Boundedness of quantum observables?

  • #26
Fredrik
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In my opinion one of the major advantages of C*-algebra approach is that it allows one to see a tower of probability theories. For every type of algebra one gets a probability theory:
Type I:
Abelian: Discrete Kolmogorov probability
NonAbelian: Quantum Mechanics
Type II:
Abelian: Statistical Mechanics
NonAbelian: Quantum Statistical Mechanics
Type III:
NonAbelian: Quantum Field Theory

I don't know what other people think of this, but it's very interesting from a conceptual level.
I find this kind of stuff very interesting, but unfortunately I'm not yet at a level where I can fully understand it. (I'm currently studying the proofs of the spectral theorems of functional analysis). I would still be interested in seeing the definitions of the three "types" you're talking about.
 
  • #27
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Have y'all considered that while an operator like X isn't bounded, other operators like arctan(X) are?

I really don't think you lose anything by laying the foundations via C*-algebras. Once you have a C*-algebra, you can use calculus (or other means) to construct additional algebras, if you so desire.
First of all, all natural observables in quantum theory (or QFT) correspond to unbounded (distributional) operators. Likwise are all natural eigenstates of these operators outside Hilbert space. Second, if you generalize away to Nevanlinna space, unitary operators also become unbounded (on some subspace of zero norm states) so here, the picture of bounded operators completely evaporates. Third, unbounded operators are more natural and in case of nontrivial deficiency indices, there are several possible self adjoint extensions which all give a different physical picture. I do not know how to appreciate this fact from a formalism which starts with bounded operators in the first place. Fourth, the composition of densely defined unbounded operators is not necessarily a densley defined operator. This gives genuine new effects (physics) which cannot be captured in a picture of bounded operators by a natural transformation (for example, I do not know of a closed formula for e^{iAB} in terms of e^{iA} and e^{iB}). So, I think C^{*} algebra's (as well as Von Neumann algebra's) are (a) not natural and (b) too limited.

Careful
 
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  • #28
Fredrik
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To define quantum mechanics (or any other physical theory) properly, we only need to define the calculus and then say how to relate the quantities that can be calculated from quantum mechanical models to the stuff experimental physicists talk about.
I often say that a theory is defined by a set of axioms that tells us how to interpret some piece of mathematics as predictions about results of experiments. It sounds like you would agree. The reason I require something more specific from a definition of "QM" is that we are already so familiar with Hilbert spaces, states, observables and so on, that we wouldn't consider a theory that uses an entirely different mathematical framework to be QM. At least I wouldn't. I don't think of Bohmian mechanics as QM either. (I think of a "theory" as being defined by its axioms rather than by its predictions, because the latter choice would make "SR + there's an invisible blue giraffe that doesn't interact with matter" the same theory as SR).

The idealization that goes into the interpretation used in your description is only a didactical trick to make definitions a bit easier to swallow. (How do you justify the C^*-algebra axiom ||A^*a||=||a||^2 from measurement??)
I know e.g. that if A is an "observable" in the sense "equivalence class of measuring devices", and k is a real number, we can take kA to be the the equivalence class of "any member of the equivalence class of A with an additional component that multiplies each result by a factor of k". So there's at least one C*-algebra axiom that can be properly motivated. I have no idea about the specific axiom you mention. I haven't even figured out why there should be an involution.

I suspect that there are no strong arguments for that identity, but I don't think is any different from why we only consider separable Hilbert spaces in the Hilbert space approach to QM. When we don't know if an assumption should be included or not, it's natural to try the choice that simplifies the math. If it appears to be sufficient, fine. If it gives us a theory that disagrees with experiment, back to the whiteboard.

In experimental physics, measurement is a very complex thing - far more complex than your ''definition'' suggests. To measure the distance between two galaxies, the mass of the top quark, or the Lamb shift - just to mention three basic examples - can never be captured by the idealistic measurement concept in your definition.

In each case, one assembles a lot of auxiliary information and ultimately calculates the measurement result from a best fit of a model to the data. Clearly the theory must already be in place in order to do that. (We don't even know what a top quark should be whose mass we are measuring unless we have a theory that tell us this.)

And the Lamb shift (one of the most famous real observables in the history of quantum mechanics) is not even an observable in your sense!
These are interesting comments that I will have to think about.

We proceed as in the modern account of the oldest of the physical sciences: Euclidean geometry, where (on laboratory scales) there is consensus about how theory and reality correspond:
...
I didn't really understand your approach, or your take on the measurement problem. But I understand that you're talking about an approach that doesn't exist, or at least hasn't (yet) been developed to the point where it can be used to state the axioms of a theory that makes the same predictions as QM.

One more thing: The main reason why I find the redefinition of the word "observable" discussed above acceptable, is that there's already a perfectly fine term ("self-adjoint operators") for what you want to call "observables", while nothing better than "thingamabobs" come to mind for the equivalence classes of measuring devices with finite size and accuracy. Since the idea of having something mathematical represent something in the real world is so fundamental in physics, I prefer to have good names for both the "things in the real world" and the "mathematical things", rather than two good names for the mathematical things.
 
  • #29
A. Neumaier
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if you generalize away to Nevanlinna space, unitary operators also become unbounded (on some subspace of zero norm states) so here, the picture of bounded operators completely evaporates.
With Nevanlinna space, do you mean what others call a Krein space
http://en.wikipedia.org/wiki/Krein_space
- like in the Gupta-Bleuler version of QED?
 
  • #30
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With Nevanlinna space, do you mean what others call a Krein space
http://en.wikipedia.org/wiki/Krein_space
- like in the Gupta-Bleuler version of QED?
It is not because Rolf Nevanlinna had strange political opinions that we should forget he considered these things way before Krein did. :wink: Actually Arkadiusz Jadczyk has his own definition and also have I (which is much more general than the one Krein gave).


Careful
 
  • #31
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I often say that a theory is defined by a set of axioms that tells us how to interpret some piece of mathematics as predictions about results of experiments. It sounds like you would agree.
Yes, though the axioms are not always clearly or fully spelled out. (I don't know clear axioms for the standard model.)

The reason I require something more specific from a definition of "QM" is that we are already so familiar with Hilbert spaces, states, observables and so on, that we wouldn't consider a theory that uses an entirely different mathematical framework to be QM. At least I wouldn't.
The C^*-algebra approach to QM doesn't start with Hilbert spaces; in any case, it has very different axioms from the standard QM axioms, but leads to essentially the same theory.

(I think of a "theory" as being defined by its axioms rather than by its predictions, because the latter choice would make "SR + there's an invisible blue giraffe that doesn't interact with matter" the same theory as SR).
Your proposal is not really cogent.

Then two versions of SR that use opposite conventions for the signature would be different theories (since the axioms differ), although they make the same predictions.

Most theories can be based on quite different axioms, depending on what you want to treat as most basic. For example, one can develop QM without mentioning state vectors in the foundations. See, e.g., the section ''Postulates for the formal core of quantum mechanics'' in Chapter A1 of my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html#postulates

I haven't even figured out why there should be an involution.
Because we want to model standard physics, and they need complex observables
(very handy for, e.g., for describing linear circuits). Useful stuff should not be sacrificed to principles of philosophy.

So at least one should be able to have complex expectation values. And conjugation ensures that replacing i by -i doesn't change the physics!


I suspect that there are no strong arguments for that identity, but I don't think is any different from why we only consider separable Hilbert spaces in the Hilbert space approach to QM.
It is very different. Without ||a^*a||=||a^||^2 you can hardly get started; the algebras not satisfying this don't resemble C^*-algebras anymore. Can't prove any spectral properties....

I didn't really understand your approach, or your take on the measurement problem. But I understand that you're talking about an approach that doesn't exist, or at least hasn't (yet) been developed to the point where it can be used to state the axioms of a theory that makes the same predictions as QM.
The approach outlined is fully developped though not widely publicized; it is the basis of my thermal interpretation of quantum mechanics. It agrees with how one does measurements in thermodynamics (the macroscopic part of QM (derived via statistical mechanics), and therefore explains naturally the classical properties of our quantum world. It is outlined in my slides
http://www.mat.univie.ac.at/~neum/ms/optslides.pdf
and described in detail in Chapter 7 of my book
Classical and Quantum Mechanics via Lie algebras
http://lanl.arxiv.org/abs/0810.1019
It is superior to the interpretions found in the literature, since it
-- acknowledges that there is only one world,
-- uses no concepts beyond what is taught in every QM course,
-- allows to derive Born's rule in the limit of a perfect von-Neumann measurement (the only case where Born's rule has empirical content),
-- applies both to single quantum objects (like the sun) and to statistical ensembles,
-- has no split between classical and quantum mechanics,
-- has no collapse (except approximately in non-isolated subsystems).


One more thing: The main reason why I find the redefinition of the word "observable" discussed above acceptable, is that there's already a perfectly fine term ("self-adjoint operators") for what you want to call "observables", while nothing better than "thingamabobs" come to mind for the equivalence classes of measuring devices with finite size and accuracy. Since the idea of having something mathematical represent something in the real world is so fundamental in physics, I prefer to have good names for both the "things in the real world" and the "mathematical things", rather than two good names for the mathematical things.
Nobody in the real world thinks about "equivalence class of measuring devices" - this is not a thing in the real world, but an abstraction invented by the C^* algebra people to give meaning to their constructs, and used nowhere else.

I prefer to have two good names for the same thing, for example ''positive integer'' and ''natural number'', if these are well established in the tradition.
 
  • #32
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It is not because Rolf Nevanlinna had strange political opinions that we should forget he considered these things way before Krein did. :wink: Actually Arkadiusz Jadczyk has his own definition and also have I (which is much more general than the one Krein gave).
Many concepts are not named after their originator.

For me the main purpose of naming is to ease communication rather than to honor the first who coined a concept. One doesn't want to rewrite articles and books each time a new historical fact turns up.
 
  • #33
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It is not because Rolf Nevanlinna had strange political opinions that we should forget he considered these things way before Krein did. :wink: Actually Arkadiusz Jadczyk has his own definition and also have I (which is much more general than the one Krein gave).


Careful
So, in my opinion, bounded operators are dead. Heisenberg thought about it in the same way and as far as my knowledge of history goes, he was playing with Krein spaces later on in his life.
 
  • #34
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So, in my opinion, bounded operators are dead.

But they are well alive and indispensible in mathematical physics.

The real thing still happens on a subspace of Krein space where the inner product is positive definite and the Hamiltonian is self-adjoint and bounded below.

Otherwise there is no well-defined and stable dynamics.

Heisenberg thought about it in the same way and as far as my knowledge of history goes, he was playing with Krein spaces later on in his life.
Even famous people play with lots of things during their lifetime. Nevertheless, they get honored not for play but for substance.
 
  • #35
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But they are well alive and indispensible in mathematical physics.
Mathematical physicists often investigate things which are not useful for physics. Give me one application which hinges upon them.

The real thing still happens on a subspace of Krein space where the inner product is positive definite and the Hamiltonian is self-adjoint and bounded below.
I am afraid you confuse negative energy with negative probability. They are different things. It is very easy to define a Lorentz covariant positive energy Hamiltonian with positive and negative norm particles. The interpretation of course happens on a sub-hilbertspace but (a) this one is dynamical and by no means invariant under the Hamiltonian and (b) observer dependent.

Otherwise there is no well-defined and stable dynamics.
Even supposing that you would allow for an undbounded Hamiltonian, I think your conclusion regarding stability is wrong. I feel this is a matter of interpretation and it depends upon the sign of the interaction terms as well (apart from the negative mass in the kinetic term). All arguments against it I know can be circumvented (including those of pair creation) so if you think you can give this a deadly blow, please go ahead.

Even famous people play with lots of things during their lifetime. Nevertheless, they get honored not for play but for substance.
Right, there is no substance behind bounded operators. The best proof is that we never use them.

Careful
 
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  • #36
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I don't think of Bohmian mechanics as QM either.

just a ψ-ontic hidden variable model (BM) of quantum theory.



about an approach that doesn't exist, or at least hasn't (yet) been developed to the point where it can be used to state the axioms of a theory that makes the same predictions as QM.
not so fast...

and why just the same predictions ? why not that predictions, and more, beyond and broader predictions.



I think of a "theory" as being defined by its axioms rather than by its predictions.
me too.

can be used to state the axioms of a theory that makes the same predictions as QM.
then ?
contradicting yourself ?
 
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  • #37
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Many concepts are not named after their originator.

For me the main purpose of naming is to ease communication rather than to honor the first who coined a concept. One doesn't want to rewrite articles and books each time a new historical fact turns up.
No need to rewrite, just use the correct terminology from the moment this fact becomes clear. I think we are obliged to honor the right persons in science, it is not only a matter of courtesy but of intellectual honesty too. And if this causes confusion in the beginning, then be it so; there are lots of confusing things in life.
 
  • #38
Hurkyl
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First of all, all natural observables in quantum theory (or QFT) correspond to unbounded (distributional) operators...
In response to the suggestion that C*-algebra is a reasonable starting point, you made an argument that C*-algebra is not a reasonable ending point. How is that helpful? It's like you completely ignored what I said in favor of what you wanted to say. :frown:


No need to rewrite, just use the correct terminology from the moment this fact becomes clear. I think we are obliged to honor the right persons in science, it is not only a matter of courtesy but of intellectual honesty too. And if this causes confusion in the beginning, then be it so; there are lots of confusing things in life.
AFAIK, the normal way is to add names -- e.g. to say "Krein-Nevanlinna space". To completely discard the common name, among other things, is obstructive to discussion.
 
  • #39
Fredrik
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and why just the same predictions ? why not that predictions, and more, beyond and broader predictions.
Because we were talking about alternatives to the algebraic approach to QM, not about attempts to find a better theory.

then ?
contradicting yourself ?
I would have contradicted myself if I had called the theory obtained using the alternative approach "QM". I said "a theory that makes the same predictions as QM" to avoid the contradiction.
 
  • #40
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In response to the suggestion that C*-algebra is a reasonable starting point, you made an argument that C*-algebra is not a reasonable ending point. How is that helpful? It's like you completely ignored what I said in favor of what you wanted to say. :frown:
No, I did not. I said that C* algebra's are not natural at all for the reasons I mentioned. A mathematical object is only any good if you can formulate the physics directly into these terms. This never happens, neither in QM nor in QFT where unbounded operators enter the calculations. Normally, people might still be inclined to use the Weyl transformation to do what you suggested but this becomes completely problematic on Nevanlinna space. Actually your trick will fail there under any circumstances because any nontrivial analytic function blows up to infinity. For example, hermitian operators can have a complex spectrum (on the ''ghost'' states) which is totally unbounded. So an arctan, e^{ix} or something like that is not going to resolve anything.

What would be interesting from the point of view of ''C* algebra's'' is that you try to extend the GNS construction to non-positive states, so that you will get Nevanlinna space representations. This requires of course a change in the C* norm identities in the first place, but it might be good to define such generalized algebra's.
 
  • #41
Fredrik
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Right, there is no substance behind bounded operators. The best proof is that we never use them.
That hardly proves a claim as strong as "there's no substance behind bounded operators". I don't know the algebraic approach well, but it seems to me that we can never make a measurement that corresponds to the momentum operator (because there's no measuring device with infinite precision). In QM books at the level of Sakurai, it is often claimed that a measurement of an observable A that gives us the result a, leaves the system in a state represented by a vector in ker(A-a), i.e. the eigenspace corresponding to eigenvalue a. For an unbounded operator such as P, this would kick the state out of the Hilbert space entirely, and leave the system in a "state" |p> of perfectly well-defined momentum. I don't think a realistic measurement, which has finite precision, can do more than to confine the state vector to some subpace of the Hilbert space. So when we actually do a momentum measurement, the mathematical representation of the measuring device won't be P. It will be a member of that C*-algebra.

I do however agree that the fact that the unbounded operators are so prominent suggests that it would be desirable to start with some kind of algebra of unbounded operators instead. Perhaps there is such an approach, that gives us a rigged Hilbert space in a way that's similar to how the C*-algebra approach gives us a Hilbert space.
 
  • #42
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That hardly proves a claim as strong as "there's no substance behind bounded operators". I don't know the algebraic approach well, but it seems to me that we can never make a measurement that corresponds to the momentum operator (because there's no measuring device with infinite precision). In QM books at the level of Sakurai, it is often claimed that a measurement of an observable A that gives us the result a, leaves the system in a state represented by a vector in ker(A-a), i.e. the eigenspace corresponding to eigenvalue a. For an unbounded operator such as P, this would kick the state out of the Hilbert space entirely, and leave the system in a "state" |p> of perfectly well-defined momentum. I don't think a realistic measurement, which has finite precision, can do more than to confine the state vector to some subpace of the Hilbert space. So when we actually do a momentum measurement, the mathematical representation of the measuring device won't be P. It will be a member of that C*-algebra.
Two quick reactions, Hilbert space is not only unsuitable because it has only positive norm but also because it cannot include distributional states. So rigged Hilbert spaces are much better. There is no problem in finding out a suitable probability interpretation for distributional states. It is not really important whether these states are measured or not, what matters is that they are the natural mathematical objects which show up in representation theory of the Poincare algebra. Since the generators of this algebra correspond to unbounded (distributional) operators (which even act well on the distributional states with a finite number of terms), the language of unbounded operators is the most natural thing. Whether energy or momentum can be measured sharply is a philosophical question (and one should not take comments of for example Sakurai too seriously); what matters is that dynamics is most naturally expressed in the unbounded (distributional) language.

Careful
 
  • #43
Fredrik
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Yes, though the axioms are not always clearly or fully spelled out. (I don't know clear axioms for the standard model.)
Agreed. A complete definition of a theory would have to include a lot more than the stuff we can list on a page of a book, including a specification of e.g. what measuring devices or procedures we should think of as measuring "energy". If we can't come up with a good axiom scheme that specifies how to identify all (or a sufficiently large class of) measuring devices with self-adjoint operators (or whatever represents them mathematically in the theory we're considering), we need a separate axiom for each type of measuring device.

Then two versions of SR that use opposite conventions for the signature would be different theories (since the axioms differ), although they make the same predictions.
Yes, this is pretty annoying. What I've been doing when I have only been thinking about these things, is to allow myself to use sloppy terminology, and call the opposite signature SR "SR" even though this contradicts my definition of "theory". This isn't very different from how mathematicians define a group as a pair (X,b) or a 4-tuple (X,b,u,e) with certain properties...and then start referring to X as a "group" the moment they're done with the definition. It doesn't confuse me, but I think I need a better system for when I explain these things to other people.

The approach outlined is fully developped though not widely publicized; it is the basis of my thermal interpretation of quantum mechanics. It agrees with how one does measurements in thermodynamics (the macroscopic part of QM (derived via statistical mechanics), and therefore explains naturally the classical properties of our quantum world. It is outlined in my slides
http://www.mat.univie.ac.at/~neum/ms/optslides.pdf
and described in detail in Chapter 7 of my book
Classical and Quantum Mechanics via Lie algebras
http://lanl.arxiv.org/abs/0810.1019
Cool, I'll check it out, but not right now. I need to get some sleep. I didn't realize that you're one of the authors of that book. I downloaded it in May, but haven't gotten around to reading it yet. I don't think I will the next few months either, because I'm trying to learn functional analysis, and it takes an absurd amount of time.
 
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  • #44
dextercioby
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In QM books at the level of Sakurai, it is often claimed that a measurement of an observable A that gives us the result a, leaves the system in a state represented by a vector in ker(A-a), i.e. the eigenspace corresponding to eigenvalue a. For an unbounded operator such as P, this would kick the state out of the Hilbert space entirely, and leave the system in a "state" |p> of perfectly well-defined momentum.
.

(bolding by me). That's the von Neumann's projection postulate which pertains to the Copenhagian view of things. It's not universally accepted and other interpretations and axiomatizations of QM completely disregard it.

I do however agree that the fact that the unbounded operators are so prominent suggests that it would be desirable to start with some kind of algebra of unbounded operators instead. Perhaps there is such an approach, that gives us a rigged Hilbert space in a way that's similar to how the C*-algebra approach gives us a Hilbert space.
This part I agree with. I haven't seen the bolded part yet, theories of unbounded operator algebras use a Hilbert space as an environment, not an RHS.

As far as I recall (but if I'm wrong, please correct me), putting an RHS into a QM problem with unbounded operators turns these operators into bounded ones, but of course, not in the original topology of the H-space, but in the topology of the antidual space in which the original operators will find their eigenvectors.
 
  • #45
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A mathematical object is only any good if you can formulate the physics directly into these terms.
:confused: At face value, this comment seems utterly absurd.

For example, hermitian operators can have a complex spectrum (on the ''ghost'' states) which is totally unbounded.
Er, so? Splitting it into the sum of real part and imaginary parts is an even more standard trick than taking the arctangent to make a real variable bounded.

... what matters is that dynamics is most naturally expressed in the unbounded (distributional) language.
Again, that hardly proves a claim as strong as "there's no substance behind bounded operators". A laundry list of reasons why you want an unbounded / distributional language does not constitute a denial of the hypothesis
The language of unbounded / distributional operators can be constructed using foundations built from of bounded operators​
 
  • #46
A. Neumaier
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Mathematical physicists often investigate things which are not useful for physics. Give me one application which hinges upon them.
But this thread is about rigorous quantum mechanics - giving logical impeccable justiifications for what theoretical physicists commonly do.
 
  • #47
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:confused: At face value, this comment seems utterly absurd.
It is clear you are a mathematician; there is nothing absurd about this comment. I would even go much further and state that mathematicians only develop the easy language and the hard one, which is useful, is left entirely to the physicists. But yeah, you need to do theoretical physics to understand why this is true.

Er, so? Splitting it into the sum of real part and imaginary parts is an even more standard trick than taking the arctangent to make a real variable bounded.
:bugeye: Errr, the only natural splitting which is allowed is by means of the natural involution dagger. There is nothing you can do here, because the operator is self-adjoint. Actually on Nevanlinna space, there is no natural algebraic criterion which gives only operators with a real spectrum. So what you propose is even bad mathematics; simply accept that your point -which any student can make- is only valid in Hilbert space.

:
Again, that hardly proves a claim as strong as "there's no substance behind bounded operators". A laundry list of reasons why you want an unbounded / distributional language does not constitute a denial of the hypothesis
The language of unbounded / distributional operators can be constructed using foundations built from of bounded operators​
Again, this is only true on Hilbert space!! You seem to be trapped here in some irrational wish for bounded operators and are willing to go through all unnatural constructions possible to save their ***. It is possible of course to define bounded operators on Krein space, but it is not the natural class of operators (since their very definition requires a Hilbert space construction!) and there would be no reason for me to even consider Krein space if I would stick to these simple animals.
 
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  • #48
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Apparently there's some work in the field of <algebras of unbounded operators> as this review article (and the quoted bibliography) shows:

http://arxiv.org/abs/0903.5446
Thanks. This is a nice paper that I didn't know before. I need to read it more carefully.
 
  • #49
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But this thread is about rigorous quantum mechanics - giving logical impeccable justiifications for what theoretical physicists commonly do.
What is not rigorous about unbounded operators ? :rolleyes: Actually, I studied quantum physics rigorously from that point of view (my master education was in mathematical physics btw).
 
  • #50
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A complete definition of a theory would have to include a lot more than the stuff we can list on a page of a book, including a specification of e.g. what measuring devices or procedures we should think of as measuring "energy". If we can't come up with a good axiom scheme that specifies how to identify all (or a sufficiently large class of) measuring devices with self-adjoint operators (or whatever represents them mathematically in the theory we're considering), we need a separate axiom for each type of measuring device.
This is why measurement (and the whole interpretational stuff) doesn't belong to the axioms. Imagine we'd have to start classical theoretical mechanics with a discussion of the classical measurement problem (it is not well settled - there are lots of unresolved issues in classical statistical mechanics).

Instead, one starts with a clean slate figuring a configuration space with an action, or a phase space with a Hamiltonian. Nobody cares there about how it relates to reality - the theory stands for itself though it is inspired by reality. And the examples used are heavily idealized compared to the real thing - they illustrate the math and physics but would get really complicated if one would have to discuss them in the context of reality.

Cool, I'll check it out, but not right now. I need to get some sleep. I didn't realize that you're one of the authors of that book. I downloaded it in May, but haven't gotten around to reading it yet. I don't think I will the next few months either, because I'm trying to learn functional analysis, and it takes an absurd amount of time.
The slides should be an easy read, though, and give the main idea of the thermal interpretation. However, that should be discussed in a new thread.
 

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