Re: Postulates for quantum mechanics (was; Is State Vector Reduction
- From: Arnold Neumaier <Arnold.Neumaier@xxxxxxxxxxxx>
- Date: Thu, 23 Jun 2005 00:30:16 +0000 (UTC)
Seratend wrote:
Arnold Neumaier wrote:
Seratend wrote:
Well, I think the postulates are clearly stated, but my comments show that they are inadequate in themselves, and in their ability to delineate formal properties from interpretation.
If you like you may comment. However, I am not really interested in discussing these postulates further, I just wanted to know the background on which I can understand your statements. (Indeed, I now understand you much better.)
Ok, I may understand. However, I am really sorry that you do not attachWhat I _am_ interested, however, is a discussion of the following, more concise set of axioms for quantum mechanics.
more importance to these postulates. These postulates are very basic
(even if not perfect) and are taught in elementary QM introduction
courses (at least in France : ).
But I explained why I find them deficient. This is just _one_ particular set of axioms, not _the_ set of axioms. I have read dozens of foundations, and they all differ somewhat. So I was looking for ewhat was common to them all, and this is much _less_ than what is in your version. And I made a rigorous version of it that really satisfies all demands of logic.
You can easily convince yourself that my version of the axioms are
implied by yours (modulo some technicalities about self-adjointness),
while the converse is not true. I made sure that my version is compatible with everything I have seen about quantum mechanics as
applied in practice, and eliminates all but the bare minimum needed
for interpretation.
However, they really contain all the modern, at least non relativistic developments and reformulations of QM. When one uses the adequate isomorphisms, projections and limits, we just recover the other formulations (e.g. replacing the observables by generic C*-algebras, etc ...).
The main importance of these postulates remains on their explicit
separation between the description choice (postulates 1-5) and the
prediction of the theory (postulate 6). This is the operational
approach of any good physical theory (it allows a formal "shut up and
calculate").
I commented each of your postulates and told you why I found them wanting. They don't achieve the separation you claim; they are _not_
just formal statements - a computer trained on recognizing formal contents would not be able to make sense of them. But it would of mine,
if properly coded in logical primitives.
Your axioms lack this clear separation:
Axiom 1 to 5, except 3 describe only the mathematical objects that may be used by the description. Axiom 3 is the prediction content of the theory (with the Hamiltonian of axiom 1).
I don't understand the qualification 'only'. A formal model _must_ be a piece of mathematics. Everything which cannot be coded in pure mathematics lacks enough structure to be called 'formal'.
Axiom 6 is not an axiom, just an intention.
It specifies that the formal content of the theory is covered exactly by the remaining axioms without anything else added, and hence tells that Axioms 1-5 are complete.
Axiom 7 who tries to build the mapping of the mathematical objects [of the description] to the reality is incomplete (explained in the point to point comments.
It is incomplete precisely because I only coded the part that is universally agreed by the community of physicists. Everything beyond A7 is controversial.
However, I am very pragmatic and at the end, I want to know what part is the descriptive one and what part is the predictive one (if possible).
The description of a particular closed system is given by the specification of a particular Hilbert space in A1, the specification of the observable quantities in A4, and the specification of conditions singling out a particular class of states (in A6). Everything else is determined by the theory and hence is (in principle) predicted by the theory.
The description of an open system involves, in addition, the
specification of the details of the dynamical law. For the basics,
see the entry 'Open quantum systems' in my theoretical physics
FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txtAs I have said in previous posts, the descriptive part of a physical theory is a [human] choice: we choose the mathematical tools and their mapping with the "reality" in order to be able to do logical propositions. Once we can make logical propositions, we can make logical predictions and vary them against reality thanks to the descriptve part of the theory. Currently, I only know this practical point of view of physics. I hope this is also your point of view. If this is not the case, please explain me, as I would like to understand how you can build your logical propositions (your assertions) on the "reality".
I more or less agree to this.
In describing reality from a physics point of view, the person modeling a system of interest makes certain choices. These consist in choosing a mathematical model of the system, and setting up a correspondence between informal objects related to the system and formal objects in the mathematical model.
More specifically, an assertion about reality is modelled as a mathematical assertion about mathematical objects in the mathematical model that carry the same names as those in the reality they are supposed to model.
As in any axiomatic setting (necessary for a formal discipline), there are a number of different but eqivalent sets of axioms or postulates that can be used to define formal quantum mechanics. Since they are equivalent, their choice is a matter of convenience.
Yes. However, we must agree on the subject: these axioms must define the description and the prediction of the theory.
Axioms A1-A7 do this.
By description, I mean the mathematical objects and their mapping with the "reality" (for practical use of the theory). In this sense, the description contains a minimalist interpretation (the mapping to the "reality").
I separated the axioms into those which are free of interpretation (apart from assigning names suggestive of reality, the objects are purely formal) and one which contains a minimalist interpretation on which I know the physics community agrees.
Increasing the contents of the latter (Axiom 7), while needed for a complete modeling, automatically will create dissent of some part
of the community.
My own attempt to complete the model is the consistent experiment interpretation.
I assume you restrict the Hilbert space to a separable one case.There are six basic axioms.
A1. 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_.
This is not needed, though it holds for the standard nonrelativistic models.
A2. A particular system (e.g., 'the ion in the ion trap on this particular desk') is characterized by its _state_ rho(t) at every time t in R (the set of real numbers).
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.
Hence I prefer to have the mixed case from the beginning. It also makes the axioms simpler.
A3. 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. (hbar is Planck's constant, and is often set to 1.) If nothing else is apparent from the context, a system is assumed to be closed.
Axiom 3 is a conditional axiom. Very difficult to see the consistency and the applicability of such axiom.
There is no condition. The axiom defines the formal meaning of the terms 'closed' and 'open' consistent with the usage in the applications.
It defines a property "closed" where the unitary evolution does apply. Therefore, any system without the unitary evolution of A3 is a
not "closed" system.
Yes. This is the conventional formal meaning of the word closed.
=> with axiom 1,2 and axiom 3, any particular system (K, rho1, H1) closed between t1 and t2 is another particular not closed system (K, rho1, H2), with H1=/=H2.
Formally yes, but in practice one would not call H2 a Hamiltonian. The axioms are like set theory - they allow certain weird structures not arising in actual applications. I have not tries to give a set of axioms such that every realization of it has physical meaning. This is neither the case with your postulates; e.g., one can define weird Hamiltonians that have no physical interpretation.
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).
In other words, in axiom 3, I do not see the utility of the closed system property and the complexity (conditional) it adds in the use of axiom 3.
The utility consists in the fact that many people who apply quantum mechanics in fact work very successfully with open systems without ever
considering a unitary embedding into a closed system. My axioms reflect
this usage.
A4. Besides the energy H, certain other (densely defined, self-adjoint) Hermitian operators (or vectors of such operators) are distinguished as _observables_. (E.g., the observables typically include several 3-dimensional vectors: the _position_ x^a, _momentum_ p^a, and _spin_vector_ (or Bloch vector) sigma^a of particles labelled a. If u is a 3-vector of unit length then u dot p^a and u dot sigma^a are called the momentum and spin in direction u.)
Ok, we just assume a set of "gentle" operators. H is also an observable.
Axiom 4 tries to define the observable property in the same loosely way as postulate 2. (i.e. we are free to choose the set of observables)
In practice, we are not really free. I have never seen an operational definition for how to observe pqp, say.
Rather, each system treated by physicists has a small, well-defined set of canonical observables. Instead of giving a full list (which I find difficult to create), I gave examples.
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.
Note: dmu_t(X,rho).A5. To every vector X of observables with commuting components, one associates a multivariate probability measure dmu_t(X) by defining integral dmu_t(X) 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.
Yes. I assumed a particular system, so rho is fixed.
Ok. We choose a restriction of continuous bounded functions that is sufficient to define the probability measure given the expectation.
Yes.
Note: once we define a probability measure (a bounded measure) we define implicitly a probability space (here the borel sets on the tensor product of the eigenspaces of vector X). Therefore, we are free to logically define the Kolgomorov probabilities (= probability measure dmu_t(X,rho) on the tensor product of the eigenvalues of vector X.
Yes. But I explicitly avoid this in the axioms since it is not needed in statistical mechanics. It is better to have this as a deduction of the theory rather than to assume it - all the more since there are foundational problems in the interpretation of probability.
This is what I have done in postulate 5 and it seems that you have not
understood what I have said (this is a formal construction always
possible):
Yes, I understand you, except that I want he formal apparatus completely separate from reality, to avoid interpretation marring the formal
structure.
The Kolgomorov probabilities are a logical tool we use to do
a practical measure of the probability law (the frequency) on the
"reality",
Interpreting probability as relative frequency is a philosophically difficult interpretation step. See the very informative books by T.L. Fine, Theory of probability; an examination of foundations. Acad. Press, New York 1973. and L. Sklar, Physics and Chance, Cambridge Univ. Press, Cambridge 1993. I want to stear clear of this bag of worms.
Therefore axiom 5 defines a probability measure on the set of eigenvalues of vector X. This is the postulate 4 without its formal mapping "eigenvalue a" "measurement result a".
Yes. Of course your postilates imply mine. Any set of postulates I have seen implies mine, but not conversely. This is why my set of axioms is minimal while the others are not.
A6. Quantum mechanical predictions amount to predicting properties of the measures defined in axiom A5 given reasonable assumptions about the states (e.g., ground state, equilibrium state, etc.)
This is not an axiom. Just a goal (we can remove it it does not change the mathematical content of the previous axioms).
It is like the completeness property ''Every set containing 0 and containing with x also x+1 contains the natural numbers''
in the Peano axioms. It tells that nothing else may be used in a formal
prediction.
In addition to these formal axioms one needs a rudimentary interpretation relating the formal part to experiments.
Yes. Currently, A1-5 only defines the mathematical objects and lacks
the logical mapping of the objects to the "reality" (for a
practical use).
This is on purpose, since it cleanly separates the formal, mathematical
part from the informal, philosophical part involving formally ill-defined notions relating to reality.
If you prefer, we need to connect the bounded measure of axiom 5 to reality. Without that, we have just a mathematical theory.
Yes.
The following _minimal_interpretation_ seems to be universally accepted.
A7. Upon measuring at time t=0 a vector X of observables with commuting components, for a large collection of identical systems closed for times t<0, all in the same state rho_0 =3D lim_{t to 0 from below} rho(t) (one calls such systems _identically_prepared_), the outcomes are statistically consistent with independent realizations of a random vector X with measure as defined in axiom A5.
Note that A7 is no longer a formal statement since it neither defines what 'measuring' is, nor what 'outcomes' are. Thus A7 is already part of the interpretation of formal quantum mechanics.
This is where your axiomatic base (axioms 1-6) does not describe a
formal mapping between your axiomatic objects and the reality.
Your axiom 7 tries to collapse in a long sentence what is defined by
postulates 2-3 and 4 and also 5 but it misses the critical part: the
formal mapping between the objects of your axioms and the reality or if
you prefer the formal description of a real experiment: its results
(and their mapping to the axiomatic objects).
This part is missing on purpose since there is no agreement among
physicsists on its modeling. (This is why I had misunderstood your earlier mails since it referred to a particular interpretation that
you assumed without my knowledge to be 'given'.)
We need to know/express formally (the logic or if you prefer the minimum interpretation set) what propositions (the results) of a given real experiment are true and how these propositions are mapped to your axiomatic objects.
Your axioms do not do it. Such a mapping can only be given if there is an unambiguous way of saying what constitutes an experiment and the taking of a measurement. Nothing universally accepted exists in this direction. Your postulates are not better than my Axiom in this respect; they are only confounding the issues.
Without these logical propositions of reality (i.e. "the measurement results") and their mapping to the objects of the axioms we cannot use the Kolgomorov probabilities results to compute [measure] the frequency nor any other results: we have no link between the "reality" and the theory.
A7 gives a sufficient link to reality, assuming someone knows what 'measuring' and 'getting an outcome' means in reality. It is much more precise than anything I have seen formulated in this direction. Usually the term 'identically prepared' is completely unspecified in its meaning, but presupposed in the interpretation.
Note that a true proposition of a given experiment may be attached formally to the outcomes of Kolgomorov probabilities (a given set of outcomes of the Kolgomorov probabilities is a given set of true propositions). This is mathematics not physics (theory independent).
The formal part is always mathematics. It becomes physics since it can be interpreted in physical terms.
Once we accept the description of an experiment is simply a set of true
propositions (the "measurement results"),
This is what I call the 'outcomes' of 'measuring'. But I am sufficiently vague to cover all existing interpretations and measurement theories.
In conclusion as your axiom 7 does not define the mapping between the objects and the "reality" (the minimalist interpretation requirement), axiom 7 is just the definition of Kolgomorov probability law associated to the bounded measure dmu_t(X,rho(t)) => it is not an interpretation.
No. It relates the computed probabilities to outcomes observable in reality.
In addition, A7 misses an important property: the *independent* identical systems. (Independent here is the Kolgomorov property).
I had implicitly assumed that if they are identically prepared and distinct they are automatically independent. But you are right, it is better to require this explicitly. I had wanted to avoid it because it introduces another ill-defined term 'independent'.
So please replace the old A7 plus its comment by
A7. Upon measuring at time t=0 a vector X of observables with commuting components, for a large collection of independent identical systems closed for times t<0, all in the same state rho_0 = lim_{t to 0 from below} rho(t) (one calls such systems _identically_prepared_), the outcomes are statistically consistent with independent realizations of a random vector X with measure as defined in axiom A5.
Note that A7 is no longer a formal statement since it neither defines what 'measuring' is, nor what 'outcomes' are and what 'independent' means. Thus A7 is already part of the interpretaion of formal quantum mechanics.
Everything beyond A7 seems to be controversial.
Yes if you add the measurement result mapping to your axiomatic objects.
What constitutes a measurement result is already controversial. E.g., you read a pointer - different readers get marginally different results. What is the true pointer reading?
Good foundations including a good measurement theory should be able to properly justify this informal consensus.)
We must at least agree on the minimalist formal description of experiments. Once we are sure on this minimalist description, we will be able to say what is missing (for practical purposes) in QM.
Yes. At present we almost agree but not quite.
To see how the more traditional setting in terms of the Schroedinger equation arises, we consider the case of a closed system in a pure state rho(t) at some time t.
See my comments on axiom 3 concerning closed systems and non closed systems. This is a point of view: do we want, formally, to describe a local state or a global state?
local and global are not defined in my setting; so I don't understand what you mean.
Personally, I do not see any difference: we recover the same results. Therefore, the choice is formal and mainly driven by the simplicity of the results (and the equations).
Then let us with my setting.
Ok. As said before, Axiom 5 is sufficient to define a bounded measure on the borel sets of the set of eigen values of the considered observable => all the results of the rest of the post come from this property (it defines the set of commutative projector or the Boolean lattice, etc ...).
Deriving the Borm rule from axioms A1-A6 makes it completely natural, while the traditional approach starting at our end makes it an irreducible rule full of mystery.
I do not understand this statement (the natural). Giving a bounded measure or the born rules where is the difference?
On first acquaintance, the squared probability law looks quite mysterious and in need of an explanation. It is much less self-evident than Axiom 5.
In other words, I do not see on how you can escape from an equivalent set of postulates 2-5, however I am really interested on the way you view experiment results (your logic when you describe a real experiment).
Let's discuss this when we agreed on the remainder. My FAQ entry on open quantum system might be what you are looking for.
This is the mapping A_phys --> A. We usually add (implicitly) the following property to this postulate: the mapping between A_phys and A is bijective
This is a highly dubious additional property, much debated in the foundations. It is certainly _not_ part of the standard consensus QM.
Well, without this formal bijective mapping we cannot logically associate the experimental results to the mathematical objects of the theory (in a verification or prediction way). Please, tell me how you can reproduce/describe (prediction of results with verification) a real experiment without such a bijection.
One needs a mapping of observables in reality to operators in the formalism, but not a converse.
Postulate 4: The measurement result "a" of a physical observable represented by the observable A of a system with a *unit* state |psi> has the *probability*:
This is no longer a formal statement since while it defines probability, it does not define 'measurement result'. Thus it is part of the interpretation.
Exactly: the postulates contain the minimalist interpretation (the
operational shut up and calculate).
And I want the formal part completely separate from the interpretation part, hence my form of the axioms.
Postutale 4 is consistent whenever we have a bijective bi-measurable mapping between the physical variable set of values and the observable set of eigenvalues. the probability law on the set of values of the physical observable infers a probability law on the set of eigenvalues and in the same for other way through this bi-measurable mapping.
Therefore, taking a set of N independent identical systems (state
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.
|psi>: postulate 1), we have: n(a)/N --> P(a||psi>) when N goes to the infinity, formally from the laws of large number.
Postulate 4 and the previous ones allow the formal mapping of the statistics of experimental results into the statistics given by an observable and a state.
The postulates do not know of 'statistics of experimental results'. Thus this is again an additional interpretation step.
No! If you do not understand this is not interpretation, but just an operational application of the postulates, we are in trouble:
What then is the statistics of experimental results'?
It requires that you have a stochastic process - reality knows not of these, and I haven't seen your postulates define one.
Arnold Neumaier
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