Experimental Tests of Projection Postulate

[slyboy]

I would like to object to the generally held belief in the physicist community that where philosophy starts, physics ends

I agree. I would like to think that the interpretation of quantum mechanics will turn out to be physics (as well as being philosophy). By this I mean that it will lead to new ways of thinking about physics, which in turn will lead to new ways of extending it. Hopefully, this will turn out to be relevant, albeit indirectly, for coming up with the correct theory of quantum gravity.

Waow Sliboy, this is almost phylosophy, where is the physics (the logical deductions and not the subjective or objective ones)

One thing that philosophers are good at is separating out the individual problems that make up the complex issues we think about as physicists. Then, we can try to analyse and solve them one by one.

My main point is that the interpretational problems of quantum mechanics seem to be very closely tied to the problem of interpreting probability theory. I would like to separate them if possible, because trying to solve one hard problem is usually easier than trying to solve two simultaneously.

The subjective theory of probability presents a problem in this regard, because we are used to thinking of quantum probabilities as the objective predictions of the theory. However, according to the subjective theory, probabilities are just not the right sort of thing to appear at such a fundamental level.

On the other hand, subjective probability does not do away with objective facts entirely. Although, probabilities are not themselves objective, things such as the possible options that an agent has to decide between, and the possible events that can occur, are taken as objective facts.

In my view, any hypothesis used to derive the Born rule should be expressible in terms of the things that are taken to be objective in any theory of probability. In the subjective theory, this means thoroughly grounding things in decision theory.

If this cannot be done, then the hypothesis is not compatible with all the major interpretations of probability, and its proponents have to go and fight the battle about the interpretation of probability before they can convince everyone that it is the correct way to think about quantum theory.

In contrast, if the hypothesis does have an equivalent formulation in all the major interpretations of probability, then we can ignore the issue and just fight about quantum mechanics instead, which is what we wanted to do in the first place.

 

[vanesch]

The subjective theory of probability presents a problem in this regard, because we are used to thinking of quantum probabilities as the objective predictions of the theory. However, according to the subjective theory, probabilities are just not the right sort of thing to appear at such a fundamental level.

I think you cannot get around this: probabilities seem to be fundamental in quantum theory, and I'd say that if you have a view on probability that cannot take probabilities as fundamental quantities, then you'll always have a problem with QM in that view. I think you are very close to the views of Deutsch, no ? I think that anyone who tries to do QM *without* somehow postulating that probabilities will appear, will run in a circle because it is the *only* thing that comes out of the wavefunction ! After all, what else is a wavefunction good for ? What does it mean to have a system in a state |psi> if there's no link to anything probabilistic ? There's a paper with critique on the MWI which explains this very well by Adrian Kent grqc/9703089. I think his critique is too harsh for MWI but it illustrates very well that *without* any probability postulate, QM is dead-empty. There is no "emergence" of probabilities simply because there's some vector wobbling in Hilbert space if that's all there is. Let it wobble. What does it mean ?

cheers,
Patrick.

 

[slyboy]

I think you are very close to the views of Deutsch, no ?

I don't think so, since I am not actually a fan of many-worlds. The main reson is that you have an object so closely related to probabilities apprearing as the "state of reality", i.e. the wavefunction, and I don't think that it can ever be made compatible with the all theories of probability.

However, if one follows an interpretation in which the quantum state is epistemic, then there is not such a big problem. We can have different agents assigning different states to the same system, so it becomes much more like a probability distribution.

Of course, the big problem with this approach is to identify what the "states of reality" are in quantum mechanics if they are not the wavefunction. I believe that there are 4 main contenders for this:

1) There is no "state of reality" - This puts us close to a Copenhagen or an instrumentalist view of quantum mechanics. We have to give up a huge chunk of realism, which I would prefer not to do.

2) Hidden variable theories - The problem here is that in most of the viable contenders, such as Bohmian mechanics, the wavefunction still enters as part of the state of reality. There is currently no plausible hidden variable theory without this property, although one can construct toy theories. However, if we drop the equilibrium hypothesis, then the wavefunction is no longer directly related to probability distributions, so this might be a viable approach.

3) Quantum Logic - Here, the only thing that changes when you go from classical to quantum is the structure of events. Probabilities are introduced in exactly the same way as in the classical theory. Events are objective in all theories or probability, so it is compatible with propensities, frequencies and subjective approaches.

4) Miscellaneous proposals that have yet to be fully worked out.

 

[seratend]

I don't think so, since I am not actually a fan of many-worlds. The main reson is that you have an object so closely related to probabilities apprearing as the "state of reality", i.e. the wavefunction, and I don't think that it can ever be made compatible with the all theories of probability.

So you are saying that the born rules is not a probability law?

For me, once you define a measure on a given set with a sigma algebra (the borel sets, in the case of the observables) and the eigenvalue-outcome link, you have what I call formally a "classical" probability space. Why asking for more than this?

Seratend.

 

[NateTG]

Well, the issue of 'quantum interpretation' is approrpiate to philosophy because it is not experimentally testable and thus not physics or science. If it is testable, then it ceases to be interpretation and becomes theory.

Physics, unlike philosophy, is not suitable for discussing how many angels can dance on the head of a pin.

 

[seratend]

My main point is that the interpretational problems of quantum mechanics seem to be very closely tied to the problem of interpreting probability theory. I would like to separate them if possible, because trying to solve one hard problem is usually easier than trying to solve two simultaneously.

I agree
However, I think one should define first the scope of the interpretation. Personally, I just need a consistent mapping between some objects of the theory and some objects of the reality in order to make some logical and practical predictions/deductions (or at least the identification some objects that may be mapped later to the "reality").

In the probability domain, what do you call the problem of interpretation? Do you mean the choice of a peculiar interpretation?

Seratend.

 

[seratend]

Well, the issue of 'quantum interpretation' is approrpiate to philosophy because it is not experimentally testable and thus not physics or science. If it is testable, then it ceases to be interpretation and becomes theory.

Physics, unlike philosophy, is not suitable for discussing how many angels can dance on the head of a pin.

Well, I think one should split the interpretation into the minimalist part (the mapping of some of the mathematical objects to the "reality") and the "philosophical" part. I need the minimalist part to describe formally experiments (apply the logic) while I can live without the second part .

Seratend.

 

[NateTG]

Well, I think one should split the interpretation into the minimalist part (the mapping of some of the mathematical objects to the "reality") and the "philosophical" part. I need the minimalist part to describe formally experiments (apply the logic) while I can live without the second part .

Seratend.

The popular 'plug and chug' interpretation, originally, I believe attributed to Von Neuman.

There is a legitemate place for interpretations as a method for deveolping hypotheses, but, Ph. D. stands for Doctor of Philosophy.

 

[vanesch]

Well, the issue of 'quantum interpretation' is approrpiate to philosophy because it is not experimentally testable and thus not physics or science. If it is testable, then it ceases to be interpretation and becomes theory.

I'd object to this, for several reasons. The most important is that the interpretation of quantum theory is the only link between the mathematical formalism and the experimental setup ; however, in many cases this reduces to something that is *intuitively clear* and we're cheating, because we switch, at a certain point, to classical physics. However, it is conceivable that in much more sophisticated setups, the intuition is NOT going to be right. Typical example: when do we have to treat the nuclear skeleton of a molecule classical, and when do we have to treat it quantum-mechanically (eg, the molecule has no structure): NH3 must be treated QM, and a protein must/can (?) be treated classically. In fact, current QM leaves rather open the question ; decoherence seems to suggest that the answers will come out the same.
But at some point, we need to know whether a "real" collapse occurs or not. This is a testable question (at least in principle, much easier to test than string theory :-) This has everything to do with the interpretation of quantum theory.

But another important objection is this: an interpretation offers a mental picture of what you are doing, and I think that such a mental picture is necessary in order to be able to devellop the necessary intuition to make progress. For instance, string theorists just take over the unitary machinery of quantum theory. But I think the first question to solve is whether gravity does, or does not, allow for the unitary evolution to continue (that's many worlds) or induces a collapse of some kind. Again, this is closely related to interpretational issues.

I'd say that if you take interpretation and mathematics away from physics, you end up with stamp collecting :-)

cheers,
Patrick.

 

[seratend]

The popular 'plug and chug' interpretation, originally, I believe attributed to Von Neuman.

Maybe popularly attributed to Von Neuman. However, surely the most difficult to understand in my opinion (i.e. we have to understand our way of thinking).

There is a legitemate place for interpretations as a method for deveolping hypotheses, but, Ph. D. stands for Doctor of Philosophy.

Yes, a typical anglosaxon point of view ; ). In other countries, there are only doctors. : )))

Seratend.

 

[slyboy]

So you are saying that the born rules is not a probability law?

Well, it's certainly a probability rule, but I hesitate to give it the title "law". As I have explained, I don't think probability statements should enter into our fundamental laws of nature.

For me, once you define a measure on a given set with a sigma algebra (the borel sets, in the case of the observables) and the eigenvalue-outcome link, you have what I call formally a "classical" probability space. Why asking for more than this?

Well, you actually have multiple classical probability spaces, one for each observable. Quantum theory says that there are events appearing in different sample spaces that are always assigned the same probability. The only way this can be justified in a subjective theory is if these events are always identified as the same. So, what you really have, is not a classical probability space, but multiple spaces pasted together, and this is essentially the probability space of quantum logic.

Well, the issue of 'quantum interpretation' is approrpiate to philosophy because it is not experimentally testable and thus not physics or science. If it is testable, then it ceases to be interpretation and becomes theory.

I used to believe this as well, but now I am not so sure. All physical theories have a verifiable part and an interpretation part - not just quantum mechanics. The verifiable part consists of the mathematical formalism, and a set of rules for relating it to the experiments. There is always underdeterminism in the interpretation part, i.e. I can always cook up bizarre ways of thinking about things that give all the same experimental predictions, but a very different picture of the world. For example, I may be able to cook up an interpretation of Newtonian mechanics that doesn't have a notion of absolute time. However, no-one would argue that Newtonian mechanics doesn't have absolute time, and that this is part of the physics rather than being just philosophy.

In the quantum case, we have cooked up this nice comforting story for ourselves, wherein there is an operational part of the theory that everyone agrees upon and understands, and an interpretation part that is just a matter of philosophy. However, there are cases where the part that we normally think of as interpretation rears its ugly head in real physics. I am thinking particularly of the debates surrounding the existence of the quantum Zeno effect, which relies on taking the projection postulate literally, and also the role of the wavefunction of the universe in quantum cosmology.

 

[Tournesol]

I thought that paper was very promising, and I would like to see soem comments from people who are less rusty than I am.

 

[vanesch]

In the quantum case, we have cooked up this nice comforting story for ourselves, wherein there is an operational part of the theory that everyone agrees upon and understands, and an interpretation part that is just a matter of philosophy. However, there are cases where the part that we normally think of as interpretation rears its ugly head in real physics. I am thinking particularly of the debates surrounding the existence of the quantum Zeno effect, which relies on taking the projection postulate literally, and also the role of the wavefunction of the universe in quantum cosmology.

I couldn't agree more

cheers,
Patrick.

 

[NateTG]

The most important is that the interpretation of quantum theory is the only link between the mathematical formalism and the experimental setup ;

I think this is primarily an issue of definitions. My notion of what 'interpretation' encompasses is narrower than yours.

But at some point, we need to know whether a "real" collapse occurs or not. This is a testable question (at least in principle, much easier to test than string theory :-) This has everything to do with the interpretation of quantum theory.

If you want to test whether a collapse occurs or not, or, for that matter, exactly what experimentally testable properties a collapse has, those are not interpretation issues.

But another important objection is this: an interpretation offers a mental picture of what you are doing, and I think that such a mental picture is necessary in order to be able to devellop the necessary intuition to make progress.

Not really. It's quite possible to, for example, look for places where the current theory has singularities, and run experiments to see what happens there.
Moreover, it's not at all clear to me that interpretation has necessarily been a historically useful for physics. Rather it seems like interpretation is a problem that physics (really science in general) keeps knocking its teeth out on. Theories that come out of 'actuarial' science - that is theories that are based on making lots of observation - and attempting to correlate the results tend to be strong, while theories that are based on 'interperation' - based on what might or ought to be - tend to be weak.

 

[slyboy]

Theories that come out of 'actuarial' science - that is theories that are based on making lots of observation - and attempting to correlate the results tend to be strong, while theories that are based on 'interperation' - based on what might or ought to be - tend to be weak.

Yes, I agree. Relativity is clearly one of the weakest theories in science :)

...but seriously, I think that these sort of sweeping generalizations are not justified by the actual history of science. It has always been a mix of effective theories based on observation, and grand extrapolations of theorists to make things fit into their grand vision of the world.

Many examples, such as relativity, Darwin's theory of natural selection, etc. could not be entirely justified by the available evidence at the time they were proposed, although they were of course guided by some observations that had been made.

 

[vanesch]

Moreover, it's not at all clear to me that interpretation has necessarily been a historically useful for physics. Rather it seems like interpretation is a problem that physics (really science in general) keeps knocking its teeth out on. Theories that come out of 'actuarial' science - that is theories that are based on making lots of observation - and attempting to correlate the results tend to be strong, while theories that are based on 'interperation' - based on what might or ought to be - tend to be weak.

There are examples of both, but some spectacular breakthroughs were based purely on "vision":
- Maxwell's equations (the d D /dt term)
- general relativity
- Dirac's equation
- the electroweak theory of Weinberg and Salam
- the expanding universe (Hubble: his data were in fact showing the opposite!)

most of these were NOT data driven at all, but based upon the vision that the authors had of how things "ought" to be.

Of course an example of a theory that was rammed down our throat by data was quantum mechanics.

cheers,
Patrick.

 

[vanesch]

If you want to test whether a collapse occurs or not, or, for that matter, exactly what experimentally testable properties a collapse has, those are not interpretation issues.

I'd say that these are extentions of the current formalism, based upon a vision that is inspired by a certain interpretation

After all, the reason why there's so much discussion and different interpretations is mostly because the current formalism of quantum theory is AMBIGUOUS. For most applications this doesn't matter for the moment, because we cheat in different ways, because we hop between classical physics and quantum theory all the time in ways which are just given by our intuition, and this works for all practical purposes. The advantage of taking up an interpretation is that it FORCES you to make choices where the actual theory is vague - so in a way, according to you, interpretational issues are in fact variant extensions of the theory. Fine.

 

[seratend]

Well, it's certainly a probability rule, but I hesitate to give it the title "law". As I have explained, I don't think probability statements should enter into our fundamental laws of nature.

Ok, I understand better what you say. For me a probability law is a measure on a sigma algebra (i.e. the mathematical definition). So you add more properties to the set of words "probability law" than me (i.e. the following of your post).

Quantum theory says that there are events appearing in different sample spaces that are always assigned the same probability. The only way this can be justified in a subjective theory is if these events are always identified as the same.

I do not understand this statement. We have a probability space completely defined for a given observable and a state. If we want we may formally connect these probability spaces by a parameter, that we may think as a context, but this is an additional external structure (such as the definition of non boolean lattice of propostions versus a collection of bolean lattices).

In addition, why do you speak about a subjective theory?

I used to believe this as well, but now I am not so sure. All physical theories have a verifiable part and an interpretation part - not just quantum mechanics. The verifiable part consists of the mathematical formalism, and a set of rules for relating it to the experiments. There is always underdeterminism in the interpretation part, i.e. I can always cook up bizarre ways of thinking about things that give all the same experimental predictions, but a very different picture of the world.

Why do you want determinism in the interpretation part and why do you think that the way of describing the "reality is unique? All what you are able to obtain is a logical consistency of the interpretation in my opinion.
And what you may think is bizarre for you may be normal for another person .

In the quantum case, we have cooked up this nice comforting story for ourselves, wherein there is an operational part of the theory that everyone agrees upon and understands, and an interpretation part that is just a matter of philosophy. However, there are cases where the part that we normally think of as interpretation rears its ugly head in real physics. I am thinking particularly of the debates surrounding the existence of the quantum Zeno effect, which relies on taking the projection postulate literally, and also the role of the wavefunction of the universe in quantum cosmology.

Here is the problem, the "literally" allows a lot of mathematical choices and the paradox comes from thinking about the collapse postulate a king of deterministic process rather than a simple description rule (i.e. we do not assume more than it is written).
It is like a person walking half the distance of the previous walk. This does not mean that the person will stop.

Seratend.

 

[slyboy]

I do not understand this statement. We have a probability space completely defined for a given observable and a state. If we want we may formally connect these probability spaces by a parameter, that we may think as a context, but this is an additional external structure (such as the definition of non boolean lattice of propostions versus a collection of bolean lattices).

In addition, why do you speak about a subjective theory?

I mean the subjective theory of probability, which as I would like to make QM compatible with, as I have said before. Within that theory, one has to derive the structure of probability measures from decision theory, making use of simple axioms about the structure of possible events and actions. These are called coherence arguments.

As far as I can see, if your sample space is a Boolean algebra, then a coherence argument will tell you that all classical probability measures on it are allowed. If you have several unrelated Boolean algebras, then you can have any comination of probability measures on them. However, QM doesn't allow this, c.f. the uncertainty relations for example.

The only way I can see to fix this is to modify the structure of events so that it is no longer a Boolean algebra. Then the coherence argument gives you the correct quantum probabilities via Gleason's theorem.

Why do you want determinism in the interpretation part and why do you think that the way of describing the "reality is unique? All what you are able to obtain is a logical consistency of the interpretation in my opinion.

I never said that I want determinism - just that there is always underdeterminism. Actually, that was a mistake, since I should have written "underdetermination". There will always be several interpretations of a theory that are "logically consistent", although I doubt that one can ever fully demonstrate logical consistency of an interpretation to the same degree as a mathematical theory. Instead, we apply principles like Occam's razor and look for explanatory power in an interpretation. This is as much a part of science as performing experiments, so shouldn't be dismissed as "mere" philosophy.

And what you may think is bizarre for you may be normal for another person .

That's what makes science fun! It is a human activity and we can debate the best way to proceed as much as in any other area of human knowledge.