On 21/07/2017 4:55 am, Russell Standish wrote:
On Wed, Jul 19, 2017 at 12:03:24PM +1000, Bruce Kellett wrote:
On 6/07/2017 5:55 pm, Russell Standish wrote:
On Thu, Jul 06, 2017 at 04:18:49PM +1000, Bruce Kellett wrote:
On 6/07/2017 2:33 pm, Russell Standish wrote:
Establishing linearity is key.
Yes, and you haven't made progress with that.
All I ask is to give me some more time on this. I have some further
ideas in this regard, but need some dedicated time to think about it.
I have been thinking about it as well. I think your problem is even
more difficult that just establishing that the sum of two observer
moments is also an observer moment. If OMs are to form a linear
vector space, they have to satisfy some further axioms:
The axioms of associativity and commutativity are fairly easy if you
have additivity, but the existence of a zero vector, 0, such that V
+ 0 = V for any vector V in the space, and the existence of an
inverse, -V, s.t. V + (-V) = 0, might be more difficult in terms of
OMs. What is a null OM? What is an inverse (negative) OM?
I think you need these properties as well as additivity in order to
have a vector space. At least, these are among the axioms for a
vector space as listed by Wikipedia.
Bruce
0 will correspond to the universal set, either the everything/nothing
object, or maybe the reference OM in the reduced space described in my
appendix.
The inverse vector corresponds to the complement.
You would be right if the zero vector corresponds to the
everything/nothing object, as that is clearly not an OM. If it is the
reference OM, then I'm not sure the Born rule derivation goes
through as given.
At first blush, the complement is unlikely to be an OM, but given the
duality between a set and its complement, we can invoke Leibniz's
indiscernibles principle, and identify a set with its complement.
In any case, I think you have a valid point that a linear combination
of observer moments is not in general an observer moment - for one
thing, the linear combination of all observer moments (ie the "3p") isn't.
What I'm trying now is considering more primitive objects, namely all
the subsets of the everything/nothing (ie the set [0,1]∞) constructed
from a set of "atomic" subsets. I have found a natural encoding of the
sets as complex vectors (well rays, really, as multiplying by an
arbitrary complex prefix doesn't change the underlying subset).
The nice thing is that this formulation naturally requires a complex
field - the real field doesn't cut the mustard. This addresses my
biggest concern with my QM derivation. It also exhibits the
relationship:
\P_A + \P_B = \P_{A∪B} + \P_{A∩B}
I haven't formally proved it, but it works on a number of nontrivial
test example I tried by hand. I'd like to see if I can shoot it down
before spending a lot of time proving theorems.
It remains to be seen how the Born rule fares - the sets in question
have a natural measure, so one ought to be able to derive a formula
describing the Born rule. However, the vector space {\P_A} does not
have an obvious inner product defined...
Anyway, I'll post more on this topic to give you guys a chance to take
your own potshots when I get time - I'm still travelling for the next
couple of weeks, so even finding time to write this stuff up in a post
is daunting ATM.
I have been reading up on Zurek's 'existential interpretation of QM.
This is an interesting attempt to understand unitary QM in an explicit
Everettian 'Relative State' model. In other words, he claims that one
can get a fully unitary model without the necessity for many worlds --
the observer defines the 'relative state' from his component of the wave
function.
The many worlds arguments of Deutsch and Wallace, among others, are very
heavy on the need for a 'realist' understanding of QM. Wallace, in
particular, seems to think that realism necessitates many worlds because
he wants a realist interpretation of the wave function itself, or at
least a space-time version of this. Zurek sees realism in different
terms, namely, as the objectivity of many observers agreeing on what the
state of the system is, so he concentrates on the emergence of classical
results from the quantum formalism, with many copies of quantum outcomes
coded in the environment -- 'quantum Darwinism'.
The relevance to your concerns is that he has an interesting derivation
of the Born rule within his approach: a derivation that avoids the
circularity inherent in the Deutsch-Wallace derivation, and the
non-uniqueness of the concept of rationality coming from their decision
theoretic approach. Zurek, on the other hand, relies on his notion of
'envariance', which is a symmetry of entangled quantum states. Zurek
starts with three axioms of quantum theory:
1. The quantum state is a vector in Hilbert space;
2. Unitary evolution (Schrödinger equation);
3. Immediate repetition of a measurement yields the same outcome.
I think if you get this far, you could perhaps use Zurek's argument from
the symmetries of entanglement to get probabilities and the Born rule.
I haven't gone into this in detail as yet, but it seems promising to me.
Bruce
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