About the rigged Hilbert space in QM

[pabloweigandt]

In Quantum Mechanics, how can you justify the use of distributions like the delta functional without introducing a rigged Hilbert space? I see that some texts do not make any reference to this subject.

 

[anuttarasammyak]

Rigged Hilbert space assures us to introduce basis vectors for continuous physical variables, e.g. |x>,|p> as Dirac did by his physical insight. Knowing that I have a confidence to keep Dirac's way in spite of a long history criticism of mathematicians who wants to limit discussions inside Hilbert space.

 

[pabloweigandt]

What I mean is why some authors avoid mentioning or treat rigged Hibert space when they suppose to be formulating QM in a rigorous mathematical context.

 

[strangerep]

What I mean is why some authors avoid mentioning or treat rigged Hibert space when they suppose to be formulating QM in a rigorous mathematical context.

Ballentine talks about this in his sections 1.3 and 1.4. In summary (p28):

We now have two mathematically sound solutions to the problem that a self-adjoint operator need not possesses a complete set of eigenvectors in the Hilbert space of vectors with finite norms. The first, based on the spectral theorem (Theorem 4 of Sec. 1.3), is to restate our equations in terms of projection operators which are well defined in Hilbert space, even if they cannot be expressed as sums of outer products of eigenvectors in Hilbert space. The second, based on the generalized spectral theorem, is to enlarge our mathematical framework from Hilbert space to rigged Hilbert space, in which a complete set of eigenvectors (of possibly infinite norm) is guaranteed to exist. The first approach has been most popular among mathematical physicists in the past, but the second is likely to grow in popularity because it permits full use of Dirac’s bra and ket formalism.

 

[anuttarasammyak]

According to P.4 of PhD thesis I referred in another thread R. de la Madrid, "Quantum Mechanics in Rigged Hilbert Space Language," PhD Thesis (2001) referred in Wikipedia https://en.wikipedia.org/wiki/Rigged_Hilbert_space

Nowadays the RHS is textbook material [12, 13, 14, 15, 16, 17, 18].
[12] D. Atkinson, P. W. Johnson, Quantum Field Theory, Rinton Press, Princeton (2002).
[13] N. N. Bogolubov, A. A. Logunov, I. T. Todorov, Introduction to Axiomatic Quantum Field Theory, Benjamin, Reading, Massachusetts (1975).
[14] L. E. Ballentine, Quantum Mechanics, Prentice-Hall International, Inc., Englewood Cliffs, New Jersey (1990).[15] A. Bohm, Quantum Mechanics: Foundations and Applications, Springer-Verlag, New York (1994).
[16] A. Z. Capri, Nonrelativistic Quantum Mechanics, Benjamin, Menlo Park, California (1985). [17] D. A. Dubin, M. A. Hennings, Quantum Mechanics, Algebras and Distributions, Longman, Harlow (1990).
[18] A. Galindo, P. Pascual, Mec´anica Cu´antica, Universidad-Manuales, Eudema (1989); English translation by J. D. Garc´ıa and L. Alvarez-Gaum´e, Springer-Verlag (1990).

I assume after this paper of more than two decades ago .many more modern texts refer RHS.

 

[pabloweigandt]

Thanks. I will review in detail all the comments and references that you posted.

 

[Demystifier]

What I mean is why some authors avoid mentioning or treat rigged Hibert space when they suppose to be formulating QM in a rigorous mathematical context.

In short, you can do QM rigorously without rigged Hilbert space if you don't use Dirac's formalism.

 

[martinbn]

What I mean is why some authors avoid mentioning or treat rigged Hibert space when they suppose to be formulating QM in a rigorous mathematical context.

Which authors do you have in mind? Also you don't have to mention rigged Hilbert spaces to use them!

 

[pabloweigandt]

That's right. But for example, if you mention the Hilbert space, and then you use the Dirac delta distribution, and you don't mention the rigged Hilbert space, I think that something is missing, and you need to justified things like in Ballentine or any other means. Of course, you can talk about "the space of kets" or just use the "wave function" and don't even mention the Hilbert space.

 

[vanhees71]

A rigoros trwatment of non-relativstic QM is much less convenient than the modern formulation using the notion of a "rigged Hilbert space". It's of course just two equivalent formulations of the same theory.

 

[pabloweigandt]

How is that "rigorous treatment" without rigged Hilbert space?

 

[mattt]

How is that "rigorous treatment" without rigged Hilbert space?

Just Functional Analysis, as usually done by mathematicians.

See for example the book "Fundamental Mathematical Structures of Quantum Theory" by Valter Moretti.

 

[George Jones]

How is that "rigorous treatment" without rigged Hilbert space?

Just Functional Analysis, as usually done by mathematicians.

See for example the book "Fundamental Mathematical Structures of Quantum Theory" by Valter Moretti.

See also "Methods of Modern Mathematical Physics: Functional Analysis" by Reed and Simon, and the beautiful book by Brian Hall. From another thread:

Even though rigged Hilbert spaces make rigourous the physicists' version of the maths used in quantum mechanics, it is not strictly necessary to move out of Hilbert space. Reed and Simon advocate remaining in Hilbert space. From Reed and Simon: "There has arisen an extensive literature on a 'rigorous' Dirac notation which attempts to capture the flavour of bra and kets more fully. ... We must emphasize that we regard the spectral theorem as sufficient for any argument where a nonrigorous approach might rely on Dirac notation; thus, we only recommend the rigged space approach to readers with a strong emotional attachment to the Dirac formalism."

If you want to study the Hilbert space approach (as opposed to the rigged Hilbert space approach), then I recommend the beautiful "Quantum Theory for Mathematicians" by Brian Hall over Reed and Simon, even though I like Reed and Simon. Hall is a really wonderful book, but it has "the basics of L^2 spaces and Hilbert spaces" as prerequisites. Most of the functional analysis presented in the book is in chapters 6 - 10, through which the author gives several paths "I have tried to design this section of the book in such a way that a reader can take in as much or as little of the mathematical details as desired."

 

[mattt]

Yes, the book of Brian Hall, the three ones of Valter Moretti, the four volumes of Red and Simon and the numerous books of Eberhard Zeidler are an endless fountain of pleasure for me...

And I must confess that even though I was absolutely obsessed with "mathematical rigour" when I was younger, now I am much more open to the "physicists way of reasoning" when it is necessary or simply useful (and more so because many times it is unavoidable).

 

[pabloweigandt]

Just above, in chapter six of Hall it says "Thus, the notion of a direct integral gives a rigorous meaning to the notion of “eigenvectors” that are not actually in the Hilbert space".
You need to recognize "eigenvectors" outside the Hilbert space anyway. Are these "generalized eigenvectors" a more rigorous treatment than the rigged Hilbert space? Am I missing something? (Thanks for all the references).

 

[mattt]

He just shows you that you obtain the exact same results with the Spectral Theorem for Unbounded Operators ( in a precise and rigorous mathematical way ) without any need of talking about a "continuous basis" of generalized eigenvectors nor the "reasoning" of the bra-ket type.

But if you understand the mathematical concepts involved and the mathematical proofs you can "recognize" that you end up with the very same end result.

 

[Demystifier]

the three ones of Walter Moretti

One of them, "From Classical Mechanics to Quantum Field Theory", has a totally misleading title. In this book classical mechanics is not treated at all, QFT is treated only at an elementary level, while most of the book is really about mathematical formulation of QM.
https://www.amazon.com/dp/B084GMNHCQ/?tag=pfamazon01-20

 

[mattt]

One of them, "From Classical Mechanics to Quantum Field Theory", has a totally misleading title. In this book classical mechanics is not treated at all, QFT is treated only at an elementary level, while most of the book is really about mathematical formulation of QM.
https://www.amazon.com/dp/B084GMNHCQ/?tag=pfamazon01-20

Yes, you're right, he only briefly treats Classical Mechanics in section 2.3.1

By the way, I just noticed that the voice dictation system of my mobile phone didn't spell correctly some of the names (I cannot edit my previous post to correct it, I don't know why).

The correct names are Eberhard Zeidler and Valter Moretti.

 

[berkeman]

By the way, I just noticed that the voice dictation system of my mobile phone didn't spell correctly some of the names (I cannot edit my previous post to correct it, I don't know why).

The correct names are Eberhard Zeidler and Valter Moretti.

(I fixed them for you. Your edit time probably expired.)

 

[dextercioby]

I am a little late for the party, but here goes.

„There is simply too much mathematics needed to set a rigged Hilbert space properly and for this reason there are only sketches in various QM texts (see the texts quoted above in RdM bibliography to PhD thesis). They only mention them for completion, and just to make the assertion "they provide a mathematical foundation to the bra-ket formalism" stand on something. So that is why we end up having Moretti or Hall, or Teschl not discussing them at all, because they should go at way too big a length to make them useful for university level physics" (or it is maybe they stick to the word of the "bible" by Reed & Simon).

But I like them and cannot imagine any quantum theory without them.

 

[GiovanniGh]

In Quantum Mechanics, how can you justify the use of distributions like the delta functional without introducing a rigged Hilbert space? I see that some texts do not make any reference to this subject.

No way. You need the RHS (Rigged Hilbert Space). Physical states belong to HS (Hilbert Space) only. Continuous states too. But the way to build those states passes through the dual of a nuclear space. Packets of vectors belonging to the dual of a nuclear space generate a HS vector; this is the foundamental result. E.g. packets of delta over a finite interval of continuous spectra generate the rectangular function, and this function belong to HS, of course

 

[GiovanniGh]

What I mean is why some authors avoid mentioning or treat rigged Hibert space when they suppose to be formulating QM in a rigorous mathematical context.

yes, you are right, and the rason us that it takes too long. RHS theory is very heavy: you need topological vector space theory, nuclear space theory, abstract kernel theorem, special vector measure theories, and so on.
From what I know the only complete reference is the book

General eigenfunction expansions and unitary representations of topological groups / Krzysztof Maurin​

 

[vanhees71]

The rigged-Hilbert-space formulation is just the modern version and to be preferred for its versatility and simplicity. It's just making the hand-waving physicists' math rigorous. There's of course also the old-fashioned rigorous version using just the usual separable Hilbert space. That was worked out in the early 30ies by von Neumann (see his famous textbook from that time, but be aware that the physics part is partially flawed, particularly the no-go theorem for hidden variables, which is circular, as already realized by Grete Hermann also in the 1930ies).

 

[GiovanniGh]

RHS and Von Neumann Theory are not equialent theories. RHS is a "continuation" of Von Neumann Theory.
Von Neumann Theory simply says that to each self adjoint operator one and only one spectral measure exsist, but it doesn't say neither a word on how this spectral measure is made and how to find it. Actually, this is done by RHS theory. So without RHS theory, we wouldn't have any practical and concrete method to go on beyond the resoltion spectral theorem of Von Neumann

 

[LittleSchwinger]

but be aware that the physics part is partially flawed, particularly the no-go theorem for hidden variables, which is circular, as already realized by Grete Hermann also in the 1930ies

This might be a difference in terminology, but von Neumann's theorem is not circular, i.e. it does not assume its conclusion. It does remove a certain "natural" class of classical theories as a model for QM.

 

[vanhees71]

Grete Hermann argues that von Neumann's assumption that $\langle A+B \rangle=\langle A \rangle + \langle B \rangle$ to be valid for the expecation values of any observables contains already the assumption that all there is to characterize the state of the system is the "wave function" (I'd rather say "the quantum state"), which was what he wanted to prove.

 

[LittleSchwinger]

Grete Hermann argues that von Neumann's assumption that $\langle A+B \rangle=\langle A \rangle + \langle B \rangle$ to be valid for the expecation values of any observables contains already the assumption that all there is to characterize the state of the system is the "wave function" (I'd rather say "the quantum state"), which was what he wanted to prove.

I know her argument, but I disagree with it.

So just some notation. $A$ refers to some observable we measure in a lab. $\hat{A}$ refers to the operator corresponding to that observable in quantum theory.

Basically von Neumann's proof consists of saying that for two Observables $A$ and $B$, we can define an observable $S$ called the "statistical sum". This observable is the one for which:
$<S> = <A> + <B>$
for all preparations. This is purely an operational definition.

His theorem then proves that if the values of $S$ are the same as the eigenvalues of $\hat{A} + \hat{B}$, then there are no states without dispersion for some observables, i.e. some observables in your theory will always have a non-trivial probability distribution and ultimately your theory will be quantum theory.

Strictly speaking what this rules out is that a classical theory "beneath" quantum mechanics cannot involve solely the quantities spoken about in quantum theory. Or in simpler words a realist theory could not merely be another classical theory involving position, momentum, energy, etc. It would have to involve new quantities that we don't directly measure, i.e. "hidden" variables.

Now many argue this doesn't actually prove that there are no realist formulations, but von Neumann didn't really set out to do that. He just intended to show that such theories involve parameters with no empirical support and that was more than enough to set them aside for scientific purposes.