Dear Michel.
I shall try to give a more detalied explanation to the problem of
addition of probabilities and relation of #quantum and classical
probabilities.#
> Dear Andrei,
> I cannot understand how the rule of addition of probabilities is
> violated. Does it mean that when A and B are events of void
intersection, the
> equality
> P (A U B) = P(A) + P(B)
> could be violated ? Or what else ?
Yes.
> In classical probability that may be false when the events are not
> defined on
> the same probability space, in which case the quantity P(A)+P(B) has
> normally
> no sense.
Well, the problem is that #the quantity P(A)+P(B) has# sense. These are
two real numbers and we can add them and if their sum is less than 1, it
can be interepreted as probability.
> Are QI probabilities falling in this situation ?
Yes. But I do not think that physicists understood this, because they
never consider the precise classical probabilistic framework with
Kolmogorov probability spaces. Therefore, if you open the book of
Feynman and Hibs, QM and path integrals, Feynman simply claim that
quantum statistical data violates fundamentally the rules of classical
probability theory (in the section devoted to the two slit experiment).
And this is just because quantum systems are such myctically unsual.
> The probabilities computed from two different quantum experiments
> should
> correspond to events in a common probability space before being
> added
> and telling that the addition lead to an unacceptable result.
> What may be this common probability space here ?
Not, of course, there is no common probbaility space. But this is never
discussed. It is typically used just the symbol P and nobody takes care
to which probability space this P belong, see my #Interpretations of
probability.#
> Detailing your example with slits would be nice (defining the
> probability space, writing the events, their probabilities,
> the sum, etc..).
> If such a space exists, and if the formula is violated, thus I would
> not
> speak about a probability space in this context. And if so, what
> could be
> the << probability >> rules indeed working in the quantum
context ?
The problem that I studied in the contextual classical probabilistic
framework: Can we work with families of probability spaces in such a way
that in particular we shall produce (just from the fact that P=P_C,
where C is a physical (or biological, or chemical) context), QM-rules,
in particular, interference of probabilities:
P (A U B) = P(A) + P(B) +2 cos theta sqrt(P(A)P(B)) ???
The answer is yes, but of course probabilities should have indexes of
corresponding contexts. But such a calculus of probabilities is more
general than QM-calculus. We can speak about contextual probability
theory as general calssical probability tehory, in particular, inducing
the quantum one.
All the best, Andrei
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Received on Mon May 29 19:39:49 2006
