RE: Observables, Measurables, and Detectors

From: Lee Corbin <lcorbin.domain.name.hidden>
Date: Wed, 25 May 2005 18:51:35 -0700

Paddy writes

> > 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.

Oh good!

> Branching is not something special in the theory, it is a macroscopic
> description that we apply to what emerges from the theory.

And so I take it that this applies to the rest of it that bothers
me: observables, observations, measurements, detectors---all those
things that you have kindly taken the time to "develop from scratch"
below.

> If you simplify your language too much, all that happens is you have
> to define all those useful approximate terms from scratch.

Very good. That *does* answer my question. At least I think it does.

We *start* with the ontology of (1) and (2), as you do below.
And actually---for a great number of philosophic purposes---
we are done. The rest is comfy language good for a little day-
to-day work in the lab. You probably realize what ammunition
this gives some of us in other debates :-)

Building up from down deep (the way you do next) is even better
than "reducing" our usual notions (even when the latter is really
understood to mean the former).

Lee

> 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? [Yes! Thanks. I be working on the last one.]
>
> Paddy Leahy
Received on Wed May 25 2005 - 21:55:36 PDT