There is a particularly interesting and surprising difference that I
am aware of between the MWI (many-worlds interpretation of quantum
mechanics) and more general multiverse models like Tegmark's and
especially Schmidhuber's.  Even though the MWI is much better known and
better accepted, it has a major unsolved problem that is actually dealt
with much more successfully in larger multiverse models.  That is the
question of probability or measure.

The MWI is basically modern quantum theory minus the concept of wave
function collapse.  Instead, the universal wave function is considered
to just evolve as a single mathematical whole under a deterministic
physical rule that is the generalized Schrodinger equation.  In fact
Everett's original work can be thought of as an attempt to investigate
the question of what the world would look like if we didn't have random
collapse and just had the smooth, deterministic evolution.  Of course
what he found was that even though there is no collapse, the universe
would in a sense *appear* to have random collapses.

Measurement-like interactions lead to effective "splits" in the
mathematical expression of the wave function, separate branches that
have essentially no more causal interactions between them.  For all
practical purposes the wave function can be split into pieces which
evolve separately.  Each piece corresponds to one possible outcome of
the measurement.  Each piece has an observer who "thinks" he sees that
particular outcome.  To him, the transition was discontinuous and random,
exactly like what is called wave function collapse in the conventional

This is the amazing feature of the MWI, that by removing wave function
collapse and simplifying the assumptions of the theory, we recover the
prediction of an illusion of wave function collapse.  It is IMO one of
the most amazing philosophical results of the century.

However, one piece is missing.  Although Everett showed that the
universe would, in effect, split and create separate observers who would
observe separate outcomes, the question remains of how that relates to
probability.  The traditional quantum wave function collapse postulate
not only says that we will observe a random outcome, it also tells what
the probability of each possibility will be, based on what is called
Born's rule.

Deriving Born's rule from Everett's analysis has been difficult.  In fact,
it has been "solved" many times over the years, but none of the proposed
solutions has really been satisfactory, which is why it keeps getting
solved all over again.  It is easy to show that the universe will split
and observers will appear to observe random outcomes; it is hard to show
that the most likely outcomes are the ones most likely to be observed.

It is this problem, ironically, which is solved much more easily
in the multiverse models we discuss.  For example, in Schmidhuber's
picture we see the multiverse as an ensemble produced by all possible
computer programs.  Each program produces output which is considered to
be a universe.

In this model, there is an easy and attractive argument for why some
universes should have higher measure than others.  If we consider only
finite-sized programs, then an n-bit program is associated with a fraction
of 1/2^n of all infinite bit strings.  This means that if we imagine all
such bit strings as the programs which are creating universes, shorter
programs will occupy a greater fraction of the total space of universes.
This leads to the most important (and perhaps only!) prediction of
multiverse models: the universe we live in should be described by a
relatively short and simple program (or mathematical description, in
Tegmark's formulation).

In principle, if we knew the program for our universe, we could calculate
its measure (at least approximately).  And likewise for the programs
for other universes.  And we have this simple and powerful argument
that explains why certain programs should have much higher measure,
and hence higher probability, than others.

Yet no such argument appears to exist for the MWI.  There, we have
to rely on a variety of different assumptions to prove the Born rule,
none of which have been widely accepted.

It's interesting that these two conceptions of a multiverse have
such different properties.  Of course the branches in the MWI have a
much closer mathematical relationship than the separate universes of a
Schmidhuber multiverse, so the problem is more complicated with the MWI.
It would be nice if there were a simple, information-based argument
similar to the one used for the multiverse, that would produce the Born
rule in the MWI.

Hal Finney

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