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Why does Quantum Mechanics have objective statistical characteristic?

  1. Apr 2, 2013 #1
    Why does Quantum Mechanics have objective statistical characteristic,but does not have subjective statistical characteristic?

    QM has two axioms:
    1-To each dynamical variable there coresponds a linear operator,and the possible values of the dynamical variable are the eigenvalues of the operator.
    2-To each state there coresponds a unique state operator.The average value of a dynamical variable r,represented by the operator R,in the virtual ensemble of events that may result from a preparation procedure for the state,represented by the operator rho,is:
    <average R>=Tr(rho.R).

    Can we deduce the objective statistical characteristic from these axioms?If it is so that,what does lead us to the second axiom?
     
  2. jcsd
  3. Apr 2, 2013 #2

    bhobba

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    The second axion follows from the first via Gleasons Theroem:
    http://en.wikipedia.org/wiki/Gleason's_theorem

    I don't know what you mean by objective and subjective characteristic. Determinism can be viewed as a special case of a probabilistic theory - it only allows 0 and 1 as the probabilities. Although it not usually presented that way the Kochen Specker theorem follows fairly readily from Gleasons theroem:
    http://en.wikipedia.org/wiki/Kochen–Specker_theorem

    It shows you can't have only 0 and 1 as your probabilities so deterministic theories are kaput - with out. There is actually one other thing that goes into Gleason - non contextuality - so it not quite true to say it follows from the first axiom. However since non-contextuality relies on being able define the expectation values of operators that depends only on the eigenvectors and not on what other eigenvectors are part of it it makes the first axiom mathematically a bit weird if its not true.

    Thanks
    Bill
     
  4. Jun 6, 2013 #3
    Despite of KS theorem,but I have heard that strictly speaking quantum mechanics keeps silent about whether microworld is deterministic or undeterministic.The probability is not only equal 0 and 1,but we can't say about deterministic or undeterministic characteristic of the world.Is that correct?
     
  5. Jun 6, 2013 #4

    bhobba

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    The KS theorem is really a corollary of Gleason's Theorem - although for some reason it's usually presented separate from it (possibly because its proof has a mystique of difficulty - but it's not too bad - a proof has been found that is not that demanding mathematically and can be followed by someone with the equivalent of first year university calculus - see for example - Hugh's - The Structure and Interpretation of Quantum Mechanics).

    Anyway its validity rests upon an assumption called non contextuality. Its a very natural assumption within the mathematical Hilbert Space framework of QM. In fact if its not true you would probably say why use Hilbert spaces in the first place - non contexuality is rather weird if you want that mathematical framework. The thing is though physically its not that weird - in fact looked at that way it's rather reasonable.

    Bottom line is hidden variable interpretations like Bohmian Mechanics exist that are contextual so its certainly possible for this to happen - but only if you assume something beyond the normal formalism ie hidden variables or something sub-quantum.

    So the situation is this - if you do not assume some kind of hidden variable theory then KS implies you cannot say it has that value prior to observation because only hidden variables allow non contextuality. If you do then yea - you can assume it. That's what I mean by some sub-quantum process.

    Added Later:

    I just realised I didn't make clear why this non contextuality is such a big deal - I merely said it was. If you choose a Hilbert space as your framework then you expect the elements to tell us something about the results of measurement - technically this is defining a measure on the states. This should not depend on the particular basis the state happens to be expanded in terms of - this is an arbitrary man made thing and the physics should not depend on it. This assumption is called non contextuality. All by itself, via Gleason's Theorem, it implies Born's rule and Born's rule means you cant define just 0 or 1 on all states.

    Thanks
    Bill
     
    Last edited: Jun 6, 2013
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