Thanks very much for engaging with my comments; I greatly appreciate it! I consider what follows here, as with my comment on the previous post, as somewhat preliminary, despite its length. I'd like to be able to understand your position more fully and respond in a more detailed fashion - both on the subject of this post and on everything else we have discussed - though I don't know when or if I'll get around to it.

I admit I am finding myself a little frustrated at this juncture, both because I am not quite sure what you mean by some of what you write and because I am not quite sure how to articulate what I mean to say in response! But I will try my best and perhaps we will eventually understand each other better.

--- Non-Locality of QM ---

First, you are absolutely right that Bell postulates a probability distribution over the states of the candidate theory to prove his theorem. I was well aware of this, and I realized upon reading this post that I conflated two things in my original comment: Bell's theorem, and the application of Bell's condition of local causality to diagnose quantum mechanics as non-local. Your view (which still doesn't make sense to me) that we cannot use probabilities to characterize uncertainty applies to the proof of Bell's theorem, but it is not relevant to demonstrating that quantum theory fails to satisfy Bell's locality criterion. That only requires reference to probabilities of macroscropic setups and outcomes, which Norsen's paper makes clear, I believe.

So quantum theory is non-local in Bell's sense. You just dispute that Bell's sense of locality is the appropriate sense, by making the substance vs event causation distinction. I would make that distinction differently than you. I think that modern philosophy made a wrong turn by focusing exclusively on events as causes (In that much, I believe I am in agreement with you). In fact, I don't think events ever serve as causes - rather, substances are causes. Events are the effects. (Sometimes, substances act as causes as a result of or in response to certain events; that's how causal chains get built up.) In certain types of events (namely, events of creation) a substance can also be viewed as the effect; substances would also be appropriately classed as the effects of God's sustaining activity.

My point here is that, from my perspective, the distinction you are trying to draw is selecting a certain sub-class of events (creation events, and maybe annihilation events) and saying "only causation of these events needs to be local". That seems ad-hoc to me. I have a bit more to say on your take of quantum non-locality below.

--- Your View of the Bell-Inequality Experiment ---

Here is my impression of your view of the usual Bell-inequality-type experiment (please correct me if I misrepresent it): you hold that the particles are emitted each with some definite, though unknown, spin state, and those states are anti-correlated with each other. This goes beyond orthodox QM, which only assigns an entangled state to the pair of particles, but no definite state to either individual particle (so it isn't exactly "taking the math literally" as you claim). It is indeterministic which definite state is emitted. If I read you right, the definite state determines probabilities for the particle to be measured spin up or spin down along any given axis (by using Born's rule on the appropriate product, e.g. <A|Y>). (I'll get back to that.) The perfect anti-correlation of measurements along the same axis is not explained by any local causes of the physical substances involved, but instead is guaranteed directly by the sustaining activity of God.

Now, I entirely agree that God upholds the universe in existence and that physics describes his regular way of doing so. But those pertain to our theology and philosophy, not, it seems to me, to physics itself. Translating your explanation into last point into the language of physics, your view is basically that it is a law of physics that the measurements of the particles' spins will always be anti-correlated, even though there is no local cause of this anti-correlation. Okay... this reads to me like a tacit admission that quantum physics is, in fact, non-local, even though you want to avoid saying so. (And, theologically speaking, the way you avoid saying so makes your view lean towards occasionalism.)

It also makes one of your criticisms of the de-Broglie/Bohm theory sound a bit like special pleading. You seem to be saying that there is no problem with God's causation being non-local, because he is immaterial and outside of space and time. Well, the pilot wave theory can say something similar: the pilot wave (which really is just the same thing as the wavefunction/quantum state of orthodox QM by another name) exists in configuration space, certainly outside of space and maybe even outside of time. And though it may be regarded as ontologically real and an object of physics, it isn't material. Particles and the things made from them are what are material; the pilot wave is not.

(I would prefer to regard the pilot wave/quantum state as an abstraction from the forms and causal powers of material things, rather than something with an independent existence, in which case that move doesn't work. But I also don't have a problem with saying that there is genuinely non-local physical causation. I have more to say about that; maybe while I'm at this I'll go comment on your A-theory vs B-theory post.)

--- Definite Spin States ---

Back to the role that the definite spin state of the particle has in your view: as far as I can tell, it doesn't work. Say particle 1 has spin up in the z-direction and particle 2 has spin down. You seem to be saying that the probability for us to measure particle 1 as spin up along some axis at an angle theta to the z-axis is a function of theta (i.e. that it is determined by the definite state of particle 1). But it isn't - if there has been (or even if there will be?) a measurement on particle 2, the probability instead depends entirely on the measurement result obtained for particle 2. And vice versa - it seems that we actually have to say that the probabilities for the measurement results are not at all determined by the actual states of the particles, but they are only jointly and timelessly determined by God's sustaining activity.

Maybe I am misreading you and you don't intend that the definite spin state actually plays any explanatory role in the theory. But this makes one of your other criticisms of the pilot-wave theory sound like special pleading. You object that the pilot-wave theory has "redundant non-observable physical objects". I'm not exactly sure by what lights you consider the pilot-wave redundant - it is the same object as the wavefunction/quantum state in orthodox QM. Nor am I sure what is objectionable about it being non-observable - we can't observe fundamental particles either, only their effects on the macroscopic reality that we can observe. But then the pilot-wave is at most one level further removed, and it isn't any greater than the leap from (indirectly) observing particles to postulating specific quantum states for those particles. But in any case, the pilot-wave is not at all redundant - it plays an important explanatory role. Contrast that to the definite spin states in your view, which seem to play no explanatory role whatsoever.

Maybe the particles must have some spin state or another, due to the requirement of realism (though why not just let them have a definite position, and an indefinite spin state?). In that case, though, I hope you don't fault the pilot-wave theory for having a preferred notion of simultaneity, which it arguably brings in for similar considerations as you bring in definite spin states.

This is getting very long, so I will break it up and put my remaining thoughts in another comment.

--- States, Wavefunctions, Superpositions, and Bases ---

One thing that I have been confused about since I first encountered your blog is the way you talk about states and superpositions thereof. Part of my confusion, especially in your earlier posts, is that you seem to import language used in quantum mechanics (states, basis, orthogonality) back into your discussion of the axioms of probability or the different metaphysical systems we might adopt. I do not find that to be helpful. Another part of my confusion is in interpreting this language in regards to your mixed ontic-epistemic view of the quantum state. You have said "the amplitude formulation is powerful enough to both" (i.e. to be both a representation of reality and a representation of our uncertainty about reality). Now, not only does it seem to me that these two jobs should be kept separate, but I am rather bewildered about what it means to mix them together. Some of the things you say in this post bring that confusion back to the forefront of my mind.

For instance, you say "I would discuss quantum states rather than wavefunctions, but I think what Matthew means is the same thing as my state." Well, as I understand things, a wavefunction _is_ a quantum state; more precisely, it is one way of representing the quantum state, which may be more abstractly represented as a vector in a certain Hilbert space. Since quantum states are (represented by) vectors in a Hilbert space, it makes sense to talk about orthogonality or non-orthogonality of quantum states; about superpositions of quantum states; and different bases in which to describe quantum states. The concepts of orthoginality, superpositions, and bases all make reference to the Hilbert space structure, so it is confusing to me when you import that language into a different context (e.g. when discussing the axioms of probability theory).

But now I really start to get confused. You go on to say "A state is a possible configuration of the particle. A wavefunction is a superposition of states, which parameterizes (as an amplitude) our uncertainty about which state the particle is in. So states are physical; wavefunctions are (primarily) epistemic." So you distinguish between states and superpositions of states; one is ontological and the other is epistemic. But this makes no sense: a superposition of quantum states is itself another (distinct) quantum state, and as you are well aware, we can change the basis we are using to write any given state as a superposition of other states.

The key point is that your distinction between states and superpositions of states has the completely untenable implication that what is ontologically real depends on our choice of how to describe it - unless, I suppose, your theory adds some kind of preferred basis to the quantum state. But you have not indicated anything like this, and it would go beyond "taking the math literally".

--- Amplitude as Representing Uncertainty ---

You also say, "When we convert from amplitudes to probabilities, we lose, or rather combine, information that distinguishes between different states. For example, we picture possible states of a system as being the different points along the circumference of a unit circle." You seem to be saying here that quantum states have something like a complex phase, and that this is why our uncertainty about them must be parameterized by amplitudes rather than probabilities. But it is still entirely unclear to me what this means, given that the amplitude is the coefficient of the state in a superposition, and a superposition of states is itself a distinct state. I don't see how you distinguish between what is ontological and what is epistemic in your view, and without a clear distinction there I simply don't understand what your view is saying about reality or our knowledge of it. What makes the amplitude a ***representation of uncertainty*** about quantum states, rather than ***part of the quantum state itself***? What makes it epistemic rather than ontic?

Also, if there are in fact states - "physical" or "ontological" ones, as opposed to the (primarily?) epistemic superpositions of them - even if these states have something like a complex phase, it is unclear to me why we can't say "there is a certain probability for the system to have this state, and a certain probability for it to have that one..." and so on. Or, for that matter, since wavefunctions are a way of representing the quantum state, why we can't say "there is a certain probability for the system to have this wavefunction and a certain probability for it to have that one". Crucially, if there are these states, then when we try doing probability calculations without accounting for possible differences in them, then of course we might end up with the wrong answers. But this doesn't prove that ***probability*** is wrong - it just means we are incorrectly neglecting relevant information.

This goes back to my original comment on your post "Why is quantum physics so weird?" Your argument there, using the double slit experiment, was (basically) that we have to represent uncertainty as an amplitude rather than a probability, because we get the wrong answers otherwise. My objection was that this ignores relevant information - the quantum state is different if you close one of the slits or if you add a which-path detector, compared to the usual case where both slits are open and you get an interference pattern. If that quantum state is ontic rather than epistemic, the probability calculation is wrong because it plugs in the wrong numbers, not because probability itself is invalid.

So overall I think your argument that we must use amplitudes instead of probabilities is circular: it only makes sense if you already assume that the wavefunction is epistemic. Your response in your recent post where you calculate the quantum state when a which-path detector is included actually shows that my objection is right (as far as I can tell): the wavefunction is indeed different. In that case, instead, you reject the possibility that the wavefunction is ontic because of the implied non-locality. I see that rejection as misguided, because the non-locality isn't really avoidable anyways.

By the way, I haven't actually seen the way you write amplitudes as analogous to probabilities (where you have amplitudes for events, or unions or intersections of events, or conditional on other events) used anywhere else. How exactly are these defined? Are there any resources that teach this way that I could look up to understand your position better?

--- Does Bell's Theorem Go Wrong? ---

You make a distinction between two different approaches: one that calculates probabilities from amplitudes and then averages to get the probability for an ensemble, and one that averages to get an amplitude for an ensemble, and then calculates a probabiltity from it. Your claim is that the second method is what is actually used in quantum physics, while Bell's theorem assumes the first. Hence, "Bell's formulation does not match what is done in the actual quantum mechanical calculations."

Forgive me if I am rather skeptical on this point, for two reasons. Bell himself was a quantum physicist; I can only assume he was well acquainted with how things were done. So how did he get this wrong? Secondly, I have never seen this response to Bell's theorem anywhere else; how are all the physicists working in quantum foundations missing it? Admittedly this response might be out there and I have never seen it; I haven't surveyed the literature extensively. But the main responses to Bell's theorem seem to be non-realism, many-worlds, retrocausality, and superdeterminism. If things as far out as retrocausality and superdeterminism are being explored, it seems like someone else should have hit upon this as well.

I'm not sure if this is a valid counterexample to your claim, but consider this possible experiment: a double-slit experiment with electrons sent through one at a time, but in any single run of the experiment one or the other of the slits is closed. Wouldn't this destroy the interference pattern? Yet if you just add up the amplitudes for each member of the ensemble, and then calculate the probability, you would predict an interference pattern, which would be wrong.

Or consider another possible experiment: a device prepares a confined electron ("particle-in-a-box" style setup) in one of several specified energy states, but it randomly chooses which one in each run of the experiment, with probability specified by its programming and method of random number generation. Then we measure the electron's position. The device records which energy state it prepared the electron in, and we can choose to post-select the data on that information or not. Isn't this a case where it is perfectly valid to say "there is such-and-such a probability for the electron-in-a-box system to have this wavefunction"? And again, it seems you would not get the correct results by adding up the amplitudes first (I don't think that would even be well-defined, since the amplitudes of the different energy states rotate at different rates, whereas the probability distributions for the individual energy states is independent of time).

All of this makes sense to me if the quantum state/wavefunction is (or represents) something ontologically real, and we can talk about probabilities for those ontologically real states (and probabilities for future states implied by them). But none of it makes sense to me when you say that we can't use probabilities to represent our uncertainty, and must use amplitudes instead. I just don't know what that means.

Thanks for your time; best regards!

"However, unlike the Copenhagen interpretation, where the particle remains as some ghostly non-physical wavefunction between measurements, I would maintain that it is in some definite state, we just don't know what that is."

After reflection, I have to agree with Matthew here: postulating that particles always have some definite state - meaning a pure state in some basis - really adds nothing to your interpretation.

Take the case of particle decay, and suppose the detectors are set to measure spin in the same direction A. Unless the fermions' hidden spins happen to point along A, measuring one particle's spin changes its spin direction to A (either up or down), which means the other particle must change to have the opposite spin (or else angular momentum isn't conserved.) That gives no advantage over the Copenhagen interpretation, where measuring one particle puts both into some definite state. Both ways, you still have both particles changing at the moment one is measured, and the same issues with defining "the same moment" under special relativity arise. The sole difference is that Copenhagen allows the zero state (which has no clear direction) to be an actual particle's state, which you would disallow.

The reasoning of your interpretation (the particles always have opposite spins if measured in the same direction because angular momentum is conserved because the action of the true physics is unchanged if the coordinate axes are rotated because God doesn't prefer one direction over another without special reason) still holds up without assigning a definite state to particles between measurements. So what's accomplished by doing so?

The other big problem with Copenhagen interpretations, the need to distinguish macroscopic objects from quantum particles, is I think resolved by hylemorphism; any substance whose matter is quantum particles defines a basis of states for particles it interacts with, by virtue of its form. If resolving that is the reason to assume that particles always have a pure state in some bases, then I don't think it's needed.

Thanks to Matthew and Michael for your comments, both here and on other posts. I'm tied up with other things this week, but I'll respond as soon as I can.

1. "I would maintain that it is in some definite state". Then the future is fixed and you should be a super determinist. Copenhagen says that the future state of the universe does not physically exist until it is encountered. That is, the future is being flexibly created (but connected to the past via entanglement) as we move through time.

2. "then there must be instantaneous communication between the two experiments". There is no instantaneous communication. The communication happened at the point of entanglement. Entanglement doesn't fit our intuition.

As I mentioned elsewhere this week, "The philosophy of quantum mechanics is primarily concerned with making Nature fit our intuitions, instead of making our intuitions fit Nature."

Then the future is fixed and you should be a super determinist. No. Indeterminism comes in with events. Particle A can decays into B, C, or D. We can't predict, based on our knowledge of particle A (and the state of the rest of the universe at that time or earlier) which of these states it will decay into. But each of B, C and D are a definite state. We don't know which state that is without experiment, or often even after experiment. But it still remains a definite state. That, to my mind, is the most natural interpretation of the mathematics.

The communication happened at the point of entanglement. I would very much like to agree with you. But the problem is that we have to confront Bell's theorem and the related theories. What you are advocating is essentially a hidden variables theorem. The particle must in some sense carry the results of every possible measurement from the decay to the measurement. But how are those results contained within it? If it is in the form of a quantum amplitude (as I maintain), then you have indeterminate orthogonal states, and some measurements will be undetermined until the event actually happens. If they are conveyed as exact results (i.e. if you make this measurement, you will get this result, if you make that measurement you will get that result, and so on for every possible measurement), then you are advocating precisely the sort of hidden variables theorem that Bell disproved. If you have any other suggestions, then I would love to hear them.

"The philosophy of quantum mechanics is primarily concerned with making Nature fit our intuitions, instead of making our intuitions fit Nature." Not entirely. The philosophy of quantum mechanics is difficult, because at some point everyone has to draw up at least one premise which is unintuitive. But everyone who does it does so with the intention of describing nature. Different people will pick a different unintuitive premise, and that choice will no doubt be partly based on their prior prejudices. But to be judged seriously, they need to find a bridge between the philosophy and the physics, and make predictions, as precise as they can make it, based on their philosophy on what the physics ought to be. That can then be tested. At the end of the day, if there is more than one philosophy which makes identical correct predictions for the theory, then we will have to have some other means to judge between them. But the work is not based on prejudice; everyone who does it tries to be guided by nature.

Matthew,

Thanks once again for your comments. Now my schedule is a bit less hectic, I'll try to respond to them gradually as I get the chance.

I'll begin with your discussion of amplitudes as a measure of uncertainty, since that strikes me as being at the heart of our disagreement.

I should say at the outset that my notation is my own, but the calculations I use that notation for are standard. It is just a different way of expressing standard physics. I'll get back to that in a moment. But first of all, I need to discuss what I mean by an amplitude.

You stated in your response

But it is still entirely unclear to me what this means, given that the amplitude is the coefficient of the state in a superposition, and a superposition of states is itself a distinct state.

But that is not entirely what I mean. By an amplitude, I mean the overlap between two distinct states. The coefficient in a superposition could be treated as an overlap between two states - if we have a basis state φ and a wavefunction ψ, then the amplitude would be <φ|ψ>. Your point that we can rotate the basis of ψ is perfectly true from one point of view, but I am discussing things from a different perspective. To me, φ represents a physical state rather than one state in an arbitrary basis. Perhaps φrepresents one possible outcome of a measurement. You can rotate the basis so that ψ is no longer in superposition. But, for consistency, you must express everything in the same basis. That means if you rotate one wavefunction, ψ, which represents something physical (namely the state of the particle or our knowledge of the state of the particle depending on how you interpret it), you must also simultaneously rotate the basis of every other physical state in the system. This includes φ (which is physical, since it represents a possible measurement value). Thus after rotating the basis, the quantity of interest, <φ|ψ> remains unchanged. Mathematically, the rotation of ψ is given by

|ψ> -> U |ψ>,

where U is some unitary operator, so

<φ|ψ> -> <φ|U^dagger U |ψ> = <φ|ψ>

When I discuss an amplitude (in the context of this discussion), I mean something along the lines of

<f|S|i>, where i represents an initial state, f the final state, and S some evolution operator that one possible path from the initial to the final time. (If the initial and final state are at the same time, or the state doesn't change over time, S can be the identity operator.) Suppose that there is only one path from the initial to the final state. |<f|S|i>|^2 is then a probability, and we both agree that a probability can be interpreted as a parametrisation of uncertainty. So why can't we also say that the amplitude <f|S|i> can be interpreted as a parametrisation of uncertainty? How does taking the mod square of a number change its interpretative status? When we have more than one path, we agree that |Σ_S <f|S|i>|^2 can be interpreted as a parametrisation of uncertainty. So why not Σ_S <f|S|i>? And if adding together a bunch of numbers gives us a parametrisation of uncertainty, then surely each individual number will fulfil the same role, especially since there is the natural interpretation that it represents the uncertainty of reaching the final state from the initial state via that particular path.

So onto my notation. I will discuss equation (13), which involves the intersection operator (∩). ψ(A ∩ B|Q) represents the amplitude that you have both measurement outcomes A and B given initial conditions Q (albeit that I expanded Q to indicate both the initial conditions and the assumptions behind the model). This would normally be expressed as <A|Q> <B|Q> if A and B are treated as independent (i.e. different measurements, or different paths etc.). I introduce a set of intermediate states Y, which together form a complete basis. Thus ∫_Y |Y><Y| represents the identity operator. So I have written

ψ(A ∩ B|Q) = <A ∩ B|Q> = <A ∩ B|∫_Y |Y><Y|Q> = ∫_Y <A ∩ B|Y><Y|Q>.

Obviously equation (13) is a bit more complex than this -- these are fermions so one has to watch out for possible minus signs -- but it is the standard formula just written in a different notation, one I choose to draw out the analogy with the earlier probabilistic calculation.

The union operator (∪) represents the or statement, so ψ(A ∪ B|Q) represents either final state A or final state B given initial conditions Q. The basic rules of quantum mechanics (again, with certain assumptions about independence) state that if we want to know the amplitude for either A or B given initial conditions Q, then we add the two amplitudes together. Thus

ψ(A ∪ B|Q) = <A ∪ B|Q> = <A |Q> + <B|Q> = ψ(A|Q) + ψ(B|Q).

True, I made the mistake of inventing my own notation, and not explaining it. Thanks for calling me out on that. Hopefully this explanation will help. But my calculation itself is standard. It is just expressed in a slightly idiosyncratic way. (Nothing wrong with that: it is often worth looking at the same calculation in a different way, because you can gain other insights from it.)

Anyway, let me know if you found the above explanation about what I mean by an amplitude helpful. If not, then we still have some discussion to do.

Hello Dr Cundy,

Thank you for your efforts in synthesizing Aristotelian-Thomistic Philosophy with Quantum Physics.

I wanted to ask if you could name a few summary books for a layman to develop enough of an idea to grasp and communicate:

1) A grasp of the pertinent principles of Quantum Field Theory/Mechanics

2) A modern summary of Aristotelian/Thomistic Philosophy/Metaphysics.

I want to purchase your book but I'm afraid I don't have the depth of knowledge to engage with it meaningfully let alone communicate the concepts to my peers; and would like to develop my understanding first.

For 2), I would recommend the works of Edward Feser, which are (I think) reasonably accessible. Start with his Aquinas, and then move onto his Scholastic Metaphysics. You can also dive into his excellent blog. Peter Kreeft has also written numerous works at an accessible level: perhaps his Shorter Summa world be a place to start.

For 1), it is a bit harder because even the layman books are perhaps still a bit hard to follow (and also I learnt the subject mainly from the academic textbooks so don't have that many popular works on my bookshelf). I have always had a soft spot for Richard Feynman's works; in particular his QED is aimed at the layman. Frank Close "The infinity puzzle" is also highly recommended.

For academic textbooks (if you can cope with some mathematics) with an approach to quantum physics similar to mine, I would recommend Binney and Skinner's work for an intermediate level (Townsend's book as a less modern alternative), and then Peskin and Schroeder for the advanced stuff. There are several decent more conventional starting textbooks; Griffiths' book is frequently recommended.

Okay, so your clarifying remarks make it seem to me that when you are thinking of an amplitude as a representation of uncertainty, you have in mind not just any basis, but a basis made up of what you call "physical states". The divide between what is ontological and what is epistemic in your view is determined by what these "physical states" are - is that a fair assessment?

If that is so, I think there is a serious problem with choosing a possible measurement outcome to be an example of one of these physical states, namely, that measurement is a highly anthropocentric concept. This is why the usual interpretation of QM comes across as instrumentalist to me: a theory for calculating probabilities of measurement results, not a theory that tells us the fundamental nature of reality (about which it really says nothing). If outcomes of measurements are among your fundamental "physical states," that seems to be tantamount to saying that there is no microscopic reality, or that observation creates reality, or such as that.

The converse is that if you want to say that your physical states are really a fundamental representation of reality, then you need a "privileged basis" which picks out the set of states one of which is physically real all the time (and about which the quantum state is a representation of our incomplete knowledge, in your view). More generally, you would need some kind of rule that picks out the set of physical states - maybe the rule depends on the Hamiltonian or on which state is presently actualized, but it really should not depend on things like measurement or observation. (I mean, maybe you could justify the dependence of the physical basis on things like measurement or observation by reference to the top-down features of Aristotelian metaphysics (e.g. form), but doing so really seems to make the status of microscopic analyses of things somewhat questionable. That's an area of Aristotelian approches to the philosophy of science that I don't understand well.)

Now if you endorse some kind of privileged basis as the set of states that can be actualized, one of these states as actual, and the quantum state as epistemic when expressed as a superposition of the basis states, then - well, I unfortunately I think I still don't know how to conceptualize your view. I was going to say I could -- sort of -- get it, but then I thought about it some more, and realized I was still conceptualizing the quantum state as more like an objective representation of the potentialities of the system, rather than a representation of our knowledge.

You (basically) asked me why a complex number can't be a parameterization of uncertainty if we both agree that the mod-squared of a complex number can be. I suppose my answer is that mods-squared are real (1-D) and complex numbers are complex (2-D). I know what it means to be more or less certain about something; I don't know what it means to be certain-in-the-imaginary-direction. I don't know how to interpret the phase information in the context of uncertainty.

(This interpretational problem doesn't arise for me if the quantum state is ontic rather than epistemic. In that case, we can just say that the objective state of the system has this phase information, or something that the phase information represents. I can sketch out, roughly, what that means on the quantum-theory-without-observers type interpretations that I am familiar with - various takes on pilot-wave theory, objective collapse, or many-worlds. But if the quantum state is epistemic, the phase is supposed to be representing something about our knowledge, and it raises the question of precisely what it is about our knowledge that it captures.)

Thanks for getting back to me, Matthew; and for this conversation. It helps me to think about how best to explain things and clarify things -- it is a difficult topic, and difficult to express oneself clearly.

Okay, so your clarifying remarks make it seem to me that when you are thinking of an amplitude as a representation of uncertainty, you have in mind not just any basis, but a basis made up of what you call "physical states". The divide between what is ontological and what is epistemic in your view is determined by what these "physical states" are - is that a fair assessment?

Not quite. As a physicist, I am going to discuss things in terms of measurement. Ultimately the goal of theoretical quantum physics is to predict experimental results, so that is how we are trained to think and express ourselves. So my apologies if I discuss measurement and experiment when I should be more general (even if I do so). When I say measurement, you should not read it as necessarily meaning the outcome of an experiment; but any circumstance where decoherence occurs and the system is forced into one state or another of a particular basis, whether or not there is a scientist at the end of it, setting the system up and reading off the results. There is some physical process that happens inside an experimental detector. The same physical process will happen in other circumstances. I discuss measurement and experiment because that's the context in which I am used to thinking about things.

Also, anyone who takes a wholly or partially psi-epistemic view of quantum physics is going to value experiment. Measurement is the point where our knowledge and reality coincide.

I don't believe that observation creates reality. There is reality between experiments. But we don't know what that reality is unless we perform an observation. But the process of observation -- and it is not the only thing that does this (for example the same effect would occur if a particle went into a detector but nobody was looking and the result wasn't recorded) -- involves forcing the particle to change into one or another state in a particular basis, a basis that is determined by the set up of the experimental detector. Observation doesn't create reality, but it does change reality when we interact with the particle we observe, and decoherence does its thing.

But when I talked about a physical basis, I had something else in mind. When we first start to construct a theoretical system, the basis we use is entirely arbitrary. There are many different formalisms which could each equally well model reality. We can choose whatever we like. We can also in principle change it midway through the calculation. For example, we can use any hermitian traceless two dimensional matrix to represent the operator whose eigenstates correspond to definite spin states along the x axis.

But our goal is not just to come up with some nice theoretical framework. Our goal is to understand things about reality. Thus if we use one particular operator to express the spin along the X axis at the start of the calculation we need to be consistent in our usage throughout the calculation. As soon as we start applying the calculation to the real world, the formalism becomes fixed. We no longer have the freedom to change it. Now, being consistent doesn't necessarily mean that we stick to the same basis throughout the calculation. There is often value in rotating the basis to make the next part of the calculation easier (for example, to rotate to an energy-momentum basis when we want to consider evolution in time). But when we do so, we have to be consistent, and rotate every operator and state in the abstract formalism in a similar way. My point was that if we had a wavefunction |ψ>, and wanted to take one particular component of it with reference to some basis vector |φ, so

<φ|ψ> = c, we can't just arbitrarily change the value of c by rotating the basis of |ψ> while keeping |φ> constant. We have started the calculation; we have already established a mapping between all our symbols and something they represent in the real world. Everything in the theoretical framework has something which exists or could exist in the real world corresponding to it. That something might be a physical particle, or a basis representing the possible outcomes of an experiment, or the possible outcomes of a possible experiment (which nobody performs), or the outcomes of when a quantum system happens to interact with something which forces it to decohere and collapse into a particular basis but isn't an experiment. We can write down a symbol to represent each basis, but my point was we have to be consistent in how we do it. Once we have the mapping between abstract states and reality, we have established our formalism; we cannot just arbitrarily change to a different formalism. We are still free to change the basis in the middle of our calculation, but we have to do so in a consistent way. We have to apply the same rotation to all the symbols in the calculation. Thus c will remain constant after such a rotation.

How can we consider something two dimensional as a parametrisation of uncertainty?

Firstly, as I stated in my original post which sparked our discussion, quantum physics is not intuitive. We can visualise Aristotle's physics. We can visualise Newtonian physics. With Einstein's physics (by which I mean relativity) it gets harder, but it still roughly follows our intuitions. Of course, we do better in each case to represent our images mathematically, but we can still back up that mathematics with a mental picture. With quantum physics, that is impossible. We can understand the mathematics, but we are not capable of formulating an accurate mental picture. It is not possible to understand quantum physics in terms we are used to, and anyone who thinks they have understood it, hasn't (as the saying goes). So in any philosophy of quantum physics there is going to be something unintuitive; something we can't visualise, which defies our common sense. In my own interpretation, this is that point.

But let me try to offer a picture, anyway. I agree the amplitude is two dimensional, and the probability one dimensional. You would have to agree that the radius of the amplitude can be used to parametrise uncertainty, since it is in a one-to-one relationship with the probability. The point where I disagree is the assumption that statements of uncertainty are necessarily one dimensional. If we have a single proposition, then I agree that a line segment would be appropriate to parametrise our uncertainty. The proposition can be either true or false, and our belief about it would lie somewhere between these two values.

But what if, instead of a single proposition, we have a set of related propositions? Maybe they depend on some parameter X, a continuous variable. Our understanding of whether each proposition is true will depend on X. So we might want to represent all of this together. So the parametrisation of our uncertainty would now be represented by a rectangle, with the vertical axis representing our certainty, and the horizontal axis X. Rather than a point on a line segment, our uncertainty of the whole system would be represented by a line on a plane.

Now suppose that X has some circular symmetry, so X=0 is equivalent to

X = 2 &pi;. In this case, we would do well to parametrise our uncertainty by a circle. The degree of uncertainty is mapped to the radius of the circle, and the parameter the phase. Our uncertainty for the whole system would be represented by a line around that circle.

Thus far we have a picture which is consistent with classical physics, and which we can imagine. Our uncertainty is parametrised on a two dimensional surface, albeit by a line rather than the point we have in quantum physics.

Now suppose that X, rather than just some random parameter, is also something that takes on a physical value. We know that some value of X is the true one. We might not be certain which value of X is correct -- it might change in time, but some value is correct. So the original uncertainty -- if X = X₁, then our degree of confidence in the proposition is C(X₁). The circle as a whole represents the entire general picture, it allows us to see our uncertainty for every value of X simultaneously. However, a single point on the circle allows us to represent two statements simultaneously: X = X₁ and our degree of certainty in the proposition is C(X₁). Of course, if we are also uncertain about X, then we might represent our knowledge of the situation as a small line segment, or add an extra dimension to display that uncertainty.

We need to keep the whole circle during the calculation because we don't know what the value of X is until the last moment.

So the radius of the circle represents our uncertainty; the phase of the circle represents an internal parameter of the system (in the case of quantum electrodynamics, the gauge). This is now where quantum weirdness comes in. The X are not independent parameters, as they would be in the classical picture. Instead, we have the superposition principle. Thus when we combine two different amplitudes (as in the or case above), we not only change the radius, but also adjust the phase of the particle.

In this sort of problem, we are chiefly interested in predicting the future. Let me offer a simple example.

Suppose that we denote our initial state as |φ>, and we work in a basis of orthogonal states |φ_i> (which corresponds to some observable; an eigenstate of the operator representing the outcomes of some experimental detector). We evolve the system for a certain amount of time, and then take another identical measurement. If |φ₁> is an energy eigenstate, then it evolves according to e^{iE₁t/&hbar;}|φ₁>. If the original amplitude at time 0 is c₁ = <φ|φ₁>, then the amplitude at time t will be

c₁e^{iE₁t/&hbar;}. So the phase (or X) changes in time, with its rate of change depending on the energy state. The probability is always c₁².

On the other hand, if the initial state is a superposition of two energy states ( cos θ |E₁> + sinθ |E₂>), then the amplitude would be something like

c₁( cos^2 θ e^{iE₁t/&hbar;} + sin^2 θ e^{iE₂t/&hbar;}). The probability changes in time. (c₁^{2 (cos4θ + sin4θ + 2sin2θcos2θ cos (E₁ - E₂)t/&hbar;)}) The system no longer remains in an eigenstate of the original basis. (Again, we have already established the mapping between every symbol and something in reality, whether a particle, or a set of possible experimental outcomes, or some other decohered system if you don't want to privilege experiment. We choose all our basis vectors such that in the initial state there was no phase. We have to be consistent, and either stick to the same basis throughout the calculation, or rotate them all in the same way.) Thus when we repeat the experiment with the same or an identical detector and the same particle but at a later time, we could get a different result.

The phase is the crucial thing here. It is an essential part of the description of the quantum state, but it is not determined by the underlying basis. Energy eigenstates change in time by a gradual rotation of the phase. Only relative phases can be measured, not an absolute phase (we can derive the zero point to be whatever we choose). But they are nonetheless key in understanding how states change in time. So when we specify a result related to a quantum state, we need to express the basis, the degree of certainty that it is in that state and that phase.

So when we come to express our degree of certainty, we can't just say that "my degree of certainty that the system is in state |φ> is c₁" because just listing |φ> is not sufficient to parametrise the physical state. We need to convey three pieces of information: the state, the phase, and the degree of certainty. A probability is a mapping between two pieces of data: a classical state and a degree of certainty. We could only use probabilities to express our uncertainty if we treated each quantum state and quantum phase together as a classical state, and then assigned a degree of certainty to each one of these pairs of data. But this would not give the correct result if we tried to combine the probabilities of two different states. The amplitude, on the other hand, maps two variables to the state, the phase and the degree of certainty. When we combine amplitudes we get the right result.

When we think about a degree of uncertainty, we need to know three things: 1) The set of possible states (which in some way will reflect the physical system of interest, and will differ from one practical usage to another); 2) our mathematical formulation that expresses the degree of certainty and whatever other information we need to know to process the system (which is an abstraction; expressed as a set of numbers; and will be a generalisation applicable to numerous different usages), and 3) how to combine degrees of certainty across different states. Without 3), our parametrisation is useless if we want to do any theoretical work. In the mathematical formulation of probability theory, 3) depends only on the information stored in 2); that's a built in assumption of the mathematical framework of probability (Kolmogorov's third axiom). In classical probability theory, everything we need to know about the system in order to perform calculations is contained within the data in 2) which represents our degree of uncertainty. And that is a useful assumption to have. We are trying to create a general framework for discussing uncertainty; we don't want to have a different set of mathematical rules we need to apply to each individual physical system, but a method we can adopt generally. But in quantum physics the probability (or its square root) is not sufficient to perform calculations, combine different degrees of certainty and so on. We need to know the phase to know how to combine degrees of certainty for two different events. 2) must therefore contain both information about both the degree of certainty itself and the phase. This allows 3) (and any subsequent calculations we make) just depend on our base abstraction, and allows it to be generalised. The amplitude conveys all the information we need to do calculations in quantum physics, and does it well. In a classical system, the phase of the state would be contained in 1) alone. In a quantum system, it needs to be included in both 1) and 2), otherwise we wouldn't get the correct results.

So the amplitude in quantum mechanics plays the same role as the probability in classical probability theory. 1) it provides us with a numerical parametrisation of the degree of uncertainty for irreducible states (which just needs the radius) or results from individual paths; 2) it provides us with the information we need to combine results from irreducible states to get a degree of certainty for a set of states (which needs both the radius and the phase); 3) it allows us to extract a numerical representation of the degree of certainty for these sets (both the radius and the phase). In classical probability, the single real number provides all we need for all of these. In quantum physics, we need both numbers. The probability isn't just a parametrisation of the degree of certainty of irreducible systems; it also provides us with all the information we need to perform calculations and deduce degrees of certainty for more complex systems according to a set of well defined rules. For the quantum amplitude, the radius alone is sufficient for the first role, but we need both the radius and the phase to fulfil the second role of the probability in providing us with the information we need to express degrees of certainty for more complex systems.

Why should we believe that Kolmogorov's axioms (or Cox's equivalent axioms) are the only way to parametrise uncertainty? I know of no proof that there is something special about those axioms; and that we cannot choose others to do the same job and build up a mathematical framework on top of them. Kolmogorov's system would hold no theoretical advantages over that other system, aside from the tie in with frequency.

To go back to my example, we have an initial state, and various states it could be in. Our degree of certainty that it is in a particular state can be expressed as c₁, which is just a real number which (at this point) we have the freedom to define to be positive. Zero represents falsity; 1 represents truth, albeit different expressions of truth. c₁ isn't a probability -- it doesn't obey Kolmogorov's third axiom -- but it is in a one-to-one relationship with the probability. But when we want to investigate how this degree of certainty evolves in time, we have to invoke the principle of superposition. There is a second question we could ask: which energy state is it in. Let's say that only the states |φ₁> and |φ₂> both contain the energy eigenstate |E₁>. The degree of certainty that we will measure it to be in state |E₁> would then be c₁ cos θ - c₂ sinθ. Again, there is no phase information -- this is just a single real number -- but again this is not a probability, in this case additionally because it could be negative. c₁ represents a degree of certainty. c₂ represents a degree of certainty. But adding them together doesn't?

Then, of course, when we evolve them in time, we introduce additional phase information. But the combined numbers have the same interpretation as at time equal to zero: they tell us how likely we are to find the state in a particular state. What other interpretation can we give to them, in a psi-epistemic view of quantum physics?

So what part of our knowledge does the phase capture? Ultimately, it is related to the physical gauge of the particle. In classical probability, we abstract out all information about the state of the particle from the number we use to represent the probability. We don't need to know anything about what the state represents. We can still do probability calculations. In quantum physics, we need to bring one additional piece of physical information into our generalised representation of uncertainty. It is not redundant. The radius gives us the degree of uncertainty (as you think of it); but the radius alone is not sufficient because it does not tell us how to combine different amplitudes. We need an extra bit of information to do that, the phase. And the phase and radius together are required if we are to perform any calculations in a quantum mechanical system, whether combining two amplitudes to get a total result in a system with interference effects, or looking at different observables.

I think it would help in explaining your viewpoint to go into a bit more detail on the divide between what is ontological and what is epistemic. You write,

"I don't believe that observation creates reality. There is reality between experiments. But we don't know what that reality is unless we perform an observation."

So I get that it is part of your view that we can't know what is happening between observations, and there is the further complication that "observation" is actually an interaction that can change the state of the system being observed. But you do think that the system is in some "physical state" in between observations, and that the wavefunction or quantum state characterizes our knowledge about the different possible states it could be in. So there is some set of possible "physical states," which are ontological, and superpositions of those physical states (different possible quantum states that we could write) are epistemic. Is that right? If not, I don't know what you mean when you say that there is reality between experiments, but it isn't just the quantum state (since that is epistemic and only represents our knowledge).

So what are those physical states? In response to my comment about the human-centeredness of the measurement concept, you write,

"When I say measurement, you should not read it as necessarily meaning the outcome of an experiment; but any circumstance where decoherence occurs and the system is forced into one state or another of a particular basis."

And,

"The process of observation involves forcing the particle to change into one or another state in a particular basis, a basis that is determined by the set up of the experimental detector."

So you seem to be using the process of decoherence in some way to distinguish the possible (ontological) "physical states" from the (generally epistemic) quantum state. Now, decoherence is something that happens to the quantum state, which is epistemic. So it can't be the process of decoherence itself that picks out the physical states; rather, it would have to be something like the structure of the Hamiltonian of the system that induces the possiblity of decoherence.

Now, I presume you want to say that experimental detectors and other such things are themselves made up of quantum objects. (It's possible to go a different route, and use the top-down features of Aristotelian metaphysics to say that it is macroscopic objects that exist fundamentally, while the particles that make them up exist only virtually. But I don't know how to make such a concept scientifically precise.) Since these detectors are made up of quantum objects, in principle our analysis ultimately is going to arrive at the quantum fields or particles that make up both our system of interest and the detectors we are using to interact with it. And that means that the Hamiltonian that determines the possibilities of decoherence (and thus picks out the "physical states" as opposed to the epistemic quantum state) is just the Hamiltonian of the Standard Model (or whatever theory ends up superceding it).

Correct me if I'm wrong, but because the Hamiltonian of the Standard Model is local (I would prefer to phrase it as "it is the quantization of the Hamiltonian of a local classical field theory," but that is another discussion - please see my latest comment on the other post), roughly speaking, the basis that decoherence occurs in is the position basis. Bohmian mechanics, for example, relies on this fact and notes that all measurements ultimately come down to observations of the position of something (e.g. position of a pointer, or spot on a photosensitive screen) to show how its actual particle positions, guided by the wavefunction, solve the measurement problem.

So it seems to me that your view should follow this line of reasoning:

-There is some actual physical state. (I take this to be implied by what you've written.)

-The set of possible physical states is determined by the possibilities resulting from decoherence. (Ditto, along with the fact that it can't be the actual occurrence of decoherence, because that is something that happens to the epistemic quantum state.)

-The possibilities resulting from decoherence are determined by the Hamiltonian.

-The revelant Hamiltonian (coming from the Standard Model or whatever supercedes it) produces decoherence in the position basis.

-Therefore, the set of possible physical states are those states forming the position basis.

So when you say that there is some objective physical state and the wavefunction characterizes our knowledge about what it is, to be more precise, it seems you should say that (more or less) there is some actual number of particles with definite locations, and the wavefunction/quantum state characterizes our knowledge about the number and locations of those particles. (Incidentally, this is a decent argument that we should take the ontology underlying QFT to be one of particles rather than fields, though I think it may be possible, again roughly speaking, to replace "particles" with "localized field states".) Maybe you would posit that the actual state includes some kind of quantum phase as well.

What are your thoughts on that? I'll come back to continue our conversation about amplitude and uncertainty at a later time...

The first thing to remember is that we don't have access to the physical states. All we have access to is the abstract representation we use to represent them. To be successful this representation will need to be in a one-to-one relationship with the physical states. [Admittedly, I have been sloppy in consistently using this language here.]

So what is the representation of the physical states? The space on which they live is the same as we use in quantum physics. So, for example, for an electron, that space would be represented by a two dimensional complex spinor on each location in space-time. For a quark, you would additionally need three indices to represent the colour. [Plus a similar number of degrees of freedom for the anti-particles.] Any self consistent set of orthogonal eigenvectors mapping this space will suffice as a basis. Any one of these eigenstates could in principle represent a physical state. The two most useful bases are, of course, that formed by the eigenstates of the Hamiltonian operator and that the location basis. [Of course, these only constrain some of the degrees of freedom: spin operators generally commute with the Hamiltonian operator or the location operator, so saying that we have chosen the location basis isn't sufficient to constrain the system; we also need to specify the spin basis.] And, of course, we immediately have superpositions: if the particle is represented by an eigenstate of the location basis, then it will be in a superposition of basis states in the Hamiltonian basis. All we know is that the physical state is represented by some state in some basis. We just don't know which one, outside of measurements. When we want to make predictions, as in my calculation in the main post, we compensate for our lack of knowledge by integrating over all the possibilities.

This is what I mean when I say that the wavefunction is part epistemic and part ontic: the same mathematical framework is used to describe both of them.

You touch onto the next issue, which is how do more complex objects fit into this system. The way I look at this is through the lens of effective field theory. To use the example with which I am most familiar, a proton is a combination of quarks and gluons. If we want to study the proton, we need to understand its energy band structure, which means we need a Hamiltonian operator that represents the interactions between bound quark states. This Hamiltonian operator will again be constructed from creation and annihilation operators, but this time they will represent the proton, the neutron, and so on. How do we get from quark and gluon creation and annihilation operators to proton and neutron operators? Through another rotation of the basis, only this time the rotation will merge together the original quark, gluon, photon, electron and so on operators. We will get an entirely new basis of operators. It is not just the sum of the parts; while we say the proton is composed of three quarks, this is a little misleading. The transformation is not a straight forward linear combination. (In practice, what people do is use symmetry principles to constrain the various terms which could contribute to the effective Hamiltonian, and then compare with experiment or first principles calculations to determine the coefficients of the expansion. The effective Hamiltonian is considerably more complex than that of the elementary particles). One cannot look at the operators for the proton, and point out the individual quarks.

Moving from quark to proton is one level. But the same process can be used to move up to the nucleus as a whole -- we have a Hamiltonian that is a further rotation of the basis from the effective Hamiltonian representing protons/neutrons. We can calculate the discrete energy levels of this Hamiltonian. Then we go onto the atom; then to molecules or the entire crystal. In each case (although it gets harder with each step; and the effective Hamiltonian for protons and neutrons is difficult enough) we get a new Hamiltonian, with creation and annihilation operators mapping between different states, and new "fictitious" particles such as phonons which carry energy and momentum around in a similar way to which photons do in the standard model. Again, you can't pick out specific nucleons in the Hamiltonian for a metallic crystal; even though they are in some sense present. This is how top-down causality works in physics; and I think there is a clear analogue with the Aristotelian notion of virtual existence.

So back to your question about the detector. The detector is obviously some large and immensely complex thing, made up of macroscopic parts. Each part will have its own effective Hamiltonian, which presumably sit alongside each other in some way (apologies for the vague language: the details are way beyond my ability to calculate). To simplify things, let's say that the detector is governed by a single effective Hamiltonian. This is not the same as the standard model Hamiltonian. Now into this comes our poor little free electron. The electron is not described by the same basis as the effective Hamiltonian describing the detector. The electron is described by a single particle Hamiltonian; the eigenstates of the effective Hamiltonian mix together all the fundamental particles. To describe interactions between the electron and the detector, when we do the calculation, we would have to split the electron up into a superposition of states in the same basis as described by the detector's effective Hamiltonian. [In practice, what we get is a new effective Hamiltonian for the detector + electron system.] For example, if the detector is used to measure spin along a particular axis, then some of these eigenstates will be spin up along that axis, and others spin down along that axis. Unlike the free Hamiltonian, which commutes with every spin operator, the effective Hamiltonian of the detector only commutes with the spin operator along one axis (this is the property that allows it to be a detector of spin). Spin states in the other directions are forbidden. The electron is forced into either a spin up or spin down eigenstate. Which one is chosen in practice us indeterminate, and the likelihood for each one will depend on the coefficients of the superposition. But it is always in a possible physical state, so interactions between the electron and the macroscopic system will force it into an eigenstate of the appropriate spin operator. This interaction between a single particle and macroscopic system is, of course, just what people refer to as decoherence; only I'm looking at it from a slightly unusual perspective (I would say a more fundamental perspective).

I think this is broadly consistent with your five steps, but maybe differing in some of the details. The main quibble I have with your steps is that "the position basis" is not sufficient to describe the particle. One needs to also specify the spin (and perhaps colour) state as well (plus whether it is a particle or anti-particle). Secondly, I also disagree that the Hamiltonian is just the standard model Hamiltonian. The standard model Hamiltonian describes the energy states of free particles (energy states being those which are meta-stable in time). When we come to compound objects, we need to use an effective Hamiltonian. This is a different basis (mixing together the quarks and gluons), with a different set of (meta-)stable eigenstates. But although it can be derived from the standard model Hamiltonian (in principle at least), it is not the same as the standard model Hamiltonian, and has different allowed energy levels and different stable states.

It does seem that representing both uncertainty and indeterminism with amplitudes (complex values of norm <= 1) instead of probabilities (real values of norm <= 1) resolves some of QM's puzzles, like interference, where we're looking at one event at a time. But I don't think it solves the puzzle of entangled particles, where we deal with two or more events at once.

To see this, let's define Amp(A|B), by analogy with P(A|B), as the amplitude for event A given that event B occurs, and assume Born's rule that P(A|B) = |Amp(A|B)|². Now a causal theory which requires that Amp(b₁|B₃, b₂) = Amp(b₁|B₃) also satisfies Bell's definition of local stochastic causality - just take the squared norm of both sides. Working in amplitudes instead of probabilities doesn't make the theory non-local in Bell's sense.

The issue, as I see it, is that the assumptions you are making seem to require that the amplitudes of entangled particles are factorizable, which we know is not the case. In the specific case of two spin-1/2 particles generated by the decay of a spin-0 boson, for instance, if we assume that the amplitude of measuring a particle's spin as "up" is fixed just by the relative angles of the detector and the hidden spin state of the particle, the equation Amp(b₁|B₃, b₂) = Amp(b₁|B₃) holds (the hidden spin is part of B₃) and Bell's inequality must hold too. To match QM's predictions (and experimental data) that equation must be violated.

In other words, even if we express hidden state in terms of amplitudes, Bell's theorem still applies.

Thanks Michael for your comment - it is certainly food for comment. Also, I agree that my interpretation was born out of considering interference effects (and particularly the path integral interpretation), and needs to be applied to entanglement.

A couple of comments on your comment, though:

Born's rule is applied when we have an ensemble of "paths"; the amplitude refers to a single path. Thus the expression should be:

P(A|B) = | int dX Amp(A|BX)|^2

where X represents the various internal variables which we don't know. If there were a single amplitude contributing to the probability, then your expression would be correct. As I stated, the proofs of Bell's theorem that I have seen rely on an integration over the hidden variables after you have calculated the probability, in effect.

P(A|B) = int dX | Amp(A|BX)|^2

So I think you are misusing Born's rule in your comment (although I am not sure that this niggle overly affects your argument). But this is important in the analysis of the effects of hidden variables; in the precise derivation of Bell's inequalities.

Secondly, my interpretation falls neither into the neat psi-ontic or psi-epistemic categories. I recognise that the fundamental states of the particle are best represented by the states used to build up the amplitude (which would be psi-ontic), but also that our representation of our knowledge of the situation is also represented by the same notation. So if we somehow had a perfect knowledge of the internal configuration of the particle, we would represent that by a wave-function. But if we have an imperfect knowledge, we can also use the wavefunction notation to express that (and, once we have an ensemble of results, can convert to a probability distribution to compare against an experimentally observed frequency distribution.) In particular, when we predict the future, our use of the wavefunction is more epistemic. When we make predictions about entangled particles, we ought to make predictions for those particles together, rather than individually. We know that if we measure the spin of the first particle along some axis as a, and the spin of the second particle along some other axis as b, then a and b are not independent. Amp(a|Bb) is used to indicate the amplitude (or our uncertainty) for result a given our knowledge of the background conditions B and that the other particle recorded a result b. In other words, Amp(a|Bb) is not equal to Amp(a|B), but this doesn't imply that anything physical passes between the two particles, because in this case the amplitude is to be interpreted epistemically rather than ontically. It is not a breakdown of locality, in the context that is in special relativity, that some physical particle or signal travels faster than the speed of light.

Thirdly, I do believe that there is a hint of non-locality present in this experiment. As stated, I distinguish between substance causality and event causality. A substance always emerges from another substance (such as in a particle decay), and this can be mapped out entirely in terms of physical causes. However events are indeterminate in physics (i.e.unpredictable), which means that the cause of events is not something physical, or representable mathematically. (The only alternative to this is to say that there is no cause; which I view as saying that the universe is ultimately irrational). I would attribute this to God (perhaps via some intermediaries). Being outside of time and space, God obviously relates to everything in the universe simultaneously in time and whatever the analogue word for that is in space. Events include those involved in the process where a particle interacts with a detector and we get a result for its spin along a particular axis. Since God controls both events, they are not independent of each other, even though there is no physical interaction between the two. In the absence of miracles, this also leads to the symmetry laws which ultimately lead to the prediction that the two particles will have opposite spin when measured along different axes. The amplitude is used to parametrise our uncertainty about God's action for each event. We know from the symmetry laws that if God causes b, then that has implications for the possible result at a.

Hope that's clear (I'm trying to think more deeply about what my interpretation implies in response to all these comments).

Hello again Dr. Cundy! Your comment regarding the "physical states" and effective field theory was very interesting, though I must admit at the end I couldn't tell if you were more or less agreeing with me, or adopting a kind of top-down pluralistic ontology (i.e. some objects exist really exist at the composite macro-level and their micro-level properties and constituents exist virtually and only appear when probed by an appropriate interaction), or something in between. I think an exposition of your view of QM would still benefit from a more detailed specification of what physically exists (fundamental particles? composite particles or objects? fundamental particles + forms of composite particles or objects? an ontological quantum state, different from the epistemic quantum state we assign based on our limited knowledge?), but let's leave that aside.

I'm going to pick up by responding to what you wrote in this paragraph:

"Firstly, as I stated in my original post which sparked our discussion, quantum physics is not intuitive. We can visualise Aristotle's physics. We can visualise Newtonian physics. With Einstein's physics (by which I mean relativity) it gets harder, but it still roughly follows our intuitions. Of course, we do better in each case to represent our images mathematically, but we can still back up that mathematics with a mental picture. With quantum physics, that is impossible."

My problem with this statement is that it is _not_ impossible: pilot-wave theory is a counterexample to that claim. Or maybe it would be better to say that it is impossible in orthodox QM, but that's only because orthodox QM is incomplete and fails to clearly specify a physical ontology, with the measurement problem being a symptom of this failure. If QM is completed by a theory specifying a primitive ontology (as in pilot-wave or objective collapse theories), or even by some more radical proposal (as in many-worlds), it is not impossible to clearly understand or even visualize what is going on. (Although in many-worlds this can only be done for systems with a very low number of degrees of freedom. It is a bit easier in something like pilot-wave theory. There we have the primitive ontology which is directly visualizable, and even the quantum state is more accessible because its role via its influence on the primitive ontology is clear, and because we can write conditional wavefunctions for subsystems in a way that is impossible in other theories.)

So I find it hard to see the statement "If you think you understand quantum mechanics, then you don't", despite its pedigree, as anything other than a cover for a refusal to acknowledge that standard QM has serious philosophical problems. (Though I'm not accusing you of using it that way personally.) "In any philosophy of quantum mechanics there is going to be something unintuitive" is much better, because that is actually true. But then the question becomes, why accept unintitive consequence A (e.g. failure of standard probability axioms) over unintuitive consequence B (e.g. non-locality)? Particularly when I can at least make sense of B, and even your view already has some of B (via God's causation to make sure the non-local correlations engtangled particles hold), and I can't make sense of A?

I appreciate your attempt to provide a picture of what an amplitude as a representation of uncertainty might look like, though it is (forgive the pun) circular reasoning. If we have a set of related propositions, paramaterized by X (mod 2pi), and _only one of those propositions can be true_ (which you indicate is the case), then the proper representation of uncertainty about these propositons seems to be a probability distribution over X, unless it is already given that something like amplitudes can represent uncertainty.

Your argument that we can't do this (treat the state+phase as a single classical state) is that it would give incorrect results if we tried to combine the probabilities of two different states. But... does it? Unless I'm grossly misunderstanding things, a mixed quantum state represented by a density matrix looks exactly like a probability distribution over different quantum states (treated as possible ontological states of the system, and which themselves produce objective tendencies for the system to arrive at different observable states upon measurement), combining in exactly the way one would expect. Nothing that I've read anywhere else indicates "beware, standard axioms of probability theory do not apply here"! You also say, "what other way can we interpret the amplitude in a psi-epistemic view of QM?" But why should I accept a psi-epistemic view of QM, which I can't understand, over a psi-ontic view?

Dr. Cundy: Granted, the order of operations is important when calculating. But I don't think it affects the argument. At least, I don't see how summing over the hidden parameters could introduce a non-local correlation, if such correlations are ruled out for each possible value the hidden parameters could take. After all, if each summand obeys Amp(a|b,X) = Amp(a|X), and each summand has equal weight in the sum, doesn't the sum have to obey the constraint, too? I know there are some counterintuitive results in probability theory, but that would be really peculiar, if true.

Matthew: In the case of interference, the data expressed in the amplitudes for possible paths just is what the "pilot wave" is supposed to convey to the particles. That is, you discover what the pilot wave for your experiment is by summing or integrating the amplitudes for each alternative. And the math for that doesn't respect all the rules of classical probability theory, because you're working with complex numbers, which aren't totally ordered. "P(A or B) > P(A)" can be easily deduced from the axioms of probability theory; "Amp(A or B) > Amp(A)" isn't even well-defined.

Density matrices are the same way. However much a density matrix may look like a probability distribution, it doesn't act like one in all respects, just because its components lack the ordering properties of the real numbers. You can take sums and products of linear operators as much as you like, but it makes no sense to say that one linear operator is greater than another. And the way you get an actual probability out of a density matrix is an operation outside the scope of probability theory.

Hi Michael, in the case of pilot-wave theory, the wavefunction is ontic rather than epistemic, so no problem of how to interpret the amplitudes as representations of uncertainty arises, and there's no surprise that they don't obey probability theory (because they aren't representing uncertainties, but properties that give rise to objective tendencies).

As for density matrices, the way they result in probabilities involves both (a) the probability distribution over the ontic quantum state, and (b) the Born rule for producing the probabilities resulting from that quantum state (the objective tendencies that arise from the quantum state). As far as I can tell, operation (b) is the only one outside of the scope of standard probability theory.

The fact that the quantum states can't be linearly ordered doesn't mean that the density matrix does not represent a probability distribution - you can have a probability distribution over a 2D plane, after all!

Hi Dr. Cundy, you write the following in response to Michael:

***

Born's rule is applied when we have an ensemble of "paths"; the amplitude refers to a single path. Thus the expression should be:

P(A|B) = | int dX Amp(A|BX)|^2

where X represents the various internal variables which we don't know. If there were a single amplitude contributing to the probability, then your expression would be correct. As I stated, the proofs of Bell's theorem that I have seen rely on an integration over the hidden variables after you have calculated the probability, in effect.

P(A|B) = int dX | Amp(A|BX)|^2

***

In my view the difference between these two cases is the following:

In the first case, the quantum state is a superposition of different terms (parameterized by X), and the whole superposition enters into Born's rule to determine the objective tendencies of the system. The whole state |int dX Amp(A|BX)> is a pure state.

In the second case, we don't know what the quantum state is; it could be one of many different states (parameterized by X). To find the probability we calculate the objective tendencies produced by each state, and average them. Each state |Amp(A|BX)> is a pure state and we're dealing with a probability distribution over them. If I'm understanding things correctly, we could represent this as a mixed state, something like int dX |Amp(A|BX)><Amp(A|BX)|.

The point here is that a probability distribution is, necessarily, a real-valued function. Whatever wild and strange things you pick the inputs from, the output has to be a real number between 0 and 1. Now, even if you write a density matrix as a weighted sum of pure states, it is not a real-valued function - it's a Hermitian matrix. You have to do something else to it to extract a probability (specifically, you take its trace.) And taking the trace of a density matrix is the equivalent of applying Born's rule to an amplitude.

In other words, calling a density matrix a probability distribution is a confusion of ideas. The two things aren't of the same type.

Similarly, "P(A|B) = | int dX Amp(A|BX)|^2" is in fact the correct way to handle mixed states, as well as superpositions - and if you use density matrices, you must integrate over the unknown parameters, then take the trace. The point in the calculation where you apply Born's rule, or take the trace of a density matrix, represents the point in the experiment where you measure the system - or, if you'd rather, where the system interacts with its environment and decoheres. The expression "P(A|B) = int dX | Amp(A|BX)|^2" represents a decoherent system, from which all quantum effects like interference and entanglement have been banished.

Michael, that's correct - I've been trying to say that density matrices can be used to _represent_ probability distributions over pure quantum states, not that they _are_ probability distributions over pure quantum states. Maybe I missed making that distinction clearly, but my above comments should be interpreted with it in mind.

Specifically, if you have some quantum states |psi_X> parameterized by X, each associated with a probability P(X), then that is represented by the density matrix which is the sum over P(X) |psi_X> <psi_X|, is it not? Then the expectation value of some observable A is the sum over P(X) <psi_X|A|psi_X>, which corresponds to the second calculation (summing after squaring) rather than the first (summing before squaring). That's according to the Wikipedia page on density matrices, at least.

My point is that taking the trace with the density matrix corresponds to _both_ of the operations (a) averaging over the probability distribution and (b) applying Born's rule on the pure quantum states. I don't believe I've said anything incorrect here.

I think you've misunderstood the Wikipedia page. You could think of a mixed state as a function from pure states to probabilities, I suppose, sending each pure state to its coefficent. But I don't see how that would be useful, either for calculations or conceptually. The mathematical object you work with is the density matrix itself, not that "function". (For instance, while the density matrix for a mixed state specifies that state uniquely, its representation as a convex combination of pure states very much doesn't. There are lots of combinations of pure states that end up as the same matrix.)

So that derivation of the expectation value of an observable is a purely formal algebraic calculation. The components p_X|psi_X><psi_X|A have no physical meaning - they are used only to prove that tr(A*psi) is the expectation value of A in the state psi. Redefining the decomposition of the mixed state's matrix (a purely mental operation) as the average of a "probability distribution", just because it looks like one, is chasing after a mirage. It doesn't help the calculation, and it doesn't correspond to anything in the physics.

In short, taking the trace of a density matrix is just applying Born's rule. It doesn't involve averaging over a probability distribution, because there is no actual distribution to average over.

I know that the density matrix itself is what is used to do calculations, and that it is more useful for that purpose than a probability distribution over pure states. I also know that different combinations of pure states can result in the same density matrix. For a psi-ontic theory, that simply represents an epistemic limitation - from experimentally observed probabilities, we can't always decompose that into probabilities resulting from quantum indeterminism, and probabilities resulting from uncertainty about the actual quantum state.

You seem to be arguing from the fact that we can't uniquely extract a probability distribution over the pure quantum states that there isn't one there at all. But that seems to me to be simply untrue: you could set up a device that randomly selected a (pure) quantum state and emitted an electron in that state. An experimenter not privy to the programming of your apparatus might only be able to determine the density matrix by doing measurements on the electrons, but that doesn't prevent there from being (or you from knowing) the actual probability distribution.

So I think it is making some assumptions about how QM should be interpreted to say that "taking the trace of a density matrix is just applying Born's rule" and nothing else.

"you could set up a device that randomly selected a (pure) quantum state and emitted an electron in that state."

Well, no, actually, I don't think you could. Consider, instead, the case of polarized light. The maximally mixed state is light that isn't polarized at all. And you can't create unpolarized light with a machine that emits either horizontally or vertically polarized light based on the flip of a coin (or any other random process.) You could split a beam of light into oppositely polarized halves, but each beam remains polarized whatever you do, and the only way to get unpolarized light out of them is to put them back together. That is, a beam of light that's known to be polarized either horizontally or vertically, with a 1/2 probability of either, is not the same thing as an unpolarized beam.

And yes, I do say that because the decomposition of a mixed state's density matrix into a convex combination of pure states' density matrices isn't unique, it's incoherent to recast that combination as "the average of a probability distribution" and suppose that the distribution means anything. It's just like deciding where to put the origin of your coordinate axes - you're deciding how to represent the physical system you're looking at, and you can pick whatever is most convenient, because that part of the representation doesn't correspond to anything real. How you decompose a mixed state's density matrix depends on what basis in the Hilbert space is most convenient for your calculations; it's an artifact of your representation, not a reflection of what's really happening.

The contrary view would be like saying that, because we can do physics under the assumption that the Earth is the nonrotating center of the universe and everything revolves around it, therefore the Earth really is just that. Which would be absurd.

On probabilistic mixtures: here is a setup which, roughly speaking, randomly selects between spin up and spin down and then emits an electron in that state. Start with an unpolarized beam of electrons with low enough intensity that roughly one electron is emitted every second. Pass it through a S-G device that sends spin up electrons down one path and spin down electrons down the other path before recombining them. Each second randomly (with equal probability) choose one of the paths to be blocked for that second. Electrons which exit the setup will be either spin up or spin down with roughly equal probability. You can do something similar with photons, if you like.

In any case, since you think I misunderstood the Wiki article, I looked up some online material from introductory QM courses, and the first two I found (one from Berkeley and one from MIT) were even more clear than Wikipedia: they defined mixed states as probability distributions over pure states, said that density matrices represent mixed states, and one of them directly stated that the same density matrix can represent different mixed states. The third one that I found took your position, equating mixed states with (non-pure) density matrices, and noting that because we can decompose the density matrix in many ways, we can't call the coefficients probabilities. But it went on to say that we can certainly go the other way: probabilistically prepare a statistical mixture of pure states and write a density matrix with those probabilities as coefficients.

So that certainly supports part of what I've been saying: that you can represent a statistical mixture of pure states as a density matrix. The only other thing I'm claiming here is that it is possible that every density matrix represents such a statistical mixture, even if we cannot in principle determine which statistical mixture it is. (In fact, I've confirmed that under a Bohmian interpretation this is precisely the case, hence my assertion that your claim to the contrary brings in matters of the interpretation of QM.)

I believe your comparison with choosing a coordinate system is disanalogous, because I'm not talking about going from a density matrix to a probability distribution, but the other way around. There's a loss of information when you write the density matrix from the probability distribution. The fact that you can't uniquely recreate the probability distribution doesn't imply that there wasn't one in the first place. In your supposed analogy, we have very good reason to think that there was no originally correct coordinate system (namely, coordinate systems are human constructs). There isn't an analogous reason for us to expect that a mixed state doesn't come from some statistical mixture of pure states.

I would say that the main reason we have for thinking that coordinate systems are "human constructs" is that the best theories of physics we have are symmetric under a change of coordinates. If, instead, our physics claimed that the results of experiments depended on their distance from a specific point, and the experiments confirmed that claim, that would be strong evidence that the universe has a God-given coordinate system.

By analogy, then, since the Lagrangian of quantum field theory is symmetric under a change of basis in the Hilbert space from which the wavefunction/density matrix that represents a state is chosen, you would need some other reason (such as prior knowledge of how the state was made) to suppose that the choice of basis is physically significant. That is, the symmetry of the "probability distribution" representation of a mixed state under a change of basis is, all by itself, good reason not to believe that a mixed state is a statistical mixture of pure states. The presumption cuts the opposite way from how you want it to.

After all, the whole reason the density matrix representation works is that the information discarded by calculating a density matrix from a probability distribution makes no difference to the outcome of any observation we can make. A distinction that makes no difference is no difference.

As an aside, this argument is quite similar to Dr. Cundy's argument that the symmetry of our physical theories under Lorentz transformations is a good reason to disbelieve presentism and accept a B-theory of time. In that case, though, I do have evidence the other way, such as our own experience of time, which is limited to a single moment rather than extending over all our lifetimes.

Rather than dispute with you about what the symmetry considerations do or do not demonstrate, I will simply claim we are in the same boat with the status of density matrices as the one you and I seem to agree we are in with respect to Lorentz symmetry and the A vs. B theory of time. Namely, orthodox quantum theory is incomplete because it fails to specify a clear physical ontology (which is most evident via the measurement problem), and what seem to me to be the most reasonable ways to resolve that problem give us good grounds for thinking that density matrices do in fact represent statistical mixtures of pure states. (E.g. pilot-wave type theories, because we don't know the precise state of the primitive ontology alongside the quantum state, and objective collapse theories, because we don't know exactly when the collapse occurs or what the quantum state collapses to, both lead to a natural interpretation of the density matrices arising from partial traces of larger systems in terms of a probability distribution over pure states of the subsystem.)

Dr. Cundy, my question to you probably got lost in the argument I had with Matthew, so...

If the amplitudes for every state a pair of entangled particles can be in don't have non-local correlations, is it possible for the process of summing the alternatives - that is, of taking the path integral - to introduce a non-local correlation in the final result? My intuition says "no", but intuition can be wrong.

Michael,

Sorry for not responding to your question. It is a good one.

Firstly, my reply (Comment 15) was in three parts, and my comment about summing amplitudes was only the first part of it. The third paragraph is probably most important part of my reply. Note also that I had the qualification: I am not sure that this niggle overly affects your argument. Which would imply even as I wrote the comment I agreed with you. I emphasised that we should be summing amplitudes rather than probabilities because I do think that this is crucial to assessing the mathematical proof behind Bell's inequalities. You might well be right that some form of Bell's theorem survives this objection; but the standard proofs have a problem.

Can summing local amplitudes lead to a non-local probability? I can only think that they would if they are conditional on different things. For example, we can consider summing psi(X|Ya) and psi(X|Yb). Perhaps this is a two slit interference pattern. The first amplitude represents the likelihood that the particle reaches X given initial conditions Y and that it went through slit a, while the second amplitude represents the likelihood that the particle reaches X given initial conditions Y and passing through slit b. Each of the amplitudes themselves is local, in that it contains no non-local interactions, but the sum of them depends on both what happens at a and b. (Of course, these are both in the past light cone of X, so perhaps this isn't the best of examples. But if we can combine amplitudes of two different events, then each one individually might be local, but the probability which is based on the combination not.)

Equally, we shouldn't forget that probabilities are compared against frequency distributions of ensembles, rather than individual events (which is what the experimental tests of Bell's theorem rely on). When considering individual events, we ask questions such as "What is the amplitude that detector a records result A given that detector b records result B?" (Or, if we don't know the result at B, what is the amplitude that we get A and B?) In this sense, Amp(A|YB) is not equal to Amp(A|Y). Is this a break down of locality? Not if the amplitude refers to our knowledge. And with regards to substance causality, no particle or particle state has been created at a distance from its efficient cause, or travelled faster than light. How does the particle at a know what happened at b? It doesn't (after a fashion -- in my philosophy God is the one who actualises the potential for it to record a particular spin state, and God is aware of what happens at a). But the result at a is not in general determined by what happens at b. It is only when we consider the ensemble that the pattern emerges.

Michael, I wouldn't call it an argument. More of a discussion!

Dr. Cundy, thanks so much for your comments! I've responded via email to avoid crowding this comment thread.

Thanks for your email, Matthew: I have read it, but haven't had time to respond properly, or complete the second part of my review of your work. I'll get on with that when I have the opportunity, but I can't say when that would be.