# Quantum Mechanics without Hilbert Space

#### StevieTNZ

This sounds like crackpot stuff. Do you have a reference to a peer-reviewed physics journal? If he hasn't been able to publish, it doesn't seem worthy of any deeper analysis.
Try http://www.neuroquantology.com/journal/index.php/nq/index [Broken] - it says its peer-reviewed, and its also a journal on physics.

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#### Fredrik

Staff Emeritus
Gold Member
So the Statistical Interpretation would be promising if it were true. But it didn't appear to be.
What makes you say that? The statistical interpretation is really just quantum mechanics with no additional assumptions. (I've read enough of the article now to see that he doesn't actually assume that every particle has a well-defined position and momentum at all times. He just claims that this wouldn't contradict QM. Maybe he's right, maybe he's wrong. Either way, it doesn't affect the statistical interpretation, since that claim isn't a part of its definition). So how could the statistical interpretation be wrong? Is your point that QM is wrong?

Now here is where its weakness lies according to this site:
You should probably select your sources more carefully. It's actually against the forum rules to post links to questionable material. This guy rejects just about everything except the idea that a mind isn't physical. (It gets funny when he rejects particles because of "no evidence", but doesn't require evidence for his claim about minds).

But such interpretations do not say what the actual world is ‘made of.’
So? What makes him think that there's a version of QM that can tell us that?

Also note in http://en.wikipedia.org/wiki/Bell_test_experiments that bell test experiments only started in 1972 (2 years after the article was written) so it didn't take into account that Bell's Thorem is violated categorically especially in light of Alain Aspect more rigorous experiment.
He was certainly aware that QM violates Bell inequalities. The experiments only confirmed that reality does too. So the experiments weren't that relevant.

So it is more likely that the Statistical Interpretation is not representative of reality at all. Or it is very incomplete.
Right, but if that's true, then the same can be said about QM itself (since the statistical interpretation is just QM without additional assumptions), so that problem can't be solved by another interpretation.

#### strangerep

I have read the [Ballentine] paper although not 100% because I don't understand all the math. [...]
I'll just say two things...

(1) Try to acquire more understanding of the math so you can understand such
papers completely. Worry a bit less about intepretation until you've done so.

(2) Regarding the following quote:

Also note in http://en.wikipedia.org/wiki/Bell_test_experiments that bell test experiments only started in 1972 (2 years after the article was written) so it didn't take into account that Bell's Thorem is violated categorically especially in light of Alain Aspect more rigorous experiment. So Statistical Interpretation is not designed to totally explain the correlation of Bell's theorem. It only mentions at the end of the 1970 Ballentine paper that it mentioned that it departed from the formalism of quantum theory but there was no subsequent updated work that would make it explain the correlation of Bell's Theorem.
One need only read Ballentine's much more recent textbook to see that this is rubbish.

#### Fra

To speak for myself, as long as we talk about the small subsystem abstraction, and complex observer - ie the cases where the statistical ensemble can in fact the realized in some principal sense, then then ensemble view is pretty close to my view. I think it's good.

But IMHO, the obvious problem is that the, the general case (ie. think cosmological models, or intrinsic measurement theory where the observer is EMBEDDED IN the much more complex system it's trying to observe), then this abstraction fails.

What I seek is the generalization, where the statistical enterpretation corresponds to one limiting case.

In cosmological models, you switch from "statistics" of similarly prepared system, to "statistics" of many measurements perhaps of the same system... but what is lacking is a coherent abstracting on this that makes sense in the general case - in particular are we switching from descpritive views to a decition theoretic view where we do not have acces to the limiting cases of "perfectly known ensembles" to beeing forced to evaluate the action based upo incomplete but rational "counting of evidence"; without reference to ensembles (because they only exists in the limiting sense, and the limit isn't at hand in cosmologilca or inside views).

My motivation is taht we need to understand this general case to solve the open issues of unification/theory scaling and QG.

/Fredrik

#### Varon

What makes you say that? The statistical interpretation is really just quantum mechanics with no additional assumptions. (I've read enough of the article now to see that he doesn't actually assume that every particle has a well-defined position and momentum at all times. He just claims that this wouldn't contradict QM. Maybe he's right, maybe he's wrong. Either way, it doesn't affect the statistical interpretation, since that claim isn't a part of its definition). So how could the statistical interpretation be wrong? Is your point that QM is wrong?

You should probably select your sources more carefully. It's actually against the forum rules to post links to questionable material. This guy rejects just about everything except the idea that a mind isn't physical. (It gets funny when he rejects particles because of "no evidence", but doesn't require evidence for his claim about minds).
Oh. I just saw that single web page. I didn't see the main index or other parts so didn't know he is a crackhead.

So? What makes him think that there's a version of QM that can tell us that?

He was certainly aware that QM violates Bell inequalities. The experiments only confirmed that reality does too. So the experiments weren't that relevant.

Right, but if that's true, then the same can be said about QM itself (since the statistical interpretation is just QM without additional assumptions), so that problem can't be solved by another interpretation.
I thought in the Solvay congress in 1927. Einstein already saw the weakness in the Statistical Interpretation (I was reading a history of the debate). Maybe Ballentine's formulation is an update. Anyway. In a single 430 atom-buckyball at a time double slit experiment.. how does the single buckyball move from the emitter to the detector? Did the Statistical Interpretation answer this or doesn't it know? If it doesn't know. Then it is not an explanation. It just smears it into ensemble of 'don't knows', isn't it.

#### Varon

Here's Gary Bowman take on the Statistical Interpretation in his book "Essential Quantum Mechanics":

"Most texts adopt—implicitly or explicitly—the so-called Copenhagen interpretation
of quantum mechanics. Like other, conflicting interpretations, Copenhagen is an attempt to tell us what’s really “going on”—to inform us of the quantum world beyond the formalism itself.

By contrast, the statistical interpretation arguably is not an interpretation, but a broad framework that describes how quantum mechanics works in actual practice. You may worry that by learning quantum mechanics from this perspective, you’ll be at a disadvantage. On the contrary, the statistical interpretation provides an understanding one can have confidence in: because it’s “just” a framework, it remains compatible with other approaches, such as Copenhagen, while avoiding many conceptual puzzles
that arise within them.

But what is the statistical interpretation? In a sense, it simply amounts to the following edict: take seriously what quantum mechanics does tell us, and don’t take seriously what quantum mechanics doesn’t tell us."

~~~~~~~~~~~~~~~~~~~~~

I wonder if there are many variants of the Statistical Interpretations (like Many worlds or Copenhagen which have many variants) or there is only one SI which came from Ballentine. Can anyone confirm?

So Statistical Interpretation is simply a framework. Not really an interpretation. It is the pragmatist framework. This means it is compatible with all interpretations and if you are serious about knowing what made QM tick at the very heart, you still have to entertain other interpretations after accepting the Statistical I. Framework. Agree??

#### strangerep

[...] take seriously what quantum mechanics does tell us, and don’t take seriously what quantum mechanics doesn’t tell us. [...]
Your post was reasonable up to this point.

But then...

[...] if you are serious about knowing what made QM tick at the very heart, you still have to entertain other interpretations after accepting the Statistical I. Framework.
No, the opposite is true. You don't need to waste any brain energy at all on fictional stories that have become grafted onto the exterior of QM theory.

#### Fredrik

Staff Emeritus
Gold Member
I thought in the Solvay congress in 1927. Einstein already saw the weakness in the Statistical Interpretation (I was reading a history of the debate). Maybe Ballentine's formulation is an update.
I don't think so. As far as I know, he only rejected the unnecessary and unjustified assumption that a wavefunction describes all features of a single system. Ballentine takes that rejection as the definition of the statistical interpretation.

In a single 430 atom-buckyball at a time double slit experiment.. how does the single buckyball move from the emitter to the detector? Did the Statistical Interpretation answer this or doesn't it know? If it doesn't know. Then it is not an explanation. It just smears it into ensemble of 'don't knows', isn't it.
That's right. (Close enough anyway). But why would you think that QM actually contains an explanation of that sort? This isn't implied by anything that's known for sure.

I agree with Strangerep's comments in the post above this one.

#### Varon

What makes you say that? The statistical interpretation is really just quantum mechanics with no additional assumptions. (I've read enough of the article now to see that he doesn't actually assume that every particle has a well-defined position and momentum at all times. He just claims that this wouldn't contradict QM. Maybe he's right, maybe he's wrong. Either way, it doesn't affect the statistical interpretation, since that claim isn't a part of its definition). So how could the statistical interpretation be wrong? Is your point that QM is wrong?
I found out Ballentine specifically claimed that a particle has a well defined position and momentum at all times! Here's the relevant quotes. Don't you agree with him? Why?

Page 8.

"This statement is often supported by one or both of the following arguments:

(i) A measurement of q causes an unpredictable and uncontrollable disturbance of p, and vice versa. [This was first proposed by Heisenberg (1927) and is widely repeated in text books].

(ii) The position and momentum of a particle do not even exist with simultaneously and pefectly well defined (though perhaps unknown) values (Bohm, 1951, p.100)"
<snip>
"Argument (ii) is easily seen to be unjustified"
<snip>
"Using de Broglie's relation between momentum and wavelength, p = h / wavelength, it is then asserted that a particle cannot have definite values of both position and momentum at any instant. But this conclusion rests on the almost literal identication of the particle with the wave packet (or what amounts to the same thing, the assumption that the wave function provides an exhaustive description of the properties of the particle)."
<snip>
"A consistent application of the Statistical Interpretation yields the correct conclusion that the division of the wavepacket yields the relative probabilities for transmission and reflection of particles. But there is no justification for assertion (ii)"

You should probably select your sources more carefully. It's actually against the forum rules to post links to questionable material. This guy rejects just about everything except the idea that a mind isn't physical. (It gets funny when he rejects particles because of "no evidence", but doesn't require evidence for his claim about minds).

So? What makes him think that there's a version of QM that can tell us that?

He was certainly aware that QM violates Bell inequalities. The experiments only confirmed that reality does too. So the experiments weren't that relevant.

Right, but if that's true, then the same can be said about QM itself (since the statistical interpretation is just QM without additional assumptions), so that problem can't be solved by another interpretation.

#### Varon

Ballentine paper assumes all particles have positions at all times. This means in Bell's Theorem. He indeed believed that the particles were connected with superluminal link? In Ballentine 1989 textbook which I studied, he mentioned:

Are the experiments conclusive?
If we accept the theoretical arguments that quantum mechanics is incompatible with locality, the next question is whether the experiments are adequate for ruling out locality. We have already seen that, strictly speaking, they are not, because of inefficiencies of the detectors and other instrumental problems. However, the fact that those photon pairs that are detected are correlated in the manner predicted by quantum theory is certainly strong evidence for the correctness of those predictions. Although it is possible to devise local models that would obey Bell’s inequality for ideal detectors, but which agree with quantum theory for the imperfect instruments presently available, such models seem rather contrived. This is especially true in view of the fact that the effect of the various systematic errors that experimentalists have studied is to reduce the coincidence detection rate. But quantum theory predicts a
coincidence rate that is greater than is permitted by Bell’s inequality.
Question. Anything wrong by assuming entangled particles exist at all times even 100 billion light years away and since Bell's Theorem is violated, they really are connected with superluminal link? This is the consequence of Ballentine's Statistical Interpretation.

Bohr arguments was the particles attributes like position didn't exist before measurements, so there was no non-local link because the particles wasn't there at all.

#### Fredrik

Staff Emeritus
Gold Member
I found out Ballentine specifically claimed that a particle has a well defined position and momentum at all times!
In section 5, titled "Joint probability distributions", he says

...quantum theory is not inconsistent with the supposition that a particle has at any instant both a definite position and a definite momentum, although there is a widespread folklore to the contrary. Of course we are not compelled either to accept or reject this supposition, but it is of interest to explore it on a tentative basis.​

So he isn't really claiming that particles have well-defined positions and momenta at all times. He's just saying that they might have well-defined positions and momenta at all times. I'm not convinced that this is true. However, if he's wrong, it doesn't have any implications for the statistical interpretation, since this assumption isn't part of its definition.

#### Varon

In section 5, titled "Joint probability distributions", he says

...quantum theory is not inconsistent with the supposition that a particle has at any instant both a definite position and a definite momentum, although there is a widespread folklore to the contrary. Of course we are not compelled either to accept or reject this supposition, but it is of interest to explore it on a tentative basis.​

So he isn't really claiming that particles have well-defined positions and momenta at all times. He's just saying that they might have well-defined positions and momenta at all times. I'm not convinced that this is true. However, if he's wrong, it doesn't have any implications for the statistical interpretation, since this assumption isn't part of its definition.
maybe not the statistical interpretation but he mentioned that when using the capital Statistical Interpretation, it's his string of ideas. So maybe we must distinguish them in the future by using smaller or capital letters when mentioning this particular interpretation to denote general or Ballentine version.

#### Fredrik

Staff Emeritus
Gold Member
I don't think it would be helpful to have two different definitions of "statistical interpretation" and "Statistical Interpretation".

I read section 1.3 again. It bothers me a lot that he says

In contrast, the Statistical Interpretation considers a particle to always be at some position in space, each position being realized with relative frequency $|\psi(\vec r)|^2$ in an ensemble of similarly prepared experiments.​

This bothered me just as much the first time I read it, but when I got to the part (section 5) where he says that it's not necessary to assume that particles always have well-defined positions and momenta, I sort of forgot how strong the statement in section 1.3 is.

What bothers me the most is that section 1.3 is supposed to be the one that defines the statistical interpretation. This means that the statistical interpretation does include the assumption that particles always have well-defined positions. I find this very strange. The main idea is to reject an unnecessary assumption, and then he goes and makes another unnecessary assumption!?

Edit: On the other hand, in section 1.2, he says

Although there are many shades of interpretation (Bunge, 1956), we wish to distinguish only two:

(I) The Statistical Interpretation, according to which a pure state (and hence also a general state) provides a description of certain similarly prepared systems, but need not provide a complete description of an individual system.

(II) Interpretations which assert that a pure state provides a complete and exhaustive description of an individual system (e.g. an electron).

So he's sort of contradicting himself. Section 1.3 says that the statistical interpretation includes the assumption that particles have well-defined positions at all times. Section 5 says that there's no need to assume that. And yet, section 1.2 talks about "the" statistical interpretation, as if there's only one. It would have made more sense to talk about "statistical interpretations".

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#### strangerep

I read [Ballentine, SI of QM paper] section 1.3 again. It bothers me a lot that he says

In contrast, the Statistical Interpretation considers a particle to always be at some position in space, each position being realized with relative frequency $|\psi(\vec r)|^2$ in an ensemble of similarly prepared experiments.​

[...]

What bothers me the most is that section 1.3 is supposed to be the one that defines the statistical interpretation. This means that the statistical interpretation does include the assumption that particles always have well-defined positions.
No it doesn't. Ballentine said "some position", not "well-defined position".

Edit: On the other hand, in section 1.2, he says

Although there are many shades of interpretation (Bunge, 1956), we wish to distinguish only two:

(I) The Statistical Interpretation, according to which a pure state (and hence also a general state) provides a description of certain similarly prepared systems, but need not provide a complete description of an individual system.

(II) Interpretations which assert that a pure state provides a complete and exhaustive description of an individual system (e.g. an electron).

So he's sort of contradicting himself. Section 1.3 says that the statistical interpretation includes the assumption that particles have well-defined positions at all times.
He's not contradicting himself. That quote from sect 1.2 is about sharply distinguishing different classes of interpretations of QM. And as I said above, sect 1.3 doesn't say what you said it does.

#### Jaisonsc

Im May, 7, Chopin sad: In addition to using a Hilbert state to describe a particle at a position, you can use it to describe a particle with a certain charge, or a certain momentum, or even something more abstract like which slit a particle goes through in the double-slit experiment.

Somebody know's how is possible to do the idea that I put in italics types?
Some article ou so on...
Thanks!!

#### IRobot

Could we say that in the path integral formalism we don't need a Hilbert space, at least to write down the functional integral, not to calculate with it.

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