Relationship between the quotient group and direct products by 
Ciro Santilli 37 Updated 2025-06-02 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-06-02 +Created 1970-01-01Although quotients look a bit real number division, there are some important differences with the "group analog of multiplication" of direct product of groups.
If a group is isomorphic to the direct product of groups, we can take a quotient of the product to retrieve one of the groups, which is somewhat analogous to division: math.stackexchange.com/questions/723707/how-is-the-quotient-group-related-to-the-direct-product-group
The "converse" is not always true however: a group does not need to be isomorphic to the product of one of its normal subgroups and the associated quotient group. The wiki page provides an example:
Given G and a normal subgroup N, then G is a group extension of G/N by N. One could ask whether this extension is trivial or split; in other words, one could ask whether G is a direct product or semidirect product of N and G/N. This is a special case of the extension problem. An example where the extension is not split is as follows: Let , and which is isomorphic to Z2. Then G/N is also isomorphic to Z2. But Z2 has only the trivial automorphism, so the only semi-direct product of N and G/N is the direct product. Since Z4 is different from Z2 × Z2, we conclude that G is not a semi-direct product of N and G/N.
This is also semi mentioned at: math.stackexchange.com/questions/1596500/when-is-a-group-isomorphic-to-the-product-of-normal-subgroup-and-quotient-group
I think this might be equivalent to why the group extension problem is hard. If this relation were true, then taking the direct product would be the only way to make larger groups from normal subgroups/quotients. But it's not.
Atoms exist and last for a long time, while in classical electromagnetic theory punctual orbiting electrons should emit radiation quickly and fall into the nucleus: physics.stackexchange.com/questions/20003/why-dont-electrons-crash-into-the-nuclei-they-orbit
In other sections:
Bibliography:
The first quantum mechanics theories developed.
Their most popular formulation has been the Schrödinger equation.
Quantum entanglement is often called spooky/surprising/unintuitive, but they key question is to understand why.
To understand that, you have to understand why it is fundamentally impossible for the entangled particle pair be in a predefined state according to experiments done e.g. where one is deterministically yes and the other deterministically down.
In other words, why local hidden-variable theory is not valid.
How to generate entangled particles:
- particle decay, notably pair production
- for photons, notably: spontaneous parametric down-conversion, e.g.: www.youtube.com/watch?v=tn1sEaw1K2k "Shanni Prutchi Construction of an Entangled Photon Source" by HACKADAY (2015). Estimatd price: 5000 USD.
Contains the clearest Bell test experiment description seen so far.
It clearly describes the photon-based 22.5, 45 degree/85%/15% probability photon polarization experiment and its result conceptually.
It does not mention spontaneous parametric down-conversion but that's what they likely hint at.
Done in Collaboration with 3Blue1Brown.
Question asking further clarification on why the 100/85/50 thing is surprising: physics.stackexchange.com/questions/357039/why-is-the-quantum-venn-diagram-paradox-considered-a-paradox/597982#597982
Quantum Mechanics: Animation explaining quantum physics by Physics Videos by Eugene Khutoryansky (2013)
Source. Usual Eugene, good animations, and not too precise explanations :-) youtu.be/iVpXrbZ4bnU?t=922 describes a conceptual spin entangled electron-positron pair production Stern-Gerlach experiment as a Bell test experiments. The 85% is mentioned, but not explained at all.
Quantum Entanglement Lab by Scientific American (2013)
Source. The hosts interview Professor Enrique Galvez of Colgate University who shows briefly the optical table setup without great details, and then moves to a whiteboard explanation. Treats the audience as stupid, doesn't say the keywords spontaneous parametric down-conversion and Bell's theorem which they clearly allude to. You can even them showing a two second footage of the professor explaining the rotation experiments and the data for it, but that's all you get. Institute of Electrical and Electronics Engineers by Ciro Santilli 37 Updated 2025-06-02 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-06-02 +Created 1970-01-01There are no stable isotopes.
Ridiculous cases of political correctness censorship by Ciro Santilli 37 Updated 2025-06-02 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-06-02 +Created 1970-01-01- Stack Exchange's censorship of "I think Trump is disgusting as a person" from Ciro Santilli's profile: cirosantilli.com/china-dictatorship/sstack-overflow-forbids-criticizing-the-character-of-genocidal-political-leaders-like-xi-jinping
Cute simple paper-cut stop motion animations videos by Mithuna Yoganathan, a PhD in theoretical physics at the University of Cambridge: www.damtp.cam.ac.uk/person/my332.
There is value in tutorials written by early pioneers of the field by Ciro Santilli 37 Updated 2025-06-02 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-06-02 +Created 1970-01-01Everyone is beginner when the field is new, and there is value in tutorials written by beginners.
For example, Ciro Santilli felt it shocking how direct and satisfying Richard Feynman's scientific vulgarization of quantum electrodynamics were, e.g. at: Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979), and that if he had just assumed minimal knowledge of mathematics, he was about to give a full satisfactory picture in just a few hours.
Other supporters of this:
- Ron Maimon: the same also applies to early original papers of the field, not just tutorials
- Dean Kamen: quick mention at: fi.edu/en/awards/laureates/dean-kamen, but a better longer mention on Dreamer (2020), nearby section from trailer: youtu.be/Cj2VKVJKf1I?t=16
Why phosphorus has multiple valencies? by Ciro Santilli 37 Updated 2025-06-02 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-06-02 +Created 1970-01-01 Partial index partial derivative notation by Ciro Santilli 37 Updated 2025-06-02 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-06-02 +Created 1970-01-01This notation is not so common in basic mathematics, but it is so incredibly convenient, especially with Einstein notation as shown at Section "Einstein notation for partial derivatives":
This notation is similar to partial label partial derivative notation, but it uses indices instead of labels such as , , etc.
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