On 3/4/2020 4:34 PM, Bruce Kellett wrote:
On Thu, Mar 5, 2020 at 10:39 AM Russell Standish
<li...@hpcoders.com.au <mailto:li...@hpcoders.com.au>> wrote:
On Thu, Mar 05, 2020 at 09:46:34AM +1100, Bruce Kellett wrote:
> The greater problem is that any idea of probability founders
when all outcomes
> occur for any measurement. Or have you not followed the
arguments I have been
> making that shows this to be the case?
>
I must admit I haven't followed the arguments either - admittedly, I
haven't read your cited material.
ISTM - probability is all about what an observer observes. Since the
observer cannot see all outcomes, an objection based on all outcomes
occurring seems moot to me.
The fact that the observer cannot see all outcomes is actually central
to the argument. If, in the person-duplication scenario, the
participant naively assumes a probability p = 0.5 for each outcome,
such an intuition can only be tested by repeating the duplication a
number of times and inferring a probability value from the observed
outcomes. Since each observer can see only the outcomes along his or
her particular branch (and, ipso facto, is unaware of the outcomes on
other branches), as the number of trials N becomes very large, only a
vanishingly small proportion of observers will confirm their 50/50
prediction . This is a trivial calculation involving only the binomial
coefficient -- Brent and I discussed this a while ago, and Brent could
not fault the maths.
The crux of the matter is that all branches are equivalent when both
outcomes occur on every trial, so all observers will infer that their
observed relative frequencies reflect the actual probabilities. Since
there are observers for all possibilities for p in the range [0,1],
and not all can be correct, no sensible probability value can be
assigned to such duplication experiments.
The problem is even worse in quantum mechanics, where you measure a
state such as
|psi> = a|0> + b|1>.
When both outcomes occur on every trial, the result of a sequence of N
trials is all possible binary strings of length N, (all 2^N of them).
You then notice that this set of all possible strings is obtained
whatever non-zero values of a and b you assume. The assignment of some
propbability relation to the coefficients is thus seen to be
meaningless -- all probabilities occur equal for any non-zero choices
of a and b.
But E(number|0>) = aN and Var(number|0>) = abN. The fraction x within
one std-deviation of the expected number is a constant
F( a-sqrt[ab/N]<x<a+sqrt[ab/N])=1/e
So that fraction become more an more sharply confined around a as N->oo.
Brent
You may counter that the assumption that an observer cannot see all
outcomes is an extra thing "put in by hand", and you would be right,
of course. It is not part of the Schroedinger equation. But I would
strongly suspect that this assumption will be a natural outcome of a
proper theory of consciousness, if/when we have one. Indeed, I
highlight it in my book with the name "PROJECTION postulate".
This is, of course, at the heart of the 1p/3p distinction - and of
course the classic taunts and misunderstandings between BM and JC
(1p-3p confusion).
I know that it is a factor of the 1p/3p distinction. My complaint has
frequently been that advocates of the "p = 0.5 is obvious" school are
often guilty of this confusion.
Incidently, I've started reading Colin Hales's "Revolution of
Scientific Structure", a fellow Melburnian and member of this
list. The interesting proposition about this is Colin is proposing
we're on the verge of a Kuhnian paradigm shift in relation to the role
of the observer in science, and the that this sort of misunderstanding
is a classic symptom of such a shift.
Elimination of the observer from physics was one of the prime
motivations for Everett's 'relative state' idea. Given that
'measurement' and 'the observer' play central roles in variants of the
'Copenhagen' interpretation.
Bruce
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