On Sat, May 16, 2020 at 11:04 PM Lawrence Crowell <
goldenfieldquaterni...@gmail.com> wrote:
> There is nothing wrong formally with what you argue. I would though say
> this is not entirely the Born rule. The Born rule connects eigenvalues with
> the probabilities of a wave function. For quantum state amplitudes a_i in a
> superposition ψ = sum_ia_iφ_i with φ*_jφ_i = δ_{ij} the spectrum of an
> observable O obeys
>
> ⟨O⟩ = sum_iO_ip_i = sum_iO_i a*_ia_i.
>
> Your argument has a tight fit with this for O_i = ρ_{ii}.
>
> The difficulty in part stems from the fact we keep using standard ideas of
> probability to understand quantum physics, which is more fundamentally
> about amplitudes which give probabilities, but are not probabilities. Your
> argument is very frequentist.
>
I can see why you might think this, but it is actually not the case. My
main point is to reject subjectivist notions of probability: probabilities
in QM are clearly objective -- there is an objective decay rate (or
half-life) for any radioactive nucleus; there is a clearly objective
probability for that spin to be measured up rather than down in a
Stern-Gerlach magnet; and so on.
The argument by Carroll and Sebens, using a concept of the wave function as
> an update mechanism, is somewhat Bayesian.
>
It is this subjectivity, and appeal to Bayesianism, that I reject for QM. I
consider probabilities to be intrinsic properties -- not further
analysable. In other words, I favour a propensity interpretation. Relative
frequencies are the way we generally measure probabilities, but they do not
define them.
This is curious since Fuchs developed QuBism as a sort of ultra-ψ-epistemic
> interpretation, and Carroll and Sebens are appealing to the wave function
> as a similar device for a ψ-ontological interpretation.
>
> I do though agree if there is a proof for the Born rule that is may not
> depend on some particular quantum interpretation. If the Born rule is some
> unprovable postulate then it would seem plausible that any sufficiently
> strong quantum interpretation may prove the Born rule or provide the
> ancillary axiomatic structure necessary for such a proof. In other words
> maybe quantum interpretations are essentially unprovable physical axioms
> that if sufficiently string provide a proof of the Born rule.
>
I would agree that the Born rule is unlikely to be provable within some
model of quantum mechanics -- particularly if that model is deterministic,
as is many-worlds. The mistake that advocates of many-worlds are making is
to try and graft probabilities, and the Born rule, on to a
non-probabilistic model. That endeavour is bound to fail. (In fact, many
have given up on trying to incorporate any idea of 'uncertainty' into their
model -- this is what is known as the "fission program".) One of the major
problems people like Deutsch, Carroll, and Wallace encounter is trying to
reconcile Everett with David Lewis's "Principal Principle", which is the
rule that one should align one's personal subjective degrees of belief with
the objective probabilities. When these people essentially deny the
existence of objective probabilities, they have trouble reconciling
subjective beliefs with anything at all.
Bruce
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