A Swift Introduction to Spacetime Algebra

[@sudgylacmoe]

To my viewers that are wanting more videos in From Zero to Geo: Good news! I'll be getting back to it right away. I just wanted to get this video out there that I think is a better SoME2 submission than the videos in From Zero to Geo. EDIT: Since people keep asking me and I realize I should have said it in the video, I said "the zitterbewegung interpretation of quantum mechanics" near the end.

[@WaluigiisthekingASmith]

I wanted to say, you should submit this for SoME. I guess that's on me for not reading the description

[@marnixklooster]

You might want to put the #SoME2 tag on this video... 🙃

[@SupGaillac]

Great work! If it’s for SoME2, may I suggest to put the hashtag in the description and/or the title, to help referencing? Edit: … just seen the other answer above ^^

[@justynpryce]

Great video! Also "zitterbewegung" is pronounced more like "tsitter-beh-veh-goong"

[@michelangelowebb4362]

Wonderful you are back ! Thank you for your time producing this videos sir.

[@JackDespero]

As a physicist: Mind=blown. I am so used to the other set of mathematical tools that it is hard for me to do anything but simple problems with geometric algebra, but I can appreciate how amazing and nice of a tool it is.

[@justynpryce]

I feel bad, I had never heard of David Hestenes until watching this video, and after looking him up I found out he works in the physics department of my university! I'll have to pay him a visit it seems. Edit: It seems he retired before I started but this was not conveyed on the university's website :/

[@yabgu79]

If you can convince him to make online videos or have an interview if you can that would be great,

[@gabitheancient7664]

did you?

[@NostraDavid2]

Did you???

[@umbraemilitos]

Can you film an interview with him?

[@justynpryce]

@@umbraemilitos thank you for reminding me of this! He actually retired as a professor a few years before I started :/ but his profile on the university website just wasn't updated, so I found out when one of my professors told me he had retired. And unfortunately I was not able to talk with him.

[@CyborusYT]

Rewatching this, the comparison of physics being unchanged by both rotation and changed reference frames sticks out as wonderful foreshadowing

[@MaxxTosh]

1. I want you to know that your intro to geo algebra was the greatest math video I’ve ever seen and I was so excited to see this 2. What was the name of the interpretation of QM, and where can I learn more? 3. I would LOVE similar videos for PGA and CGA! 4. No seriously, thank you so dearly much for the effort you put into this niche but stunning topic

[@sudgylacmoe]

I realize now that I should have written the word "zitterbewegung" on the screen or something. You can find some information on it here: geocalc.clas.asu.edu/html/GAinQM.html

[@FoxxAngel_]

I took notes during my second watch and omg, I didn't realised how much information there was. Everything just came up so naturally that I just took it in at first. I hope you continue to do videos on more complex notions in GA, It's so engaging !

[@michalbotor]

this is mind boggling. geometric algebra is the most exciting branch of mathematics i have ever encountered.

[@liammccreary2941]

I’ve been working on this topic for one year now and I can’t get over how beautifully simple the algebra is. It should be standard in physics.

[@diribigal]

This really helps me see how "hyperbolic rotations" are just like the notations I know, and why there's good intuition in doing geometric algebra stuff with the Minkowski basis/quadratic form. Thanks for such clear explanations!

[@stephanevernede8107]

The fact that time-space split can be modeled by a geometric product blew my mind. Geometric algebra will definitely become the mainstream tool to do and teach physics in a close future. keep the good work your videos are an asset for humanity !

[@nice3294]

Its so cool how simple some of these equations can get when viewed in the right lens of relativity!

[@pomtubes1205]

Actually underrated... ur zero to geo video textbook series are the works of a good samaritan... keep up the good work!

[@usopenplayer]

After watching this video, I feel like my brain has been trapped in a cage for my whole life, and this video broke me out and gave me wings. ❤

[@might_e]

hell yeah 5hr video outlining Hestenes' Zitterbewegung structure in electrons and photons when? seriously through, i love your work in making these concepts surrounding both relativity and STA easy to follow and visually intuitive whenever possible. keep up the good work :)

[@FoughtAgaisntSisera]

This chapter on GA4P has always left me quite stumped. so stoked for this video

[@Mr.Nichan]

I just realized that special relativistic spacetime is quite quaternionic. (Also, I think it's funny that the best thing my spell-checker can guess I meant by "quaternionic" is "fraternization".)

[@izaret]

Excellent work. Love how you connected VGA to STA. Very clear as always. I know you plan to push on from zero to geo but I must say I found this type of video even more important. GA is so powerful that it takes a lot of time to explore it and the books from Hestenes or Lasenby are quite dense (for good reasons) which makes it hard (time consuming) for people to go through it. With a video like this it is now much easier for me to have people watch it and then have a discussion about the powers of GA. I could see three additional videos on the 3 connections you mention in the end: mechanics, electrodynamics and quantum mechanics. One main issue I found in talking to people about GA is it takes very long to get to powerful applications and many see the elegance but ask: what do I do with this that I can’t do already? There are videos about the details of GA already, they could be improved with Manim but they exist. The three above don’t (to my knowledge). Well done. Congrats.

[@tariq3erwa]

I agree, 30-40 min videos like this are good for exploring the more advanced topics and getting a better bigger picture, while the Zero2Geo ones are more educational and slow paced.

[@Bolpat]

This is some god-tier math channel. it does not have this appearance because the audio is behind others, but really good microphones are expensive and the audio is good enough.

[@TheMindfulCraftsman]

Thanks for the clear and very welcome explanations! In order to propagate the fundamentals and applications of Geometric Algebra, such more advanced videos are desperately needed. Great work and looking forward to more videos on this topic!

[@rustybolts8953]

For me, watching your last two videos has liberated much of the known laws of physics from the rubbish bin. Much I have yet to learn and relearn. Thank you so much. I will be back.

[@vyordanov78]

I would like to quote David Hestenes: "I have been pursuing the theme of this talk for 25 years, but the road has been a lonely one where I have not met anyone travelling very far in the same direction.'" Looks like his theory is becoming more popular. Thanks for this video it is fantastic! May be some more details how we can do more calculas in STA. Just one note: Your style is very similar to 3Blue1Brown as a talk. I would recommend to select your own style of presentation.

[@adamdarx4403]

A lot of THANKS for your wonderful introduction video. Recently I've been coming across a quantum field theory textbook written by Maggiore ("A Modern Introduction to Quantum Field Theory"). In this book, the author introduces some kind of decomposition of Lorentz group generator J^{\mu u} and make into two part consisting of "inner-product"-like things and "outer-product"-like things(Levi-Civita symbol). Before I learn about geometric algebra from your video, I believe I'm still in the lack of insights about Lorentz group. However, once after seeing that you decompose the "geometric product" into two parts exactly similar with what Maggiore's done, I finally notice that the decomposition of geometric product is 100% connected with the one of Lorentz group. In that sense, we know spinor actually transform under Lorentz group, too. That explains why we can find that Spacetime algebra can be viewed as Dirac γ-algebra.

[@geoyoshinaka5251]

Your videos are a godsend! Simply amazing clarity but I shall still have to listen to the presentation several times in order to fully comprehend it all. This is so exciting and I hope to see many more on Geometric Algebra, Symmetries and Groups in the near future. Please keep up the excellent expositions!!!

[@Number_Cruncher]

I thought that I know special relativity well. But you shattered my confidence. I will look into it. Thanks for sharing.

[@erikeriknorman]

Wow this is almost beautiful in a weirdly complex but organized way

[@leodorst5841]

Well done! The whole essence conveyed in 1 accessible lecture.

[@kainatertopereira8218]

Real nice video. Thanks for that! I have a somewhat related request for future video content: I would love to see a Swift introduction to geometric calculus with some concrete examples! You are a great teacher and I think that video would be awesome!

[@sudgylacmoe]

While I agree that that would be a great video, the issue is that I'm actually not the greatest at geometric calculus. Definitely not enough to be able to teach it. Maybe someday in the distant future.

[@pedrovelazquez138]

A video about APS would be great! A fell in love with Geometric Algebra. I come from an engineering background but the math always caught my atention.

[@errantis]

Amazing, this is Amazing... Clifford's algrebras seem to have promising potential but ignored. Thank you !!. More videos like this!!

[@bobtheblob728]

this video is incredible!! geometric algebra gives such good intuition for understanding this

[@AMADEOSAM]

Very very instructive, your explanation is excellent! Keep on going please …

[@jonkirwan525]

After this and a few other videos you've made (about a year ago and more -- including the series that abruptly ends a few months back), I'd very much like to support your further work through regular payments to Patreon. I want to see you making a reasonable living out of learning more (yourself) and furthering your ability to teach/share what you develop in your mind in the process. I'll be discussing this with my wife. But slightly more than a hundred bucks a year is cheap if it allows you to progress and then share with us your mind. Best wishes, regardless!

[@AlessioAlessi]

This is mind blowing! Why is this not thought in physics courses?!? 🤯

[@MultivectorAnalysis]

This is fantastic!! Back in 2016 I made a 3-part video introduction to multivectors (using an axiomatic approach centered on the geometric product), and I had a lot of ideas for a series of follow-up videos, but then life got in the way and those plans had to be set aside. Now that I see talented creators like you making GA videos, I don't feel so bad for dropping the ball on my plans. BTW, are you planning to attend ICACGA in Denver this October? Prof. Hestenes (who is 89 years old this year) will be the keynote speaker. I'm sad I can't be there in-person, but fortunately there's also an option to attend virtually.

[@Seiffouri]

Very insightful >> 36:53 It changed my perspective about electromagnetism entirely!!! Thank you.

[@mathunt1130]

I did a postdoc in geometric algebra, it's awesome.

[@JohnFerrier]

I've been a physicist for 11 years now and I've never heard someone call it the "Lore Ints" boost before. xD This pronunciation is killing me!

[@richardchapman1592]

Hope you got to listen to Kathleen ferrier as a diversion from the stresses of solutions of non linear differential equations. What is life to me without thee? Answer: a description based on orthogonal spaces but permiting the isolation of their parametric identifiers to have probability functions of tiny interactions between them.

[@labidifaycal3185]

Perhaps is european accent.🎉

[@Bolpat]

Whenever I feel smart, I watch this video to put my arrogance in place.

[@DeclanMBrennan]

Zen like simplicity and elegance. Thank you very much. By the way, has anybody read "Dichronauts" by Greg Egan which is set on a world with 2 normal space dimensions and one hyperbolic dimension? I never realized that some of the "physical" rotations there were mathematically equivalent to Lorenz Boosts.

[@lachlanperrier2851]

I am about to watch and I’m so pumped!!!

[@alejrandom6592]

6:22 you can also just consider we are now working with c=1 lightsecond per second, instead of measuring in m/s

[@diffpizza]

This really makes relativity much better to work with. I wonder if there's a book or some open course teaching general relativity with geometric algebra as well

[@sudgylacmoe]

Near the end of Geometric Algebra for Physicists by Doran and Lasenby there's a few chapters on general relativity.

[@ericvilas]

I'm really curious about the implementation of this to relativistic quantum mechanics. Is there a way to write Lagrangians and such in a way that doesn't reference the coordinates? (And do you know where I could find videos about this?)

[@amiramaz]

Thank you so much, great video, would love to see more

[@adamgeggie-bowker9980]

Simply Fantastic!

[@filipepgoes4528]

Great way of introducing the Minkowski metric, thanks.

[@sudgylacmoe]

I know, I was surprised when I worked it out how much it made sense. I had always just been introduced to it as "Oh look this quantity we dreamed up is invariant under Lorentz transformations so let's base our whole theory on this thing we don't even understand."

[@bingusiswatching6335]

​@@sudgylacmoeidk the necessity of the minkowski metric follows quite trivially from the speed of light being equal in every reference frame

[@richardchapman1592]

None of this is set in stone. Suppose the speed of light varies as for instance a sine wave of distance divided by time would be under swift change. We would have​@@bingusiswatching6335 some kind of wave in nature that happened so fast we've never been able to observe it. Mind you, there is a dilemma about saying a speed is constant even when distance and time (the constituents of speed definition) get so distorted when it is approached. Almost a circular argument that Lorentz and Einstein found so apt in describing reality as measured. We must though try to eliminate infinities from our desciptions of nature because conservation of energy, more fundamental than inviolable light speed, is the best guide the forefathers of physics have left us. I guess magicians would say otherwise and indeed an intervening holy spirit would be a challenge regarding how it obtains its energy requirements. All this away from the point of algebraic geometry. It is to describe the mediums by which forces can happen along one dimensional lines or lines on any brane in higher spaces. This is why we need vectors and their analogues and thanks for showing us a maths relevant for them.

[@richardchapman1592]

@@bingusiswatching6335 for the theorists: starting with constant c then giving it perturbations, perhaps harmonically, may give spacetime more field structure to work on that could be relevant at Planck lengths.

[@anuman99ful]

This is video I needed so bad, thanks a lot!

[@pacificll8762]

That is an excellent video, I can’t thank you enough for all of this !

[@iamtraditi4075]

Loved every second of this :) Ty

[@ShredEngineerPhD]

I hope I will one day understand all of this. Thank you for making these great videos! :)

[@АлександрНевельский-л2з]

Most of the content is trivial, but still interesting.

[@Snoopies622]

I find this pretty abstract. How about making up a special relativity problem (with numbers) and solving it in both the conventional way and with this spacetime algebra approach? So the advantages will become more obvious.

[@laikawea4771]

What is pretty powerful about this particular system is that it allows us to talk about quantum problems and relativity problems using the same language. Maybe in 400 years somebody will see this in college. Until then... youtube.

[@tinman1943]

May I suggest an example problem: muon decay.

[@MessedUpSystem]

Seriously, why isn't this teached in mathematical physics classes instead of spending a whole semester learning how to solve a pde?

[@purewaterruler]

I mean...you still need to learn to solve PDEs whether you use geometric algebra or not.

[@MessedUpSystem]

@@purewaterruler yeah, I meant spend one semester solving ONE single pde that is what is haappening right now to me haha

[@UnitaryV]

@@MessedUpSystem I don't really see what's wrong with that. That describes courses on electromagnetism, quantum mechanics, fluid dynamics, and general relativity. I'd argue that the fact that each one of these stems from one PDE is part of what makes them all beautiful. More importantly, however, you need to learn physics as it is, and physics requires solving PDEs. Geometric algebra, while cool and elegant, does not change that.

[@rajinfootonchuriquen]

Geometric algebra is harder than linear algebra and multivariate calculus because it's framework is not euclidean geometry, so it's make sense to go from the easiest to hardest in your learning

[@logana.martinez770]

​@@rajinfootonchuriquen​I find GA way easier than linear algebra and therefore multivariable calculus. It actually aligns with one thing i've always thought about physics. That our work is more geometry than analysis. Most of the problems i had with linear algebra are calculation based more than concept based, so maybe I'm just happy to not having to use matrices. Since the time i had to waste doing a 3x3 manual matrix inversion is better spent building a geometric intuition.

[@holykoolala]

You make this seem way too understandable. Awesome job,

[@bytesandbikes]

This is going to take a few times through to get into my tiny brain!

[@TheOneMaddin]

Why is Geometric Algebra not standard in physics? I can't stand all these matrices in QFT etc. and love how neat everything looks in your notation. Is it because it has not caught on yet, or are there shortcomings?

[@sudgylacmoe]

In my opinion it's because it hasn't caught on yet. I talk about this a bit here: kzbin.info/www/bejne/aJmliHZ5ds52sLM

[@AMADEOSAM]

Great explanation! Many thanks again …

[@FlyGuyGreeny]

Spittin' facts like a G

[@NovaWarrior77]

Oh my gosh, yes!!

[@blinded6502]

I prefer to reserve "i" for the arbitrary units squaring to -1

[@SuperDeadparrot]

While I was working with those stupid tensors and co-variant and contra-variant indices I always wondered if there was an easier way to do this.

[@TheOneMaddin]

Can you make a video on how EM, QM or even QFT can benefit from Geometric Algebra? I really can't stand the contemporary physics notation. It seems so ... ugly in comparison to how beautiful these theories are supposedly are.

[@deltalima6703]

Woohoo! Love this channel!

[@thedeaner3117]

Dude im drunk as fuck and i understood pretty miuch all of this video... probably a testament to how good ur video making skills are. Congrqts brother ☝️

[@yoavboaz1078]

Great video. I still think it would have vean more clear if you marked vectora somehow. Either by using t̂ x̂ ŷ ẑ or by using e0 e1 e2 e3 or at least drawing arros over the basis vectors

[@tariq3erwa]

I almost jumped when I saw the notification

[@alikiaee1307]

Another great video, thank you

[@frogstud]

Great video, but i really don't see the usual tensor algebra foundation going anywhere sadly. Clifford algebra is Great in certain situations but tensor algebra with vectors etc is just so simple imho.

[@gammaian]

Thank you for the awesome video! What is the interpretation of spacelike rotations and timelike rotations you mentioned at the end of the video? Is a rotation in spacetime equivalent to a translation at a certain velocity in 3D space?

[@sudgylacmoe]

Spacelike rotations are ordinary rotations, and timelike rotations are Lorentz boosts. When considering them applied to the path of a particle, they correspond to rotation and acceleration.

[@SidCypher]

I also could not recognized the mangled German word, the "zit-r-BB-gone" thing was "Zitterbewegung", literally translateable as "shivering movement". The word is stated in the video description, now that I notice.

[@mastershooter64]

woah dude nice!

[@theorbization]

incredible

[@jz8741]

Very good video! Thank you! Does anyone know what software he uses to animate everything?

[@sudgylacmoe]

This is answered in question two of my FAQ: kzbin.infoUgwFByhvEg1_hD_L1Ch4AaABCQ

[@rokasha40]

0:33 "as I started studying on my own", can you make a detailed video on exactly to study on your way, like what things to study first what resources to use?

[@mistafizz5195]

Damn you have some absolute banger content

[@jperez7893]

amazing!

[@grayjphys]

I hope this is taken as constructive feedback, but I think you need to EQ your audio. Right now it sounds very grating. I think cutting some of the higher frequencies will help out a lot. It would also help with whistly/hissing sounds with words that have s's in them. Love the videos by the way!

[@sudgylacmoe]

Wow, that really does make a difference! Thanks for the tip!

[@johnchessant3012]

very interesting

[@antibank01]

Excellent

[@dean532]

Hi Just a suggestion: Have you considered coming up with solved examples on this topic- at intermediate level ofcourse! May be give a try at using GA in some problems on black-holes etc Gravity / Cosmology

[@johneonas6628]

Thank you for video.

[@gabrielmonopoli7994]

Dear profesor please could you recommend a math book to learn more about vector

[@richardneifeld7797]

Wow.

[@barryzeeberg3672]

At around 9:00, "time" is referred to as a "vector." I think it is more accurate to refer to "time" as a "component of a vector."

[@linuxp00]

It could be thought of as a vector from a vector basis, as in spacetime from classical einsteinian relativity

[@burkhardstackelberg1203]

@sudgylacmoe - here we are :-) Great video! I also followed your link to the works of David Hestenes and read over the article on real spinor fields. Still strange to me is to see the Dirac equation in a form that does not look Lorentz invariant in an obvious way, and it seems like Hestenes also preferring it in a more coordinate dependent form. Still, I also found the coordinate independent form in his article. So, in the end, the Dirac wave function seems to act as a local scale and Lorentz transform on the vierbein...?

[@burkhardstackelberg1203]

Partly answering myself -- I dove deeper into the text. David Hestenes gives us an introduction to the Dirac equation in a coordinate dependent form to relate it to the formulations one already knows from the textbooks who give the Dirac equation in matrix form. In matrix formulation, you early have to choose a representation to get things computed, so it makes sense introducing the equation this way. But the real beauty arises, at least for me, where you have the invariant formulation at hand and take your coordinate perspectives by chosen vierbeins...

[@RevolutionAdvanced1]

This is a more general question, but is division an operation which can be done on arbitrary multivectors, whether in only some geometric algebras or all of them?

[@sudgylacmoe]

Division is possible for many, but not all, multivectors. As an example of a noninvertible multivector in 1D VGA, consider 1 + e1. Because (1 + e1)^2 = 2(1 + e1), if 1 + e1 was invertible, we could cancel 1 + e1 and get 1 + e1 = 2, which is clearly false.

[@chicojcf]

wow, very informative-TU

[@romaing.1510]

The Geometric Algebra evangelist has spoken again

[@ReginaldCarey]

A great introduction. As a nube in both fields, I can’t help but feel a little dissatisfied with some of the assumptions STA makes concerning time. On one hand it’s a scalar. On the other it’s a vector. I’ll have to rewatch. Once it’s been converted to a length can’t it be treated as the other bases? I’m sure there are reasons why this isn’t so, but it’s an area that doesn’t sit well in my mental model. Thankfully it’s not a knock on GA, just the mapping of special relativity to GA. Time to pull out the pencil and paper!

[@sudgylacmoe]

Time is a vector in STA, but a scalar in VGA. When passing from STA to VGA with a spacetime split, the time vector in STA gets converted to a time scalar in VGA.

[@5hape5hift3r]

I am now convinced that velocities are timelike bivectors.

[@5hape5hift3r]

Supported by the swift introduction video mentioning that charge can be viewed as a current flowing in the time direction.

[@合同会社プラノス]

Excellent. I had realized that Lorentz boost should be a hyperbolic rotation because it is described quite simply by introducing imaginary angle. I could not go further. Now I can for the sake of GA. I wonder why only square of gamma zero is positive one while others being negative. Should this asymmetry have some fundamental reasons?

[@sudgylacmoe]

I would say the reason is as simple as the fact that space and time are not the same.

[@e.s.r5809]

"The magnetic field rotates charged particles in a spacial plane, and the electric field rotates charged particles in a temporal plane, which we perceive as acceleration." My brain: *windows XP error noise*

[@TheOneMaddin]

How comes the Dirac equation you show contains gamma 1 and 2, but not 3? This seems to break some symmetry I would expect to hold.

[@sudgylacmoe]

If I recall correctly, this comes from the spin of the electron. We have to pick some direction that the electron is spinning, and the usual convention is to say that the spin vector is in the z direction. In GA we like to use bivectors for spin, which means that the spin bivector is γ₁γ₂.

[@Bolpat]

If you wonder how the interpretation of quantum mechanics mentioned at 37:49 is spelled: It’s called Zitterbewegung (German, roughly translates to “jittery motion”). The subtitles lead nowhere close. The pronunciation is also off quite a bit.

[@sudgylacmoe]

I've been told that this is the English way to pronounce the word, which I know is quite different from the German pronunciation.

[@Bolpat]

@@sudgylacmoe I mainly commented not as a critique to you but because I wanted to read about it and it took me about half an hour to make an input into a search engine that would spit out a correction. And someone else might find my comment. I haven't encountered the “correct” English pronunciation anywhere. In fact, I added the German pronunciation to the Wikipedia article.

[@sudgylacmoe]

Yeah, that was my bad for not writing it down on the screen. I tried to write it in as many places as I could afterwards (in the description, in the pinned comment, and in an info card in the video), but I can only do so much.

[@tiagorodrigues3730]

This is something I 've been thinking of as I rewatch the video: is there a physical interpretation for STA trivectors? Scalars are scalars, vectors are inertial frames, bivectors are vanilla GA's vectors and bivectors, and the pseudoscalar is the pseudoscalar. But do the trivectors have an interpretation?

[@beanlets]

They are pseudovectors

[@yoavboaz1078]

Honestly this seems very useful but it will probably be so much easier to teach if the notation was simpler

[@tinman1943]

Thank you for this. I thought I had left a comment earlier but I don't see it now. I stumbled upon this looking for a description of Minkowski space-time, and have been blown away! You have an amazing talent for "swiftly" making the complex understandable! Now, contentwise, I'm wondering whether it would be fair to say STA is not just an alternative "algebra" of space-time but in fact introduces an alternative approach to "geometry" of space-time? Next, I'm going to look at Zero to Geo and review the Lagrangian.

[@sudgylacmoe]

I would say that the geometry of spacetime was already known before STA. STA is just another algebraic way of representing the same geometry. Also, if you understood this video, From Zero to Geo is going to be way too low-level with the stuff it currently has covered. It's currently still in the linear algebra review section.

[@jjaustin8750]

@@sudgylacmoe I got the impression that STA was a significant departure from traditional Minkowski space-time: changing the time axis to a vector and measuring it in units of distance. But maybe I'm just not that familiar with the way Minkowski space is used. As I mentioned, I stumbled upon your video while looking for an explanation of Minkowski space-time. In any event, in terms of trying to make "math" more understandable, I learned about vectors in physics class. In high school, I was mystified by the idea of vectors (magnitude and direction) until I encountered them in physics--e.g. a boat rowing across a river current. So perhaps you might try introducing physical applications sooner. I'd love to see the "rowboat crossing the river" in STA. One could even do relativity of the boat vs. the shore vs. the river. I'd also like to see the path of a decaying muon from space in STA. Perhaps it's been done--can you suggest a text or paper that does mechanics in STA? I had earlier attempted to suggest another approach to GA. It seems to me the physical "realities" are not points and vectors but the 4D "enduring objects". These are the real things conserved or transformed (rearranged). The volumes, surfaces, lines, and points are just abstract boundaries or projections of these real things.

[@alonbenjo7382]

I'm a simple person. I see a @sudgylacmoe video, I give a like

[@unitedfutures_]

Love the channel! Feeling like there’s something missing at minute 7:24 where rotation is defined on (a)with R dagger (a) R and an isomorphism between the Lorentz Boost. I’m novice and could be missing something.

[@angeldude101]

The most you'd be missing is how to get R, for which there are a few ways. Technically, there's not even much restriction on what R can be other than the product of any number of vectors or the exponential of a bivector. Once you have R, R†aR is the way to transform a using R. I guess R† could be confusing. R†, which is also sometimes noted as R̃, is the "reverse" of R, which is similar to the complex conjugate. It's the result of multiplying the original vectors is was made from in reverse order. If R was from an exponential, then it's the result of using the negative exponent.

[@porky1118]

37:48 "The ciderbibigon interpretation"? Did I understand it correctly? I couldn't find anything about it. How is it really called?

[@tariq3erwa]

"Zitterbewegung" means jittery motion it's in German

[@porky1118]

@@tariq3erwa English pronouciation...

[@tariq3erwa]

@@porky1118 zitter like jitter, bewegung like be wig oong

[@porky1118]

@@tariq3erwa I know, I'm German myself

[@tariq3erwa]

@@porky1118 I 'm Sudanese nice knowing you

[@Mr.Nichan]

22:00 You forgot to mention that 1 (the unit scalar) is also timelike.