How useful are indefinite state spaces?

[A. Neumaier]

You have a viable second option that doesn't resort to miracles?

Yes. See the reference given in #49.

 

[Maui]

Yes. See the reference given in #49.

You have a no go theorem that explicitly restricts deterministic models. My layman opinion says that if you are proposing another non-local HV theory, that'd be just another case of magic.

 

[Careful]

See Sections 7.3 - 7.5 of http://lanl.arxiv.org/abs/0810.1019

There is no way in conventional QM do to what you claim, there are actually theorems about this. If you think you have something, please summarize in a few lines. I am sure it will be great fun.

 

[A. Neumaier]

There is no way in conventional QM do to what you claim, there are actually theorems about this. If you think you have something, please summarize in a few lines. I am sure it will be great fun.

Oh, I thought you'd understand the need for more than a few lines to do something beyond what conventional QM can do:

no, I cannot explain these things even on one full A4 page since it requires many subtle considerations and additional concepts.

 

[A. Neumaier]

You have a no go theorem that explicitly restricts deterministic models. My layman opinion says that if you are proposing another non-local HV theory, that'd be just another case of magic.

It requires no magic to avoid a no-go theorem by not satisfying its assumptions or conclusions.

 

[Maui]

It requires no magic to avoid a no-go theorem by not satisfying its assumptions or conclusions.

The locality assumption or the realism(determinsim) assumption? Or do you propose some caveat? You are either too good or too naive(we most are anyway).

 

[A. Neumaier]

The locality assumption or the realism(determinsim) assumption? Or do you propose some caveat? You are either too good or too naive(we most are anyway).

Let me propose that you first look at the reference and try to understand it. Unlike Careful's treatise, it is not technical.

 

[Hurkyl]

Von Neumann's cannot be sensible since for consistency it requires a hierarchy of bigger and bigger superobservers, which is nonsense.

Accepting for the sake of argument it requires a hierarchy of superobservers, I really don't see why such a thing should be nonsensical.

I could easily imagine an argument that in reality that such a hierarchy would contain much more information than is accessible to us, but that's a rather normal state of affairs for physical theories, rather than being nonsense.

 

[A. Neumaier]

Accepting for the sake of argument it requires a hierarchy of superobservers, I really don't see why such a thing should be nonsensical.

I could easily imagine an argument that in reality that such a hierarchy would contain much more information than is accessible to us, but that's a rather normal state of affairs for physical theories, rather than being nonsense.

Let the first order observer observe a single particle, and for k>1, let the k-th order observer observe the (k-1)-st order observer.

We first fix the situation von Neumann is talking about: ideal measurements that put the system instantaneously into a well-defined eigenstate, hence measure a complete set of observables of the system.

An observer (together with the observing equipment) must therefore be much larger than the object it observes, since it must have essentially classical pointer readings for each particular observable in the complete set. If the observed system consists of n quantum particles, it has 3n continuous quantum degrees of freedom. hence the observer must have 3n classical pointers, each comprising at least N atoms, where for a reasonable accuracy, N>>100. Therefore the observer consists of a system consisting of n particles consists of at least 300n particles. But this is an extremely conservative estimate, since we ignore all the logistics that is necessary to make such complete measurements with some reliability.

The k-th order observer of a single particle (n=1) therefore contains at least 300^k particles. Assuming the number of particles in the universe to be U, we see that the existence of the k-th observer implies
k\le log U/log 300 < 33
when U=10^81. Even allowing lots of unknown dark matter will not change the resulting upper bound by much.

The only escape to this argument is to assume that the number of particles is infinite.
But even then there appear to be unsurmountable problems with the k-th observer measuring all details approximately instantaneously, given that the single particle observed by the lowest order observer is on the Earth and we know quite well the distribution of particles close enough to the Earth to be able to neglect relativistic delays.

 

[Careful]

Dear Arnold,

Your scheme needs a superobserver as well, you cannot escape that. Let me hint you were the devil is: you uncritically assume that a ''macroscopic'' ''apparatus'' behaves more or less classically. This goes straight against Schrodinger's cat argument of course; whatever way you chose to escape, I guarantuee you upfront that you won't escape Von Neumann's conclusion. It appears to me that you did not follow the logical implications of your own writings far enough.

Moreover, your reasoning about the tower of observations is ''ganz falsch''.

Johan

 

[Hurkyl]

We first fix the situation von Neumann is talking about: ideal measurements that put the system instantaneously into a well-defined eigenstate, hence measure a complete set of observables of the system.

So you're not talking about this at all?

The only escape to this argument is to assume that the number of particles is infinite.

That's to be expected from an infinite hierarchy of observers. We were expecting "nearly infinite" particles anyways anyways, so that thermodynamics could become relevant.

But even then there appear to be unsurmountable problems with the k-th observer measuring all details approximately instantaneously, given that the single particle observed by the lowest order observer is on the Earth and we know quite well the distribution of particles close enough to the Earth to be able to neglect relativistic delays.

I can't figure out what you're trying to argue here. I will make a comment that may or may not be relevant, though: how close to instantaneous "approximately instantaneously" must be depends on scale. Someone observing an experiment in a lab would probably require it to be less than a millisecond, whereas a couple hours probably counts for someone observing the solar system.

 

[A. Neumaier]

So you're not talking about this at all?

fix was meant in the sense of ''make precise'', not of ''correct''.

That's to be expected from an infinite hierarchy of observers. We were expecting "nearly infinite" particles anyways anyways, so that thermodynamics could become relevant.

The point is the exponential increase in the size of the k-th observer. Actually the factor 300 should be more like 10^8 or so, so that thermodynamics applies and produces approximately classical pointer readings. Indeed, if we consider an everyday measurement of the presence of a particle by a Geiger counter, the factor is already 10^20, and it is completely impossible to install machinery that measures the complete state of a Geiger counter. Interactions beyond the Earth are far too weak to gain significant information about the counter, except perhaps to photograph it from space, which gives < 10^7 degrees of freedom while >10^20 are needed, and no amount of namotechnology can change this significantly.

Someone observing an experiment in a lab would probably require it to be less than a millisecond, whereas a couple hours probably counts for someone observing the solar system.

Well, in this time, a decent quantum system has already changed its state so much that at completion of the experiment one can no longer claim to have measured the state at the beginning.

Thus even a second-order observer is an illusion.

Considering that many measurements on Earth are actually made, and _all_these_ would have to be collapsed by super-observers, things get even more fantastic. This is no longer physics but science fiction.

 

[A. Neumaier]

you uncritically assume that a ''macroscopic'' ''apparatus'' behaves more or less classically.

I only need that the tip of the pointer behaves classically. This is a well-known empirical fact without which we couldn't do any measurement at all. This is at the basis of my interpretation.

On the other hand, von Neumann's interpretation (and the underlying Born rule) is based on assumptions that are so highly idealized that they can't be realized except for extremely tiny systems.

Moreover, your reasoning about the tower of observations is ''ganz falsch''.

Well, where is the mistake?

 

[Careful]

I only need that the tip of the pointer behaves classically. This is a well-known empirical fact without which we couldn't do any measurement at all. This is at the basis of my interpretation.

On the other hand, von Neumann's interpretation (and the underlying Born rule) is based on assumptions that are so highly idealized that they can't be realized except for extremely tiny systems.




Well, where is the mistake?

Before we proceed, are you prepared to aknowledge that you made a mistake in case every sensible person can see that you are wrong? Because, the main problem resides there.

 

[A. Neumaier]

Before we proceed, are you prepared to aknowledge that you made a mistake in case every sensible person can see that you are wrong? Because, the main problem resides there.

Strange that you ask. I never had problems with learning.

 

[Careful]

Strange that you ask. I never had problems with learning.

First, that is a matter of perception (and I strongly disagree with your self-assesment); second, what I said is that you have problems with admitting your own shortcomings and acting rationally upon them. So, I ask again, if you are mistaken, will you as a true gentlemen admit so? I have no problems to admit when I make a mistake, but since you claim to outsmart Von Neumann and Wigner, I would think that in case you are proven wrong, a mild acknowledgment is in place.

 

[Hurkyl]

fix was meant in the sense of ''make precise'', not of ''correct''.

The reason I asked was because you don't really seem to be talking about the measurement I linked. You seem to be making four significant, unwarranted hypotheses that turn your argument into a straw-man.

The first is your hypothesis that a measurement be instantaneous. I have no idea about the original source, but it certainly wasn't required in the link I gave, nor is there any obvious reason why it should be so. What is expected is just that the joint object - measuring device - environment^* system undergoes unitary time evolution.

The second is your hypothesis that the construction and reading of the measuring device must be practical. Again there was no such hypothesis in the link, nor any obvious reason why it should be so. Even if we wanted to consider the special case of a real-world measurement in a laboratory, we still don't even require distinguishing between all states of the device to be anything resembling feasible -- many device states will correspond to the same reading.

The third is that the observer & measuring device must resemble a human and a real-life device we could call a measuring device. (or even that there must be an observer!)

The fourth one is the hypothesis that the measurement completely distinguishes the states of the object of study. While this is included in the link I mentioned, there is no obvious reason why it should be taken as a requirement.


*: I hope you don't mind me making this obvious extension.

 

[A. Neumaier]

The reason I asked was because you don't really seem to be talking about the measurement I linked.

Indeed, I didn't realize that there was an embedded link. The context didn't say there was one, and my browser indicates embedded links only when the mouse is directly upon them.

I thought that ''this'' referred to the text you had quoted.

I'll respond to the link separately, in a different thread since the topic has no longer anything to do with the title of the thread.

 

[A. Neumaier]

What is expected is just that the joint object - measuring device - environment^* system undergoes unitary time evolution.

*: I hope you don't mind me making this obvious extension.

No, I don't.

My reply to your array of comments is in post #7 of the thread https://www.physicsforums.com/showthread.php?p=3128309