Meaning of the Wave Function

The meaning of the wave function.

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    Richard Healey

    The meaning of the wave function
    Richard Healey
    University of Arizona
    October 19, 2014
    To understand the meaning of “the” wave function it is best to begin
    by asking for the use of wave functions, not what they represent. The
    current debate between psi-ontic and psi-epistemic conceptions begins with
    the shared assumption that the role of psi is descriptive: either it describes
    a physical system (directly or indirectly) or it describes “our knowledge”
    of that physical system. But psi is used prescriptively in applications of quantum theory: it’s function is to offer authoritative advice on how to
    apportion degrees of belief, in signifi…cant magnitude claims about phys-
    ical systems, to a physically situated agent not currently in a position
    to know whether they are true. A correct quantum state assignment is
    objectively true, so quantum states are real: but psi does not describe the
    momentary physical condition of the system to which it is assigned, so
    quantum states are not beables. This makes quantum state assignments
    relative to the physical situation of a (hypothetical) agent to whom they
    offer good advice. “Wave collapse” is therefore not a physical process
    but the reassignment of quantum state needed to reflect such an agent’’s
    changing situation. And we can use quantum mechanics locally to explain
    non-localized patterns of correlation in violation of Bell inequalities with
    no superluminal influences.

    (See attached file for paper.)

    Richard Healey

    Does the wave-function describe a physical system or our knowledge of that system? No.

    It describes neither. So what’s the use of wave-functions?

    The primary use of a wave-function is to prescribe (not describe!) how firmly to believe claims about the values of physical magnitudes on a physical system.
    Anyone who accepts quantum mechanics uses the wave-function to do this by plugging it into the Born rule and adjusting her degrees of belief to match the corresponding Born probabilities.

    Probabilities for what? The values of magnitudes (x-position, z-spin, energy,…). So the Born rule should be stated so as to assign probabilities to claims about the values of magnitudes, not claims about measurement outcomes.
    But don’t we know that can’t work, because there is no consistent non-contextual simultaneous assignment of values to all “observables”?
    There isn’t, but there doesn’t need to be. The Born rule can be legitimately applied only to significant magnitude claims!

    You mean claims about magnitudes in an experimental arrangement suitable for their measurement?

    In general, “No”: plenty of magnitude claims about what happened long ago in a far-away uninhabited galaxy are perfectly significant. If you are concerned about which these might be, ask whether application of a model of decoherence would pick out a “pointer basis” close to diagonal in a basis of eigenfunctions of your favorite “observable” magnitude on your favorite quantum system. If yes, it’s O.K. to apply the Born rule: otherwise, not.

    Why do this?

    1. It is consistent: Kochen/Specker-type proofs don’t rule it out.
    2. It works: beliefs formed in this way are reliably confirmed by experimental statistics: that’s why you should accept quantum mechanics and use it this way.

    What about the measurement problem? What about Bell?

    The measurement problem arises only if one mistakenly takes the wave-function completely to describe a system to which it is assigned, including the entangled system+apparatus in a measurement. Since it doesn’t describe a system at all, no problem! (Quantum mechanics can’t explain why there is a unique outcome since its application presupposes that there is. Is that a problem? Only for those who put unreasonable demands on quantum mechanics—it’s still the best theory around!)

    Bell showed that no theory of local beables satisfying a factorizability condition he took to follow from an explication of an intuitive local causality requirement is consistent with certain quantum predictions, now amply confirmed by experiment. Quantum mechanics doesn’t satisfy this factorizability condition, but Bell’s argument for that condition from the intuitive local causality requirement makes assumptions that fail in quantum mechanics. Specifically, it assumes that an event has a unique chance. But the use of wave-functions in generating Born probabilities shows why both wave-functions and chances must be assigned relative to the space-time location of a hypothetical agent applying quantum mechanics. When this is done, quantum mechanics can be used to explain violation of Bell inequalities with no superluminal influences.


    Hi Richard,

    Since I will be unable to participate in the text chat about your upcoming presentation, I give a few comments here in advance.

    In my opinion, it seems difficult for such a view to explain the results of protective measurements.

    During a protective measurement, the measured state is protected by an appropriate procedure (e.g. via the quantum Zeno effect) so that it neither changes nor becomes entangled with the state of the measuring device appreciably. In this way, such protective measurements can measure the expectation values of observables on a single quantum system, even if the system is initially not in an eigenstate of the measured observable. Moreover, the whole wave function of the system can also be measured by a series of protective measurements as expectation values of certain observables (Aharonov and Vaidman 1993; Aharonov, Anandan and Vaidman 1993).

    So, if the wave function can be measured from a single quantum system (without disturbing the system), it seems that it should be regarded as something objective about the system. For a detail argument, please see my presentation.

    I would like to know what you think about this. Thanks!


    Richard Healey


    Thank you for your post. It will take me some time to digest the different opinions I’ve seen expressed on the significance of protective measurement. I note that Max Schlosshauer disagrees with your own evaluation of its significance in his topic, and I hope to learn from this debate. I’ll let you know if and when I know what to think: at present I remain skeptical in the best Socratic tradition as one who knows that he does not know!



    I think there will be a lot of debate about the significance of protective measurements later in the workshop, so let’s perhaps put that to one side for now. I have a few comments more specific to Richard’s paper.

    Firstly, I think it is a little bit unfortunate that the modern debate about the meaning of the wavefunction uses the terms “ontic” and “epistemic”. The term “epistemic” is used because many of the physicist promoters of this view of the wavefunction, such as Spekkens, adopt an interpretation of probability in which all probabilities are in some sense about knowledge. However, this is not at all what is at issue in most of the contemporary papers on whether the wavefunction is epistemic. Instead, it is about whether the quantum state is an intrinsic property of an individual system or something more like a probability measure, however you choose to interpret the latter. It is most fundamentally about whether the explanations of quantum phenomena that draw on analogies with probability measures, as opposed to “beables” or phase space points, are on the right track. In this regard, I would categorize your view as “psi-epsistemic”, at least in the sense meant by Spekkens et. al.

    There is, of course, an important distinction to be made between interpretations that are “realist” in that they do demand a description of microphysical systems in terms of “beables”, and those that are neo-Copenhagen in that they do not think that quantum theory is a direct description of reality, but also that no such description is required. I think most people would agree that adopting a neo-Copenhagen view is always possible for someone who wants to be a psi-epsitemicist, and the relative merits of such approaches have more to do with your views on the necessity and nature of scientific realism in general than on the nature of the wavefunction specifically. Therefore, I think few would disagree that your view is logically coherent and evades the existing no-go theorems, but debating its relative merits would take us into philosophically deeper waters.

    A couple of questions to conclude:

    – What, would you say, are the key points on which your approach differs from other neo-Copenhagen approaches, such as QBism?

    – In the abstract you say:

    “A correct quantum state assignment is objectively true”

    but then later on you say:

    “When agents (actually or merely hypothetically) occupy relevantly di§erent physical situations they should assign di§erent quantum states to one and the same system, even though these different quantum state assignments are equally correct.”

    Don’t these statements contradict each other?

    – Finally, you state that, unlike phase space distributions in statistical mechanics, quantum states are not probability distributions. However, this is just an artifact of one particular mathematical formalism for quantum theory. Quantum states *can* be represented as probability distributions, e.g. the probability distribution assigned to an informationally complete POVM is isomorphic to the quantum state. Therefore, perhaps you mean to say that it is the way that the different probability distributions for different magnitudes are related to each other that is significantly different in quantum theory?

    Richard Healey

    Thanks, Matt, this was helpful.
    1. Since I do not think the quantum state is an intrinsic property of an individual system my view of the wave function is not ontic in terms of your classification. But since I do think that many wave function assignments are objectively true because of how the physical world is (not because of what anyone believes about it) I am not happy having the view called epistemic.
    2. Is my view realist? I do think quantum theory helps us make lots of true claims about unobserved features of microscopic physical things, and in this way furthers the realist goal of describing the fundamental features of the world, whether or not we are observing, or can observe them. In this sense it is a theory “without observers” in Goldstein’s sense. But I don’t think that the wave function or other elements newly introduced in quantum models themselves describe new kinds of physical features of the system to which they are assigned: quantum theory does not introduce any new beables. I would like to see a theory that gives a richer description of the physical world than quantum theory, but the desire for a more full-blooded realist theory should not influence one’s understanding of the quantum theory that we have.
    3. I depart from QBism by taking quantum state assignments and Born probabilities to be objective rather than subjective: so I reject the global subjective Bayesianism of QBists, at least in the sense that I see this as just one philosophical option one doesn’t have to take to understand quantum theory. And I take quantum state assignments to be relative, not to an actual agent’s epistemic state, but to a merely hypothetical agent’s physical situation. There are other differences (e.g. with regard to the significance of models of decoherence) , but these are the main ones.
    4. That last point is relevant to your second question: The correct quantum state assignment relative to the physical situation of one hypothetical agent may differ from the correct quantum state assignment relative to that of a different hypothetical agent. Bob and Alice can assign Alice’s photon different quantum states at the same lab. Time because Bob is in a position to know his outcome though spacelike separated Alice is not. (Bob should “collapse” his state, but Alice should not.)
    5. In answer to your last question: Yes, that is the important disanalogy: I accept that we could get rid of wave functions and replace them with probability distributions.

    Ken Wharton

    Hi Richard,

    Thanks for your interesting paper. Like Matt Leifer, I also felt a bit of a contradiction going on between objectivity and subjectivity, but as I re-read those last two pages I realized that (I think?) you’re saying that *for a given agent* there is an objectively true assignment of a wavefunction. That’s perfectly fine in my book, but perhaps the current statement at the top of page 6 is misleading as it stands.

    But then (also like Matt), I don’t see any issues distinguishing this clarified point from the psi-epistemic viewpoint in general. You say that the state does not “describe or represent the epistemic state of any actual agent”, but (again, if I’m interpreting you correctly) that’s a straw man representation of the psi-epistemic viewpoint that I’d be shocked if anyone actually held. Of course the state doesn’t describe the particular beliefs of a confused agent, for example. But so long as you’re happy with different non-confused agents using different states to proscribe ones predictions of the same system, I think that would put you firmly in the psi-epistemic camp. And then, since this approach isn’t looking for (or considering) underlying beables, I’m currently interpreting your position as essentially QBist.

    Your final paragraph, though, seems to back off on this position slightly, and made me a bit more uncertain about my above interpretation. Surely, if different agents can use different states, then the theory of QM that determines (objectively!) which state they should use *must* refer at some point to the knowledge that an agent has at her disposal…? Otherwise, how could two different agents end up with different states?


    (I was just about to post this when I saw your reply to Matt, but I still don’t see a lot of light between your position and the usual psi-epistemic viewpoint. Do you interpret “psi-epistemic” to be saying that quantum states even describe the epistemic states of *confused* agents, who have made some logical error? My last paragraph above might be relevant to this issue as well.)

    PS – I’ll be heading to the airport in mid-discussion, so will only be around for the first bit of the first text-chat. Bob and Valia; apologies for missing the live aspect of your papers, but will chime in upon my return, or possibly from the road…

    Robert Griffiths

    Dear Richard,

    I have read both your contribution to this workshop and your comment on your
    contribution twice. Here are some comments.

    First, I agree with much that you say, though I would employ a somewhat
    different language. Your assertion that the quantum psi (|psi>) is not a
    description corresponds to my calling it a pre-probability: something from
    which one can deduce a probability distribution. Your “physical magnitude” is
    what I would call a physical property associated with a subspace of the Hilbert
    space. We agree that there is no spooky action at a distance. I would add
    that much of this is to be found in my 2002 book [1], and I am glad to see
    someone else arriving independently at many of the same conclusions.

    [1] Consistent Quantum Theory (Cambridge 2002)

    But continuing the different language. I think your use of “agents” does not
    necessarily clarify things; do they have a role other than as a way of
    visualizing conditional probabilities? Or are they perhaps your way of
    getting at my approach to quantum probabilities using different frameworks
    (different sample spaces in the sense of probability theory)? The frameworks
    have the advantage that they can be expressed in a clear mathematical form;
    it is not quite so clear what is meant by an agent.

    There are aspects of your presentation which suggest that you entertain the
    opinion widespread among the decoherence advocates that all one needs to do is
    to consider an appropriate “big” wavefunction developing unitarily in a large
    closed system, and then the physics will drop out of it by applying Born’s rule
    here and there as needed. Actually, this will not work; the probabilities you
    get in this way are what I call in Sec. 9.2 of my book “one time
    probabilities”; what is lacking is the temporal correlations which are crucial
    to the physics.

    In particular, they are needed in order to relate measurement outcomes (pointer
    positions) to the properties of the measured system at a time before the
    measurement took place. Many people working in quantum foundations seem
    unaware of the fact that experimental physicists, especially those working on
    high energy experiments, quite regularly talk as if they believed that the
    muons that triggered their detectors were really there, heading towards the
    dtectors after being produced in some nuclear reaction. Some foundations folk
    take the attitude that these experimentalists are being careless and have
    forgotten what they learned in their quantum textbooks, that measurements
    create properties rather than measuring them. My attitude is that it is the
    experimentalists who are right, and the textbooks should be rewritten.

    But in order to get at the temporal correlations you need the concept of
    quantum histories–sequences of quantum properties at a succession of times,
    and for this the Born rule is inadequate, and consistency conditions have to be

    Bob Griffiths

    Richard Healey

    Ulrich Mohrhoff replied to one of your updates:

    “Hi Richard,

    You said it: “Quantum mechanics can’t explain why there is a unique outcome since its application presupposes that there is.” This cannot be over-emphasized.

    Since your upcoming presentation begins at 1.30 am my time (Indian Standard Time), I don’t think I will be in a condition to participate in the text chat. Hence a few comments here, in advance.

    You write: “Bell’s argument … assumes that an event has a unique chance. But the use of wave-functions in generating Born probabilities shows why both wave-functions and chances must be assigned relative to the space-time location of a hypothetical agent applying quantum mechanics.”

    This sounds quite similar to what QBism is saying, a critical appraisal of which I uploaded last month: You are absolutely right in denying (as you seem) that a given event has a unique chance. Since quantum mechanics *correlates* events, the probability of a given event depends on the events on the basis of which it is assigned. But this is not quite the same as making probabilities dependent on the spacetime locations of the agents assigning them. More on this in Sect. 9 of the aforementioned preprint.

    You continue: “When this is done, quantum mechanics can be used to explain violation of Bell inequalities with no superluminal influences.” Explain? Predict, yes, but that it does anyway. Rejecting an explanation (in this case, superluminal influences) doesn’t quite amount to explaining what is going on.



    Thank you for your comments, and especially for referring me to your arxiv posting.

    I have learned a lot from QBists, but I agree with you that one doesn’t need to be a subjective Bayesian to understand the significance of Born probabilities. A quantum state assignment is relative not to the epistemic state of an actual agent, but to the physical situation of a merely hypothetical agent including space-time location. This restricts such an agent’s access to information and so would make it necessary for him/her/it to aim for an ideal epistemic state that took account of all and only information about events in his/her/its backward light cone in forming beliefs about matters not so accessible. Quantum mechanics helps us actual agents to set degrees of belief using Born probabilities, among other things to derive objective chances (single case probabilities for individual events) relevant to our physical situations. But these objective chances are not physical propensities that (sometimes) make these events happen—they are not beables, and certainly not local beables.

    Certainly rejecting one explanation does not amount to having another. But quantum mechanics can explain violations of Bell inequalities as the joint of effect of a nonfactorizable common cause (whatever event(s) in the overlap of the backward light cones of the detection events backed assignment of the relevant entangled state). See my paper posted on the IJQF web site, which you can find by clicking on my name under Blogs: Members.


    Richard Healey

    Hi Ken,

    For a given agent-situation there is an objectively correct state assignment: what this is is in general different for different agent-situations. An agent-situation is not an agent: it is something physically characterized in terms of a space-time region (ideally a point) and perhaps also physical processes connecting that region to other regions relevant to the system whose state is to be assigned. So one can talk freely about (different) correct state assignments to a system in a world devoid of actual agents.

    Does this help?

    What’s confusing about the top of page 6? I don’t want to confuse people!


    Regarding this:

    “This restricts such an agent’s access to information and so would make it necessary for him/her/it to aim for an ideal epistemic state that took account of all and only information about events in his/her/its backward light cone in forming beliefs about matters not so accessible.”

    I wonder what your view of an object like “the wavefunction of the universe” would be, particularly in inflationary cosmologies in which there may be regions of the universe forever outside the observable universe of any hypothetical agent?


    Richard & Ken,

    Perhaps the distinction between QBism and Richard’s position is similar to the difference between objective and subjective Bayesianism, or between the earlier “logical” theories of probability and subjective Bayesianism. In subjective Bayesianism, probabilities represent degrees of belief and they are not to be analysed into anything deeper. Two agents in the exact same epistemic situation may still assign different probabilities. QBists say the same about quantum states. On the other hand, the other views of probability I mentioned cash out probabilities in terms of degrees of *rational* belief, i.e. they think that there is a unique probability that it is rational to assign given the agent’s epistemic situation. It seems like Richard wants to view the quantum state in a similar light.

    Robert Griffiths

    Dear Matt and Richard,

    Now having had time to read, though maybe not understand, your interchange, let me make a comment and pose a question.

    The comment: describing quantum states as “probability distributions”. Plural. That’s what I would mean by a pre-probability. You tell me what
    decomposition of the identity you are are interested, and I will assign
    probabilities to the subspaces. Give me another decomposition, maybe incompatible with the first, and I will repeat the process. Using the same wave function or density operator.

    The question. I consider myself a realist, but add that I believe that the world is quantum mechanical at any and every length scale, so reality is quantum mechanical; classical reality disappeared during the 20th century and I don’t know that it has been preserved in any museums. Quantum reality is what can be described using Hilbert subspaces, not hidden variables. So: am I neo-Copenhagen or a psi-epistmicist or some other beast?

    Richard Healey

    To Ken, again,
    “if different agents can use different states, then the theory of QM that determines (objectively!) which state they should use *must* refer at some point to the knowledge that an agent has at her disposal…? Otherwise, how could two different agents end up with different states?”

    An objectively correct state assignment relative to an agent-situation will be backed by true magnitude claims concerning events in the back light cone of that situation. Since different agent-situations have different back light cones, the backing conditions of correct state assignments relative to those agent-situations will also generally differ. That’s why different assignments are correct relative to different agent-situations.

    None of that referred to agents or their knowledge. Rather, it explained why if there are any agents in these situations they will be well-advised to assign different states as the best way they have of adjusting their beliefs on the basis of the knowledge available to them.

    We have to keep straight the distinction between quantum models and how they work, and how they assist agents in arriving at well-advised epistemic states.

    I don’t want to be disagreeing with straw QBists, so I accept their wish to consider only coherent belief states!



    Ken Wharton

    Richard; what about two agents in the same place, but only one of them (Alice) has previously programmed the computer that will be making particular measurements on a distant entangled particle. If the 2nd particle of the entangled pair is right in front of the agents, Alice should utilize her knowledge of the distant measurement setting to effectively treat the non-local entangled state as a local mixed state. Meanwhile, the other agent, at the same place and time, won’t be able to make the same predictions about the distant outcomes as Alice, and he’ll be using a different quantum state. In this example, it doesn’t seem like the spacetime-location of the agents is sufficient to “objectively” pick out the right state to use.

    But I agree with you there *is* an objectively correct state… Just that you also need to take into account what information is available to each agent.

    Richard Healey


    I can make sense of talk of the wave function of the universe if this refers to a state assignment to some restricted set of degrees of freedom of the universe—say its large scale spatial structure. Such an assignment might be useful in guiding an agent situated within that universe (or even in another one!) in forming rational degrees of belief about that universe’s large scale structure, as expressed in significant magnitude claims—significant because of decoherence with other degrees of freedom in their environment.

    I’ll have to think more about stuff going on in regions of the universe forever outside the observable universe of any hypothetical agent. As a first reaction, I see no problem, since one can certainly apply quantum theory to non-actual universes in which everything is unobservable. But this has a point only insofar as one can “project” oneself into that universe as a very hypothetical physically situated agent.

    Richard Healey

    To Ken at #774,

    Nice example! Non-optimally-informed agents can still use quantum theory as long as they recognize that they could have done better if they had made full use of all physically accessible information. I think this is what is going on with assignment of entangled states to systems that no longer exist in delayed choice entanglement-swapping experiments.


    Richard Healey

    To Matt at #771,

    I do want quantum probabilities to be objective, and I’ve said a bit about what I think that involves in my BJPS paper in the references to my posted topic paper.
    I can add this: To accept quantum theory is to grant expert status to its Born probabilities in assigning one’s degrees of belief—to treat these as authoritative. As for the wave-function that’s input to the Born rule, to treat that as objective is to undertake to be guided by relevant frequencies in outcomes of measurements. As to what counts as repeating the same measurement on the same kind of system to get those statistics, I take this to be an objective matter in that the community of physicists takes it to be a matter of fact to be resolved by argument and further experiment. Not much of this look like an extension of logic, as people like Keynes and maybe Carnap hoped!

    Ken Wharton

    Last comment before I have to leave:

    I’m certainly in full agreement with your last statement, but I would think it would have implications for your big-picture, in that all observers are non-optimally informed to some extent, and yet QM seems to still work. (In fact, if you include future measurement settings as things that *all* agents are non-optimally-informed about, then this points to a nice extension of my above example where *all* states can in principle be treated as local mixed states, if only we knew the future settings! )

    More in a few days!


    Richard Healey

    To Robert #772,

    Man physicists use the word “describes” when they really mean “applies to”. In what sense does Hilbert space (or a vector or operator in Hilbert space) describe anything? How do we use it to describe anything?

    After long rejecting Bohr’s infamous remark that there is no quantum world as rabid antirealism, I came to have some sympathy with the continuation “the point of physics is not to describe nature, but to determine what we can say about nature” (not an exact quote, sorry). I think to understand quantum theory we need both to ask what we can say about the physical world and also show why much of what we can say is true. But quantum theory itself does not say anything about the world we could not say without using waves functions, vectors, operators or Hilbert space.




    I think you are clearly a psi-epistemicist. For my purposes, this just means that you do not view the quantum state as an intrinsic property of an individual system. Whether you are neo-Copenhagen is a trickier question, which we have debated before over email. On my view, two key components of the Copenhagen interpretation are:

    – The wavefunction is not a direct representation of reality, but merely a device for computing probabilities (the words you use to describe this depend on your interpretation of probability, but I tend to mangle terminology and just call this the psi-epistemic position anyway).

    – Nonetheless, no deeper theory of the nature of reality beyond quantum theory is required.

    For the purposes of the psi-epistemic/psi-ontic debate, I often *define* neo Copenhagen to mean anything that satisfies these two requirements. It is useful to do so as it cleanly delineates a set of interpretations to which the recent no-go theorems for epistemic quantum states do not apply. In this sense, your view is neo Copenhagen.

    On the other hand, there are obviously several other important features of that vague collection of ideas that we call Copenhagen. One such feature is a broadly anti-realist stance, and some people might be inclined to view this as the main feature of Copenhagen, which all neo Copenhagen views ought to share. If so, then your view is not neo Copenhagen, or at least you do not intend for it to be viewed that way. We can debate whether consistent histories really does provide a viable realist ontology, but that is perhaps a topic for another time.

    For the purposes of the debate about the meaning of the wavefunction, I prefer to define neo Copenhagen by the two features above, and will thus include several realist views on the neo Copenhagen side.

    Richard Healey

    Thank you all!


    Robert Griffiths

    Richard, in response to 779–
    I am not sure if protocol permits discussion to continue after the bell ending the class has rung, but let me try and be brief.

    I use Hilbert space to describe a quantum system in much the same way I use phase space to describe a classical system. Granted, we have an abstract
    mathematical object (No experimentalist has ever seen a Hilbert space!) but
    by faith we physicists think that the math has something to do with the real world, and the abstract description it provides has a connection with reality (the connection is subtle, and no doubt we can learn something from the philosophers). Then we show the students lots of examples starting with spin half particles, and going on to harmonic oscillators, acknowledging that the descriptions thereof are imperfect and incomplete.

    Bohr’s comment reflects the fact that he did not understand quantum mechanics (I call on Feynman’s authority for that remark), and was trying to make the best of it. I think we are currently in a position to do quite a bit better as we stand on the shoulderes of the giants. I differ with you
    on what QT can say about the world: I challenge you to say anything in the advanced QM course without using wave functions or operators or Hilbert space.

    Richard Healey

    To Robert at #767,

    You say
    “There are aspects of your presentation which suggest that you entertain the
    opinion widespread among the decoherence advocates that all one needs to do is
    to consider an appropriate “big” wavefunction developing unitarily in a large
    closed system, and then the physics will drop out of it by applying Born’s rule
    here and there as needed. Actually, this will not work; the probabilities you
    get in this way are what I call in Sec. 9.2 of my book “one time
    probabilities”; what is lacking is the temporal correlations which are crucial
    to the physics.”

    In appealing to models of environmental decoherence as a guide to the significance of magnitude claims, I think I am considering temporal correlations in this sense. The “pointer basis” in a model of environmental decoherence is typically selected very rapidly and persists robustly for a long time thereafter. In this way appeal to such a model to underwrite the significance of claims about magnitudes whose operators are very nearly diagonal in the pointer basis does rely on the temporal correlations of the model. It must, since at any instant the Schmidt decomposition will always pick out (elements of) a unique basis.


    Robert Griffiths

    Dear Matt,

    A brief reply to your #780. Extremely clear. A pleasure to read. Thank you

    Bob Griffiths

    Robert Griffiths

    Dear Richard,

    In response to #783. I am not unsympathetic to invoking decoherence and
    relaxation times related thereto. However, I think a Schmidt decomposition is
    neither necessary nor sufficient to discuss decoherence, and histories will do
    a better job. But apart from that, how are you going to address what I call
    the SECOND measurement problem: that the measurement actually measured some
    microscopic property? I don’t see how decoherence (absent the use of
    histories) will help with that.

    Next time you’re in Pittsburgh I’ll buy you dinner and we can try and sort it

    Best, Bob Griffiths

    Richard Healey

    To Robert at #7

    “I challenge you to say anything in the advanced QM course without using wave functions or operators or Hilbert space.”

    Not accepted!
    Of course one has to talk of such things when discussing the mathematics of quantum models. And in such a context one naturally mixes such talk with talk of how the models are going to be applied to physical systems, if only to motivate what might otherwise be regarded as rather dry mathematics.
    But if one is seeking analytical clarity about how quantum models are applied it is very important to distinguish talk about the model from talk about the physical system(s) to which it may be applied.

    Unlike Max Tegmark (apparently) I don’t think the physical world is mathematical. In classical physics we use mathematics pretty straightforwardly to describe the physical world, when we say (for example) the mass of the sun is however many kilograms, or the current outside temperature is so many degrees Celsius. Actually, philosophers of science have worried about how mathematics can be used descriptively even in such simple cases. We don’t think the world contains coordinate systems, but we still use their mathematics in applying classical physics. But when it comes to quantum theory, the application of the mathematics of the models is at least more subtle. It becomes important to ask whether a particular element of a mathematical model is being used to describe or represent something in the physical world (directly or indirectly) or for some other purpose. We can associate dynamical properties of a physical system (such as z-spin up) with subspaces of Hilbert space, but can we say that a spin 1/2 system has such a property just in case its quantum state lies in that subspace? Different views of quantum theory will answer that question differently.

    Richard Healey

    To #785,

    “how are you going to address what I call the SECOND measurement problem: that the measurement actually measured some microscopic property? ”

    I think this way of talking, though often permissible, should rarely be taken literally. Entangled photons have their linear polarization measured every day in quantum optics laboratories by passing them through a polarizer and absorbing them at a detector. I think at no point does one of these photons have a linear polarization.

    There are occasions in which decoherence licenses magnitude claims about the position of a particle after detection by a detector, as when Markus Arndt’s group got their fullerenes to stick to a specially prepared silicon surface and then scanned them with an STM. Each fullerene had some pretty definite, stable, position on the silicon surface after detection, even though it had no well-defined position in the interferometer prior to detection.

    So I guess I’m with the theoreticians on this issue, and merely tolerate experimentalists’ talk of muon trajectories, unless these muons’ states were stably decohered in position basis on their way to the detector. The discussion by Mott and Bell about the formation of alpha particle tracks in a cloud chamber is relevant here.


    Thank you all for your participance in this online workshop! I hope everyone will be inspired by these intense and stimulating discussions. I will try my best to provide technical support and make the forum provide you better experience.


    Ulrich Mohrhoff

    Richard, thank you for your response to my comments (both at #768).

    I realize that our difference (if any) concerns the use we make of the wave function, and that there are different legitimate ways of using the wave function. One can even use the ABL rule (in lieu of the Born rule) to assign probabilities on the basis of both past and future outcomes, as I suggested in quant-ph/0006116 and quant-ph/0703035. I have come to realize, though, that it wasn’t consistent to consider these ABL probabilities objective, as I did at the time, for if I take all (relevant) past and future outcomes as given, then I should also take the target outcome as given, in which case the probability assigned to it is a (subjective) ignorance probability. (One could still argue that probabilities counterfactually assigned to the possible outcomes of an unperformed measurement are objective.)

    On the other hand, assigning Born probabilities with input from all relevant outcomes is equivalent to using Born probabilities with input from all outcomes within the past light cone of a hypothetical agent in the infinite or sufficiently distant future, so your understanding of the wave function includes mine and is consistent with it. As to the subjective/objective nature of probabilities in general and quantum-mechanical probabilities in particular, this depends so much on what is meant by subjective/objective (and probability or chance) that today the issue seems to me rather pointless (see Sect. 16 of my QBism preprint).

    As regards explaining violations of Bell inequalities, I gather from your paper “Local Causality, Probability and Explanation” that you require two things of an explanation: it must show that the phenomenon to be explained was to be expected, and it must say what the phenomenon depends on. The violations are to be expected because they are predicted, and they depend “counterfactually but not causally on the quantum state \Phi^+” as well as “counterfactually on that state’s backing conditions, as described by true magnitude claims”. But then counterfactual dependence doesn’t seem sufficient, for “the second requirement on explanation is met” because “the separate recording events, as well as the event of their joint occurrence, depend causally on the event o that serves to back assignment of state \Phi^+ to the photon pairs.” Not only does this seem to contradict your statement that “the theory has no resources to describe any causes mediating between o and these recording events” but the causality invoked also seems to be of a rather watered-down variety.

    What I would require of an explanation of the quantum-mechanical correlations is some physical mechanism or natural process, which there isn’t. While of course there is no law against re-defining terms, the danger in doing so is that something important doesn’t get the attention it deserves, namely the fact that good old-fashioned causal explanations simply do not work in the quantum domain.

    Two other minor quibbles. Firstly, you claim to meet Bell’s strictures against the presence of the term “measurement” in your formulation of quantum mechanics, but you seem to have added another term to the proscribed list and used it, namely magnitude claims. Secondly, you claim that an omniscient God (or creature) could describe and understand the physical world without a concept of chance. I’m not so sure. Such an entity would have complete knowledge of all events including their statistical correlations.



    Hi Richard,

    Thanks for your comments so far, I think I now understand your position a bit better. Obviously I’m more than a little late to the party, and my question isn’t about the wave function per se, but if you get a chance to respond to the following at some point, I’d be interested in your thoughts.

    You’ve said that “quantum theory does not introduce any new beables” and “quantum theory itself does not say anything about the world we could not say without using waves functions, vectors, operators or Hilbert space.”. If we’re talking about something like position or momentum then we are indeed using language that was present before quantum theory. But what about spin? Can “magnitude claims” be made about spins, and if so how can they be expressed without using quantum theory? Or am I simply reading too much in to the statements I quoted?


    Richard Healey

    In answer to Matt Pusey at #890,

    Yes, one can make magnitude claims about spin. These are all claims about angular momentum, called spin for purely historical reasons when its components take on values +/-1/2 in units of h-bar.

    One can make magnitude claims about arbitrary components of spin, and also about total spin—though these are rather dull since the only true claim is that total spin has value (root-3)/2, whenever that claim is significant.

    The claim that an electron (say) is a spin 1/2 particle, though true, is not itself a magnitude claim. Instead, it is a true claim about the mathematics of quantum models, true because if one wants to apply quantum theory to electrons (or to an electron field) one should build a model in a 2-dimensional complex Hilbert space (or other corresponding mathematical structure).

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