4 ms·

Operator algebras and the substructure of space and time

88 points by pseudolus 2y ago

ddtaylor 2y ago
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 2y ago
  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 2y ago
    Can someone tell me why quantmag’s coverage of these algebras is so different from https://en.wikipedia.org/wiki/Von_Neumann_algebra#Factors 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 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 http://www.asiapacific-mathnews.com/04/0402/0012_0018.pdf
    - rhdunn 2y ago
      Operator algebra refers to https://en.wikipedia.org/wiki/Operator_(mathematics) https://en.wikipedia.org/wiki/Operator_(mathematics). An operator can be something like the differentation operator (d/dx), the integral operator, gradient, curl, or other structure. Operators are similar to functions but work more generally, like mapping functions to other functions. See https://en.wikipedia.org/wiki/Quantum_field_theory#Canonical_quantization https://en.wikipedia.org/wiki/Quantum_field_theory#Canonical... for some of the applications/definitions in quantum mechanics. IIUC, Operator algebra is the study of the properties of operators just like real, complex, matrix, and linear algebra are the study of the properties of those objects/constructs. As such, once you have defined an operator for e.g. Schroedinger's wave equation you can manipulate and explore that using the rules and principles of operator algebra. Thus, making them easier to work with.
      - drdeca 2y ago
        von Neumann algebras are operator algebras. They are algebras whose elements are operators on some Hilbert space. They also satisfy some other conditions. What I just said is not meant as a definition.
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trhway 2y ago
>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 2y ago
  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 2y ago
    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!
galacticaactual 2y ago
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 2y ago
  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 2y ago
  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 2y ago
    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 https://terrytao.files.wordpress.com/2010/02/epsilon.pdf
twic 2y ago
Is any of this experimentally falsifiable? Or rather, where downstream of this does something experimentally falsifiable pop out?
- wwarner 2y ago
  The goal is to make a prediction about the boundaries or interiors of black holes that can be tested. It’s based on the common sense belief that information is conserved even if it reaches the center of a black hole.
  - AnimalMuppet 2y ago
    If I understand correctly, the idea that information is preserved is based on QM using unitary matrices. That's hardly a "common sense" belief. The belief that that applies even at the center of a black hole is an additional non-common-sense step.
    - wwarner 2y ago
      The black hole information paradox presents an apparent theoretical contradiction between black hole radiation, which Hawking predicted carries no information about its past, and quantum theory, which conserves information at all times. The common sense belief would be that information is conserved, perhaps because there is something measurable inside a black hole and not a singularity. Maybe I should have said that the ideas in the article are an approach to resolving the paradox, rather than invoking common sense.
      - AnimalMuppet 2y ago
        > The common sense belief would be that information is conserved I'll say it once more, since you didn't seem to understand my point the first time: "information is conserved" is not a "commonsense belief". (Unless you mean "commonsense within the quantum mechanics community".) Commonsense is "I've seen a shredder. I know information can be destroyed."

88 points by pseudolus 2y ago

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 2y ago
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 2y ago
  Can someone tell me why quantmag’s coverage of these algebras is so different from https://en.wikipedia.org/wiki/Von_Neumann_algebra#Factors 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 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 http://www.asiapacific-mathnews.com/04/0402/0012_0018.pdf
  - rhdunn 2y ago
    Operator algebra refers to https://en.wikipedia.org/wiki/Operator_(mathematics) https://en.wikipedia.org/wiki/Operator_(mathematics). An operator can be something like the differentation operator (d/dx), the integral operator, gradient, curl, or other structure. Operators are similar to functions but work more generally, like mapping functions to other functions. See https://en.wikipedia.org/wiki/Quantum_field_theory#Canonical_quantization https://en.wikipedia.org/wiki/Quantum_field_theory#Canonical... for some of the applications/definitions in quantum mechanics. IIUC, Operator algebra is the study of the properties of operators just like real, complex, matrix, and linear algebra are the study of the properties of those objects/constructs. As such, once you have defined an operator for e.g. Schroedinger's wave equation you can manipulate and explore that using the rules and principles of operator algebra. Thus, making them easier to work with.
    - drdeca 2y ago
      von Neumann algebras are operator algebras. They are algebras whose elements are operators on some Hilbert space. They also satisfy some other conditions. What I just said is not meant as a definition.
deleted 2y ago
[deleted]
deleted 2y ago
[deleted]

trhway 2y ago

>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 2y ago
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 2y ago
  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!

galacticaactual 2y ago

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 2y ago
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 2y ago
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 2y ago
  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 https://terrytao.files.wordpress.com/2010/02/epsilon.pdf

twic 2y ago

Is any of this experimentally falsifiable? Or rather, where downstream of this does something experimentally falsifiable pop out?

wwarner 2y ago
The goal is to make a prediction about the boundaries or interiors of black holes that can be tested. It’s based on the common sense belief that information is conserved even if it reaches the center of a black hole.
- AnimalMuppet 2y ago
  If I understand correctly, the idea that information is preserved is based on QM using unitary matrices. That's hardly a "common sense" belief. The belief that that applies even at the center of a black hole is an additional non-common-sense step.
  - wwarner 2y ago
    The black hole information paradox presents an apparent theoretical contradiction between black hole radiation, which Hawking predicted carries no information about its past, and quantum theory, which conserves information at all times. The common sense belief would be that information is conserved, perhaps because there is something measurable inside a black hole and not a singularity. Maybe I should have said that the ideas in the article are an approach to resolving the paradox, rather than invoking common sense.
    - AnimalMuppet 2y ago
      > The common sense belief would be that information is conserved I'll say it once more, since you didn't seem to understand my point the first time: "information is conserved" is not a "commonsense belief". (Unless you mean "commonsense within the quantum mechanics community".) Commonsense is "I've seen a shredder. I know information can be destroyed."