Unit Quaternions and Electrodynamics

Transcript

[Music] our next speaker is alexander who is going to talk about plutonium physics and the isoclinic decomposition of so4 thank you very much okay thanks for coming uh at first i would like to thank the organizers for their great hospitality it's a pleasure to be here again and i will talk about quartonians yes uh let's start how the story began with um hamilton and uh hamilton noted hamilton invented the quartonians and he noted that there is this peculiar three plus one structure and uh this is indeed something to explain i mean as a physicist you uh ask the questions why we have a three plus one dimensional space diamond he noted that this peculiar structure naturally arises in quaternions so he always thought this is relevant for physics and well here we have an expert on space time and as a physicist i apologize because i will not present any calculations or three reams or proofs i will just try to feed your intuition and uh by visualizing the quaternion multiplication all what i want to tell you is how interesting is quaternion multiplication so how we are going to visualize this a good tool is the stereographic projection and um by the way all the graphics i show are from an internet site of ben eater he has done excellent work on visualizing quaternions you just google ben eater quaternions and you find these clickable videos and i have made some examples the stereographic projection in one dimension is of course uh you need to visualize a circle and you can project it on a line the only problem is that yeah here we have the animation little animation with a rotation here the only problem is that one point uh finishes at infinity but for the rest you can with this trick you can reduce two dimensions to one dimension and of course the same works in two dimensions and i have again a little video that shows this you want to visualize the s2 in this case and you just project it to the uh plane you by the way you can you can uh put the sphere just on top the plane or let it cut by the plane but it doesn't matter it's no it's not a fundamental difference here and eventually in three dimensions uh you have this stereographic projection projection that works as well and uh yeah you can nicely visualize this you have uh the positions of a g k and in the center is the identity and you can um visualize circles that's the i uh the jk circle and and the the sphere uh that would be the sphere corresponding to pure quaternions and this straight line is also a circle but in the projection it becomes a a straight line in the real and eye direction so let's have a look at the action of right and left multiplication and now it becomes interesting because if you look closely and if you do a multiplication first from the left you see what's happening here you you're pushing the line here the green line you're pushing the line and at the same time there is a right turn in the jk direction vice versa if i pull the line the rotation is um counterclockwise again and if instead you do the right multiplication with this numbers here you have the inverse effect if you push the uh red sorry the green yellow line you have a left turn at the same time if you pull it you have a right turn so you have a different screw sense here okay these are very uh it's a fundamental difference and you can visualize this i brought this towel you have two screw sensors this is a right screw sensor and this is a left screw since don't worry i bring that back to the hotel and but this is a very interesting feature of quaternion multiplication and well here a philosophical sideline almost i mean if you think about um well real numbers you do multiplications what about two dimensions i want to do multiplications in two dimensions okay so we end up in finding complex numbers but if you want to have a field somehow naturally arises the concept of a rotation and if you have four dimensions and you want to do something you want to do a reasonable multiplication you don't have a field but you have a division algebra but now another qualitative feature arises that of screws and helicity and this of course is very exciting for physicists because uh elementary particles do have helicity and screw sense that was demonstrated in 1956 by an experiment yeah she did the experiment and the discovery and the two guys got the nobel prize unfortunately that's sometimes the work share in physics but at the end we know that positrons and electrons they have different screw sense they behave as a right and a left screw and that's this is very exciting for for physicists and uh yeah well but last year i had an argument here with a lady who said that we mathematicians do all this for fun and i i objected i said no this is serious but that now she insists it's so i would say okay it's it's like sex it's evidently it's fun but keep in mind this has existential consequences yeah so um you need in physicists need to explain the various structures of reality and one question is how do we i mean what's the necessity why do we have electrodynamics why do we have this peculiar structures this is the question to be explained and yeah i just show you now how do you how do you realize rotations here you can do that by hand and you see i have here a complex multiplication in the eye direction and now i add by hand the opposite the conjugate and i compensate for this and ending up with just a rotation okay so uh here i have a sandwich that does it automatically and if i do this sandwich multiplication with the conjugate you end up with a pure rotation and i have done this in in other directions that's uh now the line is in the real and j real in j direction so you see a rotation here if i do this sandwich multiplication and of course also i can do it in still the other direction but you you see it's a rotation and this is um this is uh what what quaternions are sometimes known for oh it's something like a rotation in three dimensions okay so you can do this very conveniently and it's nice and interesting but even more interesting is the sorry the general i hope this is the right video no i think okay so everyone knows how rotations and and quaternion multiplication is is related you can all look all this up the connection between so3 and so2 we have this double cover also i think the double cover was was at the end of the video here i don't show it again but i will will come again but yeah i should mention that the double cover because you have two possibilities to realize that state and i think it's at the very end here um if i go to minus one minus one it's the same thing then plus one plus one so for each rotation in three dimensional space we have two elements for the for the quaternions and uh now i'm realizing the general multiplication however without these two uh numbers being the conjugate of each other and i have added some more circles and uh let's first do that with the rigid rotation which is as i said uh nice but not that interesting because as long as you have these sandwiched numbers it's a rigid rotation but if i take away and i do individual multiplications first from the left and then from the right and i can play around here and i get this very interesting distortions of the three sphere and these distortions represent the general transformation that you can apply to a three sphere you can also add a sphere here just to play around and you see this sphere then transformed into a plane because if you do 90 degree rotations and it's really nice to play around with this but this is a this is a six dimensional manifold and um yeah it's really fun to play with this is a six-dimensional manifold and uh to emphasize this point here i mean if you want to rotate a usual s2 sphere okay you need so3 the rotations in three-dimensional space so to rotate a two-dimensional sphere you need a three-dimensional transformation group but if you have a three-dimensional sphere sphere s3 you need six a six dimensional group to rotate that and this is also known of course as so4 but uh this combination of left and right multiplication with a unit quaternion is is a representation of this so4 and it's called isoclinic decomposition now again why is this so exciting for um for physics because you can again group these general transformations into two sets and one set i already mentioned are the rigid rotations which are require three parameters but you can realize also so to speak pure shifts and that would mean um involving the real direction and one complex direction and i do this by hand first i'm shifting the yellow green line and i am compensating by hand this shift and compensating the rotation and i am ending up with a pure shift here so um we have the interesting feature that these six transformations can be grouped into two distinct um distortions and that's what reminds you from electric and magnetic fields obviously you have again three plus three quantities and um it's a very natural idea to relate this general transformation of the unit quaternions indeed to the electric and magnetic fields so um yeah i should mention that uh in the uh 19th century people remember open to these um ideas there is a close relation to ether theories uh i'm talking about uh 1839 um theory of the irish physicist uh mcculloch who invented a incompressible elastic solid and shows that it is indeed equivalent to maxwell's equations to electrodynamics and so but he did talk about the usual so to speak elastic solid with a boring r3 but uh the entire thing could be much more interesting if you imagine distortions generated by multiplications of the unit quaternions and but now we have a lot of problems of course and first place i'm i'm here to ask help in asking questions because how do you realize maxwell's equations you imagine the electric field everybody imagines this as a vector but if indeed the electric field is a rotation just in first approximation you could vectors you could add vectors and the superposition principle would hold but if indeed uh the electric field is is described by rotations only infinitesimal rotations add up and finite rotations do not add up so how do you do all this how do you how do you phrase maxwell's equations properly and how do you how do you describe derivatives and as i said if you have the hypothesis of left and right multiplication related to charge and soph related to the electromagnetic fields how can you define differential operators properly can quaternion multiplication of elements close to unity look like vector addition and in general what happens if you just have a group multiplication and no addition any longer i'm talking about unit quaternions of course and all these are unsolved mathematical problems um yeah and i should um going to go to a more general perspective already i'm not sure everybody can see the headline here y c and y h this is a fundamental question in physics um to uh natural constants you're not just doing mathematics in a in a space but uh in physics you have constants and these are things to be explained okay um there is no reason why mata should not should be accelerated beyond the speed of light there is no reason for the ex very existence of of the speed of light and it it demonstrates a failure of newtonian physics at the large scale and the other hand there is no reason for discontinuous phenomena in classical physics and this constant of nature age so to speak is a failure of newtonian physics at the small scale and so again here we have we see some problems of classical physics of newtonian mechanics and these are if you trace them back you end up with with space and time and there must be a problem with space and time newton just accepted it as as something three-dimensional and one-dimensional in an exomatic fashion but as hamilton said um yeah you you you should like uh to have a mathematical explanation for it and yeah well as a sideline i mean uh maybe in modern in modern physics this is not very appreciated but the the job of a theoretical physicist is not to explain three dozens of fancy supersymmetric never observed particles the job is you've got to explain the fundamental structures why do we have three plus one-dimensional space-time why do we have the electromagnetic interaction of at all why do these structures show up in nature and not the others these are the basic things which are left to explain to be explained and i addressed these other this was a little bit a repetition of last year's talk and so yeah again we end up with a lot of questions can unit quantums describe reality the peculiar three plus min plus one dimensional form arises naturally in quaternion algebra spin is something very intriguing i talked about this last year uh there is no reason why spin should exist in first place except that it naturally arises from so2 or the unit quaternions being a double cover of the rotations in three-dimensional space so um there are a lot there are a lot of hints and seemingly um interesting connections here but it's very difficult to phrase that all that in a technical language and this as i said i'm why i'm asking the help of mathematicians uh yeah alternative descriptions of reality as i said the the usual concept is we have a three-dimensional space and time and if you do quantum mechanics you have wave functions described by the complex numbers and obviously these uh complex numbers of quantum mechanics and the vector fields of electrodynamics could be encompassed by the three-dimensional unit sphere and at that point you might also think it's not talking just of replacing the fiber but also replacing um the bundle and maybe reality is best described by maps s3 mapped to s3 or also the the algebras are interesting here or even to the general transformations of s3 which are so4 yeah i will end here with a quote of hamilton somehow quaternions are a fundamental building block of the universe and that's what he thought that's what he discovered in 1843 and by the way you have may have to update your pictures because uh recently a couple of weeks ago a grave was added with hamilton at the bridge and again i apologize because this is evidently not new trends but very very old if you're interested if you're interested uh this is last year's talk on youtube and uh this is my book the mathematical reality why space and time are an illusion and you don't have to buy it everyone can get a pdf if he sends me an email so i invite you to think about these problems and to encourage you that dealing with paternals is something very essential something very important for the description of space-time thank you for your attention participants in your talk you mentioned that there is a need for the differential operators to study the unit containers so you unique products are inside the containers and we are the data operator so what is the role of the direct operator here um i i'm talking about really about the unit quaternions so uh quaternions are another thing and i i'm not sure you can associate directly these four dimensional quaternions to the four-dimensional space-time i would rather believe that unit quaternions play a special role so but because you still have the real part and the three complex parts so um i i don't see a very direct uh application of the current formalism also the the iraq equation there are ways to phrase the iraq equation also with quaternions but um i think we need quite a very new approach here because i mean everything is literally screwed up here you can't do you can't do a proper differentiation you can't do any integration anymore if you don't have a field if you just work with unique quaternions so it's really something you needed here the tangent space plays a very very interesting role because i don't know that it has that exactly answers but the tangent space is important because if i go back to two dimensions and uh the tangent space is a plane attached to this to the s2 sphere and if i walk along the plane along along the sphere i have a sequence of tangent spaces and i might interpret this sequence of tangent spaces as something three-dimensional just adding that so to speak the time damage but at the very end they have only two dimensions and in that in that sense we just might have just three dimensions not four dimensions so i doubt if this four-dimensional formal lessons on which also the direct equation uh pressure is great is is really appropriate i know that it's it's a long-held fashion physics since 1908 since minkowski introduced in general relativity this four-dimensional description but it's i think it's it might be a superficial dead end yeah okay thank you what do you think about those year's talk [Music] i mean very seriously seriously very interesting i mean um [Music] the non-associativity is is uh is something crazy somehow and um there are obvious extensions i mean we're not done here with octonions you have these seed onions and the s31 i think and every step you go you're going to lose another important property so it's it's some something like screwing up mathematics step by step and i don't know how how much physicists will go along but i agree as a matter of principle okay okay yeah i just i'm i'm still struggling with the non-commentativity and i think physicists might need to understand this first and then go ahead yeah i think chat so we can see the message if possible ah okay so i suggest i i think it was taking drake that was putting the question so i'll ask him if he wants to participate turn his microphone on or we can read yeah i'm here can you hear me okay yes we hear your thing so my thank you my question is about the signature i think you've addressed it partially but your so4 is acting on four euclidean dimensions and you're trying to make a connection to space-time which has a different signature and i'm not seeing how you get from one to the other yeah i i'm not sure that answers here but the this signature you you're talking is again arising from minkowski's original definition here and what people trying to do is to to put all this into the metric and and artificially introduce that sign convention but um i think that you you still need another more natural uh way to to describe space time by quaternions yeah but this is this is of course i mean there are there are technical issues because s s s three uh times s three isn't uh is not uh quite so4 there is a set two in between and um yeah but i don't i don't i don't see no no problem as a matter of principle if i say that all this phenomenology mathematical phenomenology if you want suggests that there is a deep connection between quartonians and space-time thank you so let's thank speaker once again [Music]