Not all things are possible, only potentially possible ones.
If you take all the numbers between zero and one, you will find an endless supply of distinct numbers but none of them are 3.
CPT symmetry isn't conserved, so that's not necessarily a given.
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.
So, what I meant to say was that using the computational gadget from physics, none of the branches will violate conservation of charge.
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.
If so, this has interesting implications for the cause of the baryon asymmetry problem.
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.
Aren't you really just talking about the level of abstraction being applied? If there are infinite possibilities between zero and one, then it's just our way of relating to it that imposes the perception of a limitation.
No. You might be reading into it a little bit too much.
My point was that finding three needs to be possible for you to find it. If you are looking on the interval 0-1, it is not possible. As other comments have kinda of suggested, you could map every integer to a real and find "three" but if you are going to do that, why not just look on an interval that actually contains the number?
It's unclear to me at least what you mean here. In term of what is logically possible, if we take for granted that all integer numbers and more are mappable from a subset of this interval, then we do have this number.
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 think you are overthinking it too. The task was to find three, not something that is equivalent to three. However you choose to represent it, it wouldn't be hard to come up with a different "three" that you won’t find within that representation. If you are changing the representation, why wouldn't you just change the interval that you are searching?
I suspect a looseness with language; this often happens with maths done in natural language, and I may be guilty of it even when I try not to be.
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.
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.
"Hilbert space is the space of all possibilities." -- Wolfgang Ketterle
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.
I love this podcast. The solo or very physics focused episodes go over my head sometimes but they're always a fun listen. I enjoyed "Jody Azzouni on What Is and Isn't Real" where they (among other things) argue about what it means for a thing to be "real"/"exist", and extend that to mathematical concepts, fictional characters etc.
Yes, and in a more general way Noether thm applies in QM/QFT as well (energy conservation is just a special case). Symmetries become operators acting on the Hilbert/Fock space (ie representation).
With the caveats:
- anomalies-> symmetries/conservation broken by quantum noise (has to be cancelled in valid theories)
- Heisenberg’s incertitude
I know, there's some tricky business with energy conservation (I'm not pretending I understand it all though). It is for me an easy analogy though. Even Hilbert space has to stick to some rules!
Surprising that he could have that intuition in 1935, before the Unruh effect was known of, before AQFT was invented, and really before most of QFT was formulated.
Hawking radiation and the Unruh effect both rely upon the fact that there is no absolute Hilbert space in which QFT unfolds, the Hilbert space used is observer dependent.
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.
You’ve probably been reading a lot of physicists that believe in the Many Worlds view. Nature did a poll of physicists last year[1] and it’s still only the third most popular interpretation, behind the Copenhagen Interpretation and information-based interpretations like QBism.
Yeah, this is also why I said many and not most :)
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.
Yes! But at some level, doesn’t it kind of get blurred?
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!
Given infinite time, is it possible to randomly draw a rational (instead of an irrational) from the interval [0, 1]? And if so, is it possible infinite times?
In this case it does. Describe any effective measure to pick a random sample from the interval and you will have cheated into choosing from the computable reals instead (of which there is a countable amount)
I don't think that's correct. You're describing a limitation of our physical computing devices (or perhaps physical reality itself) as opposed to a theoretical flaw. At least as far as I can tell.
Not all things are possible, only potentially possible ones. If you take all the numbers between zero and one, you will find an endless supply of distinct numbers but none of them are 3.
For example, there's probability 0 of worlds where charge isn't conserved.
CPT symmetry isn't conserved, so that's not necessarily a given.
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.
So, what I meant to say was that using the computational gadget from physics, none of the branches will violate conservation of charge.
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.
If so, this has interesting implications for the cause of the baryon asymmetry problem.
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.
Aren't you really just talking about the level of abstraction being applied? If there are infinite possibilities between zero and one, then it's just our way of relating to it that imposes the perception of a limitation.
No. You might be reading into it a little bit too much. My point was that finding three needs to be possible for you to find it. If you are looking on the interval 0-1, it is not possible. As other comments have kinda of suggested, you could map every integer to a real and find "three" but if you are going to do that, why not just look on an interval that actually contains the number?
It's unclear to me at least what you mean here. In term of what is logically possible, if we take for granted that all integer numbers and more are mappable from a subset of this interval, then we do have this number.
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 think you are overthinking it too. The task was to find three, not something that is equivalent to three. However you choose to represent it, it wouldn't be hard to come up with a different "three" that you won’t find within that representation. If you are changing the representation, why wouldn't you just change the interval that you are searching?
He means that 3 is not between 0 and 1, which it really isn’t.
there aren't infinite possibilities between 0 and 1; there is an infinite sequence, the possibilities of which are quite strictly bound.
on edit: maybe should say infinite set instead of infinite sequence.
Maybe you can theoretically encode arbitrary (but not infinite) information, but there are still discrete underlying states.
> encode arbitrary (but not infinite) information
What leads you to claim that you can't encode infinite information given an infinite set to work with?
I suspect a looseness with language; this often happens with maths done in natural language, and I may be guilty of it even when I try not to be.
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.
> the Hilbert space is real-valued and therefore (a vector of) ℵ₁.
But cardinality of real numbers might not be ℵ₁. It's literally unprovable in ZFC.
I had to look this up, and if I understand right: yes, all that's certain is |ℝ| = ℵₖ, k≥1? If so, same point applies.
reality does not have an infinite set though.
Infinity is an abstraction that simplifies a lot of maths, it isn't real though.
So you have proof spacetime is discrete?
Terrence Howard is that you?
And can't one do all of mathematics without 3...(just map all the numbers > 1 to points inside the interval 0-1)?
That's true for theoretical physics.
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.
"Hilbert space is the space of all possibilities." -- Wolfgang Ketterle
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
Sean Carroll (name checked in the article) has an excellent podcast “Mindscape” (https://preposterousuniverse.com/podcast/)
I particularly like the one with David Wallace.
I love this podcast. The solo or very physics focused episodes go over my head sometimes but they're always a fun listen. I enjoyed "Jody Azzouni on What Is and Isn't Real" where they (among other things) argue about what it means for a thing to be "real"/"exist", and extend that to mathematical concepts, fictional characters etc.
I also found this one to stand out, but I don't remember exactly why: https://www.youtube.com/watch?v=oLKER6zFZ0A
It’s not ‘all’ things that are possible, just things that are allowed within the laws of physics :)
Subtle difference, but energy conservation for example has to be adhered to.
Aren't the laws of physics based on repeated observations that continue to show, for example, that we always see energy being conserved?
I'm not sure we can extrapolate that out and say that can never be violated, only that we've never seen it happen.
Yes, and in a more general way Noether thm applies in QM/QFT as well (energy conservation is just a special case). Symmetries become operators acting on the Hilbert/Fock space (ie representation).
With the caveats: - anomalies-> symmetries/conservation broken by quantum noise (has to be cancelled in valid theories) - Heisenberg’s incertitude
In quantum mechanics? Energy conservation is not as strict as you might think.
It just somehow always balances out in the end.
I know, there's some tricky business with energy conservation (I'm not pretending I understand it all though). It is for me an easy analogy though. Even Hilbert space has to stick to some rules!
"I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space anymore." - John von Neumann
Surprising that he could have that intuition in 1935, before the Unruh effect was known of, before AQFT was invented, and really before most of QFT was formulated.
But then again, he was John von Neumann.
I thought QFT is using hilbert space still?
Hawking radiation and the Unruh effect both rely upon the fact that there is no absolute Hilbert space in which QFT unfolds, the Hilbert space used is observer dependent.
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.
google "Fock space"
Aren't Fock spaces still hilbert spaces?
Just a reminder.. Hilbert space is just a tool/abstraction/map... and the map is not the territory..
Not according to many worlds! I think nowadays many physicists think many worlds/an everettian view is most plausible
You’ve probably been reading a lot of physicists that believe in the Many Worlds view. Nature did a poll of physicists last year[1] and it’s still only the third most popular interpretation, behind the Copenhagen Interpretation and information-based interpretations like QBism.
[1] https://gizmodo.com/app/uploads/2025/07/favored-explanations...
Yeah, this is also why I said many and not most :)
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.
Yes! But at some level, doesn’t it kind of get blurred?
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!
No. The Hilbert space is the territory. Our perceived classical world is the map.
Prove it!
"What is a Hilbert space?"
-- David Hilbert.
The nice additional reading is that you could even know what a Hilbert space is, but not what is a Hilbert space.
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Or, there is no time. Awareness traverses the grand graph of possibility.
Disclaimer: total armchair philosophy.
What does "traversing" even mean if there's no time?
Given infinite time, is it possible to randomly draw a rational (instead of an irrational) from the interval [0, 1]? And if so, is it possible infinite times?
Your answer to that will depend on whether you think that is part of "all things possible."
Finally, a well-formed question to which the answer "42 (times)" is a possible solution.
Certainly! Probability 0 does not mean impossible.
In this case it does. Describe any effective measure to pick a random sample from the interval and you will have cheated into choosing from the computable reals instead (of which there is a countable amount)
I don't think that's correct. You're describing a limitation of our physical computing devices (or perhaps physical reality itself) as opposed to a theoretical flaw. At least as far as I can tell.
Measure 0 is different than empty.
"Randomly" drawing anything from [0,1] is already quite difficult.
You could draw a line, or a Cantor dust. ;)
is this like Wolfram's Ruliad stuff?
Superdeterminism... quantum noise and the which way paths are all correlated.