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

Arnold Neumaier wrote:

Update of the axioms

> > Axiom 1.
> > ----------
> > A generic system (e.g., a 'hydrogen molecule') is defined by specifying
> > a Hilbert space K whose elements are called state vectors and a
> > (densely defined, self-adjoint) Hermitian linear operator H called the
> > _Hamiltonian_ or the _energy_.
> >
> > Axiom 2.
> > ----------
> > A particular system is characterized by its _state_ rho(t) at every
> > time t in R (the set of real numbers).
> >
> > Note: e.g. A particular system may be 'the ion in the ion trap on this
> > particular desk'
> >
> > Axiom 3.
> > -----------
> > A system is called _closed_ in a time interval [t1,t2] if it satisfies
> > the evolution equation :
> >
> > d/dt rho(t) = i/hbar [rho(t),H] for t in [t1,t2],
> >
> > and _open_ otherwise.
> >
> > Note: hbar is Planck's constant, and is often set to 1.
> >
> > Note: If nothing else is apparent from the context, a system is assumed
> > to be closed.
> >
Axiom 4.
----------
Besides the energy H, certain other (densely defined, self-adjoint)
Hermitian operators (or vectors of such operators) are distinguished as
_observables_.

Note: e.g., the observables typically include several 3-dimensional
vectors: the _position_ x^a, _momentum_ p^a, _orbital_angular_momentum_
L^a and _spin_vector_ (or Bloch vector) sigma^a of particles labelled
a.

Note: If u is a 3-vector of unit length then u dot p^a, u dot L^a and
u dot sigma^a are called the momentum, orbital angular momentum and
spin in direction u.

> > Axiom 5.
> > ----------
> > To every vector X of observables with commuting components [of a
> > particular system], one associates a multivariate probability measure
> > dmu_t(X,rho(t)) by defining:
> >
> > integral dmu_t(X,rho(t)) f(X) := trace rho(t) f(X) =:<f(X)>_t
> >
> > for all bounded continuous functions on X. The expression <f(X)>_t
> > defined in this way is called the _expectation_ of f(X) at time t.
> >
> > Note: the probability measure is defined on the set spec(X) with the
> > adequate sigma algebra (= spec(X) intersection the Borel sets of
> > R^(dim(X))).
> >
> > -- End of the mathematical theory definition --
> >
> >
> > 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.

>
> > Therefore,
> > it is an additional parameter of the theory. Therefore, I think axiom 1
> > should also include the value of the closed property to define
> > logically a system.
>
> No. Because a system can be considered to be closed or open depending
> on the context.

Yes. The context is by itself a peculiar property. Without it, we
cannot say a system is closed or opened. In other words we cannot say
if the unitary evolution applies or not.
>
> 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)

=> 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.

>
> > 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.

Note that the minimalist interpretation, I know, is the logical choice
to map some mathematical properties/objects to some reality properties.
There is no logical problem with this procedure: one may always do
that. However, one may discover that the mapping may induce
incompatible properties and predictions => either the mapping or the
theory does not apply. This is why we have to perform tests in order to
verify the logical validity of the mapping (logical consistency).

>
> 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. Its purpose is only a
classification of these results: the prediction results and the other
ones.
However, if we use another definition, it does not change the set of
all results we may deduce from axioms 1-5 => why do we need to add this
axioms in this minimalist formulation of QM? (we can leave the
classification of the results outside the axiomatic base)

> 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.
>
Axiom 7.
----------
Upon measuring at time t=0 a vector X of observables with commuting
components, for a large collection of independent identical
[particular]systems closed for times t<0, all in the same state
rho_0 = lim_{t to 0 from below} rho(t)

The outcomes are statistically consistent with [identical] independent
realizations of a random vector X with measure as defined in axiom A5.

Note: one calls such large collection _identically_prepared_ systems

Comment: I have added "particular", to be explicitly consistent
with axiom 2.
> >
> >
> > Comment:
> > --------------
> >>From what you've said in the previous post 'identically prepared',
> > 'measuring', "getting an outcome", "independent" are
> > "reality" properties.
> >
> > 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.

> 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). The outcome value of a random variable in a
particular experiment is what you call a realization of a random
variable.
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. But this seems
to be completely false and inconsistent.

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
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.
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).

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.

Therefore the logical meaning of outcome is clearly specified by
classical probability, hence my remark to keep the classical
probability meaning when we use the word outcome. In fact this is what
we implicitly do when we use the outcome word.

>
> > What are "the
> > outcomes of measuring an observable at time t=0"?.
>
> This is undefined since it is part of reality.
>
> What does "statistically consistent" mean?
>
> This is also part of reality, and hence is unexplained.
> Informally, it means that it passes the tests physicists
> use to check the validity of a statistical theory.

> > 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?

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.

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.
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.

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.

Now, by axiom 7, what is the logical content of "the outcomes are
statistically consistent with the independent realizations (hence
outcomes) of a random variable X with a measure defines by axiom 5"?

By the axiom 5, we know that X is a real random variable (image set =
spec(X)). Therefore we know the set of outcomes of the random variable
X. However, axiom 7 does not connect the measure of random variable X
(in axiom 5) to the state at time t<0 of axiom 7. This is the first
uncertainty of axiom 7

The second uncertainty of axiom 7 comes from the impossibility to
connect logically the random variable X_logic to the random variable X,
with a probability dmu_t(X,rho(t)). This is not explained by axiom 7:

This is usually solved by the well known logical bijective mapping
eigenvalue of observable X- "real" outcomes of observable X. If you
remove it, you remove the logical consistency of the axiom 7 as the
"statistically consistent" is empty.

So does the "outcome" of a "measurement" as defined by axiom 7
is an eigenvalue of the spectrum of the observable X of axiom 5? If yes
please state it explicitly or provide a demo where we can deduce such a
logical link from the axioms.
(in other words, do we have a one to one link between the outcomes of
the random variable X_logic and X?)

> 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.

All the experiments I know rely on the eigenvalue - real outcome
formal link (the minimalist interpretation): a result of a given
experiment is logically attached to an eigenvalue of an observable.
If you remove this link, please give another one, but a logical one. If
it is just an equivalent statement, then you must provide a
demonstration where we recover this link. Above, I have tried to give
you the mathematical objects we may construct from this axiom and I
have highlighted the apparently missing logical links in order to get
an axiom.

> 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. We cannot logically decide if a
"reality" word of the axioms is valid for a given "real
system". Logic just allows one to demonstrate, if the choice in
attaching a set of "reality words/values" to "reality" is
consistent in the axiomatic set context (hence the utility of tests and
the logic given by the axioms).
Assuming some unexplained common cultural background is completely out
of the scope of any reasonable logical procedure in the reality
modelling. Therefore, for logical reasons, we need that all the
"reality" words of a given theory have a logical definition.

>
> > This remark comes from the lack of connection between the random vector
> > X and observable X: we need a formal mapping between them.
>
> The formal connection is given by the probability measure in Axiom 5.
> It defines the random properties of X given X and the state.
> And need not be separately stated.
>
I mean the "outcome" of the observable X defined in axiom 7, with
the outcome of the random variable X defined by the spec(X) of axiom 5.

Please tell me, if I may recover the one to one link between the set
spec(X) of axiom 5 and the set of possible outcomes of the observable X
in axiom 7.

> There is no formal connection, however, between X (whether random
> vector or observable) and the corresponding object in reality.
> Formal connections can exist only between formal objects.
>
Please, does we have a formal one to one link between the formal random
variable X of axiom 7 and the formal outcome of observable of axiom 7?
>
> > I suppose
> > this is the one to one mapping between the spectrum of observable X and
> > the random variable X image set (accepted by the community as the
> > orthodox QM) . If this is the case, it should be stated explicitly,
> > otherwise we cannot link the probability law of the random variable X
> > to the probability law of the observable X.
>
> I don't understand. A measure _defines_ a random vector.
> So all is already in Axiom 5.
>

Axiom 5 deals only with probability measures, not with outcomes.
Therefore the choice to define outcomes on the set spec(X) is external
to this axiom. Axiom 7 connects the statistics of outcomes of the
random variable X to the measure of axiom 5. However, it does not
connect the outcomes of random vector X in axiom 5 to the measuring
outcomes of observable X of axiom 7.

>
>
> > a) Clarification on the minimalist set of words defining the reality
> > ----------------------------------------------------------------------------------
> > I use the set of works " measurement result" while you seem to use
> > outcome, or may the eigen value of an observable to describe some
> > properties of the reality.
>
> 'outcome' is part of reality, 'eigenvalue' is a mathematical concept.
>
>
> > I have noticed that sometimes you are interpreting my "measurement
> > result" to a "measurement" (whatever it could be) as well as
> > outcomes etc ...
>
> A 'measurement' is the act of obtaining a measurement result = outcome.
> Both happens in the current setting on the level of reality,
> not on the formal level, since neither 'measurement' nor
> 'measurement result' have a formal content.
>
So why to complicate? Let's focus on the measurement result (the
physical result in a more general way). The extra property
"measurement" is not strictly necessary (all we need are
measurement results) so we can leave it for later explanations on
measurements.
However, the logical content of measurement results (or physical
results or outcomes if is sufficient), must be described by the axioms:
how we use this word in the logical context of the theory and not how
we use this word in the "reality". See as an example postulates
1-5: the logical use of "measurement result" is explained by the
axioms.

>
> >> In order to decrease the risk of confusion, I propose to use the word
> > "physical result" when we want to attach to a real object a given
> > property:
> >
> > e.g. a ball: we have a ball as a physical result (ball)
> > e.g. the speed of a ball is x m/s: we have a physical result x m/s
> > e.g. the colour of a ball is red: we have a physical result red
> > e.g. the ball is moving ~ we have a physical result path(t)
>
> I call these 'objective properties' - they are there in reality
> independent of observers (i.e., different observers are able to
> reach consensus about their properties) and hence are to be explained.
>
why do you require that? Nothing above allows you to make such a
restriction. For example I may attach the physical result "ball" to
a car or a dog.

> In complete foundations, there would also be formal objects in the
> mathematical theory such that talking about the formal objects
> and talking about the real objects is essentially isomorphic.
> We are currently far from such complete foundations.
>
I just want that you give a logical content to these small set of
"reality" words in the context of the theory (your axioms).
Currently, I do not require and do not care how one may map these words
to the reality: this is not really important as long as the mapping
chosen by someone is completely consistent with the logical results of
the theory (e.g. a car is not a ball because it is not spherical, etc
...). This is the minimalist interpretation requirement.

Therefore, ideally a good theory should be able to allow only a single
consistent mapping of the "reality" words of the theory. However, I
do not know if it is possible and if this proposition is decidable.

>
> > The outcome word is already used by the Kolgomorov probability theory
>
> No. The formal side of Kolmogorov's theory does not know of outcomes.
> But probability theory has also an informal interpretational side.
> As in quantum mechanics, the actual outcomes are part of the
> interpretational setting.
>
Please see my previous comments above.
> See, e.g., the wikipedia entry
> http://en.wikipedia.org/wiki/Probability_theory
> which I'd like to take as our consensus background on formal
> probability theory, if you agree.
>
> ''Omega is a non-empty set, sometimes called the "sample space",
> each of whose members is thought of as a potential outcome of a
> random experiment.''
> 'is thought of' signifies the interpretational level.
>
Please, as quoted before, the wikipedia seems not to be very accurate.
It makes me thought that many people seem to miss the bare logical
content of random variables and functional mappings and why we choose
them to define abstract experiments.

Taken from the introduction to probability book published by the
American mathematical society:
"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."

In other words, the outcome is logically defined by a value of a random
variable.

> >
> > 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.
This association is a logical choice. It does not require that position
x has an intrinsic signification (out of scope).
I just say that one may attach this property to a "real object". As
long as all the set of properties we attach to a "real object" is
logically consistent (in the context of the selected theory) with the
theory results, the mapping is logically OK. This is what I call a
formal mapping.

>
> > I suppose you have no logical difficulties with the approach of such a
> > formal mapping.
> >
> > If it is the case (no difficulties), you should accept, logically, the
> > formal mapping (logical choice):
> >
> > Frequency= Cardinal [subset of formally chosen physical results]/
> > cardinal [total set of formally chosen physical results] <--->
> > probability.
>
> I don't understand the formal contents of this statement.
>
I choose to attach the result Frequency to the probability law with the
same numarical value
>
> > Therefore, you should accept the additional formal definitions in the
> > context of this "experiment":
> >
> > 1) E= total set of formally chosen physical results
>
> I don't understand the meaning of 'formally chosen'.
>
It is logical choice. One decide to attach some mathematically defined
objects/properties to "reality". The way one decides the attachment
defines the context of this formal choice.
>
> > 2) A= subset of formally chosen physical results
> > 3) P(A): the probability of the event A
> >
> > Therefore, if I formally attach the Kolgomorov properties "identical
> > independent" (formal choice) to this set E, the set E may be
> > considered as the logical beginning of an infinite sequence of
> > independent identical trials leading to the probability law.
>
> This then has nothing at all to do with reality, since it is
> something entirely within Kolmogorov theory. Reality des not
> know of probabilities and hence has no notion of 'identical independent'.
>
Exactly. We need to attach/define some mathematical properties. We may
always do that: it is simply an application of bare logic. However,
from this formal attachment, we have logical results. That's all what
we basically need.

> >
> > 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 ...

But the approximation remains a logical choice.

>
> > Therefore, each time I have a set of physical results, I may formally
> > define a physical result probability law on this set by computing
> > simply the frequency (up to an error we formally estimates by the
> > Kolgomorov theorems). This formal result is independent of the physical
> > theory.
>
> I can't make sense of this.
>
> Please tell me how to formally define a physical result probability law
> given the set {0.9cm, 1.1cm, 0.8cm, 0.95cm} of physical results,
> and how to find the error, whatever these words should mean.
>
If we use the formal choice defined above, where we have the random
variable define by X: {(a,0.9cm), (b,1.1cm), (c,0.8cm), (d,0.95cm)}

We have P(X=0.9cm)= P(X=1.1cm)= P(X=0.8cm)= P(X=0.95cm)=1/4

Now, we may say (additional property): these results have no error, or
we may use some other properties to include an error (e.g. by the usual
probability estimators).

We may define other properties, such that we obtain completely
different results (and the error is infinite => any probability law is
ok).
>
>
> >>>I assume you restrict the Hilbert space to a separable one case.
> >>
> >>This is not needed, though it holds for the standard nonrelativistic
> >>models.
> >>
> >
> > 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.
>
> >>>I understand a particular system is a peculiar generic system with a
> >>>state rho(t).
> >>>Note axioms A1 and A2 is equivalent to postulate 1:
> >>>Most of the people do not see that postulate 1 already contains the
> >>>mixed states description.
> >>>Example: a mixed state rho= sum_a p_a |a><a| is given by the pure state
> >>>|psi>= sum_i p_a^1/2 |i>|a>.
> >>
> >>No. This is not a 1-1 correspondence since many different |psi> give the
> >>same rho. But statistical mechanics asserts that a particular splitting
> >>of a mixed state into individual pure states has no observable meaning.
> >>
> >
> > Note: An equivalence between axioms/postulates does not mean a 1-1
> > correspondence between the states.
>
> Then I don't mean what you understand to be an 'equivalence'.
> For me, equivalence goes in both directions.
>
By equivalence between properties/definitions, I mean the implicit
logical symbol <=>.
I hope you understand that the logical symbol "<=>" that defines an
equivalent *relation* does not define a 1-1 correspondence between
states:
postulate 1 => axiom 1 and 2 (except for the Hamiltonian part as
already said)
axiom 1 and 2 => postulate 1 (except for the Hamiltonian part as
already said)

=/=> it does not define a 1-1 correspondence between the states

>
> >>>And for some given rho(t) evolutions, we may define a new enlarged
> >>>rho_enlarged(t) where we have a unitary evolution. i.e. a new system
> >>>(K_enlarged, rho_enlarged, H).
> >>>I think with adequate working, we can prove that we are always able to
> >>>find an enlarged system where we have a unitary evolution.
> >>
> >>If the open system satisfies a Lindblad type dynamics, yes.
> >>In general, not (unless one creates a state-dependent extension).
> >>
> > That's mean yes in general?
>
> No, since traditionally, the Hilbert space is assumed to be independent
> of the state.
>
Ok, associate an artificial observable to these state dependence => in
this enlarged Hilbert space, you obtain a state independent
formulation. It therefore means yes for me, but I may be missing
something.
>
> >>Rather, each system treated by physicists has a small, well-defined
> >>set of canonical observables.
> >
> > This is rather the observables defining the configuration space.
> > Frankly, I do not know what the set of observables we are able to
> > measure is as long as I do not know if the preferred basis problem has
> > a solution.
>
> Since 'measuring' is outside the formal level, 'knowing' what it means
> only requires the common sense of a trained physicist, and not a formal
> discussion. Thus it is well-known.
>
You are therefore constraining the theory. This is a logical choice.
You can do that. However, we may define or found in the future some
other observables that are not contained by this restriction and that
describe the reality.
Restricting the set of observables to the one known currently by the
experiments seems to be very restrictive.
The last experiments of the 10 past years have shown the enlargement of
such a set observables (e.g. the squid ring experiment that
demonstrates the schroedinger cat state etc ...).

> Of course, in the ultimate foundations, there should be a formal
> measurement theory that derives it all. But this is not what we
> discuss at the moment. We discuss here only the consensus part of
> quantum mechanics.
>
Yes the logical consensus.
>
> >>However, in the quantum mechanics of N nonrelativistic
> >>particles, H and the linear combinations of the quantities I gave
> >>together with the orbital angular momentum probably constitute already
> >>the complete list of relevant observables.
> >>
> > I do not think so.
> >
> > 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?

> To check whether such a state is in fact prepared, we measure not
> the coherent state etc. but the distribution of position and momentum
> measurements, etc.
>
Exactly, we verify that the laser+PDC defines an observable with the
entangled state as eigenvector.

> In any case, one is free to add further observables to the list
> if needed.
>
That's exactly what axiom 4 allows and it is also ok for me.
>
>
> >>>>Again not a formal statement.
> >>>>It is not defined what independent identical systems are.
> >>>>This has formal meaning only in the context of Kolmogorov probability
> >>>>theory.
> >>>
> >>>This is what we are precisely doing: we choose the Kolgomorov
> >>>probability
> >>
> >>But there is no measurable space on the formal side, and no
> >>Kolmogorov probability in reality. So this hangs in the air.
> >
> > The measurable space is the observable spectrum with the borel sets
> > sigma algebra. The use of Kolgomorov probabilities is a choice, only
> > the probability measure counts as you said in your axioms.
>
> On the formal side there is the quantum state.
> Of course, once you have a measure, you have a measurable space.
> But the measurable space already depends on your choice of X.
>
Yes, this is what I call a contextual description. It is very powerful
and it reflects really what we usually do: the logical results only
exist within a given context.
Note that this does not say that a non contextual description is
impossible, but it is simpler to view the non contextual descriptions
as a subset of contextual ones.


Seratend.

.

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