Another problem occurs further down when you seem to have complex
numbers of observers observing an observer moment. Why you should
have more than one observer for any observer moment is a mystery yet
to be solved.
It's more a measure over observer moments. In a branching multiverse,
not all observer moments are equally likely, but one would expect
across a branching point, measure should be conserved.
Why the measure is complex, not real is more tricky. With the
everything, subsets naturally induce a real valued measure. But we do
know that complex measures are more general, and we need a good reason
not to choose the most general. But complex measures are not the most
general. I do say "more general division algebras cannot support
equations of the form (D.7)", but I confess, I'm still not completely
happy with that line.
But then you go on, in eq. B8 to define the inner
product in terms of the probability function. But you have merely
multiplied together two expansions in terms of projections over
possible outcomes -- assuming that there is a linear span over the
space in the process. This gives the Born rule, sure, because you
have built it into your derivation of the inner product.
By the time we get to equation D.8, we have proved that the set of
observer moments is a vector space, so yes, this construction is
allowed. We are entitled to define any real-valued bilinear operator on
that space and call it an inner product.
By using that particular inner product, you get the Born rule in the
usual form. If we'd chosen another, we'd have a different expression
that is equivalent to the Born rule.
So you know about QM from the start, and devise a strategy to get
you there. One of the problems that many-worlders face in their
attempts to derive the Born rule from within MWI is that they cannot
independently justify a probabilistic model.
Yes, but I don't start with the MWI (namely, I don't start with a
Hilbert space and unitary equation of motion - ie Schroedinger's
equation). I start with evolution in a generic multiverse.
Why a multiverse? You no doubt argue for it elsewhere, but that is
not apparent in your quantum derivation.
Yes - of course. The whole book is premised on it.
And I do not understand why the most general equation for computing
psi as a function of time is a first order differential equation.
The equation could clearly be non-linear in psi -- such things have
been postulated after all, as in general relativity and GRW for
instance.
A first order differential equation needn't be linear. Linearity comes
from assuming that the laws of physics don't change every time you
observe something, more specifically the solutions ψ_α are also
solutions.
A higher order equation can be transformed into a first order equation
by adding new variables - a trick commonly done in dynamical systems
theory.
Perhaps there's an implicit assumption that the evolution should be
Markovian. I think one could make a convincing case that it should be,
but perhaps that assumption needs to be made explicit.
Besides, you do not show that the operator H is the Hamiltonian and
the energy operator. You do not derive the basic commutation
relations between position and momentum operators -- a relation that
is central to the whole of QM.
The commutation relations between x and i∂/∂x follow
mechanically. That i∂/∂x can be identified with p (modulo a constant
multiplier) is the correspondence principle, which I discuss on page
120. Vic Stenger has an argument for it, based on Noether's theorem.
If you have a
probabilistic model in 3 or more dimensions, Gleason's theorem tells
you that the Born rule is the only consistent model for
probabilities.
My arguments go through in fewer than 3 dimensions as well, AFAIK,
although that would a relatively uninteresting world - very black and
white :). Which is why I suspect it is independent of Gleason.
But you have to say why you want a probabilistic
interpretation in the first place. Deutsch's attempts founder on the
fact that he has to assume that small amplitudes have small
probabilities, even to get started, so his argument is manifestly
circular.
Yes - I think the problem with those approaches is that they start
with a Hilbert space and unitary equation of motion (ie a classic
MWI), and then fail to generate the Born rule because there is no
observer in their mechanics.
As I said, you build a probabilistic model in at the start, so
Gleason's theorem is going to get you the Born rule automatically.
Or if you don't assume Gleason, you have an equivalent result by
another route. Assuming a probabilistic model is a very powerful
starting point......
Sure - but it is necessary. If evolution did not work the way it did,
we could only ever be Boltzmann brains, isolated observers existing
fleetingly, barely having time to consider what to have for lunch, let
alone figuring out the meaning of the universe. Fortunately for us,
evolution does work to generate complex worlds from simple beginnings,
meaning an evolved world is overwhelming more likely to occur in the
Multiverse of Everything than Boltzmann brain existences.
Why do you have to have evolution?
My 2 sentence summary is as above. If you want a more detailed
portrayal, read my book :).
It seems to me that you are
allowing enough empirical science to creep into your deliberations
to give you the results you want.
For guidance, perhaps, but I don't think I've built a just-so story
here.
I don't think Boltzmann brains are
the only alternative to evolution.
The only alternative to it arising through an irreversible process is
de novo creation.
Evolution could work in all sorts
of different ways -- such as Lamarkianism, etc. The only reasons we
rule these out are empirical.
So? - I make no comment on the central dogma of biology.
Similarly, the only reason for going
to quantum mechanics is solidly empirical -- classical physics just
does not work all the way down. So one will never be able to derive
quantum mechanics from general, non-empirical considerations. It is
just too weird for that!
If true, then point to which assumption I use that is too weird.
