[ExI] QT and SR

[scerir]

Mike Dougherty:
> thinking about photon being related to electron 
> shells in discrete units - it either exists in one 
> state or another, but there is no 'in between' - 
> or is that a probability of indeterminate states?  
> If a probability, then does the probability move 
> toward a state, or does the eventual state reflect 
> the outcome of a wave collapse?

Subtle questions. And the possible answers depend
on the specific points of view, or interpretations.
What are these states? Are they physical? Are they
mathematical? Are they statistical? Do they represent
informations carried by a quantum system? Do they 
represent observer's information? Rather, do they 
represent the 'image' of the information carried by 
a quantum system? Do they represent experimental contexts? 
Do they represent statistical ensembles? Or do they 
represent single systems? Are they subjective? Are they 
objective? Are they tendencies, propensities, potentialities? 
Are they actualities? Should we give up the possibility 
of treating the wave function as an isomorphic image 
of what is actually processed in the laboratory?

In QM the outcome of a measurement - repeated many
times - of an observable, isn't in general the same. 
So QM gives the expectation value of the observable 
to be measured. (In special cases it gives the actual 
outcome of the measurement, non just the expectation 
value).

While it is possible to say that QM does not care of
unperformed measurements, what can we say about the
value of an observable between two measurements?
Is it undefined? Is it unknowable? 

In QM the total information of a system, represented 
by the state vector, is never complete. Information
is limited. The total information of a system suffices 
to specify the eigenstate of one observable only, 
at choice. Thus, all possible future measurement results 
cannot be precisely defined.

The state vector can be said to represent our knowledge 
about the recent history of a system which is necessary 
to arrive at the set of probabilistic predictions
for all possible future observations of the system.
The set of future probabilistic predictions specified 
by the recent history of the system is indifferent 
to the knowledge collected from all the previous measurements 
in the whole history of the system. As Pauli once wrote: 
"In the case of indefiniteness of a property of a system 
for a certain experimental arrangement (for a certain
state of the system) any attempt to measure that property 
destroys (at least partially) the influence of earlier 
knowledge of the system on (possibly statistical) statements 
about later possible measurement results."

Can we say that the observable has a *definite* value between 
two measurements? No, in general we cannot say that. 
If the state is a pure state (and not a mixture) we cannot 
say there is any definite value [1].

Can we say the value of the observable is *unknowable* between
two measurements? No, we cannot say that, because QM in general
provides a sort of information, a sort of knowledge, whose nature
is probabilistic though.  
 
[1]
Imagine a spin-1/2 particle. Imagine its state described by 
the superposition psi = sqrt(1/2)[(s+)_z +(s-)_z].
There are two possibilities.
    A) That psi above is a pure state. Since we know that
(s+)_z = sqrt(1/2)[(s+)_x +(s-)_x]
(s-)_z = sqrt(1/2)[(s+)_x -(s-)_x]
(where _z, _x, are the z-component and the x-component of spin) 
we can write that psi = sqrt(1/2)[(s+)_z +(s-)_z] = (s+)_x.
Now, if the x-component of spin is measured by passing 
the spin-1/2 particle through a Stern-Gerlach with its field 
oriented along the x-axis, the particle will *always* emerge 'up' 
(that is, as (s+)_x). The experiment confirms that.
    B) But if by sqrt(1/2)[(s+)_z +(s-)_z] we mean
a *mixture* of sqrt(1/2)[(s+)_z] and sqrt(1/2)[(s-)_z],
we might also think that -before measurement- the particle 
has a *definite* value of the z-projection of spin,
say [(s+)_z] or [(s-)_z]. But in this case, measuring the 
x-component of the spin, we would find 'up' with the
probability 0,5 and 'down' with the same probability.
Experiments does not confirm that!