Boundedness of quantum observables?

Careful

Mathematical physicists often investigate things which are not useful for physics. Give me one application which hinges upon them.

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.

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.

Right, there is no substance behind bounded operators. The best proof is that we never use them.

Careful

yoda jedi

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



not so fast...

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



me too.

then ?
contradicting yourself ?

Careful

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.

Hurkyl

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.


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.

Fredrik

Because we were talking about alternatives to the algebraic approach to QM, not about attempts to find a better theory.

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.

Careful

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.

Fredrik

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.

Careful

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

Fredrik

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.

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.

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.

dextercioby

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(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.

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.

Hurkyl

At face value, this comment seems utterly absurd.

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.

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

A. Neumaier

But this thread is about rigorous quantum mechanics - giving logical impeccable justiifications for what theoretical physicists commonly do.

Careful

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.

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, 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.

A. Neumaier

Thanks. This is a nice paper that I didn't know before. I need to read it more carefully.

Careful

What is not rigorous about unbounded operators ? Actually, I studied quantum physics rigorously from that point of view (my master education was in mathematical physics btw).

A. Neumaier

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.

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.

A. Neumaier

My sentence was a response to a different statement of yours.