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Quantum Mechanics without Measurement

  1. Feb 23, 2014 #1

    stevendaryl

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    I recommend the following paper by Robert B. Griffiths on developing the theory of quantum mechanics without giving a special role to measurements:

    http://arxiv.org/pdf/quant-ph/0612065v1.pdf

    In my opinion, it does not answer all the questions about locality and realism that come up in discussions about interpretations of quantum mechanics. But what I like about it is that it removes the special role that measurement plays in some formulations of quantum mechanics, and eliminates the need for wave function collapse.

    Why was I specifically Googling for a formulation of quantum mechanics without measurements? Measurement is fundamental to some ways of presenting quantum mechanics. There is the "collapse interpretion" (which I think is due to Von Neumann) in which systems evolve deterministically according to Schrodinger's equation between measurements, but then the act of measurement causes a discontinuous, nondeterministic "collapse" of the wavefunction into an eigenstate of whatever observable was being measured. There are other interpretations that don't introduce collapse, but do make measurements the fundamental ingredient in the interpretation of quantum mechanics. For example, in the paper by Lucien Hardy
    http://arxiv.org/pdf/quant-ph/0101012v4.pdf

    Some people think that such an emphasis on measurement is appropriate, since physics is an empirical science, and empirical science is founded on measurements, experiments, observations, etc. However, I find it very unsatisfactory for measurement to play a key role in the formulation a of fundamental theory because measurements are not fundamental. A measuring device is, after all, a physical object, presumably governed by the same physical laws that govern atoms and molecules and light and gravity. What makes a particular physical object suitable to be considered a "measuring device" is pretty complicated:

    • There must be an interaction between the system being measured and states of the measuring device.
    • The measuring device must act as an "amplifier", so that microscopic properties of the system being measured can trigger macroscopic changes in the state of the device.
    • The measuring device must have states that are sufficiently "orderly" to interpret easily. Either, there must be a number of discrete states in the measuring device that are observably different, or else there must be a continuous sets of states that can readily be interpreted as a linear scale.
    • The act of measurement should result in a "record", an irreversible change that can be reliably checked later.

    My objection to using measurements as primitive terms in formulations of quantum mechanics is that measurements are anything but primitive. You have to use physics to design objects that can act as measuring devices, but the measuring devices have to already exist before you can give any interpretation to the physics. This is circular. Of course, it's not really that bad, because of the fact that we know that classical physics works approximately for macroscopic objects. So we can use classical physics to design a "first cut" at measuring devices, and then use the knowledge of quantum mechanics that we get from those devices to make improved devices, and so bootstrap our way to a self-consistent notion of physics and measuring devices.

    But it seems very messy. What I would prefer is a formulation of quantum mechanics that is about what happens in the world of particles and fields, and then use that theory to derive what makes a good measurement device in a noncircular way. I think that's the approach that Griffiths takes.
     
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  3. Feb 23, 2014 #2

    bhobba

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    Yes Decoherent Histories is a nice interpretation that avoids many of the issues with other formulations. In a sense it is Many Worlds without the Many Worlds.

    But it comes at a cost - it complicates things IMHO unnecessarily, and to some extent, with its use of frameworks, is really defining your way out of problems.

    Still many proponents call it Copenhagen done right:
    http://motls.blogspot.com.au/2011/05/copenhagen-interpretation-of-quantum.html

    Lubos is correct IMHO - it removes a slight blemish with Copenhagen. Personally though I prefer the Statistical interpretation including decoherence - it seem's a bit cleaner to me.

    Thanks
    Bill
     
  4. Feb 24, 2014 #3

    Demystifier

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    This, indeed, is the main problem with the Griffiths interpretation. To avoid EPR "paradox" and consequences of the Bell theorem, Griffiths proposes to abandon the rules of classical LOGIC, replacing them with a kind of quantum logic:
    http://lanl.arxiv.org/abs/1105.3932
    http://lanl.arxiv.org/abs/1110.0974
    Most physicists, mathematicians, and even logicians, find it very unattractive.

    In particular, let me quote from http://www.scholarpedia.org/article/Bell's_theorem
    "Thus, in CH [Consistent Histories], a "quantum measurement" is really supposed to be a measurement, simply revealing the pre-existing value of the measured observable; it is not the interaction with the apparatus that creates the observed value. That sounds a lot like a non-contextual hidden variables theory, which, as we now know, must lead to inconsistencies with the quantum predictions. ... The proponents of CH ... have imposed a rule which says essentially that arguments involving probabilities for several histories, not all of which belong to the same decoherent family, are forbidden. ... By forbidding the reasoning used to prove inequality (1), the aforementioned rule of CH prevents us from arriving at the contradiction. But a physical theory is not simply a game for which one can impose arbitrary rules about what reasonings are permitted for the propositions of the theory; if a physical theory implies both P and Q then the logical consequences of both P and Q will hold in a world governed by that theory and there is nothing that the proponents of the theory can do to prevent that. One might try to find an actual objection against the reasoning leading to inequality (1), but one cannot simply state as a "rule" that the reasoning is forbidden. ... We suspect that the proponents of CH would object to the proof of inequality (1) (within CH) by claiming that one cannot assume that all the random variables Zαi are defined over the same probability space because on each run of the experiment the value of only one among the Zα1 and the value of only one among the Zα2 is going to be observed. But if the experiments merely reveal pre-existing values then, on each run of the experiment, all the variables Zαi have a well-defined value (which may or may not turn out to be observed). ... The objection against the possibility of modeling the Zαi as random variables on the same probability space is effective only when one takes their values to be created by the experiments ... But reinterpreted in terms of values being created by experiment, CH would be pointless — it would just be orthodox quantum theory."
     
  5. Feb 24, 2014 #4
    I was particularly taken with page 13 of Stevendaryl's link. I was pretty confident that the situation at t2 would be as described, but at t1 I did a double take and had to read it a few times! Basically it says that the spin must be a certain value just before the particle enters the magnetic field of the Stern Gerlach experiment in order for everything to be consistent. However it does NOT claim that the spin is fixed at t0 (I.e. When it is emitted). That sounds not so much like a hidden variable, but more like a random variable, becoming crystallised out on entering the field.
     
  6. Feb 24, 2014 #5
    It would appear that the real cost which Griffiths talks about is that in consistent histories the time development of a quantum system is a random process, one is which the future and past states are not determined by the present state, but only related to it by certain probabilities.
     
  7. Feb 24, 2014 #6

    Demystifier

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    Jilang, you are right that Griffiths interpretation is a probabilistic interpretation, and not a deterministic one. However, I wouldn't call it a cost. Furthermore, being fundamentally probabilistic does not mean that it is not about hidden variables. Namely, hidden variables do not necessarily need to be deterministic. Hidden variables just mean that the system's properties (which may be measured if one wishes to) exist even if one does not measure them.

    The problem is that Griffiths wants to avoid the Bell theorem, according to which hidden variables (not necessarily deterministic) must necessarily be nonlocal. He avoids Bell theorem not by rejecting assumptions of the Bell theorem, but by rejecting classical LOGIC leading from the assumptions to the theorem.

    Indeed, any logical conclusion may be avoided by rejecting the rules of logic. This technique, for instance, is often used by politicians. But should we allow it in science? I don't think so.
     
    Last edited: Feb 24, 2014
  8. Feb 24, 2014 #7
    Perhaps not a cost, but as Griffiths says "all of this at what price?". Not only giving up determinism and ordinary propositional logic, but also the quantum logic proposed by Birkhoff and Von Neumann. Seems expensive! Still I'm blown away.
     
  9. Feb 24, 2014 #8
    Which rules of logic are rejected by him?
     
  10. Feb 25, 2014 #9

    Demystifier

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    Essentially, the following rule is rejected:
    (A is true) & (B is true) --> (A & B) is true

    His argument is the following: A is true in one framework, B is true in another framework, but you cannot combine statements from different frameworks. There is no single framework in which both A and B are true.

    His concept of "framework" itself remains somewhat vague, but let me present my own idea of what that might mean:
    "Madonna is not a good singer" is true in my framework, "Madonna is a good singer" is true in somebody else's framework with a different taste for music, but there is no person in the world in whose framework "Madonna both is and isn't a good singer".

    However, such a view of "framework" attributes an important role to subjective observers, which is not what Griffiths does. Therefore the "framework" is his interpretation must be something else, but it's difficult to tell what.
     
    Last edited: Feb 25, 2014
  11. Feb 25, 2014 #10

    bhobba

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    Yea - it kinda reminds me of defining your way out of problems, and that its a bit vague - so much the better.

    But then again exactly what an observation is in Copenhagen is a bit vague as well.

    When people press me about that one its really horrid I have to get so handwavey.

    Thanks
    Bill
     
    Last edited: Feb 25, 2014
  12. Feb 25, 2014 #11

    martinbn

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    I am not sure but I think that the word "logic" is used in two different ways. One as the rules we use to make conclusions and another as the algebra of certain propositions. The examples of phase spaces in classical and quantum mechanics seems to show that. If you consider a set and a family of subsets you can define "negation" (compliment), "and" (intersection) and "or" (union) and have the logic of subsets. Similarly for the subspaces of a Hilbert space "negation" (orthogonal compliment) and so on. The two logics are obviously different in the second you do not have the distributive property. But the logic by which you would go and prove statements in either situation is the same. A mathematician proving theorems in the first case, who decides to prove theorems in the second is not abandoning logic.
     
  13. Feb 25, 2014 #12

    stevendaryl

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    Unless I'm missing something, the logic and probability associated with decoherent histories seems to be exactly analogous to the logic and probability associated with a single moment in time. I guess a single moment in time would be a degenerate case of a history, but I would like to look at that case because it's particularly simple to analyze.

    With a single moment in time, we have the following collection of "atomic" statements (in the sense of "indivisible", not having anything to do with nuclei):

    For every observable [itex]\mathcal{O}_i[/itex] corresponding to a Hermitian operator [itex]\hat{O}_i[/itex] and for every eigenvalue [itex]\lambda[/itex] of that operator, there is a corresponding atomic statement: "[itex]\mathcal{O}_i[/itex] has value [itex]\lambda[/itex]". In the usual way, every statement corresponds to a projection operator [itex]P^i_\lambda[/itex], which projects an arbitrary state onto the subspace in which operator [itex]\hat{O}_i[/itex] has eigenvalue [itex]\lambda[/itex].

    If [itex]\mathcal{C} = \{ \mathcal{O}_1, \mathcal{O}_2, \ldots \}[/itex] is a collection of observables corresponding to mutually commuting operators, then we can do ordinary logic and probability in reasoning about all the atomic statements involving observables in [itex]\mathcal{C}[/itex].

    Fix an initial state [itex]|\Psi\rangle[/itex], then


    The probability that [itex]\mathcal{O}_i[/itex] has value [itex]\lambda[/itex] is given by the expression [itex]\langle \Psi|P^i_\lambda|\Psi\rangle[/itex]

    (with an easy generalization to mixed states). Then it's easy enough to define "not", "and" and "or" in terms of these atomic statements, and we can compute probabilities for compound statements and we can compute conditional probabilities, etc.

    So all of this is exactly like classical logic. The difference is that there are statements that are not expressible in terms of the atomic statements of [itex]\mathcal{C}[/itex], namely statements involving observables corresponding to noncommuting operators.

    So that's the difference with classical logic: We choose a set [itex]\mathcal{C}[/itex], and then we can use classical logic, but we can't use classical logic to reason about the collection of all statements, just the statements associated with [itex]\mathcal{C}[/itex].

    It seems to me that the decoherent histories approach just changes the focus from statements about a single moment to statements about the entire history. But the relationship with logic and probability is the same: You can reason using logic and probability, but only about a suitably compatible collection of statements.
     
  14. Feb 25, 2014 #13
    Is this the same as saying that incompatible observables cannot share the same probability space? I think I read that somewhere but I might have got hold of the wrong end of the stick!
     
  15. Feb 25, 2014 #14

    stevendaryl

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    I had heard that phrase, as well, but I didn't really know what it meant.
     
  16. Feb 25, 2014 #15
    But It looks like that's exactly what you said in your post #12 or are you pulling my leg?

    If the concept of a framework seemed sort of vague should we think of it a probability space instead?
     
  17. Feb 25, 2014 #16

    stevendaryl

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    Well, the time that I had read that phrase, I thought that the author was saying that there was something wrong with Bell's theorem, because Bell erroneously assumed that the hidden variables are defined on the same probability space. If my discussion is what was meant, then it's not that Bell made an error. This kind of "logic" isn't hidden-variables in the sense of Bell.
     
  18. Feb 25, 2014 #17
    Isn't Bells inequality just a consequence of set theory where 3 attributes A,B and C are all compatible. E.g. The number of girls over 5 foot 5 with brown hair etc? Don't we need observables to be compatible to make a statement like ( A and B ) to have any logical meaning? Perhaps I'm missing something important.
     
  19. Feb 25, 2014 #18

    stevendaryl

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    Well, yeah, that's what the "quantum logic" says. Classical logic, however, says that "A and B" is always meaningful, provided that A is meaningful and B is meaningful.
     
  20. Feb 26, 2014 #19

    Demystifier

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    Let me try to explain all this in terms of everyday-life concepts.

    Except being an excellent physicists, Feynman is also known for being a good lover. (That can also be said for Schrodinger, but let us stick with Feynman.)

    So, we can say that Feynman is a good physicist, and we can also say that Feynman is a good lover.
    But can we say that Feynman is a good scientist and a good lover?

    From the experimental point of view, no one ever seen Feynman to be a good physicist and a good lover at the same time. Under some conditions Feynman behaves as a good physicist, while under other conditions he behaves as a good lover. You don't need to know much about psychology to understand that there are no conditions under which he will show his physicist and lover abilities at the same time. So whether he behaves as a good physicist or a good lover depends on the CONTEXT. Feynman (just like any other human being) is - contextual.

    Needles to say, this psychological contextuality is very much analogous to quantum contextuality.

    All this is common sense, but we still didn't answer our first question: Can we say that Feynman is a good physicist and a good lover?

    Most people don't see any problem with saying that. Yet, someone thinking about it in the same way as Griffiths thinks about quantum phenomena would conclude that it is not consistent to say that. He would explain that "being a good physicist" and "being a good lover" belongs to different frameworks, and that one should not combine statements from different frameworks.

    So, would you agree with someone who tells you that it is inconsistent to say that "Feynman is a good physicist and a good lover"? If you would, then you might also like the Griffiths interpretation of quantum mechanics. If you wouldn't, then Griffiths interpretation of quantum mechanics is not something you might like.

    All that is nice, but for me the real scientific issue is the following. Suppose you are a psychologist who wants to explain WHY Feynman never shows his physicist and lover abbilities at the same time. And suppose that someone tells you that this is BECAUSE those two properties belong to different frameworks, so that it is not logically consistent to SAY that he is a good physicist and a good lover. As a psychologist, would you be satisfied with such an explanation? I certainly wouldn't.

    For the same reason, as a physicist, I am not satisfied with the Griffiths interpretation of quantum phenomena. The Griffiths interpretation constrains the language of talking about quantum phenomena, but for me it doesn't explain anything at the scientific level.

    P.S.
    If you liked the explanation of quantum mechanics above in terms of common-sense psychology, then see also
    https://www.physicsforums.com/blog.php?b=9 [Broken]
    https://www.physicsforums.com/blog.php?b=7 [Broken]
     
    Last edited by a moderator: May 6, 2017
  21. Feb 26, 2014 #20

    stevendaryl

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    I sort of get the analogy. However, in the case of Feynman, we could bring the framework into the question, for example:

    "If we were speaking from within the framework of physics, would you say that Feynman was a good physicist?"

    "If we were speaking from within the framework of lovemaking, would you say that Feynman was a good lover?"

    So it's possible that both of these questions can have the answer "yes" simultaneously. Feynman can't demonstrate the truth of both of these at the same time, because the demonstration of one requires a setting that is incompatible with the demonstration of the other. But it still makes sense to ask if both are true simultaneously.

    This is sort of like "contrafactual definiteness" in discussions of Bell's inequality. Measuring the spin of an electron in the x-direction is incompatible with measuring the spin in the z-direction. So we can't, with a single experiment, know the answer to the questions:

    "Is the electron spin-up in the z-direction?"
    "Is the electron spin-up in the x-direction?"

    However, you could make the questions into hypotheticals as follows:

    "If I were to measure the z-component of spin, would I get spin-up?"
    "If I were to measure the x-component of spin, would I get spin-up?"

    By analogy with the Feynman case, one might think that it makes sense to ask the two questions simultaneously, even if there is no way to determine the answers (by a single experiment). You might think that a question whose answer cannot be know might as well be meaningless. But that's not exactly true, because people can do case-based analysis. For example, in logic, you can reason

    • If A is true, then B is true.
    • If A is false, then C is true.
    • Therefore, B or C must be true.

    The violation of Bell's inequality in EPR type experiments shows, in a sense, that certain conjunctions whose truth values are unknown cannot consistently be given a truth value. It's not just that we can't know or demonstrate the truth of the conjunction, but that it really doesn't have a consistent truth value.
     
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