Eigenbros ep 130 - Introducing Geometric Algebra

Transcript

welcome back ladies and gentlemen so we've got another special episode for you today um well not really special we say it every time make sure you guys like share comment subscribe if you haven't already um patrons thank you guys so much we greatly appreciate it me and juan you know it only helps us when you guys help support the channel you know it helps us continue things on if you guys want to support the patreon just make sure you check out patreon.com eigenbros and me and one of some exclusive 30 minute podcast audio podcasts on there every single week and then we've also got the discord which you guys can access that you know will will um let you post whatever podcast suggestions you have and all that kind of good stuff and uh oh yeah and also guys make sure you check out the website so we've got eigenbros.com twitter eigenbros instagram mikey bros and then tick tock item bros 2. so if you guys haven't checked that out go ahead and do that as well and today's topic is geometric algebra wow big soul big stuff yeah geometry and algebra are the two giant things so when you think of the two sandwiched it's kind of uh it kind of leaves you like what else is there to do with it yeah like what what does that even mean like what like geometry it's like peanut butter and jelly you can mix them too right it's like i'm sure it's a similar feeling with that uh yeah i mean it's a better name than clifford algebra which is what the math is called and not not to be confused with clifford the big red dog geometric algebra and clifford algebra the same thing except apparently the emphasis on clifford algebra is the more abstract layer whereas geometric algebra is for the you know the physicists it's more of the um the uh the reality layer of things where you try to put these concepts to actual physical you know physical phenomena but actually interestingly enough and annoyingly enough algebraic geometry is not the same thing as geometric algebra just to throw that out there in case people are searching google searching and you actually waste your time like i did for like 15 minutes looking at algebraic geometry when you should be looking at geometry algebra gotcha so thanks mathematicians for that yeah so remember those phrases do not commute geometric hazard algebraic geometry don't they're not the same exactly so uh so what'd you think of the um geometric algebra when you were just studying one what what what was your uh first initial thoughts yeah when you told me i was like okay um i googled it and didn't really find a lot of stuff which was kind of surprising yeah i know um because i know this is a topic that you've i guess you've kind of like hinted at a couple times yeah we've been trying to do it for like the past three weeks or so i keep saying let's do geometry algebra 1 then you know it's like oh we gotta let's do something else other topics come up yeah and i mean this uh first impressions in case for those of you that don't know it's a new framework in which i guess physicists are proposing to make physics a lot cleaner um more powerful well would you that's a little bit loaded maybe you can maybe you could expand on that so i would say like it it's trying to make physics a lot more um less clunky because the the formalism we have now is based on like descartes uh sort of vector formalism of physics and rights yeah and then of course among others the the real um there's really the divergences between grassmann and clifford in the geometric algebra side and then um gibbs and um heavicide in terms of the vector the vector um uh multiplication and stuff that we're used to doing with like um cross products and dot products and you know of course the dell operator pearls and divergences and whatnot and by the way this all this math was like was sort of formalized in like the late 1800s like 1880s i think is usually when that was all kind of squared away well the interesting thing is so just to give a little bit of the history is the clifford algebra or geometric algebra was created before the vector the vector stuff the you know the um like what would you call it like vector calculus vector calculus stuff um it was actually created in mid-1800s like 1845-ish by grassman basically and then cliff and then um clifford you know stepped it up a notch when he he found a unification between quaternions and uh the um geometric algebra but then basically everything paused what the hell are quarters okay so let's not get too being off the trail but just to like preface the quaternions are basically like the three-dimensional version of the um well let me think how i should say this quaternions are they're not they sound like aliens to me right and we made and some of you may be even familiar with like when erica weinstein was on joe rogan he talks about the octonians too that's the next level up but quaternions are basically like this um i guess this 4d uh thing which helps you to do rotations in in 3d space and whatnot and we'll maybe hopefully elucidate more during the episode but um it's related to like complex numbers and things that says most i'll say right now but anyway um so clifford did that he he uh he unified the quaternions with um geometric algebra and then basically everybody kind of abandoned geometric algebra in terms of the physicists when gibbs and heavyside came up with the whole you know vector the vector um uh multiplication and and calculus kind of stuff with um dot products cross products you know curl laplacian um um uh what is the other one uh cross products and then because it was a lot easier and if you remember too um some of you may know like if you're if you follow the twitter for instance you know the original form of maxwell's equations was like 20 equations and those were all seeded within that whole quaternion framework but it was really complicated you know it was a bunch of equations and then when that effect when the um the curl and the formula for vector calculus was invented you know make using that dell operator made them extremely simplified and then there was only four you know four actual um maxwell's equations that you're probably familiar with to this day but then little did we know you know i think this is and the crazy thing about this geometric algebra stuff is i think this is really leading edge stuff from what it seems you know people are not really aware or using geometric algebra that much in the physical sciences at least in terms of the like classic um framework we're taught in school like were you ever aware of gms gospel when you were in school nah man it's not even mentioned in the textbooks yeah i never even heard anyone talk about it no professors or anything i've ever mentioned it but the thing is i've heard certain concepts from geometry azure mentioned right that's kind of what i was going to point to i think i saw it in landau lift shit's um i think i forget which textbook but or which book but these concepts are kind of shared um and like some books pick pick at geometric algebra concepts and then kind of use them as they go along but typically the formalism that we get is just vector calculus formulas so yeah but we'll see things like pseudo vectors and pseudo scalars i saw wedge products before um the outer product of course i didn't really know what it was yeah i love i love how one of our i remember one of our tests scarred me because the quantum exam yeah we got we got an outer product and i was like what the hell is yes and i don't know like and the professor acted like as if we had all like right this was common knowledge what are you talking about yeah and i was like okay maybe the outer product is a cross product because you know the inner product is the dot product so i'm like okay what's the only other product we know of and it's the cross products so i assumed it was the cross product but no the outer product wasn't the cross product it was some completely new thing that i just didn't even know what it meant yeah the worst part and please if you're a professor who does this kind of stuff please do you always have to what's fair is priming the student yeah and that's not you got to prime them so they know the concepts right i thought that was a little unfair but and to this day i i didn't even know what another product was until this podcast [Laughter] talking about geometry algebra not knowing it's a concept from geometric algebra yeah which is real effed up in my opinion because i'm like wow like i feel like a lot of people especially if you guys are undergraduates watching this probably have never even heard of this at least no one i've talked to in my entire undergraduate you know i think my institutions outside have ever talked about geometrical american under maybe this is a unique perspective for american undergrads because i think a lot of the international students are just like yeah i don't know though because i mean i talk to a lot of people in school and undergrads you know and we're i've never heard anyone mention geometric algebra so this to me seems like it's one of these new areas of physics where we're just kind of starting to understand how powerful this mathematical formulaism actually is and it can be translated into so many parts of physics and unifying the mathematics with a lot of things like just for instance some of the things that geometric algebra can do is it unifies um a relationship between complex numbers and rotations in space huge like crazy you've never heard of this before it also does things like i have okay but like yeah yeah yeah in grad school i have heard about this but it was like rotations like as a as as a statement of like trivia like almost a trivial statement well the thing is we knew that when you multiply something by i it was basically like a rotation on the complex plane yeah but this one makes it so that it shows you this this crazy tie where you don't even have to evoke complex numbers really to use this very similar object that people denote as i and geometric algebra to do the same kind of operations um so geometric algebra has all these crazy links and then it's also got the link between um the cross product it's basically like the wedge product is a part of geometric algebra or let's say geometric multiplication which actually is a stronger version in some sense the cross product that works better in multiple dimensions than just the cross product yeah it almost comes out a little bit naturally from like simpler accents as opposed to like just uh right and it fixes some of these weird things that you have maybe some of these questions that you've had about certain quantities with cross products like the angular momentum and then torque um i was trying to look up beyo savar as well but i didn't i didn't have enough time to really see in there but i imagine there's probably some kind of relation there with geometric algebra as well pretty much i would i would assume anything with a cross product um has some relation in there um and then there's also some in some really other insane ones like uh you can actually simplify all of um uh electric uh um maxwell's equations into one very simple equation with geometric algebra um and we'll show you guys that and you know later on um but of course you may be thinking of the tensor the tensor version like if you know if you're familiar with the max lagrangian um but this one's another version of that it's maybe even more elegant in some sense to me because it seems uh it's similar in some respects but it's very elegant um and then there's one more thing what was the other thing do you remember one what else it unified oh it also showed you um even geometry algebra is showing that a part of the mathematics of geometric algebra can explain spin in some sense like it shows you that half rotations are the way that some of these these operations are done when looking at spin quantities to where it just arises naturally and it's not some weird thing that just is you know some odd you know esoteric physics from quantum mechanics it's almost detached from the quantum and it just comes out from the geometric algebra framework yeah you know and the more you learn about these weird abstract math things you see that more often like it even reminded me of grant sanderson's video do you remember his video talking about heisenberg uncertainty uh back in the day no no but well he does this video on heisenberg uncertainty showing that the uncertainty principle is just a product of the mathematics of these wave um these wave type mathematics and it doesn't it doesn't have anything to do with like some axiom you have to invoke with quantum priority yeah it actually just comes out from the mathematics so learning about these real abstract math things can really elucidate you on the physical picture and show you that some of these strange anomalous parts of physics that we think of are actually these deeper fundamental aspects of mathematics so terence yeah is math invented or discovered are you gonna say that no but uh um no but i'm gonna say that for lex friedman's uh podcast yeah yeah no but uh yeah the uh yeah i thought this was like very a very nice way a nice a better constructed formalism that i'm surprised is not being spearheaded yeah well it's it seems really new man yeah let's let's but before we board the listeners let's let's actually show them some of the cool physics stuff that like connections yeah let me uh just shout out the guy whose book i've been using to do research on chris doron so he's a uh can you spell that yes so it's c-r-i-s and then doran d-o-r do what i can see on my thing so says he's a researcher a founder of gm geomerics fellow of sydney sussex cambridge mostly geometry algebra physics startups music and games it's his um twitter tagline so chris j doran shout uh shout out to him that's also his twitter at it's at chris j duran and this guy basically wrote this really great book on geometric algebra i would highly recommend it um and it's really i mean guys i mean you know this is one of these things you gotta check out you know i wish i would have known about this when i was um in undergrad because you know as an undergrad there's kind of these high tier mathematics that you want to take like complex analysis or geometry there's always this crossroad you get to in undergraduate you're like i kind of want to take more math but i don't know what to choose and usually those two options are going to be complex analysis and um group theory sorry um and then you can maybe say abstract algebra too but this geometric algebra is one i would highly highly recommend to people that nobody even mentions so this is again this is and i guarantee this is going to be extremely useful in the future i just see it because it has so much opportunity it seems um so yeah and the thing is it's it's really not that hard to understand as well no it's very elegant and only requires a few basic axioms yeah it's beautiful yeah it's really a beautiful mathematics and you're going to understand it fairly well if you have any kind of understanding of like dot products and cross products yeah it's really simple guys so i would definitely recommend just reading the first few pages even of chris j duran's book um because it's really an elegant uh mathematics yeah i'm surprised there's like uh it's just some more elegant formalism i think yeah because a lot of things fall out and you'll see like the basic axioms are i think they have like they define the wedge product yeah and then what else do they define they define uh they define the geometric product which contains the dot product and the wedge product yeah let's actually look at the video so yeah yeah sure let's do it you know so basically just to preference it this is the this is the geometric product and it can be derived pretty damn simply just with the property of distribution um no i don't assume and an ant com combinativity um so it's distributive not commutative and um oh the square of the vectors must equal the magnitude squared so if you just have those three axioms you can derive this oh product form yeah do they define the wedge product because the wedge product is anti-chromy yeah you can get the wedge product from those they're from those axioms okay and um the chris j duran book shows you that cool anyway this is the geometric product even though this definition might seem strange and not that useful this geometric product is the heart of geometric algebra to try to make sense of it let's work with it for a bit first let's consider the geometric product of a vector with itself we know the inner product of a vector with itself is the magnitude squared and that the outer product of a vector with itself is zero so a vector squared is just its magnitude squared this might not seem special but consider this vector when it is multiplied by u it turns out that the result is one this means that we just found the inverse of u and thus not only can we multiply vectors we can divide them as well that's pretty neat let's see what happens when you swap the order of the arguments in the geometric product the inner product is commutative while the outer product is anti-commutative so here is the formula for the geometric product when the arguments are swapped something interesting we can do with these two equations is to add and subtract them after a bit of work we now have equations for the inner and outer products in terms of the geometric product let's talk about the geometric product between so that might have been a little bit quick for people who are unfamiliar so i guess the most um the strangest part of that would be the um wedge product most likely right one would you say so next we'll show you the relationship between the um wedge product and the cross product yeah um i i i have i have a good video here um okay so here we have uh the wedge product let's so for those of you that might be confused trying to see a picture of it um this this video here does a great job of creating an illustration of what that is a geometric uh representation of it who's this video by the way this is by uh mathoma so math m-a-t-h-o-m-a in the video basically we're analyzing the best two videos on geometric options so if you type in geometric algebra on youtube these will be the ones that come up so here we go so the wedge product is going to be symbolized by you wedge v with this this upward uh triangle symbol and what you watch v is is going to be a mathematical object composed of completing the parallelogram where u and v serve as the two sides of the parallelogram so the magnitude of u y v is going to be the amount of area enclosed in this parallelogram and furthermore uh v is going to have a certain orientation associated with it uh the fact that i said u wedge v as opposed to v wedge u is going to be symbolized by a counter-clockwise circulation and it's counterclockwise because i was going in the order u v so that's kind of in the counterclockwise direction so i'm going to symbolize it like that and that's to distinguish it from v wedge u which is actually going to have opposite orientation if i were to draw in v watch u i would draw a clockwise circulation there so this u wedge of v you can think of it as an area terence doesn't like this guy said that wrong because a u to v is gonna be um clockwise v to u is anti-clockwise let's keep going which has orientation and it has magnitude so it's kind of a neat little operation and this is actually good for any number of dimensions as opposed to the cross product which is only good for three dimensions these oriented areas that i've been talking about are also called bivectors and hopefully you can see why they're called bivectors because they're formed by joining up two vectors together to form these oriented areas and now i'd like to tell you about some of the the properties of these of this wedge product one of the most important properties is that you wedged with u any vector wedged with itself is going to be zero now why is that a reasonable thing to say well remember we're concerned with the magnitude of this area enclosed within the parallelogram so if i take some vector u and another vector the same one u how much areas enclosed in that parallelogram that's zero notice too that this is very similar to the cross product too if you cross a vector by itself it's it goes to the zero vector and another property that this wedge product shares with the cross product is if i take u wedge v that's actually equal to minus v which u and those are the two important properties of the wedge product i'd like to know about that a vector wedge with itself is zero and that the wedge product is what's called anti-commutative that means when you switch the order of the two vectors you got to stick in the minus sign there anyway so that's so you're saying that yeah so that basically is what the wedge product is i know it still kind of doesn't explain it fully but just know that the wedge product is basically the cross product in two dimensions well they kind of show actually he continues on to show how that pops out which is kind of cool okay should we keep watching yeah yeah let's keep watching so yeah so uh v is minus v ygu and going back to the pictures if u wedge v was this oriented area this parallelogram here this area with a counterclockwise circulation v wedge u has the same amount of area the same absolute value it just has an opposite orientation as i it's a clockwise circulation there with these two concepts here and to show you that let's consider two arbitrary vectors in the plane u is going to be equal to again a times e one plus b times e two and v is going to be c times e one plus d times e two and we're going to do now is going to we're gonna consider uh v which just to write out is a e1 plus be2 wedged with ce1 plus de2 another assumption too that we're building into this web product is that it behaves nicely that for example now i'm going to distribute it so that's one assumption i have to build in so i'm going to do precisely i'm going to distribute and i also have an assumption that i can pull scalars out so i can take these a's and c's and move them around as i like one hour later and now we're going to use this first property that a vector wedge for itself is going to go to 0. so that gets rid of that term and that term because that was e1 yg1 and that term was e2 yg2 so that leaves me with ad e1 yg2 but now i'm going to take a look at this term i'm going to actually use this property that if i flip the order here i got to stick a minus sign in so i'm going to write this as minus bc e1 wedge e2 and i'm going to just combine the coefficients here and what i'm left with is this very interesting quantity ad minus bc so i say that this scalar that we're getting out front is indeed very interesting because it shows up in in many different ways in math you may recognize as a d minus b c as the determinant of a two by two matrix where the columns are the two vectors a b in the first column and c d in the second column so the determinant of this thing is indeed a d minus b c and you also know that one interpretation of the determinant is the what we're saying before it's the area of the parallelogram formed by the two column vectors or row vectors same determinant so that's one interpretation of this quantity a d minus b c another one is that if you really really want to take that cross product you can consider this vector to be a b 0 where 0 is just going to be the z component and a and b are the x y components and then this this vector here is going to be c d 0. if you take the cross product between those two the answer you're going to get is purely in the z direction and the magnitude or i should say the component in the z direction is a d minus b c so that's yet another interpretation of this quantity a d minus b c but the key point is that taking the wedge product its magnitude this ad minus bc that's indeed the area of the parallelogram formed by those two vectors yeah so there you go so it pretty much looks like the cross product right at least in two dimensions for sure um and there is that relation between the cross product and the wedge product so you can think of them almost as the same thing if you're in the right dimension yeah obviously this scales to three dimensions and so on and so forth so now one of the weird things about the geometric product that i've noticed that is something that we don't ever encounter really in physics or it's a it's a big new part is the fact that you know there's this wasn't shown in the video yet but what you can do with these geometric products is the dot product portion if you remember there's the u times v so u v would be the g of x product and that equals u dotted with v plus u wedge v the u dot v part would be considered the scalar portion or rank or grade zero of the um geometric product then the bivector portion which is the um wedge product part u wedge v that would be considered the bivector or um rank what is it rank zero rank one i forget if it's rank zero rank one oh yeah i think it's rank one i can have our esteemed robot assistant corrected for us if need be yeah so the thing is um yeah you have this weird thing where you can combine these quantities with different um they would have different units of physics like a scalar you cannot can you cannot combine we'll even think on different units the thing in physics is we know you cannot make a linear combination of vectors and scales right we just don't that doesn't happen in physics you can't do that you have to have them have the same they have to either be vectors all linear com combinations with vectors or scalars all linear combinations with scalars but this new object called multi-vectors which introduced in geometric algebra allows you to combine scalars vectors bivectors tri vectors even quad vectors so on and so forth so it's really weird in that sense and that's still something i'm still trying to wrap my head around exactly because um it's it's something that's if you're you know if you're a normal uh you know you're a good physicist like every physicist should be you know you um would not combine your scalars and your vectors so this is something that you may be unfamiliar with with this new concept of a multi-vector that's introduced in geometric algebra yeah it's it's a little bit like you have almost an automatic like uh recoil or something yeah yeah almost like a parasympathetic response like you're doing right reflex it's it's weird you don't you don't uh it feels uncomfortable when you're uh in physics because you're like you're disgusting how could this be yeah how dare they but um yeah that is one property you have to get used to in geometric algebra and i think even it makes sense if you go through the math enough and it doesn't feel like such a violation but um yeah you just gotta get used to that and it indeed feels the beginning yeah it indeed feels like a violation yeah but i think it even becomes understandable when you look at the math more so yeah there's a more legitimate even yeah yeah and there's a great like illustration that i'm trying to find here yeah this so this video is called the vector algebra war it kind of goes through the history here by uh by uni adele and uh and here they have a good little illustration of uh of what that linear combination terence was talking about here clifford's system does more than simply provide an elegant system of vectors it also allows a description of orientated planes given by the three bi vectors as well as an orientated volume given by the tri-vector and a point is given by the scalar we can now see that the key reason that quaternions can be mistaken for cartesian vectors is that in three dimensions we have exactly three rotational degrees of freedom as well as three translational freedoms this is illustrated in clifford's system with three orthogonal lines and three orthogonal planes this simple overlooked fact appears to lie at the heart of the disagreement between the followers of hamilton and gibbs the clifford multi vector though unifies these two descriptions thus reconciling the two sides and providing a unified and powerful formal in formalism superseding both quaternions and gibbs vectors this diagram also explains why clifford alger is eight dimensional note that we have one scalar three vectors three bivector planes and one trivector volume this totals eight elements yeah so yeah yeah yeah so basically that's a really nice illustration so for the audio listeners what it shows is this linear combination of these basis vectors so you know a lot of times in physics they'll represent the bases as e1 e2 and e3 that's pretty much the same thing as just thinking like x y and z hat um so like the i j k yeah exactly that's all the same right um but what they have is you have this one scalar part you have these three vector parts so that's e1 e2 and e3 and then you have three bi-vector portions which is e1 e2 multiplied together the geometric product then you have e2 e3 multiplied together geometric product and then you have e3 e1 multiplied together and then they have a final trivector part which is e1e2 and e3 this is for a three-dimensional space of geometric uh product so it changes like the the pseudo vector and the pseulo pseudo scalars will change depending on the dimensionality so i think we kind of got a little bit ahead a little bit there sure if you're kind of confused don't worry too much we'll go back some but um yeah it just shows you that there are there's this the thing we're trying to get to is to show you this multi-vector that is made up of these geometric products when you you know when you when you do geometric products in in uh you know higher dimensions in like two three spatial dimensions so you get this scalar vector bi-vector and tri-vector components and you can have them in linear combination which is something you probably would think of as unusual if you're used to doing physics yeah so right by now so if you're a physics listener probably you're like well okay so the math kind of makes these little things pop out that's really nice yeah but where's the utility in this right and that's good because actually it's the next video so um there's a there's some use so anything that really has a cross product in it i've always had this weird feeling where it's like what does it actually mean to have this cross product vector so if you're if you understand what i mean it's like if you imagine the torque so this next example is about the torque torque is equal to r cross f right and when you take the cross product of the radial vector and the force component let's just imagine they're completely perpendicular you get this strange perpendicular to both of them either pointing up or down and then you're kind of like what does that even mean like what does that represent physically same thing happens with angular momentum right and the thing is this is kind of one of these things that arises from the unnatural aspects of vector algebra because it actually really doesn't mean that much it just shows you okay the vector points out and shows you a directionality which can tell you the way in which the system rotates with the cross product yeah that's true but it feels less natural than what you'll see with how they do it with geometric algebra since geometric algebra when you do a two dimension when you do a vector on two or i'm sorry when you do a um a wedge product on two vectors in geometric algebra you create a bi vector which is basically a plane which just has an orient a road an oriented it's an oriented plane so it's all contained within two dimensions instead of this weird thing that happens with vector algebra where you multiply these two vectors and you get this third component with an with a with a vector in the direction of either up or down let's say normal to the surface yeah normal to the surface thank you that's even better um yes so you it's kind of like one of these questions you would have probably like what does this mean and it's kind of it doesn't really mean that much of the physical picture it's an abstraction in some sense right which you're not really used to right in physics because you know when we see force vectors that actually that force vector is relevant when you're pushing a force in a certain direction the vector shows you the direction it's being pushed in so this weird thing happens with the angular specifically these angular quantities where you get these weird pointed vectors that kind of the meaning feels less natural right right because it's com it's almost like a the resultant vector isn't it coming out of the plane of the two vectors that these that they that they span right it's it's like how do you how do you go from yeah how do you go from this to this yeah it's this two-dimensional thing to this 3d now vector that's pointing out of a direction that doesn't really even it doesn't suggest to tell you that much physically but we know mathematically that you know r cross f in that order will give you a vector that points you know let's say um counterclockwise and then f cross r would point uh clockwise so i guess that kind of gives you some meaning but it feels less natural as the is the point and that's not to say that we don't think that there is a resultant like quantity that that looks like for instance like the torque like r cross f like there is there is a resultant quantity that a resultant vector yeah there is a result there's a resultant force right sticks force no i know but okay okay yeah that's probably yeah careful i need to be careful that's a key part right because that's what you would think too that's what something that kind of tripped me up as an undergrad because it's like when you do a torque wrench right you everybody kind of knows from experience when you torque when you push a torque wrench too far you can actually get the thing to pop up along that third direction in some sense but the thing is torque is not a force and like that that vector is not really telling you anything about a force in that direction the force is perpendicular so just don't get yeah okay yeah don't get confused with that so this is this is it's showing you a little bit about the torque the connection between torque and geometric algorithm let's do it oh cool so we don't really need the outer product if we can express it in terms of the cross product i would say it's the other way around we don't need the cross product and in fact we have been held back by the cross product every time we see a cross product in a physics equation it should produce a bivector not a vector for example consider torque if you are pushing something with a force f to rotate it at a radius r the torque is traditionally defined to be the cross product of these two vectors this causes the torque to point in a ridiculous direction that has nothing to do with the rotation now watch what happens when the cross product is changed to an outer product in this case the torque is this bivector we already know that bi vectors represent rotations so now the torque is a rotation object as it should be and it is oriented in the direction of rotation this is much better than representing torque as a vector that's not even in the plane of rotation some of you may have gotten upset at me for using the terms pseudo-vector and pseudoscaler when they are already used in physics in another way well they're actually the same thing every pseudo-vector in traditional terminology such as angular momentum and the magnetic field is actually a bivector and every pseudoscaler in traditional terminology such as magnetic flux is actually a tri-vector the only reason we have been confusing them is because of a mathematical pun the fact that the vector and pseudo-vector bases and the scalar and pseudo-scalar bases have the same size so yeah so the torque section pretty straightforward right this bivector is more of a rotational object than the cross product is so it's kind of nice to use bivectors in that sense so can we can we talk about a little a little bit about why that's sort of defined as a curl or rotation why that's not i mean that that wasn't clear to me essentially like it almost seemed like because because there was an example of like the bivector forming an area the wedge product forming an area and how that's like uh two vectors forming a parallelogram almost yeah and how that also defines a curl uh how does that show rotation yeah how does that show rotation and that well that's a good question that comes from the anti-commutativity of the wedge product so when you do it's just like in the cross products so when you look at u cross v let's imagine those two as the arbitrary unique vectors right if you if you you do you cross v just like um using tail to tip notation just like anything else to create the parallelogram when you go from u to v if you can just use pretty much the right hand rule going from u to v you'll curl in a certain fashion and then when you go to v u you'll curl in another fashion you'll get a negative sign so the negative sign will tell you if it's curling one way or the other you can do the exact same thing with the cross product too except with this one the bi vector you're not actually getting another vector out actually so that's kind of the key it's kind of like this um a more it's a it's a quantity for it's an object a mathematical object it tells you the area of that spans the vectors right it tells you the area as well as the direction in which it's rotating but it's it's like a little bit different in some sense um and i know and it's kind of confusing but i looked at the definition with bivector exactly on wikipedia and they mentioned that bi vectors are different from um vectors from pure vectors in the sense that a bi vector when it's reflected does not match its mirror image whereas a pure vector when it's reflected does match its mirror image and that basically means like if you they use the example of the um current in a magnetic field so if you imagine a wire loop of current when you have the wire loop of current um let's say if your magnetic field is going um uh uh uh clockwise right that means your magnet or say your current is going clockwise your magnetic field is going to point downward right so then we reflect that over an axis that should be pointing downward if it's a vector pure vector but the thing is it's not it's a pseudo it's a pseudo vector or bi vector or slash vector right so what happens yeah the pseudo vectors and bi vectors are equivalent in um three dimensions i believe i want to say um yeah so what happens then is if you actually reflect that picture in real life what happens is your current is now moving um counterclockwise so then of course what happens to the magnetic field which way is then yeah it should be pointing up yeah right so that's the weird distinction between the bivector and vector the vector it should be completely reflected just it just looks like it's mirror image right the bivector apparently does not look like it's mirror when it's reflected so i guess that's something to hold on to if you're trying to figure out remember the distinction between bi vectors and vectors and that's a problem huh no it's not a problem it's just it's just another mathematical tool right yeah and it just also this is and the whole reason i even got into geometric algebra was the simple question of is current a vector and if you think about it we've denoted current with a vector right density at least right current density and current right like think of the coaxial cable problem where we would have current going down one part of the part of the cable and then another the current going another direction through the center yeah i guess i guess but but there is just called negative eye yeah well i see what you mean yeah yeah but we would still dinner with a with a vector line but the thing is if you think about that current does not current does not follow vector addition no right not at all like current current only follows scalar addition so that's why current is this pseudo vector this bi-vector pseudo-vector quantity and it's not actually a vector interesting um yeah but yeah so that's one do we have do we have any more examples of this yeah where the utility comes into play that that illuminates other interesting phenomena well i'm glad you asked one because i think we actually do great yeah so this next video has to do with spin spin eh yeah i hope it's not too i'm trying to fly i hope it's not too confusing but i guess it's just to show people one of the things it can do sure that may have all seemed really complicated but the gist of it is simple to rotate a vector v by an angle theta in the plane i where i is the unit by vector in that plane you use the complex exponential with half of the angle the angle is cut in half because we have to rotate twice when used to represent a rotation in this way this value is called a rotor something nice about this equation is that it works in any dimension notice one interesting thing about rotors as we increase the angle of the rotation the rotor doesn't spin as fast as the vector in fact once we have completed a whole rotation the rotor has only gone through half a rotation this is because the rotor is applied twice to the vector if we wanted to bring the rotor through a full rotation we would have to do another rotation on the vector wait a minute this looks like a spinner from quantum mechanics the way that spinners rotate is always said to be a part of so-called quantum weirdness but in fact it's just based on the fact that the best way to represent rotations involves applying the rotation twice i could keep going about rotations but i'll stop here okay so that i'm sure the viewers are like what the hell is even that mean so hopefully we can backtrack and you guys can just remember that um but i just wanted to demonstrate or show you that um there is a relation between spin and spin oh and geometry sorry i can't think of the same thing so the next thing is like what the hell so the thing is it's it's got a relation with i the complex i so i can show you this next part hopefully it elucidates you how i relates to the bi vectors yeah this is the imaginary quantity i the square root of negative one invented well was it was invented well it was discovered i'm just kidding it was uh it kind of got this treatment of like putting it into algebra right from i think was it gauss it's like x plus i x plus i y or something i thought it was um descartes no euler spoiler thank you so it was given it was given this like cartesian treatment and a lot of people try to wrap their head around like what the hell like does an imaginary plane mean so if you know euler's formula is this really powerful and you've probably seen this tattoo on all kinds of physicists calling out calling out the uh no hate but you know e to the i theta plus one equals zero is derived from euler's formula yeah so and now and now you're about to have an outdated uh you know in a hundred years when geometrical is implemented the real physicists get the tattoo of the geometric algebra version of this all right own it so they seem like scalers as we'll see in a moment they do have a few differences so we'll call them pseudoscalers we will also give the unit pseudoscaler x-hat y-hat a new name i what's so special about i and what makes it different from normal scalers the main distinguishing feature is how it multiplies with other objects let's consider multiplying a vector by i consider this vector which is two x hat plus three y hat what happens if we multiply this by i we can work out the algebra of the products like usual in the end multiplying by i ended up rotating the vector by a right angle remember that the geometric product is not necessarily commutative so let's see what happens if we multiply by i on the left so multiplying by i on the left ends up rotating the vector by a right angle again but this time in the opposite direction now that we've seen how multiplying vectors by i works let's consider multiplying i by itself we can calculate this product just like any of the other previous ones so i squared is negative one wait a second this equation looks familiar yes it turns out that imaginary numbers are actually pseudoscalers this is why i is used to represent the unit pseudoscaler also complex numbers are equivalent to the two-dimensional multi-vectors that are the sum of a scalar and a bi-vector in fact the geometric product of two of these multi-vectors is the same as the product between complex numbers in addition as we saw earlier multiplying a vector by i is the same thing as multiplying a complex number by i this fact holds in general and multiplying a vector by a complex number acts like complex multiplication rotating and scaling the vector this actually makes for a very simple way to perform rotations say you want to rotate a vector v by an angle theta all we need to do is find the complex number that represents this rotation if you know your complex numbers you know that this is e to the i theta now you might be freaking out about the fact that we're raising a number to a bi-vector power but remember bi-vectors are imaginary numbers so we can raise a number to a bi-vector power the same way that we raise a number to an imaginary power multiplying by this value rotates the vector by theta we can also try multiplying on the other side when we multiply on the left by e to the i theta it ends up rotating the vector in the other direction again by an angle of theta remember that with complex multiplication multiplying by the conjugate rotates in the opposite direction thus when multiplying a vector by a complex number on the right it is the same as multiplying it on the left by the complex conjugate so there you go so another beautiful discovery from you know geometric algebra you can treat these this geometric product of these two bi vectors as a rotational eye you know and then i guess finally let's look at the last part since we're running low on time just to show to finish off with geometric algebra how it unifies um maxwell's equations into one single equation algebra you may see descriptions such as a unified mathematical language for the whole of physics or the most powerful and general language available for the development of mathematical physics i was a bit skeptical when i first heard these claims sure it's a nice mathematical tool that i'll keep in my tool belt but it can't be this good however my viewpoint changed entirely when i saw what geometric algebra did to maxwell's equations for future reference here are maxwell's equations there are other ways to formulate maxwell's equations but this is the most well-known version some of the other ones are simpler than this but oftentimes they just hide the complexity in notation okay the geometric algebra version i am about to present might seem like it hides the complexity in notation as well but i would argue that each new bit of notation makes perfect sense first let's consider the differential operators included in maxwell's equations there are two of them the partial derivative with respect to time and the gradient which is the sum of the partial derivatives with respect to space we know from relativity that space and time are related so it would be useful to combine these two operators well in geometric algebra we can add scalars and vectors so why not just add them before you get confused by my use of the gradient symbol again notice that the spatial gradient has a vector arrow while the more general gradient does not one issue with this equation is that the units don't quite work out we can fix this by adding a factor of the speed of light to the time derivative doing this may seem familiar if you've worked with relativity enough this combination also makes sense because we now just have the sum of four derivatives so in the end it's pretty similar to the traditional gradient next let's consider the sources that create electric and magnetic fields there are two of them the charge density and the current it would be useful to combine these as well just like before we can just add them actually subtract them don't worry about the minus sign just like before i used the same letter for both the vector part and the whole value again the current has a vector arrow while the more general source does not this equation also has some units issues which can again be fixed by adding a factor of c to the charge density again this combination is seen in relativity so it's nothing new in physics it also gives an interesting interpretation of charge density as a current that is moving through time instead of space finally we want to combine the electric and magnetic fields in some way that makes sense unlike before we cannot add them the issue here is that both the electric field and magnetic field are vectors so their components mix however remember that the magnetic field is usually defined through a cross product thus the magnetic field should actually be a bivector not a vector thus our electromagnetic field will be the electric field plus the unit tri-vector i times the magnetic field remember that i turns a vector into a bivector we have a unit problem again but we can fix that with another strategically placed factor of c some people actually call cb at the magnetic field so this factor of c is not too strangely placed this combination of the electric and magnetic fields should make sense the electric field is a vector the magnetic field is a bivector and this is the simplest way to combine them anyway we now have everything we need to describe maxwell's equations with geometric algebra are you ready that's it this is maxwell's equation not equations plural but equation singular this one simple equation describes all electromagnetic phenomena and remember we did not set out to get here we initially just wanted to multiply vectors and the natural progression of things led us here okay i will admit i cheated a bit this equation is using natural units in si units the equation changes slightly with a couple constants but still this equation is incredibly simple to show that this is equivalent to the traditional set of four equations first we can expand out all of the definitions we made earlier next we can distribute this product on the left these two terms involving the gradient can be expanded in terms of our original definition of the geometric product this seems like a monster at this point but by moving a few things around something can be salvaged if you look at this equation you can see that on the left hand side there is a scalar part a vector part a bivector part and a tri-vector part and on the right hand side there is a scalar part and a vector part there is also a bivector and tri-vector part on the right-hand side as well but they are both zero for the whole expression to be equal the scalar parts must be equal the vector parts must be equal and so on thus from this equation we can make four new equations the first equation is gauss's law in the second equation let's deal with these constants first divide everything by c c squared is related to the electromagnetic constants so we can substitute we can use the equation found earlier relating the outer product to the cross product rearrange and cancel the negatives we now have ampere's law for the third equation all we have to do is write the outer product in terms of the cross product cancel the i then rearrange we now have faraday's law finally for the last equation we can just cancel ic to get the last of the four equations thus the original equation is in fact equivalent to maxwell's equations beautiful i think that says i think that says enough about geometric algebra and how useful it is wow and how much slept on it is so i think uh and judging by those quotes one was by chris duran who i shot out earlier and the other one was by um i forget his name um maybe we can go back to the quote not that far um he's another guy that's uh alive today the one that said a unified mathematical language david hastings all those guys are like fairly youngish at least from like what 30s to 60s or something so this tells me i think this is new this is new stuff i don't think a lot of people are in tune with geometric algebra just yet well the thing is uh i know that the clifford algebra stuff which is like the mathematical formalism behind all this is also from the 1800s and like right there are questions yeah i mean there are questions about like why vector calculus won out over this i think it's just because of simplicity right because geometric algebra is a little bit more tough to really elucidate all the things and it's like at that time they were just becoming familiar with the formulation of maxwell's equations in its you know in the four kind of the golden four equations that we typically think of and now we're in an era we're like looking for even more powerful simple things so it makes sense for the vector algebra to come first before geometry algebra to me in terms of adoption by physicists so this is really a nice opportunity i would say for people to really become aware of geometric algebra and even like talk to your professors about this because as far as i'm aware i don't i've never really heard anybody talking about it so i would like to get this more into the consciousness of you know physics and you know everybody in physics talking about this more often well there's a lot of uh i feel like pseudo-vectors and pseudoscalers like those concepts are all like there's a sort of nudge-nudge wink wink kind of treatment of these things so it's like it feels like it's one of those things where professors like gatekeep certain knowledge i mean why would they though i i don't know like just to seem smarter than you in some ways i think that's because like there's some professors sure but there are some professors that do kind of have like like uh like it's like oh group theory is really important but right but and it's really useful but they're like yeah you don't really need it meanwhile it's like really powerful and like really useful right right okay maybe it's just like they're trimming the fat and giving you what you need yeah because like this is this might be one of those things where it's like wow this is really powerful and useful yeah why so but they're like yeah you don't really need that maybe so but the thing is at least i hear about group theory yeah i've never heard of geometric algebra until this podcast yeah yeah same i mean it's it's really useful it's really cool um i think it's got a lot of really interesting and bold things to say um like selling you're writing a article on vogue or something well i mean looking at the maxwell's uh equation simplification of it yeah it's um yeah it's i love it it's great it's really easy to look at and then it's easier to kind of throw away a lot of terms that if you're on an exam or something you're just like oh okay i'm looking at a static problem and a lot of the dynamics goes away like it's easy to discern like which moving parts you need um because you see the whole the whole equation set the whole linear combination of of the yeah i'd like to explore it more and understanding really how you can really have these multi-vector mixing of these vectors and bi vectors and such because that's a really big concept that i still don't really think i wrap my head around i think it's just mapping like i mean that that last term mm-hmm they were just mapping like the equivalency right they were like okay the first term maps to the the row on this side but it's one thing to map it to those things and actually like manipulate these equations right and see like the bigger consequences that arise from them so that was like a great example for a first pass for sure but i think um yeah i think we need more people to elucidate on this subject and especially more physical from the more physical side of things um you know i like to see like how this gets involved with bio savar if it does at all you know and other other relationships because they have the cross product right stuff so i'm just thinking of anything with a cross product that's really interesting um and yeah i guess we'll just see you know see in the future but any final thoughts i mean i i think it's going to have a hard time because then you have to write whole textbooks of vector calculus you have to write rewrite well yeah but that could be opportunity to write a new book people are always writing books yeah there's no shortage of people who want to write books yeah yeah i mean it's so this might be the future guys i think this could be potentially the future of the mathematics of you know physics and space you know physics in like geometric space you know we usually think of vector calculus as that framework or victor algebra whatever um but i think geometric algebra could be the future yeah it just seems too powerful not to be so anyways yeah i mean what do you guys think yeah like and if you guys are more if there's people who are more like in tune with this stuff you know let us know if we're like hyping it up or let us know if you know there's what are the shortcomings yes there's some shortcomings we're not seeing or something because for what i'm seeing right now i'm only seeing all positives i don't really see shortcomings so you know if you're more hallucinated on this subject by all means please uh feel free to comment yeah yeah we'll pin we'll pin the top comment and uh and yeah just uh or if you have good resources to learn about this more if you know please let us know drop it in the comments for the the listeners and uh or the viewers and uh and yeah let us know what you thought useful is it gonna be useful is it the future let us know alrighty well with that said guys just make sure to like share comment subscribe as always check us out on eigenbros.com mygambro is on twitter i give bro's instagram my gimbals two on tick tock and thank you guys once again the patron subscribers you guys are greatly appreciated by me and juan you know we couldn't do without you guys so if you want to check out our patreon it's patreon.com bros you'll get a 30 minute audio podcast every week me and juan we'll talk about random crap i think uh was this last time we talked about music or something yeah just music yeah and then um yeah and then check out the discord as well if you're on the patreon and uh yeah i think that's it i think that's it all right i'll see you next week see you guys