From: D. Goel (deego@gnufans.org)
Date: Mon May 05 2003 - 07:43:00 MDT
my 2 cents.
>
> I think MWI is rather contrived, but you need to know a few
> technicalities to appreciate why. Naively, it might sound
> like this:
"Usual QM" has these postulates:
[1] Schrodinger's eqn.
[2]. An ad hoc "collapse" to an eigenstate of an observable whenever
an observation occurs.
--- 2 was a very imprecisely defined postulate, with it being unclear
what an observation meant. Unless the "Observation" was done by
a conscious being, no "collapse" occurred. Moreover, this
"collapse" was nonlocal and hence "disturbing". To me, the
implicitly implied need for "conscious being" was the most disturbing.
The Everett interpretation simply says:
Wait a minute, do we even need this ad hoc postulate [2]? Let's just
see what happens with [1] alone. After all, there's nothign special
about "conscious people" as far as the laws of nature are
concerned. We are made up of electrons and protons too.
So, let [1] be the only postulate, and see what happens. And surprise,
it seems that all the QM "observations" can be explained.
Everything else simply follows. Everett interp. doesn't postulate
"splitting of the universe" or anything. It has lesser number of
postulates than "traditional QM" and they are more well-defined, for
once.
>
> But here's the very first problem: there's more than
> one way of decomposing a wavefunction.
As i understand it, the wavefunction, following its natural evolution
equation, can only be interpreted to divide in certain ways. These
ways are what turn out to be what we observe as "classical reality".
Any other way of dividing the wavefunctino up is unstable to
wavefunction evolution via Schr eqn. These states are the "physical
states" we observe, and find ourselves in, and this phenomenon is
called "decoherence."
> One instance of this is illustrated by problem (a) for DeWitt's
> version of MWI, mentioned above. In the quantum mechanics of a
> single particle, exact position states are represented by "delta
> functions", formal functions which are infinitely peaked at a single
> coordinate and which are zero everywhere else; and exact momentum
> states are represented by "plane waves", the sort of functions which
> show up in Fourier analysis on multidimensional spaces. A generic
> wavefunction can be decomposed into a sum over delta functions, *or*
> a sum over plane waves (or indeed, a sum over many many other
> possible sets of functions). Which "basis set" is the real one?
When *you* "observe" position, you perform actions that end up
entangling you with the position basis. In other words, the wave
function then becomes:
[|x=2>|you_2>] + [|x=1>|you_1>] [1]
where you_1 is a version of you that observes x=1. This combined
wavefunction has only one set of allowed "natural" physical states
interpretation ---Those physical states are the position basis. In
other words:
Consider this combined wavefunction. Now, one intepretation of
this wavefunction is that that when you observe x=2, you are just the
first part of the wavefucntion.
But then, you might ask, why consider the 2 parts as above? Why not
divide up the wavefunction as in eqn. [1] differently? Why not
reexpress it as:
[ |x=2>|you_2> + 0.5* |x=1>|you_2> ] + [ |x=1> |you_1> - 0.5* |x=1>
|you_1> ]
and then consider the 2 parts as the ones in square brackets? And
then consider "you" to be the only part in the first square bracket?
What would that "you" observe? Wouldn't you observe the particle
in a weird state instrad of x=2? IOW, why do you always observe the
particle at a particular position instead of a mixture?
The answer to this, iiuc is that the above division is "not stable" to
wavefunction evolution. Only certian divisions turn out to be
"stable" as the Schr. eqn. progresses. And this phenomenon is called
decoherence.
Again, no postulates necessary. Just following the good old Schr. eqn.
Similar considerations apply if you had "observed velocity", which
would have landed the combined wavefuinction into a state:
[|v=1>|you_1>] + [|v=2>|you_2>]
Again, there's only one "stable" decomposition of the above function
from which to construct "basic observation states". That
decomposition is the one above.
Even if some things still leave one uneasy, it is certainly much
clearner than traditional qm imo. Atleast the postulates are
well-defined.
that's what i understand, i could be wrong.
The problem of measure still remains, but that remains in
"conventional QM " too,.. where measure is an ad hoc squaring of psi.
---- When i say "unstable" to wavefunction evolution, i meant that the complex coefficients in front of the various parts of the wavefunction (which i omitted above) evolve differently with time, rendering the division impossible within an epsilon second. DG http://gnufans.net/ --
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