Jacques Mallah wrote:
> >One of the Kolmogorov probability axioms is that the certain event has
> >probability 1. Conservation of probability is simply asserting that
> >the probability of the certain event does not change over time. A
> >fairly obvious corrollory one would have thought.
>
> Maybe you're trying to insult my intelligence, but in doing so you only
> insult your own.
I don't intend to insult anyone's intelligence, however if you feel
like being insulted, go ahead!
> As I said, measure is not strictly conserved as a function of time. Of
> course it is by definition true that the integral of the effective
> probability over all observer-moments in all universes (including, since we
> are presumably using the AUH, universes that are not governed by QM) and all
> times is equal to 1.
Not at all. The reason we use the term measure is that it may not be
normalisable. Probability and measure are only equivalent when the
measure can be normalised to 1. For example, the uniform measure is
only normalisable on compact sets, a property that seems particularly
unlikely of the Plenitude.
Now as to the Schroedinger equation, the question was why is the
evolution operator i times a Hermitian operator? Implicit in this
question is that we're talking about the Multiverse (with time), and that the
Multiverse is described at each point of time by a vector \psi(t) drawn
from a Hilbert space. At each point in time, all possible observable
outcomes are described by \psi(t), so as I remarked, the probability
measure of that state must be equal to 1 at all times.
It has nothing to do with measure reducing processes, nothing
whatsoever to do with your mythical absolute measure of observer
moments, nor with the Universal Prior.
> Even if it were established that the measure per unit time of an
> "observer" was equal to the squared amplitude of that "observer"'s
> wavefunction (and this has yet to be satifactorily derived from theory),
> that might (in general) not have been conserved. It may well be, too, that
> the formula for measure would have to be modified if the sum of square
> amplitudes was not conserved - which might even allow conservation of
> measure-per-unit-time without requiring conservation of squared amplitudes.
>
> >Saying that the i is there makes it easier to write misses the
> >point. The Schroedinger operator is i times a Hermitian operator. This
> >is not the most general form of linear operator, so this structure
> >ought to have some form of explanation. The one I gave is not actually
> >due to me, but I can't think where I first came across it - possibly
> >Emile Durand.
>
> It is a very well known fact that a Hermitian Hamiltonian leads to a
> unitary (squared-amplitude-sum-conserving) time evolution operator; I don't
> know who first said it but it must have been in the early days. Of course,
> if it weren't Hermitian, it would have other consequences, such as allowing
> consevation of energy to be violated.
> BTW, there are some people who attach an almost mystical significance to
> the i. I had one teacher who told the class that in classical mechanics, i
> is used only for convenience; but in QM it is fundamental; therefore QM is
> weird. That's nonsense.
>
I would concur with that. There's nothing weird about i.
> >From: Higgo James <james.higgo.domain.name.hidden>
> >Trying to derive SE from AUH is like trying to derive 'Jacques Mallah' from
> >AUH.
> >It's very easy: all universes exists, so some thoughts of the cleass "the
> >SE is -i hbar d/dt psi = H psi" exist.
> >Some thoughts of the class "why is the SE -i hbar d/dt psi = H psi" also
> >exist - and by WAP we shouldn't wonder why we *are* (not think, but are)
> >such a thought. If we weren't, we wouldn't wonder it...
>
> James, you are abusing the AP. SE is not like JM. It would be
> worthwhile to try to derive SE because the effective probability of
> observer-moments that see it is likely to be large; we might be able to get
> a prediction. In the case of JM, one name is just about as likely as
> another (of similar length), so most observer-moments probably don't think
> they have the name JM.
Agreed.
>
> - - - - - - -
> Jacques Mallah (jackmallah.domain.name.hidden)
> Physicist / Many Worlder / Devil's Advocate
> "I know what no one else knows" - 'Runaway Train', Soul Asylum
> My URL: http://hammer.prohosting.com/~mathmind/
> ________________________________________________________________________
> Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com
>
>
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Received on Thu Jun 29 2000 - 20:16:47 PDT