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From: Hal Finney <hal.domain.name.hidden>

Date: Thu, 2 Jun 2005 12:18:07 -0700 (PDT)

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

interpretation.

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

Received on Thu Jun 02 2005 - 16:22:48 PDT

Date: Thu, 2 Jun 2005 12:18:07 -0700 (PDT)

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

interpretation.

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

Received on Thu Jun 02 2005 - 16:22:48 PDT

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