Talk:Wu experiment
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weak boson coupling
[edit]in the text: "due to the W and Z gauge bosons of the weak interaction coupling only to left-handed matter particles and the right-handed antimatter particles". Z bosons do couple to left and right handed particles (with different strength), the charged W couples only to left-handed particles.
217.224.249.27 (talk) 04:52, 19 January 2013 (UTC)Hallo
But why?
[edit]Nice article, but I don't entirely get it. There must be some kind of fairly simple analysis that can be given: e.g. 'if the P operator is applied to the nucleus, then one would get ..blah', etc. I don't really understand why one can look at the asymetry and deduce parity violation. There are some kind of missing steps here. User:Linas (talk) 23:19, 30 November 2013 (UTC)
- One explanation is to put the Co60 in an electromagnet. If you change from left handed to right handed, you change the winging of the electromagnet, which changes the direction of the magnetic field. That exchanged up and down, such that beta goes the other way. Gah4 (talk) 06:30, 8 October 2020 (UTC)
The First problem
[edit]The first paragraph is unclear or contains a simple error: "it would be possible to distinguish between a mirrored version of the world and the mirror image of the current world." There are no circumstances under which one can distinguish between one thing described in two different ways, unless we are talking about distinguishing between descriptions of worlds rather than distinguishing between worlds. The options are: EITHER (1) Replace either "a mirrored version of the world" or "the mirror image of the current world" with the words "the actual world." OR (2) Explain each of the concepts clearly enough that it is possible for the reader to understand the difference between (1) and (2). — Preceding unsigned comment added by Eugene Penguin (talk • contribs) 01:08, 4 January 2015 (UTC)
Experiment Section -other way round?
[edit]In the last paragraph of the Experiment section, it is stated that:
If the electrons were always found to be emitted in the same direction and in the same proportion as the gamma rays, P-conservation would be true. If there were a bias in the direction of decays, that is, if the distribution of electrons did not follow the distribution of the gamma rays, then P-violation would be established.
I am maybe a bit puzzled, but isn't it exactly the other way round? In the results section, it is stated that If P-conservation were true in beta decay, electrons would have no preferred direction of decay relative to the nuclear spin, which seems correct to me but contradicts the other sentence. Nilpferd1991 (talk) 14:06, 22 October 2018 (UTC)
GA Review
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Reviewing |
- This review is transcluded from Talk:Wu experiment/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: Reyk (talk · contribs) 15:24, 6 December 2019 (UTC)
I'll review this; hope to have it finished in a week or so. It looks pretty good at first glance. Reyk YO! 15:24, 6 December 2019 (UTC) 1. Well written
- 1a) the prose is clear, concise, and understandable to an appropriately broad audience; spelling and grammar are correct
- 1b) It complies with the manual of style guidelines for lead sections, layout, words to watch, fiction, and list incorporation
2. Verifiable with no original research
- it contains a list of all references (sources of information), presented in accordance with the layout style guideline
- all inline citations are from reliable sources, including those for direct quotations, statistics, published opinion, counter-intuitive or controversial statements that are challenged or likely to be challenged, and contentious material relating to living persons—science-based articles should follow the scientific citation guidelines
- it contains no original research
- it contains no copyright violations nor plagiarism
4. Neutral: it represents viewpoints fairly and without editorial bias, giving due weight to each
5. Stable: it does not change significantly from day to day because of an ongoing edit war or content dispute
6. Illustrated, if possible, by media such as images, video, or audio
- 6a media are tagged with their copyright statuses, and valid fair use rationales are provided for non-free content
- 6b media are relevant to the topic, and have suitable captions
Miscellaneous points
[edit]Wu being the first Wolf Prize winner should be cited; here is a suitable sourceDone
- It's not clear from the article what is the distinction between "a mirrored version of the world and the mirror image of the current world."
- "a mirrored version of the world and the mirror image of the current world" see the clocks. Look in the mirror, and recreate what you see in the real world. For example, building a physical version of the mirror image of the clock in the real world. The expectations for most people is that of p symmetry, the hands of both physical clocks (normal and reversed) would go from 12 to 3 to 6 back to 12. Because you built a fully reversed clock (as you see in the mirror), that reverse clock goes in the reverse direction, which here is also from 12 to 3 to 6 back to 12 because these positions have reversed labels. For clocks that would would violate p symmetry, the reverse clock, built like what you see in the mirror, goes in the forward direction, or 12-9-6-3 on those reverse labels. Which is mind boggling, because nothing in our every day life behaves like that. Headbomb {t · c · p · b} 01:56, 7 April 2020 (UTC)
- The point about the Wu experiment being a way to define left and right operationally is important and should be mentioned in the lead. It's clearer to the average reader than subtle technical points about parity conservation.
Also, the bit about the emission direction of the gamma rays needs to be clarified a little. How do they work as a "control"?
- The article is coming along well. I think the Experiment section needs more sources.
- Where would those be needed? It seems covered to me by what's already there, but [citation needed] tags would help understanding your feedback here. Headbomb {t · c · p · b} 02:35, 7 April 2020 (UTC)
- It seems Wu was anxious about other scientists beating her to the punch, and therefore cancelled a holiday to do the experiment: [1] –
with proper sources and context
The images seem mostly fine. I'm not 100% sure what the point of the Feynman Diagram is, and Krea makes a good point about a possible missing image.Done
- It seems unicode isn't quite good enough, but according to some web site the closest is: Ɛ-მ-୧-ςƖ, if you still read the left to right. I suspect it is time for unicode to include a complete set of mirrored roman alphabet and digits. Gah4 (talk) 00:00, 20 April 2021 (UTC)
Comment
[edit]Coincidentally, I was looking at the page because I wanted to know the details of Wu's experiment, and I found this review. I think the article is good, but not great. Its major weakness is that it skips over the real principle behind the experiment very quickly: there's a lot of exposition about the setup and the decay, but then what you expect to see, and crucially why, is left to a sentence or two. The diagram in "The Experiment" section that endeavours to explain the theoretical setup could also be better, or at least far better explained. I can try to add a few sentences to effect these emendations and you can judge for yourself if that's any better. Krea (talk) 17:52, 7 December 2019 (UTC)
- Thanks @Krea:, you are more than welcome to work on the article. Reyk YO! 10:27, 8 December 2019 (UTC)
@Reyk: You're welcome. I've made a few edits. I tried to incorporate the theory into the existing "The experiment" section, but I found it too unwieldy. In consequence, I've added a theory section to explain the basic idea of the experiment. I'm not sure if it's clear enough to explain to the casual reader the essential points, but that's my best attempt for what it's worth. I've gone over Wu's paper, which is surprisingly short, and I've tried to add some of the points that it makes without going into the details too much. As such, I've also made a few minor corrections and added a few points. I've also removed the diagram that I was complaining about before: I think it really doesn't help because it kind of conflates Wu's setup and the general argument for why emission anisotropy implies parity invariance. My reading of Wu's paper is that the experiment doesn't depend on reversing the polarizing field, but it does do that for experimental reasons. So, a diagram that talks about flipping the polarizing field gives an incorrect impression of the experiment. What would really be nice is a diagram like the one here: [2] (p. 7) that explains the anisotropy/P-violation idea. I'll see if I can find one that is in the public domain.
Let me know if you have any comments or criticisms on my additions or changes. Krea (talk) 18:44, 8 December 2019 (UTC)
- I'm sorry for my recent absence, but a serious case of real-life-itis has cut down my time to do some serious editing. Will get this done tomorrow however. Reyk YO! 14:51, 20 December 2019 (UTC)
- I'm happy to pass this once the four reminaing bullet points are addressed. Cheers! Reyk YO! 09:13, 24 December 2019 (UTC)
- @Headbomb:- just checking in to make sure you're still aware of this. Reyk YO! 18:08, 6 January 2020 (UTC)
- I am, but I've had stuff come up in my personal life right before the new year. Nothing major, but I'm mentally occupied till the end of the week. Headbomb {t · c · p · b} 18:10, 6 January 2020 (UTC)
- No problem at all. I know how busy this time of year is. There's no rush. All the best, Reyk YO! 18:12, 6 January 2020 (UTC)
- @Headbomb: just checking in again. Reyk YO! 19:38, 25 January 2020 (UTC)
- No problem at all. I know how busy this time of year is. There's no rush. All the best, Reyk YO! 18:12, 6 January 2020 (UTC)
- I am, but I've had stuff come up in my personal life right before the new year. Nothing major, but I'm mentally occupied till the end of the week. Headbomb {t · c · p · b} 18:10, 6 January 2020 (UTC)
- @Headbomb:- just checking in to make sure you're still aware of this. Reyk YO! 18:08, 6 January 2020 (UTC)
- I'm happy to pass this once the four reminaing bullet points are addressed. Cheers! Reyk YO! 09:13, 24 December 2019 (UTC)
Thanks for the ping. I haven't forgotten about this but I've fell sick for the past few days. Should be back up on my feet within a day or two, so you can expect movement on this this week. Headbomb {t · c · p · b} 19:18, 27 January 2020 (UTC)
- Turns out this is taking longuer than expected to recover, still haven't forgot about this though. Headbomb {t · c · p · b} 01:50, 9 February 2020 (UTC)
- No problem at all. I for one am not putting up a deadline. Just ping me or drop me a talk page note when it's ready. Reyk YO! 02:07, 9 February 2020 (UTC)
A comment about the timeline: reference 4, currently used to support the statement that the experiment was carried out in December 1956
, doesn't actually state the month. It refers to a six-month work-up period
and the extraordinary jam-packed sessions
at the APS meeting in January 1957. Lee and Yang recall, The Columbia–Bureau of Standards experiment under C. S. Wu was progressing satisfactorily during the fall of 1956. She shuttled back and forth between New York City and Washington and kept us informed of the progress of the experiment. Around Christmas she finally said they were getting an asymmetry indicating that parity was not conserved in β decay, but she warned that it was very preliminary and therefore we should not spread any rumors. We didn't, but rumors somehow were generated anyway and we answered an alarming number of phone calls asking for and/or offering information.
The Gizmodo piece linked above omits the long build-up period prior to the December observations, and it actually links back to the Wikipedia article, which makes me leery of using it here. In this reminiscence, Wu says that she canceled her trip in the spring of 1956 (not just before the Christmas holidays). XOR'easter (talk) 21:27, 16 February 2020 (UTC)
Status query
[edit]Reyk, Headbomb, where does this review stand? It's been open for nearly four months, yet as best I can tell Headbomb hasn't edited the article since December 8, two days after the review opened, but has made over 12,000 mainspace edits in 2020 alone. If Headbomb can't make time in the seven days to address the issues in this review, it's probably time to close this. Thank you. BlueMoonset (talk) 18:17, 2 April 2020 (UTC)
- That's certainly fair. I'll try to make room for this by the end of the weekend. It's not really a matter of having the time (I do), but the mindset to tackle this topic again. If I don't reply by Monday, feel free to consider this withdrawn. Headbomb {t · c · p · b} 21:59, 2 April 2020 (UTC)
- Agh! Sorry to be so late replying. For some reason the ping didn't go through. I will take one last look at this tonight after work, and I suspect I will probably pass it. Reyk YO! 09:19, 21 April 2020 (UTC)
- @Reyk and Headbomb: Sorry to be that guy, but thought I would just drop a few pings to remind everyone that this is still open. AIRcorn (talk) 21:17, 9 May 2020 (UTC)
- Apologies for my absence. I was dealing with some corona crap in the family. Anyway, I read the article over a few more times, and the only outstanding issue was that I asked for a few more sources in that one section. But now having read it over again I can't remember or figure out what I was so worked up about. I am therefore passing the article. Congrats on a fine article. Reyk YO! 10:40, 10 May 2020 (UTC)
equal probability in all directions
[edit]The article says: equal probability in all directions. They don't have to be equal in all directions, only, as it says just a little farther down, the same for θ and pi-θ. Gah4 (talk) 06:34, 8 October 2020 (UTC)
gamma ray anisotropy?
[edit]Why does the results section say: gamma ray anisotropy instead of beta anisotropy? Gah4 (talk) 06:39, 8 October 2020 (UTC)
- Thanks, fixed. Johnjbarton (talk) 19:06, 8 June 2025 (UTC)
Diagrams
[edit]
@Krea: Sorry I'm late to the party. I only noticed today that you removed File:Wu experiment.jpg and suggested using a diagram like the one here (page 7) to explain the anisotropy/P-violation idea. I'd be happy to make a new image for you if you like. Just leave a note on my talk page. Cheers. nagualdesign 13:53, 16 December 2020 (UTC)
I went ahead and made the image (right). Let me know if you'd like any alterations. nagualdesign 22:11, 16 December 2020 (UTC)
- @Nagualdesign: Sorry, I've been away for quite a while so I've only just seen these messages. The new graphic looks great; well done for ignoring my silence and getting it done anyway. I can't see anything that I would want altered -- it looks fine to me. Good job. Krea (talk) 00:58, 26 March 2021 (UTC)
- No worries. Thanks for the belated reply. And welcome back! nagualdesign 01:13, 26 March 2021 (UTC)
Experimental measurements, two for gamma anisotropy, one for beta
[edit]The article says that "experiment then essentially counted the rate of emission for gamma rays and electrons in two distinct directions", when actually the gamma was measured in two directions ("polar" and "equatorial") whilst the beta only in one. The gamma measurements were used to measure the polarisation of the nuclei as its temperature changed as it warmed up from a very low temperature and the cold beta emissions compared with the warm ones. The latter differed much more than the former because of the parity violation. PeterGrecian (talk) 11:12, 8 April 2021 (UTC)
- There are two conditions in the experiment: field up, field down. In each condition the gamma ray anisotropy is measured by comparing "polar" and "equatorial" values. This gives a value for the spin polarization. In each condition the electron count is measured. This tells us about parity violation assuming spin polarization. Then the two conditions are repeated again and again as the experiment heats up and the spin polarization goes away.
- I've rewritten the content along these lines, please review. Johnjbarton (talk) 16:12, 8 June 2025 (UTC)
Article review
[edit]It has been a couple years since this was reviewed, so I took a look and noticed uncited statements, including entire paragraphs. Some paragraphs do not need paragraphs because of WP:CALC but others explaining the history or theory are probably necessary. Should this go to WP:GAR? Should I add "citation needed" templates to places that I think need citations? Z1720 (talk) 20:57, 29 May 2025 (UTC)
GA Reassessment
[edit]- Article (edit | visual edit | history) · Article talk (edit | history) · Watch • • Most recent review
- Result pending
Uncited statements, including entire paragraphs. Some paragraphs do not need citations because of WP:CALC but other statements explaining the history or experiment are probably necessary. I am happy to add "citation needed" templates if an editor pings me. Z1720 (talk) 18:15, 6 June 2025 (UTC)
- This is the difference with the version that was approved. There was no susbstantial expansions, no citations removed, and no degredation. I hardly see anything that needs improvement here, or that would be detrimental to GA status. Headbomb {t · c · p · b} 18:21, 6 June 2025 (UTC)
- @Headbomb: If editors wish, I am happy to add "citation needed" tags where I think need citations, to better highlight where I think they are needed. Sometimes another reviewer has concerns that were not highlighted by the first reviewer. Z1720 (talk) 18:29, 6 June 2025 (UTC)
- I added a couple of new sources that are especially good on the experimental side. (I apologize for not following the sfn formatting). Johnjbarton (talk) 01:04, 8 June 2025 (UTC)
There is a statement "However, Wu observed that the electrons were emitted in a direction preferentially opposite to that of the gamma rays ...", which makes no sense to me. The description of the gamma-ray emission suggests that there is no difference between the two poles in its pattern, only between the poles and the equator. Maybe it was intended to say "... direction preferentially opposite to that of the nuclear spin pseudovector ..."? —Quondum 20:38, 8 June 2025 (UTC)
- I fixed this one. Please check. Johnjbarton (talk) 22:57, 8 June 2025 (UTC)
- Yes, that bit makes sense to me now. —Quondum 11:35, 9 June 2025 (UTC)
- I fixed this one. Please check. Johnjbarton (talk) 22:57, 8 June 2025 (UTC)
- The paper could only claim a "lower limit" to the beta asymmetry of 0.7. That is, approximately 70% of the electrons were emitted in one direction, where as 30% were emitted in the other. I am left scratching my head about this: "asymmetry of 0.7" is meaningless to me (how is asymmetry defined quantitatively?) and it seems to be meant to be directly related to the "70%", but this is counter-intuitive because I would a asymmetry of 0% to mean balanced, but wouldn't this be 50% (asymmetry of 0.5)? This absolutely should be restated, preferably including an understandable definition of the quantities. Also, what does "in one direction" mean? Exactly aligned with one pole? Or emitted in one hemisphere? —Quondum 11:44, 9 June 2025 (UTC)
- I found a source this "asymmetry parameter": Wu's 2008 lecture spells it all out. I deleted the 70% bit because -.7 is only as much as Wu was confident of publishing quickly. Specifically electrons emitted into the crystal can scatter back into the detector reducing the observed asymmetry. But did not have a solid measurement for that value to use.
- Please check. Johnjbarton (talk) 22:58, 9 June 2025 (UTC)
- This is a substantial improvement; it is making a lot more sense. I have a query about Writing the emitted electron angular distribution as a function of the angle: the expression is pulled out of the air with no justification; some background would be useful to aid insight by the reader as to why it is expected. Also, from the gamma ray anisotropy has no interpretation: the expression of anisotropy must be different (with a cos2θ?), and how it relates to the polarization P would be useful for understanding. —Quondum 00:39, 10 June 2025 (UTC)
- Great questions! This angular distribution is derived in Yang&Lees paper. The gamma ray angular distribution is discussed in
- Rose, M. E. (1953). Angular correlation of nuclear radiations. Science, 118(3074), 635-642.
- The distribution is in the coincidence of the two gamma rays and is indeed cosine squared. I think the Rose/Gorter papers may have more information but I can't read them. Johnjbarton (talk) 02:16, 10 June 2025 (UTC)
- In this I see the "The angular dependence of the intensity of γ-radiation from j1 to j is: I+1 = 1⁄2C(1 + cos2θ), I0 = C sin2θ". I do not know how to interpret that (nuclear physicists seem to glide between classical and quantum reasoning, together with a hefty dose of shorthand; in any event, I find writings on the topic uninterpretable – even our articles gloss over central detail, e.g. Cobalt-60 omits neutrinos entirely). —Quondum 12:54, 10 June 2025 (UTC)
- Conway's PhD thesis
- Conway, B. J. (1961). Gamma-gamma angular correlation in the decay of cobalt-60 (Doctoral dissertation, Monterey, California: US Naval Postgraduate School).
- explains one aspect: these particular angular dependences are between the two gamma rays measured with coincidence. Thus they may not directly applicable to the Wu experiment which use gamma-to-spin axis.
- The gamma ray transitions are multipolarity of gamma radiation, E2 have 2 units of angular momentum. A (unpublished?) Master thesis
- Ganger, M. (2017). A Parity Violation Experiment for Undergraduate Laboratories.
- gives a direct analysis of cobalt-60
One thing to note about Figure 7 is that the gamma rays have a 𝑧-component of angular momenta of 2ℏ. Photons carry ℏ of spin angular momentum in the 𝑧-direction because they are spin-1 particles. In order for the gamma ray to carry 2ℏ of angular momentum, it carries both orbital angular momentum as well as its spin angular momentum.
- Unfortunately this source has no inline refs only a list at the end. Johnjbarton (talk) 16:35, 12 June 2025 (UTC)
- Conway's PhD thesis
- In this I see the "The angular dependence of the intensity of γ-radiation from j1 to j is: I+1 = 1⁄2C(1 + cos2θ), I0 = C sin2θ". I do not know how to interpret that (nuclear physicists seem to glide between classical and quantum reasoning, together with a hefty dose of shorthand; in any event, I find writings on the topic uninterpretable – even our articles gloss over central detail, e.g. Cobalt-60 omits neutrinos entirely). —Quondum 12:54, 10 June 2025 (UTC)
- Great questions! This angular distribution is derived in Yang&Lees paper. The gamma ray angular distribution is discussed in
- This is a substantial improvement; it is making a lot more sense. I have a query about Writing the emitted electron angular distribution as a function of the angle: the expression is pulled out of the air with no justification; some background would be useful to aid insight by the reader as to why it is expected. Also, from the gamma ray anisotropy has no interpretation: the expression of anisotropy must be different (with a cos2θ?), and how it relates to the polarization P would be useful for understanding. —Quondum 00:39, 10 June 2025 (UTC)
The first sentence of the Salam quote If any classical writer had ever considered giants (cyclops) with only the left eye. is grammatically incomplete. The Google books link excludes the referenced page. Is the period actually a comma? —Quondum 13:12, 9 June 2025 (UTC)
- I found the quote distracting and I couldn't decode it. Johnjbarton (talk) 15:41, 9 June 2025 (UTC)
- I concur; I have removed it. The earlier statement people at Princeton would often say that her discovery was the most significant since the Michelson–Morley experiment that inspired Einstein's theory of relativity makes the point well enough. —Quondum 16:26, 9 June 2025 (UTC)
- I found the quote distracting and I couldn't decode it. Johnjbarton (talk) 15:41, 9 June 2025 (UTC)
- The article claims that 60Ni has a spin of 4ħ, but I see that Isotopes of nickel indicates a spin of 0ħ for this isotope (in its ground state). The cited source also lists a spin of 5ħ for an "isobaric analog state". I realize that the 60Ni is in an excited state and that might not be listed in that source, but more background should be given in this article. I also cannot reconcile how a state with J = 4ħ could decay to a state with J = 0ħ through the emission of 2 gamma-ray photons, which together could carry off at most 2ħ. This makes me think that the source on isotopes is referring to total spin, and that the electron spins cancel some of the nuclear spin (quite conceivable, given Ni's ferromagnetism). This does, however, suggest that we should be clearer here that we are speaking of nuclear spins, not the spin of the isotopes as such, which the notation could suggest. Since I am reading between the lines, I am holding off on making any changes in this respect. —Quondum 14:24, 9 June 2025 (UTC)
- I see that Rose (1953) "Angular Correlation of Nuclear Radiations" and Cobalt-60 detail the decay chain: 60Co(5+) 60m2Ni(4+) 60m1Ni(2+) 60Ni(0+). I am still perplexed, though: each gamma ray would carry off 1ħ of angular momentum, not 2ħ. —Quondum 13:41, 10 June 2025 (UTC)
- The caption of the diagram with clocks is a little confusing. For example, it is not clear whether the left and right faces are two faces of the clock because of how it was "built", though my guess is that the left one is from the reference world and the right one is from the mirror world. What is meant by "built" is also unclear. —Quondum 00:39, 10 June 2025 (UTC)
- Some sense of the coupling between thermal phonons and nuclear spin orientation might be useful – basically how quickly the nuclear spins become disordered. Room-temperature NMR suggests that typical coupling is weak: precession can be measured over presumably milliseconds at least. This would also impact the difficulty of aligning the nuclear spins, though that is likely a different mechanism: radiation of microwave energy from precession might result in a quick decay to alignment in the strong magnetic field. Again, the need for the very low temperatures is not clear without a strong phonon coupling. —Quondum 00:49, 10 June 2025 (UTC)
Theory -> Design
[edit]I propose to rename and refocus the "Theory" section to "Design". The content already partly matches "Design" and it's hardly a Theory section. Could be "Experimental Design" but the article name means "Wu experiment design" would be implied. Johnjbarton (talk) 23:04, 8 June 2025 (UTC)
- The section that is currently headed "Theory" is clearly an attempt at explaining why an asymmetry in the observations would imply P-parity nonconservation. I agree that "Theory" is a poor heading. I disagree with the proposed renaming to "Design", as such a heading implies detail about the design aspects, which are not covered at all in the section. A heading such as "Motivation", "Explanation" or even "Insight" would be better. —Quondum 11:31, 9 June 2025 (UTC)
Axial vs Polar vectors
[edit]Links are confusing in discussion:
The reason for this is that the cobalt-60 nucleus carries spin, and spin does not change direction under parity (because angular momentum is an axial vector). Conversely, the direction in which the decay products are emitted is changed under parity because momentum is a polar vector.
Axial and Polar both link to Pseudovectors even though Polar vectors are not Pseudovectors at all. It's tricky because there's no real article for True Vectors though, only https://en.wikipedia.org/wiki/True_vector and the main article https://en.wikipedia.org/wiki/Vector 174.92.131.249 (talk) 18:47, 9 June 2025 (UTC)
- I agree with the thrust of this comment. The text assumes that the reader is familiar with properties of "axial vectors" and "polar vectors" under parity transformations, but we should not be sending those unfamiliar with them on a research expedition; some recap in the article explaining how spin and velocity behave under parity transformations (reflection) makes sense. The very concepts of "axial" and "polar" vectors is somewhat archaic IMO: they should be supplanted by vectors, bivectors, etc., whose behaviour under linear transformations can be intuited much more readily. —Quondum 01:01, 10 June 2025 (UTC)
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