I QFT made Bohmian mechanics a non-starter: missed opportunities?

[vanhees71]

The Bohmian trajectory (singular, not plural) for a single particle is derivable from the state, given its initial position. But the initial position is not derivable, so the initial is the additional information not present in Copenhagen and ensemble interpretations.

That's of course true. In the minimal interpretation such a situation is described by a state (preparation procedure) where the particle is well localized, i.e., you have a sharply peaked "density matrix", $\rho(\vec{x},\vec{x})=\langle \vec{x}|\hat{\rho}|\vec{x} \rangle$, which of course doesn't constrain the state itself very much.

 

[martinbn]

Then please give a reference, where your flavor of the ensemble representation is clearly stated. Obviously it's not the one, Ballentine defines in his book (and already in his RMP).

Why! Your quote doesn't contradict anything i said. Please provide a reference where in an ensemble interpretation the state describes the individual!

 

[vanhees71]

I didn't say that the state describes the individual, but I said that the state is associated to each individual in the sense that it refers to preparation procedures that allow to prepare ensembles described by this state.

I can only again refer to Ballentines book and precisely the quote I've given above. Also see the text following it:

The quantum state description may be taken to refer to an ensemble
of similarly prepared systems. One of the earliest, and surely the most
prominent advocate of the ensemble interpretation, was A. Einstein. His
view is concisely expressed as follows [Einstein (1949), quoted here without
the supporting argument]:
“The attempt to conceive the quantum-theoretical description as
the complete description of the individual systems leads to unnatural
theoretical interpretations, which become immediately unnecessary
if one accepts the interpretation that the description refers to
ensembles of systems and not to individual systems.”

 

[Demystifier]

Well, no, it is not in the ensemble interpretation.

It looks as if you are saying that the ensemble interpretation says absolutely nothing about the individuals. But individuals clearly exist (at least as individual measurement outcomes), so such a version of ensemble interpretation would say nothing about the things that exist. I'm pretty much confident that this is not what Ballentine had in mind.

 

[vanhees71]

Also have a look at the RMP paper, which has it in very precise and efficiently formulated form:

https://doi.org/10.1103/RevModPhys.42.358

 

[Demystifier]

I particularly like the following quote from the Ballentine's paper above: "Thus it is most natural to assert that a quantum state represents an ensemble of similarily prepared systems, but does not provide a complete description of an individual system."

 

[vanhees71]

The question, whether the description is complete or not, is of course somewhat mute. I think, a physical theory is as long considered complete as long there are no observational contradictions to its predictions and as it describes all phenomena. In this sense QT is of course incomplete, because it doesn't provide a satisfactory description of the gravitational interaction. It's, however, not incomplete, for the single reason that it provides "only" probabilistic descriptions for the outcome of measurements on "ensembles of similarly prepared systems". Of course, any probabilistic description says only something about such ensembles, because without ensembles you cannot test the probabilistic description, i.e., "you need enough statistics", which is why, e.g., the LHC was upgraded to "higher luminosities" and the then necessary higher DAQ rates to be handled by the detectors. However, this probabilistic description doesn't need to be a priori "incomplete". It may well be that Nature is inherently probabilistic and thus in this sense "completely" described by probabilistic laws. For me the outcomes of all the Bell experiments, in accordance with Q(F)T, are a strong indication that this might in fact be true!

 

[Demystifier]

In this sense QT is of course incomplete, because it doesn't provide a satisfactory description of the gravitational interaction.

My conjecture/hypothesis is that the key to quantum gravity might be the idea that classical spacetime symmetries are not valid at the quantum level. Bohmian mechanics served as an inspiration, but I defended this conjecture/hypothesis within standard QM, in
https://arxiv.org/abs/2301.04448

 

[WernerQH]

Of course, you must be able to say (to some accuracy at least) that each single realization of the system is prepared in this state.

@martinbn has correctly noted the sloppiness of this statement, and your reference to Ballentine is definitely not the one he asked for.

The word "state" refers to an idealization, to a "typical" or "average" particle, and certainly not to individual members of the ensemble. Its purpose is to convey the correlations that we regularly observe between "preparation" and "measurement" events. Quantum theory is about the correlations between events. On "what's going on" between events the theory is remarkably silent -- we are supposed to take all possibilities occurring in the meantime into account.

 

[vanhees71]

Then, how do you explain that in fact experimentalists can prepare with high accuracy definite states of individual systems?

 

[martinbn]

Then, how do you explain that in fact experimentalists can prepare with high accuracy definite states of individual systems?

Do you actually realise how inconsistent your statements are!

 

[WernerQH]

Then, how do you explain that in fact experimentalists can prepare with high accuracy definite states of individual systems?

Of course there are correlations, and we wouldn't have evolved without a keen sense for them. (Think of Pavlov's dog!)

"Definite" states of "individual systems" are constructions of your mind that it foists on the real world.

 

[vanhees71]

That doesn't make sense. Quantum opticians can with high precision prepare entangled photon pairs or how else do you explain the precise confirmation of the quantum-theoretical predictions? I don't see, what's inconsistent with Balentine's standard interpretation of the quantum state as being descriptions of preparation procedures of individual systems.

 

[WernerQH]

That doesn't make sense.

Exactly! Perhaps you should read Ballentine more carefully.

 

[vanhees71]

I didn't mean that Ballentine makes no sense but your claim that the standard interpretation of the state within the minimal interpretation were wrong. You also didn't clearly specify, what you think the "right" interpretation is.

 

[WernerQH]

I didn't mean that Ballentine makes no sense but your claim that the standard interpretation of the state within the minimal interpretation were wrong.

I didn't make such a claim. What doesn't make any sense to me is your reference to particular experiments in quantum optics as an argument for your sloppy use of the words "individual" and "system". Not only quantum theory needs interpretation, but the experiments too! What one person perceives as firmly established empirical facts, can be viewed as grounded in deeply engrained habits of thought by another. (Phlogiston, caloric, aether ...) But obviously you can't conceive of such a possibility.

 

[vanhees71]

So what's your interpretation of the quantum state, and how is it consistent with the obvious fact that experiments with ensembles built by preparations of single quantum systems. My photons were an example. There are more examples: Single electrons in a Penning trap, interference experiments with single neutrons, etc. etc.

 

[PeterDonis]

Then I have no idea what do you mean by ensemble interpretation

In the other thread you referenced you criticized Ballentine's ensemble interpretation. His definition (p. 46) is that, given a state preparation procedure, an ensemble is "the conceptual infinite set of all such systems that may potentially result from the state preparation procedure".

 

[PeterDonis]

Obviously it's not the one, Ballentine defines in his book

As far as I can tell, the definition I just posted from Ballentine is consistent with what @martinbn has been saying, and not with what you and @Demystifier have been saying.

I could say the same about what you posted from Ballentine, his definition of the state operator, which is based on the ensemble definition that I posted.

 

[PeterDonis]

what's your interpretation of the quantum state

Ballentine's interpretation is clear; again from p. 46: "...the primary definition of a state is the abstract set of probabilities for the various observables". The ensemble definition I quoted earlier is then introduced with: "...it is also possible to associate a state with an ensemble of similarly prepared systems." He then gives the ensemble definition I quoted. His reason for giving the ensemble definition is (earlier in the same paragraph): "The empirical content of a probability statement is revealed only in the relative frequencies in a sequence of events that result from the same (or equivalent) state preparation procedure."

 

[WernerQH]

So what's your interpretation of the quantum state

I have no doubt at all that QT is a statistical theory. But we seem to disagree on what it is about, what it is that causes the perfect correlations observed in so many experiments. I've tried to explain my view in post #309.

and how is it consistent with the obvious fact that experiments with ensembles built by preparations of single quantum systems.

Sorry, I just can't understand your question. I don't see in which sense there should be an inconsistency.

My photons were an example. There are more examples: Single electrons in a Penning trap, interference experiments with single neutrons, etc. etc.

Also experiments with single particles involve many events, taking a lot of time in the lab.

 

[Structure seeker]

So is the idea that each state in an ensemble is given by a copy of the same stochastic variable? Where the space of possible outcomes of each of these stochastic variables is (perhaps) a subspace of all possible quantum states for these specific quanta?

 

[Quantum Waver]

Freeman Dyson in “THE COLLAPSE OF THE WAVE FUNCTION” in John Brockman’s book “This Idea Must Die: Scientific Theories That Are Blocking Progress (Edge Question Series)” (New York, NY, USA: HarperCollins (2015)):

“Fourscore and eight years ago, Erwin Schrödinger invented wave functions as a way to describe the behavior of atoms and other small objects. According to the rules of quantum mechanics, the motions of objects are unpredictable. The wave function tells us only the probabilities of the possible motions. When an object is observed, the observer sees where it is, and the uncertainty of the motion disappears. Knowledge removes
uncertainty. There is no mystery here.

Unfortunately, people writing about quantum mechanics often use the phrase “collapse of the wave function” to describe what happens when an object is observed. This phrase gives a misleading idea that the wave function itself is a physical object. A physical object can collapse when it bumps into an obstacle. But a wave function cannot be a physical object. A wave function is a description of a probability, and a probability is a statement of ignorance. Ignorance is not a physical object, and neither is a wave function. When new knowledge displaces ignorance, the wave function does not collapse; it merely becomes irrelevant.”

That's one interpretation. Ignorance doesn't explain individual particles exhibiting interference patterns when the slit they go through isn't detected. Or other interesting experiments demonstrating wave and particle-like properties depending on measurement.

 

[PeterDonis]

So is the idea that each state in an ensemble is given by a copy of the same stochastic variable?

No. The idea is that the state describes the ensemble, not any individual system. That is because the state describes the probabilities of measurement results, and you can only experimentally measure those probabilities by doing statistics on the results of the same measurement on a large number of identically prepared systems.

 

[Structure seeker]

Identical and independent or identical and perhaps entangled (so that the partial state of one entity in the ensemble is a mixed state)? Usually with such experiments these are by design meant to be independent.

 

[PeterDonis]

Identical and independent

Identically prepared, with each individual preparation being independent.

or identical and perhaps entangled

The preparation procedure can involve preparing an entangled state (for example, parametric down conversion that produces two entangled photons is a valid preparation procedure). But, as above, each individual preparation would still be independent of all the other preparations.

 

[Structure seeker]

Identically prepared, with each individual preparation being independent

So we have $n$ i.i.d. (independent identically distributed) stochastic variables $X_1$, ..., $X_n$ over some space of quantum states, in case of entangled states we just take the density matrix of the whole state as $X_i$ per $i$. However we cannot know the state of any of these, they're quantum. But since they are not correlated, the stochasts that describe their measurement outcomes are also not correlated. Then the actual state $x_i$ (when it were possible to evaluate $X_i$, but the particle must have a state so outside our knowledge $X_i$ has been evaluated) is not entangled with any of the other $x_j$, right? That's what entanglement means.

 

[PeterDonis]

So we have $n$ i.i.d. (independent identically distributed) stochastic variables $X_1$, ..., $X_n$ over some space of quantum states

There are no such "stochastic variables" in standard QM. Are you referring to some particular interpretation? If so, a reference would help since there are no such "stochastic variables" in the usual ensemble interpretation (as described in, for example, Ballentine) either.

If these "stochastic variables" are something you made up yourself, please be aware that personal speculation is off topic here.

 

[Structure seeker]

If the states in an ensemble are partial states, at least they have some state, surely? It's not like we can formulate a quantum that has no state in QFT. So whatever the preparation does, it results in a state $x_i$. If it's not the same for each $i$, while the preparation is identical and independent, the mathematical way to describe this is an i.i.d. stochast set $\{X_i \}_i$ that takes values on the state space.

Let me know where the personal speculation is, if it's in here somewhere.

 

[PeterDonis]

If the states in an ensemble are partial states

I don't know what you mean by this either. In the ensemble interpretation, the state vector $\ket{\psi}$ describes the probabilities for possible measurement results on an ensemble of identically prepared systems. There are no "partial states" or "states in an ensemble".

whatever the preparation does, it results in a state $x_i$.

Not on a single system if we are using the ensemble interpretation, no. The state vector $\ket{\psi}$ describes just what I said above.

Let me know where the personal speculation is, if it's in here somewhere.

The personal speculation is that you appear to be making up a bunch of stuff that doesn't appear anywhere either in the actual math of QM or in the ensemble interpretation.