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.
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.
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?
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.
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.
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.
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?
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.
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.
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.
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.
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".
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.
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!
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.
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.
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
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.
"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.
thrill | 19 hours ago
SantalBlush | 19 hours ago
rablackburn | 18 hours ago
LPisGood | 18 hours ago
voxl | 17 hours ago
fc417fc802 | 16 hours ago
LPisGood | 16 hours ago
adrianN | 18 hours ago
taneq | 17 hours ago
hhjinks | 13 hours ago
katzgrau | 18 hours ago
Disclaimer: total armchair philosophy.
meindnoch | 15 hours ago
rozgo | 18 hours ago
kdavis | 17 hours ago
But then again, he was John von Neumann.
empiricus | 15 hours ago
senderista | 14 hours ago
freehorse | 13 hours ago
senderista | an hour ago
kdavis | 12 hours ago
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.
empiricus | 5 hours ago
ocschwar | 18 hours ago
-- David Hilbert.
cgio | 18 hours ago
djsjajah | 18 hours ago
tcdent | 17 hours ago
mathisfun123 | 17 hours ago
packetlost | 17 hours ago
fc417fc802 | 16 hours ago
What leads you to claim that you can't encode infinite information given an infinite set to work with?
magimas | 14 hours ago
Infinity is an abstraction that simplifies a lot of maths, it isn't real though.
gradus_ad | 9 hours ago
ben_w | 13 hours ago
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.
azan_ | 11 hours ago
But cardinality of real numbers might not be ℵ₁. It's literally unprovable in ZFC.
ben_w | 10 hours ago
djsjajah | 16 hours ago
psychoslave | 15 hours ago
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.
moi2388 | 14 hours ago
eth0up | 6 hours ago
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.
djsjajah | 14 hours ago
bryanrasmussen | 13 hours ago
on edit: maybe should say infinite set instead of infinite sequence.
lolakutty | 17 hours ago
dguest | 14 hours ago
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.
caturopath | 14 hours ago
ben_w | 13 hours ago
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.
TheOtherHobbes | 11 hours ago
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.
caturopath | 10 hours ago
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.
-------------
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.
gus_massa | 7 hours ago
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.
_Microft | 7 hours ago
dTal | 6 hours ago
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".
furyofantares | 3 hours ago
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.
gus_massa | 6 hours ago
(Note that the author many times is not the person that choose the title.)
inshard | 17 hours ago
rootedbox | 16 hours ago
dr_dshiv | 16 hours ago
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!
japanuspus | 16 hours ago
meindnoch | 15 hours ago
dbvdh | 15 hours ago
derektank | 14 hours ago
[1] https://gizmodo.com/app/uploads/2025/07/favored-explanations...
dbvdh | 13 hours ago
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.
tim333 | 8 hours ago
>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.
dbvdh | 16 hours ago
Subtle difference, but energy conservation for example has to be adhered to.
shiandow | 13 hours ago
It just somehow always balances out in the end.
dbvdh | 13 hours ago
immmmmm | 12 hours ago
With the caveats: - anomalies-> symmetries/conservation broken by quantum noise (has to be cancelled in valid theories) - Heisenberg’s incertitude
_heimdall | 10 hours ago
I'm not sure we can extrapolate that out and say that can never be violated, only that we've never seen it happen.
brummm | 7 hours ago
librasteve | 15 hours ago
I particularly like the one with David Wallace.
fxwin | 15 hours ago
mercer | 11 hours ago
ww520 | 6 hours ago
gjvc | 12 hours ago
sunray2 | 10 hours ago
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
pgwalsh | 9 hours ago
evanb | 7 hours ago
https://en.wikipedia.org/wiki/Totalitarian_principle