The Mystery of Spinors

hey y'all today we'll be talking about Spinners these things are really cool once you learn about Spinners your life will never be the same you'll develop a lingering Justified suspicion that this world is not as it seems and you might also develop a lifelong addiction to physics so you've been warned to set the tone I'd like to start off with a quote by the renowned mathematician Fields medalist and Knight sir Michael AA no one fully understands Spinners their algebra is formally understood but their General significance is mysterious in some sense they describe the square root of geometry and just as understanding the square root of1 took centuries the same might be true of spinners Michael was an extraordinary mathematician and if he of all people says that no one fully understands Spinners then it's okay if we're a little confused when we first learn about Spinners Spinners are confusing frankly they're disorienting and not particularly intuitive you know if you tell someone there's a thing that you have to rotate twice to get back to where it started uh they might think you're a little insane but really the insane thing is that Spinners are not insane they're mathematically coherence they show up in our most fundamental laws of physics and they refuse to go away Spinners are really at the core of fundamental physics I mean look at the wave function of an electron for example or any Fon for that matter if not for Spinners we wouldn't have the poly Exclusion Principle and so we wouldn't have chemistry matter would collapse and the world would be a very boring place so there's no way of restructuring physics in a way that you don't have spinners and therefore these strange weird little mathematical objects we do actually have to deal with and we have to work with and we have to understand where does this come from what does it mean why do these things exist and it's really mysterious okay but what is a spinner hm well in the last few videos we've been developing some of the basics of relativistic quantum physics we've looked at Klein Gordon we've looked at the D equation we've looked at an example of D equation for the igen state of a particle at rest and while we've done that we've seen that Spinners have emerged as these sort of two component complex numbers but today I don't just want to Define Spinners as a pair of complex numbers because that's not particularly geometrically insightful instead I want to show you an argument for why we should expect something like Spinners to exist in the first place and we'll do that by exploring a very subtle thing about the way objects rotate in three dimensions that'll open up a crack in reality that we can use to smuggle Spinners into our imagination then after that we can explore the algebraic properties of spinners we can see where the complex numbers come in we can see how we can draw Spinners as Flags we can look at how they transform under rotations and finally we'll tie it all back into the derac equation and we'll talk about how this relates to electrons this video will hopefully answer many questions and it'll surely raise many more Spinners are are one of those things where the more you learn about them the more wonderful and magical they become before we talk about rotations there are a few vocabulary words that we need to borrow from topology these are homotopic Simply Connected and homotopy class so imagine we have a whiteboard and we've drawn a squiggly loop the loop is alive and sensient and is feeling its way around the Whiteboard as it goes along in each moment it takes on a different shape but it doesn't break apart it can stretch and Shrink but it can't be cut and from one moment to the next its topology stays the same the fancy way to say this is that it's evolving into different Loops that are all homotopic homo is in same and topic as in topology so homotopic means that they have the same topology oh whoa little guy's getting nervous H that's okay stage fright you know it's understandable oh there it goes shrinking down to a point it's a defense mechanism that's fine you know I'm a bit shy myself so who am I to judge anyway the reason our Loop was able to shrink like that is that there wasn't an obstacle or a hole or something weird that got in its way instead it just had a nice smooth simple space so it was able to shrink on down to a point and that is our next vocabulary word Simply Connected a space is Simply Connected if all the loops that live in the space can shrink down to a point whenever they want to but now what if we take our whiteboard and we drill a hole in the middle so now as the loop explores its environment it goes around and it realizes that it can't go into the hole and it can't escape the board but it can still shrink if it wants to so no problems there but now consider this little guy which was born into quite predicament it's wrapped all the way around the hole so it can't shrink down to a point when it's feeling shy at best it can only become a small circle not a point so that tells us that this space whiteboard with a hole in it is not Simply Connected so now think about this the loop on the left can evolve into a variety of different shapes that are all homotopic to each other all the different forms that it can take on are all a loop that lives between the hole and the edge of the Whiteboard likewise the loop on the right can also evolve into many different homotopic shapes but those are all wrapped around the whole and so you see here we have two different homotopy classes class in the sense of like a set of entities that all have something in common so we can say that the loops fall into different classes depending on whether or not they wrap around the hole and actually it's a bit more complicated than that because in this case you could also have loops that wrap around the hole multiple times and so we actually have a different homotopy class for each number of times it wraps around and so the winding number matters the number of times that the curve is wound around the hole that matters but those are all the homotopic classes of loops on the Whiteboard uh you can organize them just by specifying the so-called winding number and a curve that has a certain winding number cannot be continuously deformed into having a different winding number so you can stretch them and Shrink them but you can't change the winding number unless you cut the loop which is not allowed this hurts the loop do not hurt the loop okay by the way in this context the loop can pass through itself so we don't have to worry about the loop getting tangled up in knots and uh what are the different kinds of knots and all that no no it doesn't matter the purpose of these Loops are not to model like physical rubber bands but just to tell us something about the nature of the space in which they live oh one more thing on this topic imagine we repair the Whiteboard so there's no more hole but now it has the magical property that the left and right edges of the board are identified with each other such that the loop can appear to teleport from one side to the other so there's no longer a hole in the board but there is this weird teleporting situation so now is this space Simply Connected to address that question we have to think about what it means for the curve to teleport like this from one side to the other without breaking that means that each point on the left side of the the board and the opposite point on the right side of the board must be literally the same point so we realize that this space is actually the surface of a cylinder without the top and bottom circles and so from that point of view no the space is not Simply Connected because in addition to Loops that can shrink to a point we also have the possibility of Loops that wrap around the cylinder and those cannot shrink to a point now when we look at those loops on the Whiteboard they look like weird teleporting lines not Loops but when we're looking at the three-dimensional view that really shows the nature of the space without any cuts and teleports but we see the real space the real for real like how it actually is then we can see that indeed this is a loop and it cannot be shrunk to a point by the way here again it's the same exact situation as before where we can also have different winding numbers and it turns out that all the homotopic classes of loops on a cylinder can be organized simply by their winding number it's not a coincidence that the hole in the Whiteboard And the cylinder have the same set of homotopy classes because if you think about it a whiteboard with a hole in it well that's a CD and a CD is a lampshade and a lampshade is a cylinder so therefore for whiteboard with a hole in it is a cylinder I love topology you know the difference between geometry and topology in Geometry you take your ruler and your protractor and you draw lines and angles and shapes in topology you take your ruler and your protractor and you throw them out the window okay so now we know about homotopy Simply Connected and homotopy class we also know that if a space seems like it has teleports from one side to the other this might actually be a totally continuous space with no teleports but embedded in a higher Dimension such that you can actually pull those points together now we're going to apply these Concepts to the group of rotations in three dimensions but first we have to specify what even is the group of rotations in three dimensions let's say we have a cube and let's go ahead and stick a flag in it to make it really easy to see how the cube is oriented now let me ask you this if we rotate this Cube what are all the different ways it can be oriented and how can we represent that set of all possible orientations well there's a couple different ways of going about this and they're all basically the same but probably the easiest way is to notice that every orientation can be defined with some axis of rotation and some amount of rotation around that axis and this is known as the axis angle representation so what this means is that let's say we have some reference flag that's say pointing up and in a certain direction then we can get to any other flag orientation just by rotating the reference flag some amount around some axis this fact is not obvious but it is true uh it follows as a direct consequence of Oilers rotation theorem okay now let's introduce the concept of an axis angle rotation Vector so what this Vector is is it points in the direction of the axis of rotation and its length is the amount of rotation so here I'm showing a cube rotating around a single axis and you can see that as the vector grows and shrinks the cube is rotated more or less around that axis and so if you think about it what is the space of rotations in three dimensions if you wanted to categorize everything how can we possibly come up with some kind of space that encodes Every Which Way a thing can be oriented well it's the space of all directions multiplied by the space of all lengths but it's a little bit tricky because it's not really the space of all lengths you know if you rotate 180° around some axis that's the same as rotating 180 around the opposite axis so actually what you want to imagine is like a sphere where if the vector pokes through the sphere by the way the sphere has radius of 180° or Pi if you want but when it goes through the sphere then it ends up teleporting to the exact opposite side of the sphere and it ends up coming back in if you keep rotating it see cuz what happens here and here we'll just go along a single axis to make it easy to see but as the vector grows and grows the thing rotates it gets to 180 and at that point we're going to map its rotation Vector onto the opposite axis 180 and then we're going to keep rotating and that's going to keep coming in and coming in which is like a lesser and lesser rotation around that opposite axis until finally it ends up back where it started and so in summary the space of all possible rotations in three dimensions can be represented as a ball a ball meaning a sphere and also the volume contained within it of radius 180° where every point on the spherical boundary of the ball is identifi IED with the exact opposite Point such that if you go outside of the sphere you seem to get teleported back through the other side of the sphere but remember this is not actually a Teleport there's not actually anything discontinuous happening here um this apparent teleport is just an artifact of the way we're drawing the space in reality the space of rotations in three dimensions ironically doesn't really fit in three dimensions and that's not that weird because if you think about it what we have here is an abstract space that represents every possible kind of rotation so that's a different kind of thing than the space that we're rotating within so it's actually not that weird that this space ends up having this kind of topologically interesting character but a lot of weird things follow as a consequence of that as we're about to see what I want to show you now is that there are different kinds of ways to rotate an object back to its original orientation and that seems weird because you'd think any way you rotate an object back to itself it should all be the same right there shouldn't really be a meaningful distinction between how you go about doing that but that's not actually the case so first let's recognize that if we rotate an object back to its original orientation what that means in our axis angle space is that we're starting with the zero Vector at the origin meaning no rotation then the vector is going to go out and grow in some direction and it can move around and in general it can do whatever it wants to do it can teleport if it goes more than 180 whatever the case may be but in order to return to the original orientation it has to come back to the zero Vector so the space of all possible ways to orient an object back to itself is exactly the space of all possible closed Loops in this axis angle representation there is one caveat here which is that when we look at Loops in the axis angle representation we should think of it only as going around the loop once for reasons that we'll see in a moment and so what we're going to do actually tell you what we're going to take a circular cross-section so not the full three-dimensional Ball but actually just a two-dimensional dis now this is just a slice it's just a cross-section but it's easier to look at than the full three-dimensional thing and what I want to show you about there being different ways of rotating things all of that we can see by examining this cross-section and then we'll see that the same exact Concepts easily generalize into the full three-dimensional space now when we look at these Loops I'm also going to going to use a color map that winds around the rainbow and then comes back to the starting color and what that's going to do is that's going to highlight for us the continuity of the loop and in a similar Spirit I'm also going to put a little dot on the loop that just goes around and around and so when we're looking at teleports the ball is going to help us visualize which parts of the curve are connected to which other parts of the curve but remember each of these Loops only corresponds to rotating an object once around this particular Loop okay and that's important because as we'll see there's a difference between going around once or twice or three times or four times and so there are some subtleties there so in our diagram whenever we have a loop think about it as that represents rotating the object one time around this path in the space of rotations all right well with all that said let's go ahead and start to explore some of the different kinds of Loops we have in this axis angle space the first kind of loop that we'll look at is maybe the simplest kind it's just a small Loop that stays totally within the boundaries of the diagram and this corresponds to a wiggle where the cube never rotates more than 180° around any axis it just sort of Wiggles around for a bit and then returns to where it started as you can imagine this Loop is homotopic to any other wiggle that stays within the boundary and passes through the origin because you can always just imagine stretching it and shrinking it and exploring the entire homotopic class of Wiggles and again these Loops can pass through themselves so we don't have to worry about the question of knots and the different kinds of knots so aside from Wiggles what other homotopy classes might there be well consider a cube that rotates once along some axis so it does a full 360 that means that its Loop is going to start at the origin and then it's going to go out to 180° then it'll come back in along the opposite axis and finally it'll return to the origin again having completed a full 360° rotation now that's a rotation right that's not just a wiggle that's a full rotation and by the way if you look at the Loop it might not look like a loop it looks more like a weird teleporting line but remember the cylinder and remember that the apparent teleportation here is just an artifact of the way we're drawing the space but it doesn't represent any real discontinuity in the space you have to imagine that the space is curvy in a way that doesn't really fit in the dimensionality of your screen and so the points that the curve appears to teleport through are actually literally the same point and in this diagram all pairs of opposite points are literally the same point now this kind of loop is a genuine rotation I mean look it goes all the way around it's way more than just a wiggle it's a full 360 and you can see that it can't be reduced to a wiggle because there's no way of continuously deforming this Loop into a loop that doesn't cross the circular boundary without cutting it okay so then then now we have at least two different homotopy classes of Loops in the axis angle space Wiggles and full rotations so the space of rotations in three dimensions is not Simply Connected which is really surprising right you wouldn't necessarily expect that but here we can very clearly see that there are at least two totally distinct homotopic classes of Loops in the space of rotations in three dimensions and you know that's just a fundamental fact of reality all right well this feels similar to the cylinder example where you had Loops that didn't teleport and you also had Loops that teleported and then you had like multiple winding numbers of teleportations and so we might expect to find infinitely many homotopy classes here for the rotations depending on how many times the loop goes through the boundary but surprisingly that's not the case there are actually only two different homotopy classes consider for example the fact that a wiggle is homotopic to an octopus that's got to be a brand new sentence I don't think anyone's ever said that you're hearing it first here folks freshly minted sentence okay so the wiggle crosses the boundary zero times but the octopus crosses many times and yet the two are totally homotopic as you can see as we're going back and forth between these two you see they're homotopic it's all continuous Transformations here there's no cutting so we don't have one homotopic class for each winding number like we did with the cylinder because now it's a different kind of teleporting boundary the boundary is round and has a different geometry and so the loops can slip around and slip through the boundary and what that ends up doing is it means that you can always add or subtract two boundary crossings by continuously deforming the loop see for another example let's say we have a rotation that teleports once and by the way each pair of points here we're counting as one Crossing notice they're the same color they're really the same point it looks like two points on our diagram but that's an artifact of the way we're drawing our diagram okay so this is one Crossing but you can show that this Loop is homotopic to a loop that teleports multiple times but when you do that with this Loop because it started off Crossing once you can only ever get to an odd number of Crossings and so as it turns out we only have two homotopy classes and they're based on whether we have an odd number or an even number of teleports you can see this by noticing that you can always pull part of the loop through the circular boundary but when you do so you'll always pull two line segments at a time because we're grabbing onto a closed loop and that's just how it is and by the way you can see that this argument generalizes not just to the circular cross-section but also to the whole spherical situation because it doesn't matter whether you're pulling a loop through a circular boundary or a spherical boundary because either way the loop is going to come through two line segments at a time when you pull on it so this thing about the two homotopy classes even though we've demonstrated that for this uh circular cross-section it actually does hold true for the entire space of of rotations uh in three dimensions oh and here let's go ahead and introduce some terminology so we'll say that a loop is class one if it has an odd number of teleports and we'll say it's class two if it has an even number this is easy to remember because one is odd and two is even you know we've now arrived at a checkpoint like in a video game when you get to a certain level and your progress is saved you'll want to make sure that you're really comfortable with all of these ideas discussed so far because we're about to get algebraic and it makes all the difference in the world if you're able to see and to feel in your soul that the rotations in three dimensions are not Simply Connected but actually have two different homotopic classes of Loops I really want to emphasize that this is a fundamental Insight a pivotal transformation of your worldview that you have to go through if you want to study Spinners you even though we haven't used any equations yet there's a lot to think about here and it may take a while to digest these ideas I know it took me a while so it's worth meditating on these Concepts until they become second nature because seeing that the rotations aren't Simply Connected but have rather two homotopy classes is really the difference between your intuition holding you back and pulling you forward if you get this concept about two homotopy classes if you really feel it then instinctively you'll suspect that maybe there might be some mathematical object that is sensitive to the homotopy class of rotations and so it would necessarily have to double cover the space of rotations and it might not exactly return to itself after a single rotation but instead might pick up some phase factor that alternates depending on whether it's a class one or a Class 2 and so if you really get this then your imagination is going to pull you in the direction of spinners you'll yearn for them even before you've seen what they are and you'll be prepared and excited when it comes time to actually write an equation all right let's get algebraic so we have this idea of an axis angle represent ation of a rotation and now the question is how do we actually work with this what do we do with this let's say we have some rotation vector and we want to say Okay I want to rotate an object in accordance with that axis angle Vector what equation do I write what code do I put in the computer what do I do and I want to tell you about two different ways of rotating an object the vector way and the spinner way so I want to focus on the vector way first and this involves rotation matrices and then later I'll talk about the spinner way and this involves su2 matrices so starting with the vector way of rotating things um actually you know what here first we're going to go into two Dimensions I'll show you how to rotate vectors I'll show you the implications for this Visa rotating objects and then we'll expand up into three dimensions but I'll use the two Dimensions as an example to lay out some of the mathematical framework all right so imagine we're in a two-dimensional space and we have a unit Vector pointing along the xaxis and you know a unit Vector along the x-axis is kind of the simplest thing you can put in a vector space so we're going to use that as our reference object because you know no need to over complicate it now in order to rotate this Vector what we can do is we can construct something called a rotation Matrix in two Dimensions the rotation Matrix about some counterclockwise angle Theta has the form cosine of thet sin Theta sin of thet cosine of theta and if you investigate the properties of this Matrix you'll find two things first the inverse of the Matrix is the transpose of the Matrix so if you transpose The Matrix and if you take the inverse that's the same thing and another way of writing that is that the Matrix times the transpose equals the identity Matrix so that's the first thing and that means that the Matrix is orthogonal so when the inverse of a matrix equals its transpose that is an orthogonal Matrix and another thing we can see by inspection is that this Matrix has determinant 1 because you know cosine squ + sin square is 1 so our Matrix is orthogonal and it has determinant 1 and so the fancy word for that is it is a special orthogonal Matrix in particular it's a special orthogonal 2x2 matrix and so what we say is it is an S SO2 Matrix so a rotation Matrix in two Dimensions is also known as an so SO2 Matrix oh by the way even though we're talking about SO2 in terms of matrices the term SO2 can also refer to the group of rotations in two dimensions in a more General sense whether or not we use matrices to represent them for example instead we could represent 2D rotations with unit complex numbers and complex multiplication because the group of e the I thetas under complex multiplication is isomorphic to the group of2 matrices under Matrix multip application and that's a neat little connection there between the complex numbers and the SO2 matrices we'll see in a moment that there's a similar connection between the querian and vs2 matrices but more on that later but for now I just wanted to let you know that there are many equivalent ways to reformulate all of these Concepts but I like the Matrix representation because it's very mechanical and transparent now if we look at this blue thing swinging around we realize that we can use the set of all rotated blue vector to visualize the set of all SO2 matrices because if we Define the unit Vector along X as our reference Vector then each s SO2 rotation Matrix corresponds to one and only one rotated unit Vector so if you tell me hey here's a reference vector and now here's a rotated Vector then by looking at how rotated the rotated Vector is I can figure out what so SO2 Matrix was responsible for the rotation likewise if you give me an SO2 Matrix and a reference Vector then I can draw you the rotated Vector so there's a one: one correspondence here and this is true regardless of what reference Vector we pick no matter its length or orientation which you can prove to yourself by skeptically tilting your head and leaning back so anyway we can see that we can Loop through the SO2 matrices in a periodic way passing through each of them in order of increasing angle and then ending up back where we started and so we can see that the topology of so SO2 is just a circle and that's why SO2 is also called the Circle Group so in two Dimensions we can represent an SO2 Matrix in other words a rotation with just a vector but when we go to three dimensions because of the extra degree of Freedom uh we're going to have to use a flag and the reason we're using a flag is because if you just had a vector in three dimensions then that kind of points in a certain direction but then there's the question of how is the vector rotated along its axis and so with a flag we can see that flag rotation angle and so we can use that to represent the rotations in uh three dimensions but before we get into three dimensions I want to show you how to rotate a flag in two dimensions and basically the way to do it is just imagine the vector that points from the origin to each of the interesting points on the flag so the zero Vector points to the flag pole base then you have a vector that points to the top of the flag pole couple of vectors that point out to the sides of the flag and then another Vector that points along the flag pole to the point where the flag meets the flag pole so basically we just treat those Five Points as vectors and you apply a rotation Matrix to those vectors and as it turns out the whole pattern of the flag will rotate when you do this and that's because rotation matrices preserve the relative lengths and angles of vectors that they're transforming and so any lines drawn between two vectors which is also the difference of those two vectors will transform in the same way as all of the other vectors now here's a view where we offset the flag from the origin so here this really drives the point home that if you have some shape made out of vectors and you rotate all the vectors you end up rotating the shape there's a couple of other Concepts I wanted to touch on while we're here for example what would happen if we scaled up our rotation Matrix so that its determinant were greater than one well now it would no longer be a special orthogonal Matrix but it would still be a matrix and it would still transform a vector field and as you can see from this animation if we take a rotation Matrix and scale it up so that its determinant is greater than one then that's just going to expand all of the vectors likewise if we scale down a rotation Matrix so that its determinant becomes less than one then that's going to shrink the vectors and the reason I bring this up is so that you can see the connection between a matrix having determinant one and a matrix doing a rotation transform because with the rotation we don't want to expand or Shrink vectors we want to keep them the same length and by the way when you're transforming a vector space with a matrix it's not always a uniform expansion or contraction uh sometimes you can also get a skew and a matrix that skews a vector space is not an orthogonal Matrix because it doesn't preserve the relative lengths and angles of the vector field you know it squishes them and so anyway that's another reason why we want to use a special orthogonal Matrix for rotations because they don't skew the vector field oh and also if we go ahead and allow the determinant to go negative you can see that an inversion of the vector field happens and by the way I know this looks kind of three-dimensional but it's actually two-dimensional and it's getting squished along one axis until it flips around and goes out on the other axis while at the same time I'm rotating it but anyway the main thing I wanted to show you there is that an inversion can happen when the determinant goes negative so I hope these examples of growing and shrinking and skewing and inverting vectors gives you some sense of why we want to use special orthogonal matrices to do rotations okay so now that we're experts on so SO2 and rotations in two Dimensions we'll go ahead and pop up into the third dimension a lot of what we just talked about in 2D will transfer directly over to 3D but of course now we're in an expanded environment so some of the concepts will be adjusted accordingly but something I want to emphasize is that all of this special orthogonal terminology gives us a precise way to describe what kinds of matrices we're looking for regardless of the dimensionality of the space that we're in because all of the same intuitions apply in 3D as in 2D about not wanting to grow the vectors or Shrink or skew or invert anything and so what we're looking for now are s SO3 matrices that is special orthogonal 3x3 mat matrices where the same constraints apply about the inverse being the transpose and the determinant being one but now there's just an additional Dimension and so the essence of special orthogonality is the essence of rotation and yes by the way this also applies in higher Dimensions but we'll save our tears for another day so uh anyway earlier we saw that given some rotation angle Theta we can construct the 2D rotation Matrix as cosine negative s sin coine but now in three dimensions instead of just an angle for the rotation we now have an axis angle Vector like we talked about earlier and so we're faced with the question of how do we write the SO3 Matrix that goes with any particular axis angle Vector this is actually quite a difficult problem but don't worry it was solved in 1840 by Al Linde Rodriguez and the solution is called the Rodriguez formula the Rodriguez formul formula is just a recipe for converting an axis angle Vector into the3 Matrix that does that rotation so you take an axis angle Vector put it into the Rodriguez formula and out pops your s SO3 Matrix and then you take that S3 Matrix you just slap it onto a vector and it rotates the vector and that's how that works that's how you rotate vectors in 3D it's pretty neat So in theory in principle according to everyone's intuition and Imagination that should be good enough if we want to represent rotating any object you should totally be able to just list out a bunch of position vectors of points on the object rotate all your position vectors with special orthogonal matricies and then that should be like all you need to rotate things in math and physics like you should only ever have special orthogonal matrices for rotations like we have it already it's good enough right this is fine well not exactly we got to talk about su2 su2 I don't even know where to begin with su2 well here how about this we'll start off by thinking about su2 as a complex generalization of s SO2 let me clarify what I mean by that so think about an SO2 Matrix as we've seen you can write an SO2 Matrix as cosine Theta minus Sin Sin cosine or equivalently we can say that the general form of an SO2 Matrix is a b ba a with the constraint that A2 + b^2 = 1 and you can see that this is the same thing by the substitution a is cosine of theta and B is sin Theta and uh when you write it in terms of a and b and a s + b s = 1 you see that we have a unit circle of possible SO2 matrices okay now with that in mind let's look at the definition of an S U2 Matrix so an su2 Matrix involves complex elements will symbolize them as Alpha and beta and it has the form Alpha negative beta conjugate beta Alpha conjugate where Alpha and beta are complex numbers and the magnitude of alpha squ plus the magnitude of beta squ equal 1 you can see that in the special case that Alpha and beta are real numbers then this Matrix has the form of an S SO2 Matrix but because Alpha and beta can be complex numbers su2 is actually a more General kind of Matrix than an so SO2 Matrix it has more degrees of freedom and whereas an so SO2 Matrix acts on real valued two component vectors an S U2 Matrix being complex naturally is going to act on complex valued two component vectors those complex valued two component vectors are called Spinners and this is one way of thinking about what a spinner is it's the vector like thing that an su2 Matrix acts upon by the way there are ways of generalizing the concept of spinners using other kinds of matrices but I'm not going to worry about that today today we're just focused on two component complex Spinners which are the most common kind of spinner you'll encounter in physics okay now something you can see by exploring the properties of these su2 matrices is that they have determinant one and that the inverse of the Matrix is the complex conjugate transpose of the Matrix so this is very very similar to a special orthogonal Matrix but technically the correct word now is a special unitary Matrix where unitary refers to the fact that the inverse of the Matrix is the complex conjugate transpose of the Matrix and that complex conjugation comes in because the dot product of two complex vectors involves a complex conjugation and so in order to preserve lengths and angles between complex vectors we have to preserve the dot product and so there's a complex conjugation that comes in in order to preserve dot product but anyway I'm not going to go into too much detail on that just know that these are the constraints on su2 matrices okay so what we've looked at so far is a very algebraic definition of su2 and what we should look at now is the picture of what do these matrices actually do so the way to do that is to First draw some reference spinner and this will be analogous to the reference Vector we looked at when we looked at s SO2 and then we're going to multiply it by an su2 m Matrix and we're going to see how it transforms so first of all how should we even draw a two component complex Vector well there's a lot of different ways of drawing it but today I'm just going to keep it simple we're going to look at one complex plane and then I'll put two dots in the plane corresponding to the two components of the complex vector and I'll draw a line between them just to highlight the point that these two dots are actually two parts of the same object there's another way of looking at all this math in for Dimensions but we're not going to get into that today anyway so when we multiply our reference Spinner by su2 matrices we can see that it kind of goes around and it it spins and it transforms in this rotational kind of way and so naturally we can ask the question of what are the degrees of freedom here as we're Transforming Our spinner how does it evolve and how many degrees of freedom does it have well naively it should have four degrees of freedom right because there is two complex numbers and each complex number has two degrees of freedom in the complex plane but we're going to take out a degree of Freedom by virtue of the fact that the su2 Matrix has determinant one and is unitary and so what that means is that the spinner is not changing in size and that means that the magnitude squar of the first component plus the magnitude squar of the second component is always going to equal one so we actually have three degrees of freedom here and these three degrees of freedom them are isomorphic to the hypersphere which is the locus of points in four dimensions that are all equidistant from the origin which you can see when you think about it right you start off with four pure degrees of freedom and then constrain them so that they have to be constant size what you end up with is the hypersphere interestingly the hypersphere is a Simply Connected space that'll be really important later but for now just think of these Spinners as rotating around and spinning around in a way that is Simply Connected okay so here we have three degrees of freedom and it involves some kind of spinny rotating thing so the question arises does this have any connection to S SO3 that is the threedimensional rotations maybe it will maybe it won't right I mean if we're just thinking about it from a mathematical perspective that's a legitimate question to ask and as it turns out shockingly there is a very very close connection between su2 and S SO3 and in particular su2 double covers s SO3 let me explain what that means so imagine we have S SO3 which acts on some three-dimensional vector and rotates it and we also have su2 which is going to act on some two component complex Vector that we call a spinner and it's going to transform it and kind of spin it around what the double cover means is that we can draw a connection between any possible rotation in three dimensions and any possible su2 transformation on these Spinners in this two complex dimensional space but the connection is not 1: one it's actually 2: one for every rotation in three dimensions we have precisely two possible su2 Transformations and moreover those two su2 Transformations are related to each other in that they only differ by a minus sign so you know at the beginning of the video I mentioned that quote by Michael AA where he was talking about how Spinners are in some sense the square root of geometry and you really see this in the way that su2 double covers s SO3 cuz if you think about it what is the square OT of 4 well it's two but it's also -2 and what is the sare root of 9 well it's three but it's also -3 so when you take the square root of something you get an answer but you actually get two answers that differ only by a minus sign likewise if you look at any rotation in 3 Dimensions which corresponds to a single SO3 element and you ask what is the corresponding su2 element you get an answer but you actually get two answers that are related by a minus sign okay now let's talk about the spinner way of rotating things so because su2 double covers s SO3 there's an intimate connection between every su2 transformation and every rotation in three dimensional space therefore if we wanted to we could mathematically deal with rotations using the framework of su2 matricies instead of s SO3 but we would just have to ignore the redundancy that comes with a double cover but that's not really a problem I mean we can literally just ignore the redundancy right we can just project it out let me give you an analogy so suppose that there is a math problem and you have some computer program that calculates the answer but it's kind of slow and it takes a while to run so you run your program you wait a moment while it calculates and then let's say you see that the answer is four well okay if you're not in a rush that method works fine but let's say You're really in a hurry or maybe you have to do a bunch of these calculations super fast so you look for a way to do it faster and then you discover a faster method but the catch is that it doesn't give you the answer but instead it gives you the square root of the answer so if the answer is four then sometimes the fast method gives you two and sometimes it gives you -2 okay fine that's actually not a problem at all because you can just Square it and the ambiguity washes out and you've got your answer the same reasoning applies to using su2 to do rotations because it turns out that su2 lends itself to faster more numerically stable calculations than S SO3 and even though there are two possible spinners for each s SO3 orientation they only differ by a minus sign which can easily be washed away don't get me wrong SO3 is pretty fast too I mean I used like a million SO3 matrices while making this video but then again I also rendered these animations overnight I wasn't in a hurry so I just took the easier route in terms of thinking about SO3 and axis angle vectors and Rodriguez formula and that worked fine the only time I really had to use su2 for 3D rotations is when making these colored flag animations those are actually based on su2 because I wanted it to match the Spinners anyway with animation there's not really a time constraint so you might as well just think in terms of axis angle vectors Rodriguez formula SO3 rotation matrices because it's easier on the brain and uh usually there's no benefit to rotating a vector like a microsc faster so why torture yourself if you don't have to but if you were riding software for video game Graphics or a vision system for a self-driving car or the attitude control system for a rocket then you'll for sure want to use su2 because in those cases you're going to have to rotate a lot of vectors very quickly and the only cost of using su2 is that it's harder to think about when you're writing the code because your imagination has to roll around on a hypersphere and it's kind of dizzying but at the end of the day you'll end up with a code that's faster uses less RAM and is more Zen than a code that uses s SO3 the numbers flow more gracefully when you use su2 su2 also lets you avoid the problem of gimbal lock which affects SO3 I don't have time to go into it today today but Google Apollo 11 gimbal lock for a cautionary tale about the Perils of SO3 also when using su2 for rotations in three dimensions people often use the querian representation because it's clean and simple we saw earlier that there's an isomorphism between s SO2 and the unit complex numbers and similarly there's an isomorphism between su2 and the unit querian and by the way this isomorphism makes manifest the hyperspherical topology of su2 so anyway whenever you hear people talking about a rotation algorithm that uses su2 or Spinners or querian it's all the same the basic idea is that you're doing your rotational math in this weird Simply Connected hyperspherical kind of redundant group that somehow lives like kind of underneath the vector rotations it's uh it's it's very bizarre but it works somehow it works and so su2 is actually used ubiquitously in applications that demand Max maimum performance and so su2 is more than just some fanciful abstraction it's actually very practical and you know it'd be one thing if it were just practical it'd be one thing if su2 was just like a more efficient way to do rotations and you just ignore the spinner phase fine I mean okay that's cool but that's not particularly profound but the thing that's really a trip is that in theoretical physics the spinner phase actually matters so su2 is not just another way of doing rotations but that redundancy that you wash out if you're just doing like video game Graphics that actually is something that's relevant to the wave function of a relativistic electron for example and as we'll see later in the video that is a very very deep mystery as to why that's the case Okay so let's look at one more double cover animation and then we'll talk about the physical mystery of spinners so here we have a very very complicated rotational path and it's all like and it's all like swooping around and doing all these moves and the point I want to illustrate here is that even when the rotational path is super complicated there's still this connection between the flag rotating an S SO3 and the spinner rotating with an su2 Matrix there's still that 2:1 connection such that if we go around a class one curve we end up with the phase switched then if we run that class one curve again Bring It full circle we end up doing a class two and we end up back to where we started so let's take a closer look at this and we can see exactly how that happened okay so first it comes in it's like and then look it's about to teleport right there do you see that okay cool let's keep it going now it comes around and look we'll pause here okay now it's come back to where it started but the phase is reversed as indicated by the fact that the flag is red and also if you look at the plot you can see that the spinner is the negative of what we started out with all right and continuing it's going to swoop around and it's going to teleport right now and and it's going to come back through and it'll endend up back where it started so we've done two class ones and that's a class two and that brings us back to the starting point isn't this neat I think it's neat the thing I really want to impress upon you is that su2 double covers s SO3 and it's like a perfect double cover it's like a hand in a glove or like I guess a hand in two gloves you know so like that's really cool you know like what what does that mean right I mean the algebra I think as you can see the algebra is formally understood I mean look at all the numbers on the screen if you pause it at any moment in time you can check for yourself that the matrices are right and you know all the equations we've looked at that's all true it's all good but why like what does this mean you know the funny thing is when making these videos the equations are the easy part because an equation is just a recipe and I can put it in the computer and make an animation and I can just tell you how it is and how it works the hard part is conveying the sense of Mystery I mean aside from the fact that it's just kind of confusing and like wo what are these things but how can I share with you that sense of awe and wonder that so many people have when they work with spinners especially in the context of physics and you know that's much more of a storytelling problem that's the objective there is to really share a state of mind or an attitude and so I just want to tell you about it from my point of view so please take everything I'm about to say with a grain of salt this is just my perspective but I hope it resonates with you as well okay so you might have heard of the five stages of grief well I like to think that there's a similar five stages that you go through when learning about Spinners uh hear me out so except it's not as bad instead of anger it's astonishment and instead of depression it's existential shock now I don't think anyone's ever gotten to the fifth stage but I can tell you about the first four so starting off with the denial stage if you tell someone that there's something that you have to rotate it twice to get back to where it started they're just going to tell you no you're wrong like that can't be like have you ever rotated an object everyone knows all you need is to do a 360 rotation in any axis and then you get back to where you started this is like a basic fact of reality and look I get it I mean I understand I've rotated objects too I've actually personally rotated back in the days when I used to jump out of planes I spent a lot of time rotating atmospherically so I'm not naive when it comes to rotations I've been around a time or two sorry I couldn't resist the pun anyway if there's one thing that I know and you know and everyone knows it's that a rotation of 360° is all you need to get back to where you started you don't need a 720 rotation so what is this spinner stuff it makes no sense so that's the denial stage and what breaks you out of the denial stage is when you become really familiar with the math so when you get to know SO3 and su2 and you see that su2 double covers s SO3 you see that s SO3 is not Simply Connected but su2 is and so now you have two different ways of looking at rotations and one of them is kind of more elegant but in some sense more redundant than the other and it's sort of like okay there's something here now when you become familiar with the math it's very astonishing so I like to call this the astonishment phase and this is when you might be very curious about the math and you might study it a lot more you might explore it you might write some python codes you might investigate Every Which Way You Can rotate these different things and you might make pretty plots and it's very wonderful and beautiful and you find that there's a very rich structure here that you can spend a lot of time exploring and after having done that there's no way that you can go back to the denial stage but even still you feel the need to contextualize the math you feel the need to say okay but what does it all mean and how literally should I take it and what are the physical implications if any and the part of you in your heart is not going to want there to be any physical implications I mean you want to think it's just the product of a mathematician's overactive imagination you don't want to think there's anything in the world that actually does have to rotate twice to get back to where it started because that's deeply unsettling and so we enter into the bargaining stage where we say okay fine mathematically sure yeah Spinners are a thing but it's not physics this isn't real this isn't like tangible and so you can sort of argue the math in physics and say no it's just a useful Fiction it's just the structure of the equations but it doesn't have any physical implications and the the problem is that is a losing battle because the more you learn physics the more prominent and essential Spinners become you know for example look at the durac equation look at the wave function of an electron it's a b spinner field and it's this thing that's written in terms of spinners and it's at the heart of quantum physics so okay but you still want to bargain some more you still want to say well our understanding of physics is incomplete and so the derac equation maybe there's a way of writing it without Spinners so you look into geometric algeb and these different ways of writing the D equation and you find that nope no matter what you always have spinners or something equivalent to Spinners and so then you say okay but maybe the D equation itself is wrong in some sense and then so okay it's like what is the D equation well it's three ingredients it's Loren in variance it's the quantum mechanical energy momentum operators and it's the requirement that the equations be first order in space and time now to the first one of those things no one has ever observed a violation of Loren and variance and in my opinion in my lifetime probably no one ever will now do I think that Loren invariance is literally true at every scale no personally I don't but I also don't think we're ever going to be observing any deviations from it anytime soon so if you want to try to undermine the importance of spinners in physics by arguing against lenen variance it's not going to work out likewise with the quantum mechanical energy momentum operators I mean there's really not much room for flexibility here because they describe so well pretty much like everything I mean you want to look at the black body Spectrum you want to look at like the hydrogen energy levels so you're not going to be doing away with the quantum mechanical energy momentum operators but if you look at the third ingredient the fact that the equations have to be first order in space and time here we do have a little bit of leeway because with the benefits of historical hindsight we can say that dck was a little bit misguided in his original approach you know I think it's fair to say that he was right but perhaps for the wrong reasons or maybe better uh for reasons that were not fully appre appreciated or understood at the time anyway with the development of quantum field Theory we now know that you can have particles that aren't Fons and so there's no reason for us to think that the electron must be a firmon it just so happens that they are and it just so happens that the way to write that way function is to say that it's a first order equation in space and time take the square root of the mass shell Spinners pop out you know that whole thing so it's right but it's not obvious that it's like mathematically necessary for an electron to be a firon aha so the way we're going to bargain out of having to deal with spinners is to say that the electron is not actually a firmon instead we can just choose to model it as a spinless particle right so instead of a by spinner field we could just use a complex scalar field for its wave function sort of like schinger or Clin Gordon and there are kind of ways to do that if you're dealing with one electron and they okay making an approximation but the problem is if you take that approach while you're trying to deal with multi-electron systems like if we're doing quantum chemistry or condensed matter physics then that approach doesn't work because we get slapped in the face with something called the spin statistics theorem and it's such a hard slap that we get knocked out of the bargaining phase and into the existential shock phase the spin statistics theorem deserves its own video and actually for now I'll just say there's an excellent book written on it called poly and the spin statistics theorem by Ian duck and ECG Saran it's a very interesting book and I highly recommend it if you want to good overview of the history of this theorem and its profound implications and what is known about it and what is still not known about it but anyway for today we just have to know what the theorem says in a nutshell it says that particles with half integer spin are Fons and their wave function will pick up a minus sign when two particles are exchanged and that leads into FID derac statistics on the other hand particles with integer spin are bosons and their wave function will remain the same when two particles are exchanged that leads into Bose Einstein statistics in short the spin statistics theorem says that there is a direct and mysterious link between the minus sign or lack thereof picked up by a particle under 360° rotation and the minus sign or lack thereof picked up by a multi-particle wave function under the exchange of any two particles this fact is going to lead to radically different Behavior depending on whether you're dealing with an ensemble of Fons or an ensemble of bons bans don't mind being all up in the same Quantum State at the same time it's all good but ferons will never share the same Quantum State they do not like to be together there's a principle that every high school chemistry student learns called the poly Exclusion Principle it says that no two electrons can occupy the same Quantum State at the same time this is a profound principle with far-reaching consequences it plays an absolutely fundamental role in chemistry it's the reason for example that atoms have different orbitals that the electrons have to stay in you know cuz you might wonder why isn't it that uh an atom at rest why can't all the electrons collapse into the ground state together what is it that keeps some of them propped up into higher energy states well it's the poly Exclusion Principle you know the lower orbitals are already filled up so chemistry students take for granted is sort of one of the axioms of chemistry but with physics we go a bit deeper and right now we're actually logically one level underneath the poly Exclusion Principle because it follows as a consequence of the more fundamental spin statistics theorem to see this just consider a system of n electrons whose multi-particle wave function can be written as s is a function of X1 X2 dot dot dot x a dot dot dot XB dot dot dot xn where X subn in this context refers to all the quantum numbers of the nth particle including spatial coordinates and spin now suppose we switch electrons A and B according to the spin statistics theorem because electrons are firion the new wave function is just the minus sign of the original one so s of the original quantum numbers is going to equal negative s of the same quantum numbers but with the numbers for A and B switched around so all we've done so far is basically just stated generically what the spin statistics theorem tells us about Fons but suppose now that we force electrons A and B into the same Quantum state with the power of imagination then by the symmetry of what it means for two things to be the same we conclude that there is no difference between the original and swapped wave function by definition there can't be if a and b are the same state then you switch them nothing can change right because they're the same well now here we have two apparently contradictory things when particles A and B are exchanged the wave function both does and doesn't pick up a minus sign how can that possibly be because we're saying that s has to equal negative s well that can only be the case if s is precisely zero and because s is a probability amplitude we conclude that it is absolutely impossible for two Fons to occupy the same Quantum State at the same time there is zero probability of that happening and that is the poly Exclusion Principle notice that this argument does not apply to bons with bons there's no minus sign picked up under particle exchange and so there's no logical tension when you put two particles in the same state switch the identical particles around all you want it's all good and that's why the poly Exclusion Principle does not apply to photons for example photons are bosons and they can share the same Quantum State no problem and in fact they often prefer to do so you can see that the spin statistics theorem is quite profound and plays a major Ro in shaping the world all around us but why is it true where does this all important principle come from well we know that the spin statistics theorem is true every moment is bursting with an abundance of empirical evidence supporting the theorem as electron orbitals continue to exist and photons continue to be able to occupy the same state Additionally the theorem follows logically from the assumptions of quantum field Theory and it's been proven rigorously in that context but despite all that nobody really understands in a geometric or visual way why there is this mysterious connection between Spin and particle exchange if you think you do understand it because you've seen fineman's topological argument or some similar intuitive proof read chapter 20 of the spin statistics book long story short those so-called proofs are actually analogies without any explanatory power but they are fun to think about okay fine now there is one last battle where we try to work our way back into the bargaining phase and that is to say well maybe the spin statistics theorem is not actually logically underneath the poly Exclusion Principle maybe it's just some kind of cosmic coincidence so maybe spin doesn't play a fundamentally consequential role in the spin statistics theorem but instead maybe it just so happens to be that the particles in our universe that happen to be Fons happen to obey the poly Exclusion Principle whereas bons do not now this is unlikely but we're really grasping at straws here to try to retain some semblance of our natural tuition you know we don't want to feel like the world is much more bizarre than it seems but but it is but anyway the final nail in the coffin is to say okay look what if we take an electron and we change its spin so that it no longer is a half integer kind of thing but actually behaves as a bon then if that electron is set free of the poly Exclusion Principle we would know that we actually are logically underneath the poly Exclusion Principle right now because that experiment would give us a definitive proof that the poly Exclusion Principle actually does arise from the spin statistics theorem in a really meaningful way now it's actually not super easy to convert electrons into bosons but it can be done in some materials because of the way electrons mutually interact with the crystal lattice you can occasionally get a spin up electron pairing up with a spin down electron and that pair of electrons has a net spin of zero and so the pair of electrons is collectively a bosonic entity so in the material you can have a great many of these and they're able to condense into something that's spiritually very similar to a Bose Einstein condensate and they genuinely are occupying the same state at the same time which electrons hardly ever do but when you convert them to bons all we're doing is modifying their spin we're not changing their charge we're not taking away any mass or any of that we're just pairing them up so the spin cancels out out and now we set them free from the poly Exclusion Principle they condense into a very low energy State and can slip through a material like a ghost without interacting with the atoms along the way and this is the phenomenon of superconductivity and so superconductivity gives us an undeniable proof that the spin statistics theorem is genuinely underneath the poly Exclusion Principle by the way we'll have a lot more to say in the future about superc conductivity it is an incredible phenomenon and you know I've personally seen what a superconductor can do my colleagues and I have sent enormous electrical currents through superconductors many thousands of times the intensity that would melt copper and I've watched it stay cool as a cucumber it doesn't heat up at all because all that current is being carried by these super electrons that are behaving as bons and so it's not even like regular physics like we know and love it it's like this weird surreal like crazy version of reality where the poly Exclusion Principle only you know doesn't always apply to every electron it's amazing it's a wonderful thing so we'll talk about that a lot in the future we'll get into Ginsburg landow and flux quantization and all that good stuff but anyway today I just bring it up to highlight that this whole spinner thing you know it's real it's mysterious and if you take away the spinner nature of the electron wave function you'd get some pretty bizarre and surreal effects that actually are real this is not just Theory this is not just socres philosophies and hypotheses no this is like real like technology like engineering people are building deves with this stuff so you know so that's something to think about anyway is it possible to reach the acceptance stage and to really understand Spinners in a way that would satisfy even Michael AA maybe but we're not there yet among other things the lack of an intuitive but rigorous proof of the spin statistics theorem shows that we don't fully understand the significance of spinners in physical reality we're aware of spinners we're able to work with them we know how their algebra works but we still do not know why spinner valued wave functions are anti-symmetric under particle exchange now you might be thinking Rich come on now you're just dramatizing your own ignorance this isn't actually a mystery you just don't know the answer well it's true that I don't know the answer but I'm not the only one who thinks that this is a mystery let me share with you this quote from the spin statistics book this quote is in a passage about whether or not physics can explain the spin statistics theorem or is just consistent with spin statistics we should modify the meaning of understand and at the same time reduce our expectations of any proof of the spin statistics theorem what is proved whether truly or not whether optimally or not in an acceptable logical sequence or not is that the existing theory is consistent with the spin statistics relation what is not demonstrated is a reason for the spin statistics relation to belabor the point it is difficult to imagine a fundamental mechanism for the poly Exclusion Principle upon which all depends which predicates it upon the analyticity properties of vacuum expectation values of products of quantized field operators did God for lack of a better word build a series of failed worlds which sputtered and died or exploded and disintegrated before discovering the stabilizing effect of anti-commutation relations for half integral spin Fields was this before or after imposing the requirements of Loren and variant are we the lucky winners of a Monte Carlo simulation in which every choice was tried and one survived must we reduce our demands on physics to require only consistency does an understanding of the why of the spin statistics relation have no direct answer in physics or must physics be formulated to include it the poly result does not explain the spin statistics relation and cannot the noinch wanders and the findin of the world must remain unsatisfied because the consistency of relativistic quantum mechanics and Quantum field Theory with the poly Exclusion Principle has every reason to be as complicated as these subjects are not as simple and direct as the poly Exclusion Principle itself so in short there uh there sure is a lot of mystery here and it's confusing but it's kind of cool to think about right but you know honestly I can only think about spinners for so long before it starts to hurt it's like diving underwater you try to go as deep as you can but at some point you have to come back up for air and it's all so disorienting that sometimes when you stop thinking about Spinners it's all so confusing and you feel like you're right back to square one but my friends if there's one thing that Spinners have taught us it's that just because you returned to where you started doesn't mean you have and picked something up along the way so in that Spirit I'd like to reiterate part of the quote from the beginning of the video no one fully understands Spinners their algebra is formally understood but their General significance is mysterious I hope today I've given you some sense of the algebra but most importantly an appreciation of the mystery if you enjoyed watching this video as much as I enjoyed making it then please consider supporting my channel on patreon on there you'll be able to download these beautiful magnificent PDFs with the equations and diagrams discussed in this video and other videos too also if you sign up on patreon your input will help determine the future direction of this channel I know what I'm going to do for the next few videos at least but beyond that I don't know I need your advice what what do you want to see where do you want to go you want to go into electromagnetism as a gauge Theory you want to go into Super conductivity do you want to go into whatever the case may be so anyway your support is not obligatory but it is greatly and sincerely appreciated all right well hey thanks for watching and I'll see you in the next one

Channel: Richard Behiel

Views: 743,526

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Id: b7OIbMCIfs4

Length: 69min 42sec (4182 seconds)

Published: Wed Mar 20 2024