# Mathematical basis of QM

## Main Question or Discussion Point

Someone posted this on another forum, and, not knowing enough about it to supply a satisfying answer, I figured I'd ask you guys.

I've never studied physics, but I've read a bit of basic stuff on quantum mechanics and chatted to a few physicists about it and the following question has always bugged me.

Apparently in the standard formalism for quantum mechanics you represent (pure) states as elements of a Hilbert space (or rank one projections on a Hilbert space, I guess, but whatever) and observables as densely defined Hermitian operators on this space. In the finite dimensional case every Hermitian operator has a spanning eigenspace and measuring a state p with respect to some observable A just collapses that state to some eigenstate of A, where the probability of getting any given eigenstate is just the inner product of that eigenstate with p.

However, I'm very hazy as to what happens in the infinite dimensional case. I've seen various sources pretend that it's basically the same as the finite dimensional case but this can't be right as Hermitian operators on infinite dimensional Hilbert spaces need not have any eigenvectors at all. A basic example of this is the position operator acting on L^2(R), the square integrable functions on the reals. This has no eigenvectors, so how does measuring position work? Apparently in the C* formalism you represent pure states as measures and so I guess <vague bullgarbageting> you could determine whether the position of some state lies in a particular range by integrating a suitable function over that range with respect to the state </vague bullgarbageting>. However, surely you don't need anything as abstract as this to define something as simple as measuring position, and even if you do use this formalism I have no idea what happens to the state of the particle as a result of performing this measurement.

Any explanation of this would be greatly appreciated.

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martinbn
You are right, the infinite dimensional case is more difficult and requires more care. Unfortunately in almost all books that is not covered as to save space, time, make things easier to see the general ideas and so on. Often physicists don't worry about these things. I cannot direct you to a suitable source and it is frustrating that it is very hard to find mathematically satisfactory expositions. You could try to search for Rigged Hilbert spaces. It is also possible that these matters are treated in the Reed-Simon four volume set. I know it is bad to give vague references and point to a few thousands of pages to search in, but as I said nothing comes to mind right now.

A. Neumaier
2019 Award
You are right, the infinite dimensional case is more difficult and requires more care. Unfortunately in almost all books that is not covered as to save space, time, make things easier to see the general ideas and so on.
It's not covered in any book - because there is no satisfactory theory for it. Yes, there are generalized eigenvectors for position in rigged Hilbert space, but that doesn't help the least in the position measurement problem.

What one usually says is that position cannot be measured exactly anyway, so instead you measure presence in some tiny box. This yields a well-defined projection operator. But of course, given a real position measurement, nobody can tell you which box was measured - everything remains very fuzzy.

But the explanation is appeasing enough that most people don't inquire further. The few who do come up with POVMs http://en.wikipedia.org/wiki/POVM for approximate position measurements - but this is already far away from the textbook dogma.

We are currently discussing some related things at

martinbn
A. Neumaier, that may be the case, but the issue is not even mentioned in the books. I mean not even a remark that there is something to be careful about. This of course may be simply because I haven't looked at enough books.

dextercioby
Homework Helper
A. Neumaier, that may be the case, but the issue is not even mentioned in the books. I mean not even a remark that there is something to be careful about. This of course may be simply because I haven't looked at enough books.
Indeed, the combination of the tricky measurement problem for observables with continuous spectrum and the rigorous mathematical formulation in terms of rigged Hilbert spaces is not addressed in textbooks, perhaps in some articles (that I don't know about).

Due to the nature of the continuous spectrum and its parametrization to a subset of R, we can't speak of <the probability to find the electron in point x of the real axis>, but rather of <the probability to find the electron in the closed interval [x_1,x_2] of the real axis> and here we may use a modified version of Born's rule for observables with (partly) continuous spectra.

A. Neumaier
2019 Award
A. Neumaier, that may be the case, but the issue is not even mentioned in the books. I mean not even a remark that there is something to be careful about. This of course may be simply because I haven't looked at enough books.
The issue is not mentioned since the whole initial discussion of probabilities is purely didactical and so far removed from actual measurement (except for some very simple standard examples) that it is completely irrelevant for most of quantum physics.

This is also born out by the fact that the foundational issues haven't been resolved after more than 80 years of QM. If it would matter, there would be agreement as in most other scientific matters that matter.

martinbn
A. Neumaier, I don't think that there is a problem (mathematically) here. It seem to me that it is a matter of style of exposition and preference of focus.

A. Neumaier
2019 Award
A. Neumaier, I don't think that there is a problem (mathematically) here. It seem to me that it is a matter of style of exposition and preference of focus.
I know. Given the present obscure state of the foundations of quantum mechanics, it is good style to concentrate on the essence of QM (which is in the ''shut up and calculate'' part) and to be short on (i.e., mostly avoid the messy and controversial part about) measurement. The latter can be delegated to a later course on real measurement - where things are discussed in a very different way than in introductory textbooks.

cgk
In this context one must also mention that physics is primarily an empirial science, after all. And so far predicting measurements has worked fine without an extensive functional analytic treatment, not in the least because in pretty much every case where you actually want to predict something which can be measured, it does essentially boil down to the finite-dimensional case. It is quite possible that the real physics lies in the handwaving rules emerging from the treat everything as if it was finite dimensional[1]"-assumption, and not in a proper functional analytic treatment of riggend hilbert spaces, positive measures, distributions, convergence measures, C* algebras and so on!

[1] or at most countably infinite and sufficently mathematically friendly

martinbn
cgk, are there Hilbert spaces used in QM, which are not at most countably dimensional?

A. Neumaier
2019 Award
cgk, are there Hilbert spaces used in QM, which are not at most countably dimensional?
Yes. While in earlier times, nonseparable Hilbert spaces were considered to be unphysical curiosities, they were shown more recently to be unavoidable in some cases of great physical interest: All infrared problems in quantum field theory are related to nonseparable spaces defined by nonregular representations of Weyl groups - as described e.g. in
Acerbi, F. and Morchio, G. and Strocchi, F.,
Infrared singular fields and nonregular representations of canonical commutation relation algebras,
J. Math. Phys. 34 (1993), 889.

martinbn
Yes. While in earlier times, nonseparable Hilbert spaces were considered to be unphysical curiosities, they were shown more recently to be unavoidable in some cases of great physical interest: All infrared problems in quantum field theory are related to nonseparable spaces defined by nonregular representations of Weyl groups - as described e.g. in
Acerbi, F. and Morchio, G. and Strocchi, F.,
Infrared singular fields and nonregular representations of canonical commutation relation algebras,
J. Math. Phys. 34 (1993), 889.
Interesting! I'll take a look.

A. Neumaier
2019 Award
the issue is not even mentioned in the books. I mean not even a remark that there is something to be careful about. This of course may be simply because I haven't looked at enough books.
There is some readable discussion about position measurement in the book by
A. Peres, Quantum Theory: Concepts and methods (look it up in the index).

Re: Mathematical basis of QM

Someone posted this on another forum, and, not knowing enough about it to supply a satisfying answer, I figured I'd ask you guys.
QM can be settled on a pure algebraic ground, no Hilbert space necessary.

strangerep
QM can be settled on a pure algebraic ground, no Hilbert space necessary.
In the "Boundedness of quantum observables" thread, i.e.,

specifically, starting at my post #125 therein, but more especially post #133
and following posts, I raised the question of how one can derive
the usual SO(3) spectrum for (intrinsic) angular momentum in
that Hilbert spaces are indeed necessary, even though the algebra
takes center-stage.

If you still stand by your statement above, then please show me how
to derive said spectrum without using a Hilbert space anywhere.
(But please do so in the other thread I mentioned above, since this
is a bit off-topic here.)

Leslie Ballentine has a quantum book I've read parts of that considers some of the foundational issues. In particular he spends a fair amount of time on Rigged Hilbert space and has a number of examples/exercises on this topic. He has also written a number of papers (referenced in the text on those moments when the mathematics makes a difference. Though you can google scholar him if you're curious.

https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

Last edited by a moderator:
martinbn
In the "Boundedness of quantum observables" thread, i.e.,

specifically, starting at my post #125 therein, but more especially post #133
and following posts, I raised the question of how one can derive
the usual SO(3) spectrum for (intrinsic) angular momentum in
that Hilbert spaces are indeed necessary, even though the algebra
takes center-stage.

If you still stand by your statement above, then please show me how
to derive said spectrum without using a Hilbert space anywhere.
(But please do so in the other thread I mentioned above, since this
is a bit off-topic here.)
Usually when people say algebraic in this context it means the use of C*-algebras. Which is misleading, because this belongs to analysis (mathematical). If some one reads just a brief popular text he may get the wrong impression.

A. Neumaier
2019 Award
Usually when people say algebraic in this context it means the use of C*-algebras. Which is misleading, because this belongs to analysis (mathematical). If some one reads just a brief popular text he may get the wrong impression.
The boundaries between algebra and analysis are at times fuzzy, such as here. Usually one refers to something as algebraic if there is no need to directly look at objects in infinite-dimensional spaces as functions - even though limits and hence topology play a role. (They play a role even in number theory - consider what is done with p-adic numbers.)

In the above case with SO(3), the Hilbert spaces are needed to define unitary representations. But all you do with them is purely algebraic!

martinbn