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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