Is the phrase "discrete spectrum" in rule #5 a contradiction in terms?

[BohmianRealist]

When conversing informally about QM, there is often wonder about the apparently acausal nature of processes that we may call "quantum leaps" between physical states. It is often said that the purely mathematical foundations of QM give no reason for such wonderment, i.e., that the math, in itself, simply forces this state of affairs upon us, and that all interpretive questions are just the legacy of the all-too-human tendency to ask overly philosophical questions concerning the ultimate nature of our shared experience that we call "reality".

Before I dive into the topic at hand, I want to thank the members of this forum who took the time to distill the vast area of human knowledge called quantum mechanics into 7 Basic Rules.

In terms of those rules, everything seems to be remain comfortably within the mental framework that we may call "classical logic" for the first four of them, and for the first sentence of the fifth. Rule #4 speaks of an operator "with real spectrum" acting on the Hilbert space. All thinking is situated entirely within the domain of mathematics up to here, at which point the physically significant notion of "measurement" is introduced (in rule #5) by speaking of the resulting values of the application of the operator. But only the case of a supposed "discrete spectrum" is considered, and the two remaining rules follow from this case.

The obvious question that is begged is, "What about the case of a continuous spectrum?" But an even deeper mystery is the conceptual leap that is necessary in thinking that there is some sense whereby something like a "spectrum" can even exist without being placed wholly within the realm of things that are characterized by the quality of continuity. Taking this tack, we may replace the word "spectrum" with the phrase "continuous range of values", and then wind up with the second sentence of rule #5 saying, "In the case of a discrete continuous range of values...".

The mind halts here. It doesn't simply "wonder" at the mystery of a new kind of science: it just cannot continue this train of thought, because the kind of thoughtful consideration that we call "logic" has terminated. And given that all sciences are merely particular applications of logic, what possibilities are left when thinking about how to proceed in the development of the "inner workings" of foundational (or non-interpretive) QM?

 

[PeterDonis]

only the case of a supposed "discrete spectrum" is considered, and the two remaining rules follow from this case

Only the case of a discrete spectrum (not "supposed"--the concept is perfectly well-defined) is explicitly discussed. That's because that is the easiest case for most people to grasp conceptually, and doesn't involve the technical complications that come into play for operators with continuous spectra.

The obvious question that is begged is, "What about the case of a continuous spectrum?"

That is what QM textbooks are for. The Insights article you reference is certainly not intended to substitute for a QM textbook or to give a comprehensive treatment of QM.

an even deeper mystery is the conceptual leap that is necessary in thinking that there is some sense whereby something like a "spectrum" can even exist without being placed wholly within the realm of things that are characterized by the quality of continuity. Taking this tack, we may replace the word "spectrum" with the phrase "continuous range of values", and then wind up with the second sentence of rule #5 saying, "In the case of a discrete continuous range of values...".

This is all nonsense. The concept of a "discrete spectrum" in QM has a perfectly well-defined meaning. If you don't like the word "spectrum" in this connection, just substitute something like "family of eigenvectors and eigenvalues". The physics will be the same.

The mind halts here. It doesn't simply "wonder" at the mystery of a new kind of science: it just cannot continue this train of thought, because the kind of thoughtful consideration that we call "logic" has terminated.

This is not a problem with QM. It's a problem with your erroneous understanding.

 

[weirdoguy]

Taking this tack, we may replace the word "spectrum" with the phrase "continuous range of values"

Googling up what "spectrum" means would have saved you a lot of time:
https://en.m.wikipedia.org/wiki/Spectrum_(functional_analysis)

 

[BohmianRealist]

To all, I grant that it may be true that "I just don't understand QM"!

Well, in my defense, I think I only need to quote the distinguished Professor Feynman who once said something to the effect of, "Nobody understands QM". (Probably the second most quoted phrase concerning QM, just behind you-know-who! )

It would be quite easy to start getting into fine-grained arguments about the precise meanings of the natural language terms that we use when setting up our more rigorous arguments, whether they be purely mathematical or involving statements that predict the outcomes of scientific experiments. And I do think it might be interesting to get into discussions over the "legitimate" use of the word "spectrum", and whether it has been potentially misused so as to allow for the existence of a dizzying array of interpretations in regard to the ultimate nature of physical reality. But it would not be honest of me to do so, because I have quite concrete ideas that I have been playing around with for over a decade in terms of the application of operators involving continuous spectra to the Hilbert space.

To Peter, I thank you again for the thoughtful response! Since you are the one that wields the administrative power here, I now want to address you directly to ensure you that I intend to tread lightly in terms of the question of overly speculative argumentation. My issue is that there are things that can be done in the realm of pure mathematics to the Schrödinger picture, that are not at all advanced, and in fact they are trivial to explain. I believe even bright third-graders would be able to get the basic idea. But I will not attempt to do so here out of abundance of respect for the rules of the forum.

Back to everyone, what I hope to do in this thread, is to try to honestly start thinking through what kinds of possibilities that the actual mathematical language opens up for us. In particular, my thinking lies along the lines that the concept of an operator on a Hilbert space depends crucially on that of "transformation".

From: https://en.wikipedia.org/wiki/Hilbert_space

Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.

My concern with the usage of operators involving discrete spectra (which can also loosely be called "eigenthings") is that the intended meaning of "transformation" simply no longer holds when they are applied to the Hilbert space. In other words, I see that the "stretching" that must be done is so extreme that the result is not one involving "stretched out" (or "bunched up") local distortions vis-a-vis the original picture (such as what we see when looking at ourselves in funhouse mirrors), but rather a hopelessly "broken" (or "shattered") one. Interpreters of the act of applying this type of operator must therefore resort to the particularly violent notion of "collapse". While I grant there may be some sense to the idea that this case (the discrete one) is the "simplest" one to convey to novices, it is also, in my opinion, the precise reason why the good professor quoted above said what he said.

So now, my situation is this. Like I stated in the original post, the QM literature is indeed vast, which is why the public depends on the kinds of services, such as "7 Basic Rules", provided here at PF. I've spent quite enough time perusing the available literature to be pretty darn sure that there is nothing, of which I am reasonably aware, that approaches close enough to my own thinking that enables me to cite an acceptable publication, and therefore have the ability to hash out precisely what I have in mind. Of course, I am also aware that trying to get into a journal is always an available option, but the politics (or whatever you call it) involved in that approach is very much a rabbit hole that I am simply not currently willing to entertain.

I want to wrap this up by reiterating the importance of the place of pure mathematics that I see in all of this. It is the one discipline that we have available that on the one hand allows us to be amazingly productive, and that on the other requires no outlay of capital other than what is given to us by our very nature; or: the proof is in the "pudding" that is the human mind. Up to the point of "collapse", all of QM is contained within the realm of this kind of purely constructive thinking, of the kind that Schrödinger himself was engaged in while developing the precise form that his wave picture was to ultimately take.

In light of the reasoning I have laid out here, I want to rephrase the question at the end of the opening post in a slightly more pointed (and perhaps more hopeful) manner: What new possibilities are open to us when we remain faithful to the purely mathematical notion of transformations that stretch the space* within which the Schrödinger picture is given to us?

* The phrase quoted above, taken from the Wikipedia entry for "Hilbert space"

 

[PeterDonis]

It would be quite easy to start getting into fine-grained arguments about the precise meanings of the natural language terms that we use when setting up our more rigorous arguments

Such an argument would be irrelevant to this thread, since you have not offered any rigorous arguments.

I want to wrap this up by reiterating the importance of the place of pure mathematics that I see in all of this.

If you think pure mathematics is so important, perhaps you should try framing a question using mathematics instead of vague ordinary language.

 

[PeterDonis]

I've spent quite enough time perusing the available literature to be pretty darn sure that there is nothing, of which I am reasonably aware, that approaches close enough to my own thinking that enables me to cite an acceptable publication, and therefore have the ability to hash out precisely what I have in mind.

And based on that statement, this thread is closed.