Re: Observables, Measurables, and Detectors

From: Patrick Leahy <jpl.domain.name.hidden>
Date: Wed, 25 May 2005 13:24:35 +0100 (BST)

>
> It looks as though you advocate a role for each of these:
>
> observables
> measurements
> detectors
>
> and for all I know
>
> observers
>
> It seemed to me that MWI allowed me to get away with a considerable
> simplification. Gone were observers and even observations. Even
> measurements, I discard. (After all, who can say that a measurement
> occurs in the middle of a star? And yet things do go on there, all
> the time.)
>
> Now *some* of that language perhaps returns when decoherence is
> discussed. I mean, I'll grant that *something* significant starts
> off a new branch, and so it's okay for it to have a name. :-)
>
> But here is what I'd like to be able to say:
>
> A new branch starts, or decoherence obtains, or an irreversible
> transformation occurs, or a record is made. They all seem the
> same to me. Why not?
>
> My main motivation is to get as far away from Copenhagen as possible,
> and so thereby get free of observers and observations, and anything
> else that seems to afford some pieces of matter a privileged status.
> Do you think that such simplified language leaves out anything important?
>

I don't think we disagree much about the physics. The trouble is, the
physics is even simpler than you suggest. Branching is not something
special in the theory, it is a macroscopic description that we apply to
what emerges from the theory. If you simplify your language too much, all
that happens is you have to define all those useful approximate terms from
scratch.

Just for fun, here's how it would go:

The framework of QM in the MWI is that

(1) The state of the "system" (universe) can be represented by a
time-dependant, normalized vector, say |S>, in a Hilbert space.

(2) Time evolution of |S> is linear.

That's it! (1) implies that time evolution is also unitary, so the vector
stays normed. (1) + (2) imply the Schrodinger equation, including the fact
that the generator of time evolution ("Hamiltonian") is a Hermitian
operator. (2) causes all the trouble.

A full (non-framework) description requires you to (a) specify the Hilbert
space (b) specify the Hamiltonian (c) specify the initial state. None of
which are known exactly for the universe. (And in fact for the universe as
a whole we had better adapt this description to relativity somehow, since
you can't just take time as a given.)

Now to introduce some more specific terms so we can relate the theory to
everyday reality.

"Observable": In a simple system, the set of values of an observable are
simply the labels we attach to elements of a basis, i.e. a set of
orthogonal unit vectors (defining a "coordinate system"), in Hilbert
space. We can freely choose any basis we like, but some are more useful
than others because they relate to the structure and symmetries of the
Hamiltonian. Let's call a basis {|o>} where o is our variable label. The
set might be finite, denumerable, or continuous, depending on the size of
the Hilbert space. For convenience, and to make the transition to
classical physics as seamless as possible, the labels are usually chosen
to be real numbers.

To put my previous answer to Serafino into this context, note that
observables (e.g. position) play a very different role in the theory from
time.

For each basis, we can construct a linear operator on Hilbert-space
vectors whose eigenvectors are the basis vectors and whose eigenvalues are
our "observable" labels. If our labels are real, the operator will be
Hermitian. With suitable choice of labels, the algebra of some of these
operators approximately maps onto the algebra of variables in classical
physics, which explains why classical physics works, and also how QM was
discovered. (In particular, since the Hamiltonian itself is hermitian it
has a set of real eigenvalues which we call "Energy").

"Wave Function": The inner product of a basis vector with the state
vector, written <o|S>, is "geometrically" the length of the projection of
the state onto that basis vector, and so the "cartesian coordinate" along
the axis defined by |o>. In conventional QM it is the probability
amplitude for "observing" o. If the basis is continuously infinite, as in
position or momentum, <o|S> is a continuous function of the real variable
(observable) o. This is what we call the "wave function" in o-space. (e.g.
o = position, or momentum).

"Subsystems": In a complex system, we have to be a bit more careful. What
physicists call observables certainly don't parameterize a complete basis
for the universe. Such a complete basis would be characterised by a
"complete set of commuting observables". Commuting because their
characteristic operators commute. In effect, we factorize the Hilbert
space into subspaces (corresponding to quasi-independent subsystems).
Practical observables correspond to bases on some subspace.

"Branching": In *some* bases of sufficiently complex systems (appropriate
basis and needed complexity depending again on the Hamiltonian), the
time-structure of the wavefunction approximates a branching tree. In those
bases, the observable o corresponding to a particular branch (to within
microscopic uncertainty) is the value that a suitable detector would
record in that branch (as proved by Everett).

"detector": a subsystem which can interact with another subsystem and
permanently correlate its state with the state the other subsystem
had at the time of the interaction. The change in state of the detector
is the "record".

"observer": left as an exercise to the reader. In particular, why do
observers find themselves more frequently in branches with high "measure"
(= integral of |<o|S>|^2 over the o's corresponding to the branch)?

Satisfied?

Paddy Leahy
Received on Wed May 25 2005 - 08:28:49 PDT