The Time Symmetry Debate in Quantum Theory

[stevendaryl]

I think you are missing my point for some reason. I was relating the measurement problem precisely to the insistence in analogies with the classical particle model.

Maybe you could say a little more about the connection between the two (the measurement problem, and classical particle model). The basic axioms of quantum mechanics don't mention particles or classical mechanics. They are something like:

1. The set of possible states of a system are the normalized vectors of a Hilbert space.
2. Each observable/measurement corresponds to a self-adjoint linear operator.
3. The results of a measurement corresponding to an operator O always produces an eigenvalue of O.
4. The probability that a measurement of an observable corresponding to operator O produces outcome o_i is given by:
   | \langle \psi | P_i |\psi \rangle |^2, where P_i is the projection operator onto the subspace of the Hilbert space corresponding to the states with eigenvalue o_i.
5. Immediately after the measurement, if the result was o_i, then the system will be in state P_i | \psi \rangle.
6. Between measurements, the system evolves according to equation:
   \dfrac{d}{dt} | \psi \rangle = -i H | \psi \rangle (in units where h-bar = 1), where H is the Hamiltonian operator, the operator associated with the total energy of the system.

I'm not worried too much about whether I've got these exactly right, or whether there is disagreement about them, or whether some axioms can be proved to follow from others, or from more basic considerations. I'm just putting a straw man collection out there so that you can say which one has to do with assuming a classical notion of "particle". I don't see it.

 

[bhobba]

[*]Between measurements, the system evolves according to equation:
\dfrac{d}{dt} | \psi \rangle = -i H | \psi \rangle (in units where h-bar = 1), where H is the Hamiltonian operator, the operator associated with the total energy of the system.

I think at least one of his concerns is the form of the Hamiltonian (ie H = (p^2)/2*m + V(x)) is the same as classical physics. I suspect he believes that is because of classical analogies. My point is that the form follows from symmetry considerations (that's what Ballentine proves in Chapter 3) - not classical analogies. The same is true in classical mechanics which is the point of my reference to Landau's classic (he proves that form from symmetry considerations as well). That Classical Mechanics is like this is because QM is like this - not the other way around.

Thanks
Bill

 

[TrickyDicky]

Sigh. It looks like there is a schism here that's difficult to overcome. For example 'classical symmetries' - there is nothing classical about the POR that these symmetries derive from - its true relativistically, QM, classically, all sorts of ways.
It is these symmetries that determine the dynamics of QM - not an analogy to anything. Even if you know nothing of classical physics exactly the same equations result.

The reason I recommended Landau is it shows classical mechanics is really about symmetry so its no surprise QM is also about it. In fact the reason classical mechanics is about symmetry is because QM is about symmetry. Its not the other way around - QM is not based on classical analogies - it based on the implications of symmetry and that implies that classical mechanics is also. In QM the symmetries are in the quantum state and observables - in classical mechanics its in the Lagrangian. But as Feynman showed the Lagrangian follows from the rules governing states and observables.

Thanks
Bill

It's no use to keep circling around this. I'm not disagreeing with this rather with the way you present it anyway.
Instead and since I think you favor the statistical interpretation, QFT is also related to statistical field theory by a Wick rotation. I sometimes entertain myself relating ensemble properties to field properties.

 

[TrickyDicky]

Maybe you could say a little more about the connection between the two (the measurement problem, and classical particle model). The basic axioms of quantum mechanics don't mention particles or classical mechanics. They are something like:

1. The set of possible states of a system are the normalized vectors of a Hilbert space.
2. Each observable/measurement corresponds to a self-adjoint linear operator.
3. The results of a measurement corresponding to an operator O always produces an eigenvalue of O.
4. The probability that a measurement of an observable corresponding to operator O produces outcome o_i is given by:
   | \langle \psi | P_i |\psi \rangle |^2, where P_i is the projection operator onto the subspace of the Hilbert space corresponding to the states with eigenvalue o_i.
5. Immediately after the measurement, if the result was o_i, then the system will be in state P_i | \psi \rangle.
6. Between measurements, the system evolves according to equation:
   \dfrac{d}{dt} | \psi \rangle = -i H | \psi \rangle (in units where h-bar = 1), where H is the Hamiltonian operator, the operator associated with the total energy of the system.

I'm not worried too much about whether I've got these exactly right, or whether there is disagreement about them, or whether some axioms can be proved to follow from others, or from more basic considerations. I'm just putting a straw man collection out there so that you can say which one has to do with assuming a classical notion of "particle". I don't see it.

Well if in axioms 4 and 5 you associate the observables with properties of a classical particle (considering a classical particle an object with a trajectory) you hit the measurement problem, otherwise you don't.
Besides if one thinks about the reason one chooses a linear space as axiomatic in the first place one realizes it demands point particles, just like Euclidean space in classical mechanics demands classical point particles.

 

[TrickyDicky]

I think at least one of his concerns is the form of the Hamiltonian (ie H = (p^2)/2*m + V(x)) is the same as classical physics. I suspect he believes that is because of classical analogies. My point is that the form follows from symmetry considerations (that's what Ballentine proves in Chapter 3) - not classical analogies. The same is true in classical mechanics which is the point of my reference to Landau's classic (he proves that form from symmetry considerations as well). That Classical Mechanics is like this is because QM is like this - not the other way around.

Thanks
Bill

Certainly Schrodinger derived the Hamiltonian from classical mechanics whether one considers it a historical accident or not.

 

[bhobba]

Besides if one thinks about the reason one chooses a linear space as axiomatic in the first place one realizes it demands point particles, just like Euclidean space in classical mechanics demands classical point particles.

I don't get that at all.

For example check out the link I posted before:
http://arxiv.org/pdf/0911.0695v1.pdf

Precisely where in that axiomatic development is point particles assumed?

Thanks
Bill

 

[bhobba]

Certainly Schrodinger derived the Hamiltonian from classical mechanics whether one considers it a historical accident or not.

The point though is a lot of water has passed under the bridge since then and its real basis is now known. Like I said Wigner got a Nobel prize in part for figuring it out.

Thanks
Bill

 

[TrickyDicky]

The point though is a lot of water has passed under the bridge since then and its real basis is now known. Like I said Wigner got a Nobel prize in part for figuring it out.

Thanks
Bill

Thus my "whether one considers it a historical accident or not".

 

[stevendaryl]

Well if in axioms 4 and 5 you associate the observables with properties of a classical particle (considering a classical particle an object with a trajectory) you hit the measurement problem, otherwise you don't.
Besides if one thinks about the reason one chooses a linear space as axiomatic in the first place one realizes it demands point particles, just like Euclidean space in classical mechanics demands classical point particles.

I don't see that, at all. To me, the "measurement problem" is the conceptual difficulty that on the one hand, a measurement has an abstract role in the axioms of quantum mechanics, as obtaining an eigenvalue of a self-adjoint linear operator, and it has a physical/pragmatic/empirical role in actual experiments as a procedure performed using equipment. What is the relationship between these two notions of measurement? The axioms of quantum mechanics don't make it clear.

I don't see that it has anything particularly to do with particles.

There is a pragmatic issue, which is that we don't really know how to measure arbitrary observables. There are only a few that we know how to measure: position, momentum, energy, angular momentum. It might be correct to say that we only know how to measure observables with classical analogs. But you seem to be saying something different, that the measurement problem only comes up because we're insisting on classical analogues. I don't see that, at all. We can certainly pick an observable with no classical analog. The measurement problem doesn't go away, it becomes worse, because we don't know how to measure it.

 

[TrickyDicky]

I don't see that, at all. To me, the "measurement problem" is the conceptual difficulty that on the one hand, a measurement has an abstract role in the axioms of quantum mechanics, as obtaining an eigenvalue of a self-adjoint linear operator, and it has a physical/pragmatic/empirical role in actual experiments as a procedure performed using equipment. What is the relationship between these two notions of measurement? The axioms of quantum mechanics don't make it clear.

I don't see that it has anything particularly to do with particles.

There is a pragmatic issue, which is that we don't really know how to measure arbitrary observables. There are only a few that we know how to measure: position, momentum, energy, angular momentum. It might be correct to say that we only know how to measure observables with classical analogs. But you seem to be saying something different, that the measurement problem only comes up because we're insisting on classical analogues. I don't see that, at all. We can certainly pick an observable with no classical analog. The measurement problem doesn't go away, it becomes worse, because we don't know how to measure it.

Well if you reduce the discussion to abstract observables without attributing them to any particular object be it a particle, a field or whatever, you don't have a way to connect it with the physical/pragmatic side so no measurement problem for you, but as Ballentine warned in the quote posted by devil then you don't really have a physical theory but just a set of abstract axioms without connection with experiment.
Quote by Quantum Mechanics - A Modern Development, Leslie E. Ballentine (1998)
Kinematics and Dynamics:
"The results of Ch. 2 constitute what is sometimes called “the formal structure of quantum mechanics”. Although much has been written about its interpretation, derivation from more elementary axioms, and possible generalization, it has by itself very little physical content. It is not possible to solve a single physical problem with that formalism until one obtains correspondence rules that identify particular dynamical variables with particular operators."
It is in those correspondence rules that the problem arises, and depending on how one interprets the Born rule, for instance, you might have a smaller or bigger problem.
I would say the Born rule was devised having the particle picture of classical mechanics in mind. Don't you?
When you were talking about observables were you thinking about them in terms of properties of particles, fields...?

 

[bhobba]

It is in those correspondence rules that the problem arises, and depending on how one interprets the Born rule, for instance, you might have a smaller or bigger problem.

I can't follow you there. I know Ballentine pretty well and what he shows is based on the invarience of probabilities the dynamics follows.

I would say the Born rule was devised having the particle picture of classical mechanics in mind. Don't you?

What do you mean by devised? Historically - probably - but so? We now know it follows from much more general considerations having nothing to do with particles eg Gleason's theorem.

Thanks
Bill

 

[stevendaryl]

Well if you reduce the discussion to abstract observables without attributing them to any particular object be it a particle, a field or whatever, you don't have a way to connect it with the physical/pragmatic side so no measurement problem for you, but as Ballentine warned in the quote posted by devil then you don't really have a physical theory but just a set of abstract axioms without connection with experiment.

I think that's what I intended to say: that the "measurement problem" is about the connection between the notion of "observable" that is a primitive in the quantum theory, and the "observable" that is something that requires a measurement apparatus and a measurement procedure. But I don't see how that supports the claim that the measurement problem has anything to do, intrinsically, with classical properties of particles.

I would say the Born rule was devised having the particle picture of classical mechanics in mind. Don't you?

No, I don't see much of a connection between the two. The Born rule about probabilities, or something like it, is forced on us by the assumption, or empirical fact, that an observation always produces an eigenvalue of the corresponding operator, and that operators don't commute (so it's not possible for all observables to have definite values simultaneously). I don't see that there is anything particularly particle-like about any of this.

When you were talking about observables were you thinking about them in terms of properties of particles, fields...?

They are properties of a system, as a whole. The electric and magnetic field at a point in space is an observable. The mass, position, momentum, magnetic moment of a lump of iron are all observables. Yes, those observables are all macroscopic sums of observables associated with individual atoms of the iron, but the observables don't require particles to make sense of them. So I really don't understand the point you are making about the relationship between observables and particles.

 

[Jano L.]

No. Kolgmorgrov's axioms are clear on this point:
http://en.wikipedia.org/wiki/Probability_axioms
'This is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space.'

If something has probability 1 it must occur.

I think it will be easier to explain this by example. Consider random process that produces series of red points somewhere in a unit disk with uniform probability density. The probability of the event that the next point will concide with any point A of the disk is equal to 0.

However, after the event occurs, some point of the disk will be red. At that instant, an event with probability 0 has happened.

Actually, all events that happen in such random process are events that have probability 0.

So "event has probability 0" does not mean "impossible event".

Similarly, "probability 1" does not mean "certain event". Consider probability that the red point will land at point with both coordinates irrational. This can be shown to be equal to 1 in standard measure theory. However, there is still infinity of points that have rational coordinates, and these can happen - they are part of the disk.

In the language of abstract theory, all this is just a manifestation of the fact that equal measures do not imply that the sets are equal.

 

[Jano L.]

I would say the Born rule was devised having the particle picture of classical mechanics in mind. Don't you?

Very good point. As far as know, there are actually two Born rules, although people tend to think they are the same.

The first rule, well working in scattering and quantum chemistry, is the assumption that

$$
|\psi(\mathbf r)|^2 ~\Delta V,
$$

gives probability that the particle is in the small volume element $\Delta V$ around $\mathbf r$.

This really refers to particles and their configuration.

The second rule, I think proposed after the first one, is that

$$
p_k = |\langle \phi_k|\psi \rangle|^2
$$

gives the probability that the system in state $\psi$ will manifest energy $E_k$ (or get into state $\phi_k$ in other versions) when "measurement of energy" is performed (or even spontaneously, due to interaction with environment in other versions). This is more abstract and does not require particles.

We should really distinguish these two rules. The first one is easy and does not depend on the measurement problem, and is gauge-invariant.

The second is difficult to understand, because it is connected to measurements and is gauge-dependent - if we choose different gauge to calculate $\psi$, we get different $p_k$.

 

[lugita15]

No. Kolgmorgrov's axioms are clear on this point:
http://en.wikipedia.org/wiki/Probability_axioms
'This is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space.'

If something has probability 1 it must occur.

It's true that if an event will definitely occur, then it must have probability 1. But it's not the case that if an event has probability 1, it will definitely occur. See this wikipedia page.

 

[TrickyDicky]

I think it will be easier to explain this by example. Consider random process that produces series of red points somewhere in a unit disk with uniform probability density. The probability of the event that the next point will concide with any point A of the disk is equal to 0.

However, after the event occurs, some point of the disk will be red. At that instant, an event with probability 0 has happened.

Actually, all events that happen in such random process are events that have probability 0.

So "event has probability 0" does not mean "impossible event".

Similarly, "probability 1" does not mean "certain event". Consider probability that the red point will land at point with both coordinates irrational. This can be shown to be equal to 1 in standard measure theory. However, there is still infinity of points that have rational coordinates, and these can happen - they are part of the disk.

In the language of abstract theory, all this is just a manifestation of the fact that equal measures do not imply that the sets are equal.

It's true that if an event will definitely occur, then it must have probability 1. But it's not the case that if an event has probability 1, it will definitely occur. See this wikipedia page.

Good points that simply go to support Jano L. posts #71, #74, #78... IMO showing that Bill's reliance on Gleason's theorem can not be used in the general case for what he thinks it can, but only for discretized, lattice models of physical systems, a very strong assumption in the light of what we know, or at least I think most physicists still favor a continuous picture of nature as exemplified by successful theories like GR.

What do you mean by devised? Historically - probably - but so?

Probably no, certainly, you just have to read Born's original 1926 paper.

We now know it follows from much more general considerations having nothing to do with particles eg Gleason's theorem.

I wouldn't be so sure we know that. See above.

 

[stevendaryl]

I think it will be easier to explain this by example. Consider random process that produces series of red points somewhere in a unit disk with uniform probability density. The probability of the event that the next point will concide with any point A of the disk is equal to 0.

However, after the event occurs, some point of the disk will be red. At that instant, an event with probability 0 has happened.

Actually, all events that happen in such random process are events that have probability 0.

So "event has probability 0" does not mean "impossible event".

That's certainly true, mathematically. On the other hand, in the real world, we never measure real-valued observables to infinite precision. We never really observe: "The particle's momentum is P", we observe something like "The particle's momentum is somewhere in the range P\ \frac{+}{-} \ \Delta P. For this reason, if we have two states | \psi \rangle and |\phi \rangle such that \langle \psi | \phi \rangle = 1, they are considered the same state, as far as quantum mechanics is concerned. Adding or subtracting a set of measure zero does nothing.

 

[audioloop]

Complete:

.-brought to an end : CONCLUDED

.-fully carried out : THOROUGH

.-To bring to a finish or an end: FINISHED

.-having all parts or elements; lacking nothing; whole; entire: FULL
...

in any order
http://oxforddictionaries.com/definition/english/complete?q=complete
http://dictionary.reference.com/browse/complete
http://www.merriam-webster.com/dictionary/complete
http://www.thefreedictionary.com/complete

 

[Jano L.]

That's certainly true, mathematically. On the other hand, in the real world, we never measure real-valued observables to infinite precision. We never really observe: "The particle's momentum is P", we observe something like "The particle's momentum is somewhere in the range P +− ΔP.

Yes, but we really discussed theoretical difference between probabilistic and deterministic description. I think the limitations of observations have no bearing on that argument.

...if we have two states |ψ⟩ and |ϕ⟩ such that ⟨ψ|ϕ⟩=1, they are considered the same state, as far as quantum mechanics is concerned. Adding or subtracting a set of measure zero does nothing.

It does nothing to probability. That was my point - in probabilistic theory, the probability is 1 both for certain and almost certain event. We cannot adequately describe the difference between the two in such a theory. Ergo deterministic theory is not just a special case of probabilistic theory. They are different kinds of theories constructed for different purposes.

 

[DevilsAvocado]

The electron in orbit was given as an example of a possible path. I have read about this argument before. That an classically orbiting electron should emit radiation presumably because it is a charged object and an accelerated charged object (changing direction) will emit electromagnetic radiation. Yet if you were to put opposite charge on two spheres, one light and one heavy, and then set the lighter in orbit (in space) around the heavier would the two spheres not act just as the two body problem for gravitational force.

... but electromagnetism is 10^39 times stronger than gravitation ...

I think the Bohr model is pretty dead, there are incompatibilities to empirical spectral lines, and it also violates the uncertainty principle, and even if you magically could fix all that – where is your single localized particle in the double-slit experiment?

It doesn’t work...

 

[DevilsAvocado]

This is in fact the defining property of an inertial frame - the Earth isn't exactly inertial but for many practical purposes such as this experiment it is.

I’m sorry bhobba, I’m completely lost... are you saying that the inertial frame of Earth has anything to do with the double-slit experiment??

 

[DevilsAvocado]

QM does not insist on analogies with a classical particle model. All it assumes is position is an observable - which is a fact.

In QM the symmetries are in the quantum state and observables - in classical mechanics its in the Lagrangian.

I don't see that, at all. To me, the "measurement problem" is the conceptual difficulty that on the one hand, a measurement has an abstract role in the axioms of quantum mechanics, as obtaining an eigenvalue of a self-adjoint linear operator, and it has a physical/pragmatic/empirical role in actual experiments as a procedure performed using equipment. What is the relationship between these two notions of measurement? The axioms of quantum mechanics don't make it clear.

I don't see that it has anything particularly to do with particles.

Well if you reduce the discussion to abstract observables without attributing them to any particular object be it a particle, a field or whatever, you don't have a way to connect it with the physical/pragmatic side so no measurement problem for you, but as Ballentine warned in the quote posted by devil then you don't really have a physical theory but just a set of abstract axioms without connection with experiment.

I think that's what I intended to say: that the "measurement problem" is about the connection between the notion of "observable" that is a primitive in the quantum theory, and the "observable" that is something that requires a measurement apparatus and a measurement procedure. But I don't see how that supports the claim that the measurement problem has anything to do, intrinsically, with classical properties of particles.

We should really distinguish these two rules. The first one is easy and does not depend on the measurement problem, and is gauge-invariant.

The second is difficult to understand, because it is connected to measurements and is gauge-dependent - if we choose different gauge to calculate $\psi$, we get different $p_k$.

Guys, it’s very interesting to read this discussion, and this stuff is always hard to talk about. Still, let me give you something to chew on while the ‘battle’ continues.

http://arxiv.org/abs/0707.0401

“The beables of the theory are those elements which might correspond to elements of reality, to things which exist. Their existence does not depend on ‘observation’. Indeed observation and observers must be made out of beables.”​

Or as he explains elsewhere,

“The concept of ‘observable’ ... is a rather woolly concept. It is not easy to identify precisely which physical processes are to be given the status of ‘observations’ and which are to be relegated to the limbo between one observation and another. So it could be hoped that some increase in precision might be possible by concentration on the beables ... because they are there.”​

Bell’s reservations here (about the concept “observable” appearing in the fundamental formulation of allegedly fundamental theories) are closely related to the so-called “measurement problem” of orthodox quantum mechanics, which Bell encapsulated by remarking that the orthodox theory is “unprofessionally vague and ambiguous” in so far as its fundamental dynamics is expressed in terms of “words which, however legitimate and necessary in application, have no place in a formulation with any pretension to physical precision” – such words as “system, apparatus, environment, microscopic, macroscopic, reversible, irreversible, observable, information, measurement.” As Bell elaborates,

“The concepts ‘system’, ‘apparatus’, ‘environment’, immediately imply an artificial division of the world, and an intention to neglect, or take only schematic account of, the interaction across the split. The notions of ‘microscopic’ and ‘macroscopic’ defy precise definition. So also do the notions of ‘reversible’ and ‘irreversible’. Einstein said that it is theory which decides what is ‘observable’. I think he was right – ‘observable’ is a complicated and theory-laden business. Then the notion should not appear in the formulation of fundamental theory.”​

As Bell points out, even Bohr (a convenient personification of skepticism regarding the physical reality of unobservable microscopic phenomena) recognizes certain things (for example, the directly perceivable states of a classical measuring apparatus) as unambiguously real, i.e., as beables.

[...]

The unprofessional vagueness and ambiguity of orthodox quantum theory, then, is related to the fact that its formulation presupposes these (classical, macroscopic) beables, but fails to provide clear mathematical laws to describe them. As Bell explains,

“The kinematics of the world, in [the] orthodox picture, is given by a wavefunction ... for the quantum part, and classical variables – variables which have values – for the classical part... [with the classical variables being] somehow macroscopic. This is not spelled out very explicitly. The dynamics is not very precisely formulated either. It includes a Schrödinger equation for the quantum part, and some sort of classical mechanics for the classical part, and ‘collapse’ recipes for their interaction.”​

There are thus two related problems. First, the posited ontology is rather different on the two sides of (what Bell calls) “the shifty split” – that is, the division between “the quantum part” and “the classical part.” But then, as a whole, the posited ontology remains unavoidably vague so long as the split remains shifty – i.e., so long as the dividing line between the macroscopic and microscopic remains undefined. And second, the interaction across the split is problematic. Not only is the account of this dynamics (the “collapse” process) inherently bound up in concepts from Bell’s list of dubious terms, but the very existence of a special dynamics for the interaction seems to imply inconsistencies with the dynamics already posited for the two realms separately. As Bell summarizes,

“I think there are professional problems [with quantum mechanics]. That is to say, I’m a professional theoretical physicist and I would like to make a clean theory. And when I look at quantum mechanics I see that it’s a dirty theory. The formulations of quantum mechanics that you find in the books involve dividing the world into an observer and an observed, and you are not told where that division comes... So you have a theory which is fundamentally ambiguous...”​

The point of all this is to clarify the sort of theory Bell had in mind as satisfying the relevant standards of professionalism in physics.

Don’t know why I love this paper, but I do – it’s ‘crisp & clear’...

 

[bhobba]

I’m sorry bhobba, I’m completely lost... are you saying that the inertial frame of Earth has anything to do with the double-slit experiment??

It has nothing to do with it per se.

My comment was in relation to the claim the measurement problem had something to do with QM holding the particle picture as fundamental. QM doesn't do that - the dynamics are, just like Classical Mechanics, determined by symmetry arguments. There is no particle assumption other than position is an observable which is an experimentally verified fact.

For many practical purposes the Earth can be considered to have these symmetry properties - that was my point.

Thanks
Bill

 

[bhobba]

Consider probability that the red point will land at point with both coordinates irrational.

Well since the rationals have Lebesque measure zero and there is no way to observationally tell the difference between a rational and rational point, since that would require an infinite measurement precision, it's not a well defined problem physically.

If you seriously doubt a probability of 1 does not mean a dead cert then I think this thread is not the appropriate place to discuss it. I think the Set Theory, Logic, Probability and Statistics statistics subforum is more appropriate so I will be doing a post there.

Thanks
Bill

 

[bhobba]

Gleason's theorem can not be used in the general case for what he thinks it can, but only for discretized, lattice models of physical systems, a very strong assumption in the light of what we know, or at least I think most physicists still favor a continuous picture of nature as exemplified by successful theories like GR.

Gleason's theorem holds for infinite dimensional Hilbert spaces:
http://kof.physto.se/theses/helena-master.pdf

I have zero idea why you would think otherwise.

It even holds for non-separable spaces - not that that is of any value to QM.

The issue with Gleason's theorem is its physical basis is a bit unclear - mathematically what's going on is well understood, it depends on non contextuality, and, again mathematically, contextuality is a bit of an ugly kludge, but exactly, from a physical point of view why you require it is open to debate. This is the exact out Bohmian Mechanics uses and its a valid theory. But the Hilbert space formalism is ugly if you don't assume it - you can't define a unique probability measure so the question is - what use is using a Hilbert space to begin with - and indeed for BM the usual formulation is secondary in that interpretation.

My point is Born's rule is not dependent on a particle model - its basis is non-contextually in the usual formulation, or specific assumptions in other formulations like BM.

Thanks
Bill

 

[bhobba]

Adding or subtracting a set of measure zero does nothing.

Exactly. This is bog standard stuff from more advanced probability texts that take a rigorous approach. Finding probabilities associated with determining rational or irrational numbers is not a well defined problem since the rationals have Lebesque measure zero.

I think a discussion on exactly what probability 0 and 1 means is best dome on the probability subforum and I will do a post there.

Thanks
Bill

 

[TrickyDicky]

Guys, it’s very interesting to read this discussion, and this stuff is always hard to talk about. Still, let me give you something to chew on while the ‘battle’ continues.

http://arxiv.org/abs/0707.0401


Don’t know why I love this paper, but I do – it’s ‘crisp & clear’...

Because it brings us Bell in his deep and intelligent own words, contrary to the tradition of misinterpreting him that abounds in QM literature .

 

[TrickyDicky]

The issue with Gleason's theorem is its physical basis is a bit unclear - mathematically what's going on is well understood, it depends on non contextuality, and, again mathematically, contextuality is a bit of an ugly kludge, but exactly, from a physical point of view why you require it is open to debate. This is the exact out Bohmian Mechanics uses and its a valid theory. But the Hilbert space formalism is ugly if you don't assume it - you can't define a unique probability measure so the question is - what use is using a Hilbert space to begin with - and indeed for BM the usual formulation is secondary in that interpretation.

My point is Born's rule is not dependent on a particle model - its basis is non-contextually in the usual formulation, or specific assumptions in other formulations like BM.

Thanks
Bill

Bill, I agree with the quoted part.
Non-contextuality is a strong assumption IMO. But yes it makes the Hilbert formalism "ugly" not to adopt it. But Gleason's theorem assumes non-contextuality and that was the sense of my comment about lack of generality of the theorem as there are QM interpretations that don't assume non-contextuality (you mentioned BM but there are also the modal interpretations and others).
I have a doubt about this because I've seen quantum non-contextuality defined in two ways that I guess are equivalent, maybe you can help me connect them: as referred to independence of the measurement arrangement and as basis independence of the probability assigned to a vector.

 

[Jano L.]

I think a discussion on exactly what probability 0 and 1 means is best dome on the probability subforum and I will do a post there.

I am looking forward to it. However, the argument was about something different: that deterministic theory is a special kind of probabilistic theory. I am quite interested what others think about this.

 

[audioloop]

Bill, I agree with the quoted part.
Non-contextuality is a strong assumption IMO. But yes it makes the Hilbert formalism "ugly" not to adopt it. But Gleason's theorem assumes non-contextuality and that was the sense of my comment about lack of generality of the theorem as there are QM interpretations that don't assume non-contextuality (you mentioned BM but there are also the modal interpretations and others).
I have a doubt about this because I've seen quantum non-contextuality defined in two ways that I guess are equivalent, maybe you can help me connect them: as referred to independence of the measurement arrangement and as basis independence of the probability assigned to a vector.

http://arxiv.org/pdf/1207.1952v1.pdf

..."The concept of contextuality states that the outcomes of measurement may depend on what measurements are performed alongside"...