Monday, July 2, 2007

The many-worlds interpretation

In 1957 H Everett I11 wrote an article entitled “‘Relative State”
Formulation of Quantum Mechanics’ (Reviews of Modern Physics
29 454) which introduced what has become known as the ‘manyworlds’
interpretation of quantum theory. He began by noting that
the orthodox theory requires wavefunctions to change in two
distinct ways; first, through the deterministic Schrodinger equation
and, secondly, through measurement, which causes the reduction
of the wavefunction to a new wavefunction which is not uniquely
determined. It is this second type of change that causes problems;
what is a ‘measurement’?, what are the non-quantum forces that
cause it?, how can it occur instantaneously over large distances?,
etc. Everett was in fact motivated in his work by yet another
problem: he was interested in applying quantum theory to the
whole universe, but how could he then have an ‘external’ observer
to measure anything?
The solution that Everett proposed to the problems of wavefunction
reduction was to say simply that it does not happen. Any
isolated system can be described by a wavefunction that changes
only as prescribed by the Schrodinger equation. If this system is
observed by an external observer then, in order to discuss what
happens, it is necessary to incorporate the observer into the system,
which then becomes a new isolated system. The new wavefunction,
which now describes the previous system plus the observer, is again
determined for all times by the Schrodinger equation.
To help us understand what this means we shall put it into
symbolic form. To this end we return to the barrier experiment, in
particular as this was discussed in $3.5. We write the wavefunction,
after interaction with the barrier, in the form:
PR( w ’ D ~ ~ +D P~T(W ~‘D~RO )FFD ~ON ) ,

This is not really as complicated as it might appear. The Ws
describe the particle, with the arrows indicating the direction, and
the Ds the two detectors. The first bracket is then the wavefunction
of the reflected wave and the second that of the transmitted wave.
Each of these wavefunctions is taken to be 'normalised' so that it
corresponds to one particle. Then the PR and PT are the parameters
that give the magnitudes of the two wavefunctions. The squares of
these numbers give the probability for reflection and transmission
respectively. We notice that this wavefunction correctly describes
the correlations between the states of the detectors and those of the
particle, e.g. that if the right-hand detector is ON then the particle
has been reflected, etc. This correlation exists because, as noted in
$3.4, the wavefunction is not simply a product (in fact in this case
it is the sum of two products).
According to the orthodox interpretation of quantum theory
such a wavefunction reduces, on being observed, to
w'DON OFF with probability Pi
R DL
or to
w-DOFF ON with probability Pf. R DL
(See figure 15.)
In the interpretation due to Everett, however, this reduction does
not occur. The true reality is always expressed by the full wavefunction
containing both terms. This is all very well, we are saying,
but did we not convince ourselves previously that the reduction had
to occur; that deterministic theories are not adequate to describe
observation? We certainly did, so we must examine the argument.
It relied on the fact that we, or more properly I, do not see both
pieces of the wavefunction. To me, either reflection or transmission
has occurred, not both. Clearly then, in order to understand what
is happening, it is necessary to introduce ME into the experiment
and to include ME in the wavefunction. Although my wavefunction
is very complicated the only relevant part for our purpose here is
whether I am aware of reflection or transmission. We denote these
two states of myself by ME"' and ME"^^ respectively. Thus the
complete wavefunction, according to Everett, is:
P~(WDD)-MEI~+* Pr(W DD)+ME"~"~
where we have simplified the notation in an obvious way. Notice that again the wavefunction contains the correct correlations: if the
particle is transmitted then I have observed transmission, etc.
Previously we argued (e.g. in 54.1) that, since we are only aware
of one possibility, one of the terms in the above expression must
be eliminated. Everett would argue instead that there are two MES,
both conscious but unaware of each other. Thus, through my
observation of what happens in the barrier experiment, I have split
the world into two worlds, each containing one possible outcome of
the observation.
Similar considerations apply to other types of observation. In all
cases the Everett interpretation requires that all possible outcomes
exist. Whenever a measurement is made we can think of the world
as separating into a collection of worlds, one for each possible
result of the measurement. It is through this way of thinking that
the name ‘many worlds’ has arisen. Such a name was not, however,
in the original Everett paper, and in some ways it is misleading. The
key point of this way of interpreting quantum theory is that
measurements are not different from other interactions; nothing
special, like wavefunction reduction, happens when a measurement
is made; everything is still described, in a deterministic way, by the
Schrodinger equation.
How can we reconcile this with our previous belief that
measurements were special? The previous argument was basically
as follows:
I am only aware of one outcome of a measurement, therefore there
is only one outcome.
Now we would argue differently:
I am only aware of one outcome of a measurement because the ME
that makes this statement, is the ME associated with one particular
outcome. There are other MES, which are associated with different
terms in the wavefunction, and which are aware of different outcomes.
The wavefunction given above for the barrier experiment
illustrates this: both of the terms exist, there are two MES but they
are not aware of each other.
It will be seen that, from the point of view of the many-worlds
interpretation, the ‘error’ we made earlier was that we inserted a
tacit assumption that our minds were able to look at the world from outside, and hence to conclude from our certainty of a particular
result that the other results had not occurred.
The ‘branching’ of the world into many worlds is therefore an
illusion of the conscious mind. The reality is a wavefunction which
always contains all possible results. A conscious mind is capable of
demanding a particular result (this is what we mean by making an
observation) and thereby it must select one branch in which it
exists. Since, however, all branches are equivalent, the conscious
mind must split into several conscious minds, one for each possibie
branch.
Is this then the answer to the problem of reality in the quantum
world? At first sight it appears more satisfactory than our previous
ideas where consciousness seemed to have to affect wavefunctions;
now this is not required. Nevertheless the general view of the
theoretical physics community has been to reject the many-worlds
interpretation. This of course is not in itself a strong argument
against it, particularly when we realise that many writers have
rejected it on grounds that suggest they have failed to understand
it. Here I should admit that the above discussion was an attempt
to describe what I think is the most plausible form of the Everett
interpretation. The original paper, and others mentioned in the
bibliography, contain mainly the formalism of orthodox quantum
theory with little comment on the interpretation.
It is probably fair to say that much of the ‘unease’ that most of
us feel with the Everett interpretation comes from our belief, which
we hold without any evidence, that our future will be unique. What
I will be like at a later time may not be predetermined or calculable
(even if all the initial information were available), but at least I will
still be one ‘1’. The many-worlds interpretation denies this. For an
example to illustrate this lack of uniqueness (some would say rather
to show how silly it is) we might return to the barrier experiment
and suppose that the right-hand detector is attached to a gun which
shoots, and kills, me if it records a particle. Then after one particle
has passed through the experiment, the wavefunction would contain
a piece with me alive and a piece with me dead. One ‘I’ would
certainiy be alive, so we appear to have a sort of Russian roulette,
in which we cannot really lose! Indeed, since all ‘aging’ or ‘decaying’
processes are presumably quantum mechanical in nature, there
is always a small part of the wavefunction in which they will not
have occurred. Thus, to be completely fanciful, immortality is guaranteed-Z will always be alive in the only part of the wavefunction
of which Z am aware!
It is important to realise that the fact that another observer does
not see two '1's is not an argument against this interpretation. As
soon as YOU, say, interact with me so that you can discover
whether I am alive or dead, you become two Yous, for one of
which I am dead and the other I am alive. In wavefunction
language, using the previous notation, we would have:
PR(WDD)*ME~~~YOU' + WDD)~ME"~~~YOU'.
Neither of the two YOUS is aware that there are two MEs.
Two final remarks in favour of the many-worlds interpretation
should be made here. It has long been known that, for many
reasons, the existence of 'life' in the universe seems to be an incredible
accident, i.e. if many of the parameters of physics had
been only a tiny bit different from their present values then life
would not have been possible. Even within the framework of
'design' it is hard to see how everything could have been correct.
However, it is possible that most of the parameters of physics were
fixed at some early stage of the universe by quantum processes, so
that in principle many values were possible. In a many-worlds
approach, anything that is possible happens, so we only need to be
sure that, for some part of the wavefunction, the parameters are
correct for life to form. It is irrelevant how improbable this is,
since, clearly, we live in the part of the wavefunction where life is
possible. We do not see the other parts. Thinking along these lines
is referred to as using the anthropicprinciple; for further discussion
we refer to articles listed in the bibliography.
The other remark concerns the origin of the observed difference
between past and future, i.e. the question of why the world exhibits
an asymmetry under a change in the direction of time when all the
known fundamental laws of physics are invariant under such a
change. One aspect of this asymmetry is psychological: we
remember the past but not the future. (Note that it is because of
this clear psychological distinction between past and future that we
sometimes find it hard to realise that there is a problem here, e.g.
it is possible to fool ourselves that we have derived asymmetric
laws, like that concerning the increase of entropy, from laws that
are symmetric.) The many-worlds interpretation gives an obvious
explanation of this psychological effect: my conscious mind has a unique past, but many different futures. Each time I make an
observation my consciousness will split into ‘as many branches as
there are possible results of the observation. Some readers may
wish to note that this might allow vague, shadowy, probabilistic,
‘glimpses’ into the future-thus, a prophecy is likely to be fulfilled,
but only for one of the future MES.

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