Wikipedia:Reference desk/Archives/Science/2024 September 18

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September 18

Is there any physical theory, claiming that every elementary particle can turn into some other elementary particle?

HOTmag (talk) 13:05, 18 September 2024 (UTC)

Quarks are the only known particles whose electric charges are not integer multiples of the elementary charge. Therefore, in physical theories that accept both the Standard Model and the law of charge conservation, a quark cannot turn into another particle but a quark. But the types of quarks all have different masses, so all such quark–quark changes violate the law of conservation of mass.  --Lambiam 17:57, 18 September 2024 (UTC)

If you are referring to a single elementary particle, so why didn't you mention the electron, besides the quark?

If that's because an electron colliding with a positron turns (together with the positron) into a pair of photons, then also a quark colliding with an anti-quark turns (together with the anti-qurk) into a pair of gluons.

Anyway, in my question I allow a given elementary particle to collide with its anti-matter for becoming another elementary particle.

More important: My question is theoretical, so it's not only about known particles, but rather about all possible particles, including those which haven't been discovered yet. HOTmag (talk) 18:21, 18 September 2024 (UTC)

If someone claims all swans are white, it suffices to debunk the claim by finding one purple swan. Maybe there also blue, brown or black swans, but it is not necessary to search for further counterexamples. Likewise, if some physical theory claims every elementary particle can turn into some other elementary particle, it suffices to debunk the claim by finding just one elementary particle that cannot turn into some other elementary particle. I just started with the top line of File:Standard Model of Elementary Particles.svg. There may be many other counterexamples (like the Higgs boson), but why bother to keep searching?

The as of yet undiscovered bunkon and trashon, whose properties are still unknown except that they are postulated to be different elementary particles, can turn into each other. A difficulty in finding them is that their properties are unknown, so experimental physicists don't know where to look. There may be many more such pairs, which may never be discovered.  --Lambiam 10:17, 19 September 2024 (UTC)

By mistake, I thought you meant the quark was the only particle that couldn't turn into another particle but a quark, but now I see this was not what you meant, so I take my first sentence back.

However, I still emphasize that my question allows a given elementary particle to collide with its anti-matter for becoming another elementary particle.

Re. your senetnce: "The as of yet undiscovered...different elementary particles, can turn into each other": Is there any physical theory claiming what you've claimed in that sentence? Actually, this was my original question... HOTmag (talk) 10:44, 19 September 2024 (UTC)

I'm pretty sure that that sentence was a joke. Look at the names: BUNKon and TRASHon. --User:Khajidha (talk) (contributions) 11:25, 19 September 2024 (UTC)

No "maybe" about black swans. They were recorded by Europeans in 1697, possibly earlier. 2A00:23D0:F6F:1001:2D07:A712:8909:7D91 (talk) 11:56, 19 September 2024 (UTC)

All right. HOTmag (talk) 12:35, 19 September 2024 (UTC)

See also black swan and black swan theory. -- Jack of Oz ^{[pleasantries]} 18:37, 19 September 2024 (UTC)

OP may have in mind something more like the particles created from particle-antiparticle annihilation, such as those in the chart at Annihilation § Electron–positron annihilation, as opposed to something like the weak decay of quarks.

It seems to me that OP could mean either: 1) a single elementary particle can spontaneously become another single elementary particle (with the help of another particle that remains unchanged), in which case I think the answer may be no for any particle; or 2) for two given particles, there's an interaction in some condition where it's meaningful to say that one specified particle is in the input, and it becomes in the output the other specified particle [Edit: which may include any number of other particles in the reaction doing anything else]. Not sure (I didn't do particle), but I think (2) might be considered more or less accurate (to the extent the fuzziness of the wording necessarily allows). SamuelRiv (talk) 16:10, 19 September 2024 (UTC)

I adopt your option 1# if it means Particle decay, and I adopt your option 2# if it means annihilation (i.e. by colliding with the anti-particle).

But contrary to the way you divided you answer: option 1# for all particles, or option 2# for all particles, I didn't exclude a third option which is: option 1# for some particles, and option 2# for the rest of the particles (without excluding particles that satisfy both 1# and 2#).

All agree, that some particles satisfy option 1#, and that some particles satisfy option 2#.

My question is about whether there is any physical theory claiming, that every particle (including any particle that hasn't been discovered yet), satisfies either 1# or 2# (or both). HOTmag (talk) 10:25, 20 September 2024 (UTC)

I was trying to interpret your question literally. #1 is one single particle becoming one different single particle with everything else unchanged -- this does not happen at all afaik, nor in general in theory (User:Lambian gives a simple example for quarks in their answer above). #2, the way I worded it, afaik can (and does) happen for all particles in the Standard Model.

What I'm trying to convey with this #1/#2 description is that it's not a particularly meaningful one, if you can claim "every elementary can turn into some other elementary particle" just by comparing one reactant to one product in a complex interaction.

(As maybe a sorta-ok example, consider the chemical reaction of a strong acid + base into salt + water: HCl + NaOH -> NaCl + H2O. Would you say the HCl (reactant) turns into salt? turns into water? Or does it change nothing at all because both the reactants and products are largely remain just free ions in aqueous solution? This is why I'm not sure what you're trying to ask is very meaningful.) SamuelRiv (talk) 17:24, 20 September 2024 (UTC)

I'm not asking about a single particle becoming one different single particle with everything else unchanged. Let's put it this way: Is there any physical theory claiming that all particles in the dark matter can turn into other particles? (whether by a decay or by annihilation or by any other way). HOTmag (talk) 01:30, 22 September 2024 (UTC)

There can never be such a theory. Why? Dark matter is named so because it never interacts with ordinary matter, except by gravity. As soon as it interacts in any other way it's not dark matter anymore but a new kind of ordinary matter. By interaction by gravity no distinction of particles is possible, only sum and distribution of mass is measurable. So dark matter can decay all it want, no human physicists are able to prove it or disprove it. By definition of the word. It may be that some particles, that we now subsume in dark matter, are later discovered to be not dark. And then we would know the conditions where they participate in the normal decay of ordinary matter. But that can never tell us about the real dark matter, as long as there's real dark matter. 176.0.153.105 (talk) 18:20, 23 September 2024 (UTC)

Most particles decay into other particles. The only "stable" ones are Electrons, Protons, Photons and (to a degree (see here)) Neutrinos. But even these can (under the right conditions) either combine or "destroy" and "create" with each other. 176.0.165.39 (talk) 12:20, 20 September 2024 (UTC)

These interactions can all be depicted in a Feynman diagram as lines meeting in a vertex. The lines correspond to particles. In a Feynman diagram, a vertex is always the meeting point of three lines. A particle → particle change would correspond to a Feynman diagram in which just two lines meet in a vertex.  --Lambiam 22:47, 20 September 2024 (UTC)

Or two lines merging in two vertices that are connected by two lines in the form looking like an eye. 176.0.153.105 (talk) 14:24, 23 September 2024 (UTC)

Yes, I know that. but you're talking about partciles of the Standard Model, while I'm asking about a theory that claims that all particles, including those which haven't been discovered yet, can decay turn into other particles. HOTmag (talk) 01:30, 22 September 2024 (UTC)

In your question and first response you talked about a particle that "can turn into some other elementary particle", and people are trying to clarify what that can mean. But now you're asking about decay: "Particle decay" is where only one particle goes in, and some other number of different particles come out, but as others have said there are stable particles that do not decay. (There are of course experimental bounds to what we currently know of this, and there are interesting subtleties in the theory for example why a photon does not decay.)

There's no theory which can claim anything meaningful about all particles that have not been discovered yet. A theory predicts new particles, and the theory becoming successful may lead to building experiments to verify empirically the particles ("discover" them, although a "discovery" was equally done when the theory was written).

You could imagine another type of theory that might say all particle physics theories are really at a fundamental level part of X-theory, and in X-theory everything decays into X-dust in 20 billion years, but that's not a particle theory. SamuelRiv (talk) 05:48, 22 September 2024 (UTC)

Re. your first paragraph: I'm sorry for not being clear in my recent response. Thanks to your comment, I will make it clearer, as I've alraedy made it in my first post.

Re. your second paragraph: I mean, something like the supersymmetric theory, claiming something about all particles, including those which haven't been discovered yet. So, again, I'm asking whether there's a theory that claims that all particles, including those which haven't been discovered yet, can decay turn into other particles, whether by a decay or by annihilation or by any other way. HOTmag (talk) 12:45, 22 September 2024 (UTC)

Although the above discussion may imply that the words "hypothesis" and "theory" can be used interchangeably, a scientific hypothesis is not the same as a scientific theory.

An answer is No, there is no scientific theory that just claims anything. A scientific theory offers a generalized explanation of how nature works described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction ("falsify") of it.

The OP's question using the word "theory" to mean a claim that at best will remain unproven or speculative actually looks for a Hypothesis meaning an educated guess or thought about something that cannot satisfactorily be explained with the present scientific theories. The OP seems to be thinking aloud[1] a new hypothesis. Philvoids (talk) 10:37, 22 September 2024 (UTC)

See Supersymmetric theory. It claims something about all particles, including those which haven't been discovered yet, and it's still called a "theory". Anyway, I'm not focusing on terminology but rather on an idea: Is there any theory, or a well known hypothesis, or a well known conjecture, or whatever, claiming that all particles, including those which haven't been discovered yet, can turn into other particles, whether by a decay or by annihilation or by any other way. HOTmag (talk) 12:45, 22 September 2024 (UTC)

Wikipedia editors are careful about terminology. By "Supersymmetric theory" you link to the article titled Supersymmetry. Its first line clarifies that it refers not to a theory but a theoretical framework. Read further to see how it anticipates what might characterise a supersymmetry theory without specifying any one for attention. Merely calling a supposed bosonic superpartner to the electron a selectron hardly amounts to a falsifiable hypothesis and obviously says nothing about undiscovered particles. Beside our care with terminology, we are not in the business of prediction. Philvoids (talk) 09:17, 23 September 2024 (UTC)

I've linked to our article [page] Supersymmetric theory, which does exist in Wikipedia, so when I used the term "supersymmetic theory" I used a term used in Wikipedia as well. Indeed, it passes to the article Supersymmetry, but also this article does point out - in its second paragraph - that "Dozens of supersymmetric theories exist", whereas the first paragraph of this article - does point out that the suppersymmetry "proposes that for every known particle, there exists a partner particle with different spin properties. There have been multiple experiments on supersymmetry that have failed to provide evidence that it exists in nature."

Anyway, my original question was not about the terminology used in Wikipedeia, but rather about the very idea, and I allow you to call it: theory, theoretical framework, hypothesis, conjecture, proposal, suggestion, idea, or whatever, but the main idea still remains, as long as I understand you and you understand me (I guess this is the case). HOTmag (talk) 10:07, 23 September 2024 (UTC)

Anyway your claim "I've linked to our article Supersymmetric theory, which does exist in Wikipedia" is a false claim indeed. Philvoids (talk) 11:48, 26 September 2024 (UTC)

Try now to click again on Supersymmetric theory, and this time you will see that this article [page] does exist in Wikipedia, giving a link to the other article, but without passing to the other article. Actually the article [page] Supersymmetric theory exists in Wikipedia since 14 January 2006, as you can see in its history page. Generally, every article [page] that passes to another article [page] must exist in Wikipedia: If it hadn't existed, it couldn't have passed to the other article [page]. HOTmag (talk) 22:31, 26 September 2024 (UTC)

Please see WP:REDIRE. A Wikipedia redirect is a page that automatically sends visitors to another page, usually an article or section of an article. It aids navigation and searching but a redirect page is not itself a Wikipedia article nor is its mere existence a reliable source. The excuse of an honest mistake won't justify an attempt to argue black as white by misrepresenting links. Philvoids (talk) 11:47, 27 September 2024 (UTC)

My only innocent mistake was my replacing "page" by "article". Anyway, when I wrote "article" I really meant "page", and I was sure this would also be what you would interpret when you read the word "article" in my responses. But since I'm realizing now this was not what you interperted, I'm striking out every "article" and replacing it by "[page]".

As for what you call "a reliable source", I've already quoted the second paragraph of our article Supersymmetry: "Dozens of supersymmetric theories exist". This quote proves that there is no fundamental difference between linking "UK" to "United Kingdom" and linking "Supersymmetric theory" to "Supersymmetry": All of these four terms are legitimate. HOTmag (talk) 12:46, 27 September 2024 (UTC)

Now you only have to define what you mean by particle. For instance in Standard Model you have the everyday particles (Electron,Proton...). Nothing of that is stable in your definition. Then you have Quark. No quarks are ever single. So it is a question of the neighbouring quarks which reactions are possible. But even then there is no stability in your definition. But never is one particle turned into exactly one other particle. Even if that were possible you only have to look at a different level of abstraction and a group of particles would turn into a different group of particles. 176.0.158.114 (talk) 09:34, 23 September 2024 (UTC)

By elementary particle I mean what physicists mean by that term: quarks, leptons, gauge bosons, and also elementray particles that haven't been discovered yet, like axions. Anyway, I really meant what you suggested in yout last sentence: "a group of [elementray] particles would turn into a different group of [elementray] particles". HOTmag (talk) 10:07, 23 September 2024 (UTC)

Then I can give you a definite answer. Any group of ordinary matter particles (whether known or not) can turn into another group of particles if they encounter the right conditions (even if the conditions could not be achieved in a laboratory or somewhere near Earth). For dark matter particles that can not scientifically be said. And never will be possible to say with science. If someone says something about changing about dark matter it is and never will be science. That is part of the definition of the word "dark" in "dark matter". If that definition changes nothing about the future I have written will continue to be valid. 176.0.153.105 (talk) 19:28, 23 September 2024 (UTC)

Memorandum: My question was about all elementary particles, and by elementary particle I mean what physicists mean by that term: quarks, leptons, gauge bosons, and also elementray particles that haven't been discovered yet, like axions.

To sum up: The question is whether, for every group of elementary particles, including those which haven't been discovered yet, there exist (what you call) "right conditions", under which this group can turn into another group of elementary particles. HOTmag (talk) 13:10, 24 September 2024 (UTC)

Then as I said all the way before, with my "#2", then if it can be called a "particle", yes you can always make it turn into stuff.

But you should understand what people here are trying to say: this becomes rather meaningless as your understanding of what a theory means, in the sense of the "discovery" of particles, is not very accurate, so when you're trying to force these incompatible notions into your question it makes it difficult to give meaningful answers. (Mine, as I said before, is not particularly meaningful.) SamuelRiv (talk) 13:28, 24 September 2024 (UTC)

Theories don't necessarily discuss discoveries, and a theory about elementary paricles doesn't necessarily discuss what you call "discovery" of elementary paricles, it can also say something about elementary paricles that haven't been discovered yet, e.g. gravitons, axions, electrinos, gravitinos, axinos, and the like. Anyway, my question is about all elementary particles, including those which haven't been discovered yet. Are you referring to all of them in your first sentence? HOTmag (talk) 18:26, 24 September 2024 (UTC)

YES! Even for particles that are not discovered yet, the right conditions are possible to determine. For an example see the article about the Axion. Even although the particle is not discovered yet, the right conditions for some way to turn into another group of particles are already known (strong magnetic field). Of course there will probably be other ways but one is enough to answer your question. And there is even a general answer. Most particles will turn at the surface of a Neutron star. If you are in doubt, there will almost always be the right conditions. Is there some particle that does not turn at the surface of a neutron star? Of course the neutron. But that turns at enough distance from the surface. And it is possible that the Alpha particle is a special Quantum state of twelve Quarks. Should that be the case a neutron would even be turned on the surface of the neutron star into an alpha particle. 176.0.147.163 (talk) 13:03, 26 September 2024 (UTC)

Have you got any source for your claim, that all elementary particles (other than neutrons), including those that haven't been discovered yet, e.g gravitons (and gravitinos and electrinos and axinos and likewise), will turn at the surface of a nuetron star? HOTmag (talk) 22:13, 26 September 2024 (UTC)

First my claim was for almost all particles, what you acknowledge by excepting the neutron. Second, no I don't have a verifiable source, because it is based on a personal communication. But it is easy to verify. Elementary logic and quantum mechanics are sufficient. Every physicist knows that quantum mechanics and high gravity or strong electric or strong magnetic don't fit together. As soon as a quantum particle interacts with something that needs relativity to describe, the equations begin to change. Of course the particles will change with them. I can give you an example, that does not need an exotic environment. If you read any Wikipedia article about a heavy element you will read about relativistic effects, that affect the electrons. And at the same time the core will go more and more unstable. Sometimes the elements are only "observationally stable". And the more affected elements are really radioactive. Until it comes to U238. There's a rule. Every number of charge has an associated ideal atomic weight. If the weight is too large neutron are not stable anymore and decay. If the weight is too small, coulomb force wins and an alpha particle, the least energetic particle in the core, will be ejected. Now to U238. According to its pattern of decay it's both too heavy and too light. Of course that is relativity playing havoc with the quantum mechanics. And now imagine what will go on where relativity really comes into play. Such as at the surface of a neutron star. 176.0.144.177 (talk) 02:36, 27 September 2024 (UTC)

PS. The claim about the surface of a neutron star is meant here. For this claim exceptions are possible. The surface of a neutron star is an example for the right conditions and a good bet,but not guaranteed. Further above was another claim about the right conditions. That claim is universally true, because there is always an environment where the wave function can be distorted beyond recognition. Which is the whole point of the claim. 176.0.144.177 (talk) 02:49, 27 September 2024 (UTC)

I deliberately asked about what you call "exotic environment" (e.g. gravitons, electrinos) because, while your example regarding U238 is about matter quite known to us - experimentally speaking, you're using a kind of generalization, from what we have already encountered (e.g. U238), to what - we haven't indeed encountered yet - but we can assume something about in (your own?) theoretical framework stating that: since (as you begin) "as soon as a quantum particle interacts with something that needs relativity to describe, the equations begin to change", so (as you conclude) "of course the particles will change with them". Of course your conclusion is possible and legitimate, but you haven't proved that it's the only one possible - theoretically speaking - mainly as far as (what you call) "exotic environment" is concerned.

To sum up: It seems like you're using your own theory. This is legitimate of course, but when I asked for a "theory" (see the title), I mainly meant: a (well sourced) theory - although I didn't add this condition in the title (but only in my previous response). HOTmag (talk) 08:49, 27 September 2024 (UTC)

How I said, there's private communication involved, so it's possible that through misunderstanding a private theory emerged, but the goal was to talk about the normal quantum physics. I never claimed that quantum physics is the only theory where universal decay is possible/mandated. That quantum physics and general relativity are mutually incompatible is well sourced. The example of U238 was only to show that the equations begin to change in the region of both relativity and quantum physics. And that the trend in this regard goes to less stability instead of more. Of course there are exceptions, particles that are normally not stable but can be stabilised by the exotic environment, better the changed equations in the exotic environment. Of course every exotic environment changes the equations in its own way. So for every needed change a matching exotic environment can be (theoretically) constructed. And then there is the well sourced Hawking radiation. If you want a random change an appropriately sized black hole will provide an exotic environment that is guaranteed to destroy any quantum state (well sourced No-hair theorem) and emit another random quantum state. 176.0.167.84 (talk) 00:16, 28 September 2024 (UTC)

I let you use any (well known) theory, including Quantum physics, for claiming that as soon as a quantum particle interacts with something that needs relativity to describe, the equations begin to change. I only asked whether your conclusion, that if the equations begin to change then the particles will change with them, must be concluded from this (well known) theory.

As for the option of black holes, yes, this is what I'd thought, and I'm asking now a further question in my following thread, regarding this option. HOTmag (talk) 18:14, 28 September 2024 (UTC)

See Richard Feynman as he talks about Path integrals. If the formula changes, the sum will change. That is clearly mathematics. I don't know if somebody has written about these mathematical relations, because it is only article worthy if there is a whole theory in the conclusion, but that is not how far we know the formulas yet. But even as you begin to try it, you will see immediately that there is change imminent. And the example of U238 shows that the change is not an artefact of the mathematical formulation but is reality. And the ultimate weapon (the black hole) drives the point home. Only the exact relations are unknown today. 176.0.164.155 (talk) 20:24, 28 September 2024 (UTC)

"If the formula changes, the sum will change". IMO, it's still too obscure, because:

As long as the particles haven't been discovered, we don't know what properties those particles carry, so we don't know how the equations involving those particles look like, so we don't know how those equations are going to change, so we don't know whether this change may influence the issue of whether the particles involved in those changing equations may turn into other particles.

As I've already said, IMO it's still too obscure, as long as we don't know what particles we are talking about. HOTmag (talk) 22:39, 28 September 2024 (UTC)

Every particle, discovered or not, must obey the Schrödinger equation. Part of the solution of the Schrödinger equation is an Integral. One way to do it is the Path integral formulation. That formulation has the advantage that the Spacetime is part of the explicit input. Relativity Theory modifies space. Modified input results in modified output, according to Mathematics.

That is essentially the whole point of the "theory". 176.0.164.155 (talk) 00:15, 29 September 2024 (UTC)

"Modified input results in modified output". For claiming that, you must make sure that this equation reflects a one-to-one correspondence between input and output. HOTmag (talk) 01:05, 29 September 2024 (UTC)

That's not true. In the pure mathematical sense there may be a function where that applies. But never in physics. Even for the Sine and cosine function it's not true, if you really think about it. In the Schrödinger equation there is a double Differential operator. So for the same output you need the same value, the same first differential and the same second differential. And now you can imagine a function that is at least two times differentiable, has at two inputs the same output, has at least a third input with a different output (constant function do not count) and is not (theoretically) reversible in the limits of our universe. 176.0.164.155 (talk) 12:05, 29 September 2024 (UTC)

You did not prove that the Schrödinger equation reflected a one-to-one correspondence. You have only presented some kind of intuition for claiming that. I still wonder if this intuition is rigorously provable. HOTmag (talk) 17:41, 29 September 2024 (UTC)

No, I didn't, because it is not true. But I argued (intuitively) that it's not necessary. And some functions (like sine) are theoretically not one-to-one, but really they are,because there are only 111 digits for all values in our universe. And the Wave function has some of the properties of the sine function (because they are both solutions to second derivative differential equations), but here the value is dependent on the location. That means, even when a value at some location does not change, at another location the value will change. 176.0.159.38 (talk) 13:22, 30 September 2024 (UTC)

You are pointing at some analogy between the Schrödinger equation and the Sine/Cosine, but I suspect I don't even accept what you're claiming about the Sine/Cosine, so how can I figure out your analogy? Let me explain:

While you're claiming "some functions (like sine) are theoretically not one-to-one, but really they are, because there are only 111 digits for all values in our universe", our article Sine and cosine explicitly states: "sine and cosine are not injective", i.e. they are not one-to-one correspondences. This is true, both theoretically and in reality: Here is a concrete example taken from our real universe: $\sin(0)=\sin(\pi ),$ so modified input in the Sine function does not necessarily result in modified output - as opposed to what you'd claimed about the Schrödinger equation you'd compared with the Sine function.

Unless you think that zero or pi does not exist in the universe as an exact numeric value, but if so, then no value exists in the universe as an exact numeric value, so all physics may collapse. But it doesn't, while it uses both zero and pi in so many equations, including the well known equations in Quantum physics we're talking about. HOTmag (talk) 19:49, 30 September 2024 (UTC)

Pi exists in our universe. But not really that value, but rounded to 111 digits. And now you insert that value into your equation. Of course you need to round the results to 111 digits too. I don't know if the sine has not (by random chance) points where both roundings are cancelling each other, but theoretically there shouldn't be such points. The wave function has better behaviour in this case because the number of potential equivalents is finite. In contrast to the sine, where they are infinite. 176.0.159.38 (talk) 00:09, 1 October 2024 (UTC)

I don't know if the sine has not (by random chance) points where both roundings are cancelling each other. Me either, and this was exactly my point, without any rounding. For being sure there aren't any points like those you're talking about, we have to trust a reliable theory claimig there aren't, and this is the theory mentioned in the title. HOTmag (talk) 08:42, 1 October 2024 (UTC)

Without any rounding? That is not possible in our universe. And that's the difference between physics and mathematics. 176.0.159.38 (talk) 09:56, 1 October 2024 (UTC)

When I wrote "without any rounding" I was referring to my exact "point", i.e. to my exact argument, which involved no rounding, i.e. no round corners. HOTmag (talk) 11:18, 1 October 2024 (UTC)

And my argument is that, because of Planck units and Heisenberg uncertainly relation, no physics is possible without roundings. 176.0.156.195 (talk) 23:38, 1 October 2024 (UTC)

I'm fine with roundings, mainly for practical purposes (and for other reasons). I'm only claiming, that for being sure there aren't any points where - both roundings are cancelling each other - giving identical output for modified input, we have to trust a reliable theory claimig there aren't, and this is the theory mentioned in the title. HOTmag (talk) 10:54, 2 October 2024 (UTC)

Even if there are some points in the function where identical output comes from different input, we don't have to hit them. If one of the inputs is physically impossible (half an elementary charge, for example) it's still one-to-one. And part of the inputs is space. That in turn lets us avoid such points by designing the curvature of space. That is part of the right conditions above. 2A02:3032:308:A7D3:6020:6890:D467:113B (talk) 21:10, 2 October 2024 (UTC)