QED Prerequisites Geometric Algebra 25 Relative Reversion

Transcript

[Music] thank you welcome back we are going to continue our study of the space-time algebra using this wonderful paper and we have made a lot of progress we have gotten all the way into section 3.62 para vectors today we are going to finish pair of vectors and hopefully begin or maybe even finish relative reversion so let's begin now we left off last time with this new form of expressing a multi-vector and it is it involves only pair of vectors with a bunch of scalar parts and a bunch of relative Vector Parts they call them Vector Parts but we got to be careful now vectors meaning at least four three different things right vectors are the four vectors of special relativity vectors are the uh the proper vectors of the algebra and now vectors are three relative three vectors so if you just say vector and it there's actually a fourth one right anything in a vector space is a vector so any member any multivector is also a vector right so the vector is like the most overloaded word in this mathematical physics language I'm absolutely convinced of it I can't think of another word that can have that many meanings so let's understand what we're after here we're talking about three relative three Vector Parts because you see these arrows in this paper when you see these arrows you're dealing with relative three vectors so we've learned about this new Partition and uh which is now manipulated the we the way we can get with only two things the way we can get away with only two things a bunch of scalars and a bunch of relative three vectors is by using these Duality factors we everything's related in the geometric algebra through these two dualities and the space-time algebra to be specific through these dualities so let's have a quick look at how we got here so remember that a multi-vector m this is the way a multi-vector m is expressed you'll see that this is the scalar piece right here right and the vector there's the vector basis right and this guy here is a vector in the multi-vector and it is a we're going to call that a proper Vector but it just it's a bunch of components it's a four-dimensional Vector space of in its own right and it's got four components and there it is right there then the same uh the same can be said for the bi vectors right the bi vectors are all right here and the bi vectors they can be expressed in these this uh they have six basis vectors Gamma One Zero again the two zero three zero two three three one and one two where I'm choosing the basis vectors from the paper and I'm pretty sure that the way they did it is they went with Gamma One Z now we know why they chose gamma one zero two zero and Gamma three zeros because they wanna they want to use the bonus the benefits of Sigma 1 Sigma 2 and sigma 3 later so that's so that explains why they choose these as our basis vectors for the general algebra and these guys here if I'm not mistaken should be Sigma 1 I Sigma 2 I and sigma 3 I they should be the duals of them uh actually I think I think they went with this though I think they went with the inverse dual the the in the inverse tool regardless though they the paper chose it and it's convenient for whatever that reason is the key point being they definitely chose these to be the relics so they could use this relative basis vectors so there are all the bifectors right there and then the tri vectors are likewise constructed with component and there's four tri-vector basis vectors right one two three four and then ultimately there is one pseudo scalar basis Vector so every multi-vector can always be written like this and this is where our geometric algebra any geometric algebra has that structure and just to simplify it we say scalar vector by vector tri-vector and here we allow ourselves instead of coming up with another symbol for for the pseudoscaler like we could have come up with some weird symbol like like if we use Greek letters for scalars we could say we use uh maybe some sort of Hebrew letter for pseudoscaler I don't even know what a Hebrew letter looks like Let Me Maybe maybe we would use say uh I don't know some weird little hieroglyphic right that would look like you know some sort of that symbol would be a pseudoscaler right and uh but we don't do that what we do is we we allow ourselves to just to throw down this casuto scalar basis Vector as I and then allow beta to be a real number so this first step is actually a bit of a simplification from this overall General massive representation in the full basis we allow ourselves to borrow a real number and multiply by I and call that product the a pseudoscaler right so we're already using a duality relationship right in the beginning because that is the that pseudoscaler is clearly the Dual of a scalar so we're exploiting Duality even in this early stage of writing down a general multi-vector symbolically like this and we kind of go long on this and we say but there's another way of organizing it we can take the scalar part and the pseudoscaler part and bind them together into a complex scalar and we can take the vector part and the tri-vector part and bind them together into a complex vector and so in principle we could write this as a complex scalar Z plus a complex Vector which we might call bold V plus a bi Vector right and so that might be yet another way of writing down a general multi-vector and all that is is just rearranging sort of rearranging the order of these guys and getting rid of Tri vectors and in favor of duals of vectors so we always need now a vector W and its dual will be a tri-vector but we're now going to wrap all of these tri-vector basis vectors up into Duels of of these basis vectors here so that's another way of tightening up our work but then we can go even farther and we can say oh if I know what gamma zero is right so so far everything above this line that's all proper right that means it doesn't matter at all what reference frame you're in but below this line it does below this line I do need to know what reference frame I'm in because I'm going to now take a this vector and I'm going to split the vector up into a relative a relative 3 vector and the Dual of a relative 3 vector and I'm going to distribute that into these uh two parts so actually oops wait a minute that's not right scratch that I'm taking the bi Vector right the bi vector I'm taking the bi vector and I'm splitting it up into its relative three Vector part and the part that's left over is the is the Dual of a relative three Vector so I need to find two relative three vectors one of which absorbs half absorbs the first three basis vectors right so the first three of these actually end up Landing here in E and then the last three of these have to land in b i right bi but I have to make sure that the basis is is correct right so um so I've I'm using but I need to know what gamma zero is to do this right because I need to be able this split is very dependent on gamma zero if I don't have gamma zero down I can't do this split at all so um so what I've basically gone is I've taken this bi Vector now and split it in half but uh you'll also notice though this does not include the complex this is a progressive right I think I've this complex formulation I've left it behind because here you know you see I have Alpha and beta I but here I is all over here so I've sort of separated beta I from alpha in this kind of expression so all I've done here is I've used R gamma zero and broken up the bi Vector into a relative three vector and the Dual of a relative 3 vector and then the last step is we went really long on that we said well I've already taken this bifector and I've split it up well how about taking now I'm going to take this vector and I'm going to split it up into a para vector right multiplied by gamma zero that's exactly the same thing right V 0 V the proper the proper vector v is a some pair of vector right multiplied by gamma zero so it is the relative dual of this para vector and we defined a para vector as a scalar plus a relative three vector so this is getting really fascinating because we're just we're getting closer to things we actually use in regular physics right in regular physics we use electromagnetic field vectors we use magnetic field vectors we use momentum vectors and then Delta is the time component of a momentum four Vector this object here is the geometric algebra or the space-time algebra equivalent of a true four Vector of regular space time um uh regular space-time calculations well actually I gotta say no I think that I think this proper Vector is probably more the GM the space-time algebra equivalent of a regular four Vector this is when you're trying to do special relativity uh without four vectors right you would have something like this you would have a Time piece and a vector piece and you have to keep track of how to keep them all together and but likewise uh this pseudo Vector right that's a pseudo Vector right there well I take this the vector piece the the Dual of that pseudo Vector is w i take that and I break it up into its para Vector part with right multiplied by gamma zero and of course I just floats out on the far right side so this whole product here is where it is is the migration of Omega I or gamma wi goes to this this new para vector this new para Vector time this the Dual of this new pair of vector times I phew so you can look at all these things and this is about as far as we can go but now the difference is of course these two and this one in particular is fully determined by your reference frame if you don't have your reference frame locked down and chosen this split is hard to do or you can't do it but what's important to really understand isn't so much that it's that um is that this partitioning completely depends on gamma zero so if you have a different reference frame with a different gamma zero none of these every one of these components is going to change well that's actually not completely true Alpha won't change for example and E won't necessarily change this this um well Alpha will change Alpha I'm sorry Alpha's we're going to go through this in detail actually Alpha won't change e will change because remember e that split for E depends on Gamma 1 Sigma 1 Sigma 2 and sigma 3 and each of those depends on gamma zero right that's Gamma 1 0 for example and Gamma 2 0 and Gamma 3 0. so if you change gamma zero you're going to change those basis factors which means you're going to change e right so e does not necessarily change I'm so e n e does necessarily change sorry e doesn't necessarily change but a is the only thing in this whole list that isn't going to change at all in fact let's go through that I think that's the next part of of the the material so let's uh let's look at how these things depend on gamma zero and also how they depend on the handedness of the reference frame right in fact that's the next topic of the paper so we've got this fully frame frame dependent form usually when you see frame something it's always frame independent frame independent frame dependent now is what we're talking about here it's all frame dependent so we've broken everything down to a bunch of scalars and a bunch of three vectors A bunch of scalars and a bunch everything in here is a scalar or a three Vector right gamma zero and I are these Duality factors now you think of them more as Duality factors so but scalars and three vectors um so we've kind of really we've used the dualities the two dualities we have there's this relative space Duality a relative frame Duality and this Hodge Duality as fully as we can to decompose M into this form so the appropriate Duality factors right these pair of vectors are the quantities usually considered in frame dependent treatments of relativistic systems that use traditional Vector analysis and that's true every traditionally we choose a reference frame and we do all our work in a certain reference frame and because it's intuitive and it's it involves a little bit more elementary math and you certainly don't need to learn space-time algebra to do it I suppose arguably we should teach space-time algebra earlier in the curriculum and then then you would use space-time algebra to do it but I can't imagine a world where that's the way it's done but that might just be because I'm so entrenched in the world that I I developed and trained through so it's hard to know how Young's younger students are going to think in the future right anyway these pair of vectors are quantities usually however the geometric origins of the quantities involved become obscured when they are lifted outside the space-time algebra so we're going to talk about that a little bit we can recover in the next pair in this paragraph We can recover some distinguishing information about the scalars by examining their behavior under frame Transportation Transformations and spatial Reflections which we summarize in table six so let's have a look at that that's what I was about that's what I was doing a moment ago let's consider Alpha right if I were to change if I were to change gamma right if I were to change gamma 0 to gamma 0 Prime that is a frame transformation I'm going to a new frame where it's uh where the stationary Observer is has this different relative state of motion than uh this the state the then the stationary observer in the first frame right but if I do that what happens well this gamma zero will be changed right it will be transformed we're going to have to transform that m0 and there's a gamma zero hidden in there and this scan is zero changing affects those two and this gamma zero changing certainly affects this term it definitely affects that there's a gamma zero buried in here right as a gamma zero buried in here so um but none of those gamma zeros are going to affect this term over here so Alpha doesn't change if you if you change frames Alpha doesn't change so so that's kind of interesting and what does it say the scalar Alpha is invariant under frame Transformations and spatial reflection so it's a proper scalar so what about spatial Reflections well spatial Reflections are introduced wherever you see gamma I where I is not zero so a spatial reflection definitely affects this I so everything in here is affected by spatial Reflections this guy its basis vectors are based you know Sigma I always has a gamma I gamma zero so spatial Reflections affect e and um so well the point is it doesn't affect there's no there's nothing that affects a for spatial Reflections because that's where they're talking about the scalar a you you don't see if you change gamma I a remains unaffected is the point so it's so frame Transformations are what we're going to call changing gamma zero so this is related again a zero and spatial Reflections are related to gamma I so it's a proper scalar the scalar B flip signed under spatial Reflections but does not change under frame transformation so it is a proper pseudoscaler so I think the way to understand this is you go up to Beta here right and let's look at where it sits in this big picture well ultimately beta's real representation is beta I now when you do a frame transformation I never changes I sort of a constant thing it's a frame and variant idea however if you do a spatial reflection right if you do a spatial reflection what's going to happen is remember I equal this is beta uh a spatial reflection either one or all three of these vectors are going to change and so I is going to change sign to minus I right so beta the beta eyepiece that's the part that is the true pseudo scalar and it changes under spatial reflection but if you transform all the components together you're just you're just going to end up with gamma zero Prime Gamma 1 Prime gamma 2 Prime gamma 3 Prime and uh beta the number beta won't change so the scalar beta flip sign under spatial Reflections but does not change under frame transformation it's a proper pseudo scalar we call it a proper pseudoscaler and they say all right well let's look at all the rest of these things right now the game is to understand um how all of these different pieces transform in different ways and Delta is invariant under spatial Reflections but changes under frame Transformations that's Delta here well Delta's a scalar there's no multiplication by any of the gamma I's but there is gamma zero and if gamma zero changes then it's going to affect Delta so it changes is a relative so they're calling this now this is a sort of a new word a relative scalar that's a new word but it's easy to understand it is the time part of a four Vector right so it is the time like part of a four Vector that's they're calling it a relative scalar and uh sure that's exactly what we understand to be Delta rho right when we write it in when we write this para Vector in four Vector form it looks like Delta rho so that's easy then the scalar W now less more obscure pseudoscalers are already pretty obscure right this beta is already a little bit of a weird thing because it doesn't really show up in elementary physics you don't see it much see plenty of scalars out there but you don't see a lot of pseudoscalers um they do exist you know in particle physics you have pseudoscaler type particles um but uh in element in electricity and magnetism a pseudoscaler would represent a uh a magnetic charge which we presume is zero in elementary magnetism um the scalar Omega flip sign under spatial reflection and changes under a frame transformation so it is a relative pseudoscalar part of a four pseudo Vector so that's this guy here right well that gamma zero is what drives its change under uh frame Transformations and flipping the sign of any one of those uh spatial directions or all three will give you a give you its parity right so it and it changes parity so that's all good and likewise you can do the same you can play the same game with e with the um with the bi vectors right this clearly will change sign under a parity under a reflection because uh it has gamma I's in it right that that we've already discussed how how a full Vector how how e which is dependent on Sigma 1 Sigma 2 and sigma 3. those all have gamma I zeros in it so you start screwing with gamma I you're going to screw with e so e changes sign under uh reflection through the origin you would be itself all alone so does B but the quantity b i the two the two Reflections cancel right the reflection in I will cancel whatever reflection is in B and so ultimately the quantity b i is a pseudo3 vector and you combine that with beta and you get a pseudo4 vector right and then P the which is a classic example that they're trying to Signal here is momentum momentum clearly changes under a frame transformation but momentum uh also changes sign under a parity flip but this thing a for which they don't have an example but I think they're trying to lean into angular momentum here uh we'll change will change with a frame transformation and will not change will not change under a reflection a spatial reflection because the I the change in I will cancel with the change in a So reading on we can distinct similarly we can distinguish three vectors by looking at spatial Reflections and transformation properties the vectors e and p flip sign under spatial reflection so they are polar three vectors but b and a do not flip sign under spatial Reflections so our axial relative three vectors however while the combination of components Delta P transforms as a proper form four vector and Omega a transforms as a proper Force pseudo vector the combination pause of the components in E and B so the if you take all of the bi vectors it transforms and was as what is usually considered as an anti-symmetric rank to tensor which is a bifector as we indicated in 3.1 so we've already talked about that right that e and B together are just one big bifector and as being a bifector it's got all the anti-symmetric properties of a rank two tensor we did that we discussed that a few lessons ago for convenience we summarize the various expansions of General multi-vectors in table seven comparing the proper expression of 3.7 to the frame dependent expression of 3.47 makes the geometric origins of all these transformation properties clear so what they're saying there is they're saying their what are they saying they are this is this is the 3.7 right if you look at all these things this is the straight geometry right this is little pieces of volume little pieces of area little lines a point and a little pieces of hyper volume right and so you take this stuff and you compare it to this just like we did and now we can say oh the reason that you know e is a polar Vector is because it's a relative three Vector built off of these little planar objects and that that's how it works right so this is where we sort of attach geometry to the things we knew in standard physics is by understanding this and by understanding this this is the geometry actually lives in here ultimately I'm pretty sure that's uh seems to be what they're getting at um and uh the proper combinations of relative quantities correspond to distinct geometric grades of a space-time multi-vector right and uh so the proper combinations of relative quantities what they mean there is they mean things like T and uh T and and I guess I guess X is a good example right T and X this is the this is a combination of relative quantities that really forms a proper object X right so they correspond to distinct geometric grades of a space-time multi-vector so this is a grade one object despite the fact that this combination is a grade zero and a grade two object where we can we know that they're related to this grade one object through the uh this Duality transformation so that's that's what they're saying there and distinct the geometric structure is left implicit in standard Vector tensor analysis methods even though the transformation properties remain so the space-time algebra makes this structure explicit intuitive and easily manipulated in calculations so what I think they're getting at there is that if you look at if you look at this right um if if you're working just in our classic you know classic Elementary physics if you're staring at elementary physics you're looking at you're looking at uh scalar objects you're looking at things like Alpha you're looking at electric Fields you're looking at just magnetic fields right just the magnetic field you're not not this I is not part of it it's just a magnetic field you're looking at momentum vectors or if you're doing somewhat intermediate work at least you're looking at at least the momentum four Vector right P mu and nowhere are you really looking at pseudoscalers but with all of that you have to remember oh this is a polar Vector this is an axial Vector with these maybe I'm giving it this arbitrary little signal this is a this is a invariant scalar this is a pseudo scalar right and you but but we don't tag it with these singles we just have to remember right now what they're trying to say is no you don't have to remember anymore because uh this structure here is really designed just to make contact with the world that we've worked in and to see why things are the way they are fair enough right I mean that's that's sort of the point I guess is to make contact with this abstract Clifford algebraic structure with the regular physics that we use and this statement right here is that connection all right so I like that what next you know after all that writing I just did the paper of course has a beautiful chart because they understand how important this is but the graded form that's what we in the complex form and the relative form all of this is fully embedded in the space-time algebra and if we had done the historical stuff we would have seen that you know historically you were dealing with these quantities and there was a historical Trend to use these quantities in the past and um so it's just kind of interesting to see how this is all laid out but this gives you a nice complete list of all of these things the pseudoscaler the relative axial 3 Vector part of X right so um so the only I guess the only thing I want to call attention to is this part here like let's look at B as written right here right the expectation is that b equals B1 Sigma 1 plus B2 Sigma 2 plus b 3 Sigma 3. so if you just look at b as literally written right there forgetting about this I it is a relative three vector and a relative 3 Vector just like this e here um does uh it is all by itself is mapped into our physical space as a polar vector so this B here as written is really a polar object in the sense of how it it handles itself under reflections I'm not quite sure why they didn't write b i here to capture the axial nature right until you multiply by this I the axial nature doesn't become apparent right so but I think what's going on here is what they're trying to say is that when you work in regular physics you deal with a magnetic field B that looks exactly like an electric field e right there's no mathematical distinction in the in the way these quantities are written down and and worked with right they are the same type of vector there is only one type of Vector in regular three-dimensional gibbsian style geometry but we remember we keep in our little p brains oh but remember B is axial not polar so we we carry that around with us in our brains so I think that's what they're trying to get at here is B is what we normally think of in regular physics as an axial three vector right but uh and it's and uh but it's the relative axial three Vector part of F Well F only has one relative three Vector part and that's e the axial 3 Vector part has to refer to b i right so I found that a little confusing I'm not sure maybe there's some other way of understanding this but um uh just keep that in mind when you when you look at this so now they're going to introduce A New Concept called relative reversion so let's study that relatively quickly I don't want to get bogged down in this one too much just because um you know it's it's it's sort of obvious now I mean these things shouldn't be too hard to see I don't think we have to do an example of every single thing anymore but this symbol that we normally use in physics for adjoint uh they use for in this paper relative reversion now sometimes be careful I've seen papers use the dagger for reversion straight reversion right and uh I've seen different symbols for reversion I've seen even other symbols for reversion but relative reversion you don't see not a lot of papers uh well there are papers that don't even mention relative reversion but let's get a handle on it relative reversion is simply reversion both right and left multiplied by gamma zero so if you take if you were to take our favorite new completely relative form of M and execute relative reversion you would just see a couple key sine flips you'll see a sine flip here a sine flip here and a sine flip here so the uh the relative three Vector part um of both sections of both partitions of M changes sign and the pseudoscaler and the the pseudoscaler and the pseudo vector or the I guess the Dual relative Vector part flip sign so you know we I guess we could work that all out but just remember you got to move this I in here because we're taking the relative version of the entirety of M right well so for example let's let's work out this one so we consider beta I which is beta gamma zero one two gamma three we take its reversion the reversion of this whole thing is beta gamma 3 gamma 2 Gamma 1 gamma zero we just reverse each one of those then we slap two gamma zeros on both sides this guy squares to one so that goes away and then I see I count one flip two flips three flips and three flips three flips and that so you get a minus gamma three two one zero and I equals I reversion and this is I reversion here so it's just minus beta I so beta I goes to minus beta I under this operation of of uh reversion and pre and post multiplication by gamma zero so that's where this negative sign here comes from and you can do the same for beta yourself right so you can kind of work all of these exercises and it's a good training right prove to yourself This is like I don't give homework in this thing I just assume you guys are struggling with this at the level you care about but you should be able to demonstrate each of these negative Signs by just the similar work that I had just done so if we compare with 3.47 where's 3.47 what was that so 3.47 is just our expression for the fully relative form of a multi-vector that's what 3.4 so without these sign changes right so I guess we're going to compare they want to compare the relative reversion of M which is gamma zero M adjoint gamma zero they want to compare that with this and let's see we should see the minus sign here see if I've memorized this right we put a minus sign there put a minus sign there I think those are the three places that we showed it let's see if I got that right so we're comparing this to 3.47 we note this gamma dependent involution has the effect of proper complex conjugation for the complex scalars so uh but the effect has conjugation with respect to the space-time flip for the complex bi-vectors okay so first of all gamma dependent involution well clearly this is gamma dependent right if you have different Gammas this this process is going to change if you change reference frame so it's a gamma dependent involution what's an involution well if you do this thing twice right if you go with M dagger dagger what will you get you'll get gamma zero M reversion gamma zero and then you'll have to reversion the whole thing gamma zero gamma zero and the reversion of this guy inside is going to be well it's going to go gamma zero M reversion reversion gamma zero so it's going to go gamma zero m double reversion gamma zero that's because the double reversion or the version of a vector is just a vector right and then you have gamma zero out here and so these gamma zeros combine to give you one so you have M reversion reversion which equals n right so that's why it's an involution right it's an involution because uh repeated application Returns what you have somebody asked in the comments what's the difference between an involution and something that's item potent idem potent right right there's a difference right involution and item potent are the same are different um item potent means if you apply it once and you apply it again nothing happens so if I have an operation on x o x equals y but o y is going to equal y again right and forever so you apply it once and you and it and it never never changes that's my recollection of what item potent is so what it's saying here now is that uh this operation of relative reversion on a complex scalar is effect is the same as complex conjugation of the scalar okay fair enough I mean it's pretty easy to see we know that Alpha doesn't change and we know that beta I does change so that makes sense and but it has the effect of conjugation with respect to the space-time split for complex bi vectors so if you take a a bi Vector F right f is always going to be split into this relative 3 Vector plus another relative three vectors dual right they're saying oh this relative reversion is going to change the sign of this thing right [Music] um okay and so that that's that's pretty easy but that's not equal to f hanji the complex conjugate of f and this point is a little bit subtle because the idea of remember the idea of a complex conjugate of a bi vector right was f equals the its canonical form times e to the I Phi right where this has a fixed signature of minus one right so the complex conjugate of f is equal to f e to the minus I Phi right and what they're saying is that that obviously this is proper in the sense that it doesn't change with it doesn't change with reference frame but this thing this thing here right this space-time split is does in fact change with reference frame now if you go back to the complex scalar right what saves that well why why is it that the reversion of the complex scalar does in fact equal its complex conjugate well Alpha and beta as we learned before are unaffected by frame Transformations right so so these two coalesce to be the same in this case in this case they don't coalesce right because you've got a the space-time split is totally frame dependent and then this Vector case is a somewhere sort of in between I guess um the relative reversion of a four Vector only flips the sign of the relative three Vector part so you actually have to take your four vector write it in this relative form right that you take this para Vector you write it as a form of a dual of a para Vector that's what's going on right here and then you take its relative reversion and the relative reversion of this object will just flip this one sign I guess we could work that out actually this is a good exercise I just ran through it but we're looking for Delta plus P this para Vector the Dual of these relative dual of this para Vector adjoin so we're trying to solve this expression right here right so so we run through the definition right so we pre and post multiply by gamma zero and we take the full on reversion right and the full-on reversion is going to put gamma zero over here and then gamma Delta plus the relative Vector P will go on the right but we have to reverse Vector P remember as I keep saying the relative 3 Vector p is a bi Vector right so being a bi Vector it has a reversion right the proper Vector P its reversion is equal to itself so P reversion equals P but the relative 3 by Vector its reversion is actually significant because its basis vectors Sigma 1 equals gamma zero Gamma One well the reversion of that is the reversion of that which is Gamma 1 gamma zero right which changes sign so with that in mind we got to do this whole reversion so here's the pre the uh the left the left geometric or space-time product with gamma zero the right space-time product with gamma zero this is the reversion so I've got to calculate the reversion of P but I just showed you that each of the each of the basis vectors Sigma I reversion is going to equal minus Sigma I so this so you immediately get this sine flip I took these two gamma zeros together and squared them to one so you just don't see it anything on the left and then I distributed this in from the right and I got um Delta minus P gamma zero I guess that was sort of a wasted step right because right there if I had not done that it would have been Delta minus p gamma zero and that's what we're trying to show we're trying to show that this is true but for some reason I distributed it in I distributed it in and then I notice that this gamma zero is going to cancel the gamma zero for each of the Deltas or each of the sigmas in p and you're going to get with gamma I and that's just sort of another this is the proper form of it so that's kind of interesting and then if you factor out a gamma zero you end up with this and that factoring out is critical because um gamma inverse gamma zero inverse equals gamma zero which is what allows you to factor it out so easily so there there we've demonstrated this uh interesting statement right here all right and what next so then then they basically give you these formulas these are fun formulas right I can now use these relative reversion formulas to pull out all of these Elementary physics quantities you know I if I if I know the complex scalar form uh of M so if m is written in this complex scalar form I can do this relative reversion pull out a I can pull out E I can pull out the pseudo vector and the vector I can pull out you know the the [Music] axials the the pseudo vector the three pseudo Vector type so uh anyway so these are some facts that we verified enough of these already so but uh it's a bunch of formulas that we probably won't be using much but it the point of all this is relative reversion is what's so critical uh to in order to make this evident all right so in the next lesson now we will get to Lorenz transformations in the next lesson commutator bracket and the lorentz group and lorentz Transformations that'll be a lot of fun and I'll see you next time