We don't know the mechanism which generates the gauge symmetry that makes charge discrete or generates conservation laws from symmetries, so it's risky to be too prescriptive about what is and isn't possible.
In the limit you can imagine a meta-technology which manipulates the mathematical structures that generate reality instead of just assuming the rules are fixed. That might or might not be possible, but the rules have to come from somewhere and there's no absolute guarantee they're hardcoded.
However, my post sucked and I didn't get around to emphasizing that. In quantum mechanics, it's possible for some state to have zero probability, and it's those probabilities that we're talking about everything being possible.
There's also the epistemic side where we know that there's a chance our theories are wrong in any given way. Those can't go to zero based on our model since the whole point is we're not restricting to a model. BTW, I really like your example as a good, clean limiting case.
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I'm sure people have come up with theories that violate conservation of charge, but they'd have to break U(1). I don't know what conservation CPT would break, if any; Noether's theorem shouldn't apply to it as a discrete symmetry.
The C in CPT is not only a change f charge but also a change of electron <---> positron, and quark <---> antiquark, and ...
The conservation or charge comes from the U(1) gauge symmetry. That is almost obvious if you ignore the mixing with the weak interaction, but after the mix you still have a U(1) gauge symmetry anyway.
Either the baryon number started non-zero and was conserved; or non-conservation of baryon number (protons vs antiprotons etc) have to be paired with equivalent non-conservation of lepton number (electrons vs positrons etc); or it can happen without simultaneously balancing the lepton number, and charge isn't conserved.
Unfortunately (for me) those implications are above my level. I can't usefully converse with an LLM about it, as this isn't a domain I know well enough to detect if it's making stuff up.
Now, on logistical level, the mapping procedure is not free, it does have a physical/energetic/mental cost to handle it, and a more trivial representation of integer has a lower cost on this regard.
But the simpler integer representation can represent less many situations. If all one need to deal with in a certain frame is covered by a simpler shipper representation, from an economical perspective it can be argued it worth the risk of hitting a point where the representation will be insufficient as the situation evolved with cases it can't handle.
My impression is that your underlying premises are conflating all of this distinguishable aspects. But of course my impression is limited by my own narrow representation of the world, no matter how open and curious I can be about learning from experiences it makes me go through.
I sometimes wonder how much we allow the map to become the territory.
That said, I accept that literally, 3 never occurs between the ∞ of 0-1 -- unless, we are assuming symbolically, rather than actual, that in our particular ∞, something does not transcend entirely, which we really cannot know, because the closest we have ever come to a real infinity is a representation or symbol of it.
Edit: I realized that the question appears, or is, stupid. But when I see 0.3, I tend to see the 0.0, 0.1, 0.2 between. This is not what 0.3 is supposed to mean though. But on the testability of any real infinity, I do not budge.
on edit: maybe should say infinite set instead of infinite sequence.
What leads you to claim that you can't encode infinite information given an infinite set to work with?
If spacetime is infinite in extent yet discrete at the Plank scale, it would be ℵ₀; the Hilbert space is real-valued and therefore (a vector of) ℵ₁.
The Bekenstein bound, in practice, would do limit how much information can be actually encoded, even if all quantum states have an underlying real value and therefore even just one quantum value of one particle is ℵ₁ all by itself.
But cardinality of real numbers might not be ℵ₁. It's literally unprovable in ZFC.
Infinity is an abstraction that simplifies a lot of maths, it isn't real though.
There's a possible future where you're President, but no possible future where you're a talking cucumber. This intuitive fact gets overlooked in pop culture explorations of Many Worlds, such as Everything Everwhere All At Once, which has ridiculous parallel universes which obey the rather more relaxed rules of "anything the writers can think of that looks silly", rather than "anything which can conceivably happen".
Ignoring that you have to give up a sense of personal identity at this point, there is nothing in the laws of physics that says the matter that makes up what we're presently calling me can't morph into a talking cucumber with a brain with my memories in it. It's just absurd number of extremely specific things that have to happen out of an ever more absurd number of possible things to happen.
(Note that the author many times is not the person that choose the title.)
Experimental physics takes it one step back: it's all about setting bounds on what is potentially possible. A model of the universe might say that the numbers are between zero and one, but there's some other model that says there could be 3s.
Note that this epistemic uncertainty is not the same thing that the article is focused on. The article is mostly about one specific model, which itself has some randomness baked on.
I particularly like the one with David Wallace.
He gave this pithy summary of his answer to a memorable discussion in a Nobel Laureate panel about what it actually means to explore Hilbert space, from a more practical perspective. Their discussion may be a good complement to the title article of this thread.
Others in this thread already correctly noted that just because something --- 'thing' more precisely defined as some action that results in an observable outcome --- exists in Hilbert space doesn't mean it's possible. But what Ketterle and others were getting at is that, as we make new theories and experiments, we can explore more of Nature and what's possible within, guided by the structure that Hilbert space could give to our questions' outcomes.
That being said, Ketterle explains it much better than I do.
Ketterle's answer in context: https://youtu.be/DNhmYkTW_jI?t=4220
Subtle difference, but energy conservation for example has to be adhered to.
I'm not sure we can extrapolate that out and say that can never be violated, only that we've never seen it happen.
With the caveats: - anomalies-> symmetries/conservation broken by quantum noise (has to be cancelled in valid theories) - Heisenberg’s incertitude
It just somehow always balances out in the end.
But then again, he was John von Neumann.
For example, in the Unruh effect an inertial observer sees the vacuum in their Hilbert space as having no particles. However, a uniformly accelerating observer sees that same vacuum as containing particles.
This state of affairs is addressed via algebraic quantum field theory (AQFT), which doesn't take a Hilbert space as the primary thing one starts with, but views Hilbert spaces as derived concepts.
[1] https://gizmodo.com/app/uploads/2025/07/favored-explanations...
But I think if you ask experts (philosophers of nature, people into foundational quantum mechanics) most of them would be many worlds proponents.
I finished my masters in a quantum computing lab. The reality is that most physicists just don't care about foundational QM, and stick to the Copenhagen interpretation. To me the Copenhagen Interpretation does not even count as a complete theory. It just stays very fuzzy on when wave function collapse happens, which makes it impossible to falsify.
>Questions: 'Which of the following, in your opinion, provides the best interpretation of quantum phenomena and interactions?'
in what do they mean by 'best'. For calculating and doing practical physics then sure Copenhagen Interpretation. But for what is the underlying reality of the universe you might get a different answer.
Copenhagen has an observer - your particle goes all ways through both slits deterministically following Schrodinger type stuff until 'the observation' when a dice is effectively rolled to choose one. But how do the laws of physics know the bunch of atoms far away is in the form of an observing physicist having a look? It would be cleaner laws of physics wise for them to ignore that, resulting in multiple observers seeing different outcomes. But for calculating, rolling a dice is easier.
That said probably Hilbert spaces are tools.
There are other problems with the Copenhagen Interpretation, but this isn’t one of them
That is, we are trying to come up with rules describing the underlying rules, right? The world isn’t made of stuff, it’s made of rules, right? Or math? Just not the same notation, at least.
Help! I feel like I’m falling off a metaphysical cliff!
-- David Hilbert.
Disclaimer: total armchair philosophy.
