aap_

I slightly edited my post while you were writing yours. I'm aware of the article but I still think it could be clearer. I'm a big fan of constructive proofs but that does not seem the case for most mathematicians, so i find a lot of stuff in math very unsatisfying and have to rediscover it myself via different routes. Educational for sure but not always so easy.

And yes, of course there are many possible matrix reps, sorry i was not being precise. The ones i was referring to are the ones given here: https://en.wikipedia.org/wiki/Classification_of_Clifford_alg...

Unfortunately there are slightly different but related notions of what spinors are. One key idea is indeed how they transform. A spinor ψ transforms with a transformation S ∈ Spin(n) in a one-sided way: ψ -> Sψ. A vector v in contrast transforms in a sandwich way (with the inverse on one side): SvS^-1. Intuitively this explains why spinors transform "half as much" as vectors, e.g. the 720° vs 360° rotational symmetry that shows up in physics. So for any S ∈ Spin(n) the sandwich-action (S|S^-1) gives you the corresponding element of SO(n). Because a negative sign on S squares away in that case, S and -S map to the same SO(n) action, and therefore Spin(n) is said to be the double-cover of SO(n) (personally I think it would be better terminology to call SO(n) the half-cover of Spin(n)).

So a spinor could be said to be anything whose symmetries are a Spin group. Spin groups are easily constructed in clifford algebras and it turns out that they have matrix representations. Whenever you have matrices (linear maps) you may wonder what the vector space is that they act on (i'm now using the term "vector" abstractly, not in contrast to spinors as above). Well, those are the spinors (technically pinors)! Another definition of spinors is that they live in a minimal left ideal of a clifford algebra. This does not sound very intuitive at first, but it can be understood easily as simply taking the matrices with only one non-zero column. These are really not very different from colunm vectors then. There seems to be some confusion about pinors and spinors in that perspective though...it just seems to be a somewhat confusing concept in general.

The spinors/vectors relevant to the article are those of Spin(8), which has something to do with triality (still need to understand all of this better myself). The basic idea is that in Cl(8) the vectors and spinors are both 8-dimensional and the algebra can be generated by left-multiplication of octonions. So there are some interesting symmetries occurring. The Baez-article goes into that too but it could have been a bit more explicit for my taste.

I hope some of that made sense, i don't know your background. I'm still trying to wrap my head around this topic myself and have been for about 2 years now.

Maybe check out the "spinors for beginners" series on youtube. It's very good and quite extensive.

What i think is super cool about h3(O) is that you have scalars, a vector and two spinors in one object with the jordan-product giving you all the products between them that have to do with triality. See section 3.4 of John Baez' paper on the octonions.

Also not sure I'm understanding you right :) My view is this: the wave function is mere bookkeeping and not anything ontologically fundamental. However the fact that such a seemingly bizarre concept lets you do quantum physics (even if it's not the only way) points to some fundamental questions about the nature of...well, nature.

Of course this is not the only valid view...just one that makes sense to me. Thinking about these sorts of questions is a very fun endeavour.

I morally agree, but not quite: think of the wave function as not more than a bookkeeping device. It does get the job done but be careful to ascribe it too high an ontological status! The path integral formulation seems a lot more natural to me and it does not need a wave function, instead you can derive it and treat it as a bookkeeping device. The way I think about it is that it's an attempt to deterministically model non-deterministic behavior: you "pretend" that the system is deterministic by keeping track of all the possible ways it could have evolved in time. sure enough, once you make a measurement this probability distribution "collapses" and you find out what is actually the case.

For a single particle they are easy to confuse. A wave function ψ(t,x) for a single particle gives a probability amplitude to find the particle at coordinate x at time t. In this case one can imagine an amplitude at each point in space and time, like a field. This interpretation however completely breaks down once you introduce a second particle: the wave function ψ(t,x1,x2) gives a probability amplitude to find particle 1 at x1 and particle 2 at x2 at time t. This no longer admits an interpretation of assigning some value to locations in space. Intuitively one might think you get one amplitude for each particle at some location but that's not how QM works, so we shouldn't think of the wave function as living in physical space.

Funny coincidence. I've written a small 3D rendering engine for the ESP32 as well last week. It has standard 3d math, clipping, and rasterization of points, lines and triangles. all in ~1500 lines of code. The repo is not polished yet but you can find it here: https://github.com/aap/psiop

And 3x3 matrices are 9 dimensional, yet usually you can interpret them perfectly fine with a 3d perspective. The dimension of the algebra is usually not very meaningful if you're trying to gain some intuition about it.

I'm curious about your perspective on exterior algebra. So far I've mostly seen it as a special case of GA so my view is probably GA-tinted. I'm just curious how you even do any sort of transformations since the exterior product doesn't allow these sorts of things. It seems like it only gives you the "things" in your algebra with few ways to do anything with them. You seem to agree with the author of the article you linked to in that the most useful aspects of GA come from EA. I find this very hard to see, so maybe you can shed some light on it? E.g. how do you rotate a bivector in EA?

No example of such a crackpot is given. The author just claims this without evidence. I've heard this sort of argument before but it's not very clear what it refers to.

I think one of the coolest examples is probably classical mechanics. See the SIBGRAPI 2021 videos on https://bivector.net/doc.html

The article reads more like a trolling attempt. Geometric/Clifford algebra is incredibly useful and by throwing away its product you lose a lot of the power of the algebra. It's like saying matrix multiplication is not useful and you really want to be multiplying and adding numbers in various ways. After all GA/CA elements can always be mapped to elements of a matrix algebra. To get rid of the idea of linear transformations and that they should compose just doesn't sound well thought through.

I don't know what sort of crackpots he's talking about, personally i haven't heard of them, only the accusations. If the author can't separate the math from the people who developed and/or popularized it, too bad. Does GA magically give intuitive explanations for all sorts of weird things? no. Can you formulate a lot of stuff much more efficiently and concisely, and does it help gain new perspective on some things? yes, absolutely. It provides a wonderful framework for expressing geometric ideas.

Because quaternions are so cool yet under-appreciated. Unfortunately not many octonion posts.

Compulsively checking HN i suppose :D

good luck though. my emulator is not particularly user friendly, as in, it has no user interface. i recommend simh (although perhaps not for this thing in particular).

XOR is also the basic element-wise multiplication in clifford and cayley-dickson algebras. the way that they differ and which gives them their individual character is all in the sign calculation.

I agree. PDP-11 is so much more pleasant, it's not even close. One could make the argument that the addressing modes are conceptually more complicated than what you'd have on a RISC, but the 6502 addressing modes are probably actually harder to understand than the PDP-11's.

Unfortunately i can't. I've been thinking about spinors and related concepts as a hobby for maybe two years now. But i never found an explanation like mine above anywhere. Good geometric intuition about these things does not seem to be highly valued in physics education so people don't seek it. It turns out you can represent a dirac spinor (as the sum of left and right weyl spinors) extremely well just with your two hands. the dirac matrices simply tell you how to rotate your hands or mirror them (which you can do because you have two!). I should make a video about this actually...

To understand spin it's good to consider a gyroscope. when it has a lot of angular momentum there are two stable states for it in a gravitational field: aligned or anti-aligned with gravitation, up or down. in all other cases the gyroscope precesses. a spinor doesn't spin quite like a gyroscope but it is spinning in a sense (after all spin is angular momentum). but just like the gyroscope, you can think of it as having two stable states: in alignment with a magnetic field, or in anti-alignment. and because the magnetic field is a measure of some kind of rotation it can add to or subtract from the angular momentum of a spinor. this difference is "felt" as a negative or positive potential difference. this you can think of as two opposite forces on the spinor that split it apart into up and down components. the interesting thing is that a spinor with an arbitrary axis can always be written as the sum/superposition of an up and down spinor for some chosen direction. turns out quaternions have precisely the properties that you need to model this. i hope this was intelligible, it's a bit hard to put the geometry into words.

Interestingly showing spherical trig identities was how Hamilton demonstrated his quaternions to the royal irish academy when he found them. See (D) through (K): https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quater...

When i first read this i found it very hard to understand, because i was unfamiliar with spherical trigonometry, but there's quite some beauty to be found there.

Wow, pomega is such a terrible name for it!

CAR here and CDR just below: https://texdraft.github.io/lisp-1.5/listing.html#line-7626

2.11BSD is still a popular choice if you have a PDP-11 (or are emulating one).

Why even advertise with "beer" when the link just takes you to a page saying there will be no beer?

There are only 7 bits in ASCII. An 8th can be used for parity when transmitting data but a regular program will never see it. Anything above 0x7F is simply not a character.

Thanks for "Imaginary Property". I will use this from now on.

I think that's a really interesting point. I don't think there are many high level languages where code and data are fundamentally the same kind of thing. I'm aware of only lisp and tcl (funnily there's a thread about tcl on the front page right now). Also people probably don't think about machine code all that much and even if they do they probably don't consider self-modifying code. The only exception to this may be JIT compilation? Anyway, it seems to be a concept that's rare in high level languages so I understand the surprise.

I would say Ken Thompson and Steve Johnson are the closer.

Yeah, no idea how to run the code. There are links to the final demo at the end, but everything else just links to this editor :/

Why octonions though? Quaternions are nice and simple, octonions are much stranger. Still hoping we find interesting uses for them sometime.