[HN Gopher] Operator algebras and the substructure of space and ...
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Operator algebras and the substructure of space and time
Author : pseudolus
Score : 60 points
Date : 2024-09-25 16:46 UTC (6 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| ddtaylor wrote:
| I think this should be retitled to specify it's a story about the
| key players or the history of this subject, not so much subject
| matter itself.
|
| Often I click something related to Quanta and it seems to occupy
| a very strange target audience. It starts with a title or link
| that seems lures a reader with in-depth knowledge of some
| subjects. I think most readers see that title and get excited
| about all kinds of things like a simplified introduction to
| subjects they weren't able to grasp in the past or someones
| unique perspective that might help them better understand the
| topic.
|
| Instead the article is actually the story and history of some
| people involved. That's an interesting article as well, but I
| think it's a different title.
| kkylin wrote:
| Agree with your point. The actual title, "If the Universe Is a
| Hologram, This Long-Forgotten Math Could Decode It," has
| another issue: physicists may have "forgotten" operator
| algebras, but it's certainly been a very active part of
| mathematics.
| gradschoolfail wrote:
| Can someone tell me why quantmag's coverage of these algebras
| is so different from
|
| https://en.wikipedia.org/wiki/Von_Neumann_algebra#Factors ?
|
| Wikipedia calls them by a different name! As for history,
| Tomita is said to be the most important guy* behind the
| elucidation of Type III but gets no credit...
|
| Feels like this is a classic case of blind men and elephant
|
| * https://en.wikipedia.org/wiki/Tomita-Takesaki_theory
|
| > _These were further developed later by Takesaki, and the
| theory is called the Tomita-Takesaki theory. It has great
| influence in statistical mechanics too. That was the
| beginning part, but in Tomita's papers, he didn't write
| proofs. I: Mathematicians usually like proofs. Is Tomita a
| mathematician? A: [Minoru] Tomita is a pure mathematician.
| There are a lot of algebraists in Japan, including
| [Masamichi] Takesaki, but Tomita is a completely different
| kind of person, very "singular"._
|
| http://www.asiapacific-mathnews.com/04/0402/0012_0018.pdf
| trhway wrote:
| >Physicists have taken this to mean that the contorted space-time
| fabric of a black hole may be made of atomlike parts, just like a
| gas.
|
| as far as i see the "singularity" at the center of black hole is
| just a mathematical artifact of the smoothness of the GR. And
| while that smoothness is a valid approximation at macro scales,
| by all the accounts the world isn't that smooth at the micro
| scales, and similarly to white dwarfs and neutron stars it seems
| naturally for a black hole core to be some next step of
| degenerate matter, something like quark-gluon soup.
| sigmoid10 wrote:
| It's more tricky than that, because inside a black hole things
| turn really weird. Of course things might get unpredictable
| close to the singularity if you really assume it is just a ton
| of matter squished into a tiny point as seen from the outside,
| but the overall spacetime geometry inside black holes would be
| untouched by that. Especially if you consider large,
| supermassive black holes that even have comfortable tidal
| forces at the horizon. If you now look at black hole geometries
| in GR, you'll find that the singularity is not a point in
| space, it is actually a moment in time. Once you cross the
| event horizon, that moment becomes part of your future, which
| means that there is absolutely nothing you could do to escape
| it. So a more accurate description of a singularity would not
| say "super dense point in space" - it would literally describe
| it as "the end of time." As in the opposite of the big bang.
| mystified5016 wrote:
| I really like Penrose's new theory that the singularity is a
| torus within a bubble of normal space inside the event
| horizon. The point-like singularity predicted by GR seems
| obviously implausible, it makes more sense for it to be
| smeared throughout some volume of space. If I understand
| correctly, he proposes that the singularity is a compact
| object rotating about the center so fast that it's more or
| less a solid torus.
|
| Intuitively, it also makes sense that space within a black
| hole could be more or less normal. Or at least have a
| consistent curvature that approximates normal space.
|
| Conversely, I find the notion of space-time equivalence as
| illustrated by Penrose diagrams to be quite unconvincing. All
| we really have is high order approximations of how GR might
| behave in extreme conditions. I believe that if it ever
| becomes testable, we'll see some kind of limit to how far
| space can translate to time. Or that time is more complex
| than we think.
|
| No one really knows for sure, but it's a lot of fun to
| speculate about!
| jiggawatts wrote:
| Personally, I'm not convinced that there is an "inside". From
| the outside there doesn't appear to be one -- you never see
| anything cross the horizon, including the substance that
| originally formed the black hole! Conversely, observers see
| the black hole evaporating from just above the surface, with
| all infalling matter recovered in this way. World lines
| approach the horizon, hug it closely, then leave. They don't
| fall "in".
|
| In my mind scientists filled in a blank with their
| imaginations, but the blank they are filling in may not even
| exist _as a location_. It's like complex (imaginary) numbers,
| you can talk meaningfully about solutions to equations: but
| ordinary +, -, *, / arithmetic "can't get you there". A
| black hole could be like a pinhole stretched out into a
| larger hole in a stretchy sheet of fabric. From the point of
| view of an ant walking on the surface the hole is a boundary
| that can be approached, but not entered, and around it the
| fabric is highly distorted.
|
| There have been some good papers published recently on
| related topics. For example, Penrose diagrams as typically
| drawn use a simplification that black holes extend forever
| forward and backward in time. This allows infinitesimals to
| add up over infinity, which is a non-physical sleight of
| hand. Real black holes form over time, and worldlines don't
| enter them -- they just approach the horizon and then "boil
| off" in the distant future due to Hawking radiation.
|
| There is this obsession in modern physics of clinging to
| overly simplified models and then treating the edges of their
| capabilities as _real things_ to discover instead of
| modelling failures to get past with better models.
|
| You can't climb the mountain range represented by the fold
| crease in your map.
| galacticaactual wrote:
| In the article they describe a 2D wrapper that can represent 3D
| objects in the 3D bulk space between it with mathematical
| equivalence. Can we not extrapolate this to mean that it is
| possible our 3D universe encapsulates a 4D bulk space?
| Vecr wrote:
| That's how it would generally work, assuming you don't need
| tiny curled up dimensions to stuff unwanted particles into.
|
| Assuming you mean 3D boundary, 4D bulk.
| enasterosophes wrote:
| I don't know the answer, but a couple of things to keep in mind
| with such speculations:
|
| * Properties of lower dimensions don't always extrapolate to
| higher dimensions. An example that comes to mind is the result
| in probability theory that a 2D random walk will always return
| to the home position an infinite number of times, whereas a 3D
| random walk has a 2/3 chance of never returning.
|
| * Physics is interested in what is observable and testable. In
| your grant application, what are you saying are the testable
| aspects of this 4D bulk space which would validate your theory?
|
| On the other hand, we can already perfectly represent 4D
| objects in 3D space. Just write down a bunch of 4-vectors. If
| this seems like it's trivializing what you're saying, then it
| means you need to provide a clearer definition of the objects
| which you have in mind, and what it means to represent them.
|
| So overall I think you'd want to be careful with how you're
| defining the objects you're interested in, and what is the
| mathematical form of the claim you want to make about those
| objects, and how to test whether it has any physical relevance.
| lanstin wrote:
| For example, the Banach-Tarski paradox can't happen in two
| dimensions, it needs at least three. Terence Tao's 2nd book
| on analysis explains it, tho I can't yet understand the
| explanation. (It (the weird measure expanding rearrangement)
| can happen for countable # of sets in 2, even 1 dimension,
| but not for finite # of sets).
|
| https://terrytao.files.wordpress.com/2010/02/epsilon.pdf
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