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I am aware that physicists are trying to derive born rule from unitary evolution

  1. Aug 15, 2012 #1
    I am aware that physicists are trying to derive born rule from unitary evolution. Has there been any success? What is the current status of that program?
     
  2. jcsd
  3. Aug 15, 2012 #2
    Re: Born Rule

    Practically all attempts to derive the Born Rule from unitary evolution are in the context of relative states and/or decoherence, both realist views of the quantum state. The Born rule in pilot wave theories is not a result of unitary evolution but rather a choice of a certain initial distribution of the particles according to the so called quantum equilibrium. They don't count as emergent Born Rule derivations.

    The derivations of an emergent Born Rule from unitary evolution falls in roughly three categories:

    1) Logically incomplete, like Everett's derivation in his thesis. He only shows that if there is a probability distribution that meets certain requirements, then it must be the Born Rule. But he fails to show how the requirements are necessary or relate to actual physics. This derivation is considered to be incorrect by practically all theorists these days.

    2) Circular: Many derivations introduce the Born Rule that they attempt to eventually get through the backdoor. Most do that by postulating the existence of ensemble descriptions in terms of density operators. That construction already contains the born rule, because only then the density operator is a valid ensemble representation and simplification of the canonical representation as a list of state and classical probability pairs. There are derivations where the circularity is hidden better, particularly in an MWI context.

    3) Use of additional assumptions: It can be shown (quite easily in fact) that a result that depends on the amplitude of certain components cannot be derived from a linear mechanism. So it seems the Born Rule is not derivable without further assumptions that introduce some kind of nonlinearity to the theory while preserving the evolution. These assumptions can be for state counting for example. If you want to count the significant branches for an outcome in many worlds you need to establish some kind of counting rule that allows you to ignore branches with low amplitude. For example you can postulate that you only are interested in the limit of infinitely many measurement iterations where certain amplitudes vanish and are excluded from counting. Other possible assumptions include the ones that Zurek uses in his derivations. His quantum darwinisn postulates that only states with a certain robustness and stability are relevant to our observation. Then he counts those states. These assumptions are questionable, because they add something to basic quantum theory that cannot be derived from it, however plausible it appears to be.

    To sum up, there is not a single generally accepted derivation of the Born rule that just starts with a unitary state evolution and nothing else. There's one paper however that has not yet appeared in a journal and that follows a new approach, and it could be what you're looking for. The preprint is on arxiv (1205.0293).
     
  4. Aug 15, 2012 #3
    Re: Born Rule

    I'd like to mention the following work: http://arxiv.org/pdf/1107.2138v2.pdf (see references to their journal articles there). They do derive the Born rule from a rigorous solution of the equations of unitary evolution for a specific system, but only as an approximation, not as a rigorous result.
     
  5. Aug 15, 2012 #4

    Hurkyl

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    Re: Born Rule

    Only if you believe kets are "fundamental".

    If you take the probability distributions are "fundamental", then so long as you avoid a ket-based presentation of QM the Born rule simply doesn't enter the picture at all. The Born rule, then, simply becomes how we define the use of kets to represent states; if we wanted a different rule, we can! But the laws of physics would have a different form.

    To extend your door analogy: sure, you have to go through the "backdoor" to get from kets to density operators. But this POV said you already had to go through the door to get to kets in the first place. But you never really had to go through the door just to walk back out again: you could have stayed outside the entire time.

    A physical theory gets one "here are the laws of physics" 'for free' (meaning that 'we believe this because it's been empirically verified). For QM, the thing we get for free is not "the Schrodinger equation along with a Hilbert space": the thing we get for free is "the Schrodinger equation along with a Hilbert space and the Born rule". Asking for the Born rule to be 'justified' is really the same question as asking for the Schrodinger equation to be justified.

    I'm always confused when people insist this is an additional assumption: by the frequentist interpretation of probabilities, this is what probability means. Showing that the probabilities defined by QM correspond to frequentist probabilities is what we've been trying to do all along!
     
    Last edited: Aug 16, 2012
  6. Aug 16, 2012 #5
    Re: Born Rule

    You have to distinguish two uses of the density operator. As a state representation for the evolution of (sub)systems, and as an ensemble description. Only if you assume the measurement postulate both can be identified. So, of course you can say that I want my quantum theory with measurement postulate and only regard it as a statistical theory, but that's not the point here. A lot of people find that view entirely unsatisfying because it leaves so many questions open (namely exactly what we call the measurement problem). So if you want to *derive* the Born Rule from unitary and implicitly deterministic evolution then this view is not a possible starting point.

    That's not the point. Of course we need to talk about frequentist probabilities in the limit of large numbers. But state counting does a lot more than that. The problem is to eliminate states of zero amplitude from the counting process, but the zero amplitudes are only established as a limit for infinitely many measurements. So that way of state counting can have no influence whatsoever on a single measurement. That's how it is interpreted however.

    A real derivation of the Born Rule must allow the Rule to be applicable to a single measurement process, but with the statistics being defined over many measurements. You can only get that from state counting if you introduce some kind of cutoff nonlinearity, like from assumung that complex amplitudes are in fact discrete and truncate towards zero if they're small enough.

    Just to make it clear what we talk about here. The goal is to start with a deterministic description of the universe and a unitarily evolving state representation of this universe. That state representation can in fact be a vector or a density, or if you come up with something more general then even that. You have then to show that the observations made in a subsystem appear to be indeterministic and follow the Born Rule for single observations already, with the statistic being defined over many measurements. You will also have to specify what exactly an observation and a measurement are, explain why the result is not deterministic, and most importantly, why it appears to be nonlinear despite a global linear state space and dynamic.

    I'm arguing in precisely this context, because I don't consider anything else a derivation of the Born Rule in the sense of the quantum measurement problem.
     
  7. Aug 16, 2012 #6

    bhobba

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    Re: Born Rule

    Its an entirely different question with an entirely different justification - justifying Schrodinger's equation seems to require Borns rule. The Born rule follows from Gleason's theorem with its basis being non-contextuality - it really shows how powerful the innocent looking non-contextuality assumption is. Schrodinger's equation follows from the Galilean invariance of probabilities defined by Borns rule (see Chapter 3 Ballentine).

    Thanks
    Bill
     
    Last edited: Aug 16, 2012
  8. Aug 16, 2012 #7
    Re: Born Rule

    For me this seems problematic because no one has ever done infinitely many measurements. Therefore showing that QM predicts a certain thing in the limit of infinitely many measurements doesn't actually predict anything about any experiments we have done or will do.
     
  9. Aug 16, 2012 #8

    bhobba

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    Re: Born Rule

    Probability can be defined by Kolmogorov's axioms having nothing to do with infinitely many measurements. The assumption of the law of large numbers interpretation of probability, itself derivable from Kolmogorov's axioms, is as the number of trials increases it approaches a stable limit. Its the same as any other limit used in physics such as for mathematical simplicity assuming the limit at infinity of a function is zero - of course you can never go to infinity to find out but its a conceptualization which at that level is fine. Conceptualizations abound - eg in Euclidean geometry a point is defined as having position and no size - ever seen something with position and no size? Does that invalidate Euclidean geometry? The same with the law of large numbers.

    Thanks
    Bill
     
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