Dear Andrei, Hans and all
I agree with Andrei. And why make quantum theory more complex than it is? One 
may use all  kinds of mathematical tools in a scientific theory, and the more 
these tools simplify calculations the better. I see no reason to avoid using 
amplitudes or  matrices in quantum theory. Using a mathematical concept for 
making calculations doesn't entail that I accept that that concept represent a 
physical property.

To Hans: Where exactly did Einstein wrote that one should avoid unmeasurable 
concepts in the description of Nature? I can't remember having read that.

The issue is how we should interpret quantum theory, in particular the wave 
function, i.e., probability amplitudes; are they just mathematical tools, or do 
they describe real physical features of quantum systems? I believe the latter 
alternative is true and so did Schrödinger. But there are formidable 
difficulties to give a realistic interpretation of wave functions, and 
Schrödinger didn't succeed. But I think the difficulties can be overcome and I 
have published my views about these things (Lars-Göran Johansson: Interpreting 
Quantum Mechanics. A realist view in Schrödinger's vein, Ashgate, Aldershot 

22 jan 2014 kl. 10:59 skrev Andrei Khrennikov 

      Dear Hans,

I would like just to point that 99,99% of people working
in quantum theory would say that the complex amplitude of
quantum probability is the main its intrinsic property, so
if you try to exclude amplitudes from the model
you can in principle do this and this is well known
long ago in so called quantum tomographic approach of Vladimir
Manko, but in this way quantum theory loses its simplicity and
clarity, yours, andrei

Andrei Khrennikov, Professor of Applied Mathematics,
International Center for Mathematical Modeling
in Physics, Engineering, Economics, and Cognitive Science
Linnaeus University, Växjö-Kalmar, Sweden
[<>] on behalf 
of Hans von Baeyer [<>]
Sent: Wednesday, January 22, 2014 12:21 AM
Subject: [Fis] Probability Amplitudes

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!


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Lars-Göran Johansson
filosofiska institutionen
Uppsala Universitet

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