From: Mitchell Porter (mitchtemporarily@hotmail.com)
Date: Mon May 05 2003 - 06:40:41 MDT
Lee Corbin said:
>Also, I too think that Einstein would have felt relieved
>if he'd lived to hear Everett's explanation. The whole
>issue for him was determinism, I believe, and the famous
>dice remark refers to the universe taking just one of the
>possible random branches. MWI is, of course, completely
>deterministic.
I think MWI is rather contrived, but you need to know a few
technicalities to appreciate why. Naively, it might sound
like this:
1) Quantum mechanics says that some things happen completely
at random: one out of a set of possibilities occurs, for no
reason.
2) The many-worlds interpretation says that every possibility
is realized: whenever such a choice-point is reached, the
universe 'splits', with one offspring for each possibility.
This is MWI according to DeWitt, and it has some problems
which are widely recognized:
a) Complementary observables. A quantum particle can't have
definite position and definite momentum at the same time.
When the split occurs, does the particle start out in each
branch with a definite position, or with a definite momentum,
or in some other state?
b) Relativity. Is the whole universe copied at each split?
If so, what reference frame are we using? Alternatively,
one could suppose that the split propagates outwards at
lightspeed, with the parent universe dynamically cleaving
in superspace, but no-one has expressed this mathematically.
c) Interference effects. Quantum effects such as the
interference fringes which show up in the double-slit
experiment require *recombination* of separated wavepackets
(in that example, the recombination of wavepackets that
have passed through the separate slits). If the universe
splits before recombination is to occur, recombination would
seem to be impossible, because the two wavepackets are now
in separate universes. So either universes can recombine,
or splits don't occur between measurements, both of which
pose further problems.
All this is well-known amongst MWI advocates, and so most
of them prefer MWI according to Everett, which is a more
subtle approach. Instead of splitting universes, Everett
talks about "relative states" of subsystems of the universe.
Formally, this involves taking the total wavefunction of
the universe, and expressing it as a superposition of
"product states" of parts of the universe:
z1|a1>|b1>|c1>... + z2|a2>|b2>|c2>... + ...
(Just to explain: a,b,c,... are physical systems.
|a1>,|a2>,... are distinct quantum states of system a.
|a1>|b1> means 'a is in state a1, b is in state b1', etc.
z1,z2,... are complex numbers, coefficients in the overall
superposition.)
The basic task of the MWI is to connect the particular
universe we see with the ensemble of universes implicit
in the supreme wavefunction. Everett does it through this
algebraic decomposition: you-in-this-world are one of
those physical systems, in one of those distinct quantum
states appearing in the sum above.
But here's the very first problem: there's more than
one way of decomposing a wavefunction. 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?
The same consideration applies to finite-dimensional
quantum states, such as the state of a qubit. Qubits are
normally thought of as existing in a superposition of
|0> states and |1> states, but they can equally well be
thought of as being in a superposition of |0>+|1> states
and |0>-|1> states.
In the many-worlds FAQ, which is the most popular
exposition of MWI available online, a particular basis
is singled out for attention, but it's also explicitly
stated that this basis is preferred only for purposes
of calculation, it's not ontologically special in any way.
There's a further step in the MWI according to Everett:
the construction of a probability measure. Again, I think
this involves the postulation of extra structure, rather
than being implicit in the wavefunction. I'm in the middle
of discussing these issues with the author of the FAQ,
and we haven't got to this step yet, so I won't say more.
But I just wanted to point out *why* the MWI is not as
natural as it might sound.
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