Re: Is State Vector Reduction a 'Process'?

Seratend wrote:

Arnold Neumaier wrote:

Seratend wrote:

Note that in order to recover the probability law in the frequency of
outcomes we must have the independence of identical systems (hence, we
need a "preparation" to select the systems).


But this is not satisfied in many experiments analyzed by quantum
mechanics. For example, in an ion trap, one has the continuous
measurement of a single system, in which the observation at different
times can by no means considered to be observations of independent
systems.

Where is the problem (logically)? (I may miss something). 

The problem is that ensembles require _identically_ prepared
_independent_ systems. An ion trap has (over a macroscopic
time interval) a _single_ system whose measurements are _not_
independent.


Do you mean
the collapse postulate is not satisfied in continuous measurements?


Of course it isn't; but this wasn't my point.

(Continuous measurement requires a description as a quantum stochastic process.)


Please note that when you say a continuous measurement over time of a
single system, you mean a single measurement that spans over time of
one instance of this system (the measurement has to be specified over
time and space or the equivalent observables to have a meaning). 

No I mean a measurement of the single instance of this system at
many closely spaced instances of time.


For an example of what I mean, please see "When the unitary evolution
is derived from the collapse postulate in QM" in
http://www.physicsforums.com/showthread.php?t=72181


What you say is that the Schr"odinger equation can be written as
(H-E) psi = 0 in the extended Hilbert space. Your interpretation
of this as a measurement is arbitrary, and contrary to the
measurement concept in QM.

Formally one can do many things, but meaningful is much less.




Thus your conceptual framework for interpreting quantum mechanical
experiments is too restrictive.

I do not understand on how it is too restrictive.
I am using the "shut up and calculate" view.


Which predicts a lot but explains nothing.


When we just have a single experiment trial in all the universe, we
just have the only interesting result: the unique set of results of
this experiment. There is no way to make predictions (this is also true
for CM).

One wants to (and can) predict some subset of results (e.g. those for tomorrow) from another subset (those of today). Just as one can predict
the path of a classical particle tomorrow if you know that of today.
Or as you can predict the acceleration if you know position and velocity.


In addition, it raises an interesting question when we consider the
continuous measurement (we suppose all the interactions may be modelled
into interaction Hamiltonians, including the interactions of the
measurement apparatus):
Is it the state of the system that changes upon measurement due to the
collapse ("action" of the collapse) or is it the measured values
that change continuously upon time ("acknowledgement" of the
collapse)?


The state changes and the measured values change, due to the dynamics
of the combined system plus environment.



A physical theory is mainly a choice of description (formally, we are
free to choose what we want)


..but only if we don't care about the quality of our predictions.
If we want to have good predictions, we must choose what quantum
mechanics tells us to choose.

Every physical theory is a description choice. Good predictions is
somewhat a matter of taste (i.e. a practical choice).


But physicists have the taste to want good, accurate predictions.
Otherwise they could have stayed with Aristoteles.


For example, take My god determinist theory: the collection of all the
"measurable" properties of the universe we may ever know. I call it
determinist because, labelling all the properties (and assuming the
collection of labels and properties is a ZF set: the only restiction),
it defines an implicit function. Does it make good predictions?


If there is a compact description of it, yes.

This is precisely what theory does - give a compact description of the
collection of all properties of a system (small or large or the whole universe).

For example, for a classical point particle over some time, one can take as observables all functional of the trajectory x(t). Theory tells that
given six independent such observables (sufficiently well chosen),
we can predict all others. _Therefore_ the theory has high predictive value.




What is a true outcome in a world described by quantum mechanics/

I have outcomes, hence they are true otherwise I cannot say (logical
meaning) 

So your theory says very little.

My demands on good foundations are much higher:
_Everything_ physicists talk about in real life must be
faithfully represented in the mathematical model.


Note we have an analogue problem in classical mechanics. How can we say
the proposition "a particle at position q at time t" is true? The
theory does not explain that, it uses it as QM does.


The theory specifies what is true in the model, and the
experimentalist can therefore check the model's adequacy.



In other words, I may choose to view the "reality" by different
mathematical concepts, this does not change the reality, just the
description of the reality. The only required property is the
conservation of the logic (as I do not know what to take to replace the
usual logic).


There must be an informal but quantitative correspondence between
what is real in the model and what is real in Nature.
This correspondence is comminly called the interpretation.



Question, why does the computed frequency of head/tails of a coin
flipping is 50/50?
Is it due to a mysterious ontic property, or is it due to the coin
flipping experiment and its description choice (by saying p= n(a)/N)?

It is due to the formal definition of a fair coin-flip in the mathematical model. The probability is 1/2 because we _define_
it that way. It has a priori _no_ relation to reality.


You can read about my view of probability theory in my theoretical
physics FAQ at
        http://www.mat.univie.ac.at/~neum/physics-faq.txt

I am surprised because, we seem to say almost the same (math results)
(except may be for your comments on Bayesian probability).
(Note: I have implicitly selected my preferred interpretation of your
words : )).

However, I have a suggestion: you should explicitly say that the sample
space of the sequence of trials of a random variable of more than one
value (i.e. P=/=100%) is uncountable. 

Only for an infinite sequence.

This property explains most of
the problems with probabilities and the "uniqueness"
(reproducibility) of each infinite sequence.


The problems are already there for large finite sequences.
No one ever observed an infinite sequence of trials.


What I want as a basis of physics is a mathematically defined
model of the world in which one can give unambiguous descriptions
of all that matters in physics - physical systems, detectors, observers,
individual observations, statistics about these observations,
error analysis, etc. in such a way that it mirrors reality.

I understand, however, there is more than one model. 

So far, there is _no_ model coming close to what I want.


Just as in matheamtical logic, one models the whole logical process
in a concise mathematical framework.

This is a sort of compression process of the "god determinist
theory". Therefore, I hope you should accept loss of information
(description) in this process.


No. Mathematical logic loses no information in this modeling process,
and Physical logic shouldn't either.

Loss of information amounts to dissipation
and should be explained rather than bbuilt in.


I think the QM theory is a concise one (may be too). The main problem
seems to be the preferred basis prediction. However, If we look at
general relativity we encounter an almost analogue problem: the
preferred frame to describe the events.


The preferred frame in GR is determined by the matter distribution
in the universe. Only empty space has no preferred frame.
This is why we can speak unambiguously about the age of the universe.


Although not very clearly separated in many discussions,
these two processes happen never simultaneously but context
dependent, and are of course only approximations to more
realistic measurement situations.

For example, in a Stern-Gerlach experiment, the system (silver atom)
moves from the source along the magnet towards the screen with very
good accuracy in a unitary (and indeed reversible) way. But a few
split moments before it hits the screen it feels its interactions,
and describing it as a closed system becomes hopelessly inaccurate.
Instead, since the interaction time is very short, it can be
described very accurately by an instantaneous collapse.

Why do you say it becomes hopelessly inaccurate? 

Because the closed system in this setting contains >10^20 degrees of
freedom, and we cannot model such systems accurately. We need the
thermodynamic approximation, and with it an unavoidable inaccuracy
in the response to the microscopic particle state.

You can describe the stern-gerlach experiment with an excellent
approximation as a closed quantum system. 

No. In a closed quantum system, no collapse ever happens.


And How can you really
apply a collapse to a non closed system? In this case, don't you
think the collapse result (the outcome) should be independent of the
partial system description versus the whole system (including the
universe if necessary)?


Look at the corresponding classical situation. A classical particle
encounters a classical screen (say, a thin foil through which
the particle will most likely escape) involving a huge number
of classical particles bound by (and interacting with the
incident particle) by empirical forces. It ends up in some state
that is determined only probablilistically, once you ignore the
detailed structure of the screen. But it ends up in a _definite_
state.

If you say it ends up in a _definite_ state you are implicitly
_defining_ a true property for this system instance 

Of course: This is a _classical_ system!

I just wanted to say that the classical situation is already precisely
analogous to the quantum situation, and needs also a description
in terms of collapse. So collapse has nothing to do with
acknowledgment by anyone, but only with lack of control over the
unmodelled environment.

There is no need to indroduce an additional level of observers,
and neither is there a need in the quantum case.


To describe it, however, without reference to the state of
the screen, necessitates a probabilistic description and a collapse.

But you have a property for the screen, otherwise you cannot apply the
collapse ("the don't care property or if you prefer, the Identity
projector).


In this analogy, everything computable from the classical deterministic
state is a definite property of system, screen, or environment,
depending on which part of the state variables are used to compute it.


The quantum system is - in the consistent experiment interpretation -
completely analogous, except that the dynamics differs in detail
significantly from the classical dynamics.

I am still working on this (I need to understand the logic). 

Take your time. It took me several years to gradually adapt to this
new point of view. Of course, I had to find out the hard way what
could be asserted and how far it reached, while you get everything
spelled out. So you can take some short cuts...


Arnold Neumaier



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