The Biggest Ideas in the Universe | 17. Matter

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll we're here for idea number 17 which is matter by matter I mean not what matters in life but the matter that you and I are made of the ground beneath our feet the planets the Sun the stars things like that the stuff out of which we are made and I'll admit that this represents a shift in the biggest ideas series for a couple of reasons you know we're not gonna have an infinite number of biggest ideas videos sorry there's a finite number I had a plan when I started the plan has shifted a little bit as we've gone on but there is an ending point in sight we were closer to the end than we are to the beginning sorry about that so we're gonna be finishing up with a slightly different kind of ideas we've done a lot of work recently on building up quantum field theory and the standard model engage theories and things like that and we have other fish to fry as it were so we're gonna be less about that this video will still be about quantum field theory but it'll be about a more tangible aspect of quantum field theory so the focus and the biggest ideas will always be conceptual it will always be the big underlying ideas not the nitty-gritty of this and act but now we're gonna get into things that are a little bit more relevant a little bit more not relevant familiar let us put it that way easily grasped to us so even in this video on matter so what I should say is having said that even though this video is about matter you might think well okay it's gonna be about materials and things like that and you know it's it's really not because I don't know anything about that stuff you should go to other people who know more about that for those topics I want to connect some of the conversation we've been having about quantum field theory and the underlying laws of physics to the features of matter that we that make up you and me okay in particular most of today's idea is going to be on a single feature of matter which is that it can be solid right that a table is solid you can't just put your hands on it and fall right through this might not seem like a question but we'll ask it anyway you know why is matter solid you might remember back way back in one of the early videos we were talking about the fact that just before quantum mechanics came on the scene you could have imagined in the late 1800s you could have imagined a version of physics where there were two different kinds of stuff there were particles that made up matter you know the particles that we would now know is the electron the proton the neutron and so forth and there were forces that pushed and pulled the matter together that's not exactly right but there's a very close relative to that and so today we're gonna explore the matter side when we talked about gauge theories the other day we're really talking about the forces side of things so why is matter solid at all now you might think well once you know about atoms and particles this shouldn't be difficult question you know if you have some collection of atoms you can just stack them on top of each other or maybe you could be supposed to be atoms not just bowling balls why can't you just stack things on or of course atoms can stick to each other right with chemical bonds and you would imagine there is some solidity there that that makes sense but then you remember we're drawing atoms as little spheres we know better right atoms are not little spheres atoms are quantum objects in particular the size of the atom as we talked about in the video on scale is set by the Compton wavelength of the electron it's really set by the Bohr radius which is the Compton wavelength divided by the fine-structure constant so the electron has a wave function right the electron wave function is wavy it's sort of smooth and you could easily imagine deforming it so the question is why can't why aren't they squishy why aren't atoms squishy this is one question by which I mean that looks like squash II let me make it look more like squishy by which I mean you could easily imagine didn't look like squash II it was like squishy if I have two atoms like this one right on top of each other and I push them on the bottom and on the top why don't they just squeeze down to more elliptical shapes so that they're much smaller okay well well they're much smaller in one direction maybe wider in some other direction why don't they just deform they're just waves after all and the answer is sort of implicit in what we talked about when we were talking about wave functions and the shape of atoms in the first place namely that the electron in a hydrogen atom or even more complicated atom its wavefunction has the shape it does because it's trying to minimize its energy right is trying to minimize its energy subject to certain constraints may be there are other electrons in the way it's being attracted by the nucleus and so forth so if you you can certain you can imagine taking an atom and squishing it there's no problem with imagining doing that but its energy would change and since its energy is already at the minimum given the constraints it's subject to the energy has to go up you have to put energy into it you might imagine that's not much energy right like how much energy can it take you might be a squishy kind of springy thing but in fact it's a lot of energy if you changed you know in a table if you changed all the electrons energies by one percent that turns out to be an enormous amount of energy I didn't actually calculate it that's a good homework problem I predict it would be something like an atomic bombs worth of energy if you wanted to squish all the atoms in a table to change their electrons energy by one percent so there's a reason why electrons want to remain spherical in atoms it's because they're in their lowest energy states already it would take a lot of energy to change them but there's another question why don't they fall on top of each other why don't atoms go on top of each other why are they solid individually so we say the tables are solid because they're made of little atoms right so that presumes that the atoms themselves have some ability to take up space right to occupy volume so why can't we just imagine going from two atoms pushing them together and then this goes into one atom right on top of another one right what's what's the danger of that it's not quantum mechanics per se that is the problem there there's no problem with having two electron overlapping in their wave functions I mean electrons in an atom do that but they don't do it arbitrarily there are rules that they follow and that's basically what we're going to be talking about today in today's video to give away the secret which I like to do we're gonna be talking talking about the properties of fermions fermions are a certain kind of particle named after Rico Fermi famous italian-american physicist and they are in Contra distinction to bosons and roughly speaking bosons named after professor Bose whose first name I can't pronounce who's an Indian professor so Bose hooked up with Einstein to invent Bose Einstein statistics which are particles of a certain kind roughly speaking particles that like to pile on top of each other particles that actually prefer to be in the same place if you have identical particles of the same species or bosons fermions obey what are called Fermi Dirac statistics after Fermi and drak and they'd like to take up space from yawns do not like to be on top of each other so the glib answer to why atoms don't want to be on top of each other is because electrons which are giving you the size of the atoms are fermions they take up space they don't want to be on top of each other but we want to get closer to why that's actually true okay so the name of this principle that says why not is the pout that that says that this doesn't happen rather is the Pauli exclusion principle so again this is a the fact the the motto electrons take up space is a very cute little retelling of the Pauli exclusion principle which is a precise quantitative statement which says that two electrons cannot be in exactly the same quantum state at the same time now even though exclusion principle is is less than there is to the story there's more to it than that but this is a nice simple way of thinking about what's going on if you imagine the shape of an electron wave function in an atom if you tried to put two electrons in exactly the same state which works out to be both the same the same shape for their wave function and also the same spin you can't do that according to the pal the exclusion principle so that is what we want to get to two fermions cannot occupy the same state quantum state of course it's a quantum state what else would it be well you want to talk a little bit about why that's true and also to give away some of the surprise at the end you may have heard fermions defined as particles that have a spin of one half or three halves or five halves half-integer spin as we call so bosons have spins of zero one two integer spins that's true but that's not the definition there's a separate fact about fermions which is that they have a spin of one-half three-halves etc but the defining principle of them is this they do not take up the same quantum state the connection between the taking up space and the having half integer spin is called the spin statistics theorem and we'll talk a little bit about that in a somewhat unsatisfying way okay so what is a fermion that's what we got to get into here and fermions well we could just list all the ones we know what is a firm young they are either quarks the ones we know are either quarks so that's up down charm strange top bottom or what we call the leptons which are the electron the muon the Tau the muon and the Tau or just heavier cousins of the electron they're the same in every other way except for their mass and then there are the neutrinos and there's sort of one neutrino for each charged leptons is an electron neutrino a muon neutrino and a Tau neutrino these are all the fermions that we know about let me think I hope not skipping any it's always hard to actually remember when you're a working physicist because there's so many hypothetical fermions that we almost take for granted they exist but these are the ones we actually know these are the fermions of the standard model of particle physics and what makes them fermions okay consider two particles so we're gonna consider the quantum mechanical wave function for two particles and part of the whole discourse we're gonna have through this video is that we're going to remember that deep down we're talking about quantum field theory okay but we're not going to really do any quantum field theory so we're gonna work with particles as if their original official particles all by themselves but we remember in the back of our minds that what these particles are are a certain way of talking about quantum fields in what we call low energy states right the lowest energy state is zero particles the slightly higher energy state is one particle and there are states with two particles three particles etc so we're talking about particles but they come from quantum fields and that will actually matter so what is the wavefunction for two particles look like and forget about spin and things like that we're not gonna go into those quite yet we'll get there eventually so the wave function capital sci for two particles just depends on where they are particle one is that x1 and particle two is that x2 let's say for example okay there you go we're not to say what it is yet but we're gonna drive some properties of the wave function for two quantum particles and in fact let's imagine these are identical particles oops identical why would we imagine that well um they don't have to be there are particles that are not identical a proton and electron are not identical but two electrons are identical particles now why are there identical particles we're gonna refer back to the fact that they come from quantum field theory two electrons in two different places are two different vibrations in the same underlying field so of course they are identical they're their particle Ness is not what makes up nature field this is what makes up nature and it's the same feel that is giving rise to these two electrons the same exact properties of the field so of course the particles that we get by looking at I braised in electron field will be identical particles so that is something that we inherit from the fact that it's really quantum fields going on and now we want to ask well what happens if we exchange the two particles with each other the thing to remember here is that x1 and x2 are not cific locations in space right there what we call the argument of the wavefunction so there are two different labels on our three-dimensional space one of which says what is the likelihood of finding particle one there and the other one says particle two so you see that already something interesting is going on if the particles are identical I mean how do you know which one you've seen if you actually measured have you seen particle one or particle two so there's probably some relationship that relates x1 and x2 to each other in the wavefunction or rather what we could say is there's got to be some feature of this function sy that relates x1 and x2 to each other because we don't know when we measured a particle which one it is we're actually measuring because they're just vibrations in the field so a way to think about this is to imagine interchanging what if we what if we exchanged the two particles so let's send sy of X 1 comma X 2 to something which we're going to call the interchange operator so you know this is something that takes the wavefunction and switches the two particles so the interchange operator acting on Phi of X 1 and X 2 which guess what is going to be equal to sigh of X 2 comma X 1 as a function so again mathematicians are gonna be a little bit annoyed by this notation but you know what I mean it's the hope okay and we want to figure out what this actually means what this what properties they should have so there are two things we know about this just on the basis of quantum mechanics okay one thing that we know is that if you interchange the two particles and then you interchange them again you get back where you started okay nothing actually happened so one thing is that the interchange operator the act of switching the two particles if that were to act on what you get by acting on the interchange operator if this reminds you of our group Theory discussion that's probably not a coincidence this is a set of transformations you can do on this wave function then you act that on sigh of X 1 comma X 2 you better get back where you started so that is sigh of X 1 comma X 2 you didn't do anything you switched them and you switch them back that seems kind of obvious and not worth noting but believe me we're gonna have to note it the other thing that we know is that if you only interchange once so if you just do the interchange on sigh X 1 and X 2 because they are the same because they are identical particles because they are vibrations in the same underlying quantum field there can't be any observable consequence of doing this ok and this is where quantum mechanics comes in because quantum mechanics distinguishes between what is real and what you can observe you can't observe the wavefunction directly right what can you observe you can actually measure to see is the particle at X 1 or X 2 so basically what you can measure is the probability of measuring particles there when you look that's when you can calculate that's what you care about that's the quantity that gives you observable predictions is the probability so this better equal the same physical situation because they're identical particles interchanging the two shouldn't change the physical situation and what that means is that the probability of observing a particle of X 1 and X 2 is of course capital psy of X 1 comma X 2 magnitude squared so this thing therefore psy squared doesn't change let's let let me let me write it in a little bit more formal way of saying it size squared equals whatever you get by interchanging sigh and then squaring that this is the requirement that the physical situation the predictions the observable outcome does not change if you interchange two identical particles with each other so then you have two possibilities there are two ways to satisfy these two things we know one is if you do interchanging twice nothing happens what the other is of you interchanging only once you better make the same physical predictions so there's two possible two ways to satisfy these two criteria one is the obvious way okay if you interchange I was going to write int know along rather than the whole word interchange if you interchange Sai x1 and x2 maybe you just get back Sai x1 x2 maybe nothing happens to it at all okay this would be the intuitive non quantum mechanical thing if you had two identical particles and you switch them for each other and you know so you did someone did that while you weren't looking you wouldn't know right it looks exactly the same maybe it looks exactly the same because it is exactly the same that is absolutely a possibility and this possibility is realized in quantum mechanics and recall particles that obey this relation bosons that's what bosons are bosons are particles for which when you have two identical ones if you switch them in the wave function the wave function doesn't change at all so just to make that clear for example eg for some size of one particle okay so imagine now someone gives you a wave function of just a single particle I can instantly create a wave function for two boson ik particles that satisfies these criteria right I can let capital sigh of x1 x2 be just sigh of x1 times 5x - this is a perfectly good boson equate function note that if this first wave function were different than that one then in general this would not be a good wave function for a boson ik particle because if I switch them it would not get the same answer so more generally for two different functions little side one and little side two I can make a boson equate function capital sign by taking side 1 of X 1 site 2 of X 2 but then adding plus psy 2 of X 1 psy 1 of X 2 ok and these are just numbers so I want X 2 their numbers for every value of X 1 and X 2 so they commute and everything like that there's nothing funny business going on but by construction by the way that we made this particular wave function it will always be the case that we've satisfied these two criteria you interchange twice and you don't change anything you're gonna change once and you make the same probability exactly because in fact you just get the same exact wave function we interchange that's automatic no matter what the specific function side one side two here are so bosons are allowed physical particles and the bosons of the standard model are photons and gluons and W and Z bosons and the Higgs boson and the graviton there you go graviton you can debate whether accounts but accounts so what this means is that there is no objection to two different identical bosons having the same exact quantum state and in fact what Bose showed is that they'd like to have the same quantum state given one boson just sitting there in some quantum state another one actually likes to cuddle up with it you know they like to be sociable they like to be right on top of each other doing exactly the same thing that's what bosons like to do so bosons can pile on to each other and therefore this has a very very important dramatic effect in macroscopic world physics they can give rise to long range macroscopic forces okay the reason why you could discover the gravitational field or the electromagnetic field as big classical fields long before you had discovered gravitons or photons is because gravitons and photons are bosons and they can pile on top of each other to give you a strong detectable field in the classical regime so they can give rise they don't always necessarily but they can give rise to macroscopic classical fields gravity and electromagnetism are mediated by bosons now the strong force is also mediated by bosons gluons are very happy to pile on but because of confinement they don't ever escape the proton of the neutron so we don't see them the weak force W and Z bosons also happy to pile on just like the Higgs boson also happy to pile on but those particles are so massive they just decay away very quickly so the massless non confined bosonic fields can give rise to long-range macroscopic force fields and guess what that's exactly what they do you shouldn't be surprised okay there's another way to satisfy these two criteria here are the criteria interchange once and you get the same probability but not necessarily the same state and change twice and you get the same quantum state well you're gonna guess what it is you're gonna guess correctly the other way to satisfy it is that interchanging sigh of x1 x2 gets you - sigh of x1 x2 if you only knew this second criterion that you interchange once and you get the same probability then you would say well ok any complex phase e to the I theta could multiply the wavefunction when I exchange two particles but you really want the wavefunction to be exactly unchanged when you do it twice so it really needs to be multiplying by minus one is the only possibility by the way footnote here nothing that we say in this video is rigorous sometimes I'm able to take a rigorous argument and give you a version of it that is understandable and transferable and make sense this whole subject is one in which there are rigorous arguments but they're just difficult they're just intrinsically difficult even to say in words so I'm offering plausibility arguments more than anything else so sticklers will know that you know whether or not bosons and fermions are the only kinds of fields depends on the number of dimensions of space-time for each which you know not mentioning here but if you if space-time we're three-dimensional set of four-dimensional you'd be telling a very different story but we're we're just taking the shortcuts here to give you what we want you to understand why this makes sense not necessarily to rigorously know that it's been proven true okay those are our standards so this this is another way that you can satisfy the criteria that we settled and this way is called fermions this is what fermions do fermions are particles for which in a multi particle wave function when you exchange two of them the wave function picks up a minus sign that's the real definition of what fermions are okay so that means they can't be in the same state so for example you could make a fermionic state sigh X 1 of X 2 and X 2 which is equal to some function psy 1 of X 1 times some other function side 2 of X 2 minus sy 2 of X 1 Phi 1 of X 2 okay so now you notice just by this version of this formulation of how I wrote the wave function that if I let X 1 go to X 2 the wave function as a whole picks up a minus sign that's what fermions are supposed to do and then you also notice that if sy 1 equals sy to as functions then this capital sy would have to equal 0 you can't make a wave function by multiplying to one particle wave functions together and then antisymmetry Singh in this way if the two wave functions are identical unless the whole thing is just exactly 0 so this is the way that we have to derive the fact that fermions that this is it gives us the poly exclusion principle let me just say it that way this says that two particles cannot be in the same state Palli exclusion principle probably by the way is just one of the heroes of early quantum mechanics he didn't quite get there to invent matrix mechanics or the Schrodinger equation but he mends a lot of stuff he was also like really difficult to get along with you know he was very notorious for telling other people how stupid they were sometimes the other people were actually rekt and then he would go on and develop their ideas and get credit for them so like there was Polly's students I think who first suggested the idea of spin and he said that they were idiots and they shouldn't do that and someone else show that spin was really there and probably developed the whole spin matrices and everything about spin that we now know and love today so he was smart but you know even smart people are not always right anyway the exclusion principle very very important so this is anti symmetry right this is an anti symmetric wave function it implies that side one cannot equal side two but they can be different so you can have two fermions in two different states and then the total wave function would look like this and if you want to say oh that that looks like they're entangled with each other yeah that's exactly right you know two identical particles have to be entangled in this way two fermions have to be entangled in this way to bosons can just be right on top of each other in exactly the same state which would not be entangled but fermions always got to be entangled okay okay so this means in other words that fermions take up space that's what we've learned because they cannot be two fermions cannot have exactly the same wavefunction one way or the other and that is why atoms are solid because you know this is again simplification because there's something like spin that we're gonna talk about in a second but roughly speaking if you imagine two hydrogen atoms okay two protons with a single electron around them and you try to put them right on top of each other those two electrons would want to be in the same quantum state but the Pelley solution principle says they can't okay it's a better example of helium atoms but you know we've done hydrogen more than helium so let's just talk about that okay so this is usually called so bosons versus fermions that that classification that classification is known as the statistics of the particles they this distinction first became noticeable or noticed when people Bose and Einstein bear me and Dirac we're actually thinking about large collections of many particles and then the fact that the particles either want to take up space or pile on top of each other clearly effects the statistical likelihood that they're going to be in different quantum mechanical States okay so Bose Fermi statistics the word statistics is what is used to say whether or not a particles of boson or a fermion does it obey Bose statistics or does it obey fair mean statistics and bosons is a little bit redundant but I'm saying it anyway just to stick in bosons give rise to forces fermions give rise to matter now what we said was that to obey our two rules for what happens when you interchange two particles there were two options either be boson or fermionic there's no in between there's no rule that says that a theory needs to have both but of course the standard model particle physics certainly does supersymmetry is a hypothetical symmetry that tries to relate bosons to fermions it says there is a symmetry transformation in nature that Maps a boson ik field to a fermionic field and vice versa the standard model is clearly not super symmetric there is no mapping of the fermion fields on to the boson fields that that satisfies any symmetry but you could imagine that like the electroweak symmetry supersymmetry is broken spontaneously and so for every Fermi on there is a partner which is a boson and vice-versa but they're just too heavy because remember symmetry breaking often gives mass to particles so maybe all the superpartners are just so heavy that we haven't observed them yet and then in nature you could actually be super symmetric now we were hoping to see super practicals at the Large Hadron Collider we didn't so therefore is a good Bayesian your credence that supersymmetry is correct should go down but you know if you didn't have hopes and dreams for the electroweak scale and Large Hadron Collider there's nothing to prevent you from saying that supersymmetry is broken at a much higher energy scale so the fact that we haven't seen supersymmetry at the LHC is evidence against supersymmetry existing but it's nowhere near definitive evidence it's very very easy to come up with supersymmetric theory with the supersymmetry is hidden from us and we'll never see it within my lifetime anyway so we don't know about that but the standard model certainly does not have supersymmetry in it anyway you often hear you were often told that bosons have spin of 0 1 2 dot dot some integer and fermions have spin of 1/2 yeah I was thinking while writing again three halfs five halfs etc okay so as I said as I foreshadowed this is this is a true fact but it is not the definition of what it boson and Fermi on is bosons pickups like to pile on top of each other fermions take up space fermions have the Pauli exclusion principle bosons actually like to be are more likely to be where other bosons are so we want to relate we want to figure out why this is true right why is it the bosons or spin zero one two fermions are spending 1/2 3 X 5 halves so I guess I will have to tell you what the definition of spin is well to talk a little bit about spin now we've been fearlessly talking about spin already because it's one of those words you've heard before let me pause to say something that is really not in the textbooks but I think is true so people talk about spin as a version of angular momentum okay elementary particles like the electron or quarks or the photon or whatever have spin and you can transform angular momentum in the good old-fashioned sense like an electron moving with angular momentum around proton in an atom to spin and vice versa so what we call orbital angular momentum good old angular momentum that you would measure by something moving around is this part of the same conserved quantity that spin is okay so spin is definitely unambiguously clearly a kind of angular momentum otherwise in government it would not be conserved so you should think of spin as angular momentum however people will tell you do not think of spin as something actually spinning do not think of an electron as actually having a rotation around some axis and if you say well why not why shouldn't I think of that they will say well look the electron is a point particle okay it would have to be spinning literally infinitely fast in order for it to have any angular momentum at all by the formula for angular momentum okay but it it can't be spending infinitely fast because that would be faster than the speed of light that would make no sense in other words the idea that the electron is a point particle seems to be incompatible with it having intrinsic spin another way of saying it is well even if you don't believe the electron has is a point particle or even if you if you say well the electron really is a wave function not a point particle the wave function has a sighs you know the Compton wavelength for example and if you have if you make a sphere a solid sphere the Compton wavelength of the electron and you let it rotate then you can figure out that it would again have to be rotating faster than the speed of light to have the spin that the electron really has spin one-half but neither one of those neither one of those arguments should be convincing to you at all because you know that really the electron is an excitation in a quantum field and you might ask you know Ken fields have angular momentum even if they're not literally moving can the configuration of a field have angular momentum the answer is yes in fact magnetic fields I'll just sitting there all by themselves can have angular momentum so what you really should be asking is can be configuration of the electron field have the angular momentum of 1/2 that we would give to it by the ordinary way of talking about spin so the answer seems to be yes my colleague at Caltech chip Stevens who I wrote a paper with about everybody in quantum mechanics that is completely orthogonal to this has been working on understanding feel spin in quantum field theory and his understanding which seems completely correct to me is that you can think of spin as literally the angular momentum of the field describing the electron or the quark or whatever so my slightly heterodox opinion following what chip has done is to say that you should think of spin as angular momentum because it is because there literally is something spinning but it's the field it's not some notion of an elementary particle that is point like in any way okay so if you know in other words remember neither any nerd er oops ah not a good day all right she said that conserved quantities oh sorry she said it the other way around she says that symmetries imply conserved quantities so for example the you one gauge invariance of electromagnetism implies conservation of electric charge translation invariance in one direction of space implies conservation of linear momentum time translation invariance implies conservation of energy and it is rotational invariance that implies conservation of angular momentum and when you go through the details of what's going on you can show that fields can have transformation properties under rotations and you sort of know this if you just have a scalar field like the scalar fields just a number at every point in space under a rotation nothing happens to it it's a number at that point I do a rotation around that point the field doesn't care it's just sitting there but if I have a vector field okay a vector field points in a direction so if I'm sitting at one point and I rotate my coordinates the components of the vector better change to compensate for the fact that I'm changing my coordinates so when we think about how a field changes under rotations of the coordinates different kinds of fields can be different and that's a question you can ask what are the transformation properties under rotations of the particular field we're looking at and the answer tells you the spin okay so spin is the transformation or comes from let's put it this way it's put an arrow that way the transformation properties of fields under spatial rotations as you might guess or fear or anticipate we're not gonna go into details about this okay but I'm telling you this is the true thing so you can derive equations for what the spin of different fields are using the properties that they have under rotations in space and you can relate that and you can show that spin plus orbital angular momentum is the thing that is conserved according to northers theorem it is the total conserve angular momentum so let's go through some examples they're scalars scalar fields which I just said scalars don't change I just said that in words and I move my hands but let's just draw a little picture okay so what I mean is here is space so this is x and y and let's focus in on the value of some scalar field at this point right there by okay and I undergo a rotation rotate by anything at all it doesn't matter what angle I do and at the end of the day I suppose I should draw my axes rotate it a little bit Phi is this thing thing Phi is the same it hasn't changed so the there's a slight inconsistency in the labeling here but but it all works out for reasons that we're not going to go into we call this spin zero it doesn't transform at all we don't we don't pivot any spin and that is what pops out of the mathematics as well okay so that's one possibility another possibility is vectors so it's a little complicated we'd have to go into matrices and so3 and things like that if we wanted to tell you in all glory how vectors change under rotations but let's simplify our lives and look at rotations by 360 degrees okay rotation all the way around right you're gonna you're gonna anticipate this is could be boring because when you rotate by 360 degrees nothing should happen but aha you're going to be surprised so vectors don't change for a vector you're right okay for vectors they don't change under 360 degree rotation that's 2 pi if you're a radians person ok go I'm trying to be user-friendly to the degrees people under they're out there so here is again space here's my little vector I'm going to draw my vector pointing up and I'm gonna rotate so this is a rotation by 360 degrees and what happens nothing at all I mean something happens in between but once I've done the entire rotation by 360 I go back to exactly where I started ok so we call this spin 1 it has unit spin so really when I say 1 I really mean 1 equals H bar H bar Planck's constant which we set equal to 1 turns out dimensionally if you don't use natural units H bar has units of spin and a spin one particle the amount of spin that it has is exactly equal to H bar that's why the 2pi is there that's why we define H bar is H over 2 pi this is this way you get a nice formula for the relationship between H and spin so you can talk about spin one rather than I don't know spin 1 over 2 pi it would be something like that yeah no spin 2 pi I guess yeah it's been one is much easier to say ok there are fields like let's say the gravitational field ok the remember we talked about the metric tensor field we're not going to go into details here again but there are fields that have the property there exist fields that have the property that they're invariant immediately be consistent with why I said before they don't change when you under when you rotate by just 180 degrees which is just PI radians ok so how could that be if I'm drawing my little picture here if I draw anything other than a dot like when I drew my vector clearly if I rotated that by 180 be pointing down now so what can I do well what I can do is just draw something that is symmetric right so I'm drawing something that looks like it points both up and down and then I'm gonna rotate that by 180 degrees and if it's truly symmetric up and down then that's gonna look exactly the same before and after okay and in fact that is more sorry this is called spin too and you can go on higher things 360 degrees for spin 180 degrees for spin to 360 degrees divided by 320 degrees for spin 3 etc and gravitons our spin to photons and W and Z bosons or spin 1 Higgs bosons or spin 0 so all these are actually realized in the real world ok and interestingly you can actually see this behavior or sort of a reflection of this behavior into classical influence of these fields you know if you compare I said spin 1 photons would be an example electromagnetism spin 2 gravity right so think about what happens to particles as one of these classical waves goes by a photon electromagnetic field goes by electromagnetic wave or a gravitational wave goes by well what happens is for an en em wave electromagnetic if I imagine that the electromagnetic wave is coming at you or coming at me coming out of the board okay so and here are here's a charged particle like an electron and here is the X and y axis and so the particle the wave rather is moving in the Z direction so it's coming at you what's gonna happen well the electron the electric field the electric field is going to be pointing up and then a moment later it's gonna be pointing down if that's so this is e2 if that's c1 so the electrons motion is up and then down it vibrates in a vertical direction and this we know this if you if you know a little bit about E&M this is the polarization of the electromagnetic wave look if you wear polarized sunglasses they will let through electromagnetic waves of one polarization but not the perpendicular polarization so there's a relationship between the fact that the field is invariant under 360 degrees which means it's either pointing up you have to go all the way around before it's pointing down again and the fact that when the field oscillates a particle under the influence of that goes up and down so it's it's invariant under rotation by 180 degrees whereas for gravitational waves what happens and you might think if I have a little test mass sitting there X Y and the gravitational wave goes by in the Z direction well it will also move the particle up and down that's what a wave should do right turns out that's not right it's a little bit trickier than that and the reason why is because remember the principle equivalence when we talk about gravity in a last video if I have a bunch of particles I can't compare them to each other if they all just move in the same direction all I can do is ask how they move relative to each other so if the particles did move up and down in lockstep then I wouldn't have a gravitational wave at all that's equivalent to nothing happening that's equivalent to just changing my point of view right whereas with the charged particle I could put a charge particle next to an uncharged part look like I literally see it move up and down but because of the principle of equivalence for gravity they all move up or they all move down so that's not really motion at all what I can do is imagine that I have a series of particles again these imagine these are like test particles floating in outer space okay so they're not pulling or put pushing on each other they're just floating out there a gravitational wave goes by and what can happen is the shape can be distorted that is something that is measurable if all the particles were to move in lockstep that wouldn't be measurable but if they change their shape with respect to each other that is measurable so what actually happens is this kind of arrangement will change over time into something that is more stretched out like this okay and then it will oscillate back to this circular configuration and then it will oscillate to being stretched out this way so in other words the overall effect of the gravitational wave passing by can be thought of as forming a plus pattern right you see the plus sign there that is the actual pattern not the most beautiful drawing of this pattern okay let me draw let me do that again you know this is gonna be on YouTube for like a million years right I'm never gonna take this down so I better get it make it pretty so here's a circle yep they're making a circle now I can make an ellipse look at that now make another ellipse look at that okay not the prettiest lips as ever but still not not bad okay move it yeah there we go so this is supposed to represent the set of particles moving in this plus pattern up and down like that and there that's one polarization of the gravitational wave the other polarization is called the X pattern or the x pattern where here's a circle so this is the position of some test particles and they vibrate oops and they vibrate this way into this kind of ellipse tilted and then into this kind of ellipse tilt to the other way I made a square yeah without wanting to about that design ellipse yes good can I move it yeah there you go good enough so this is vibrating in an X pattern there are still polarizations two polarizations of the gravitational wave but look they're related to each other by looks like just a 45-degree offset rather than the 90-degree offset of the two polarizations of the electromagnetic wave so this these features all of which is to say in a very long-winded way these two features these features of the classical waves and electromagnetism and gravity are a reflection of the fact of the underlying quantum fields our spin one for electromagnetism spin-2 for gravity okay there you go you could also imagine a polarization where a bunch of test particles under gravity just sort of expanded and contracted they didn't get deformed at all that would be measurable that would correspond to a spin zero graviton which we don't think exists but you could imagine you could have theories of big scalar fields that actually would pull particles a breathing mode for the particles going in and out okay all right all that see I can enthusiastic and I get distracted from the actual matter at hand the matter at hand is fermions right so that was just spin zero spin one spin too those were all the bosons where the fermions going to come from I mean if if a boson with spin one already has the feature where is it that it doesn't change it's left invariant under 360 degrees and higher spins are invariant under fewer and fewer degrees then less been less than spin one is going to be invariant under more than 360 degrees so of course if it's something is invariant under a rotation by 360 will also be invariant under rotation by 720 which is twice 360 but how in the world can something be invariant only under a rotation by 720 without also being invariant under rotation by 360 right so it seems that spin 1/2 should be invariant under if we go by the pattern you've already established under a rotation by 720 degrees 4pi but not by a rotation under just 360 degrees so how is that even possible okay well so let me just state that it is possible and it's going to happen so let sigh of X the what we call a spinner field spinner with an O R at the end so this is in contrast with this is a vector field this is a tensor field a particular kind of tensor field the graviton there is something called a spinner and this is a spinner field that has the property that a rotation by 360 degrees acting on Phi of X gives us minus sy of X there's a family resemblance there between this and fermions right but it is different number one the fermions statement was about quantum wave functions not about classical fields number two they were interchanging two separate particles and this is a statement about the single classical field what is it's happening under rotations and then of course rotation by 720 degrees acting on sy banks we know that had better give back the original sy of X because it's just two rotations by 360 all right so how is that even possible is the question how is it possible to have a thing a geometric object which is invariant under rotations by 720 but not by 360 so we're gonna have a visual aid we almost never have visual aids here on the biggest ideas in the universe but this is a special occasion so here's my furnitures cat mug okay you can get these yourself I think is the unemployed philosophers guild yes unemployed philosophers guild sells the front of your cat mugs they come in pairs you put hot water in them and you see whether you have the mug where the cat is alive or the cat is dead I was not in charge so it's not sleeping and awake cats but anyway you don't know it doesn't really matter for the purposes of this particular visual demonstration here is the demonstration you can see this is a famous demonstration I didn't invent it Richard Fineman has done this very famously and other people have done in the demonstration is the following what we want to do is if you just consider the cup the mug then clearly if you rotate it around the vertical axis by 360 degrees it comes back to the state it was in originally okay there's nothing we heard about that but we're trying to challenge the idea that everything in the universe has the property that if you rotated by 360 degrees it comes back to the state of sin are there things that you have to rotate them by 720 and the answer is yes and the thing that you have to the thing that remains invariant under rotation by 720 degrees is not the but the relationship between the cup and me okay this is an abstract concept but this is what you're paying for the relationship between the cup and me changes if I only rotate the cup by 360 but does not change goes back to where it originally was if I rotate it by 720 okay the demonstration of this is me holding it in my hand and rotating the cup around a vertical axis so I'm not allowed to do this that's cheating cup is empty don't worry but I'm not allowed to try to hold this words visible not allowed to tilt it so I need to keep it vertical and I rotate it so let's see what happens so I can rotate it without breaking my arm keeping it vertical and you see that is now rotated by 360 degrees but it's clearly in a different relationship but I can keep rotating it then it goes back to where it started okay so you can wind the video back I rotated it in the same sense I did not unwind it that would have been cheating I rotated it by 720 degrees and I got back to where I started and it turns out we're not going to go into the details here it turns out this is a reflection of the following math fact PI 1 of s 0 3 equals Z 2 see you thought that all that work trying to understand topology was just sort of a mathematical diversion it was not the so3 is the group of rotations in three dimensions PI 1 of so3 is the set of ways that we can map a circle that is say a continuous path that is a closed loop into the set of rotations and what this is saying is that here's a rotation by you know 1 degree 2 degrees 3 degrees etc go you can go all the way back to 360 degrees and then keep going it again and come back to where you started and that is a topologically non-trivial thing you can do so I guess the way to say it is again this is not very clear sorry but it's not going to be clear the point is that what you might have thought was topologically non-trivial thing in fact is not do you have to go around twice to make it topologically trivial that's what this is saying there our ways of doing rotations that if you do them twice you get back to where you started that's that's this particular mathematical statement okay so here is a field that is invariant there there could exist fields that are invariant under rotations by 720 degrees we would call them spin 1/2 in comparison to what we already said about spin 1 and spin to okay and they exist in nature electrons and quarks and so forth all have spin 1/2 we've never discovered a particle with spin three-halves an elementary particle you can make them by combining you know spin one particle with a spin 1/2 particle but that's cheating a little bit of the elementary particles that we know of they're either spin 0 1/2 one or two if you count the graviton in supersymmetry if there is a supersymmetric partner for the graviton it will be called the gravity no and it would have spin three-halves but again no evidence for that actually existing right now so let me say a bit more about how spin works and then we'll relate it in a triumphant conclusion we will relate it to fermions and bosons so the other thing I want to say is that you can measure spin and classic example is an electron if you send an electron through a properly stretched magnetic field it will be deflected upward if the spin is spin up and downward if the spin is spin down that is called the stern-gerlach experiment so when you measure spin there's a feature of quantum mechanics that says you're only going to get quantized answers you're not going to get any answer and this is this is what was so weird about the stern-gerlach experiment right so you have a magnetic field let me draw the string Gerlach experiment this is how you measure the spin of an electron or another charge spinning particle you can find more details in my book something deeply hidden about the stern-gerlach experiment so you have a magnet which is in homogeneous so I'm gonna draw it so here is the North Pole the magnet and it comes to a point and here is the South Pole of the magnet which is sort of roundish and then the magnetic field does things like this so it's squished at one end and then I send an electron through it ok and you might think well okay if the electron has spin it will be deflected so the the electron is basically a little magnet okay because it's a charged particle that is spinning when you start charged of spinning you get magnets so you're not surprised electron is deflected and if you say it's spin 1/2 you might say well okay it'll be either if its axis is exactly aligned with the magnetic field would be deflected one way if it's anti aligned it'll be deflected the other way and if it's somewhere in between it'll be deflected somewhere in between but what actually happens is it's only ever deflected up or down never in the middle so this is what we call spin up and this will be called spin down and these are the measurement outcomes for this particular electron so the real electron before you measure it could very well be in a superposition of spin up and spin down but just like Schrodinger's cat the electron once you look at it will only be seen to be in one state or another spin up or spin down so that's a feature more generally of spin that you will only observe certain quantized answers and the answers that you get will be separated by spin one by one unit of spin so these are different by one unit of spin of angular momentum if you like because this spin up is plus one half the electron has spin one half this spin down is minus one half okay so that's more general that's true for any set of any spinning particle you will observe its spin to be certain quantized numbers and the numbers you're allowed to get are separated by units of one units of h-bar units of the fundamental Planck constant okay so for spin zero what you can get you'll always get the same answer there'll be no deflection in the magnetic field so well we'll draw is just this is the spin that we're measuring it's zero and you will always get that so there is the thing that you will measure compared to 0 it is 0 good no surprise there's only one allowed value of the spin for spin 1 sorry let's do spin 1/2 which we're just doing up here now this is spin 0 but you never get that he never observed spin 0-4 has been 1/2 particle you either observe plus 1/2 or minus 1/2 so that is two possible values for the spin for spin one now you can measure zero in other words the part the the spin might be along so I should say perpendicular to the magnetic field and so it's not gonna be deflected at all or could be deflected up or down so you can measure it up here down there you get spin zero plus one or minus one three allowed values for the spin okay there you go so this is what you actually measure in an experiment this is how spin works all that long ring role was to say there's something called spin it's a variety of angular momentum it's quantized in quantum mechanics and it's quantum values are such that the allowed values that you can measure come in half integer units for any one kind of particle the allowed values you will ever actually see are separated by units of one h-bar okay so good now I haven't mentioned fermions or bosons right and a long time I've been mentioning spin and so I wanted to drive home the idea that what makes you a Fermi on is whether or not you obey the Pauli exclusion principle whether or not you take up space and what makes you a particle with spin is that there's some value the environment and that you can measure and it's quantized but there is a relationship okay the spin statistics theorem it's called you know what spin is and remember statistics is are you a boson or are you a Fermi on and the spin statistic theorem is bosons have spin zero one two double dot fermions are half integer spin one-half three-halves five hands okay why is there this theorem so this is another example of a true fact in physics that you can prove mathematically and this sort of notoriously difficult to explain why this is true you can prove it through very dense mathematics but boiling the essence of the proof down to something people can understand it turns out to be very very difficult and people have tried and as soon as you come up with a clever explanation someone who really understands the proof says well here's a counter example to your proposed explanation so I'm not going to try to prove it now I'm gonna try to give you reason behind it the proof what I'll do is what everyone else does and I will provide you with an argument that will make you think it's not surprising okay so all I'm trying to do here is to make you think that it is plausible there's a relationship between this fact which has to do with the interchange of two particles and this fact which has to do the rotations of single particles right bosons and fermions say what happens when you change two particles with each other interchange spin is says what you doing take a single particle and rotate it there clearly in the math that we did seems to be some resemblance there but they're also clearly independent concepts so how do we relate them to each other so we have a second demonstration look at this two different visual aids in the same big idea video okay so here is and this is again famous I think this all goes all the way back to direct but certainly wheeler talked about this and Fineman and plenty of other people who did this demonstration so here is a belt okay a little strip let me try to get it in the video need to remember to make the video big enough so you can see this but it's a band okay that has two ends and these two ends represent the positions of two identical particles okay so now we have a two particle system and you can see that I'm holding them in a certain orientation so there's no twists in the band I'm not doing anything not nothing up my sleeve it looks like a little bit if I hold them like this but there's no twist in it like that okay what I'm going to do is not rotate either particle but I will interchange them okay so I'm not allowed to do this all right not gonna do that I'm just gonna interchange the two particles with each other boom and what you notice is that now there is a twist in the band that it's a topological fact and I could go back into the topology of that hopefully it's visible against my shirt Det cetera but there is now a twist in the band that you can see it okay and what I want to argue is that I can undo the twist by rotating one of the particles so this this twist got in there when I interchanged the two particles okay so I did the thing that if they're fermions connects them to each other and now I'm saying that by taking one particle the one in my right hand here and by rotating it by 360 degrees I can undo the twist so watch this I do it 180 it's still a twist 360 degrees bang there is now no longer a twist okay if you can see that so what I claimed proved or at least you know hand-wave ly argue in favor of is that for fermions for things that obey the rules of the belt over here interchange the by which I mean the effect on the quantum wave function of interchanging two identical particles can be undone by rotating one particle by 360 degrees that's what the belt is supposed to show I can undo the topological twist I put in the belt by rotating one of the particles not both of them just one by 360 degrees but we know that for fermions interchange picks up a minus sign there for the the fermions are going to be particles where when you rotate one of them you better pick up a minus sign when you rotate one of them by 360 degrees and guess what that is I hope I wrote this down yep I don't know if I did write this down yes there we go the the field which has the property that when you rotate it by 360 degrees you pick up a minus sign is the spinner is a spin 1/2 particle so therefore again don't don't think about this too hard you know just be inspired by it don't don't lean on it in times of trouble therefore fermions are spinners I'll write spin one-half and which I mean I'm going to include three-halves I didn't talk about spin three houses cetera but you could do the same thing for those that's the spin statistic theorem see that wasn't so hard also the completely bogus proof that is that it's offered you but that's the spirit of it it really is that that's the kind of argument that goes into it that's the ultimate reason why deep down it is and that in turn is the reason why I can build a table out of atoms because atoms get their size and shape from electrons electrons are spin 1/2 therefore they are fermions therefore they take up space it takes too much energy you squish them that is why tables are solid so this again you know just to again take things back to previous videos there's this pernicious idea out there that atoms are mostly empty space that can't be true if atoms for mostly empty space tables would not be solid like if atoms were really little solar systems you could totally squish them now the reason you can't is because atoms are not empty space because the electrons wavefunction defines the size of the atom and it's not empty it has a wave function there and it's taking up space and something cannot squish right on top of it there are subtleties here of course because what I proved I didn't have anything at all but what I what I said was the exclusion principle that you can't have two electrons in exactly the same quantum state what I'm deriving what I'm implying from that is that if you get them close to the same quantum state they push back in a kind of force in fact this is well in certain contexts this is known as Fermi pressure in some cases it's even fermi degeneracy pressure if you're in a neutron star or white dwarf or something like that but this is the pressure you get from fermions pushing back on each other because of the exclusion principle and people want to say you know is it is this a new force of nature does this count and as I as I alluded to in a previous video who cares is the answer to that it's perfectly you know that the idea that we separate the things going on in nature into particles and forces an antiquated idea should have gotten rid of that you know decades and decades ago well we have our quantum fields obeying the equation of motion which is ultimately the Schrodinger equation with the Hamiltonian for the standard model of particle physics ok so it's a forced like thing the Fermi pressure it is the thing that pushes back on you when you pushes your hand when you push your hand on at the table but what I like is you know it's just another example a really nice example of how these abstract ideas in quantum field theory filter up into the everyday world we've sort of this is another payoff video in the sense that we have closed the circle it was many videos ago we talked about the fact that smart physicist in the year 1899 could have thought that soon we would understand all of nature in terms of particles and forces acting on them we know better now it's all just quantum mechanics in one big wave function but that wavefunction can be thought of as describing fields that are either fermions and bosons and there's a close relationship between the fermions and quantum field theory and the matter particles that are 18th century physicists would have guessed and the bosons in quantum field theory and the forces so the universe does ultimately make sense if not always for the reasons you might initially have guessed

Channel: Sean Carroll

Views: 73,003

Rating: 4.9143877 out of 5

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Id: dQWn9NzvX4s

Length: 69min 42sec (4182 seconds)

Published: Tue Jul 14 2020

Reddit Comments

[u/Valdagast]

So something can have mass and yet not be matter?

There's something that feels deeply wrong about that.