I attach a brief PDF of what I have so far. It shows how observer moments, modelled as sets of bitstrings classified by looking at a finite number of bits naturally map to vectors in a complex vector space. There are some lemmas, proofs and conjectures (theorems I haven't managed to prove yet, but think are plausible).
Cheers
On Fri, Jul 21, 2017 at 04:55:02AM +1000, 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.
>
> --
>
> ----------------------------------------------------------------------------
> Dr Russell Standish Phone 0425 253119 (mobile)
> Principal, High Performance Coders
> Visiting Senior Research Fellow [email protected]
> Economics, Kingston University http://www.hpcoders.com.au
> ----------------------------------------------------------------------------
--
----------------------------------------------------------------------------
Dr Russell Standish Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellow [email protected]
Economics, Kingston University http://www.hpcoders.com.au
----------------------------------------------------------------------------
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quantumDerivation.pdf
Description: Adobe PDF document
