Pierz wrote:
I wonder if someone with a good understanding of QM can answer this
question, which has been troubling me. Let us imagine the case of a
single particle, and let us imagine we know its position at time 0. Now
my understanding of the evolution of the wave function for the position
of the particle, according to a Deutsch-ian version of MWI, is roughly
as follows:
The position wave function of the particle represents the distribution
of universes in which the particle is at a certain position. As an
observer, I don't know which of those universes I am in, until I make a
measurement of the particle's position. There is one eigenvalue for each
universe (or better, branch in the multiverse), and the probability of
that eigenvalue is a measure of the underlying universe 'count', or
proportion.
This may be Deutsch's take on MWI, but it is idiosyncratic to the point
of incoherence. The wave function for a particle is, in general, a
superposition of eigenstates of the position (or momentum) operator. But
this does not represent a distribution of universes. The problem with
such an account is that the basis states for the superposition are
undetermined. For a single particle, there are an infinite number of
possible bases for position space, and an equal infinity of possible
bases for the representation in momentum space. This is known as the
basis problem -- why is it that when we make a measurement, one
particular basis is picked out, and the same basis is picked out
consistently every time. If it were merely a matter of self-location in
a multiverse, there would be no way of ensuring that the same basis
eigenfunctions were selected every time.
The basis is actually determined dynamically by the process of
einselection, where the preferred basis is that which is stable against
environmental disturbances. Separate universe do not emerge until
decoherence has dissipated the interference terms into the environment
in an irreversible way. Thus universe emerge only after interaction
(measurement); they are not present beforehand. So the terms in the
superposition do not represent separate universes -- they are just
elements of a mathematical representation of the quantum state.
As an aside, quantum probabilities as given by the Born Rule do not come
from branch counting -- the probability is not a measure of the
underlying universe count.
So far so good. The explanation seems coherent, even if we
haven't explained the distribution of those universes. However, I am
puzzled by the case of spins. Consider a set-up in which a photon is
polarized in the z direction, so that we know that the particle will,
with probability 1, pass through another polarizer also oriented in the
z direction. However what of the situation where the second polarizer is
oriented at 45 degrees to the first one? In that case, the probability
is 0.5 that the photon will pass through. If it does, then obviously the
probability is 1 that it will also pass through a third polarizer also
oriented at the same angle.
This is a good illustration of the problem with thinking of the elements
of the superposition as separate worlds. Every state, even an eigenstate
of momentum or spin, is a superposition of eigenstates in different
bases -- in fact, an infinite number of different superpositions. When
you see it this way, you realize that the components of the
superposition can't be worlds, because then you would have to say that
the world in which we find ourselves is actually an infinite number of
different superpositions of different worlds -- a set for each possible
basis of the Hilbert space. The notion of a "world" then becomes incoherent.
The photon polarized in the z-direction is a superposition of
polarization states in any other direction. It is an equal mixture of 45
degree polarizations, but unequal mixtures of polarizations at any other
angle. It is these coefficients of the expansion of the state in a
particular basis that give the probabilities (as absolute values of the
coefficients squared). Since any real value is possible, branch counting
is ruled out.
So what is going on in the multiverse in this scenario? Clearly, prior
to the photon hitting the 45 degree polarizer, it can't be the case that
half the universes have photons polarized at 45 degrees to the z axis,
because in fact 100% of them are polarized in the z direction. Yet after
the hitting the polarizer, half do. So in this case the discontinuity
between quantum state and measurement, which MWI saves us from in the
case of a continuous variable like position, seems to persist. What is
going on at the point of the photon's interaction with the polarizer in
an MWI account? Clearly, the multiverse differentiates into two branches
corresponding to the two spin eigenvalues, each with measure 0.5. But
does MWI have anything to say about the the weirdness of the jump
between the z polarization and the 45-degree polarization?
You have seen the difficulty in Deutsch's account, which is that he
makes worlds into effective hidden variables -- there is a hidden
variable which tells which world we are actually in. Deutsch is trying
to recover the idea that a quantum particle has definite values for each
observable before measurement, one value in each of the superposed
worlds. The "world" variable is then the operative hidden variable.
It has long been understood that such hidden variable approaches to
quantum mechanics fail to account for the date. The notion of a "world"
cannot be regarded as a non-local hidden variable.
Bruce
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