In Hilbert space, all things are quantumly possiblequantamagazine.org
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?
Finally, a well-formed question to which the answer "42 (times)" is a possible solution.
Certainly! Probability 0 does not mean impossible.
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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)
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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.
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You could draw a line, or a Cantor dust. ;)
Your answer to that will depend on whether you think that is part of "all things possible."
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?
"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?
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google "Fock space"
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Aren't Fock spaces still hilbert spaces?
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Yes, they are complete inner product spaces.
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.
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reading about AQFT, found this: "no examples are known of AQFT-compatible interacting field theories, and in particular the standard model cannot at present be made AQFT compatible". Should I continue to read?
"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.
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.
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.
Terrence Howard is that you?
Maybe you can theoretically encode arbitrary (but not infinite) information, but there are still discrete underlying states.
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>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?
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reality does not have an infinite set though.
Infinity is an abstraction that simplifies a lot of maths, it isn't real though.
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So you have proof spacetime is discrete?
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.
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>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.
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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.
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?
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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.
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He means that 3 is not between 0 and 1, which it really isn’t.
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What is the difference between 0.3 discrete units and 3 discrete units?
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.
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?
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.
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.
For example, there's probability 0 of worlds where charge isn't conserved.
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.
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.
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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.
Charge IS conserved!
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.
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I thought that in an expanding universe, mass isn't conserved. I don't see why charge would be conserved in that case.
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Mass isn't close to being conserved. If an electron and positron annihilate, they become two massless photons, for example. You don't need to invoke an expanding universe.
In an expanding universe, energy isn't conserved. (Unless we do the trick we always do with energy and just define it to be conserved https://youtu.be/lCgKTpIzjRk?t=174 )
I don't know exactly why you think this applies to charge conservation. The U(1) symmetry is where charge conservation comes from and it isn't affected by an expanding universe.
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The photons have energy, so through mass-energy equivalence, they aren't massless, I thought.
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Sorry for the too late reply...
There is "rest mass" and "relativistic mass". The GP is talking about "rest mass" that is not conserved even in simple examples.
Most modern books try to eliminate "relativistic mass" because it's misleading, for example is good to calculate acceleration perpendicular to the movements (for example in a cyclotron) but it's not good to calculate acceleration in the direction of the movement.
So most modern books they call the same idea just "energy" and use that "energy" to calculate gravitation. "Energy" is conserved in special relativity and almost conserved in general relativity.
But the "relativistic mass" is useful in some problems and is widespread in popular culture and don't ask about the weird parts of the mass of the proton, so it's difficult to kill the idea.
When an electron flies beyond the cosmological event horizon, it disappears for good.
I just read the article and can not make any sense of your comment. It does not relate to anything of the actual article content (but only superficially to the title itself).
Many Worlds Hypothesis is the extrapolation of taking Hilbert Space at face value, and dumping concepts like "wave function collapse" which privilege a particular future as the "real one that happened".
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".
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I think there is a possible future where "I" am a talking cucumber, though?
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.
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Indeed such a strange configuration of matter would arguably no longer be "you", but I think an even more immediate point of philosophical incoherence is that it would certainly not be a cucumber. [Cucumber != cucumber] => false, QED.
If talking cucumber was too pedestrian for you, we could add stipulations until your intuition collapses. Perhaps, starting next week, the Earth begins to spontaneously turn into raspberry custard on Tuesdays. Perhaps the entire universe turns to custard! Just think of it - billions of lightyears of custard in every direction. Whoops, did I say billions? Too grounded! Let's make it a googolplex. And a single lone piece of rhubarb, floating somewhere in the middle of it all...
...sorry, couldn't resist. Joking aside, while we can quibble about the boundaries of the possible, the fact remains that there are boundaries - at least, if we're still talking about known physics. No evolution of the universal wave function will lead to a world where pi suddenly equals 3.
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I love custard as much as the next man, but that ratio of rhubarb:custard would ruin my enjoyment of the Universe and leave me very grumpy.
It reminds me of that scene where the guy is upset about the ratio of ingredients in his wife's dessert:
The article is probably good but the title is a mix of clickbait and nonsense. I'm tired, I'll skip the article.
(Note that the author many times is not the person that choose the title.)
Superdeterminism... quantum noise and the which way paths are all correlated.
Just a reminder.. Hilbert space is just a tool/abstraction/map... and the map is not the territory..
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!
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...
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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.
There are issues with how that question was worded
>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.
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That’s a misunderstanding of what an observer is. An observer is better thought of as an interaction which distinguishes between states; there need not be a conscious observer. For example, long wavelength light interacting with the electrons in the double slit experiment does not affect the interference pattern because long wavelength light interacts with both states of the superposition in the same way. Short wavelength light has different physical characteristics than long wavelength light and interacts with one state of the superposition differently than the other, resulting in decoherence and the wave function “collapsing”.
There are other problems with the Copenhagen Interpretation, but this isn’t one of them
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.
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!
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
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.
Energy is not always conserved. In the universe as a whole, energy is not conserved because we do not have time symmetry.
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Sure, that could be a reason why it isn't conserved. My point wasn't that we know ways it isn't conserved, only that theoretical understanding isn't universal or necessarily true.
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
Mindscape is one of the best podcasts out there.
is this like Wolfram's Ruliad stuff?
Not even slightly.
"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
Couldn't resist. A little listening while reading. Ott - Adrift In Hilbert Space. https://www.youtube.com/watch?v=o6Zn_XuEdYs&list=RDo6Zn_XuEd...
Gell-Mann's totalitarian principle: in quantum mechanics anything not forbidden is compulsory.
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