if we look at the magnetic Field’s energy density this also doesn’t change because when we transform a we add to it the gradient of a scalar field but the gradient of a scalar field does not curl and so therefore the amplitude of the curl of a prime is the same as the amplitude of the curl of a and another way to think about this is remember what is the curl of a well it’s the magnetic field B and the strength of the magnetic field is an observable quantity so therefore that also can’t depend on our choice of gauge so the whole Ginsburg landow free energy equation is completely independent of whatever gauge we’re working in all right so now we’re finally ready to look at the Anderson higs mechanism first what we want to do is imagine a superconducting condensate with a constant amplitude of Sai but that hosts these nambo gold stone modes in the form of these little phase oscillations so the phase is going to get kind of wavy like how we talked about before and at first what we can do is suppose for the sake of argument that there is no Vector potential yet okay so we’re just thinking about a superconductor on its own without being influenced by the vector potential and by the way everything we’re about to talk about you could also think about in a more General context but for now we may as well at a equal to Z well then you know another way to look at this situation with the wavy waves is to say let’s hop into the unitary gauge where we apply a local phase transformation e to the II where fi is the phase of SII and so by applying that transformation e to the negative II we’re transforming the wave function SII which is complex valued and has these phase waves into a real valued scalar field S Prime that has no phase and no phase waves so we’ve gauged away the nambo gold stone modes oh that’s fantastic but you know what happens when you apply a phase transformation to S we also have to apply the corresponding phase transformation to a so we’re going to add H Bar C over etimes the gradient of the angle that we’re transforming by in this case the angle is going to be 5 at each point we’re twisting the phase and S by an angle of NE 5 and so we actually get Negative H Bar C over e star times the gradient of F and there’s actually a sneaky double negative here because bear in mind the charge of a super electron is negative and so a is actually going to pick up a quantity that is directly proportional to the gradient of the phase of sigh before we go into the unitary gauge so when you look at these two pictures which by the way are gauge equivalent descriptions of the same situation you can see that in the bottom picture what were previously gradient in the phase of the condensate now show up in a as the gradient vector and so for example if we pause this real quick and if you look in the direction of increasing phase somewhere so that’s going to be where the phase angle is turning counterclockwise well the gradient in the phase there is going to be pointing uphill in the direction of the phase increase but if you look then at the bottom part in the unitary gauge that’s what a is going to be and so that’s all we’re saying we’re saying that when we hop into the unitary gauge what were previously these nambo Goldstone modes in the condensate now get absorbed or eaten by the vector potential a so this is very interesting because earlier we talked about naambo Goldstone modes we talked about Symmetry and broken Symmetry and these waves should be able to contain energy because of the phase rigidity but now we’re seeing okay yeah that is true or that seems to be true but we have to be careful here because there’s an extra level of neons which is that in a superconductor these nambo gold stone modes these phase waves have a form which can be completely gauged away and accounted for instead entirely in the vector potential and so to the extent that the condensate can contain energy because of its phase rigidity we might as well account for that energy in terms of a modification of the usual behavior of the vector potential a and it turns out that what that modification is going to do is going to give a mass and in so far as a photon is a disturbance in a that means that photons are going to acquire an effect effective mass in a superconductor so let’s take a look at how this happens well first of all we have to calculate how much energy is contained in the nambo Goldstone modes and the answer to that can be found in the kinetic term of the Ginsburg landow free energy density which as we’ve seen is h^ s over 2m star time the amplitude squared of the covariant derivative acting on S and for a simple analysis let’s suppose that the vector potential is zero so that we’re really focusing purely on the energy contained in the phase waves well then in that case when you write out the covariant derivative you see that the term involving the vector potential goes away and the kinetic energy contained in these phase waves is simply hr^ s over 2m startimes the amplitude squared of the gradient of the wave function and then when you think about like the amplitude of s is constant everywhere you see that the gradient term really just depends on the gradient in thi and that has to do with this idea of the phase rigidity of the condensate containing energy you see it in this expression but now then when we hop into the unitary gauge we gauge away the nambu gold stone modes so that the gradient of s is zero and then the question is wait a minute does the kinetic energy become zero too because we just solved kinetic energy is proportional to the gradient of sigh and so if that’s zero then kinetic energy is zero right well no because energy is observable and so therefore it cannot be affected by our choice of gauge and so that kinetic energy term involving the gradients of s is going to have to be replaced with the term involving the vector potential a and okay so we can do that by again looking at the covariant derivative this time the gradient of s is the zero vector and so the S term goes away we’re left with just the term involving the vector potential and s in those constants and when you plug that into the kinetic energy equation we get an expression which is e^ 2 over 2m star c^2 amplitude of s^ 2 time the magnitude of a^ s and that is another way of accounting for the energy in those phase waves and it’s totally equivalent to the expression that we derived earlier but this is the correct expression when we’re in the unitary gauge and so what this all means is that because we have the freedom to choose our gauge and because we think that a condensate should be able to host these nambo Goldstone modes with the phase waves well one way to look at that is that in the condensate there’s going to be some energ that arises from this irrotational contribution to the vector potential a notice that the contribution to a from the namu Goldstone modes is an irrotational field because it’s based on the gradient of the scalar field fi and the gradient of a scaler doesn’t have any curl now normally when you’re dealing with the vector potential a the energy of a is contained in its solenoidal part thinking in terms of hemh hold’s decomposition and the irrotational part of a is usually not exactly real because you can make it go away by hopping into the coolum gauge where your wave function sigh takes on whatever phase angle it has to in order to accommodate a diverg less Vector potential but the situation is more nuanced in a superconductor because the phase rigidity of the condensate means that there are energetic consequences that come in when you try to Simply erase the irrotational part of a because if you try to gauge away the irrotational aspect of the Vector potential a it’s just going to manifest energetically as nambu Goldstone modes inai to develop an intuition for what’s going on here it’s helpful to imagine something that I like to call the masochist gauge this is not a technical thing you won’t find it in a physics textbook but it is a genuinely legit concept the idea is just to imagine some nambu Goldstone modes and then hop halfway to the unitary Gauge by subtracting out half the phase angle of sigh so it’s like half a hop we have one foot in each world and so we get to deal with the worst of Both Worlds because in the masochist gauge the nambu Goldstone modes are only half eaten by the vector potential and so we still have some phase waves in the condensate but we also have some irrotational contribution to a that contains energy you’d have to be an enjoyer of suffering and misery in order to ever work in this gauge but it is useful for helping us think about the Anderson higs mechanism because all three of these pictures are gauge equivalent and so you can imagine your perspective shifting around between gauges as a continuous thing so if you put the Nabu Goldstone modes in the left hand of your mind’s eye you know what I mean and then if you put the unitary gauge in your right hand it’s not like you have to choose between your left hand or your right hand you can shift around mentally in the space of gauge equivalent descriptions of the situation and you’ll find the masochist gauge as like a halfway point between your left and your right hand and you don’t want to spend any time there but by noticing that it exists it kind of gives you a feel for the continuity of the different gauge choices that we’re allowed to make because one of the questions that comes up with the Anderson higs mechanism is what’s so special about the unitary gauge and the answer is it doesn’t matter what gauge we’re working in but so therefore you might as well work in the gauge where everything’s the simplest where all of the shenanigans about phase waves and get absorbed on up into the vector potential a and so you might as well work in the unit gauge in which the phase waves are accounted for as part of the upgraded massive Vector potential now if we look again at the equation that we just calculated for the free energy density contained in these waves in a in the unitary gauge we see that it’s an expression that depends on how much a we have somewhere if you think back to the last video we saw that the masslessness of the four potential a mu is precisely the statement that the amount of energy in the system does not depend at all on how much or how little a we have but now the situation is different in the unitary gauge there is an energy term which depends on how much a we have and so therefore a has mass something to notice here is that even though we started our analysis with zero Vector potential before hopping into the unitary gauge and so all of our A’s the irrotational contribution from the nug Goldstone modes had we instead started with a nonzero a then we could apply the same kind of reasoning and we would get the same result about energy density being proportional to a^ S But the A in that context would be a more General Vector field not just an IR rotational field but also a field that can have a solenoidal component and so even though the N Goldstone modes are eaten specifically by the irrotational component of a in a more holistic sense all of a with all its degrees of freedom have become upgraded by the consumption of the nugal Stone modes regardless of how exactly a is Rippling around in a superconductor in the unary gauge there is energy that depends on A’s magnitude squared and therefore a has mass notice also that the strength of this Mass term depends on the density of the superconducting condensate s squ and also the charge of a super electron and that makes sense because imagine you’re a photon and you’re interacting with a condensate well how much your resistance intera acts with the condensate depends on the density of the condensate and also how strongly you’re coupled to its constituent particles and so the thicker the superconducting condensate is the more massive the photon becomes so how should we calculate the mass of a photon in a superconductor in the unitary gauge we’ll invoke a principle from Quantum field Theory which is that if you have a term like this where the energy density depends on the magnitude squared of a real valued Vector field then that’s going to be a mass term and it’ll have the form 1 8 piun * MC h^ 2 * the magnitude squ of the vector potential a now we’re not going to be deriving that principle today in the interest of time today we’ll just take it as a given and later on we’re going to use essentially the same principle again when calculating the w and z masses okay so to calculate the mass of a photon in a superconductor we just have to match terms and solve for the mass so first if we multiply by 8 piun * h^ 2 / c^ 2 we end up with an expression for the mass of the photon squar and then all we have to do is take a square root and we solve that the mass of the photon is twice the square root of pi over the effective mass of a super electron time the effective charge of a super electron time H bar / c^ 2 time the absolute value of s notice how the mass of the photon is directly proportional to the amplitude of s later on we’ll see that the masses of the w and z bosons are directly proportional to the higs vev and for precisely the same reason but we’ll get to that later for now let’s focus on the vector potential a now as you can imagine an a that has mass behaves differently than an a that has no Mass all right well there’s a way of looking at this math that you’ll often hear people talk about which is that when a becomes massive in a superconductor it gains the ability to be longitud Ally polarized and this gives the vector potential a in a superconductor an extra degree of Freedom or an extra way that it can be excited that is physically meaningful so the blue arrows on the left Show an example of a vector potential in the coolum gauge so the coolum gauge condition is that the Divergence of a is zero so then we end up with an a that is purely solenoidal now remember the magnetic field B is the curl a so when you think about the way that the magnetic Vector potential encodes magnetic energy you realize that really it’s only the solenoidal part of the fields that you have to worry about and the IR rotational part is not physically significant however in a superconductor when you’re dealing with the unitary gauge as we’ve seen in the unitary gauge the nambu gold stone modes in s are going to manifest as an irrotational contribution to a and so on the right what we’re seeing is the the vector potential in the unitary gauge and so you can see that this Vector potential is no longer purely solenoidal but also has a non-zero irrotational part corresponding to those NAU gold stone modes so in addition to solenoidal and irrotational there’s also the terminology transverse and longitudinal and you know a transverse wave is something where whatever direction the wave is propagating the thing that’s actually oscillating is oscillating in a perpendicular direction to that so for example here you have this wave that’s kind of going upwards at a 45° angle but the thing that it’s comprised of the vector oscillations that make the wave wavy are going 90° relative to that 45° you see what I mean they’re kind of going up and to the left and down and to the right so that’s a transverse wave now the connection between a transverse wave and a solenoidal field is that if you take an arbitrary Solen field and say I’d like to look at this from the perspective of 4A analysis where we decompose the vector field into a sum of plane waves well the fact that your vector field is purely solenoidal means that those constituent plane waves themselves have to be solenoidal meaning that they don’t diverge or converge and in order for a plane wave to be divergenceless it has to be transverse so that whatever way it’s moving it’s oscillating perpendicularly to that and you’ll notice that that wave does not contain any Divergence or convergence and so when you stack a bunch of arbitrary transverse waves to construct some general Vector field that field is always going to be solenoidal because none of those transverse plane waves are going to add any Divergence or convergence to the field and the Divergence is a linear operator so you know you add a bunch of diverg less plane waves and you end up with a Divergence list AKA a solenoidal AKA a swirly Vector field okay now on the other hand a longitudinal mode is a wave where the direction of propagation is in the same direction as the thing that’s sloshing around and comprising the wave so here we have a longitudinal wave going up and to the right and we see that the vectors themselves are sloshing around up into the right and down into the left and so that is a longitudinal mode and this mode doesn’t have any curl so when you stack a bunch of longitudinal modes you’re going to end up with an IR rotation field and the reasoning here is very similar to what we just saw but instead of adding up a bunch of transverse waves which are Divergence less now we’re adding up a bunch of longitudinal waves which don’t have any curl because for a longitudinal wave the amplitude vector and the wave Vector are parallel so you can stack up as many longitudinal waves as you want each one has no curl and because curl is a linear operator the resulting sum is going to have no curl and so that’s why from a forier analysis perspective if you’re dealing with an irrotational field you can write that as the sum of longitudinal waves now what I’m showing here is an example of two waves superimposed on each plot and in fact they have the same wave vectors but the waves on the left are purely transverse and the waves on the right are purely longitudinal and you can see just by stacking two waves that we end up with a swirly situation on the left and a diverging and converging but not rotational situation on the right so when you generalize that to a more General situation you can see the connection between transverse waves and solenoidal field and longitudinal waves and an irrotational field so now if we return to the picture we were just looking at where on the left we have a solenoidal A in the kolum gauge and on the right we have a more General a in the unitary gauge just to emphasize that a in the unitary gauge is not purely irrotational no it can contain a idal component as well it contains both and so if you wanted to decompose it into a sum of plane waves you’d have both transverse and longitudinal modes in that sum so this is what people mean when they say that because of the Anderson higs mechanism the vector potential a is able to be longitudinally polarized in addition to the transverse polarizations that it can usually host and so a gains these extra degrees of freedom that are physically meaningful but of course as you now know those extra degrees of Freedom don’t just come out of nowhere those are secretly just the nambu Goldstone modes in Sai but you know a mode By Any Other Name Is Still uh I don’t know I thought that was going somewhere but it’s not so all right hey let’s move on something else that’s cool about the Anderson higs mechanism is that it gives us a fresh perspective on the Meisner effect so let’s say we’re looking at the surface of a superconductor with the vector potential shown here in Orange and we’re going to try to excite the vector Potential from outside the superconductor by applying a magnetic field to it well in the unitary gauge because the vector potential a has mass that means that there’s going to be some energy density which is proportional to the magnitude squar of a and so therefore a is going to want to minimize its energy by relaxing itself out of existence to the zero Vector in the bulk and you can personally relate to how a feels in a superconductor because it’s sort of like Monday morning when you have to wake up and you’re like oh man I’d rather just be the zero Vector right now I don’t want to do anything but of course in a superconductor there are other kinds of energy pertaining to gradients and all that sort of thing and so if you try to excite the superconductor by applying a magnetic field to it well a is going to be aroused a little bit near the surface of the superconductor up to some distance Lambda the penetration depth which by the way is inversely proportional to the effective mass of a photon in a superconductor beyond the length scale of the penetration depth Lambda a is just too lazy to play along and so out there it would rather be zero now because a wants to be the zero vector and the curl of a is the magnetic field B well the curl of the zero Vector is also the zero vector and so there you go that’s the misner effect inside a superconductor the vector potential at the nomu Gold Stone modes in Sai that gave it mass and now it’s all massive and lazy isn’t this a beautiful way of looking at the Meer effect I love it I think this is awesome it’s great it’s fantastic now of course you might be thinking hey well hold on now this is only in the unitary gauge but suppose we chose another gauge and we wanted to look at it in that situation and we’re always free to do that you can always hop into a different gauge if you want but bear in mind that the vector potential in any other gauge is only going to differ from the vector potential in the unitary Gauge by the addition of the irrotational component that comes from the gradient of the phase angle of our phase transformation and so the vector potential we’re looking at in the unitary gauge which wants to be the zero Vector in some other gauge it’s going to want to be a purely irrotational field and what do we know about purely irrotational Fields they don’t curl and so in any other gauge it’s still true that the magnetic field goes to zero inside a superconductor Okay so we’ve been talking about n Goldstone modes and the vector potential and N Goldstone modes getting absorbed into the vector potential in the Anderson higs mechanism but we can also Imagine amplitude modes in sigh so as the name suggests these are waves in the amplitude of Sai now these waves are going to be inherently massive it’s going to cost energy to make the condensates magnitude deviate from where it wants to be so if you think about the amplitude of the condensate at some point increasing and decreasing and you think about the somero part of the Ginsburg landout potential you know the Alpha and the beta when the amplitude of s is inherently not going to want to deviate from the value that minimizes that part of the potential and so therefore any excitation in this mode requires energy regardless of the wavelength and that gives rise to a mass energy in these excitations now when you factor in the quantum nature of the condensate it turns out that you can only excite this mode in discrete energy packets and so therefore there is going to be a mass Gap in the system where if you want to excite an amplitude mode you have to to put in at least the mass energy required to excite that mode now it’s very difficult to excite an amplitude mode and it’s a very unstable form of energy so right after you excite an amplitude mode its energy is going to scatter away into some more stable form but still there are ways to experimentally confirm that superconductors can host amplitude modes also the amplitude mode of a superconductor is often called the higs boson of a superconductor not to be confused with the higs Bon of the superconductor of the universe but it is exactly the same idea and that’s why the discovery of the higs boson was such a big deal because it showed that the superconductor thing is more than just an analogy because it can be directly jiggled if you followed along with everything so far then you’re now in a great position to learn about the higs field and to understand it in a genuinely deep and meaningful way that goes beyond the Molasses analogy now that you know what is super conductor is we can use the superconductor analogy okay so when upgrading our mental image from a superconductor to the higs field there is one extra bit of algebra that it’s worth talking about for a moment and that is su2 now some of you are already familiar with su2 and we talked about it in the spinner video but let’s just reflect on it for a moment and also there’s a way of framing su2 that segs nicely into the electro week model so even though it’ll seem like we’re just talking about su2 right now we’re actually setting the stage for Electro week unification all right so the easiest way to imagine su2 is to start off by imagining U1 which we talked about at length in the last video U1 is all the ways of changing a complex number without changing its magnitude so it’s all the ways you can swing around a complex number in the complex plane algebraically we can encode these transformations as e to the I Theta which is a unit length complex number with a phase angle of theta counterclockwise from the real axis and because complex multiplication involves adding the phase angles of the numbers being multiplied multiplying a complex number by the unit length e to the I Theta simply changes the numberers phase angle by Theta we’ve used this idea a lot in previous videos so that shouldn’t really come as a surprise but here’s a cool observation we can get from U 1 to su2 just by changing one word su2 is all the ways of changing a complex dublet without changing its magnitude a dublet is a fancy way of saying pair if you’ve ever seen two peas in a pod then you’re familiar with the concept of a dublet it’s just two of a thing that are to be regarded collectively as the same thing I like to draw a dublet as two points in the complex plane representing the two PS with a little line connecting them representing the Pod so you see a dublet is a cute thing it’s not a scary thing there’s no need to fear doublets doublets are our friends and if you want to you can think of a complex dublet as a vector in C2 but we don’t even need to overthink it like that a dublet is just twps in a pod and for understanding su2 it’s helpful to think with that level of structureless generality you can of course use Duets to encode different things things for example a spinner field can be written as a doublet valued field but the spinorial essence of a spinner comes not from it being writable as a dublet but also the space-time transformation properties of the field that are governed by for example the Dr equation but anyway all that’s to say a spinner field can be written as a doublet field but a dublet field is not necessarily a spinner field it could just be like a scaler field in terms of how it transforms under Loren boosts but the scalers have a form of a dublet that’s okay and in fact as we’ll see the higs field is a complex doublet scalar field anyway I find that at first it’s helpful not to get too hung up on the ontological meaning of a dublet just think of it as a kind of number and usually we write a dublet as a colum of complex numbers so if we have a double it named fi we can write that as a stack of the complex numbers f 1 and 5 2 so what is the size of a dublet well let’s use the same idea as for a complex number one of the fundamental rules of complex numbers is that the magnitude squared of a complex number fi is given by fi conjugate fi right F star fi so then the magnitude of f is the square root of f star five we can apply exactly the same idea to a doublet but instead of just conjugating fi we also have to turn it over on its side so that it becomes a row Vector because then we can take the dot product and have the dimension fun ality work out correctly and so actually the magnitude squar of a douet is given by fi dagger fi where the dagger symbolizes the transpose conjugate which just means that we flip F on its side and conjugate it why we use a dagger to symbolize flipping a thing on its side and conjugating it I don’t know maybe I don’t want to know but it is what it is so the magnitude of I is the square root of f dagger okay so let’s return to our definition of su2 all the ways of changing a complex dublet without changing its magnitude well by inspection a complex dublet has four degrees of freedom two for each complex number so then if we want to change a dublet while holding its magnitude constant how many degrees of freedom do we have four degrees of freedom minus one because the magnitude is fixed leaves us with three phas like degrees of freedom and so this su2 group is going to have three generators that is uh three fundament distinct ways of rolling around in su2 now when dealing with U1 we use e to the I Theta to rotate a complex number by the angle Theta similarly when dealing with su2 we can use three Expressions to rotate a double it around by three different angles in three different ways and you can combine this into a big exponential of e to the I Theta 1 T 1 plus Theta 2 to 2 plus theta 3 T 3 where the thetas you can think of is like phase angles and to 1 to 2 and to 3 are the poly spin matrices you often see these matrices written as Sigma but in the electro week model it’s actually more common to write them as too okay so just by looking at these matrices it’s not obvious why they are what they are or what exactly they do and if this is your first time with Matrix exponentiation you might also be like hold on how do you exponentiate a matrix but that’s all just algebraic stuff that you learn along the way the important thing is to see what su2 Transformations actually look like so the animation here shows some example doublets being transformed on the left we have tow one in the middle we have tow two and on the right we have to 3 and you can see by looking at this that these are all like different ways of kind of rotating the P’s in the Pod but it’s not obvious how to make sense of this and so it would be nice if we had a better way of visualizing doublets a way that makes these su2 Transformations more easily digestible by the imagination because uh the complex plain peas in a pod picture is nice because we can directly see each part of the dublet so it’s a very direct and transparent window into the algebra at least in terms of what the doublet is at each moment but the problem is when we look at su2 transformations in this picture the Transformations are confusing and hard to describe I mean you can kind of of get a sense that there are these three independent ways of swirling the Duets but the motion doesn’t quite click into the imagination in the way that we might like it to fortunately there is a way of visualizing su2 which makes the Transformations themselves much much easier to understand but the cost is that the objects they transform become a little more abstract and this is the flag picture that we saw in the spinner video and in the last video here’s how the flag picture works any dublet can be drawn as a flag in accordance with some equation so there’s some equation which you put in the dublet and it gives you the coordinates of the flag and how it’s oriented now the details of that equation are actually not so important for now all you really need to know for today’s video is that a dublet can be depicted as a flag and the how is not so important okay of course if you’re really getting into this topic then you should also learn the how and you can find that in the paper in introduction to Spinners by Andrew mste Link in the description the flag picture applies equally well whether you’re dealing with spinners or doublets more generally so spend some time with that paper if you’re interested in the details but we don’t need to linger on those details today okay so the reason I bring up the flag picture is that in this picture the generators of su2 that is the poly matrices TOA are very easy to visualize TOA one rotates the flags around the xaxis to 2 rotates the flags around the Y AIS and Tow 3 rotates the Flags around the Z axxis so it’s as simple as that by representing the doublets as Flags we can imagine the poly matrices in terms of our natural intuition about threedimensional rotations that makes the math way easier also a cool thing to know is that in this flag picture a U1 transformation rotates each flag around its own axis so you see that U1 doesn’t go beyond what su2 can do in the sense that for any given U1 transformation on any given flag we could also write that in terms of the appropriate su2 transformation that gets you from the first flag to the second flag and so U1 is just kind of a different way of transforming a flag it’s more narrow you know su2 goes beyond U1 because su2 can and usually does change the flag pole orientation whereas U1 only changes the flag orientation around its flag pole and by the way what is the deal with the flag you know it’s kind of a weird mathematical object right well it’s just a thing that represents an orientation in three dimensions if we instead used a line segment well then there’s the question of what is the role of the line you know so a flag is the kind of minimal object that lets us visualize a 3D orientation it doesn’t come as too much of a surprise that the three phase-like degrees of freedom of a complex dublet can be related somehow to the orientational degrees of freedom in three dimensions what is surprising is that the phase-like degrees of freedom of of the dublet actually double cover the 3D rotations so that when a dublet does a full su2 rotation and gets back to where it started in the complex plane P’s in a pod picture it will have rotated twice in the flag picture that’s what people are talking about when they say that su2 double covers s SO3 where s SO3 the group of 3x3 rotation matrices is the group that’s typically associated with rotations in three dimensions but we already covered that in the spinner video so let’s not double cover it here oh I’m the worst hey while we’re here let’s notice that su2 Transformations generally don’t commute meaning that if you apply two consecutive su2 transformations to a dublet it matters which transformation we do first starting with the same dublet doing transformation a then transformation B will in general yield a different final dublet than doing transformation B then transformation a this is different than U1 where the order doesn’t matter because the phase angles just add and order doesn’t matter when you add two numbers but because of the connection between su2 and three-dimensional rotations we can see that su2 matrices do not commute that the order does matter because that’s a natural property of 3D rotations you know in math there are different kinds of proofs proof by deduction proof by induction proof by contradiction proof by exercise for The Reader Pro proof by it came to me in a dream proof by putting so many kinds of proof but what we’re looking at now is my favorite kind of proof the most Zen flavor of proof proof by just look at it and see how it is suppose you have some object and you rotate about the x-axis than the Y AIS If instead you had rotated it first about the Y AIS then the x- axis it would end up at a different final orientation and so right there you can see an example of how 3D rotations do not commute and we just picked the X and Y axis as an example but you can imagine any other axis of rotation and you’ll find that the only time two 3D rotations commute is if the rotation axes are the same or 180° opposite that’s okay too because then you’re just rotating in the same plane and that’s spiritually a 2d rotation which does commute anyway the reason I bring this up is just to show that it is totally natural and intuitive that su2 matrices do not commute this is not a bizarre thing no having seen that S2 is related to rotations in three dimensions we would expect these Transformations not to commute and you know on the topic of commutation for what it’s worth su2 and U1 do commute with each other if you do an su2 transformation followed by a U1 transformation that’s the same as first doing the U1 transformation and then the su2 transformation and the reason this works is that an su2 transformation rotates a flag by a particular axis X Y or Z or some combination thereof whereas a U1 transformation always rotates a flag by its own axis so when you combine an S2 transformation with a U1 transformation you’re just swinging the flag and then twirling the flag and whether you do the swing in then the twirling or the twirling then The Swinging you end up at the same place so su2 and U1 commute with each other so as you see the flag picture teaches us a lot about the group theory of su2 as well as U1 and very important ly as we’ll see in a moment about the interplay of su2 and U1 you can just think of it all in terms of how the flags rotate and again for now you don’t even have to be an expert on how to draw the dublet as a flag just know that it can be done so that we have these two different Windows into the algebra on the one hand we have the P’s in a pod picture where the complex components are readily visible but the Transformations are confusing and we also have the flag picture where the Transformations are easy to understand but the complex components are obscured and confusing so you want to think in terms of both because each has their advantages also the fastest way to truly digest su2 into your imagination is to get your hands dirty with some actual code math is a contact sport and when you’re actively wrestling with the equations rendering animations with various parameters and seeing how it all plays out that is when the essence of the math really soaks into your soul and becomes a part of you So to that end on my patreon you can download the animation codes that make these animations with the transforming duets with the su2 and the U1 and the P’s and the Pod and the flags and you can put in your own dublet or list of doublets and see what happens and that’ll turbo boost your learning experience now these codes are Primo content exclusively available to those who have joined the club on patreon I hope you’ll consider signing up and uh hopefully this can be a mutually beneficial way to fund the channel a sort of like selling books you know if I write a book that you want to read you can buy the book well here’s some codes that’ll install in your mind a familiarity with su2 and U1 acting on doublets that’s a great thing right well if that’s something of value to you then uh please consider joining the club and getting in on this mathematical Adventure okay so why have we been spending so much time talking about su2 and U1 and doublets great question it’s because the gauge group of the electro week model is su2 cross U1 and the higs field is a complex double scaler field that breaks down su2 cross U1 into the U1 of electromagnetism that we know and love so these insights about transforming doublets are exactly what we’ll need in just a moment as we dive into Electro week unification but first let’s take a step back from all this math and zoom way out and get philosophical and meditate on the nature of reality the universe is a complicated thing there’s a lot of stuff going on and there’s also a lot of different kinds of stuff going on but amazingly when you really zoom in on this wild economy of cause and effect you find that the Dynamics of nature are governed only by four forces every push or pull that’s ever been observed falls into one of these four categories first off we have gravity that’s when SpaceTime gets curvy and time is no longer purely perpendicular to space fate picks out a preferred Direction stuff just goes in that direction as if Enchanted by Destiny we call that falling and on a cosmic scale you have stuff falling all around itself and swirling around and all that so that’s gravity it’s the way SpaceTime gets weird and curvy under the influence of matter and energy then we have electromagnetism good old local phase Symmetry and this comprises most of what we know in love not only electricity and magnets but also things like chemistry and friction and contact forces and basketball and even your own DNA and the way molecules behave is an electromagnetic phenomenon Common Sense can be summed up as a pragmatic understanding of electromagnetism and gravity for all intents and purposes those are the only two forces that we interact with in a meaningful way the other two forces the nuclear forces are not nearly as familiar to the imagination so first up we have the weak Force based on the name you might think that this Force force is weak but it’s not exactly weak it’s more like a turned off Force if two particles are very very close together like within 10us 18 M which is about a thousand times smaller than the nucleus of an atom then the weak force is active actually at that very close range the strength of the weak force is suspiciously similar to the strength of electromagnetism but then if the particles are even slightly farther apart say just a tenth of an atomic nucleus away from each other then the weak force is completely turned off it’s as if there’s something in the vacuum of space itself which swoops away the weak Force so that its effect can’t penetrate more than some distance Lambda away from a particle H H and that’s a little something called foreshadowing okay but hold on how do we even know that the weak Force exists if it only acts over such a short length scale well nowadays we have particle accelerators that can directly probe that length scale by Smashing particles together with extreme energy smooshing them right up next to each other but even without those fancy accelerators we see traces of the weak force in nature so the weak force is mediated by the w+ W minus and z bosons and we’ll talk more about those later on but for now just think about those as sort of like photons except they’re really massive so they’re not often created and they don’t like to exist exist for very long and as the name suggests the w+ boson carries a positive electric charge W minus carries a negative charge and the Z boson carries no charge and one of the ways that we see the weak force in nature is when an atom undergoes beta Decay so what’ll happen during beta Decay is that a neutron in an atom converts into a proton and emits an electron and an electron anti-neutrino but behind the scenes what’s really happening there is that within that Neutron you have a down cork that is turning into an up Cork and a w minus boson and then that W minus boson is decaying right away into an electron and an electron anti neutrino and you know atoms sometimes they undergo beta Decay that’s just a way that elements be transmuted sometimes and it’s an indication that the weak force is a thing and you can also have beta plus Decay which is a very similar process but involving a w+ bon so this has been observed in nature it’s a known effect and so that’s how we know that the weak force is a thing but also while we’re here this is kind of a tangent but it’s really cool you should look up the super konde detector this is a nutrino detector in Japan and it’s really cool it’s like a kilometer underground and the way that it works is that it has 50,000 tons of Ultra Pure Water and it’s way underground so there’s no light or anything it’s super like relaxed and you’ve got these neutrinos and a nutrino you can think of as like an electron that has basically no mass and no electric charge and you have a bajillion neutrinos flying through all the time like flying through you right now so like in this cave underground with all the water there’s a bunch of neutrinos going through neutrinos don’t interact electromagnetically so they normally just pass right through ordinary matter but they do interact via the weak force and so every once in a while a neutrino comes through and it interacts with the water molecule via the weak force and then it emits some light and the light goes into the detectors and based on the pattern you can figure out where the neutrino came from and so in that way this cave full of water serves as a kind of camera that sees with neutrinos instead of light and this is really cool so here’s a picture of the Sun that was taken with neutrinos at the super commond detector and the crazy thing is these neutrinos came through the Earth because the Earth is almost entirely transparent to neutrinos oh you know what there’s one more thing about neutrinos you have to know and this is going to blow your mind it makes no sense but it’s true every neutrino that’s ever been observed has been left-handed meaning that if you take your left hand and you point your thumb in the direction of motion of the neutrino then your fingers are going to curl around like the spin angular momentum of the neutrino now this is super weird and it should keep you up at night and it should give you kind kind of an existential crisis because we would expect for neutrinos to be half of the time left-handed and half of the time right-handed right like photons but that’s not the case they’re all left-handed I don’t know it makes no sense like it’s literally a glitch in reality that does not have an explanation as we’ll see later in the video we can model this we can bake this mysterious chirality into our description of nature but that doesn’t explain where the chirality comes from oh by the way the W bons only inter interact with left-handed particles the zons interact with both left-handed and right-handed well anyway there’s some very weird very strange chyal stuff going on in the Electra week sector as we’ll talk about later so anyway that’s a little bit about the weak force and last but not least we have the strong force this is the force that binds quirks into protons and neutrons a proton is two up quirks and a down cork bound into a trio of quirks by theu Force likewise a neutron is two down quirks and an up Quirk and when you have multiple protons and neutrons together as in the nucleus of an atom the strong force from within each proton and neutron sort of leaks out and the residue of that force is what binds the protons and neutrons together into the nucleus and that is what makes the nucleus stable because if it were just electromagnetism the protons would all repel themselves and the nucleus would go flying apart but it’s the residual strong force that holds the nucleus together now that effect starts to weaken if the nucleus gets too big and things get too far apart then electromagnetism takes over and then the nuclei go flying apart and so that’s why the heavier elements are unstable well anyway there’s a lot to be said about the strong force much like electromagnetism blossoms from U1 symmetry the strong force blossoms from local su3 symmetry su3 has eight generators and so you end up with eight gauge fields the gluon fields but let’s not get into that today okay in fact for today let’s not worry about gravity or the strong force don’t float away and don’t explode but let’s just focus on electromagnetism and the weak Force so we can see what’s going on with them and what secret relationship they have and what that can tell us about the higs field and the nature of things actually we’ll get back to electromagnetism and the weak force in just a moment but first there’s a thought experiment that we should perform in the laboratory of the mind and that is what would it look like if space itself were superconducting well suppose we have a bunch of particles floating around and bumping into each other and interacting electromagnetically and we’ll use this blue glow as a kind of way of visualizing the electromagnetic interaction so maybe that represents the electric and magnetic field strength and we’re being a little bit vague on what exactly it means just you know it’s poetic license okay but the point is now if we make space superc conducting then the electric field is going to get shooped away the magnetic field is going to get Meisner affected away and all of a sudden now this blue glow of electromagnetic field it goes away because photons are massive now they can’t propagate very far and so the electromagnetic influence of a particle on its environment it just doesn’t extend anymore I mean okay maybe it extends some distance Lambda and then it gets attenuated and so maybe two particles that happen to be very close range with one another could still interact electromagnetically you know if they’re within a Lambda or so but aside from that I mean the long range electromagnetic interaction of the particles would have been turned off because the medium in which the particles exist the space in which the particle exists is now superconducting and so it does not allow for a photon from one particle to travel over to the other particle right it’s a whole different kind of medium so the reason I bring this up is to say that if our universe were superconducting then electromagnetism would be turned off and by turned off what I mean is that it would acquire some very Clos range length scale Beyond which the force effectively wouldn’t occur but if you’re within that length scale then the force still exists and at that close range you know ASM totically it’s still as strong as it ever was it’s just that you know you go out a little ways and the superc conductor turns it off now given this thought experiment and given the fact that electromagnetism in our universe is a long range Force we know that we are not living in a superconductor but what do you make of the fact that the weak nuclear force has been turned off in precisely this way oh ho ho ho you you picking up what I’m putting down you see what I mean we are living in a superconductor it’s just not exactly the same superconductor as a superconductor that is a superconductor it’s the superconductor that is the universe in which we live do you see it isn’t it incredible I mean it just gives you like a moment of wo and it changes your life forever I mean it’s a subtle thing but just to sense that there is a water in which we swim it’s remarkable so what do you make of that huh I’ll tell you what I made of it as soon as I had that realization I’m like oh I got to make a video on this right away I read Lance’s paper and I’m like oh yeah no this is the next video right here hydrogen 3 can wait I got to tell the world about super connectivity in the higs field okay so our electromagnet ISM and the weak nuclear force secretly related somehow well let’s do a little compare and contrast so with electromagnetism the force carrier is the photon the photon is a vector boson it’s an excitation in the four potential a mu photons are massless and so they have infinite range so that lets us see galaxies that are like super far away because the photons are able to travel all the way from there to here also the photon itself carries no electrical charge you know charged matter has charge but the photon itself does not have charge and the principle of local phase symmetry blossoms into electromagnetism on the other hand if we look at the weak Force we find three Force carriers we have the w+ W minus and z bosons and these are all vector bosons and they have spin one just like a photon but unlike a photon they are massive like really massive and because these particles are so massive they’re really unstable and so they just don’t propagate very far before they Decay and how massive are they well we’re talking for the W’s 80.4 GS the z’s are like 91.2 gevs and just for context this is like over 100,000 times the mass of the electron so that’s quite different than a photon you know there’s a big difference between zero mass and a lot of mass and to make matters even more confusing the W bosons have electric charge w+ has positive charge and W minus has negative charge so that’s confusing because photons don’t have charge so ostensibly that’s a difference between the weak force and electromagnetism but then also it’s like wait a minute they have electric charge and that seems like an electromagnetic thing so maybe there is some kind of underlying Unity between electromagnetism and the weak nuclear force and then you look at the Z boson and it’s electrically neutral it doesn’t have any charge at all one way to remember that is think of Z means zero charge I don’t know if that’s actually why they named it that but hey it works and then what about the gauge group of the weak Force because when you look at electromagnetism and you see that the whole thing is reducible to this local U1 symmetry naturally the question arises of maybe we can account for the weak Force using similar principles but in that case then what kind of symmetry group would we need in order to generate the three bons of the weak force it have to be something in involving three degrees of freedom right because you would have like these three different ways of messing up your wave function and then you’d have to have three fields that can come in and clean that up so maybe there’s something where like there’s some three degree of Freedom symmetry group that might give rise to the weak Force well actually the answer is a little bit more complicated than that but it’s complicated but it’s also simple so it’s it well it’s it is what it is it’s it’s elegant and interesting so let’s take a look at how electromagnetism and the weak nuclear force are indeed two sides sides of the same coin so in our observed reality we see electromagnetism and the weak Force as like two totally different things electromagnetism is the manifestation of local phase symmetry it involves a massless gauge field and that enables infinite range on the other hand with the weak Force we have the W’s and the Z very short range cut off length scale of 10us 18 M seems like it’s totally different but the key to understanding how these things are related is to understand the higs mechanism very similar to the Anderson higs mechanism but underneath the higs mechanism the source of both electromagnetism and the weak nuclear force is the electro weak interaction and this is based on the Symmetry group s2l cross U1 y that we’ll go into in more depth in just a moment but that symmetry group gives rise to four massless gauge Fields W1 W2 W3 and B the W’s are the gauge fields of weak isospin and B is the gauge field of hypercharge and if not for the higs field all four of those gauge Fields would be massless but because of the higs field which happens to have three phas likee degrees of freedom you’re actually going to end up with three massive gauge fields and one gauge field that remains massless and it’s not as simple as the B just getting split off from the W’s but it’s actually more complicated because the higs mechanism actually mixes together W3 b as we’ll see in a moment but this is the big picture of how the electro we interaction gives rise to electromagnetism in the weak Force it’s all about the higs mechanism and that’s why we’re talking about Electro weak unification because if you want to understand the higs field you have to understand it in the context of Electro week unification the higs field is really the star of the show when it comes to seeing how electromagnetism and the weak nuclear force are actually two sides of the same coin or two P’s in a pod or actually I guess four P’s in a pod one p from the electromagnetism and three okay I’m overdoing this moving on all right let’s take a look at the gauge group of the electro week interaction this is s2l cross U1 y first of all starting with the U1 as we’ve seen this is just all the ways of rotating around a complex number in the complex plane and su2 is all the ways of swinging around a complex dublet without changing its amplitude so what’s the deal with the subscript L and the subscript y well in the Electro week model this U1 is actually not the same U1 as electromagnetism this is actually the U1 of something called hypercharge and so we use y to symbolize hypercharge I think because y rhymes with Hyper it’s a mid rme according to the haters anyway what this does is when we put a U1 gauge symmetry into the model this is going to give rise to the hypercharge gauge field and in your mind you can imagine this as exactly the same as the electromagnetic four potential a mu except we’re not in our observed reality yet we’re in the let’s call it a hyper reality before we put in the higs field okay so if you followed along with the last video then you’ll be well aware of what happens when you give your model a local U on symmetry you end up with a four Vector field that can augment the energy and momentum of particles within your model and so the idea of the electro week model is that part of it is a kind of local U1 symmetry like we saw for electromagnet ISM but that’s actually not the full story and that’s actually not the U1 of electromagnetism as we’ll see in a moment the U1 of electromagnetism is kind of like a slice through the space of s2l across U on y we’ll articulate that more precisely in just a minute but I think the more interesting part of the electro week gauge group is su2 L so the L means left-handed and one of the most unusual things about the electro week model is that it treats left-handed particles and right-handed particles differently at a fundamental level so a particle is Left-Handed if imagine it’s moving away from you and it’s turning counterclockwise and it’s right-handed if as it moves away from you it turns clockwise one way to remember this is it’s sort of like a football if you throw a football with your right hand it’s going to spiral like a right-handed particle and if you throw it with your left hand well then it’ll be like a left handed particle now in the electro week model at first none of the particles have mass so there’s no Mass involved and so we don’t have to worry about the difference between helicity and chirality everything’s massless everything’s moving at the speed of light and so each particle has a welldefined left-handedness or right-handedness and so we take it as a first principle in the electro week model that this su2 symmetry only applies to left-handed particles and how does it apply well the way it applies is that if the particle is Left-Handed it’s going to be able to transform as a dublet under su2 so for example consider this dublet on the left we have an electron neutrino up top and an electron on the bottom and even though the neutrino wave function and the electron wave functions are themselves spinner Fields you can imagine this doublet as sort of encoding a complex superposition of those wave functions and then the idea of local su2 symmetry is that you can transform that superp position of the neutrino and the electron arbitrarily under an su2 transformation and that’s not going to change the size of the dublet that’s not going to change you know the overall amplitude of the superp position or anything like that it’s just going to mix around or kind of spin around the neutrino and the electron in the dublet and the same goes for the up Quirk and the down Quirk so if you have a left-handed up Quirk and down Quirk those also are going to be able to transform under su2 as a complex dublet so what is the consequence of putting an su2 symmetry into the model well what’s going to happen is it’s very similar to local U1 symmetry giving rise to a four Vector field and in fact it is actually quite similar instead of one degree of Freedom giving rise to one gauge field we have three degrees of freedom giving rise to three gauge Fields these are the so-called weak isospin gauge Fields W1 W2 W3 and and as a good starting point you can imagine that these are very much like the gauge field b or a but with the important caveat that because su2 does not commute the weak isospin gauge Fields W1 W2 W3 are also going to be able to interact with each other so when you’re looking at a lren that involves the weak isospin fields in addition to the usual contributions from the field strength tensors which we saw in the last video we’re also going to have a self- interaction term but now today we don’t have to go into too much detail on that um oh and by the way let me just say this uh the electro week model is a very complicated thing and there’s a lot to it and I know I’m probably not going to do it justice today in terms of like you know giving you a comprehensive overview of the electro week model it’s like a big thing what I’m hoping to do here is to show you enough of it that you really get a good sense of what the higs field is and what it does and then also to the extent that uh this part of the video is going to raise more questions than it answers I hope that those questions are going to Lead You In a productive direction as far as what to study next if you’re interested in the electro week model and you want to learn more about it definitely check out that book by Chris quig it’s a fantastic book okay so back to the electro week model in the electro week model and actually in the standard model more generally we have three generations of matter so all the matter that we’re familiar with is from the first generation our atoms are made up of electrons and up quirks and down quirks but in addition to all of these particles you also have the second generation and the third generation which is just a copy and paste of the first generation but the second generation is more massive and the third generation is even more massive but other than that it all seems like a copy and paste and we still don’t know who ordered that we still don’t know why that is or whatever but you know anyway it’s just we have to factor that into our description of nature so that is what it is so in the second generation we have the muon neutrino and the muon we have the charm Cork and the strange cork and in the third generation we have the to nutrino and the TAA as well as the top Cork and the bottom cork you know back in The Groovy days they used to call that the truth Quirk and the beauty Quirk but then it became hip to be square so now we have to be serious and we have to call it the top quk and the bottom Quirk you’ll notice in this model there is no such thing as a right-handed nutrino and that matches what has been observed in nature you know all of the neutrinos observed so far have all been left-handed and it’s an open question as to whether right-handed neutrinos exist or not if they do maybe they’re sterile and they don’t interact for some reason for now I just want to point out that based on the fact that no one’s ever observed a right-handed neutrino in the electro week model they simply do not exist and uh similarly the other right-handed particles the electrons and corks and uh muons and to if it’s right-handed it does not transform under su2 and so it doesn’t couple to the W gauge Fields it only couples to the hypercharge gauge field if it’s right-handed so you can see that we build in the mindblowing chirality of the weak interaction into the first principles of the electro week model so it’s not an explanation of where the chirality comes from but it is a description of the situation it remains a big mystery even to this day what the deal is with the chirality and the weak interaction this is like a super deep mystery and when you think about it it’s hard to even imagine what an explanation for that would look like it’s one of those mystery that seems to defy not only explanation but imagination so we’ll save that for future generations to figure out but in the meantime let’s just work with what we know oh and one more thing while we’re here is notice that uh when you have these doublets that transform under su2 what we’re saying there when we’re putting in an su2 symmetry is that we should be able to arbitrarily apply an su2 transformation to our left-handed particles and that’s going to mix the components of the doublets around so so suppose we have just an electron well we can transform that into an electron neutrino and that doesn’t change the description of the situation so long as we also transform these W fields in the corresponding way so that’s a very very mysterious thing because it tells us that in some sense the neutrino and the electron are sort of the same particle I mean it’s super weird it’s like what do we even make of that it’s hard to say but it is what it is so we’ll take that as a starting point for today so the electro weak model it’s very mysterious and it’s very like wo what are what is this how can this be and in particular there are actually two problems with the model or two apparent problems that are like really problematic the first is that the electro weak gauge Fields the isospin fields W and the hypercharge field B these are massless and do we know that they have to be massless because it’s the nature of a local symmetry that you can mess up your wave function by an arbitrary amount here versus there now versus later and so when you apply this arbitrary s2l cross uny transformation and you’re messing up your wave functions you can do it in a very scrunchy very floppy way or you can do it in a less scrunchy and more relaxed way and depending on the intensity of your transformation you’re going to have to pull in more or less W and B and so therefore there cannot be an ener cost or benefit that depends on how much of the isospin fields or how much of the hypercharge field you bring in this is the exact same reasoning we saw in the last video when we demonstrated that the photon has to be massless and so if we want our Electro week model to be able to give us the massless photon field but massive w and z Fields this is like a major problem because in reality the weak gauge Fields have mass but in the electro we model very clearly they don’t have mass and and to make matters even worse we have another major problem with the electro model which is that only the left-handed particles transform under s2l and so only the left-handed particles feel the isospin gauge Fields so why is that a problem well aside from being weird it’s okay that it’s weird but the problem is one observer’s left-handed particle might be another observer’s right-handed particle depending on their velocity if the particles had Mass cuz if the particle has mass you can always Loren boost into a reference frame where the particle is going the other way but still spinning the same way and so one observer’s right-handed particle could be another observer’s left-handed particle if the particles have mass and then the question becomes well okay the two different observers are going to disagree on whether or not that particle transforms under Sul and whether or not the particle feels the weak isospin gauge fields and so if our matter particles have mass then there’s this inherent incompatibility with the idea of an s2l gauge field in the context of special relativity and different inertial frames it just doesn’t work and the only way to solve that is to say okay all of our matter particles are massless so therefore there is no reference frame where you can boost it the other way and have it going the other way no a massless particle is going to be left-handed or right-handed and all observers are going to agree on that and so if we want the electro week model to be logically coherent all of the matter content has to be massless but of course in reality matter has mass so what’s up with that now these two problems are very severe I mean you would think just throw away the model like this doesn’t work at all like not even remotely right but instead of just throwing it away we’re going to do something even crazier we’re going to spill a superconductor into it and this is going to be a special kind of a superconductor okay this is the higs field we’re talking about so we’re going to use the symbol fi instead of SII so notationally it’s a little different but it’s kind of similar also the Hig field fi is going to be a complex dublet field so that it can transform under both s2l and U1 y it has to be a doublet because if it were just a single complex valued field it wouldn’t be able to transform under su2 so the higs field is going to be a complex valued doublet field and it’s going to transform under Loren boosts as a scaler so it’s not going to be a spinner field it’s just going to be like a relativistic lorenzian variant scaler field now when we do this when we put our superconductor into the electro weak Model A Miracle happens the higs field fi is going to transform the massless isospin and hypercharge Fields into three massive gauge Fields the w+ the W minus and the Z of the weak force and one massless gauge field am mu the photon field and that breaking down we can understand entirely in terms of superc conductivity all you do is you imagine some nambu gold stone modes in the higs field then you hop into the unitary gauge and you calculate the amount of energy in the fields and that gives you the mass term and you get the mass and you can show that it’s massive and then you can show the photon has no Mass we’ll get into that in just a second it’s really an amazing thing the higs mechanism and this is so cool but okay so we bring in the higs field to fix the electro week model to give the W’s and Z Mass to leave the photon massless and while we’re doing at we bring in a prediction because if the Hig field is like a superconductor then it should have an amplitude mode and that’s the higs boson that’s what the higs boson is it’s the amplitude mode of the superconductor that is the higs field so that is a really fantastic prediction because it’s highly non-obvious that SpaceTime itself should be excitable like a superconductor is excitable but if that turns out to be true and okay spoiler the higs Bon was discovered in 2012 so it is true then it’s like whoa this thing about unifying electromagnetism in the weak Force led to this prediction about SpaceTime itself having a superconducting kind of essence then the amplitude mode was discovered it’s like wow that’s really cool oh and uh one more thing while we’re here is the Hig field fi is also going to give Mass to the matter particles in the electro week model via their yukawa couplings now how exactly the yukawa couplings work is still deeply mysterious there’s actually not a good answer and so it kind of goes beyond this superc conductor analogy but long story short the yukawa couplings provide a way for the gauge group of the electro week interaction to be compatible with special relativity so we’ll briefly look into that in a little more depth towards the end of the video all right let’s get existential so imagine some space with nothing in it whatsoever where all that’s there is that it’s there no dust no atoms not a thing at all the darkest silence a perfect vacuum how Dreadful aane person thinks nothing of it a physicist stares into the void and writes a lrin because as you now know empty space is not empty it’s a superconductor in hyper reality we can no longer take comfort in the naive innocence of Youth now we have seen too much so let us write to lran about nothing we’ll start with the idea that the vacuum of space is not empty and that we’re living in a superconductor that we’ll call Fi now if fi is a superconductor then it should have the kinetic and potential energy of a superconductor the kinetic energy term in the lran is going to be the magnitude squared of the gauge covariant derivative acting on fi in essence this is exactly the same as just writing out the amplitude squared of the gauge covariant derivative acting on fi but to be technical because fi is a doublet we have to do the thing with the transpose conjugate with the dagger and also because fi is relativistic we have to take our derivatives in a way that makes Einstein happy so we have the product of a loren covariant gauge covariant derivative and a Loren’s contravariant gauge covariant derivative so every inertial Observer can agree on this now the gauge covariant der for the higs field is not the simple gauge covariant derivative that involved the magnetic Vector potential because remember the higs field lives in the electro week hyper reality hyper reality is not a technical term but it’s a term that fits really well so I’m going to keep using it anyway the gauge covariant derivative is the same basic idea as what we saw before where you have a four gradient and then you’re adding in some gauge Fields so in the case of the electro week model we have the weak isos spin Fields W corresponding to the poly spin matrices TOA and we also have the hypercharge field B corresponding to the hypercharge assignments y the hypercharge Y is just a scalar quantity and we’ll get into what the hypercharge means later on but for now capital Y just know that for the higs field that’s just the number one so for now you can really just ignore that capital Y and then you can see that the essence of this hyper charge term is very much similar to the electromagnetic four potential term in electromagnetism by the way everything we’re doing from now on in this video is going to use natural units when you’re dealing with the standard model and stuff you get a lot of terms and a lot of characters and we don’t want to be carrying around H bars and C’s all over the place so H bar equals C equals 1 from now on also you’ll notice G and G Prime these are the Sul coupling constant and the U1 y coupling constant respectively and those coupling constants set the strength of the weak isospin and hypercharge gauge Fields respectively so G and G Prime are very analogous to the electric charge e and in fact they’re deeply related as well okay so that’s the gauge covariant derivative and the cool thing about what we’ll be doing today is when we’re looking at how the higs field gives Mass to the w and z bons all we have to do is eval at the magnitude squar of the gauge covariant derivative acting on fi as we hop into the unitary gauge so we don’t even have to go into that much detail on how exactly the electro week gauge Transformations work as we’ll see in a moment the higs mechanism and the w and z mass calculations are actually a really nice introductory exercise in sort of Dipping a toe into the electro week model without getting too too technical all right so the lran is kinetic minus potential energy lrin density and energy density if we’re being Technical and so in our expression we’re also going to have some kind of potential energy which is a function of fi and because fi is like a superconductor we would expect that to be the Ginsburg landout potential energy so you know Ginsburg landow is Alpha * magnitude si^ squ plus beta/ 2 * magnitude of side of the fourth well when you translate that into higs language the same exact idea is written as Mu ^ 2 dagger 5+ Lambda * F dagger 5^ 2 and although it looks different it’s actually exactly the same thing as what we saw before with the Alpha and the beta because if you look at the green term that I put up there for reference the potential of a superconductor you can see that Alpha is mu^ 2 beta/ 2 is Lambda and the magnitude squar of s is the magnitude squar of the higs dublet which is f dagger fi so therefore the magnitude of the fourth of s is f 5^ s and so even though the Ginsburg landout potential is dressed up differently here it’s still exactly the same idea and it’s still going to give us a sombrero potential in the magnitude of the higs field though of course now that fi is a doubl it it has three phas likee degrees of freedom but never mind that for now first thinking purely in terms of the magnitude we can see that the higs field is going to take on a non-zero vacuum expectation value oh by the way when people talk about vacuum expectation value they call it vev that’s what everyone says vev Okay so let’s calculate the vev if we take the mass parameter of the higs field that’s that mu Factor experimentally we know that mu is about 88.4 GS and there’s also the self-coupling constant Lambda and experimentally that’s going to be about 0.129 and then as we talked about before you want to think about negative mu^ 2 as being Alpha think about Lambda as being beta/ 2 and think about F dagger 5 as being the magnitude of s^ squ and then remember before way back when we calculated the density of a superconducting condensate and we found that it was the square root of the absolute value of alpha over beta well if you apply the same reasoning here then you find that the magnitude of f is going to be the square root of mu^ 2 over 2 Lambda and you can take the mu^ S out of the square root so you can write that is Mu over the < TK of 2 * Lambda so yeah that’s how the magnitude of the higs field in the vacuum State relates to the mass parameter and the self-coupling constant conventionally we write the higs dublet as 0 v/ < tk2 where the square root of two is just a conventional normalization factor and V is the higs vev the vacuum expectation value but we call it a vev CU we’re cool and cool people say vev so the vev is going to be mu over the root of Lambda which you can see by looking at the equation we just found for the magnitude of F and so if you plug in the value of 88.4 gevs for mu and 0.129 for Lambda you’ll find that the higs vev is 246 gevs let’s take a graphical look at what the higs dublet looks like in both the P’s in a pod picture as well as the flag picture so the higs vacuum state is a double that we write as fi f+ is an electrically charged scalar field and ph0 is electrically neutral but never mind that for now we’ll come back to that later for now the thing to know is that this doublet is 0 V over < tk2 so if we plot this dublet in the complex plane P’s in a pod picture you see that the top component is going to be at the origin and the bottom component is going to stick out along the real axis by an amount V over < tk2 now as we’ve seen the magnitude of fi and so the value of V is determined by the Ginsburg landow potential part of the higs you know that somero potential but then there’s still the question of what are the phase like degrees of freedom you know cuz a dublet has three phase-like degrees of freedom well by convention we set the top component of the higs dublet to zero and we make the bottom component of the higs dublet real and so of all the possible doublets that have have the same magnitude that conventional choice is why we’re dealing with 0 V over < tk2 in particular and as we’ll see later on in the video the facts that we’re defining the top component to be zero is intimately related to the facts that the top component represents a Charged scalar field and the bottom component represents a neutral scalar field and so you don’t have to wonder like why is it that the higs field spontaneously broke with a phase that just so happens to have a zero top component no no that’s just matter of convention we could reformulate the entire Electro week model with an arbitrary higs dublet but then everything just gets more messy and there’s no point in doing that all right well because our dublet has zero in the top component when we look at it in the flag picture it’s going to be pointing straight down and so when you imagine su2 and U1 y Transformations acting on the higs field one of the ways we want to imagine that is rotating this downwards flag okay and so the higs vacuum state is this dublet all throughout SpaceTime so it’s a nice constant dublet field imagine a field of these doublets the same exact thing everywhere all over the place that’s the higs field but now the question is hm hold on can we have nambu gold stone modes in the higs field you know like the phase waves we saw in the superconductor can we have those kinds of oscillations in the higs field I mean we’re going to have three degrees of freedom instead of one but but other than that it’s the same basic question right like we kind of would expect to have nambu Goldstone modes here and that is a very interesting line of reasoning and as we’re about to see that line of reasoning is going to let us calculate the masses of the w and z bosons so you know before when we looked at a superconductor that was just chilling with no magnetic field we realized wait a minute there could be phase waves potentially but then we thought like well okay yeah but those phase waves can be gauged away and then you’ll end up with an irrotational part of the vector potential and that gives you longitudinal modes and so then the photons become massive in the superconductor okay now we’re going to take exactly the same idea but apply it to the higs field now it’s a little bit different because the Hig field is like an upgraded superconductor you know it has the two complex numbers and so instead of one phase like degree of Freedom it has three and okay so that’s a little extra complexity and maybe we can symbolize that with like an RG color space or something you know you can imagine having nambo Goldstone modes in the higs field and it’s just sort of more complicated more degrees of freedom but still if we hop into the unitary gauge well then we can erase those nambo gold stone modes by definition but their energy still has to exist somewhere and if we let the nambo Goldstone modes drag us around in the space of gauge equivalent descriptions of the situation in other words we say okay we’re going to go into the unitary gauge so we’re going to demand that the higs field phase is locked to a certain value fine but the energy contained in those nambo gold stone modes is now going to pop out as IR rotational additions to the three gauge fields that we need in order to put the higs field into the unitary gauge so in essence this is exactly the same idea that we looked at earlier when we were looking at the Anderson higs mechanism in a superconductor the long and the short of it is the nambo gold stone modes in the superconductor get eaten by the gauge fields and give them an irrotational components that manifests as longitudinal polarizations and once your vector field is able to be longitudinally polarized then it’s no longer massless now it has that third degree of freedom and it corresponds to a massive Vector field okay so you start with the assumption that the higs field is like a superconductor and then you follow exactly the same thought process of all right it’s going to have nambo Goldstone modes but wait those waves are exactly the kind of thing that can be gauged away so you hop into the unitary gauge then Ambu Goldstone mod go away but then your gauge Fields pick up a kind of energy that gives them mass and we’re going to dissect this in more mathematical detail in just a moment but first I just want to emphasize that these two pictures on the left and right these are gauge equivalent they both describe exactly the same physical situation by the s2l cross U1 y gauge symmetry of the electro week model we’re allowed to hop into the unitary gauge that’s totally fine and so when you think about it if you have a complex doublet superconductor filling all throughout space and you equip that field with the gauge symmetry of the electro week model well that’s going to be indistinguishable from an empty vacuum with three of those gauge Fields having acquired Mass corresponding to The higs Field’s three phas likee degrees of freedom and there’s some nuances about the mixing of the third component of the su2 group with the U1 Y and we’ll get to that in just a moment but the thing I want to emphasize is that when you see the masses of the w and z Fields one of the ways you can read that one of the ways you can interpret and imagine that is as space being filled with a superconductor and this is like a gauge equivalent way of thinking about the nature of things but it gets even better because actually these two pictures are only mostly indistinguishable because the thing about the higs field is yeah it has three phase-like degrees of freedom and yeah those can give Mass to the three Vector fields that they correspond to but the higs field is a complex dublet it has four degrees of freedom total three that are like a phase and one amplitude and so if this higs field really exists and is not just some kind of accounting trick or mathematical construct then it has to exist all the way and so it should also be able to host amplitude modes and so you don’t see it in this picture of in ambo Goldstone modes and the unitary gauge and the gauge bosons acquiring Mass but when you think about the consequences of what has to happen if we’re really to take this seriously you realize okay well this model seems to very clearly pred predicts that empty space itself should be excitable like a superconductor is excitable in terms of the amplitude mode and that brings us to a little something called the higs Bon but let’s not get ahead of ourselves I believe we uh have to calculate some w and z Mass okay okay so to do these W and zmass calculations we’re going to split up the problem into two parts so rather than considering the higs field with its full three phas likee degrees of freedom and the w and z Fields together first we’re going to look at the higs field with just the two degrees of freedom corresponding to TOA 1 and to 2 which corresponds to the W gauge fields and that’ll let us calculate the W mass by hopping into the unitary gauge and calculating how much energy is associated with how much w we have and then after we do that we’re going to consider a higs field with nambu gold stone modes corresponding to just TOA 3 which corresponds to the Z field and the B field is going to emerge as a combination of W3 and B as we’re doing the mass calculation and then once we see the combination of W3 and B that becomes the Z field we can take the orthogonal combination and we can associate that with the photon field which is going to remain massless so okay so I just bring this up to say that we’re going to treat the W’s and Z’s separately although it should be pointed out we don’t have to do this if we wanted to we could keep everything together and we could do this the W’s and Z’s all at once it would just be a bit algebraically Messier if we do it that way so let’s go ahead and split it up into W’s and Z’s oh there’s something I have to say about the w+ and W minus bons so these are fundamentally the same kind of thing the same degrees of freedom as W1 and W2 but actually w+ and W minus are the definite charge states that emerge from W1 and W2 so this is very much like how you can have circularly polarized light by adding up some outof phase combinations of linear polarizations so if you’ve worked with circularly polarized light then you’ll recognize this formula but if you haven’t for today it’s really not that important in your mind if you want to imagine w+ and W minus as being synonymous with W1 and W2 but only as it pertains to mass then I think that is okay for today’s video when it comes to calculating the W masses this new about definite charge States isn’t super super important so I do bring it up in passing and I have the equation here and I have some information but as far as the higs mechanism is concerned in your mind you can think of the w+ and W minus mass as basically being the same as the W1 and W2 mass then once you learn about the higs mechanism and you see the galman and N shishima relation that’s when you’ll want to go back and think about w+ and W minus and diagonalizing T3 and you’ll be like oh yeah okay the isospin projections are plus one and minus one and so you’re going to have a positively charged w+ a negatively charged W minus so if this is your first time seeing the definition of the w+ and W minus bosons I know it seems kind of arbitrary at first but if you spend some time with the electro week model eventually it’s all going to click okay but for today we’re just going to take this w+ W minus definition for granted all right let’s calculate the mass of the w bosons first we write out the magnitude squar of the gauge covariant derivative acting on the higs field and we get this expression involving the four gradient of the higs field as well as the isos spin fields and the hypercharge field and for notational Elegance we’ll go ahead and use the indices A and B so that we can write the isospin terms a little bit more concisely using the Einstein summation convention okay so remember what we’re imagining now is that the higs field is hosting some nambo Goldstone modes in the degrees of freedom generated by TOA 1 and TOA 2 so rotations of the flag about the X and Y AIS and when the higs field is vibing in this way then when we hop into the unitary gauge that energy is going to be transferred from those naambo Goldstone modes into the W1 and W2 terms in the covariant derivative so in the unitary gauge the gradients that were in the higs field from the nambu Goldstone modes go away and likewise we can ignore the hypercharge for now because we’re just dealing with W1 and W2 and so we can go ahead and unpack the W terms writing out just W1 and W2 and ignoring W3 for now we’ll come back to W3 later when we calculate the Z Mass okay so now we have this expression that is going to give us energy as a function of how much w we have and so this is going to be the W Mass term in the lran and to evaluate this all we have to do is crunch the algebra so first we can go ahead and pull out a couple factors of G over2 so we end up with G ^2 4 in front of the equation and then we can go ahead and take the transpose conjugate of the left side of the expression when we do this we’ll end up with the transpose conjugate of fi and we move that over to the left because bar mind it becomes a row Vector now and then the W terms what those are going to be is a scalar time a poly Matrix and if you look at the definitions of to 1 and to 2 you can see that they’re equal to their own transpose conjugate so therefore when we take the transpose conjugate of the W Expressions it’s just the same W expression as before all right and so now what we’re going to do is we’re going to go ahead and foil through the Expressions involving the w use in the TOs and when you foil this out you end up with Expressions involving to 1 * to 1 and to 1 * to 2 and to2 * to 2 and we can simplify this by looking at the properties of the poly matrices what we find is that to 1 * to 1 as well as to 2

• to 2 these are both the identity Matrix and likewise if we look at to 1 * to 2 or negative to 2 * to 1 we see that those are also both equal to the same thing and so therefore the to ones and the to ones we can ignore because those are just the identity and the to 1 to 2 plus to 2 to 1 cancels out to zero so that middle term gets deleted from the expression now then to clean up the FI dagger and fi terms we can go ahead and write out the higs double at F which is 0 V over < tk2 and when you take F dagger fi you end up with 12 V ^2 so substituting that into the equation we end up with G ^2 V ^2 over 8 now what about this expression involving W1 and W2 well here let’s go ahead and invoke the definitions of w+ and wus that we saw earlier you’ll find that if you multiply together w+ and wus and multiply by 2 you end up with W1 * W1 + W2 * W2 with which is exactly the expression that we find multiplied g^ 2 v^2 8 and so therefore if we substitute in to W + wus we end up simplifying our expression to Simply G ^2 V ^2 / 4 * w+ wus now w+ and wus are conjugates of each other and so we can interpret this expression as being of the form m^2 * the magnitude squ of the W field and by symmetry you can think of this as w+ or W minus there’s no difference for the mass term in the lran w+ and W minus have the same mass now this term m^2 magnitude of w ^2 we recognize this as a mass term and you can see quig equation 2.2.4 for justification of why that is this is the form of the mass term in the lran for a complex valued four Vector field and then simply by identifying the constant in front of the expression we can see that the mass of the W boson is going to be GV / 2 so the mass of a w boson depends on the su2 coupling constant G as well as the higs vacuum expectation value and that kind of makes sense I mean you know what else would it depend on so that’s really cool that’s a nice uh satisfying expression because all we really did was the same thing we did when calculating the effective mass of a Photon in a superc conductor but instead of using a complex scalar field sigh and the vector potential a we used the Loren invariant complex douet field fi and the isos spin gauge Fields W1 and W2 and you know what if we want to calculate the mass of the Zeos on it’s exactly the same idea the only difference is now instead of imagining nambu Goldstone modes in the higs field in terms of the degrees of freedom generated by to 1 and to 2 we’re going to think about it in terms of the degrees of freedom generated by to 3 and the hypercharge U1 transformation so again we follow the same procedure we write out the magnitude squar of the gauge covariant derivative acting on the Hig field we hop into the unitary gauge where the four gradient of the Hig field goes away and we look at the terms involving W3 and B now this time we have both the isospin coupling constant G as well as the hypercharge coupling constant G Prime so we’re not going to pull out a factor of G /2 instead we’re going to pull out a couple factors of 1/2 and we end up with a 1 qu on the outside of the expression so now let’s go ahead and take the transpose conjugate of the left half of this expression while foiling it too in one Fell Swoop we can do this by recognizing that the W term is not affected by the transpose conjugate because to 3 is its own transposed conjugate and the hypercharge term is not affected by the transpose post conjugate because it’s a scalar quantity so all we have to do when we foil this out is move over the FI dagger onto the left side and then just go ahead and foil out the other terms all right and now in order to simplify this expression let’s make the following observations first of all just like we saw before if we write out the higs double at F we find that 5 5 is 12 V ^2 and that’s good to know but inside this expression we also have to 3 and to 3 * to 3 and y^2 and B and B so we have to evaluate some other things too and one of the things we have to evaluate is the expression fi dagger to 3 F to evaluate this all you have to do is put the to 3 Matrix between the FI dagger and the FI and you end up picking up a negative 1 from the to 3 so that expression evaluates to-2 V ^2 we also make the observation that the higs field has a hypercharge of one so the Y and the y^2 go away and so we can rewrite the expression like so and next what we want to do is looking at this expression with the w3s and the B’s we recognize that it has the form of a square and so we can Factor it out like so but now if we look at this equation we recognize hey wait a minute this is like some constant times the magnitude squar of some four Vector field now what exactly is this four Vector field well it’s the combination of W3 and B that has mass like by definition right I mean we’re not just saying that arbitrarily we find while doing this calculation that there is this field this particular combination of W3 and B that has energy associated with how much of it we have and so by following the same reasoning as the superconductor and the W fields we can see that this field right here is the massive combination of W3 and B but you know what if we look at this field we can see that it has the coupling constants G and G Prime multiplied onto W3 and B and it would be nice if we could divide those out somehow so that this combination of fields were more similar to W3 and B but then what do we divide by do we divide by G do we divide by G Prime well how about this imagine a right triangle where we have G and G Prime as the two legs of the triangle so that the hypotenuse has a length ofare < TK of g^

• G Prime s so let’s go ahead and divide out our field by the square root of G ^2
• G Prime s the Z field is the massive combination of W3 and B where the ratio of how much W3 and how much B comes right out of the covariant derivative that we’re analyzing to look at how much energy is in the W3 and B fields in the situation with the higs field and the N Goldstone modes and all that and we get this expression involving G W3 and G Prime B and then we divide out by this Factor < TK of G ^2 + G Prime s that lets us sort of normalize those coupling constants now then we can rewrite our expression in terms of the Z field by pulling the denominator out front and factoring it out and we get the expression shown here V ^2 8 * G ^2 + G s time the magnitude of z^2 and we can recognize this as the mass term in the lran for a real valued for Vector field you can see quig equation 2.2.1 to see where that comes from and now all we have to do is simply match the terms and solve for the mass of the Z field and we end up solving that the mass of the Z field is V /2 * the < TK of G ^2 + G Prime 2 and so if you look at this you can see that it’s actually very similar to the mass of the W but with an additional contribution from the hypercharge coupling constant G Prime in fact imagine setting G Prime equal to zero so if hypercharge weren’t a thing then the mass of the Z and the mass of the W would be the same but because hypercharge is the thing it adds an additional contribution to the mass of the Z you know there’s another way of looking at this in terms of something called called the Weinberg angle or the electro weak mixing angle and what this is as shown in the triangle with the purple text this is the angle in the right triangle involving G and G Prime so by definition the tangent of theta W is G Prime over G and if we want to reframe everything we’ve been talking about in terms of theta W then we can easily Define not only the Z field but also the photon field a mu as the rot Matrix acting on the w and B fields so rotation Matrix has the form of cosine of thet sin Theta sin Theta cosine Theta and if you Analyze This Matrix acting on the W3 and B Vector you find that Z is exactly like what we just solved for and then a mu is the orthogonal combination of W3 and B that doesn’t contribute at all to the mass term and again the Z field is the combination that has mass and and the a field is the combination that has zero Mass so these are both Mass igen states of the combinations of W3 and B Okay so we’ve calculated the W mass and the Z mass and we’ve seen how those get their Mass from the Anderson higs mechanism and when we calculated the Z Mass we saw this thing about the mixing of W3 and B and the mass igon States and z and a but here’s the thing all of this should make us wonder how is it that the photon escapes Mass free because you know electromagnetism is based on local phase symmetry so what we need to find in this picture of this higs field and the su2 and the U1 and the whole Electro week model what we need to find is a circular degree of Freedom that doesn’t affect and is not affected by the higs field and in the vacuum State we can imagine that simply as the complex douet z v rout 2 which in the flag picture is the downward pointing flag so the question is we know that the higs field can transform as a dublet under su2 L and can also transform under U1 y so is there any of those Transformations or any combination of those Transformations that doesn’t do anything to the downward flag that is the higs field hm if so it’s not obvious because if you think about it any su2 transformation is going to rotate the higs flag by definition right that’s what an su2 transformation does in the flag picture but at the same time any U1 transformation is going to rotate the flag about its own axis so you can’t leave the higs field alone with purely an su2 or a U1 transformation what we need to do instead is to combine a transformation from su2 with a U1 transformation in such a way that they cancel out for the dublet that is the higs field but at the same time they allow a circular degree of Freedom a circular group of Transformations a U1 group of Transformations on every other dublet and in order to see what that might look like imagine we have a dandelion of flags and all this dandelion is is a way of getting a feel for the transformation so this arrangement of flags is not important this is just designed to be like a good sampling of the space you know a good representative sample of Duets and that kind of fills in the space so we can see what this transformation looks like and how it transforms Duets now if we look at this transformation and we look at the bottom flag we can see that it’s not moving so what kind of a transformation is this anyway well what it is is we’re rotating every flag about the Z axis using to 3 right using an su2 transformation we’re rotating all of those flags around the z-axis in the flag picture but at the same time we’re using a U1 transformation to rotate each individual flag about its own axis cuz that’s what a U1 transformation does and it just so happens that when you do this and you look at the flag that’s pointing straight down that’s the flag of the higs field you see that it’s not moving so this transformation does not bother the higs field in any way you end up with this circular degree of freedom where you can go around and around in circles and it has a different effect on each kind of dublet but at the same time it has the same effect for every dublet it rotates the flag around the z-axis and it also rotates the flag around itself it’s just that it’s only for the flag of the higs field where those changes cancel out so that the higs field is totally immune or totally insensitive to this kind of transformation whereas every other dublet sees this transformation as a kind of circular transformation and so that’s the flag picture but now if you look at the complex plane in the dublet picture you try to figure that out and like what’s going on there it’s a little confusing at first but then it’s like oh wait a minute the first component of each duet that is those circular dots is spinning around like a U1 phase transformation in the complex plane and the second component those squares are not changing at all so that’s how to look at it in the complex plane picture it’s just that this transformation does a U1 transformation on the first component and it leaves the second component invariant and then because of the fact that the higs field is defined to have a first component of zero you can see then in that picture that the Hig field is insensitive to this transformation and of course both pictures represent the same set of doublets it’s just that in one we’re looking at the flag picture and the other one we’re looking at the complex plane picture now let’s take a closer look at exactly what’s going on with this transformation [Music] well that was a bit much but uh here we are so in the left column here we have the transformation that we were just looking at which rotates every flag around the Z axis while also rotating it around its own axis which in general is going to rotate and twirl some arbitrary flag but because the motion cancels out for the downward flag of the higs vacuum State this transformation does not affect the vacuum State and therefore this transformation will remain a symmetry of the electro week model even after we’ve spilled our superconductor into hyper reality so this transformation is the U1 transformation of electromagnetism the middle and the right columns show a TOA 3 and a U1 transformation respectively and for all these trans Transformations we can think in terms of the peas and a pod picture or the flag picture because those are two different Windows into the same algebraic structure now as I mentioned earlier we use the letter Y to symbolize hypercharge and that’s effectively a multiplier on how much or how fast the flags are going to rotate around themselves for the same U1 transformation so for example all the doublets shown here have a hypercharge of one like the the higs field so for all intents and purposes we don’t really have to worry about hypercharge for today but suppose we wanted to understand how a dublet behaves if it has a hypercharge of negative one well in that case the transformation on the right would just go around the other way with the flags twirling around backwards and then when you combine that with the to 3 transformation you would find that the top flag doesn’t rotate and we see this with the lepon Duets where those doublets have a hypercharge of negative 1 and so you end up with an electrically neutral top component the neutrino and then you end up with an electrically charged electron muon and too having a charge of negative 1 where the negative sign arises from the negative hypercharge and in fact in general the electric charges of particles are determined by how they transform under both to 3 as well as their hypercharge and this is the essence of the galman nishima relation which we’ll get into in just a moment but first let’s take a closer look at this transformation which is the U1 symmetry group of electromagnetism as it’s the only subgroup of su2 cross U1 which is independent of the higs vacuum State well actually there is one thing we should adjust if you look at the P’s in the Pod picture you can see that to 3 rotates the top components those circles counterclockwise and it rotates the bottom components the squares clockwise on the other hand the U1 transformation rotates both components counterclockwise in the complex plane since we’re multiplying the whole doublet by e to the I Theta so when we combine these two Transformations the motion of the bottom components cancels out and so those don’t move in the combined transformation but then the motion of the top components adds and so those end up moving around twice as fast so now let’s go ahead and adjust our definition of the transformation by dividing the phase angle by two just to slow down the motion so that the combined transformation turns around at the same rate as the TOA 3 and the hypercharged Transformations now this is a more properly adjusted transformation which is still fundamentally the same degree of freedom but now it goes around at the same rate as the weak isospin and the hypercharge Transformations okay so let’s look at the specific formulas for these Transformations like the actual equations that are making things go well in the middle column the to 3 transformation the transformed dublet F Prime is calculated by multiplying the untransformed dublet F by e to the I Theta to 3 which swings the dublet around in the to 3 degree of freedom in the right column we have a U1 transformation and so that’s just e to the I Theta * the doublet and you can imagine that either as e to the I Theta where Theta is a scalar quantity or if you want you can imagine Theta as Theta

• I2 with I2 being the identity Matrix either way it’s the same also bear in mind for now we’re assuming that the hypercharge is one because we’re dealing with the higs field but in general that equation would actually be e to I * y * Theta where Y is the hypercharge a scalar quantity okay so to arrive at the equation for the combined transformations in the left column we can just multiply those two equations together noticing that the order doesn’t matter because an su2 transformation commutes with a U1 transformation and remember we’re also going to divide the phase angle Theta by 2 to slow down the transformation to the appropriate rate so we end up with the equation shown here but we can simplify that equation by absorbing things together into one exponential to do this we just have to add the to 3 and the U1 terms and we can slip a 2x2 identity Matrix onto the U1 term so that the Matrix dimensionality works out and that’s a legit move because scalar multiplication does the same thing as multiplication by a scaled identity Matrix now if we add these two matrices we find that the top terms add up to one and the bottom terms cancel out so we end up with the simple Matrix 1 0 0 0 and it turns out when we exponentiate e to the I Theta time that Matrix we end up with the Matrix e to the I Theta 0 01 and that Matrix is kind of like a mermaid but instead of a woman and a fish it’s a U1 transformation up top and an identity Matrix down below so now we see that our combined transformation acting on a doublet of hypercharge 1 is nothing more than a U1 transformation e to the I Theta on the top component of the dublet and it does nothing to the bottom component and so no wonder this doesn’t affect the higs vacuum State because by definition the top component of the higs is zero and zero doesn’t change under a U1 transformation you can see now why we associate the top component of the higs with a Charged field and the bottom component with a neutral field hey you know what there’s one other thing we should know here so in the electro week model we often use T3 instead of to3 where T3 is defined as 1/2 to 3 the generators of su2 are often normalized with this 1/2 Factor so that the transformation has igen values of plus or minus 1/2 and we see the same normalization convention in a different context when thinking about the spinner wave function of an electron which also transforms under su2 but in a different context and it also has the igen values of plus orus 1/2 since electrons are spin 1/2 particles now the su2 of the electro weak gauge group pertains to weak isospin not spin but these two ideas share the same underlying group theoretic structure and so we often use T3 instead of T3 because it has that baked in normalization factor of 12 so anyway with that little Nuance in mind we can write our transformation as the combination of T3 and y/ 2 now this equation looks more complicated than it really is because bear in mind that this is just the idea that The Unbroken U1 subgroup of the electro week gauge group is generated by rotating all the flags about the higs flag pole axis and then rotating all the flags about their own particular axis by the same angle and then slowing it down by a factor of two so it has the right speed people often wonder why is T3 singled out in particular instead of T1 or T2 or some combination of the t’s but that’s just because of our choice that the higs vacuum state has zero for the top component If instead we started with the convention that orients the higs vacuum State differently we would arrive at basically the same equation but dressed up with different T terms and so you see there’s nothing really cosmically fundamental about T3 in particular that’s just a reflection of the that the higs flag is conventionally vertical people also often wonder why the hypercharge transformation is divided by two it’s like does hypercharge only matter half as much as isospin well remember there’s a hidden one half built into the definition of T3 and so really we’re just averaging the su2 in the U1 Transformations it’s not like hypercharge is half as important as isospin no these hypercharge and isospin Transformations balance each other equally when applied to the higs vacuum state okay so we have now unlocked a very deep and a very important equation are you ready for this wait for it ha Q the generator of the U1 transformation that gives rise to electric charge and electromagnetism is a combination of weak isospin and hypercharge this equation is the galman nishima relation you know this is often taught as like one of those rules that you just have to believe and you just have to take as an axiom but actually it’s not arbitrary it exists for a reason and you can see that reason if you just think in terms of a dandelion of Flags Okay so if we look at the particles of the standard model their properties are encoded with a lot of numbers and it’s not clear where all of those numbers come from or in other words it’s not obvious why we have the particles we have in our universe but there are patterns in some of the numbers and so it’s not a total mystery for example as we now know the electric charge of a particle is the sum of its T3 igen value and half its hypercharge so for the su2 douet circled here in pink the T3 igen values are just 1/2 for the top component and negative 1/2 for the bottom component and this mirrors the structure of spin 1/2 systems in quantum mechanics where for example an electron can have a a spin of plus or minus 1/2 and that is encoded in the top and bottom components respectively of the electron’s spinner wave function now for the particles that don’t transform under su2 their T3 igen values are zero sort of like a particle that doesn’t have any spin metaphorically speaking physically weak isos Spin and spin are different things but they share a lot of the same mathematical structure and then for the W bosons which are not matter particles but are gauge bosons those T3 values are a little weird those aren’t exactly igen values but are rather a reflection of the fact that w+ and W minus can flip a doubl it over in the flag picture because bear in mind w+ and W minus are comprised of rotations of the flag about the X and Y AIS as well as a little twirling action that comes from their complex definitions but we’re not going to get into that for now and so anyway what a w boson can do is it can transform an electron into a neutrino or vice versa and so when you’re flipping around a doublet you’re going to change the particles T3 igen value by plus or minus 1 depending on whether it goes from plus 1/2 to - 1/2 or - 1/2 to+ 1/2 okay now if we look at the hypercharge column the hypercharge assignments are more mysterious than the isos spins let’s not worry today about where the hypercharge numbers come from that is a long story and a topic for another day there is an easy but unsatisfying explanation for where the hyper charges come from which is just that they’re whatever they have to be in order for the charges of the particles of the standard model to align with experimental observations but of course we want a deeper more fundamental explanation of why those hypercharged numbers are what they are and for that we would have to get into anomaly cancellation and that’s a whole thing but hopefully we’ll come back to it someday oh speaking of hypercharge I want to give a shout out to Martin Bower Martin thank you for taking the time to talk to me about hypercharge and anomaly cancellation our conversations have given me a lot to think about and I look forward to discussing the topic further and maybe even collaborating on a video one of these days you’ve been very generous with your time and I just want to let you know I really appreciate it so thank you all right let me make a couple more comments on this table before moving on so in addition to the matter particles shown here we also have the second and third generation particles the muan and the TOA and the charm and the strange and the top and the bottom and those have identical properties to the first generation they’re just much more massive and very unstable but they do exist for some strange reason oh and if we look at our gauge bons we have the W’s and the Z and the photon and we now know that those are born of the electro week gauge Group after it’s been broken down by the higs field so the ideas we’ve talked about today give us a really profound insight into where most of the gauge bosons of our universe come from but then we also have the gluons the gluons arise from the su3 C gauge group of the strong force which we haven’t talked about today because it would have taken us too far a field gluons do not transform under su2 l or U1 Y and so they have no weak isospin or hypercharge and therefore no electric charge but they do have color charge which pertains to the strong force and s3c but that’s a topic for another day today we’re just looking at the electro week sector of the standard model okay well I have to mention the yukawa couplings at least briefly but this is a complicated topic that goes beyond the scope of today’s video so we’re not going to go into it in detail I just want to mention it in passing okay so you know before we put the higs field into the electro week model we saw that all the matter particles had to be massless because otherwise different Observers might disagree on which particle is Left-Handed and which particle is right-handed and whether or not it transforms as a doubl it or a singlet under su2 and whether or not it feels the W Fields so we made everything massless but now that the higs field is here and has a non-zero value technically we could have these yukawa coupling terms in the lran which allow us to mix together the right-handed and left-handed components of our matter particles in a way that’s compatible with relativity as well as the gauge group of the electro week interaction the way to think about this is now different observers can disagree on whether a particle is Left-Handed or right-handed and in reality when you look at the wave function of a particle you actually have a left-handed and right-handed component and different inertial observers can disagree on how much a particle is right-handed versus left-handed but all inertial observers are going to agree on the value of the lran term shown here because you can see how it kind of mixes the right-handed and left-handed components together in this equation L is the left-handed component and R is the right-handed component and this Zeta factor is a coupling constant so for example if we think about the electron in the higs field it’s ukaa term is going to have the form shown here with Zeta sub e being the ukawa coupling of the Ron and v/ < tk2 coming in from the higs vacuum State and the net result of what this term does when you think about like the Dynamics of the physics and what happens when you have a term like this is it ends up giving the electron mass and the mass of the electron is the product of the electron’s yukawa coupling constant times the higs vev / the < TK of two when you learn about the ukawa couplings the first thing everyone wants to know is what is the nature nature of this what is going on here what is like the microscopic description of this interaction and the answer is there is not yet a good answer so the way to think about the ukawa couplings is it’s more like the lifting or the resolution of a paradox rather than a good explanation so if not for the higs ffield we really would have to have massless particles but with the higs field there acting as this scalar field that Fons can interact with well then it’s possible to have these yukawa coupling terms in vran but it is a deep mystery and also it’s possible that the true story is more complicated than what we’re showing here so what I’m showing here is kind of like the vanilla standard model but there are extensions to the standard model involving multiple higs fields or other kinds of fields in which the ukawa couplings have a more Nuance to nature but again we’re not going to go down that rabbit hole for today I just wanted to mention it in past so that we can see that this apparent contradiction between special relativity and the chyal electure week gauge group can actually be resolved by the presence of a scalar field that the Fons can interact with okay well based on everything we’ve seen so far you can start to appreciate that there really is something to this Electro week model even though it starts off kind of weird once you see it it’s like hold on wait this actually kind of works you start off with the U2 L cross uy symmetry group you break it with the complex dublet and you end up with the correct masses for the w and z bons and you get a massless photon and you get yukawa couplings giving Mass to the Fons and it’s like okay cool I think we might be on to something here but of course these ideas rely on the fact that we live inside of something that’s very much like a superconductor which is very difficult to accept and something that we should be very skeptical of but also something that if it turns out to be true would validate everything else we’re talking about like if we had a way of directly probing the superconductor the higs field then we would know that we’re on the right track and this is it and this is what’s going on in our universe and that would be amazing and so when you think about what are the different ways of directly probing the higs field this complex dublet field well you think about the phase-like degrees of freedom the nambu Goldstone modes and you have to recognize that that’s equivalent to looking at the w and z Fields right as we’ve seen I mean based on the higs mechanism the w and z Fields get their Mass from the nambu Goldstone modes in the higs field so you can interpret measurements of the w and z field as telling us something about the higs field but if we’re being skeptical about this concept of a higs field you could always imagine interpreting those measurements as something else right so if we want to really directly probe the higs field what we would have to do is actually excite an amplitude mode in the dublet because that can’t just be gauged away that’s a real thing that’s the higs field really being itself and if we can observe such an amplitude mode also known as a higs bon then we know that this superconductor thing is not just Theory it’s real like it’s really for real it’s like actually real and so that’s why you know there was all this hype about like let’s discover the higs BOS on that would be like the best thing ever so let’s talk about what the higs Bon would look like and in particular how much mass we would expect it to have so the higs on is an excitation in the amplitude of the higs field fi so that instead of a constant vev we have V plus h where H is a scalar field that’s Rippling around in SpaceTime so imagine V the vev which is like 246 gevs and then a small h on top of that that can go up and down and so a higs boson is like a quantized jiggle in h okay so writing out the higs field in terms of 1/ < tk2 V + H we can calculate the amplitude squared of the field as 12 V + h s and now the basic idea is that the mass of the higs boson arises from that Ginsburg low Sombrero part of the higs lran that we saw earlier so let’s write that out with the MU and the Lambda and the F dagger fi and when we substitute into fager 5i 12 v+ h^ squ we get an equation which tells us the energy that corresponds to having these ripples in the amplitude of the higs field now what follows is a bit of algebra and it’s a little bit tedious but it’s not so bad so let’s just work through it real quick the first thing we’re going to do is we’re going to multiply out the quadratic and the cortic terms in v and H so just multiply everything out there and you see the expression shown here next up we’re going to collect some terms into this V KN parameter and that’s just going to be negative m ^ 2 2 V ^2+ Lambda 4 V 4 so that corresponds to the potential of the higs field with just the vev so that’s like a reference point of a constant higs field without any H Rippling on top of it and what we can also do is we’ll use a small H approximation to neglect the H cubed and H to the 4th terms the idea here is that because H is going to be small especially relative to V we can get away with just a expanding it to second order so when we do that we get a slightly simpler expression shown here and next up what we want to do is we want to substitute in V = mu over the < TK of Lambda and remember that equation comes from calculating the vev of the higs field it’s very much the same as calculating the density of a superconducting condensate and when we do that we end up with the expression shown here you’ll notice that two of these terms cancel each other out and then we can absorb the other two together -2 + 3 is 1 so we end up with v+ mu ^2 h^2 and now at this point we may as well use V = mu over theun of Lambda again so that we can write the expression as V + Lambda V ^2 h^2 and now what we can do is we can interpret that term as a mass term inran of a real valued scalar field in this case the field is H which is a field of amplitude ripples on the higs field and so a term like that in the lran is going to have the form 12 m^2 h^2 and so we just match the terms and we solve for the mass of the higs boson we end up getting that the mass of the higs boson is the higs vev V * the < TK of 2 * the self-coupling constant Lambda before the higs boson was discovered people had done a bunch of measurements which constrained the values of what V and Lambda could be so Lambda was something between 0.1 and 0.3 and V the higs vev that was known with more accuracy that it’s about 246 gevs So based on that if you look at this equation you can estimate that the mass of the higs boson should be somewhere between about 110 and 190 G depending on the value of Lambda which wasn’t known with a ton of precision until after the discovery of the higs boson but there was a pretty decent estimate you know so the higs boson should have a mass of like about 10 and something gevs now a particle of that mass would be very unstable but if you could create it you could detect it through its Decay products and so when you think about like what do you have to build in order to actually make higs bosons you need extraordinarily intense particle collisions but in order to make that happen you would have to build the largest machine ever built with the largest and most powerful superconducting magnets this would be insane you would need to collide protons at like 99.999 lot of NES the speed of light which is very difficult but in theory it can be done and so people did it and you know if you look at all the things in history that people have done a lot of bad things a lot of stuff that wasn’t great but on this one people did a great job building the large hydrant collider that was really quite something and you know I think it’s worth pointing out this video is called super conductivity and the higs field and so far we’ve been very theoretical and it’s all been like theoretical physics but super conductivity in the higs field is also the story of the greatest experimental physics achievement in recent memory because this involved more Super conducting material than anything that had ever been built and it was built to discover the higs Bon so anyway I think it’s just beautifully poetic how the story of superc conductivity in the higs fields isn’t just a theoretical thing but also spills out dramatically into the real world it’s really amazing well anyway when you’re smashing protons together you’re creating all kinds of particles and it’s really a mess but you have these detectors and these detectors tell you what comes out of the collisions and in order to do good science you want to have a couple of detectors not just one so that you can compare their results so what we’re about to look at are Atlas and CMS these are two of the detectors on the large hydron collider and their job is to measure like what is it that’s coming out of these insanely intense collisions all right so the first way in which the higs boson was observed was in terms of its Decay into two photons so the higs boson is very unstable if you create it it’ll Decay almost right away but one of the ways it can Decay is in two photons and what’s nice about the two Photon Decay mode is that it’s very clean so what I’m showing here in these plots are data from the two different detectors Atlas and CMS and in both cases on the x-axis we have the combined energy of the two photons that are produced in these collisions and the top of each plot shows the background as well as the signal for the higs boson that’s that little bump and the reason there’s a background is that when you’re smashing particles together with crazy intensity this is giving off all kinds of light like you’re going to have photons coming out of that all over the place but statistically you’re going to have a nice clean curve for those photons like if it were just random photons coming out flying out you can calculate what that distribution is going to look like in terms of how often you get a low energy Photon versus a high energy Photon and that’s going to give you a nice smooth background curve but then you see that there’s a bump at a particular energy scale of about 125 gevs and so what the bottom of each plot shows is the bump with the background curve subtracted out and in both Atlas and the CMS you find that there’s a bump at the same energy scale of about 125 gevs and so you think about like what is the kind of thing that can create a bump well if you create a particle that has mass energy of 125 gevs and it doesn’t have to exist for very long just the shortest little blip of time but then it decays well when it decays into two photons that’ll add a little blip onto your chart and you’ll say hey yeah we got a new particle at about 125 gevs this was extremely exciting cuz bear in mind based on the theory we’re expecting the higs Bon to be a new particle somewhere between maybe 110 and 190 GS so here’s a new particle at 125 gevs like that’s it right like that totally seems like this is the higs Boson and of course this detection got people very excited because it seemed like they had found what they were looking for but technically how do you know that this is the higs boson as opposed to just some other new particle with a mass of 125 gevs well I mean that would be pretty unlikely but technically you know if you’re doing science you have to have a lot of confidence and you have to have a really sound set of experimental data but fortunately there’s a way to do even better than just the two Photon measurement because the mass of the higs BOS on is not the only prediction that the standard model gives us we can also predict all the possible ways that the higs BOS on can Decay and how probable each Decay mode is and so if you look at the chart here I’m showing the nine most common Decay modes for a higs boson so a higs boson is most likely to Decay into a matter antimatter pair of bottom corks the second most likely Decay mode is into a couple of w bans except well technically the second W has an asterisk and what that means is that only one of the W bans is real the other one’s virtual off shell and it decays into something else before it really has a chance to exist but that’s like a Nuance that we’re not going to get into now the third most likely thing is to Decay into a couple of gluons the fourth most likely is to Decay into a couple of tow particles Tow and an anti-tau fifth most likely is a couple of charm corks sixth most likely is a couple of zons again one real and one off Shell One virtual the seventh most likely is into a couple of photons as we just looked at with the background curve in the bump so that is actually not a very common Decay mode but it is a very clean signal and so the cleanliness of that signal is very nice from an experimental perspective even though that decay mode is pretty unlikely and then the eighth most common Decay mode is into a zebon and a photon and the ninth most likely is into a couple of muons a matter antimatter pair and beyond that there are other possibilities but they’re extremely unlikely like for example technically a Hig boson could Decay into four photons but that only happens like one in a bajillion times and then some Decay modes are completely ruled out completely impossible like for example a higs boson cannot Decay into three photons because that’s just not allowed by the conservation laws of the standard model well anyway when you take all of this into consideration you see that the standard model actually gives us a very specific fingerprint for what the higs Bon should look like in terms of what it decays into we should expect to see that particle that mysterious particle of 125 gevs in all of these different Decay modes and we shouldn’t see it in the Decay modes for which we shouldn’t see it you know so you’re not just measuring the prediction you’re also measuring that it’s not the thing that the prediction doesn’t predict it’s like uh you have to think about the dogs that aren’t barking if you know what I mean and on top of all of that the Decay probabilities that is the branching ratios the relative percentage likelihood of each of these Decay modes should also match the standard model and if that’s the case if you can observe this mysterious 125 gev particles across even just a few of these different Decay modes and it matches then okay then that’s the hix boson so I looked on the internet and I found a paper from 2015 that showed Atlas and CMS measurements of five of these Decay modes and as you can see it’s a match you know within some error bars because given the state-of-the-art of present technology and you know these numbers can only be calculated so precisely at the present time but hopefully in the future we’ll have better Precision on this in fact hey maybe since 2015 there’s already better Precision I don’t know I just I found a paper that kind of worked so I might be a little bit out of date on this already but anyway yeah as you can see the higs boson has been observed in a number of different Decay modes the branching ratios match the prediction and so we can say with a very high degree of confidence that the higs boson does actually exist that is to say we are actually living in a basically a superconductor and we know this not just because it’s conceptually very nice and very pragmatic and it works and it lets us unify electromagnetism in the weak nuclear force no but now we know that it also exists in its own right like it’s been directly observed so that is just an incredible thing it’s amazing and astonishing in and it’s just wonderful I mean it’s like wow so let me comment a little bit on how the Hig field might not be like a superconductor like what are some of the ways in which the metaphor breaks down well first of all everything we’ve been talking about so far is at the level of Ginsburg landow you know so with superconductors we also have the BCS theory that tells us the microscopic story of how electron pairs form into the condensate that we can then analyze with Ginsburg landow with the higs field we don’t yet have that BCS level of description so we only have the phenomenological math of the Ginsburg landout potential and that means that there is a deep mystery as to what the higs field is and where it comes from and like what even is it and why are we living in a superconductor like that’s very strange right and when you think about possible answers to that mystery there’s a very beautiful line of reasoning which involves extending the superconductor analogy one step further and saying okay here’s what we’re going to do let’s assume that the higs field is a condensate and better yet let’s assume that there’s something about the standard model maybe su3 in the strong force that naturally forms into a condensate cuz you know one of the things with the strong force is you have color confinement where if you try to pull quirks apart you’re just going to end up creating more Corks And so you have this really cool situation where corks are always bound with other corks and so maybe there’s another property of the strong force which is like Universal condensation or something you know and so then if you could show based on looking at the Dynamics of the strong force that a condensate naturally arises maybe if we explore the properties of that condensate we’ll find that that is the higs field and then we no longer have to invoke the higs field as this like arbitrary complex double it scaler field but rather at that point we would have a microscopic explanation for what the higs field is and also why it exists so this is a very beautiful idea but it doesn’t work and the reason it doesn’t work is it predicts a Mass for the higs Bon which is a few thousand times smaller than the measured value and you know that’s kind of a big deal so there goes that theory but it’s such a beautiful idea that maybe we want to try to preserve it anyway maybe we just have to kind of warp it and make it work some other way but to do that we would have to postulate the existence of as yet undiscovered Fons interacting in accordance with some as yet undiscovered force and so the model becomes a lot more speculative now if you’re interested in this idea you can look up Technicolor models and that’s the name for this General category of models in which the higs field is actually secretly a condensate now Technicolor it is a very beautiful idea and it was popular back in the day when we didn’t have as much experimental data as we have today but nowadays it’s not really as fashionable as it used to be because the experimental data seems to be indicating that Technicolor is probably not true there’s probably something else going on with the higs field so then like what even is the Hig field we don’t know it’s a beautiful thing you know in this time period in which we’re living in the long Arc of History we’re not at the end so we still as of today we do not know the fundamental nature of the higs field we just know that there’s this phenomenological model for it that can be interpreted pretty much in terms of super conductivity though of course you know that analogy it shouldn’t be constrictive it should be constructive so an analogy is just a thing that gets here from point A to point B where point a is what you know and point B is what you don’t know yet and so you can kind of discard the analogy once you see the thing so having used super conductivity as a kind of gateway to the Hig field of course we don’t want to think that all the higs field could potentially be is strictly a superconductor like no it might surprise us and there’s still a lot of open questions we have speaking of open questions here’s a question how many higs fields are there well before the Hig field was discovered as far as we knew there were zero Hig Fields but now we know there’s at least one and you know I’ll say this kind of loosely so don’t take this too literally but a superconductor in a way is a place where there are at least two Hig Fields see here I’m reversing the analogy I’m saying the superconductor is like a hfield cuz if you were a sensient being that lived in a superconducting thing and you were a scientist you’d be like aha I know that photons of mass I’m so smart but then someone outside the superconductor would be like hey superconductor being actually photons don’t really have mass that’s just the property of the material in which you live so then the superconductor being can like break out of their Paradigm and be like oh yeah wow actually it is a circumstantial thing and that photons don’t really have mass but then of course you can go another level up because you can think like well the W’s and Z’s have mass but it’s like not really not if you factor in the higs field and kind of zoom out a level and be like who actually the W’s and Z’s don’t really want to have mass it’s the higs field that gives them mass and so now we’re finally at this exalted level where we can see the standard model in its full glory and we’re like okay su3 cross su2 cross U1 is like the gauge group of the standard model and that is the fundamental thing but it’s like wait a minute how do we know that is the farthest out meta level of analysis like what if even when you account for the higs field what if there’s another higs field that we just haven’t OB oberved yet and this could totally be the case cuz if you think about it imagine another higs field but with a way more massive higs BOS on well that would give a lot more mass to whatever gauge bosons it’s interacting with so it would suppress them a lot better and there could potentially be other forces out there other super massive gauge bosons other higs fields that we just haven’t been able to observe yet because they exist at like this super high energy scale now on one hand aam’s Razer says well okay let’s not assum there’s like a whole bunch of higs fields but on the other hand if you do allow for the possibility that maybe there’s another higs field or even multiple more Hig Fields then it’s possible that the gauge group of the standard model might actually be a broken down remnant of a more elegant more fundamental gauge group like for example su5 or s so10 s so10 is the way of rotating things in 10 dimensions and that one is a suspiciously good fit for the standard model in a variety of ways that we won’t get into now but anyway that’s the idea of grand unification so all that’s to say there’s a lot about the higs field that we don’t yet know and so even though the Hig field is very very much like a superconductor think of it as potentially something that really goes beyond a superconductor in an as yet unappreciated way all right before wrapping things up here I just want to make one final comment which is technically we don’t yet know whether the higs field is stable or just meta stable so the higs field originally condensed I think like a Pico second after the big bang and ever since then our universe has been in this superconducting state right this higs vacuum state but in theory it might be possible for the higs field somewhere to fluctuate into a different vacuum State and then for that vacuum state to propagate out in a bubble at the speed of light and as it goes along it’s changing all the physics in its path it’s like a bubble of Annihilation that goes along dissolving atoms like sand castles in a tsunami just completely changing the nature of reality and everything and you end up with a cosmic Amnesia and a whole new operating system for nature and it would be like it would be a terrible catastrophe at least for anyone who dreams of leaving behind an enduring Legacy in this universe it would just wipe out all of our attempts to have any connection to the Eternal like it would just be the worst thing ever um so it’s possible we don’t know so far so good I mean it’s been 13 whatever billion years and we’re still okay but you never know the vacuum state of the higs field could change one of these days now there’s a silver lining which is that because this bubble of annihilation would expand at the speed of light you wouldn’t see it coming although that silver lining has a Touch of Gray because if you can’t see it coming then for all we know it’s already on the way oh man but let’s hope that’s not the case oh jeez ah I need to go for a walk hey thanks for watching and I’ll see you in the next one assuming the higs field doesn’t decay [Music]