# Continuous eigenvalues

## Main Question or Discussion Point

Dear all,
in basic QM books the position and momentum operators (continuous eigenvectors) are introduce by means of the dirac delta and some analogies are made with the infinite dimensional, but discrete case in order to provide some intuition for the integral formulas presented. My knowledge is limited, but it seems to me that thee formulas apply in the distribution sense. Can anybody explain to me how the position and momentum operators are treated in a rigorous way? An outline will do.

Thanks for any help!

Goldbeetle

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Can anybody explain to me how the position and momentum operators are treated in a rigorous way? An outline will do.

Thanks for any help!

Goldbeetle
This is not an easy subject. The keywords to search for are "http://en.wikipedia.org/wiki/Rigged_Hilbert_space" [Broken]" and "Gelfand triples"

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Fredrik
Staff Emeritus
Gold Member
You need the RHS if you insist that these operators should have "eigenvectors", but you can also just accept that they don't, and study a spectral theorem for unbounded operators in a book on functional analysis.

Remember that infinities in physics are simply shorthands for limit procedures. Add to this a physicist's lack of need for mathematical rigour, and loose reasoning with infinities, and you should conclude that whenever such things arise, you should start looking for what that limit procedure actually is...

Fredrik: the second option you mention is what Geroch does in chapters 55-56 of his book "Mathematical Analysis"?

To everybody: thanks!

Fredrik
Staff Emeritus
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Fredrik: the second option you mention is what Geroch does in chapters 55-56 of his book "Mathematical Analysis"?
Yes, that looks like the sort of stuff I had in mind. I still don't know much of it myself, so I don't have an opinion about Geroch's presentation. My own plan for learning the rigorous treatment of both bounded and "not necessarily bounded" operators, is to study "Functional analysis: Spectral theory" by V.S. Sunder, but I've been planning to do that for a long time now.

(A comment mainly for other people reading this: Geroch's book is titled: "Mathematical physics", not "Mathematical analysis").

That's for the correction, Fredrik! By the way, the book you mention is available online for free on the author's webpage.

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George Jones
Staff Emeritus
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Another very readable exposition of functional analysis and spectral theory is given in chapters 1 - 3, 7 - 11 of Introductory Functional Analysis by Erwin Kreyzig.

For a rigourous overview of rigged Hilbert spaces (Gelfand triples) and Dirac notation, I recommend highly sections 11.2, 11.3, and 12.2 from Quantum Field Theory I: Basics in Mathematics and Physics (A Bridge Between Mathematicians and Physicists) and subsection 7.6.4 from Quantum Field Theory II: Quantum Electrodynamics (A Bridge Between Mathematicians and Physicists) by Eberhard Zeidler.

George,
thanks. I have the book Kreyzig and it's extremely readable. I'm also reading in parallel two introductions that make use also of Lebesgue integration.

I'll check out Zeidler's material because it looks very promising.

Fredrik, George: the material you suggested is wonderful! Thanks again!

George: have you read also the two books on functional analysis by Zeidler? How are they?

George Jones
Staff Emeritus
Gold Member
George: have you read also the two books on functional analysis by Zeidler?
No, I haven't looked at Zeidler's functional analysis books.

The first place you should always check for questions of this nature, in my opinion, is Landau. If you are not aware of it, Landau wrote the books on every subject in theoretical physics and his proofs are often quite rigorous and beautiful. In this case, the reference you are seeking appears only on page 15 of vol3, 3rd edition, section 5, "the continuous spectrum". In this section, all of the relevant details are discussed exhaustively.

I would dare to say that Landau does not warn the reader strong enough that formal manipulation with "eigenstates" for the continuum spectrum can lead to apparent paradoxes. In my opinion he should. Many other textbooks and course of QM are similarly guilty.

George,
Where exactly in Zeidler's QFT I 12.2 is the proof of the completeness relation for continuous eigenvalues?
Thanks a lot!
Ciao.
Goldbeetle

George Jones
Staff Emeritus
Gold Member
George,
Where exactly in Zeidler's QFT I 12.2 is the proof of the completeness relation for continuous eigenvalues?
Thanks a lot!
Ciao.
Goldbeetle
I don't have Zeidler home with me, but, if I remember correctly, Zeidler doesn't prove this. I think Zeidler gives a reference to Gelfand's nuclear spectral theorem. This is why I wrote "overview"; sorry.

Thanks. If you have good references for Geldfand's spectral theorem, please post them.

George Jones
Staff Emeritus
Gold Member
There is a nice, short overview in Chapter 8.4 of "Nonrelativistic Quantum Mechanics" by Anton Z. Capri, World Scientific (2002). Chapter 8 is about distributions and Fourier transforms.

strangerep
Goldbeetle said:
which is

"Dirac Kets, Gamow Vectors and Gel'fand Triplets"

which was a reasonable review back in 1989 when it was written.
I have a copy somewhere, but (iirc) it doesn't go into full
proofs, but merely quotes the results.

This book should contain the proof of the nuclear spectral theorem

Maurin, "GENERAL EIGENFUNCTION EXPANSIONS AND UNITARY
REPRESENTATIONS OF TOPOLOGICAL GROUPS"
but this review:http://projecteuclid.org/DPubS/Repos...ams/1183533390
is not very comforting...
I have this book too, and I must agree that it is not a well-written
book. There seem to be quite a few typos/errors. There was an
errata sheet with the book, but it also seemed to contain some errors
involving page number and/or context (iirc).

The best book is still therefore Gelfand & Vilenkin, but there's
an error in the proof of this particular theorem, and
it's essential to read this paper:

G. G. Gould (J. London Math. Soc. 43 (1968) 745-754)

But none of these sources are an easy read unless you're proficient
in functional analysis.

(BTW, note that even the proof of the spectral theorem for the "easier"
case of compact hermitian operators in infinite dimensions still
requires significant functional analysis. See, e.g., Lax's book.)

I get the feeling most physicists just content themselves with
knowing the results, never understanding the detailed proof (and
all the other functional analytic stuff that's needed to set it up).

I'm learning functional analysis. Thanks for the reference to the Gould's article.

I think quite often we use distributions in order to calculate efficiently. Then, once we know, more or less, what to expect, we can try to justify the end result by using bounded operators spectral measure techniques and the concepts form von Neumann's algebras direct integral decompositions.