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A Building on QP from 5 reasonable axioms

  1. Dec 7, 2017 #1
    Lucien Hardy's Quantum Theory From Five Reasonable Axioms has deepened my understanding of QP foundations, and motivated me to write a paper. The essence of my paper is that "connectedness" of state space (or the acting Lie group), need not be assumed, but can be deduced. Before linking to the paper, here, I want to confirm that this is OK from a forum etiquette perspective.
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  3. Dec 7, 2017 #2


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    Per the PF rules, your paper must be published in a peer-reviewed journal before we can discuss it here. What journals are you considering submitting it to?
  4. Dec 7, 2017 #3
    The paper is a contribution, but not "cutting edge", so I am somewhat in a quandry of where to submit the paper.
  5. Dec 7, 2017 #4


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    What is the exact mathematical nature of the "state space" you consider?
  6. Dec 7, 2017 #5
    The state spaces satisfy axioms very similar to the ones that Hardy proposes e,g, level 2 state spaces sit in R^4 and level 3 state spaces sit in R^9. If you postulate connectivity of state space, you get QP, as Hardy demonstrates. If you do not postulate connectivity, it turns out the level 2 state space can consist of two parallel circles, of the same radius, on the Bloch sphere. My paper further shows that this disconnected state space does not admit a level-3 counterpart in R^9. My paper also shows that, although Lorentzian geometry needs to be considered, it does not support even a level-2 state space.
  7. Dec 7, 2017 #6

    Vanadium 50

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    What journals does it cite? Which ones are most likely to publish it?
  8. Dec 7, 2017 #7
    I'm trying to locate a peer-reviewed version of Hardy's paper, but could only find it on arXiv so far. If it hasn't been published as a peer-reviewed work, maybe it's out of bounds for us, too.
  9. Dec 7, 2017 #8


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  10. Dec 7, 2017 #9


    Staff: Mentor

    You can also publish it as an insights article here

    I know that paper well and people have been asking me to do one for quite a while, but I never seem to get around to it.

    The thing is it must not contain your own personal ideas, but rather be an analysis of the paper - you can of course have things that could be investigated, further thoughts etc at the end of the article that we can discuss here, and that is where you can mention your lie group ideas.

    In doing that, and maybe helping in answering some of your 'issues' the following may help:

    Basically all the paper does is justify in a rigorous way the following. In mathematical modelling there are generalizations of ordinary probability - they are called generalized probability models or theory. A generalized probability model makes only very simple and quite general assumptions. You have:

    1. Something unspecified called states. All you are doing here is saying whatever you are modelling can be in something called a state without specifying in anyway what a state actually is. Not much of an assumption really.

    2. This is the main assumption - the space of states is convex - which simply means you can apply ordinary probability theory. Specifically it says any state a can be written in the form a = ∑pi ai where ai are other possible states of the system and the pi are all positive and sum to one. Of course that sum can just contain one element so you have a = a which is trivial. If that is the only way a state can be written as such a sum then it by definition is called pure. The interpretation of the pi is as a probability ie if the system is in state a then when you do something to it, without even specifying what that something is, the pi gives the probability in will be found in state a1.

    Ordinary probability theory easily fits this - in fact it's the simplest generalized probability theory. The pure states are the possible outcomes of what you are modelling and the sum ∑pi ai where the ai are other pure states ie the event space as per the Kolmogorov axioms, then pi is the probability of getting ai. In this view the pure states are usually thought of as a vector where the i'th element is the i'th event of your event space.

    Now in Hardy's paper all he is noticing is in ordinary probability theory you can't continuously go from one pure state to another. But if you want to model physical systems by pure states then this is something you want to do. If a system is in pure state a at time t=0 and state b at time t = 1, then it went through some other pure state at time t=1/2. Imposing that and you are inevitably lead to QM which Hardy's paper gives the technical detail of. That's all QM is really. Formally QM is just the simplest generalized probability model where systems continuously change to other pure states.

    That's why I always say formally we know very well what QM is, and why it is that way, - what it means, or even if what it means is worth pursuing ie the math is all that's required, is another matter, and we have all sorts of answers to that.

    Last edited: Dec 7, 2017
  11. Dec 7, 2017 #10


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    That's good summary. I just read (today) up to part 6 of Hardy's paper and enjoyed it a lot. One of thoughts was 'I wonder what BHobba thinks of it'.
    Now I know.
  12. Dec 7, 2017 #11


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    Oh yes - its not really that hard.

    Interesting what the OP has discovered.

  13. Dec 7, 2017 #12


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    Deleted as answered in my next post.
    Last edited: Dec 7, 2017
  14. Dec 7, 2017 #13


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    I have just had it pointed out to me our forum rules allow that:
    References that appear only on http://www.arxiv.org/ (which is not peer-reviewed) are subject to review by the Mentors. We recognize that in some fields this is the accepted means of professional communication, but in other fields we prefer to wait until formal publication elsewhere. References that appear only on viXra (http://www.vixra.org) are never allowed.

    Since this paper has been cited in so many reputable sources discussing it is fine.

    Now the issue is simply discussing the paper the OP has come up with. I will leave that to the mentors to decide - but if the OP, or anyone else, wants to discuss Hardy's paper its perfectly ok.

  15. Dec 8, 2017 #14
    Thanks for all the interesting comments, and clarification of what can be discussed in this forum. My paper references, of course, Hardy's paper, another paper published in Studies in History and Philosophy of Modern Physics, and three books. I will take V50's suggestion and see if the journal is interested, and let you guys know how it turns out. In reading Hardy's paper, several questions arose. For example
    1. Why is state space a closed subset of the ambient vector space? I am referring to states that actually exist in reality -- not just mathematical constructs. In terms of measurements, this can be rephrased as:
    Given a sequence (Sn) of actual states, and a candidate state, S, such that for any measurement, f, f( Sn ) --> f(S)
    Then S is an actual state.
    2. Why is the probability of state transitions symmetric? i.e. P( S transitions to T) = P( T transitions to S ) ( = Tr(S*T) in QP operator notation)
  16. Dec 8, 2017 #15


    Staff: Mentor

    Well actually it isn't. The physically realizable states are all really finite dimensional, but sometimes of very large dimension. You can see this by thinking of an observation as being displayed on a digital readout rather that a meter etc - if it was of infinite dimension then you would need a readout of infinite length.

    Hardy's paper, and many papers on the foundations of QM skirt around it by just considering the finite dimensional case.

    I had given this a lot of thought many many moons ago, and couldn't really resolve our use of Hilbert spaces etc until I started investigating Rigged Hilbert Spaces. During that sojourn I came across a little book, QM for Mathematicians or something like that written in the 1950's stuck deep in the dust covered recesses of the ANU library in Canberra where I lived.

    It talked about something called a Dirac Space. Its quite simple really - you take as a space all the vectors of finite length. These are the physically realizable states. Now purely for mathematical convenience you consider it dual - it in fact turns out to be any vector, finite or of infinite length and the book called it a Dirac Space - but I have never seen the terminology anywhere else. This is the archetype of Rigged Hilbert Spaces. It is often more convenient to work in the Dual than the original space - in fact there is no way to tell the difference, physically from a vector of very large dimension and of infinite dimension. So that it OK from a modelling viewpoint.

    Contained in the Dirac space you have all sorts of things - a Hilbert Space, a Schwartz space - you name it, it's probably there.

    But for various reasons its too big eg not everything in there can be mapped to a Schwartz space hence you cant take a Fourier Transform of it.

    So what you do is cut down its size depending on your problem. For example in the Dirac space you have good functions - you can take that as your test space. Doing that its Dual is the Schwarz space so Fourier transforms are on - that in fact is the space usually used in QM. This is expressed in the so called Gelfland tripple T* ⊃ H ⊃ T where T is a test space of some sort (eg good functions), H a Hilbert space and T* the dual of the test space. This triple is the basis of Rigged Hilbert Space's - rigged not as in a game of chance - but like the rigging on a ship it provides a way to 'climb' above these issues. But you must always keep in mind all of them are just subsets of the Dirac Space that is introduced purely for modelling purposes.

    You can, as Von-Neumann did in his classical treatise - Mathematical Foundations Of QM - work entirely in Hilbert Spaces, but it is neither as convenient or as beautiful as Dirac's treatment in his equally classical - Principles Of QM. Word of caution however - do not do what I did - read those books without a though understanding of modern QM as found in Ballentine. I know - it causes all sorts of issues - not the least of which is Ballentine explains, not in full rigorous detail mind you, the whole Rigged Hilbert Space vs Hilbert Space issue. You can get further detail on it all here:

    But again not until you are comfortable with Ballentine.

    So after all that whats the answer to your question - simply mathematical convenience. The real space is the space of vectors of finite dimension which isn't complete or any of those nice mathematical functional analysis things - we just extend it to have them by means of RHS's.

    Last edited: Dec 8, 2017
  17. Dec 8, 2017 #16


    Staff: Mentor

    That my friend, is tied up with the important, but not often discussed Gleason's Theorem - see post 137:

    It's the reason for Born's rule which is the underlying principle for these things.

    But I get the feeling what you are asking is more than just the Born rule. - it transition probabilities in general eg spontaneous emission.

    Here is the paper that expalians that:

    The calculation of the probabilities is an example of Fermi's Golden Rule:

  18. Dec 10, 2017 #17
    Thanks for the detailed responses to my two questions. Here, I comment on the response to the one question
    Why is the probability of state transitions symmetric? i.e. P( S transitions to T) = P( T transitions to S )
    I read through the discussions on POVM, Gleason's Th, QFT and What is a photon, and Fermi's Golden Rule. I feel remiss in that I did not emphasise that the question was asked in the context of Hardy's paper, of deriving quantum physics from basic axioms/assumptions. For example, the concept of modelling states and measurements by operators on a Hilbert space, is a main goal of the paper. So, the formula Tr(S*T), which is clearly symmetric, can not be used to "explain" the symmetry of state transitions. I am looking for something more fundamental. For example, time reversibility does not seem to fit the bill. A special case of the question is:
    If S cannot transition to T, why cannot T transition to S?​
  19. Dec 10, 2017 #18


    Staff: Mentor

    Ahhh - got you now.

    That I don't think anyone really knows other than the obvious - similar arguments in classical physics ie disorder is much more probable than order leads to an arrow of time. At least I don't think anyone knows - could be wrong of course..

    Hardy doesn't go into it either - he simply notes what some reasonable axioms used to analyse the standard QM observational setup - preparation, transformation, measurement leads to.

  20. Dec 10, 2017 #19
    Again, thanks for the detailed response. I appreciate the time it takes to do this. Here, I am commenting on the response to my question:
    Why is state space a closed subset of the ambient vector space? ​
    I do not understand what is meant by, "Well actually it isn't". For example, the Bloch sphere is a closed subset of R^4. I suspect I may have, again, not been sufficiently clear in my question. The question is asked in the context of Hardy's paper, which is concerned with finite dimensional QP (although Hardy does briefly discuss infinite dimensional QP in h is paper).
  21. Dec 10, 2017 #20


    Staff: Mentor

    Sorry - I was getting my terminology confused. Yes closure - for sure - I was thinking of other things like completeness.

    Closure is a trivial deduction from the principle of superposition. The space of all vectors of finite length is closed but not complete.

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