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Quantum Mechanics Born to be Linear?

  1. Jul 24, 2010 #1
    http://www.sciencemag.org/cgi/content/abstract/329/5990/418" [Broken]
    Two pillars of modern physics, quantum mechanics and gravity, have so far resisted attempts to be reconciled into one grand theory. This has prompted suggestions that theories about either or both need to be modified at a fundamental level. Sinha et al. (p. 418; see the Perspective by Franson) looked at the interference pattern resulting from a number of slits, to test the "Born rule" of quantum mechanics. They verified that Born holds true—that the interference pattern is built up by the interference from two paths, and two paths only, with no higher-order paths interfering. The result rules out any nonlinear theories of quantum mechanics; thus, any modification of theory will need to take into account that quantum mechanics is linear.

    I wonder about the inherent non commutativity of operators acting on the same degrees of freedom of the wave function in qm. It just seems like that lends itself to a development of relevant tools in non commutative geometry to describe these physical interactions in qm (via Connes). But I don't normally think of non commutative geometries and linearity in the same breath. And to clarify I think they are saying that the operators act as linear maps on the eigenkets in qm. Is this supposed to be globally true? Are there not other spaces where operators do not act as linear transformations on the eigenkets?
    Tell me how my thought process is in error please.
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jul 24, 2010 #2
    A space is a shorthand for vector or linear space. By definition, the operations defined in it are linear. It is a mathematical construct that proved useful in explaining the laws of Quantum Mechanics.

    If you want to criticize something, I suggest you focus on a particular topic, make your ideas intelligible and give the arguments supporting your stance logically. Only in this way can you expect some feedback from others. As it is, I have no idea what you are asking or trying to convey to us.
  4. Jul 24, 2010 #3
    I think you misunderstood me, I was asking for someone's input not criticizing the paper just trying to understand the last sentence of the abstract in context.

    Better question, how do operators act on wavefunctions in non commutative geometrical context?
    Last edited: Jul 24, 2010
  5. Jul 24, 2010 #4
    I am sorry, I don't understand what you mean by this. Would you care to elaborate a bit more.
  6. Jul 24, 2010 #5
    Okay to clarify, I think about performing mathematical operations on structures that have certain characteristics. As a kids we always dealt with set elements that satisfied the commutative law. But in introductory qm we are faced with examples where operators don't commute with one another and there is physical significance to this non commutation because the two operators that don't commute cannot be simultaneously measured.

    In this sense, it seems as though the geometrical interpretation of the two operators that don't commute could be described using non commutative geometry. And I know that Alain Connes has done a lot of work on developing non commutative geometries and integrating them into physics, so I ought to start reading some of his old papers. If you can suggest some, it would be appreciated btw.

    Anyway, let me clarify what I mean with a concrete example that you might have seen before. An old Math GRE problem has elements of a set where each element is equal to its negative, i.e. s=-s For some number t which is also part of the set, (s+t)^2 does NOT equal s^2+2st+t^2 because since s=-s, the two middle terms cancel out and in THIS structure, (s+t)^2=s^2 + t^2. So how do rules of qm look in this type of structure and how do operators behave?
  7. Jul 24, 2011 #6
    Pretty shocking I'd find this out, despite what was written in the Science magazine,

  8. Jul 24, 2011 #7


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  9. Aug 21, 2011 #8
    there is recent evidence that nature is non linear


    ......"A value this close to the Planck length means that quantum gravity models in which there's a linear relationship between photon energy and speed are "highly implausible." That leaves other quantum gravity options open, including those in which the the relationship is non-linear. Hopefully, theoreticians will be able to devise real-world tests for some of these...."

    by me at :
    post 5

    Last edited by a moderator: May 5, 2017
  10. May 14, 2012 #9
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