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.

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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