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Is this question acceptable?: "mathematical justification ... that the universe contains a finite amount of information"

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    • CommentAuthorpeterwshor
    • CommentTimeJan 8th 2011 edited
     

    The question: Is there a mathematical justification/proof for the claim that the universe contains a finite amount of information? was recently closed. There's an interesting question (or at least one I'd like to get the answer to) related to this: under what circumstances does a state of a quantum field theory contain a finite amount of quantum information?

    Just because a question is conveyed in the language of physics doesn't mean that it's not interesting mathematically. Do all questions on MathOverflow have to be stated in the language of mathematics? I voted to reopen, but I'd like to see what the community thinks.

  2.  

    I think the question could use some cleaning up by someone who knows some information theory. For example, the bits about data compression look like they could be safely removed. I'd vote to reopen if my vote weren't all-powerful.

    • CommentAuthorpeterwshor
    • CommentTimeJan 8th 2011 edited
     

    Agreed: it could use a lot of cleaning up. The data compression stuff is completely irrelevant, and the lack of any mention of quantum mechanics (which is the only thing makes the information content finite) is a serious flaw.

  4.  
    Peter, maybe you could post your own question on this?
    • CommentAuthorpeterwshor
    • CommentTimeJan 8th 2011 edited
     

    I should do that ... this question is probably too badly formulated to save. I guess what upset me was the comment "You cannot prove propositions about the universe," which got a number of upvotes, and pretty much implies that no physics can be in scope for this site. I hope that not too many people agree with that.

    • CommentAuthorMariano
    • CommentTimeJan 8th 2011
     

    I cannot read that comment in that sense---and I surely did not write it in that sense, either!

    My comment was a reaction to the title, mostly, and I'd say the same thing is someone asked the less elaborate "how can we prove that things fall?". My, hmmm, epistemological nerve got hit.

  7.  
    Peter, my take on Mariano's comment is that you can't prove things about the universe, you can only prove things about models of the universe. Metaphysics gets in the way.
  8.  

    I agree with Ryan. I am very far from an expert here, but my take on "You cannot prove propositions about the universe" is that it is rather self-evident. As I understand the terms, what you can prove propositions about are mathematical objects. I am not aware of any philosopher or physicist within the last 100 years who has believed that the universe is a mathematical object.

    Professor Shor asks another question: "Do all questions on MathOverflow have to be stated in the language of mathematics?"

    Speaking personally, this is the only language I am comfortable using when discussing mathematical questions. I tend not to even read questions which sound like physics (or some other science) rather than mathematics, so I rarely vote to close them. But for the sake of discussion: is there anything wrong with requiring all MO questions to be stated in the language of mathematics? What desirable content would be excluded by doing so?

  9.  
    @Pete: You'd be surprised! http://arxiv.org/abs/0704.0646v2
  10.  
    Dear Peter,it is obvious to everybody that you don't care a fig about MO reputation but it might be different for Dan Tao, who after all thought of the original question, warts and all. So maybe a fair compromise would be to severely edit Dan Tao's question ( you said it was *probably* too bad to be saved: there is a glimpse of hope in that adverb "probably" !) and to delight us with your insights in an answer.
  11.  

    @Mariano: sorry for misinterpreting your comment. I'm too busy to fix the question (or write my own question) right now, but maybe in a week or so.

  12.  

    @Dan Petersen: as you might have suspected from my phrasing, I consciously left room for someone to point me to an academic who felt differently. Thank you for doing so.

    I still think it is fair to say that this is a decidedly minority view.

  13.  
    One can make a mathematical question out of this with a precise definition of information, asking about a bounded subset of a spacetime instead of the whole universe, and stating which framework is to be used like e.g. quantum field theory. Then the question is about a rigorous proof of the Bekenstein bound, for example.
  14.  

    You probably aren't going to get a rigorous proof of the Bekenstein bound, because nobody has a mathematically well-defined theory of quantum gravity. However, some quantum field theories are mathematically well-defined, so given a quantum field theory, it seems to me you should be able to bound the amount of information that exists in a finite volume with a given energy. That's what I'd like to see.

    • CommentAuthorAnixx
    • CommentTimeJan 9th 2011 edited
     
    This is completely physical/natural sciences question. It would be good it there was a similar physical site, and the question then would be exactly appropriate there.
  16.  

    Yes, wouldn't it be good if the stackexchange people could be persuaded to set up a site for Physics questions. Maybe they could call it http://physics.stackexchange.com.

  17.  

    Any luck in persuading the stackexchange people yet, Andrew?

    :-)

    Anixx said: "This is completely physical/natural sciences question."

    Right, but as peterwshor said, there are rigorous frameworks for quantum field theory, for example the Haag-Kastler axioms. A quantum field theory is described by a net of operator algebras on a suitable spacetime. Physical states correspond to certain states of the algebras, and the concept of entropy has been generalized to the noncommutative context, too, for example by Connes et alt. A mathematical question that is related to the original question would therefore be if there are any results about the entropy of operator algebras of Haag-Kastler nets (like a bound of the entropy of states with finite expectation value of the Hamiltonian depending on the volume where the state differs from the identity), although I bet that such a question would not be answered :-)

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