On 27/06/2017 10:21 am, Russell Standish wrote:
On Tue, Jun 27, 2017 at 08:52:15AM +1000, Bruce Kellett wrote:
On 26/06/2017 3:57 pm, Russell Standish wrote:
On Mon, Jun 26, 2017 at 11:50:45AM +1000, Bruce Kellett wrote:
That is not what is normally meant by the '+' symbol. You have
simply defined a conjunction to be a disjunction!
We are constructively defining +. I would not be so cruel as to use +
if the end point were not the usual group operation.
Yes, the endpoint is that the '+' is simple addition. It seems to me
that if you actually wrote
psi_a v psi_b,
where 'v' stands for disjunction, or 'or', you would not have got
very far with your derivation. By writing the sum
psi_a + psi_b = psi_{ab}
you have, in fact, simply assumed linearity.
Not at all. The latter equation is identical to the first with the
symbol v replaced by +.
A significant property
of linear systems is that if you have two solutions, the sum is also
a solution. If you are dealing with sets, the the operation is the
union of sets, which is different. But you specifically state that
your projection operator acting on the ensemble produces a single
outcome psi_a = \P_{a}*psi, so you are dealing with addition of
numbers or functions, not the union of sets.
No, you are just dealing with a function from whatever set the ψ and ψ_α
are drawn from to that same set. There's never been an assumption that
ψ are numbers or functions, and initialy not even vectors, as that
later follows by derivation.
psi(t) is an ensemble, psi_a is an outcome. The projection produces the
outcome from the observer moment psi. It maps the ensemble to one member
of that ensemble. Certainly, psi is not a number or a function, it is a
set of possible outcomes: the psi_a are single outcomes, be they
numbers, functions or vectors, but the are not just further ensembles.
Thus, for the sum to make sense you must assume linearity.
If you are objecting that the use of the symbol '+' implies linearity
where no such thing is assumed, then feel free to replace it with the
symbol of your choice. Then once linearity is established, feel free
to replace it back again to + so that the formulae following D.8 have
a more usual notation. Fine - that is a presentational quibble. My
taste is that it is unnecessarily cumbersome, but if you find it helps
prevent confusion in your mind, please do so.
Now
linearity is at the bottom of most distinctive quantum behaviour
such as superposition, interference, and entanglement. It is not
surprising, therefore, that if you assume linearity at the start,
you can get QM with minimal further effort.
Except that I don't assume linearity from the outset.
There seems to be some confusion between outcomes of observations and
sets of possible outcomes. The \P_A*psi is actually defined as a
superposition in (D.2), ad you then seek to determine the probability of
this superposition? You define the probability of a set of outcomes by
P_psi(\P_A*psi), which is P_psi(\Sigma psi_a). I find it hard to
interpret what this might mean -- the probability of a superposition of
measurement outcomes (with equal weights, what is more)?
You then talk about this as though you were still partitioning sets, but
the probability is not defined on a set, only on a superposition. If it
is a set, then (D.2) makes no sense.
You then introduce, quite arbitrarily, multiple observers for each
observer moment. This then gives you a measure, which is then made to be
complex!! The number of observers for each observer moment, even if
there can be more than one, which is not proved, cannot be complex. So
your introduction of a measure, or weight for each superposition really
does not make sense. You then conclude that V, the set of all observer
moment, is a vector space over the complex numbers.
I remain baffled. You start with an observer moment as a set of
consistent possibilities. But there is no specification of what
'consistent' might mean. There is also no particular structure imposed
on this observer moment, and you conclude, after a number of obscure
manipulations, that the set of all observer moments is a vector space
over the complex numbers. I look in vain for the magic that converts an
unstructured ensemble into a linear vector space. This is surely a
non-trivial restriction on the nature of observer moments, but you do
not restrict the possible generality, you only project particular
(unstructured) results from this observer moment. What, exactly, has
gone on to extract linearity?
That is why I think that you have actually built linearity in from the
start -- there is no mechanism, engine, or procedure that extracts this
linearity. And what happens to things in the ensemble that are not linear?
This does not, to me, pass the sniff test (or, in the Australian
vernacular, the pub test). Can you wonder that I am sceptical that you
have actually proved anything?
Bruce
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