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QM and the continuum

  1. Aug 9, 2014 #1

    bobie

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    Relativity and continuum

    I have read contrasting things about the issue.
    Can you explain briefly why the assumption of discrete space and time is absolutely incompatible with relativity?

    What is the main problem? Lorenz invariance or what?
    Thanks
     
  2. jcsd
  3. Aug 9, 2014 #2

    bobie

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    I have read that although the main feature of this theory is discreteness, yet the assumption that space and time is discrete is absolutely incompatible.

    I read about HUP, the field equations etc...
    Can you explain what are the real problems to consider them discrete.

    I read about integration , differential equations etc , is it time the real problem?
    Can we imagine a world where space is discrete and time is not?
    Can't we consider the continuum a continuum of discrete building blocks?
     
  4. Aug 9, 2014 #3

    atyy

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    The main feature of QM is not its discreteness. The discreteness of some quantities like energy levels in atoms is a secondary aspect particular to some quantum systems. The analogy here is like a violin string, which is continuous, yet has discrete harmonics due to its being tied down at both ends.

    However, one can largely imagine that the matter we see is made from a discrete lattice, where the lattice spacing is fine enough so that we cannot see it with current experiments. There is one difficulty with this view, which is whether chiral interactions can be put on the lattice, which I think is still being researched. You can read more about this point of view in http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf:

    "Often, authors forget to mention the first, very important, step in this logical procedure: replace the classical field theory one wishes to quantize by a strictly finite theory. Assuming that physical structures smaller than a certain size will not be important for our considerations, we replace the continuum of three-dimensional space by a discrete but dense lattice of points. ...

    If this lattice is sufficiently dense, the solutions we are interested in will hardly depend on the details of this lattice, and so, the classical system will resume Lorentz invariance and the speed of light will be the practical limit for the velocity of perturbances. If necessary, we can also impose periodic boundary conditions in 3-space, and in that case our system is completely finite. Finite systems of this sort allow for 'quantization' in the old-fashioned sense: replace the Poisson brackets by commutators. Note that we did not (yet) discretize time ..."

    At the end of that article, 't Hooft discusses taking the lattice spacing to zero, which at the moment cannot be done, because of the "Landau pole" of QED. He then says the problem is probably not so important on its own, because the Landau pole is above the Planck scale, and one would presumable need to solve the problem of quantum gravity too.
     
    Last edited: Aug 9, 2014
  5. Aug 9, 2014 #4

    Dale

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    I am not aware that it is incompatible, just there is no experimental evidence for it.

    Do you have a reference asserting the incompatibility?
     
  6. Aug 9, 2014 #5
    The main problem with space and time quantization arises when we try to divide spacetime in a "crystal" manner. That is, regullary. Such quantization explicitly violates Lorenz invariance and also translation invariance.

    One solution is to attempt a "glass" quantization. If the quanta are aligned irregulary, there is no preferred point, direction and frame.
     
  7. Aug 9, 2014 #6

    bhobba

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    You might like to investigate Lattice Field Theory
    http://en.wikipedia.org/wiki/Lattice_field_theory

    Thanks
    Bill
     
  8. Aug 9, 2014 #7

    jtbell

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    Note: two threads have been merged.
     
  9. Aug 9, 2014 #8

    bhobba

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    There is no incompatibility, at least as far a I am aware.

    But it is often thought stuff like virtual particles etc may simply be an artefact of taking the continuum limit in QFT and then using perturbation theory.

    Evidently they do not occur in non perturbative approaches in lattice theory.

    Thanks
    Bill
     
  10. Aug 10, 2014 #9

    atyy

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    Lattice gauge theory is typically not exactly Lorentz invariant. However, since we have no experimental proof of exact Lorentz invariance, lattice gauge theory is consistent with observations.

    One form of "spacetime" discreteness that is compatible with Lorentz invariance is causal set theory. http://arxiv.org/abs/0709.0539
     
  11. Aug 10, 2014 #10

    Barry911

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    I thought that the trend was toward a Leibnezian universe i.e. S-T independent and wholly relational. It would
    seem that a fundamental description of the universe must exist at the level of the Planck length. At this
    level (10 minus 16th below the quantum level of fermions) there would exist correlations and correlation
    vertices (misleadingly termed spin nodes the units of which are h, the quantum of action. Q.M. is relevant
    at this level but cannot be thought of as the fundamental level of the universe. At this level almost all is speculation and all interpretation premature.

    thanks,

    Barry911
     
  12. Aug 10, 2014 #11

    Barry911

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    About discreteness:
    With Bell's theorem and the experimental results of Alain Aspect the concept of discreteness seems to have
    lost any real meaning for me. However I cannot accept the idea of action at a distance Big Al's "locality"
    remains valid.
     
  13. Aug 10, 2014 #12

    bobie

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    Can you expand on that or give me some useful links?
    I read that when calculus was discovered there was a debate and Leibniz supported discrete integration and calculus but Newton's view prevailed.
    Are you referring to that? what is a S-T independent universe?
    Thanks
     
  14. Aug 10, 2014 #13

    bhobba

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    He is just chewing the fat about some conjectures floating about.

    We have zero idea at this juncture if space-time is in some manner discreet or we can always apply the methods of the calculus which assume a continuum amongst other things.

    The kinds of questions you are asking relate to mathematical analysis:
    http://math.univ-lyon1.fr/~okra/2011-MathIV/Zorich1.pdf [Broken]

    In particular the so called completeness or least upper bound axiom.

    Thanks
    Bill
     
    Last edited by a moderator: May 6, 2017
  15. Aug 11, 2014 #14

    Barry911

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    some conjectures floating about??
    Space-time independence! hardly a "conjecture" Newton's universe contained such notions as "action
    at a distance", "the rigid ether" and the idea that somehow both space and time are absolutes.
    Space - time independence was established by Albert Einstein and is an essential part of Special
    relativity theory. It is also essential to invariance in relativity theory.
    I thought that this was a physics not a math forum.
     
    Last edited by a moderator: Aug 12, 2014
  16. Aug 11, 2014 #15

    Barry911

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    bobie:
    Leibniz was a true genius and co-discoverer of Diff and Integral calculus. But he was much more.
    Remarkably, he offered the insight that absolute space and time are unacceptable in any notion of the
    universe. Further he felt that continua were impossible. He therefore posited the notion of a universe
    of fundamental elements with correlations (?)
    I would agree with bhobba that there are some really strange conjectures floating about e.g. Maldecena
    and his "holographic universe", "the landscape", "string theory" (uh-oh now I'm in trouble) but certain
    speculations seem more substantial than others i.e.- Lee Smolin's loop quantum gravity requires space-time
    independence and relational structure. Roger Penrose also speculates on the validity of Smolin's theory and
    offers the addition of "spin-networks" mathematical objects at the Planck level". I can't help but believe that
    "quasi-discrete" is the best we can do since Bell demonstrated essential quantum correlations.

    Respectfully,

    Barry911 (911...the car not the emergency)
     
  17. Aug 12, 2014 #16

    bhobba

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    The conjecture bit is if space-time is a continuum or not.

    It's obvioius, in view of the context of this post, that's not what I was referring to. What I was referring to is:

    When we model something using real numbers we subsume its axioms such as the LUB axiom.

    Thanks
    Bill
     
    Last edited: Aug 12, 2014
  18. Aug 12, 2014 #17

    Barry911

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    I find an essential bias in most "conjectures" i.e. if one can fit a mathematical model to an assumption
    the assumption becomes real physics as opposed to "speculations" about physics. e.g.- in a misguided
    effort to conserve information in an absolute sense (What would Amy Noether have said?) Juan Maldecena
    came up with a mathematical model ADS/CFT its interesting but irrelevant. I have in my old age become
    more and more impressed with the insights of the truly great physicists: Einstein, Heisenberg, Schrodinger
    et.al. Special Relativity was almost entirely conceptual and Einstein,s derivation of the Lorentz transformations
    was independent of the ether (not so Lorentz) and followed his realization of the invariance (in S.R.) of the
    constant C. I'm (I would guess like most of you just a very interested amatuer . I majored in physics in
    college as a pre-med but went on to become a surgeon). I really like math but I love Physics. In my retirement
    it has been my one great pleasure. I would guess that I resent a smattering of incomplete mathematical
    renderings as being acceptable but all serious efforts at gauging "concepts" being unacceptable.
    1. Mathematics though essential to Physics is not physics.
    2. There are however certain mathematical structures that have great "physicality" i.e.- Euclidean, Riemannian,
    Lobachevskian geomentry, Tensors and Tensor densities that the mathematician Liv Lieberman described
    as the "facts for physics", Cartan's Spinors and perhaps Penroses's Twistors?? However, remember that
    Newton's conceptual dynamics preceded his discovery (invention?) of the Calculus. Heisenberg independently
    invented a form of matrix algebra that satisfied his concept of duality, just because he needed it! Einstein was
    of course completely dismissive of mathematics in 1905 "unnecessary pretension" . He of course changed his
    views after discovering Riemannian geometry and Tensors. However it took enormous insight on his part to
    recognize this missing formalism as the "correct" mathematics.
    Well I have been recently upbraided by the forces that be for insulting a member in response to his allegation
    of my "chewing the fat about a few conjectures floating around" I have received 3 "bads" (Is big brother going
    to come to my house and blast me with his gamma laser? uh-oh.
    In any case I do sincerely apologize to those I've offended.
    Barry911
     
    Last edited: Aug 12, 2014
  19. Aug 12, 2014 #18

    Barry911

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    bhobba:
    We don't model QM with real numbers?
    Barry
     
  20. Aug 12, 2014 #19

    Barry911

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    bhobba:
    Since when can we assume a continuum in the fundamental theorem of calculus!
    By its very nature it is not continuous cf.-Leibniz. What I'm trying to say is that the concept of the
    infinitesimal "ds or dt" cleverly avoids continua by the definition "ds is the smallest number that is
    not 0" i.e. teen-weeny but discrete.
    Respectfully
    Barry
     
  21. Aug 12, 2014 #20

    atyy

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    @Barry911 and bhobba, the Leibniz thing has nothing (or very little) to do with calculus. It's a point of view advocated by Smolin http://arxiv.org/abs/hep-th/0507235, and does not have anything necessarily to do with fundamental discreteness.
     
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