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QM avoiding representation

  1. Jun 24, 2011 #1
    Does someone know a textbook that treats QM without relying on representations much? I mean like saying "lets show that this commutator (momentum and position etc.) is reasonable and then derive everything from that without ever talking about representation much".

    Moreover is it possible to justify these fundamentals in this form??? I only know the vague Schroedinger argument for the wave function exp(ikx), but this is an argument about a representation and not the operator?!
     
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  3. Jun 24, 2011 #2

    dextercioby

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    Essentially the Dirac formalism uses an interpretation-independent formalism which replaces both matrix mechanics and wave mechanics.
     
  4. Jun 24, 2011 #3
    Of course, it provides the tools for representation-free treatment, but so how do you set up QM with it?

    I mean something more explanatory than just quoting Schroeding again and saying <x|p>=exp(ikx). Maybe rather a statement about the operators themselves rather than the basis functions? I suppose anything in QM is just guess, but I prefer guesses about operators rather than basis functions. And ideally a total treatment of everything else starting from operators.

    (btw: similar to the algebra of ladder operators!)
     
  5. Jun 24, 2011 #4

    dextercioby

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    Quantum mechanics can be formulated/presented axiomatically using this representation-free formatlism and this is done in serious books like Galindo and Pascual or Prugovecki who put a lot of mathematical rigor in their statements.

    Or perhaps you mean a formulation purely in terms of operators, like the algebraic one of Segal ?
     
  6. Jun 24, 2011 #5
    I don't know these more advanced formulations, but it's quite likely that I mean one of those. What's the names of these books and do you know some more? I might have a glance at them in the library. Thanks!!!

    I don't need mathematical rigor really, but I'm fine with what's common to physicist by saying "it works and is well behaved" :)
     
  7. Jun 24, 2011 #6

    SpectraCat

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    Sakurai's "Modern Quantum Mechanics" doesn't rely much on any particular interpretation of QM. In particular, he explains how the relationship between momentum and position operators can be understood in terms of momentum being the generator of rotation.
     
  8. Jun 27, 2011 #7

    Demystifier

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    The thread is about (absence of) REPRESENTATION, not interpretation. :tongue:
     
  9. Jun 27, 2011 #8

    Fredrik

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    The algebraic approach is presented e.g. in "Mathematical theory of quantum fields" by H. Araki. But I doubt that's the sort of thing you want. You would just be replacing stuff about representations of groups with stuff about representations of C*-algebras.

    Perhaps you could explain in more detail what it is that you don't want to see? Don't most books already avoid talking about representations of groups? Did you mean something other than "representations of groups" when you said "representations"?

    Did you perhaps just mean that you don't want the book to emphasize a specific Hilbert space? In that case, maybe Sakurai is a good choice.
     
  10. Jun 27, 2011 #9
    Thanks for the suggestion!
    Maybe you are right. This book could be rather mathematically rigorous, which isn't my primary goal. It would probably be too tedious for me.

    Thinking of the Dirac representation, I don't want to see statements about the wave vectors, but rather about the operators infront of them. Like the ladder operator derivation where you tell the commutator and everything else follows from that once you postulate a creation operator. Ideally you never mention the specific basis functions.
    Best would be if QM would follow from some sensible axioms which explain the position and momentum operator physically and mathematically.

    I hope to find such treatment in the books of Dexter. But what are these books called? :)
     
  11. Jun 27, 2011 #10

    SpectraCat

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    What you are describing is pretty much exactly how Sakurai handles the subject of QM. This is why I suggested it, because I found it to give exactly the treatment that you are asking for .. focus on the operators and underlying physics, rather than the eigenvalues and wavefunctions. Reading it really helped fill gaps in my understanding of QM that I didn't even know were there.

    Have you looked at that book?
     
  12. Jun 27, 2011 #11
    I dont have easy access to a library at the moment.
    I remember learning from Sakurai a lot. Not quite sure if I missed some part.
     
  13. Jun 30, 2011 #12

    A. Neumaier

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    You can find a fully developped operator approach to quantum mechanics in my book

    Arnold Neumaier and Dennis Westra,
    Classical and Quantum Mechanics via Lie algebras,
    2008, 2011. http://lanl.arxiv.org/abs/0810.1019
     
  14. Jun 30, 2011 #13

    DevilsAvocado

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    Gerenuk, one thing you should definitely avoid, is the paper by Arnold Neumaier. It’s not peer reviewed, and it’ll never be.

    Please take a visit to http://www.mat.univie.ac.at/~neum/" [Broken]:

    "I learnt to see chance as the innovative potential of the Creator, both in small things (Prov. 16:33) that may be insignificant and in decisive situations (Ps. 139:13). I learnt to see that (when one looks at the bottom line of arguments of chance, where they can be checked through calculations that can be compared with experiments) chance is simply the scientist's way to account for influences beyond one's control, that cannot be modeled by deterministic laws but appear to have a degree of surprise. (And indeed, one can find technical definitions of the concepts of `innovation' and `surprise' in some treatises of statistics. I recently wrote a formal paper about surprise.)"

    On the same page Neumaier declares:
    "The way protein molecules fold into their biologically active state is determined by the same mechanism: different possible conformers compete for survival, and the law of large numbers guarantees that most of the molecules will be in the shape corresponding to the smallest energy - forcing the shape to be in a well-defined form that allows the molecule to be used as a biological machine.
    ...
    The possibilities of life are built in into the delicate mixture of deterministic and stochastic laws of nature; which possibilities are realized are determined by the creativity of God, combined with the properties of the environment that carries out the plans of God through self-organization processes governed by the second law of thermodynamics."

    Besides the complete crackpot nonsense about "protein folding" and "the law of large numbers" (since it requires a time http://en.wikipedia.org/wiki/Levinthal%27s_paradox" [Broken] to arrive at its correct native conformation), Neumaier has his own QM interpretation called the "Thermal Interpretation", and the rest you can figure out by yourself...

    Some have http://www.askwhy.co.uk/truth/810NeumaierCoprolalia.php" about Arnold Neumaier, and personally it feels a little bit awkward he got the affirmation as a PF Science Advisor...? :bugeye:
     
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  15. Jun 30, 2011 #14

    SpectraCat

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    Personally I don't care about his motivation, and I doubt that the rest of PF's "Inner Circle" cares much either. What I *do* care about is that he clearly separate his personal beliefs from his science, and in my experience he has never combined the two. He clearly delineates the interpretative aspects of the thermal interpretation from the hard math and science, which agrees with mainstream physics. Furthermore, his "Thermal Interpretation" is really only discussed on the thread in "Independent Research", which also happens to be a perfectly acceptable forum for such discussion.

    This is the second time I have seen you post these statements about Arnold. It's fine if, in the interest of "full disclosure", you wish to make others aware about statements he has made publicly (i.e. in print) about his personal beliefs. I don't think the mocking tone you have asserted in the above post is appropriate however. If Arnold were proselytizing about his personal beliefs in these forums, then he would be sanctioned by the PF staff. The fact that he has instead been nominated and confirmed as a PF Science Advisor should be taken as a vote of confidence from the PF staff that Arnold understands how to appropriately maintain a separation between his personal beliefs and his scientific theories. I guess you should respect their judgment as well.
     
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  16. Jun 30, 2011 #15
    Well said SpectraCat. I don't always agree (and assume others don't always agree either) with all of the discoveries or claims or like certain approaches of Dr. Neumaier. But one thing is certain. His approach to QM is thorough, rigorous and adventurous. Rather than being someone who is content to regurgitate lessons learned by someone else (where usually something important gets lost) he is very actively and energetically attempting to build a cohesive new platform for understanding old and new problems. And he is providing all of the details of the building blocks. That should earn anyone's respect who appreciates how science advances.

    By the way, if the last site you posted was supposed to represent criticism, in fairness you should warn readers of the bizarre and incoherent nature of its content.
     
    Last edited: Jun 30, 2011
  17. Jun 30, 2011 #16
    I don't get it. I thought either the Maths is correct or it is wrong?! How can there be a dispute. I don't get how science can be a religious war, but then having done a PhD in condensed matter physics I might guess. When an there is an argument usually both sides are wrong and know nothing.

    But thanks for pointing out some paragraphs. Such wording indeed makes a bad impression on me.

    So anyone else with a suggestion for a book? :)
    Ideally one which isn't too mathematical. I don't need theorems and rigor, but rather essentials.
     
  18. Jun 30, 2011 #17

    Fredrik

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    I still don't fully understand what you're asking for. It sounds like you just want a standard textbook that's written for people who have studied some QM before. How about Ballentine?

    The books that really emphasize operators are much more mathematical than the standard textbooks. How about reading chapter 2 of Weinberg's QFT book? It's "mathematical", but not rigorous, and it's probably the best place to understand the concept of particles.
     
  19. Jun 30, 2011 #18

    DevilsAvocado

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    Check out his professional http://www.mat.univie.ac.at/~neum/" [Broken] at the Fakultät für Mathematik, and you’ll get a slightly different experience. On 123 pages Neumaier refers to God, Jesus, Christianity or the Bible in an unambiguous mix of religion and science.

    Mainstream physics?? 500 not peer reviewed pages is mainstream physics?? wow...

    It looks like you are somewhat 'thunderstruck' by the "hard math"; however it doesn’t impress me that much. I listen more to what the man has to say, and he says (among a lot of strange things) that "conservation of energy" explains what goes on in the double-slit experiment. When 'explaining' Entanglement, he uses (only) the Stern–Gerlach Experiment as an example, but never mention that spin angular momentum is a purely quantum mechanical phenomenon. Instead, he talks about the "magnetic moment" and the fact that the silver atoms are heated, and concludes – "This explains why the two blobs in the Stern–Gerlach experiment are equally bright." ...

    In the same chapter, in a footnote about 'coincidences' and 'acceptances', Neumaier also takes the opportunity to 'explain' that the acceptance of general relativity was a coincidence based on "noise" in agreement with the theoretical results...
    "As more often in the history of physics, it was a coincidence that determined the acceptance of a theory. Another such example was the measurement of the deflection of rays of the stars that can be seen close to the sun during a solar eclipse done by Eddington in 1919, thereby verifying the general theory of relativity of Einstein. The actual deflections are too small to be measured and hence the deflections found by Eddington have to be ascribed to noise; luckily the noise gave a pattern in agreement with the theoretical results."

    The actual 1919 "Noise":
    300px-1919_eclipse_negative.jpg

    (And I could go on and on about this paper, but that would be too tedious)

    Maybe this is just me not understanding... or it’s just this paper... or just one little 'mistake'...

    I don’t think so. Neumaier has been active in this 'regime' for years. In 2003 he published the paper http://arxiv.org/abs/quant-ph/0303047" [Broken] in which he included a revised analysis of Heisenberg's HUP in terms of Bohr's Complementarity. I’m sure you’re familiar with the 'concept', which we discussed intensely in another thread (from abs):
    "... The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical realization and a quantum realization."

    (Almost identical 'thoughts'...?? :bugeye:)

    Though Neumaier gets harsh critics at http://www.natscience.com/Uwe/Forum.aspx/physics/12383/Neumaier-s-Modification-of-Heisenberg-Uncertainty-Principle-HUP" [Broken]:
    In my world, this is not mainstream physics; this is "classical madness", most probably motivated by "personal beliefs".

    Sure, but in my browser he’s now in QM and promoting his own personal not peer reviewed stuff.

    Thanks SpectraCat, for your support. I’m sure you agree that when one of Neumaier’s (now permanently banned) followers states that; "Neumaier who is equal to von Neumann in mathematical ability" and "He gets a Nobel" etc, someone has to react.

    Since most of the content in my post is quotes from Neumaier’s homepage, I guess you are referring to my words "complete crackpot nonsense", right? I’m able and willing to get down on my knees and make a deep and sincere apology – if you can show me one trustworthy paper that solves protein folding solely by the law of large numbers (without any help from 'above').

    True.

    If the PF staff is truly aware of the 'science' that Arnold Neumaier is advocating, and they sanctions this as perfectly okay "mainstream physics" – the case is real easy – I’m out of here in a second.

    However, I don’t think this case is as simple as that. Arnold Neumaier is a very brilliant man, as you may have noticed, and he has great skills when it comes to mathematics, and apparently he helps a lot of users, and he should of course have all the credit for this. And this is what he got from the PF staff.

    In my view, the problem starts when he stops helping users and starts to delude them...

    I would be really surprised if the PF Science Advisor status gives Neumaier carte blanche in posting anything he likes on PF, like recommending his own not peer reviewed papers to users, who only sees his advisor status.

    But I could be wrong, maybe it’s me who will get an infraction or banning... we shall se...


    (By the way, Neumaier and his delude followers are http://www.natscience.com/Uwe/Forum.aspx/physics/44445/Double-Slit-Experiment-Mystery-Solved" [Broken] spreading the 'great news'...)
     
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  20. Jun 30, 2011 #19

    strangerep

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    It's blindingly obvious such "followers" don't understand the maths of standard QM.
    Just because someone yells "follow the sandal!" or "no, follow the gourd!" doesn't mean we should take notice of any of them without independently testing their assertions.

    Arnold's only "mistake" was that he tried to dumb down some of the material in his book.
    The venom responding against such dumbed down explanations is both astounding and unwarranted.

    I sometimes wonder whether anyone but me and Meopemuk who participated in earlier photodetector/double-slit threads involving Arnold actually studied the detail in Mandel & Wolf (mentioned at length in threads here and in IR) on which Arnold's explanations are based. None of those who spit venom have bothered to contradict specific calculations in Mandel & Wolf, but such calculations are all that matter in the end -- for comparison with experiment.

    For my part, I'm happy as long as the scientific material remains clearly delineated from other stuff.
     
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  21. Jul 1, 2011 #20

    DevilsAvocado

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    Apparently, a lot of things are "dumb down" at the moment, including rhetoric.
     
  22. Jul 1, 2011 #21

    A. Neumaier

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    He didn't quote from my book, where you find only the math.
     
  23. Jul 1, 2011 #22

    DevilsAvocado

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    I have been perfectly clear when quoting from Arnold Neumaier’s homepage. When talking about "the paper" (or 500 not peer reviewed pages), it is of course Neumaier’s own link that I’m referring to:
    Classical and Quantum Mechanics via Lie algebras,
    2008, 2011. http://lanl.arxiv.org/abs/0810.1019

    To avoid any misunderstanding, everything below refers to the paper above:
     
  24. Jul 1, 2011 #23

    A. Neumaier

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    The cited statement is fully correct. My book of course only gives an abbreviated account of the full story. The latter can be found in the paper

    H. von Kluber,
    The Determination of Einstein's Light-deflection in the Gravitational Field of the Sun,
    Vistas in Astronomy 3 (1960), 47-77.

    To check the accuracy of my account, it is enough to look at the figures on pp. 60-63 to see the amount of noise to be expected in the data. Eddington's scarce observations are given in the first of these plots and look (because of their sparsity) much less noisy compared to the observations of later eclipses.

    von Kluber writes on p.64:
    and in the conclusions on p.73, he writes:
    This was the situation in 1960 ---- 40 years after Eddington!
     
  25. Jul 1, 2011 #24

    DevilsAvocado

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    This is just amazing and completely new info, I had never heard of H. von Kluber before.

    I’m trying real hard to understand this... you and von Kluber are saying that the measurement in the picture below is just noise, right?

    300px-1919_eclipse_negative.jpg

    Since you are highly skilled in mathematics, what is the probability for the photons to (completely at random) form this kind of trail (in an exposure of 28 seconds), to be in perfect agreement with the general theory of relativity? That must have been a major fluke, doesn’t it??

    And If I understand this correct; the "next photo" would have shown a completely different trail, since we are talking noise, right??

    Wow... just wow...

    I tried to find out more about H. von Kluber, and I found discussions on http://forums.catholic.com/" [Broken].

    Then I saw that Harry Collins and Trevor Pinch had made references to him in the book http://books.google.com/books/about/The_golem.html?id=ijtA0JLYlooC":
    "Through a series of intriguing case studies including the study of relativity, cold fusion, the "memory" in worms, and the sex life of lizards, this book debunks the view that scientific knowledge is a straightforward outcome of competent theorization, observation, and experimentation."

    Interesting indeed.

    Finally I found a better source at the http://www.ast.cam.ac.uk/about/history/biographies.e.-.l", where one gets the quite stunning facts that Sir Arthur Eddington and Harald von Kluber worked in the same institute! Eddington as Plumian Professor/Director of the Cambridge Observatory (1914-1944) and von Kluber as Assistant Observer (1949-1960).

    This is developing into a thriller...

    If I have gotten this right – you are claiming that an Assistant Observer, in the same institute as Sir Arthur Eddington was active during the 1919 total solar eclipse, 41 years after the measurement finds out the astonishing fact that it was all "noise", but he doesn’t tell anyone at the University of Cambridge, but instead he publish his findings in an astronomy journal?? Hence, on the http://www.ast.cam.ac.uk/about/history/biographies.e.-.l" at the institute it still says (on the same page as Harald von Kluber):
    Sir Arthur Eddington (1882-1944) Plumian Professor: 1913-1944, provided evidence for Relativity in 1919 eclipse expedition, popularizer of Relativity. Director of the Cambridge Observatory 1914-1944

    I feel really really sorry for Harald von Kluber, someone needs to restore his good name...

    Don’t take it wrong, I do not question your authority, but don’t you find this story a little bit 'peculiar'...?

    And finally, what’s your view on the (probably faulty) information at Wikipedia?
     
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  26. Jul 1, 2011 #25
    Neumaier's book is like the movie "A Bridge Too Far", an ambitious and bravely attempted project, but if only the real evidence was taken into account something less ambitious might have succeeded.

    The "thermal interpretation" has no hope of explaining modern multi-entangled state experiments, let alone the incredible experiments of Zeilinger and co with single photons.

    The Quantum world is so non-classical that Neumaier's ideas can clearly be seen to be in a long history of fruitless failed attempts to show otherwise.

    The Lie Group/Algebra stuff looks good though. Should have stuck to that and maintained credibility.
     
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