Re: Is State Vector Reduction a 'Process'?

Aaron Bergman a =E9crit :
> In article <1117183732.077481.57630@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
>
> Why do we observe outcomes with probability |<a|psi>|^2? QM has no
> answer for this question.
>
This is the born rules, a postulate of QM. I think we are at the centre
of our minor problem of comprehension. QM formulation provides a very
formal statistical description of the system as Classical statistical
mechanics does. This is a formal choice: We describe the statistics of
outcomes rather than the description of the evolution of individual
outcomes.

For me at least this is clear. I am not sure if it is clear for you
too. In my comprehension, your question is just a mathematical
question: why the probability of an event (P(A=3Da)) is the frequency of
independent outcomes with identical probability (the statistics of
outcomes). This is simply the law of large numbers. Formally, I may
always do such affirmation: the hypotheses of the experiment where I
have this equality. This is not different from the coin tossing
experiment. (You can always question the validity of the independence,
but this is the problem of the experiment realization and not the
validity of the results).
In QM, we choose the observables (the set of outcomes is the
eigenvalues of the observable) and states (or generalized states) to
describe, formally, a probability law of the outcomes of the observable
(the random variable). The choice of observables and states to describe
the probability law is more related to the time evolution of the
probability law where we have a simple expression (e.g. the difference
between the observable expression of QM and the random variable
expression in Bohmian mechanics).

> > I can only understand this sentence as the quest for a deterministic
> > (causal) description of outcomes compatible with the statistics of QM.
>
> QM is deterministic and causal. That's the problem.
>
I do not understand where the problem is. I have a deterministic time
evolution of the probability law. I may also have a deterministic
evolution of the set of eigenvalues (the Heisenberg representation).
Where is the problem?

> > If this is the case, we already have such a description, bohmian
> > mechanics for the position eigen basis (and equivelent formulations i=
n
> > different eigenbasis).
>
> Bohmian mechanics has no relativistic generalization that I know of.
>
I think you should rather say bohmian mechanics has not a simple
relativistic generalization (the mathematical expression becomes more
difficult has the particle number in not conserved).
See for example Trajectories and Particle Creation and Annihilation in
Quantum Field Theory, quant-ph/0208072, 2002.
With bohmian mechanics, we must separate the equivalent formulation of
QM with contextual random variables from the interpretation. As long as
we stay with mathematical expressions (and knowing their hypothesis of
validity), there is no problem.


> [...]
>
> > > I'm not sure I understand what you're asking. Let me try to answer
> > > something, then. The question of what why we observe what we observ=
e is
> > > completely unanswered by quantum mechanics.
> >
> > So you are saying that QM theory does not explain the preferred basis.
>
> You'll have to communicate to me better what you mean by 'preferred
> basis'.

Ok, I will try to explain what I mean. I have no problem with the
preferred basis (the basis where I have the experiment outcomes). I
just have a problem with the prediction of such a basis by QM. I mean,
where in the QM theory do I have results inferring the preferred basis?
Up to now, I just know the preferred basis after I have done the
experiments. For example the interference pattern of double slit
experiments observed on the screen. I know, from this experiment, if I
place a screen, I will do a position measurement. But I have not seen
anywhere in QM, where the mathematics can predict such a basis as every
basis is possible from the theory point of view.

I can live with such a result. However, I prefer if everybody can
acknowledge such a result: the QM formulation does not predict the
basis of outcomes in an experiment (out of scope). It is important (at
least for me) to understand the scope of a physical theory (what it may
predict and what it does not predict).

> Given an experimental setup with a classical measuring device
> (where classical means large numbers of microstates per macrostate), I
> can describe to you a preferred basis in that setup.

Ok, let's play with a simple toy model and you will try to tell me
the proffered basis of this experimental setup.
Let's take the double slit experiment with electrons (formally
simpler in the terms of interaction description at the slit plate, but
we can change and choose photons as outside the local interaction the
free propagator or the photon and electron is equivalent).

a) Let's assume that the plate with the slits is a reflective plate
such that we can describe it through an interaction: Hint_plate:
|plate><plate|(x) V(r)
Where |plate> is a state of the free plat hamiltonian (no interaction
with the environment: we neglect it during the period of observation).
V(r) is null outside the plate and at the slits and infinite within the
plate (hypothesis model of the energy conservation). We just model the
plate with a quantum wall (with a non null thickness).
b) At a distance of the plate (z direction), we place a screen. Let's
describe this screen by another Hamiltonian: Hint_screen=3D
|Screen><Screen|(x)Vdiff(x) where Vdiff is a scattering potential null
outside the screen (such that photons or electrons arriving at the
screen are transferred into another direction: how we may see
externally the interference pattern from another direction). |Screen>
is an eigenstate of the Hamiltonian.

We have for this system: H=3D Ho+Ho_screen+H_int_plate+H_int_screen
We also have [Ho_screen,Hint_plate]=3D [Hint_screen,Hint_plate]=3D
[Ho_plate,Hint_plate]=3D0

Assume that the intial state is |state(o)>=3D|psi(o)>|plate>|screen>
where |psi(o)> is the state of the electron or photon. Before the
plate, |psi(t)> may be described by a wave packet centered on a
momentum |p_z> parallel to the z axis (free propagator of the electron
or the photon).

We may add or remove the properties we want to this toy model. The
important fact is the ability, by thought, to decrease the interaction
with the environment and to check if we still are able to predict an
eigenbasis or if the shcmidt eigenbasis is always correct for this
experiment (depending on the different hypotheses).

Now how can you deduce (from QM theory) the preferred basis and what we
"really" measure in this experiment?


Seratend.

.

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