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    • CommentRowNumber1.
    • CommentAuthorIan_Durham
    • CommentTimeMar 31st 2010
    Added page quantum state.
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2010
    • (edited Mar 31st 2010)


    maybe you should mention the word "Hilbert space" somewhere.

    I am a bit unsatisfied with what you have under "Definition". I think that under that section name we should have stuff thatt really qualifies as a definition in the mathematical sense. What you write there is more like an extended Idea-section.

    Also, I don't understand the remark about states and objects in a category. And the link to 2-vector spaces is a bit unmotivated. You might want to tell the reader what you think 2-vector spaces have to do with quantzum states.

    If you don't want to write down precise statements yourself, maybe you can point the reader to some literature where details are given.

    • CommentRowNumber3.
    • CommentAuthorIan_Durham
    • CommentTimeMar 31st 2010

    I should have added "under construction" at the top, but had to run off to teach a class.

    Actually, I was trying to come up with a precise definition of quantum state. Despite what we, as physicists, may publicly claim, there is some disagreement on what that means. In fact, I think potentially thinking about it categorically might help us to come up with a concise definition. The definition I gave was more like the "list of numbers" definition of a vector.

    I pointed to the 2-vector space because quantum states are usually written in terms of vectors, a definition that is debated itself a bit, something that is highlighted on the 2-vector space page.
    • CommentRowNumber4.
    • CommentAuthorTim_van_Beek
    • CommentTimeApr 1st 2010
    Hi Ian,

    I inserted a few comments on the page, this is meant as a building block for further discussions.
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2010

    Despite what we, as physicists, may publicly claim, there is some disagreement on what that means.

    Well, there are different proposals for formalizations of quantum theory, which are known in parts to be related, in parts not, and they have different precise notions of what a quantum state is.

    One should start with plain quantum mechanics, where states are rays in a Hilbert space. One should mention the special case where that Hilbert space is square integrable functions on a Riemannian space, or square integrable sections of a Hermitian vector bundle on that space, more generally.

    Then one could talk about AQFT, where states are certain linear functionals on the operator algebra.

    Then one could talk about extended TQFT, where quantum states are the generalized elements of whatever the TQFT n-functor assigns in some codimension.

    All these definitions are a little different, but there are some known and some conjectured ways of how they relate. We don't have to give the full picture in one go, but should try to describe th aspects that are known.

    But an incomplete entry is okay, of course. But le't not write "Definition" if then no definition of anything is given. I think that gives the wrong impression.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2010

    I added some words at the beginning of the idea-section in an attempt to set the scene.

    Then I moved what was under "Definition" to the end of the "Idea"-section and instead put at "Definition" an empty template list of subsection headlines that indicate the types of definitions that I think should eventually be included here.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeApr 1st 2010
    • (edited Apr 1st 2010)

    One should start with plain quantum mechanics, where states are rays in a Hilbert space.

    This is true in quantum mechanics of anything including in QFT. Of course sometimes there is more fine structure, but still at the bottom is a projectivization of a Hilbert state or an equivalent formalism (e.g. stated in terms of C-star algebra without a concrete representation fixed, what can be done via GNS-construction).

    Of course, in statistical mechanics one can have in addition states which are not pure states, but are rather weighted sums of pure states.

    Urs, I do not like that pejorative referal you use about general quantum mechanics calling it "plain quantum mechanics". Quantum mechanics is about a Hilbert space and a Hamiltonian operator. QFT is a special case, at least from the physicist' point of view. There is a finite-dimensional case (QM of finitely many degrees of freedom) when the mathematical problem with well-definedness etc. are typically simpler than for the quantum field theories. When you talk about specifical features of finitely many degrees of freedom, then why not just saying QM with finitely many degrees of freedom. Talking about Hilbert space does NOT confine to this setup but is rather the feature of general QM. Substationally different formalisms are when one allows non-pure states in statistical (quantum) mechanics and in study of open quantum systems.

    Another thing is axiomatic quantum field theory. One should distinguish axiomatic quantum field theory as a general subject dealing with rigorous approach to QFT to the biggest subset of it usually called algebraic QFT which is based on C-star algebras.

    • CommentRowNumber8.
    • CommentAuthorTim_van_Beek
    • CommentTimeApr 1st 2010
    Hi Zoran,

    I tried to lay down some of the points you address on the pages quantum state, operator algebra and AQFT and the ones linked to from there.
    I hope we don't spend too much time discussing if quantum physics is a superset of qm is a superset of qft, if (Hilbert space + Hamilton operator) = qm and is a more general setup than qft or if axiomatic qft is more general than algebraic qft etc.
    I try to set one definition per page and explain that there are different concepts used in the literature, too. For example, for Haag we have "axiomatic QFT" = "algebraic QFT" = "local QFT", whereas he does not like "axiomatic" because that implies that the theory is finalized, nor "algebraic" because that does not capture the basic idea, and goes for "local". Now, QFT (from every viewpoint) is all about locality (localizing quantum states and observables), so most other physicists will probably disagree with Haag on this.
    Just a suggestion: Let's explain this where appropriate and be done with it.
    All in all I think the outline that Urs provided is a useful one.
    • CommentRowNumber9.
    • CommentAuthorIan_Durham
    • CommentTimeApr 2nd 2010
    Thanks, all. Some useful things here that I will have to get to tomorrow (long day). I just want to say that I'm reading my friend Chris Fuchs' paper on quantum Bayesianism and the QBists (as they call themselves these days) claim there is no such thing as a quantum state, at least as it is normally understood. Nevertheless, I think I see the specter of a possible lnk between what they are doing and category theory, but it requires some more thought.
    • CommentRowNumber10.
    • CommentAuthorIan_Durham
    • CommentTimeApr 2nd 2010
    Let me just add that I don't necessarily buy the QBist argument, by the way.

    And I should also add that many of the foundational quantum people don't think about quantum things in field theoretic terms, which isn't to say they should or shouldn't, but is just a warning that people like myself who work in quantum foundations and information have a very particular point of view and it may take some getting used to a different view.
    • CommentRowNumber11.
    • CommentAuthorTim_van_Beek
    • CommentTimeApr 2nd 2010
    @Ian: All I know about the role of probability in connection with interpretations of quantum mechanics comes from the book by Roland Omnes ("The Interpretation of Quantum Mechanics") and a superficial aquaintance with Gleason's theorem - neither is mentioned in the paper about QBism cited in the references on the quantum state page, so obviously there is a gap between my knowledge and the position where this paper takes off. I hope this hints at what kind of material could be added to that page :-)
    • CommentRowNumber12.
    • CommentAuthorIan_Durham
    • CommentTimeApr 3rd 2010
    Well, I shirked my responsibilities and did not get to it today either. Perhaps this weekend. (I went fishing instead.) Anyway, the gist of the QBist view is that what we call quantum states are just really probabilities, i.e. even in their most complete form they are nothing more than states of knowledge or belief. What really matters to a QBist, it appears, is the transformations between quantum states. Now, upon reading this in Fuchs' paper, I was immediately reminded of a quote I'd heard somewhere that said what really matters in category theory are the arrows (morphisms). So it seems to me that category theory might be reconciled with the QBist worldview after all.