Copenhagen: Restriction on knowledge or restriction on ontology?

[DarMM]

I don't know what you mean. What are you saying is impossible?

The usual argument about Bell's inequality is not that it is impossible for a stochastic model to reproduce the statistics, but that it is impossible for a local model to reproduce the statistics. There is no claim that the stochastic model is local in the sense of Bell.

Much simpler than that. How do you define $B_2$ on each outcome so that it is a random variable on this sample space. Ignore Bell's theorem as such for now.

 

[stevendaryl]

Much simpler than that. How do you define $B_2$ on each outcome so that it is a random variable on this sample space. Ignore Bell's theorem as such for now.

I don't understand your question. There are 16 possible events. Each event has 4 components: $A, a, B, b$. You give a probability to each of the 16 combinations so that the probabilities add up to 1.

 

[DarMM]

I get that. How do you define $B_2$ as a random variable, i.e. a function on this space. So that you can compute things like $\langle B_2 \rangle$?

 

[stevendaryl]

I get that. How do you define $B_2$ as a random variable, i.e. a function on this space. So that you can compute things like $\langle B_2 \rangle$?

I didn't make any claims about $B_2$ as a random variable. I don't know what that means. There is one random variable, which is which event out of 16 possibilities. It can take values 1 through 16. If the value is 1, then in that history, the setting of $A$ is $0$, the setting of $B$ is $\frac{\pi}{2}$, the outcome $a$ is $up$, the outcome $b$ is $up$.

 

[stevendaryl]

I didn't make any claims about $B_2$ as a random variable. I don't know what that means. There is one random variable, which is which event out of 16 possibilities. It can take values 1 through 16. If the value is 1, then in that history, the setting of $A$ is $0$, the setting of $B$ is $\frac{\pi}{2}$, the outcome $a$ is $up$, the outcome $b$ is $up$.

You can certainly look at the subset of histories in which $B = \frac{3\pi}{4}$. For those histories, you can compute the expectation value of the result $b$.

 

[DarMM]

I didn't make any claims about $B_2$ as a random variable. I don't know what that means.

Really? $B_2$ is an observable quantitiy and will be encoded in a probability model as a random variable. Thus if you are using Kolmogorov probability $B_2$, $B_{\frac{3\pi}{4}}$ as we are calling it, has to be a random variable on that space. That's just how probability theory works.

You can certainly look at the subset of histories in which $B = \frac{3\pi}{4}$. For those histories, you can compute the expectation value of the result $b$.

Exactly. There is only a subset on which $B_{\frac{3\pi}{4}}$ is defined as a random variable. Thus in truth these are four separate Kolmogorov models with their own individual sample spaces.

 

[stevendaryl]

Really? $B_2$ is an observable quantitiy and will be encoded in a probability model as a random variable.

Okay, suppose that I take 16 cards. On each index card, I write down 4 quantities: $(A, a, B, b)$ where $A$ is either 0 or $\frac{\pi}{4}$ and $a$ is either $up$ or $down$ and $B$ is either $\frac{\pi}{2}$ or $\frac{3\pi}{4}$ and $b$ is again either $up$ or $down$. So I play lots and lots of rounds where I pick a card according to some probability distribution. Tell me in that situation what it means to say that $B_2$ is a random variable.

The random variable is which card I choose. There are 16 cards. The values of $a$ and $b$ and $A$ and $B$ are functions of this one random variable.

 

[DarMM]

Tell me in that situation what it means to say that $B_2$ is a random variable.

What you're missing here is that $B_2$ is a physical quantity. It's something you can observe. In a probability model it should therefore be a random variable. Note though what you are saying isn't entirely wrong. See the next the question below, this might serve the discussion better.

The usual argument about Bell's inequality is not that it is impossible for a stochastic model to reproduce the statistics, but that it is impossible for a local model to reproduce the statistics. There is no claim that the stochastic model is local in the sense of Bell.

Where is the nonlocality in your table?

 

[stevendaryl]

What you're missing here is that $B_2$ is a physical quantity.

In my card game, what corresponds to $B_2$?

 

[DarMM]

In my card game, what corresponds to $B_2$?

A value on the card I assume. Let's just focus on the nonlocality. In your table how does it manifest?

 

[stevendaryl]

A value on the card I assume. Let's just focus on the nonlocality. In your table how does it manifest?

The probability for each result for one particle depends on the setting for the measurement of the other particle.

 

[stevendaryl]

A value on the card I assume.

I'm asking you what $B_2$ means. You're saying it's an observable. I don't know what you mean by $B_2$. The way I set things up, there are two different things: the choice of what measurement to perform on the second particle, and then there's the result of that measurement. What do you mean by $B_2$?

 

[PeterDonis]

We have 16 possibilities, which we can characterize by 4 numbers: $A_j$, $a_j$, $B_j$, $b_j$ where $A_j$ is the $j^{th}$ measurement on the first particle, $a_j$ is the $j^{th}$ result of that measurement (either "up" or "down"), $B_j$ is the $j^{th}$ measurement on the second particle, and $b_j$ is the $j^{th}$ result of that measurement.

I think this is misstated. You are only making one measurement on each particle, so there is no need for an index $j$ saying which measurement. For $A$ and $B$, you have two possible orientations you can choose for the one measurement.

 

[PeterDonis]

take the observable $B_2$.

There is no observable $B_2$. There should not be an index on $A$ or $B$, since only one measurement is being made on each particle. See my post #153 just now.

 

[PeterDonis]

There is only a subset on which $B_{\frac{3\pi}{4}}$ is defined as a random variable.

I think the random variables for the measurement settings are just $A$ and $B$, both of which are defined on the entire sample space of 16 alternatives that @stevendaryl gave. What you are denoting $B_{\frac{3\pi}{4}}$ would be one of the two possible values of the random variable $B$, i.e., one of the two possible results of the random choice (coin flip) being made to determine the setting of the measurement $B$.

What I think you are trying to say here is that the random variables describing the results of the measurements for a given setting are not defined on the entire sample space. For example, the random variable that I would describe as $(b | B_{\frac{3\pi}{4}})$, i.e., "the result of the measurement on particle $b$ when the measurement device $B$ is set at angle $\frac{3\pi}{4}$", is only defined on a portion of the sample space, namely, the 8 out of 16 cases where that is indeed the setting of measurement device $B$.

 

[DarMM]

Yes you are correct @PeterDonis that's what I'm trying to say. Another way of saying it is that $S_{\frac{3\pi}{4}}$, just to indicate it is spin, is supposed to be a physical quantity and like momentum in statistical mechanics should be a random variable over the whole sample space.

Here it no longer is, but rather is the outcome of another random variable: "Device Setting". So perhaps we can agree that you can have a single sample space, but at the "price" of forgoing $S_{\theta}$, with $\theta$ any angle, being a physical quantity. The only physical quantity is "device setting", thus rendering the account non-representational.

 

[DarMM]

The probability for each result for one particle depends on the setting for the measurement of the other particle.

How is that enough to establish nonlocality rather than just correlation, i.e. isn't that just $P(R_1|R_2) \neq P(R_1)$

($R$ being the results)

 

[Lord Jestocost]

Hey, could you tell me what book/paper that quote is from?

It is from the book “Physics and Philosophy: The Revolution in Modern Science” by Werner Heisenberg (the book is an outgrowth of his Gifford Lectures at St Andrews University).

 

[Lord Jestocost]

But he forgets the collapse, which is not part of the minimal statistical interpretation ...

As Jan Faye puts it (https://plato.stanford.edu/entries/qm-copenhagen/):

"Second, many physicists and philosophers see the reduction of the wave function as an important part of the Copenhagen interpretation. But Bohr never talked about the collapse of the wave packet. Nor did it make sense for him to do so because this would mean that one must understand the wave function as referring to something physically real. Bohr spoke of the mathematical formalism of quantum mechanics, including the state vector or the wave function, as a symbolic representation. Bohr associated the use of a pictorial representation with what can be visualized in space and time. Quantum systems are not vizualizable because their states cannot be tracked down in space and time as can classical systems. The reason is, according to Bohr, that a quantum system has no definite kinematical or dynamical state prior to any measurement. Also the fact that the mathematical formulation of quantum states consists of imaginary numbers tells us that the state vector is not susceptible to a pictorial interpretation (CC, p. 144). Thus, the state vector is symbolic. Here “symbolic” means that the state vector's representational function should not be taken literally but be considered a tool for the calculation of probabilities of observables."

 

[stevendaryl]

How is that enough to establish nonlocality rather than just correlation, i.e. isn't that just $P(R_1|R_2) \neq P(R_1)$

($R$ being the results)

I don't actually care whether it's local or nonlocal. The most obvious way to implement such a transition rule is:

1. Check to see which measurement is being performed on one particle. That's $A$.
2. Check to see which measurement is being performed on the other particle. That's $B$.
3. Use those two pieces of information to compute the probabilities for the pair $(a,b)$

 

[A. Neumaier]

As Jan Faye puts it (https://plato.stanford.edu/entries/qm-copenhagen/):

Faye identifies the Copenhagen interpretation with Bohr's views alone, whereas it is usually considered to be the whole spectrum of views in Bohr's and Heisenberg's writings. Note that Faye acknowledges that Heisenberg endorsed collapse:

it was Heisenberg's exposition of complementarity, and not Bohr's, with its emphasis on a privileged role for the observer and observer-induced wave packet collapse that became identical with that interpretation.

Moreover, I doubt that Faye is a completely reliable source. For example, in the same article he also claims that

Heisenberg, in contrast to Bohr, believed that the wave equation gave a causal, albeit probabilistic description of the free electron in configuration space.

This is in direct opposition with the quote from Heisenberg given in post #21, where he denies causality.

Finally, to show that Faye's position is not the consensus on the issue of collapse in the Copenhagen interpretation, note that

• Henderson, J. R. (2010), Classes of Copenhagen interpretations: Mechanisms of collapse as a typologically determinative, Studies in History and Philosophy of Modern Physics, 41: 1-8.

writes in the abstract:

There are, however, several strains of Copenhagenism extant, each largely accepting Born’s assessment of the wave function as the most complete possible specification of a system and the notion of collapse as a completely random event.

 

[stevendaryl]

Here it no longer is, but rather is the outcome of another random variable: "Device Setting". So perhaps we can agree that you can have a single sample space, but at the "price" of forgoing $S_{\theta}$, with $\theta$ any angle, being a physical quantity. The only physical quantity is "device setting", thus rendering the account non-representational.

This has probably gone on long enough, but the point of this "toy" model is to realize the idea that the only thing that's "real" is macroscopic observables. Microscopic observables are calculational tools, merely. It's a toy model, not to be taken seriously. BUT I claim that it is empirically equivalent to standard quantum mechanics. I really think that the minimal interpretation basically amounts to this.

 

[stevendaryl]

I think this is misstated. You are only making one measurement on each particle, so there is no need for an index $j$ saying which measurement. For $A$ and $B$, you have two possible orientations you can choose for the one measurement.

What I meant is that there are 16 possible "situations": Which measurements were performed and what results attained. The index $j$ is over possibilities.

 

[DarMM]

This has probably gone on long enough, but the point of this "toy" model is to realize the idea that the only thing that's "real" is macroscopic observables. Microscopic observables are calculational tools, merely. It's a toy model, not to be taken seriously. BUT I claim that it is empirically equivalent to standard quantum mechanics. I really think that the minimal interpretation basically amounts to this.

I think that's fine and we've already agreed that essentially only one spin (let's say) observable gets to become a macroscopic observable in each round.

All that really matters here is the core of contextuality which can be seen in your table. If all spin measurements, $(S_x,S_y,S_z)$ say, could be amplified up to being macroscopic observables, then you could define them as random variables over the whole space. This would permit you the notion of random microscopic observables.

This is the core of the difference between QM and theories where nature is random but classically stochastic. The contextuality/counterfactual indefiniteness means you can only consider macroscopic observables to be real, where as in a classical stochastic theory you would be able to consider microscopic observables as real even if random.

Essentially it's not so much that I think what you are saying is wrong, but I don't see how it supports your original contention that contextuality has no real physical import in QM and doesn't present anything new beyond classical stochastic theories. As the entire literature and even your own example seem to say that it is exactly the core difference.

 

[vanhees71]

Faye identifies the Copenhagen interpretation with Bohr's views alone, whereas it is usually considered to be the whole spectrum of views in Bohr's and Heisenberg's writings. Note that Faye acknowledges that Heisenberg endorsed collapse:Moreover, I doubt that Faye is a completely reliable source. For example, in the same article he also claims that

This is in direct opposition with the quote from Heisenberg given in post #21, where he denies causality.

Finally, to show that Faye's position is not the consensus on the issue of collapse in the Copenhagen interpretation, note that

• Henderson, J. R. (2010), Classes of Copenhagen interpretations: Mechanisms of collapse as a typologically determinative, Studies in History and Philosophy of Modern Physics, 41: 1-8.

writes in the abstract:

One must be a bit careful, because in the early writings particularly by Heisenberg there hasn't been made the very important difference between causality and determinism.

The modern view is, e.g., according to Schwinger, Quantum Mechanics, Springer Verlag (see the very concise Porlogue of this book about a very detailed exposition of a physical heuristics for QM) as follows:

(a) Causality (time-local form): If the state of a system is precisely known at time $t_0$ and the dynamics of the system is precisely known, then the state of the system is precisely und uniquely known at any later time $t>t_0$.

(b) Determinism: In any state any observable of a system takes a definite value.

Obviously in this sense QT is causal but not deterministic (at least in any probabilistic interpretation of the state, which is in my opinion so far the only consistent interpretation; an exception is Bohmian mechanics which is a deterministic interpretation, but so far not satisfactorily generalized to relativistic Q(F)T).

 

[A. Neumaier]

One must be a bit careful, because in the early writings particularly by Heisenberg there hasn't been made the very important difference between causality and determinism.

The modern view is, e.g., according to Schwinger, Quantum Mechanics, Springer Verlag (see the very concise Prologue of this book about a very detailed exposition of a physical heuristics for QM) as follows:

(a) Causality (time-local form): If the state of a system is precisely known at time $t_0$ and the dynamics of the system is precisely known, then the state of the system is precisely und uniquely known at any later time $t>t_0$.

(b) Determinism: In any state any observable of a system takes a definite value.

Obviously in this sense QT is causal but not deterministic

Well, the notion of state also changed with time.

In 1927, Born, Bohr and Heisenberg always equated ''state'' with ''stationary state'', and not with a general wave function. See the quotes given in post #21. These states preserve energy and momentum exactly (as all three authors emphasize) but don't respect causality in the form (a), which was also the notion used by Born, Bohr and Heisenberg.

In contrast, the modern notion of state respects causality in the form (a), but preserves in the statistical interpretation energy and momentum only on the average.

In my thermal interpretation, the q-expectation of total energy and total momentum are exactly conserved beables, and the collection of all q-expectations is causal and deterministic in your sense (a) and (b), since the observables are not the operators but their q-expectations.

 

[lodbrok]

But I refuse to think of it as formalism, with or without $\lambda$. I insist on thinking about it as possible events in the laboratory, which is quite Copenhagenish in spirit. In the case of spin measurement by SG apparatus, the possible outcomes are macroscopic dark spots on the screen at 4 possible positions (corresponding to 2 possible spins in direction $d1$ plus 2 possible spins in direction $d2$). Those 4 possible positions of macroscopic dark spots on the screen can certainly be placed in a single sample space.

No. DarMM is correct. You can't combine outcomes from two different experiments into the one sample space if the experiments are incompatible. The sample space corresponds to all possible outcomes from a single experiment. $d1$ is a different experiment from $d2$ and since you can't do both at the same time, you can't combine the their outcomes into one sample space.

 

[lodbrok]

As for why nature lacks counterfactuals, that goes unexplained.

Nature lacks counterfactuals because that is what it means to be counterfactual. It is an epistemological concept that can never exist in actually measured experimental data.

 

[zonde]

No. DarMM is correct. You can't combine outcomes from two different experiments into the one sample space if the experiments are incompatible. The sample space corresponds to all possible outcomes from a single experiment. $d1$ is a different experiment from $d2$ and since you can't do both at the same time, you can't combine the their outcomes into one sample space.

In post #76 I gave reference to an experiment where photon polarizations are measured in different bases as part of a single experiment.

 

[zonde]

Nature lacks counterfactuals because that is what it means to be counterfactual. It is an epistemological concept that can never exist in actually measured experimental data.

This of course is true, but when we discuss general properties of nature (rather than results of particular experiment) we are using models. And counterfactuals is the essence of models. Basically it is meaningless to use counterfactuals when we speak about facts (nature itself) and it is meaningless not to use counterfactuals when we speak about models (properties of nature).

 

[Demystifier]

No. DarMM is correct. You can't combine outcomes from two different experiments into the one sample space if the experiments are incompatible. The sample space corresponds to all possible outcomes from a single experiment. $d1$ is a different experiment from $d2$ and since you can't do both at the same time, you can't combine the their outcomes into one sample space.

What if the experiment is enriched with an additional piece of equipment that randomly decides whether the measurement will be performed in $d1$ or $d2$ direction? In this case all 4 outcomes are possible results of the measurement, so all 4 belong to the same sample space.

And by the way, in the consistent histories interpretation, which perhaps is the most ontological version of "Copenhagen" interpretation, those 4 outcomes correspond to 4 histories that belong to the same consistency class.

 

[Demystifier]

an exception is Bohmian mechanics which is a deterministic interpretation, but so far not satisfactorily generalized to relativistic Q(F)T

It depends on what one means by "satisfactorily". I am quite satisfied with the idea proposed in my signature below.

 

[vanhees71]

Well, the notion of state also changed with time.

In 1927, Born, Bohr and Heisenberg always equated ''state'' with ''stationary state'', and not with a general wave function. See the quotes given in post #21. These states preserve energy and momentum exactly (as all three authors emphasize) but don't respect causality in the form (a), which was also the notion used by Born, Bohr and Heisenberg.

In contrast, the modern notion of state respects causality in the form (a), but preserves in the statistical interpretation energy and momentum only on the average.

In my thermal interpretation, the q-expectation of total energy and total momentum are exactly conserved beables, and the collection of all q-expectations is causal and deterministic in your sense (a) and (b), since the observables are not the operators but their q-expectations.

A state cannot preserve energy and momentum or not. Energy and momentum are conserved if the Hamiltonian is not explicitly time-dependent and momentum if the Hamiltonian doesn't depend on position. In the formalism an observable $A$ (maybe explicitly time dependent) is conserved iff the operator for its time, derivative,
$$\mathring{\hat{A}}=\frac{1}{\mathrm{i} \hbar} [\hat{A},\hat{H}]+\partial_t \hat{A}=0.$$
A stationary state is, for a not explicitly time-dependent Hamiltonian, of the form
$$\hat{\rho}_{E}=\sum_{\alpha,\alpha'} \rho_{\alpha \alpha'} |\alpha,E \rangle \langle \alpha',E \rangle,$$
i.e., it's a state with definite energy. Here $|\alpha,E \rangle$ are the eigenstates of $\hat{H}$ with the eigenvalue $E$.

Of course, the quantum state obeys (a). If it's initial value at $t=t_0$ is known, it's known for all time due to the von Neumann equation,
$$\mathring{\hat{\rho}}=0.$$
QT is however not deterministic, because not all observables can take definite values for any state (within the probabilistic interpretation of QT a la Born and its modern generalizations).

Again, you repeat the WRONG statement that q-expectations are "beables" (in normal language "observables"). I don't need to expose this very early error of the founding fathers since I've done this at length at least twice within this very thread! The identification of q-expectations as the observables is not the common practice in the application of QT to real-world experiments, and it was your declared goal to formulate an interpretation meeting the modern application of the QT formalism to real-world experiments!

 

[A. Neumaier]

Energy and momentum are conserved if the Hamiltonian is not explicitly time-dependent and momentum if the Hamiltonian doesn't depend on position.

This is the case for every closed system, in particular for the universe as a whole. Of course I don't claim conservation in open systems.

Of course, the quantum state obeys (a).

Only the quantum state as interpreted today, which is what I had already asserted. But in 1927, the notion of state was different!

 

[Demystifier]

This of course is true, but when we discuss general properties of nature (rather than results of particular experiment) we are using models. And counterfactuals is the essence of models. Basically it is meaningless to use counterfactuals when we speak about facts (nature itself) and it is meaningless not to use counterfactuals when we speak about models (properties of nature).

Well said! There is no theoretical physics without counterfactuals.