Dear Srinandan,

Your question about teh difference in statistical data for commuting and
noncommuting observables is extremely important for probabilistic
foundations of QM. First I recall my and yours points:
> On 20-May-06, at 11:13 AM, Andrei Khrennikov wrote:
>the real problem is not in some distinguishing features of so  
> called quantum systems, but in combining of statistical data from
> a few different experiments.

Srinandan Dasmahapatra:
> However, this procedure/algorithm must have features built in which 
 > distinguish between classical modes of combining the results and  
> quantum ones.  For instance, in quantum systems, the results of  
> measuring observables A and B which commute will have different rules
> for aggregation than those which do not commute.  Is there a way of 
> seeing this clearly in your formulation?

Roughly speaking we can formulate the problem in the following way. We
have two different observables A and B. We have no idea either their
classical or quantum.  Is it possible to find some statistical invariant
that would say us that these observables could be represented in the
Hilbert space by commutative or noncommutative operators? Yes, it is
possible to proceed in this way, such a coefficient, denoted by lambda
was proposed and the simplest introduction can be found in (Brain as quantum-like computer).

This approach to noncommutativity gives us the possibility to apply the 
Hilbert space formalism outside the conventional domain of QM. 
Complementary observables observables A and B can be found in different
domains of science, e.g., cognitive sciences, 
see (A Preliminar Evidence of
Quantum Like Behavior in Measurements of Mental States) or economy.

I really think that we have not yet explored the quantum formalism. We
found just one special application, namely, in the microworld.

With Best Regards, Andrei Khrennikov

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