RE: Observables, Measurables, and Detectors

From: Brent Meeker <meekerdb.domain.name.hidden>
Date: Wed, 25 May 2005 08:34:36 -0000

>-----Original Message-----
>From: Patrick Leahy [mailto:jpl.domain.name.hidden]
>Sent: Wednesday, May 25, 2005 12:25 PM
>To: Lee Corbin
>Cc: EverythingList
>Subject: Re: Observables, Measurables, and Detectors
>
>
>>
>> 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

Nice summary! But I'm not satisfied because of:

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.

The meaning of "probability" is unclear. I know there have been attempts to
derive the Born axiom within MWI, identifying the probability as a relative
frequency, but I don't think any have been successful.

I also feel little uneasy with:

"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".

My unease is that all detections are, in principle, reversible and there are no
permanent records. But I know that they are irreversible FAPP - so maybe it's
not a problem.

Brent Meeker
Received on Wed May 25 2005 - 12:28:32 PDT