# 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?