User talk:Rschwieb/GA Discussion/Comparison of methods: rotations

Sundry questions

Extended content

R, did you really just mean "find a rotation"? (any rotation)? In 3D there is a family of rotations that will do this. If you are wanting to compare procedures, the question should either be fully specified (to have a unique answer), or you should ask for a parameterized family of rotations. I suspect you mean the former. — Quondum ^☏✎ 15:41, 8 February 2012 (UTC)[reply]

I'm sorry that my original questions are not very well posed :) Most of it is my unfamiliarity with the capabilities of these formalisms. I was under the impression that we were thinking only of rotations about the origin, but I made no mention of that, so it's possible the solutions coming in are for rotations about lines that do not go through the origin. Any clarification about this in the solutions would be welcome.

Maybe, for simplicity, all of my questions should be about vectors bound at the origin.
A slightly more complex question would be to consider two vectors bound at different locations, and ask for a rotation which fixes the source of the second vector and turns it to be pointing in the same direction as the first. Rschwieb (talk) 16:06, 8 February 2012 (UTC)[reply]

Second paragraph on page 13 of Macdonald's paper, by the way, is what I mean in my edit summary. Rschwieb (talk) 16:10, 8 February 2012 (UTC)[reply]

I too am referring to rotations around the origin only. So we still need to get to the bottom of our confusion. I'll look up the reference and get back to you on this. — Quondum ^☏✎ 16:41, 8 February 2012 (UTC)[reply]

Macdonald is talking about given a rotor R (specified in terms of a bivector i and angle θ) acting on a given vector u, find v. When he says "Only in 3D does a rotation have an axis", he is not saying that the rotation given two vectors is unique, he is saying that only in 3D does the subspace unchanged by the rotation have one dimension. In general, simple rotations have an unaffected subspace of dimension n−2 (and for general rotations, n−2k for some integer k, 0≤k≤⌊n/2⌋), and only in 3D does one refer to this as an axis. To get back to the phrasing of your question: you are dealing with a different problem, where you are given u and v, and want to find R. So my position remains unchanged. — Quondum ^☏✎ 17:25, 8 February 2012 (UTC)[reply]

No, you are wrong. If you directly quote a person say "there, that is what I want to express", there is no room for misinterpretation. This is entirely about my entirely-too-vague comment on underspecification. In 3-D I gather there is a handy solution with a nominal amount of alternatives. In n-D, I'm saying the problem becomes underspecified because "there are more than enough alternatives to choose from, so we should require more of our solution." Explain to me what you want to convey by "underspecified", and maybe this will solve the problem. Rschwieb (talk) 18:30, 8 February 2012 (UTC)[reply]

I'm sorry, I'm not seeing the wood for the trees. Macdonald's statement corresponds to problem C, in n-D, where the plane is specified by a unit bivector blade. There is no underspecification for any n. — Quondum ^☏✎ 18:43, 8 February 2012 (UTC)[reply]

The rephrasing is good. It's very natural and concise, and is a simple thing to use for that page. Let's try to figure out what I was thinking of here, rather than on that page. In 3-D, we can find a rotor to rotate the plane of the vectors and all is well, every vector perpendicular to this plane will remain perpendicular after rotation, because there is only one direction perpendicular to the plane. But in 4-D, a rotor which aligns the two vectors by rotating their plane might have weird and different side-effects outside of the plane, right? My impression from what jheald had up was that you might execute the rotation and find some vectors which used to be orthogonal to the plane not orthogonal after the rotation. An ideal solution (if possible) would satisfy the extra requirement of preserving the space orthogonal to the plane. (PS I was not the originator of the word "underspecification" here, I adopted it into the question from jheald's original phrase. Maybe if you read what he originally wrote it will help determine what it means?) Rschwieb (talk) 21:17, 8 February 2012 (UTC)[reply]

Okay, now you've got me thinking. And perhaps I was a bit hasty in saying there is no underspecification (it depends on interpretation). Specified as any rotation that keeps vectors within the given plane, it will never rotate a vector that was orthogonal to the plane into one that is not (I think you will see that easily enough: as a linear function, this violates the specification). However, it is possible to have a compound rotation (whatever the correct term is) that simultaneously rotates two planes that are orthogonal to each other (for which you need 4D or higher), each rotation being "within its plane". Anyhow, you can have in general up to ⌊n/2⌋ orthogonal planes of vectors, each with its own independent angle of rotation. Each such subspace remains orthogonal to the others upon rotation. Of course, a vector non-orthogonal to both of two simultaneous planes of rotation will be affected by both. So the effect one would see for a compound rotation is that a vector neither in nor orthogonal to the (primary specifying) plane of rotation will generally not move as expected from the unique (up to a double-cover) simple rotation that fits the specification. — Quondum ^☏✎ 05:17, 9 February 2012 (UTC)[reply]

So a "simple rotation" is one which changes directions only in a 2-D subspace, and preserves directions in "the other" n-2 dimensions? That definitely sounds like what I was interested in. More easy questions (I'm sure the answers are there, if I had time to read thorougly): is the quaternion treatment just a special case of the GA rotor treatment? In Minkowski space, are we only interested in spatial rotations, or does the time dimension interact? I don't know how to interpret a mixture of time and spatial dimensions experiencing rotation. Rschwieb (talk) 14:53, 9 February 2012 (UTC)[reply]

Yes, a "simple" rotation is generated by a simple bivector (i.e. a bivector blade). A compound rotation (multiple planes of rotation) is generated by a bivector that is not a blade, via e^−B/2. And yes, quaternions form the even subalgebra of Cℓ_3,0(ℝ), and the vectors to be rotated are mapped into that subalgebra as their duals (by multiplying by the pseudoscalar), so the entire quaternion treatment is exactly expressible within G(3,0). In Minkowski space, all four dimensions get in on the rotation. Space-only rotations correspond to normal spatial rotations. Those that involve the time dimension correspond to changes of velocity. Collectively they form the Lorentz group. You cannot separate time and space, just like you cannot separate the x and y axes in a way that does not inherently depend upon your frame of reference. It is best pictured as a hyperbolic rotation in 2D (t vs. z) on a graph (see Lorentz transformation). — Quondum ^☏✎ 15:55, 9 February 2012 (UTC)[reply]

Thanks, the Minkowski space comments are incredibly helpful. I want to make another comment, using "orthogonal" in an extended sense. In the theory of bilinear forms, transformations which preserve the bilinear form are called "orthogonal", since they preserve the "angles and distances" dictated by the bilinear form. My understanding is that the Lorentz group is the orthogonal group of Minkowski space. The ordinary orthogonal group has two components, the matrices with determinate 1 and the matrices with determinate -1. I've gathered that there are actually four distinct components of the Lorentz group, and I haven't found yet if the determinant flags these also. For physical applications it looks like you can restrict attention to the transformations preserving orientation of space and direction of time. Are you aware of any applications where the orientation changing transformations make sense? Rschwieb (talk) 17:43, 9 February 2012 (UTC)[reply]

If I had to guess, I would think the determinants would be ±1 and ±i. Rschwieb (talk) 17:59, 9 February 2012 (UTC)[reply]

Hm, found it on the page, looks like only ±1 occur... strange... Rschwieb (talk) 18:23, 9 February 2012 (UTC)[reply]

Incidentally this (IMO) is a particularly cute, intuitive introduction to thinking about hyperbolic orthogonality and hyperbolic angles. Written quite naively and aimed at high-school students, but highly recommended. Jheald (talk) 18:35, 9 February 2012 (UTC)[reply]

What planet is it where mathematics education conferences are held by high schoolers? Rschwieb (talk) 18:59, 9 February 2012 (UTC)[reply]

Um... It was a conference about the mathematical education of high schoolers, and this was some sample course enrichment material. Or am I taking your cmt more seriously than intended? Jheald (talk) 19:13, 9 February 2012 (UTC)[reply]

It's semisarcastic. Directly speaking, I wanted to express disbelief that this material could be aimed at high school students in general, even in Europe. If it's for an accelerated enrichment program, I could believe it. Rschwieb (talk) 20:43, 9 February 2012 (UTC)[reply]

Have you ever read QED – The Strange Theory of Light and Matter, by Richard Feynman? Aimed at a nontechnical audience ("an intelligent and interested public"). Amazing stuff. — Quondum ^☏✎ 21:05, 9 February 2012 (UTC)[reply]

I've come close to picking up Feynman's QED book many times... based on his texts and a few transcribed lectures, his writing style is very pleasant reading. Rschwieb (talk) 21:48, 9 February 2012 (UTC)[reply]

There were also two accompanying example sheets [1] [2] exploring some parallel constructions in Euclidean and Minkowskian geometry. Jheald (talk) 21:25, 9 February 2012 (UTC)[reply]

The later two documents look useful, however I could not recommend the "cute-intuitive-naive" document to anyone other than a geometer. Anyone who has read it can see it is not aimed at high school students but at teachers of high school geometry, as one would expect at an educational conference. It is completely the opposite of naive: it requires a fair amount of geometry background from the reader. Flatland is what I call a naive introduction... maybe there's a Minkowski Flatland out there? Rschwieb (talk) 20:34, 16 February 2012 (UTC)[reply]

Yup, det = ±1 only, corresponding to rotations (R being of even grade) and reflections (R being of odd grade) respectively. The other distinction is orthochronous and non-orthochronous, meaning the time axis ends up parallel or anti-parallel (direction within the light cone, not actually parallel) respectively. They are topologically unconnected because the light cone has two parts. The combination make up four topologically distinct components of the Lorentz group (proper orthochronous, improper orthochronous, proper non-orthochronous, improper non-orthochronous). Add another time-like dimension and they're connected: the orthochronous/non-orthochronous disconnect becomes connected, and leaves only two components corresponding to the determinant. Applications: the time-reversal (called T) and space-reversal (called P) symmetries are important in physics, and of course combinations. Add to this charge reversal (called C, or matter-antimatter interchange), and you get various combinations. The universe breaks all these symmetries but one (the combination CPT) as far as I know. It surprised the hell out of physicists when they discovered the violations; very subtle they are. Will look at the ref. — Quondum ^☏✎ 18:50, 9 February 2012 (UTC)[reply]

Would love to see the stuff that surprised the physicists, in case you get around to looking for it :) Rschwieb (talk) 18:26, 13 February 2012 (UTC)[reply]

Bear in mind that quantum mechanics had developed phenomenally accurate equations describing the evolution of processes (in cases to thirteen decimal places, though at the time accuracy was less), and that these equations were all symmetric (invariant under reversals of time or space). Included was Dirac's equation that successfully predicted antiparticles, by virtue of them being mathematically indistinguishable from time-reversed particles and these being unavoidable solutions to the same equation that perfectly described the electron and many other particles. Remember too that the theories at heart were developed upon a myriad symmetries; e.g. special relativity has Poincaré group symmetry as its premise. I can really only quote Roger Penrose (The Road to Reality p634):

In 1956 there was great shock to physicists when Tsung Dao Lee and Chen Ning Yang made an astonishing proposal, concerning β decay—and concerning weak interactions generally—that they should not be reflection-invariant, this proposal being startlingly confirmed experimentally by Chien-Shiung Wu and her associates shortly afterwards, in January 1957. According to this, the mirror reflection of a weak interaction process surprisingly would not generally be an allowed weak interaction process, so weak interactions exhibit chirality. In particular, Wu's experiment examined the emission pattern of electrons from radioactive cobalt 60, finding a clearly mirror-asymmetric relation between the distribution of emitted electrons and the spin directions of the cobalt nuclei. This was astounding, because never before had a reflection-asymmetric phenomenon been observed in a basic physical process!

He goes on (p638) to speak about an even more subtle (though closely related) symmetry violation that was subsequently observed. This one has been observed only for very limited situations, generally only in particle accelerators.

... ordinary weak interactions are not invariant uner either P or C separately, but it turns out that they are invariant under the combined operation CP (= PC). We may regard CP as the operation performed by an unusual mirror, in which each particle is reflected as its antiparticle. There is a famous theorem in quantum field theory, referred to as the CPT theorem which asserts every physical interaction is invariant if all three of the operations C, P, and T are applied to it at once. ... dependent on the physical validity of its assumptions. ... Accordingly, the CP invariance of ordinary weak interactions implies their invariance under T (time-reversal symmetry) also.
A very few observed physical effects are known to violate CP-invariance. The most long-standing example (a 'non-ordinary' weak process, first observed by Fitch and Cronin in 1964) is a particle decay seen to be non-invariant under CP. ... This is the decay of the K⁰ meson ...

So, there you have it. — Quondum ^☏✎ 20:09, 13 February 2012 (UTC)[reply]

Great, I might have to snag this book. I think I saw a cheap copy on one of my trips to the bookstore... I'll probably have a few questions soon about what's said above (the physics accent is very heavy). I'll need some time to figure out what to ask though. Rschwieb (talk) 21:13, 13 February 2012 (UTC)[reply]

I'd certainly recommend it. It starts as a whirlwind tour of a broad swathe of ungrad math topics (some that were new to me), very refreshing and readible. It progressively becomes more physics-oriented, with the last quarter of the book a bit beyond me. Aside from one or two ideas that have not been widely received well, he has produced a lot of very clear thinking, and is able to express it very well. It is one of my most valued books, often referenced. It's not guaranteed to be pitched right for everyone, though. — Quondum ^☏✎ 16:45, 14 February 2012 (UTC)[reply]