Re: Postulates for quantum mechanics (was; Is State Vector Reduction

Seratend wrote:

Arnold Neumaier wrote:

Comments:
-------------
a) Axiom 3 defines one property: closed (open=[closed]*). However, it
does not define what systems are neither closed nor opened.


It defines 'open' as 'not closed'. Thus your third category is empty.


Oups! I am sorry (dyslectic error : )), I meant it does not say what
system is closed or opened.


This is on the same level as that thermodynamics does not say which
system is an ideal gas and which is only a van-der-Waals gas.


Theories of physics do not say what a system (such as an
electron, a star, an ideal gas, a crystal) is in reality.
Nevertheless, it is possible to check the reality contents
of a physical theory. How does this come about?

Let us consider thermodynmaics. Thermodynamics does not say which
system is an ideal gas, which is only a van-der-Waals gas,
which is a liquid, or a solid.

Indeed, such questions need not be answered by the theory.
Instead, they are answered by checking how a system behaves:


If a real system behaves as the theory for an ideal gas (a solid,
a crystal, an electron) requires, a physisict will say it 'is'
an ideal gas (a solid, a crystal, an electron); if not, it is not.

While this definition may seem circular, it isn't once it is recognized
that one can check some characterizing properties of systems that have
a particular label (such as 'ideal gas') by a small amount
of measurements, and then deduce many more properties from the theory
that can be checked subsequently.

Engineers call this process 'system identification'.

Thus the task of theory is to provide models with just enough
flexibility that they cover the range of relevant possibilities,
while being still restricted enough so that one can identify
the system with a limited amount of data. Exactly in this case
a theory has predictive value.



For example, the system 'hydrogen atom' can be closed
or interacting with another system.


Yes. In other words we have:
Case A: (K, H_hydrogen, closed)
Case B: (K, H_hydrogen', open)


Yes, and a deeper level model for case B would be to add the context
as part of a bigger model which perhaps can be considered as closed.



=> the closed/open property should be associated to the Hamiltonian of
the generic system for a better logical definition of the system
handled by your axiomatic set.
In other words, you use the words "a generic system is defined" in
axiom 1 and the closed/open is necessary to define the system.


I rather consider this as a subsequent refinement of the theory,
so it does not need to show up in axiom 1. It is like considering
first verctor spaces in general, and then requiring something extra
such as finite-dimensional. In our case, the extra is 'closed'.

Mathematics would be clumsy if all later conceptsd would have to be
introduced on the level of basic axioms. I use the freedom customary
in mathematics to get a good formal voew of physics.


b) For, me " spin_vector, momentum etc ... of axiom 4 are just labels
of mathematical objects in these axioms (no connection with
"reality" without a minimalist interpretation.


They are (like energy) mathematical objects in the theory. At the same
time they suggest an interpretation, without forcing anyone to
make this interpretation precise. Like numbers as part of Peano's
axiom system and as the size of real collections of objects.


Yes, but the natural numbers of peano are not necessary the natural
numbers of other theories. They just share some common properties.  One
must not forget this point, especially when building its own
interpretation.


Of course. I was just comparing 'number theory' with 'physics', as
far as their formal foundations are concerned. To apply numbers
to reality one would need an interpretation of what a number should
mean outside the formal context. This step is usually omitted since
in number theory no one finds it problematic.



Note that the minimalist interpretation, I know, is the logical choice
to map some mathematical properties/objects to some reality properties.


This mapping is, in my view, wholly outside logic.

To bring it into the realm of logic, one needs a formal model
of reality. Such a formal model is indeed what I am aiming at
in my consistent experiment interpretation.


I do care. For I want to be able to represent ultimately _everything_
physicists do in their work on the formal level, so that it can be
discussed with the same rigor as formal logic discusses what
mathematicians do. Thus every concept used by physicists must
ultimately get a formal version.

Yes, it allows to construct logical sentences with the word
"prediction" in the context of this axiomatic set.

However, this is not important for the foundations: we can choose this
definition for "prediction" or another one: it does not change the
results we may deduce from axioms 1 to 5. 

But it delineates its use in reality.
I put it in for clarity, since I see no benefit in having a minimal
set of axioms which is difficult to understand.


I think that Axiom 6 serves this purpose for 'prediction', while
Axiom 7 does not and just codifies the current minimal consensus,
while a future complete theory would have to replace this by a set
of formal axioms defining on the mathematical level what is meant
by 'measuring', 'outcome' etc. This would require a theory of
multiparticle detectors.

A current minimalist consensus must be a logical one in order to be
useable => We must before solve the implicit formal mapping that may
exist in your axiom 7 or clarify it.


As I said before, there cannot be a logical use of objects in 'reality'
unless reality itself is given a formal meaning.

On the other hand, a formal correspondence is not needed to have
practical interpretation. See the section
''What is a system (e.g., an ideal gas)?'' (currently S16i) in my
theoretical physics FAQ at
        http://www.mat.univie.ac.at/~neum/physics-faq.txt

However, there is a confusion between "the outcomes of measuring an
observable" and "outcomes" of a random variable.


''Outcomes of a random variable'' is meaningless. Random variables
are mathematical objects for which the notion 'outcome' is undefined.

Please, an outcome is explicitly defined in the usual probability
context. There is no ambiguity. Any mathematical introduction book to
probabilities deal with outcomes and random variables.


This is perhaps again a difference between different traditions?
That's why I proposed to use Wikipedia as a reference.
For me, 'outcome' is an informal interpretational concept in
probability theory, while 'realization' is the formal object.


Random variables have 'realizations', which is a formal version
of 'outcome' but it has a different formal meaning - Probability
theory makes no assertion at all about single outcomes, but only
assertion about the ensemble of _all_ outcomes.

A random variable is a measurable function between the set omega (the
sample space) and another set, the set of some possible outcomes (that
can be the set omega). 

This only defines the concept 'possible outcome'. For me, the possible
outcomes of a random variable are complex numbers though some or most
of them might occur with probability 0. Thus I don't need this concept
on the formal level.


The outcome value of a random variable in a
particular experiment is what you call a realization of a random
variable.

No. In my usage, a (scalar) random variable maps Omega to C (the set of
complex numbers); the elements of Omega are called the realiziations,
and f(omega) is the value of f in the realization omega.
(Note that Omega must be a really huge set for most constructions of measure theory. Only in useless toy examples, it is a small set like {1,...,6}. The law of large numbers cannot be formulated with such a
small set.)

In the informal interpretation, one would identify each realization omega with an experiment, and the value f(omega) as the outcome of f
in this experiment.


This is the short definition of the random variable (usually in
introduction courses the random variable function is restricted to the
set of real numbers) as well as the outcome.

Now take wikipidia: http://en.wikipedia.org/wiki/Random_variable

[Quote]
Unlike the common practice with other mathematical variables, a random
variable cannot be assigned a value; [end Quote]

And later:
[Quote]
As noted above, a random variable is essentially a function mapping
events to numbers. [end Quote]

I do not know how to write a correction to wikipedia. 

There is a link on the top of each page for suggestions for editing
the page. But make sure your version is really better before you do so.


But this seems
to be completely false and inconsistent.

No. This is fully correct (apart from that 'event' should be more concisely 'elementary event').


Now, let's take the book introduction to probability, Grinstead and
Snell, published by the American Mathematical Society as a more
consistent reference. It is freely available in pdf format at:

http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html

Let's quote the beginning of section 1.1

[Quote]
-------------------------------------
In this chapter, we shall first consider chance experiments with a
finite number of
possible outcomes omeg_1, omeg_2, . . . , omeg_n. For example, we roll
a die and the possible outcomes are 1, 2, 3, 4, 5, 6 corresponding to
the side that turns up. We toss a coin with possible outcomes H (heads)
and T (tails).
It is frequently useful to be able to refer to an outcome of an
experiment. For


This is not formal enough. What is an experiment?

I want foundations that don't refer to reality but remain on the
formal level.


example, we might want to write the mathematical expression which gives
the sum
of four rolls of a die. To do this, we could let Xi, i = 1, 2, 3, 4,
represent the values
of the outcomes of the four rolls, and then we could write the
expression
X1 + X2 + X3 + X4
for the sum of the four rolls. The Xi's are called random variables.


This is a very informal explanation, not a mathematical treatise.

A random variable
is simply an expression whose value is the outcome of a particular
experiment.
Just as in the case of other types of variables in mathematics, random
variables can
take on different values.

[End quote]

I hope, you accept the definition of the outcome as a particular value
of a given function (a random variable).


No. I don't understand what a 'particular value' is on the formal level.
Only the set of 'possible outcomes' is well-defined, but not a procedure
to select one of them as particular.


In other words, a random variable A defines an outcome a through the
functional property a=A(o) where "o" is an element of the sample
space omega: this is the logic behind the probability framework. In
other words the functional mapping defines the true properties of the
experiment realizations. Without this procedure, speaking of an outcome
of an experiment looses its logical meaning.


In the concise and formal terminology I am using, o is the realization
and a the value of A in the realization o. (But I prefer to write omega
for the realization and f for the random variable.)

Probability theory makes no statements at all about values a=A(o) of particular realizations o.


If you do not explain that, Axiom 7 has no meaning. 

I agree to the statement that 'Axiom 7 has no mathematical meaning'
That's why I call it part of the interpretation.


???? what is an axiom without a logical signification? 

Axiom 7 is meaningless mathematically since the terms it uses are
not formally defined. So let us drop the name 'axiom' and call what was
till now Axiom A7 'statement MI' (for minimal interpretation),
clearly exhibiting its non-formal nature.


Please, the axiom is used in a mathematical context and must be
consistent. If you do not explain, for example, the logic behind the
words "statistically consistent" axiom 7 is simply an empty (of
logic) sentence.

I will try to help you:

Let's do again some basic logic with the axiom 7. 

Statement MI is not for formal logical reasoning but for
informal rerasoning in the traditional cultural setting that defines
what a trained physicist understands by reality.



Let's label each system of the large collection of independent
identical systems. Let's call this collection of labels the set O
(for Omega), the sample space.


I prefer writing out Greek letters (with exception of the
abbreviation eps for epsilon)

Let's give a label to each different outcomes of the given large
collection of independent identical systems. Let's call the set of
these possible outcome labels F_X.

I hope, you accept this logical construction is allowed by your axiom 7
and does not change the results of the axiom 7.


Many different interpretations of statement MI are possible and allowed
since i do not fix its formal contents. And different people _will_
interpret it differently.

The lack of precision in statement MI is on purpose, since it allows
the statement to be agreeable to eveyone in its vagueness; different
philosophical schools can easily fill it with their own understanding
of the terms in a way consistent with the rest.

So I won't agree on fixing the contents of statement MI any more
precisely than I did.


On the other hand, I think it is worthwhile to create a formal
model reality in which statement MI can be given a formal meaning.
In such a model reality one can discuss the consequences of MI
and analyze (formal) experiments, and thus analyze the consistency
of quantum mechnaics in the same way as formal logic analyzes the
consistency of mathematics. Indeed, my consistent experiment interpretation contains (in my German FAQ) germs of such a
model reality.


Now, lets define the function X_logic by the couple (o, x), for each o
in the set O,
where o is the label of one independent identical system and x, the
outcome associated to "upon measuring at time t=0 the independent
identical system" labelled o.

=> X_logic is the function: O --> F_X
                                        o --> x= X_logic(o)

This is logic. We still haven't changed the axiom 7. I hope you
accept this logical construction. Now X_logic is a genuine random
variable defined in the context of measuring a large collection of
identical independent systems.

I don't deny that one can formalize MI. But then one postpones only the interpretational problems to the stage where the formal parts of this
formulation must be related to reality. There is no escape from this
dilemma.

On the other hand, a formalization of MI that meets the demands of
understanding the measurement process is much more difficult than
what you presented above.


However, it has
informal meaning - everything stated is understandable by every
practicing quantum physicist.


?? by a statistical consistency? With what?
Frankly, I have tried to give a logical content to this axiom and up to
now, I have failed.


There is no logical content but informal content.

Physicists know how to check a statistical prediction
against experiments. This knowledge defines statistical consistency.




All the experiments I know rely on the eigenvalue - real outcome
formal link (the minimalist interpretation): 

No. This only applies to ideal von-neumann measurements. read the book
by Braginski and Khalili that I had quoted to see why it is hopelessly
inadequate in realistic cases.


Interpretational axioms necessarily have this form, since they must
assume some unexplained common cultural background for perceiving
reality.


??? Interpretational axioms define the logical content/use of the
"reality" words, that's all. 

'reality' is an empty word if not filled by a social context.

It seems we differ very much in our understanding of the nature of reality.




There is a difference between interpreting and the logical construction
of a formal mapping between physical results and outcomes (what I mean)
on a given "experiment" [i.e. a contextual formal mapping]


I don't see at all how one can give formal mappings between
formal and nonformal entities.


Well take a ball. A formal mapping is the association, for example, of
the property position x, where position and x are completely defined in
the context of a given theory.

Neither the ball nor its position are formally defined, hence your statement hangs in the air.

Formal logic can be taught to a computer. Your example cannot, hence is
not formal.





I therefore may logically say that frequency= Card(A)/Card(E) is an
approximation of the probability law p(A)


'approximation' is a formally empty word.


Approximation may be well defined by the definition of a topologic
space. It may be well quantified by the definition of a metric space.

Is 1 an approximation to 0, or to 0.999, or to 0.999999?
Who decides?

The properties we formally attach to the "reality". For example, 1
may be an excellent approximation to 10^10000 if we decide such a
property (i.e. a is a good approximation of b iff |a-b| <
10^100000000000000). Generally we try to make more general quantifiers
such  |a-b|/b when b=/=0, etc ...


I agree that |x-1|<=0.001 is a formal statement.
But 'x is an approximation to 1' is not; it is formally empty.





Well, If I have a non separable space, a discrete non degenerated
spectrum observable does not exist in a non separable Hilbert space
(problem of the non countable basis).


This is not true. Any projector to some ray is selfadjoint and has
the spectrum {0,1}.

?
This projector is highly degenerated. Ker(P) has an uncountable basis
if the eigenspace 1  is finite or countable dimensional.

This doesn't matter. It is selfadjoint and has
>>the spectrum {0,1}, hence has all the properties required of an observable in typical textbooks.


The entangled state of a PDC crystal, the squid states, coherent states,
etc ... All of them may be attached to specific observables otherwise,
we cannot used them in defining the preparation of well known
experiments.


These are states, not observables.


? A laser + a PDC does not define an observable that has the entangled
state (as an approximation) as an eigenvector?

If no, how can we do an experiment where I may say I have an entangled
state?

Since certaion properties of states can be observed by measuring certain observables in these states.


Arnold Neumaier

.

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