On 25/06/2017 9:25 pm, Russell Standish wrote:
On Sun, Jun 25, 2017 at 04:25:07PM +1000, Bruce Kellett wrote:
On 24/06/2017 8:36 pm, Russell Standish wrote:
On Sat, Jun 24, 2017 at 06:29:54PM +1000, Bruce Kellett wrote:
On 24/06/2017 5:23 pm, Russell Standish wrote:
OK, it was possibly the case that you gave arguments earlier in the
book. But I was going on the basis of the Appendix "Derivation of
Quantum postulates".
But the problems only begin with the assumption of a probabilistic
model. Psi(t) is the set of possibilities consistent with what is
known at time t. But how do you limit this set? At the moment, I
could go to the pub for a drink, could open a bottle of wine at
home, stroke the cat, turn on the telly, talk to my wife, etc,
etc,..... The possibilities consistent with what is known at this
time is not a well defined set, or limited in any way.
The everything is the set of all infinite length strings, each of
which describes a universe to infinite detail. Some of these strings
will describe universes compatible with our current observer moment -
an infinite number even, as the information content of our OM is
finite. Others will not. It is a well defined subset of the everything.
What does "compatible" mean? Is this linked to our current moment by
law-like behaviour, or just any any string which happens, by chance
to contain our present moment? If the latter (and also in compete
generality, given your definitions), then my characterization of the
list of possibilities is correct, and the string could contain any
future indeed. If the former, how do you know that there is not just
a single string that contains the only law-like continuation of your
present state?
There is an interpreter of the strings. In Solomonoff-Levin's theory,
the interpreter is some given universal machine U, but in this case,
the only possible interpreter is an observer of the system.
Compatible just means that the string is interpreted as giving the
current observer moment by the observer.
At this point, there is no theory about what likely successor strings
might be. The most obvious model (ie just read some more bits of a
given string), doesn't work.
What I propose in the appendix D work is to apply an evolutionary
process to it.
You mean your statement about the variation upon which anthropic
selection acts? Does this mean that the continuations that are
anthropically allowed are those the permit the observer's continued
existence? Or is something more implied?
Because you then go on to define projection operators in terms of a
sum over the members of this set of possible outcomes. That is
meaningless unless you are already assuming the the outcomes are
just possible results for a well-defined measurement, and that this
measurement process can be defined in a linear vector space.
Summation of the projection operators is defined in equation D.1 for
disjoint observations a and b (ie where it is impossible to observe a
and b simultaneously). Linearity is not assumed at this point.
That is where you have an enormous problem, following on from the
previous point that you have not limited the possible continuations
in any way. You define a projection as occurring when the observer
applies an operator A (again undefined and unlimited) to the
observer moment, which operator divides it into a discrete set of
outcomes, psi_a. Note that 'a' is only an index, not an eigenvalue
or any such. Note that you explicitly state that \P_{a} is not
assumed linear at this point. (I use \P as a notation for your
script P.) You then *define* addition for two distinct outcomes a
=/= b as:
\P_{a} psi + \P_{b} psi = \P_{a,b} psi.
This is, of course meaningless if psi_a is 'taking the dog for a
walk,' and psi_b is 'stroking the cat'. You can define a '+' sign as
anything you like, but such a definition does not ensure that the
result has any meaning -- as in my example, which follows completely
logically from the definitions that you make.
Note that {a,b} is just {a} ∪ {b}, so that \P_{a,b}ψ is is just the
observation "taking the dog for a walk _or_ stroking the cat". That is
a perfectly acceptable classical observation.
That is not what is normally meant by the '+' symbol. You have simply
defined a conjunction to be a disjunction!
Bur that does not work for the equations immediately following, where
you simply sum over all the possible outcomes of the operator 'A'. Why
use the symbol \Sigma if you mean a disjunction? And you then go on to
say that if the outcome is a continuous set, you replace the sum by an
integral with uniform measure. It is difficult to avoid the conclusion
that you do actually mean that '+' implies standard addition.
Since psi_a is in general just a set of continuations selected by the
operation 'A', it is by no means clear that such a summation has any
meaning in general. Of course, there may be some sets that can sensibly
be summed, but that does not seem a reasonable proposition for sets of
possible continuations such as the one I have given in terms of cats and
dogs, walking, and talking.
In order for the development you outline to make sense, you would have
to specify the operator 'A' in a lot more detail, so that it only
selected things over which summation was meaningful. IOW, you actually
want 'A' to be a measurement of a quantum state. And you specifically
want a quantum state, because you want there to be more than one
possible result for the measurement 'A'. If 'A' is a classical
measurement of position, for example, then there is only one possible
outcome, and your further development of the situation becomes trivial,
not giving you quantum mechanics at all.
It seems, therefore, that in order to get quantum mechanics out, you
have to essentially assume quantum mechanics right from the start. You
have, at least, to assume that there is variability in the results of
operation 'A', and that this variability can sensibly be superposed.
It is clear that your are trying to introduce the concept of a
quantum superposition by the back door, without doing any work, and
relying on the inherent ambiguity in the '+' operation.
Why is this ambiguous?
Se the above. Does '+' mean 'or', or not?
If you have
nothing but classical outcomes from your observer moment psi(t),
then you cannot simply add these outcomes as if they were separate
eigenfunctions of a quantum operator. There are no such things as
superpositions in classical physics.
There are, it's just that they're not particularly interesting from a
physical point of view.
A superposition of a blue ball and a red ball just means that we don't
know what colour the ball is.
That is a matter of ignorance, not a superposition of different colours.
Sorry, but the whole procedure is nonsense on stilts. It does not
get any better from then on in, but I refrain from analysing further
-- my blood pressure will not stand it!
I think what you're reacting to is the transition from D.6 which
describes what happens in a classical world of a single observer, with
distinct outcomes, to equation D.7, which is the full linear
superposition. The reasoning I give goes in two steps - consider a
whole number of observers of each outcome - eg 2 versions of me decide
to observe a, and versions decide to observe b. Then the combined
projection 2\P_A + 3\P_B plausibly describes this (multiverse)
situation.
I think that rather than imagining an arbitrary number of observers,
what you are actually wanting is to count the number of continuations of
your observer moment that give the each particular result, psi_a = \P_A
psi. In other words, you are using a simple branch counting algorithm to
get quantum weights, measure, or probabilities -- whichever you are
trying to derive. It is known that branch counting is not a good way to
derive quantum probabilities.
Then the next move is to consider the sets of observers to
be drawn from a complex set. Quite what this means is a little hard to
wrap your head around (since we're used to reasoning about whole
numbers of people), but a) it is clear that we don't end up with QM
wihout it,
Yes, there's another rub! You simply assume what you need to get the
result you want. That scarcely counts as a derivation from general
principles.
and b) complex measures are more general than positive real
(or whole number) measures. One had better have a good reason to
impose a more specialist measure, and being comfortable with our usual
notions of discrete persons is not good enough. Particularly when the
universe doesn't agree.
Well, the universe is quantum, and you are not going to get quantum
behaviour unless you either assume it, or take the empirical approach
that science has used in the past.
The problem is why don't we have an even more
general measure, such a quaternions. The answer I give in my book
doesn't satisfy me, even though true, and probably part of the solution.
Pragmatism is actually the answer -- you use what you need to get
results that fit with observation. And that does not count as a
derivation from first principles.
Bruce
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