Undergrad Classical chaos and quantum mechanics

[stevendaryl]

Reading the posts in this thread I thought that i could ask the following question.

Do bell inequalities need explicit experimental verification in special experiments aimed to check the inequalities?

The violation of the classical CHSH<=2 inequality for two spins 1/2 is based on calculations of QM correlators like <A.B>, <A.B‘>,<A‘.B‘>,<A’.B> where A and B are the the spin operators based on Pauli matrixes, <> is an average over singlet w.f. It is then easy to show that CHSH can be 2.sqrt(2)>2. The calculations are based on the rules of QM and are exact.

Now, if we think that CHSH<=2 should be preserved and try to make complicated experiments, we somehow implicitly assume the the rules of calculations that we used to calculate 2.sqrt(2) are not exact. But if it so, how then we have SM of particle physics which is a very precise proof of QM?

If it was found after 1964 Bell’s paper that CHSH is always <=2 in test experiments, this would mean that the rules of QM are not completely correct in contradiction to all other experiments in particle physics, solid state physics, ...

If I understand correctly what you're saying, then you're right. QM predicts a violation of Bell's inequality (and the CHSH inequality), so if experiments didn't find a violation, that would show that QM is wrong.

 

[vanhees71]

I think you missed what i tried to say. (Except that determinism is different from causality i agree with what you say).
/Fredrik

It is very important to understand the difference between determinism and causality before entering any sensible (i.e., science based vs. philosophical gibberish) discussion of QT.

Definition 1: A theory is deterministic if and only if at any time all observables of a system have determined values.

Definition 2a: A theory is causal if and only if the state of a system is given for $t<t_0$ then the state of the system is determined at any time $t \geq t_0$ either (weak form).

Quantum theory is indeterministic, because never all observables of a system can take a determined value at once, but it's causal, even in a stronger sense (locality in time): If the quantum state is given for $t=t_0$ it is determined at any later time $t \geq t_0$.

 

[read]

If I understand correctly what you're saying, then you're right. QM predicts a violation of Bell's inequality (and the CHSH inequality), so if experiments didn't find a violation, that would show that QM is wrong.

I mean that the only fact that CHSH>2 calculated by QM, purely theoretically, is enough to prove nonlocality of QM. There is no need for specific experiments with entangled photons to see if this is experimentally confirmed.

 

[stevendaryl]

I mean that the only fact that CHSH>2 calculated by QM, purely theoretically, is enough to prove nonlocality of QM. There is no need for specific experiments with entangled photons to see if this is experimentally confirmed.

I would say that the theoretical prediction of QM is enough to show that it is nonlocal in Bell's sense. Actual experimental tests of the inequality are tests of QM, not demonstrations that QM is nonlocal in Bell's sense.

 

[Fra]

Definition 1: A theory is deterministic if and only if at any time all observables of a system have determined values.

This is not the definition I used, which resolves our disagreement.

What i had in mind:

A theory is deterministic iff the future state is implied (by a deductive rule) from the current state.
(The alternative to this, is a theory that is inductive, stochastic or evolutionary)

(Note the distinction of state and single events, this is the gap in the connecting the probabilistic foundation to reality, because we do not directly observe distributions as single events)

Definition 2a: A theory is causal if and only if the state of a system is given for $t<t_0$ then the state of the system is determined at any time $t \geq t_0$ either (weak form).

This is a strange definition to me? Your definition of causality implies also determinism if you by "determined" mean exactly and uniquely determined.

You are excluding general non-deductive causations with this definition.

If we can replace the word "determined" by inferred i can agree.

I think of a theory as causal, when its inferences of the future states only depend on the current and past states. But the inference need not be deductive!

So QM is causal and deterministic in my sense. The fact that individual observations of events are only probabilistically determined by the state even if the past is known precisely, is noted separately, as single events are not what defines the state space in QM anyway. The state space is defined by (according to interpretation) P-distributions, ensembles or "information states", and the theory defines a causal flow on this space which is deterministic in QM.

About that all possible observables does not commute, in my eyes has nothing todo with indeterminism. It has to do with dependence of the underlying observables. Ie. conjugate variables (if we defined them as related by the Fourier transform) are statistically dependent.

/Fredrik

 

[read]

I would say that the theoretical prediction of QM is enough to show that it is nonlocal in Bell's sense. Actual experimental tests of the inequality are tests of QM, not demonstrations that QM is nonlocal in Bell's sense.

Still, I would like to ask further. More specifically, the correlators for CHSH are just -cos(angle(a,b)), and this is just because of Pauly matrix and singlet w.f. Now, for an angle like 135 degrees we get 2.sqrt(2), so 70% more than in classics. Why should we check CHSH inequality? If we think that we can have 70% of accuracy, then other more precise and developed experiments in particle physics also should see this.