QED Prerequisites 30 Geometric Algebra - SPINORS

Transcript

okay so I've decided that we're just going to go right back into our paper there's this paper is actually really really long and we've made some pretty good progress right we are finally at the end of where well let's see we're at the end of the space-time algebra section and we're about to do this section right here section 3.71 the spinner representation so with that let's begin all right so here we are we are about to study this notion of spinner representation we've just gotten off the idea of figuring out how to rotate right we're rotating General bi-vectors in the space-time algebra and then we learned how to apply this to any Vector in this any multi-vector in space-time algebra so we had this object here that we converted into this double-sided product and we've now write that as U as RF R reverse and that is how we create the rotated version of f right and very convenient we've worked through that whole thing and we learned how to do it now we are going to move into a section what we call spinner representations Spinners we learned from quantum mechanics as a mathematical object to represent states of spin one-half in elementary non-relativistic quantum mechanics and a representation is a matrix is a collection of matrices that mimic some uh behave that mimic the behavior of a group or a vector space or anything that you're representing if you if you take some algebraic object like a group or a vector space or a ring and you replace the algebraic elements with matrices and those and matrix multiplication mimics that group that's a representation of the group so we're trying to figure out somehow we're going to use Spinners to do to represent something that in our space-time algebra system so that's what we're expecting to learn here so let's begin reading recall that in sec in equation 3.57 we converted the exponentiated commutator bracket into a double-sided product right that's we actually I just showed you that didn't I that was this whole thing right here exponentiated commentator bracket is right here there's the commutator bracket and notice this was just on the left side but we converted it to this thing we did actually quite a long I spent quite a bit of time demonstrating why this was the case and but this becomes a two-sided product of regular exponentials and this is the form we use to represent the rotation of f we never use this form we use this form all the time that's what they're saying here since the double such a double-sided product such a double-sided product is a more natural algebraic representation of a group transformation that can be extended from bifectors to the entire algebra so it's a an algebraic representation so using the bi vectors that in itself is a it's a it's a way of representing the group which is the lorentz group right that's the one we're talking about and the lorentz group is a transformation each element of that group is a transformation that will transform our multi-vectors and so we're using the algebra the space-time algebra we have pieces in it that represent those Transformations that's what these U's are these exponentiated forms of Sigma 3 and sigma 1 and sigma 2. those are elements of the space-time algebra right just to be a reminder of course it's a sign minus sine Alpha over 2 Sigma 3 and then it's cosine Alpha over 2 minus sine Alpha over 2 Sigma 3. right that is what this thing is right and this is just a real number and this is a bi Vector boom it's inside the algebra so we have an algebraic representation of the lorentz group transformation because that's what this is this is a rotation in the Lowrance group or a boost I should say so specifically if we notate product factors like this then the transformation has this form which we just went through where we have used sigma 3 reversion is minus Sigma 3. now to make sure that we understand and remind ourselves how that works it's just you've got to remember sometimes we're thinking in the full four-dimensional space-time algebra with gamma mu and sometimes we're thinking in the three-dimensional geometric algebra with Sigma I right but Sigma I regardless those even if we're thinking here Sigma I is understood to be a bi-vector right and the reversion of a bi-vector of Sigma 3 so Sigma 3 is actually gamma 3 0 in gamma 3 0 reverse so Sigma 3 reversion is gamma zero three but that equals minus gamma 3 0 and which equals minus Sigma 3. so Sigma 3 reversion is minus Sigma 3 right so you know you've got to be a little flexible because sometimes it's very nice to think of just terms of 3D geometric algebra where these guys are the basis vectors where where Sigma I are the basis vectors and that's great but every now and then you've got to remember no it's all these are these Sigma i's are actually bi vectors and then they just point out that these Factor satellites satisfy the normalization condition hence this form of the transformation makes it clear how the transformation preserves the product structure of the algebra so in order to be a representation of the algebra that's really the key you every algebra has a product and the product structure of the algebra has to be preserved in Matrix terms but in this case we're using the algebraic representation where U represents these Transformations so the product structure of the algebra is represented by this right you take two multi-vectors G times H the product G the which is the geometric product of G and H but uu reversion is one so you can slap it right in the middle and then you can take the whole thing and put it on the outer ends and that looks like ug u reverse u h u reverse which looks like the transformations of G and H which this is just that this becomes G Prime and this becomes H Prime right which is the same as g h Prime which is what this is so you can multiply so when you so you could multiply and then transform and you'll get the same thing as if you transform and then multiply and that's what we mean by preserves the product structure then they remind us we call the quantity u a rotor to emphasize that Lorenz Transformations have the geometric form of a rotation through space time and we studied that a lot in our last two lessons just remembering that rotation has to be understood in a a rather General way to include boosts which are these hyperbolic rotations right typically the rotation generators S and K the rotation generator squared to -1 and produce spatial rotations while the Boost generator is K squared to plus one so here I'll just remind you that this idea here first of all we know that um the the boosts are produced by the by our relative vectors in the space-time algebra so if Sigma 1 Sigma 2 and sigma 3 produce the Boost and that's Gamma 1 0 gamma 2 0 gamma three zero rotations are generated by by these s's now s remember I squared equals one right the the pseudo Vector the basis of the pseudo vectors squared is negative one negative one so I inverse is just minus I so you have I times minus I equals one so I inverse is minus I so what you want to write is you want to write s i equals minus Sigma I I but they don't like to use that minus sign in front so they end up writing minus they end up writing Sigma I I inverse that's that's why they that's why you have this inverse right here right up here you have this inverse because they don't want to put the minus sign out front but let me flash back and remind you that this these were the commutation rate relations that ended up looking just like the lorentz group right but in order for this to be a case S1 gamma 2 3 S2 gamma 3 1 S3 Gamma 1 2 in order to get these guys from our sigmas which is Sigma 1 is Gamma 1 0 well I guess I'll go Sigma K is gamut K zero right in order to do that you have to Define SK is Sigma k i inverse right that's how that's how it's done so just you know these are little conventions we have to keep in mind okay so specifically the rotation generator Square to minus one the space while the Boost generators Square to plus one and produce hyperbolic rotations in space time since the rotor representation of a Lorenz transformation produces a geometrically meaningful rotation the bi Vector F can be replaced in 360 with any geometric object that can be rotated indeed any multi-vector n can be Lorenz transformed according to m u m u inverse now if you go back to where we study rotors and rotations we proved this we proved that any multi-vector M can be can be Lorenz transformed or rotated according to this prescription right so this is not something that you need to infer based on geometrically meaningful rotation uh I don't think you know it's it's not clear to me why because this is a geometrically meaningful thing therefore you can replace this with any multi-vector any geometric object can be rotated that's like English that's not math but we did do that and we did prove that before in a previous lesson the Rota U is an example of a spinner PSI which is an element of the even graded sub algebra of the space-time algebra that may be decomposed as PSI equals Zeta plus F where Zeta is a complex scalar and F is a bi vector okay so this is where we start getting a little heavy right a rotor is an example of a spinner side right and what they're saying is I guess what they're saying now is PSI is defined because it says PSI comma which is an element of the even graded sub algebra of the space-time algebra now if you looked at this you might think wow that doesn't look like the whole even sub algebra right because that's a scalar that's a bi-vector but the even sub algebra should also include pseudoscalers right so because pseudoscale that's grade four that's grade two and that's grade zero so that's the even sub algebra and this confused me for a while but of course I wasn't reading clearly because Zeta is a real number plus the pseudoscalar right it is a complex scalar so you can see how this notation can really get out of hand if you're not careful if you don't realize that PSI is a complex scalar in this prescription you're wondering where is the where is the pseudo Vector that is supposed to be part of the even graded sub-algebra so anyway but this uh this is the even graded sub algebra and what they're saying is a spinner is some element of the even graded sub-algebra now what you and I know Spinners are are totally different we know Spinners are as two-dimensional well as column vectors of 2D complex column Vector right that's a spinner and it's you know and it's got its row Vector counterpart so evidently there's some correspondence between these things right but as far as this paper is concerned all we need to know is that a spinner is an element of an even graded sub-algebra so the obvious question is what is this connection because they don't talk about column vectors anywhere around here so let's read on just a little more this even graded sub algebra is closed under the full Clifford product meaning you take the geometric product of any two even graded elements of the algebra even graded multi-vectors or multi-vectors which do not have vectors or Tri vectors where I could write Tri vectors as a vector times uh the pseudo vector so if you take any geometric product of it of of multi-vectors that only have even grade components you'll always end up with another multi-vector with just even graded components uh as an example of this decomposition the Boost rotor this is a boost rotor of in the long can be expanded in terms of hyperbolic trigonometric functions as this now we've done tons of that in our last selection which is the sum of a scalar and a bi Vector similarly the spatial rotor can be expanded in terms of circular trigonometric functions Okay so why is that an example that's an example of a decomposition of the even graded sub algebra so I guess what they're saying here is that if you have a rotor U it's definitely in this even graded sub algebra and therefore by definition it is a spinner because a spinner is by definition an element of the even graded sub algebra so we know that our rotors clearly have real parts and bifector Parts real part and bifector parts okay but um but look I want to know what this connection is between a spinner and this even-graded sub algebra and so that is our next subject of of diversion because if you read the rest of this paragraph it doesn't really bubble up and and now all of a sudden we've got this word spinner floating around we need to understand the connection what is the connection between this and and is that relevant to Quantum Mechanics turns out the answer is yes so now we're going to take a diversion and sort of understand the connection between this and Spinners that we have always known about from quantum mechanics okay one moment please all right so to make this connection as best as I can um let's go back to our elementary quantum mechanics and we're just going to talk about Spinners so this is the spinner this would be this just the spinner part of a wave function so we're ignoring any spatial part um in the momentum space or in regular space we're just going to look at pure Spinners if you do a lot of quantum computation this is all you do you only think of qubits in terms of pure it's two level Spinners right so qubits are a very popular subject now you know everything's taught everything's talking about the block sphere this is all about the block sphere but we can just take it to its Bare Bones right where we're looking at a quantum mechanical space is just broken down into two parts the uh it's a superposition of the zero State and the one state the Lower State and the Upper State and the superposition has two complex numbers Alpha and beta these are complex numbers now right and so this is sort of the bra KET notation very typical used in in this subject this sort of two level system subject but we can also create this spinner object which is a column Vector with Alpha and beta in the two positions now this thing has four degrees of freedom the real part of alpha the real part of beta and the complex part of Alpha and the complex part of beta so don't make it's two dimensional in the sense of complex numbers but it's four real numbers so that is how you write down a spinner in elementary quantum mechanics now also in elementary quantum mechanics you have the observables The Operators the spin operators and almost always these are these three poly operators Sigma three which we put first because it's the diagonal it's all we choose Sigma 3 to be the diagonalized operator and sigma 2 and sigma 1 are not diagonal right they have this form and actually this is kind of the interesting this is one of the things that geometric algebra promises to deal with is well look at this you've got these eyes here right you know quantum mechanics this is all pretty arbitrary it seems like like what is this eye all about and and the math all works out it's all very tight when you get your head around the whole thing but the promise of geometric algebra is to sort of explain These Eyes in terms of geometry which is interesting um but right now we're just reviewing the basic quantum mechanics so we've got these three palette matrices which are essentially all of the spin operators can be constructed from these poly matrices and then you have the uh the spinner itself which really represents the wave function of the of the atom or the molecule or whatever I guess whatever qubit you're working with so I'll call I'll write down qubit that's not standard quantum mechanics language right that's the language of of the subject of quantum computation but nowadays more people know about qubits than they do about two level systems right which is um or spin one-half systems and I'm serious about that right qubits people don't necessarily talk about spin one-halves when they study qubits and Quantum computation at all they're they don't care they completely abstractify what these two states could be but uh when when I look at this I think immediately of just you know an electrons with spin one half what else could it be I mean that's the most archetype that's the basic system that you you would consider here these are angular momentum operators and these you know this would be SX or SZ Sy and SX um this the actual spin operators there's a factor of Planck's constant over two that will probably I can't remember if I put it in here eventually or not but the point is is this is the spinner right this guy here and these matrices are the are the representations of the angular momentum anyway so if I want to find for a given spinner if I want to find the expectation value of Sigma 3 I literally am going to do a matrix process and that Matrix process is going to look like Alpha conjugate beta conjugate one minus one and Alpha Beta right and if you execute this multiplication you should get this expression this number I used a different I'd use the different uh when I when I Drew this up I used a different form for the conjugate so the kanji gets actually like that and you end up with this expression right and if same thing with with Sigma 2 if I want to find the expectation value of Sigma 2 I throw in here the Matrix Sigma 2 which has that minus sign these off-diagonal terms and the the little I's and you end up with this expression a beta Alpha Beta conjugate minus Alpha conjugate beta all times I and likewise for Sigma 1 I get Alpha Beta conjugate Alpha conjugate beta right so these this is how you would calculate these expectation values uh using the regular spinner algebra so let's see if we can connect this I want to connect this ultimately to the geometric algebra so so now we take a look at these things here and we say well this is the expectation value of an angular momentum in each of these directions now what we're going to do is we're going to give each of these a name we're going to call this N3 in two and N1 and we're going to kind of Imagine an angular momentum Vector where the components of the vector in three-dimensional space are these three numbers are these three numbers right here and if we did that the magnitude of this Vector right would be basically the magnitude squared would be the sum of the squares of these three objects so let's Square these three objects add it together and find the magnitude of this titular pseudo-angular momentum Vector I'm nervous about calling an actual angular momentum Vector of course because the poly Exclusion Principle I'm not I'm not not the uh not the palette Exclusion Principle but just the Heisenberg uncertain now which one um the just the principle of complementarity that you can't measure all these things at once but you can still speculate about this Vector right this is ultimately related to the notion of the block sphere but what we're going to do is we're going to write down this number right here by squaring each of these terms and adding them together so when we do this process we take this first one here and we Square it to get these these three terms right and the second one when we Square N2 when we Square N2 this I squared becomes a minus one out in front but uh you know we the inside is just easy to square but the interesting thing is this will cancel with this right you'll get a cancellation right here when you add them together this will cancel with that so that's good and then when you take the third term which is this guy and you square it you get these three terms now here you get cancellations here here with those two so the only surviving terms are this term this term and this term and when you take the survivors down and write them out together you realize that that is just a a conjugate plus b b conjugate squared which is PSI again PSI squared which is one so the point is is this magnitude N squared is going to equal 1.

and that tells us that we can take this magnitude and and we can create this notion of a block sphere of radius one and this arrow is going to or the the uh the arrow representing n is going to have two coordinates coordinate Theta and a coordinate Phi so it only have two coordinates it's going to be pointing somewhere on this sphere it's a vector in space right with magnitude one so it must have only two degrees of freedom and we can just represent it as an arrow pointing on a sphere and that's typically called The Block sphere right another very well-known concept from Quantum information science okay so with that with that idea that where that we can we're taking these three expectation values and throwing it in onto a block sphere with two coordinates now we have to kind of worry about now what we're going to do is we're going to work on these two coordinates to identify this Vector n which represents these expectation values so here's my little picture of things let's see if I can um label it well right this angle here will be the angle Theta it'll be the I guess we call we would call that the the polar angle and then this would be the asymmetal angle right here okay so that's the Polar angle the projection of and this of course is the vector n the projection of n on the uh this would be the X3 axis that projection would be of course cosine Theta and then the projection uh actually this is kind of this kind of looks a little skewed doesn't it yeah I think this red line here in order to actually kind of do my projective geometry right it should look something like that right yeah that's better so cosine Theta and then cosine Theta doesn't really look as good there as it should look it should be it should be here let me see hold on it should be here cosine Theta and it should represent this distance right and uh likewise these projections here are going to be uh in this case this particular projection is going to be cosine Theta or cosine Phi right that's this projection and then this projection here should be cosine Theta wait wait wait wait wait wait wait wait that cosine should be sine Theta cosine Phi right sine Theta right because this this distance here is sine Theta so it should be sine Theta cosine Phi and this just this part here will be sine Theta sine Phi okay that'll be this distance here okay so we can break in up this is our Vector n we can break this guy up into these component parts right I've done that where did I do that I did that right here so we're going to say that the N3 component which is that the component along the three axis that's just going to be cosine Theta right but we know what N3 is it's these two it's this product of the complex numbers and their kanji gets subtracted from each other so that we're going to call Alpha squared and beta squared because alpha alpha conjugates just our Alpha squared beta squared and what we're going to realize is cosine of theta has to equal the difference of two squares but we know what that is from trigonometric considerations we know the double angle formula I think it comes from the double angle formula so it goes cosine 2 theta equals cosine squared theta minus sine squared theta so what we're doing is we're sort of turning this into a some kind of reverse half angle Thing by replacing two Theta with Theta and Theta with Theta over two right so we can replace Alpha and beta with cosine squared theta over 2 minus sine squared theta over 2. so that's exactly what we do so we have sine Alpha the value of cos Theta over two and beta sine Theta over 2. now already you see these half angles coming in which are exactly what we get in our rotor form so we're kind of leaning already into the rotiform the problem is that when you take a complex number and multiply by conjugate if there's an overall phase you can't really detect it if all you're looking at is these two real numbers so we have to include an overall phase as a possible part of alpha so Alpha ultimately it's proportional to cosine Theta over 2 but there's another phase factor in here that'll disappear when you take this conjugation same with beta and those phase factors are obviously independent so now we're going to replace Alpha everywhere we see Alpha with cosine Theta over 2 e to the I gamma and sine Theta over 2 e to the I Delta so then we just go to N1 which we know is Alpha Beta conjugate plus beta Alpha conjugate but we know what Alpha and beta are they have these real Parts which are going to come out front and it's going to leave behind just these phase pieces which I've organized to be gamma minus Delta so the phase of alpha minus the phase of beta is the form so gamma minus Delta is sort of its own Factor now but you see I've done it in order to be constructive here right because I have e to the I some phase plus e to the minus I some phase and if I just throw that with a 2 over 2 you can see that this piece right here is going to look like a cosine of gamma minus Delta right so that's good and then I do this for N2 as well right same thing but now we have this I out there and what are we going to see well we're going to see the same thing the cosine half angle cosine and half angle sine but we also have now the difference between these exponentials which uh is going to and then there's this I up front and so this is going to be I times the sine of these two things so I now redefine the difference between these two phases as Phi right and then I say okay our our cat PSI which we are expressing in the representation of of a spitter representation which is this two-dimensional thing we now have something we can put in these spinner slots this is Alpha and that is beta and now I can take Alpha and beta and I can multiply both I can multiply the whole Spinner by e to the I gamma where I get end up getting this difference down here in this denominator or not denominator in the lower part of the spinner right so you see this gamma minus Delta well here I have Delta minus gamma and then I can keep working this to extract um a uh a gamma minus Delta from everything so now I have the gamma minus Delta in both components of the spinner and of course it's ends up being multiplied by gamma plus Delta on the outside so now this is beginning to look because you know I'm about to replace this with uh Phi and sure enough I get cosine Theta over 2 e to the I Phi over 2 because that's going to be 5 and then e to the minus I Phi over 2 and then I ignore this because now this is a global phase of the entire spinner and we ignore those global phases right so we are left with a spinner that is modeled like this and that is exactly how we model things on the Block sphere right this is quantum mechanics right this is actually familiar for those of us who've done quantum mechanics and even those who have worked Quantum information science you actually derive this thing so now what have we done we've basically just we've done no geometric algebra right we've just taken this and we've created this sort of block sphere spinner form that encapsulates all of that information so I guess it's worth taking a moment to understand how we interpret this so if we have let's say we started with a vector n on this block sphere right it's block sphere because n has always got unit length right so we know Theta and we know PSI uh Phi because we that's how we know n right so I can write down I can calculate cosine of theta over 2 and sine of theta over 2 and I can calculate these two numbers as well and I'd end up with a complex number here but I'd end up with with a version of you know this would be Alpha real plus I Alpha imaginary and beta the real part plus I beta the imaginary part right and those imaginary Parts come from the I that lives in here where this gets replaced by uh this gets replaced by cosine Phi over 2 minus I sine Phi over 2 for example right that because that comes in here right and so you end up when you multiply everything together you end up with a real part in an imaginary part so you can calculate all of these numbers using these formulas are using the formula for the spinner that we just figured out and this block this situation on the Block sphere this particular value of n if that was the state of our two level system and I chose to measure the angular momentum or the I measure the state of the system with a in the basis X3 the probability of finding it uh is is going to be or finding it in the state zero will be related to this length and if my basis was in X I guess X One it would be related to this length and X2 the probability would be related to that length so that is uh basically the quick and dirty analysis of the block sphere now we need to somehow connect this to geometric algebra so let's first take note that we still have these coordinates that we've laid out now notice that this is not symmetrical right N3 has a certain kind of favoritism in the terms of Simplicity right this is nice and easy in order to write this expression down we have to choose sort of an axis an axis to measure these coordinates off of so this breaks sort of the mathematical symmetry in the sense that not all directions are easy to calculate but it doesn't break any significant symmetry because this is generally true regardless of what axes you choose so there will be this interesting asymmetry will emerge in our calculations and it all can be traced to choosing these three axes but say you've chosen those axes so now once I've chosen those axes I could actually write them down in a sort of a vector form right I can say well this is the x-axis so I'll call that sine Theta this is the component on the x-axis so sine Theta cosine Theta Sigma 1 will be the basis Vector in the X Direction and two sine Theta sine Phi will be the basis Vector in the y direction this is sine Theta cosine Phi in the X Direction sine Theta sine Phi in the y direction and then just cosine Theta in the Z Direction now you should see where this is going right these are just basis vectors sort of on our block sphere right with this would be the sigma 3 Sigma 1 Sigma 2 . let me see no this would be Sigma one I think but regardless uh you know you have the three basis vectors those are going to be what jumps into our geometric algebra these are going to be bi vectors in the space-time algebra or they're going to be regular vectors if we're looking at this in terms of the three-dimensional geometric algebra that is embedded in the space-time algebra right whether they're the relative vectors or whether we think of them as bi-factors but we'll think of it as the relative vectors for now but then I can take this structure here and I can actually break apart the inside by pulling out this sine Theta right and What's Left Behind is cosine Phi Sigma 1 sine Phi Sigma 2. and then of course cosine Theta Sigma 3 is left over here but what I should recognize right away is that this piece here is actually the rotation of Sigma 1 around the sigma 3 axis right which would be in the plane well the rotation of Sigma 1 about the sigma 3 axis which and it's a spatial rotation of Sigma 1 about the sigma 3 axis and I recognizing that I can write that down as Sigma 1 rotated around Sigma 3 using our prescription for rotors right so this is the rotor this is the reversion of the rotor I want to move it an angle of PSI right no Phi Phi so I write it as Phi over 2 here but that rotates it by when I use this prescription that rotates it by a full angle of Phi so just to drive it home again if this this is a rotation let's execute right so this exponential can be broken down using Euler's rule this way I Sigma 3 because remember we're rotating around the axis Sigma 3.

we recognize this as rotation around the axis Sigma 3. so that means we have to rotate in the plane of Sigma 1 2 which is I Sigma 3. that's where these i's are coming from in this I guess some uh to be clear that's not standard in the rotors right rotors have bi vectors up here and here we have a bi Vector times um a uh well okay so I guess if we're looking at in terms of the space-time algebra right we have a bi Vector which is this relative Vector times the pseudoscaler times the uh the the grade four pseudoscaler so a bi Vector times a pseudoscala this should be another bivactor right but if we look at it in terms of the three D geometric algebra that's embedded right then we think of this as a one-dimensional vector and this is now a three-dimensional or grade three pseudoscale I shouldn't say one-dimensional I say a grade one vector a regular vector and this is a Grade Three pseudoscaler which would be written as Sigma 1 Sigma 2 Sigma 3. it turns out you look carefully at all this I'll just remind you it's all equivalent right of it has to match because remember that three-dimensional geometric algebra is embedded in the space-time algebra but that's why you see this I Sigma 3 here we're thinking about rotation around the axis Sigma 3 but remember all rotors really refer to a plane of rotation so you've got to get this I in there okay so that uh and there you see we you know here we have our I Sigma 1 Sigma two this minus sign comes down here uh the sigma 1 drops down here like that without much trouble um and then likewise this uh rotor be using its Euler form becomes this thing right here notice sine Sigma Sigma over two sine of PSI I keep saying PSI it's Phi sine of Phi over 2 is a real number so whether it's on the left or the right of Sigma 1 2 is irrelevant so the fact that I've got it on different sides here is doesn't matter but this expression here is this rotor is this not the rotor it's it's the rotation of Sigma one in the plane which which caught our eye when we notice that uh it it when we wrote down the full Vector form of our block Vector we notice this little embedded rotation so we're kind of flushing it out so if you take this and you finish all the multiplication you multiply Sigma 1 to the left you get cosine Phi over 2 Sigma 1. and then you change this to a plus sign because you're going to exchange Sigma 2 and sigma 1 in their orthogonal and the anti-commute so you pick up a minus sign when you do that flip but then PSI 1 geometric product PSI 1 is just one so that's what that comes from and then you have this product execute and this is just a cosine Phi over 2 Sigma 1 cosine Phi over 2 sine Phi over 2 Sigma 2 plus these this cross term oh yeah see you have these two cross terms and then you have sine squared the two cross terms uh how did we do this oh yeah we just jump into the trigonometric relationships right the cosine squared and the sine squared of Phi over 2 together wait there's a minus sign missing here right because this the cosine squared Phi over 2 plus sine squared Phi over 2 that would just be one you need the difference between these two to end up with this double angle formula that we used earlier but these two I think do work out to be sine five let me find out where my minus one went oh okay I think I found it when I multiply this term by this term right you end up with a sigma 2 Sigma 1 Sigma 2 right and then you have to flip these two to get minus Sigma 1 Sigma 2 Sigma 2 which equals minus Sigma uh one right so this here should be a minus sign right and now you have cosine squared Phi over 2 minus sine squared Phi over 2 and that combines to give cosine e uh of Phi and then this one and this one combined together to give you sine of Phi okay the point being this is that this final result here is actually what we had here so it just shows that this is in fact the rotation and this rotation does in fact get you this thing that was just a long demonstration to show how that worked okay so now that I believe that this is given by this rotor form right where we're taking this vector and we're just rotating it I can now make the replacement just straight make that replacement you know I'm replacing this with its rotor form right here and I've replaced I Sigma 3 with Sigma 1 Sigma 2 just to make it very clear and what do I do next then um then I pull out this exponential this rotor I pull out the rotor now it's easy to pull it out to the left because sine Theta is a number so I can just move that number in and it's and it just sits right inside the question is what about pulling it out to the right because what I've actually done is I've kind of taken this and thrown it inside here so it's almost like I've pulled it all the way out and how can we do that well what you have to realize is that Sigma 3 commutes with Sigma 1 Sigma 2 right if you flip Sigma 3 through you've got to pick up a minus sign then another minus sign so Sigma 3 commutes to Sigma 1 2.

so that's another way of saying rotating in this plane doesn't affect Sigma 3. so because of that the two exponentials that sandwich cosine Theta Sigma 3 this guy is going to commute through it will not pick up a minus sign and it will cancel with this and you'll end up with cosine Theta Sigma 3. so I can put it in there and if I expand this out I get sine Theta Sigma 1 sandwiched between the rotor and its reverse but when I get cosine Theta Sigma 3 it's also sandwiched between the rotor and the reverse but it comuts with the reversion and then the two rotors cancel out and you're just left with this term so this is one of those Maneuvers where it kind of is elegant in geometric algebra because I'm using the geometric algebra rules here but is it all that easy to see to see insights to simplify stuff in geometric algebra that you have to kind of know you have to be pretty familiar with this it's a new way of you have to understand everything that commutes with everything and you have to keep that in the front of your mind to simplify this stuff and so I think proponents of geometric algebra just like to show look how easy this all is it all comes together but you know if you have to work through each and every step in your little mind you have to track you have to track different things than you're used to tracking and I can't say that those things are harder to track it's certainly it's like learning a new language right because you know it's it's still mathematically clear but instead of just immediately realizing that things can be distributed you have to realize that oh this commutes therefore I can distribute it so that's just a personal reflection on that but nevertheless here we are and now once I'm in here I realize I'm in the same situation as I was back here when I realized that this was a rotation now I realize that this is a rotation and it's a rotation of Sigma 3 in the plane of Sigma 3 1. which is about the axis of Sigma 2. right so now I can introduce and I won't go through all the math like I did before because it's the same thing but now we have this I wrote this down as like the rotor is this that's not true the rotor well okay okay it is true the rotor here is e to the minus Sigma 3 Sigma 1.

Theta over 2. and now if I take these two together right these two together the actual rotor for the whole thing is this expression here right and so I can now write the vector n right I'm realizing remember remember remember where we started right we started this is the vector n right this is n as a vector I call it the block Vector from two level system language right and end Quantum information science language too now I guess um and so I'm all of this is just rewriting the vector n so when I'm all done the vector n is just this rotor and here's you know uh uh Phi and here's Theta right so those two those two numbers are that Define n are actually embedded in this rotor and now n is just Sigma 3 rotated with this rotor right so I've now written n using the geometric algebra language right it's still a vector right it still comes out to be this guy right here which is still derived from this which is derived from you know these quantum mechanical calculations right so so but it's now using geometric algebra of rotors to Define n so that's pretty cool I guess so so our last step is to compare this rotor R which basically has all of the information for the spinner end remember this Sigma 3 is an artifact of our the fact that Sigma 3 shows up there is not particular to this uh block Vector n right if all the information about n is contained in in Phi and Theta right this Sigma 3 is there because of sort of the asymmetric mathematics of choosing the polar coordinates that Define sine Theta so this is a formula for n and sigma 3 belongs in there all the time as Sigma 3. so now we need to take this rotor n and connect it up to the literal spinner that we're so familiar with From the Block sphere and so let's we can kind of do that right because we know that this part represents the road I Sigma 2 right Sigma 3 1 is I Sigma 2. Sigma 1 2 is I Sigma 3 and and sigma 1 2 Sigma 3 1 is I Sigma 1 when you work all that out so these coefficients that exist in front of these bi-vector terms can be matched with the expansion of this block vector and if we expand this block Vector into its real and complex parts we get these four real numbers cosine Theta over two cosine 5 over 2 cosine Theta over 2 sine Phi over 2 and these pairs down below so what we see is that this a0 term which I'll call a0 not Alpha zero but a0 cosine Phi over 2 cosine Theta over 2 that can be found right here and we will call that a0 also so those match and then we see okay the let's look for the I Sigma 1 term which would represent rotation around the sigma one axis sine Phi over 2 sine Theta over two well here's sine Phi over 2 and here's sine Theta over 2 but there's a factor of I here so this guy we're going to call uh we'll call this A1 and we'll call this I A1 and then we'll look at this term here we have minus cosine Phi over 2 sine Theta over 2. well here we have cosine Phi over 2 sine Theta over 2 but without a minus sign so we'll call this um just the the real number part right we'll call this with a minus sign we'll call that a 2 and then we'll have to call this minus A2 right and then the last piece we have to match up is the I the dual of a i of Sigma 3 and that's sine Phi over 2 cosine Theta over 2 with a minus sign well here we have the minus sign but we don't have I so if I call this A3 then I will call this I A3 and then in the end of course this is what the final spinner looks like so now if I now I'm good because if I start with a spinner if I start with a spinner here just give me an arbitrary spinner from quantum mechanics it's a legit spinner so I have a real part an imaginary part a real part an imaginary part I can break up these real and imaginary parts and I can plant them in this expression right with the uh the the the real part or the grade Zero part and this is the grade two part and I look for the I look for the number I take this one here and I put it in front of the I Sigma one I take this number here and I put it in front of you know I figure out what A2 is so I take the opposite of the number that's living in that I take opposite of the real part of the second component and I call that A2 and I put that right here in front of Sigma 3 Sigma 1 and you know I do it for all three and now I have taken an arbitrary spinner this was my goal right I've taken an arbitrary spinner and I've converted it to a rotor which is in fact an element of the even sub algebra of the space-time algebra which is the whole thing I was trying to show so that is the connection between spinners in quantum mechanics and even sub-algebra elements and in fact it's a little more specific it's not only just part of the this connection isn't just the even sub algebra as a whole because you don't see any grade four parts in this right so you know it goes to just a real part in a bivactor and furthermore you know it's actually going to be a rotor so Spinners in quantum mechanics translate into these rotors in the geometric algebra and just to put a fine point on it um if if I take the spinner one zero right well I go with my prescription a0 is one and all three of these guys are zero this so if I put in a if I put in one for this value right that becomes a0 these guys go away and what I'm left with is the the prescription for that rotor is going to the rotor is now one so the block Vector is going to be represented in the algebra as one Sigma three one reversion which is just Sigma 3.

but notice that this step is sort of Superfluous almost it's the one that matters the most right because all the information is contained in the one likewise if I wanted the the down spinner well the down spinner I have to go back to this and I say well now I have this slot is held by minus two I'm sorry minus 1 right is the real part of this this is minus one I'm sorry this spot is held by one right because it's this is zero that is zero so the whole top is zero that's zero so the imaginary part is zero but this is one so minus A2 equals one so A2 equals minus one right so if A2 equals minus one then you have um minus I Sigma 2. and so what I'm expecting is r R is going to be this is minus I Sigma 2 is the row is the uh the spinner and if I calculate minus I Sigma 2 I get Sigma 1 Sigma 3 so I have now Sigma 1 Sigma 3 Sigma 3 Sigma 3 Sigma 1 which is the reversion these two guys disappear to one so I have Sigma one Sigma 3 Sigma one that ends up being minus Sigma 3 which is sort of the opposite of this so now this represents the block Vector in the geometric Algebra I guess I should make this an arrow right like that because it's not really equal the question is how do we find expectation values and other things in the geometric algebra and we're not going to go into that but I have shown the connection which answers this question back here right about Spinners being elements of the even graded sub-algebra this FaceTime algebra so that's the whole point of what we're trying to do today okay so it's a pretty long lecture to get to there but it does make this connection and so now we can move on so I'll see you next time