Dear Joseph,
you are going toward quantum probability theory where 
probabilities are determined by vectors; moreover, the vectors
belong to complex Hilbert space, i.e., roughly speaking each probability
has not only the direction but even the phase, andrei

Andrei Khrennikov, Professor of Applied Mathematics,
International Center for Mathematical Modeling
in Physics, Engineering, Economics, and Cognitive Science
Linnaeus University, Växjö-Kalmar, Sweden
________________________________________
From: fis-boun...@listas.unizar.es [fis-boun...@listas.unizar.es] on behalf of 
Joseph Brenner [joe.bren...@bluewin.ch]
Sent: Wednesday, January 22, 2014 8:54 AM
To: Dino Buzzetti; Hans von Baeyer; fis
Subject: Re: [Fis] Probability Amplitudes

Dear Hans and Dino,

This is a direct question to both of you, to which I have not found a clear 
answer: are value and amplitude the only parameters that have been assigned to 
probability?

In my theory, the changing value of actuality and potentiality of specific 
antagonistic process elements are probability-like in not including 0 and 1, as 
I have said. Can, in addition, probabilities have some vector-like properties, 
that is, include a /direction/?

This concept would be moving toward (and past) Dino and away from Hans . . .

Your comments and those of others would be welcome.

Best wishes,

Joseph
----- Original Message -----
From: Dino Buzzetti<mailto:dino.buzze...@gmail.com>
To: Hans von Baeyer<mailto:henrikrit...@gmail.com> ; 
fis<mailto:fis@listas.unizar.es>
Sent: Wednesday, January 22, 2014 3:53 AM
Subject: Re: [Fis] Probability Amplitudes

Dear Hans,

Thank you for your explanation about probability amplitudes,
that clarifies a lot.  My only worry was about the *epistemological*
implications of quantum mechanics in its standard formulation,
that in my opinion point to a paradigm shift, which is felt not only
in this domain, but in all fields where *emergent* phenomena are
accounted for—a process that I thought was hinted to by Wheeler's
famous words "It from Bit," that I remember reading for the first
time precisely in your book on information.  That's the ground for
expressing my worry that reverting to classical probability theory
might entail a drawback to this decisive epistemological turn.

But I might misunderstand the whole story, that is certainly not
over yet  :-)                              -dino



On 22 January 2014 00:21, Hans von Baeyer 
<henrikrit...@gmail.com<mailto:henrikrit...@gmail.com>> wrote:
Dear Dino and friends, thanks for bringing up the issue of probability 
amplitudes.  Since they are technical tools of physics, and since I didn't want 
to go too far afield, I did not mention them in my lecture.  The closest I came 
was the wavefunction, which, indeed, is a probability amplitude.  In order to 
make contact with real, measurable quantities, it must be multiplied by its 
complex conjugate. This recipe is called the Born rule, and it is an ad hoc 
addition to the quantum theory. It lacks any motivation except that it works.

In keeping with Einstein's advice (which he himself often flouted) to try to 
keep unmeasurable concepts out of our description of nature, physicists have 
realized long ago that it must be possible to recast quantum mechanics entirely 
in terms of probabilities, not even mentioning probability amplitudes or 
wavefunctions. The question is only: How complicated would the resulting 
formalism be?  (To make a weak analogy, it must be possible to recast 
arithmetic in the language of Roman numerals, but the result would surely look 
much messier than what we learn in grade school.)  Hitherto, nobody had come up 
with an elegant solution to this problem.

To their happy surprise, QBists have made  progress toward a "quantum theory 
without probability amplitudes."  Of course they have to pay a price.  Instead 
of "unmeasurable concepts" they introduce, for any experiment, a very special 
set of standard probabilities (NOT AMPLITUDES) which are measurable, but not 
actually measured.  When they re-write the Born rule in terms of these, they 
find that it looks almost, but not quite, like a fundamental axiom of 
probability theory called Unitarity.  Unitarity decrees that for any experiment 
the sum of the probabilities for all possible outcomes must be one. (For a 
coin, the probabilities of heads and tails are both 1/2.  Unitarity states 1/2 
+ 1/2 = 1.)

This unexpected outcome of QBism suggests a deep connection between the Born 
rule and Unitarity. Since Unitarity is a logical concept unrelated to quantum 
phenomena, this gives QBists the hope that they will eventually succeed in 
explaining the significacne of the Born rule, and banishing probability 
amplitudes from quantum mechanics, leaving only (Bayesian) probabilities.

So, I'm afraid dear Dino, that the current attitude of QBists is that 
probability amplitudes are LESS fundamental than probabilities, not MORE.  But 
the story is far from finished!

Hans



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