What are the properties of Rigged Hilbert Space compared to Hilbert Space?

In summary: In particular, they give a QFT definition of a RHS in terms of a self-adjoint linear operator and a Hermitian inner product. They also mention that the RHS is a quantum system that is stable under the action of a Hamiltonian.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.
  • #1
piareround
79
0
So I was recently learned that for some square integrable position wave-functions in Hilbert Space have the momentum function is not square integrable. Thus the momentum function are not in hilbert space. However, due to "Fourier's Trick" Dirac discovered for quantum mechanics, the momentum functions behave just like if they where square integrable.

Being a curious physics student, I asked my professor more if there was a "special" space these momentum function where in even though they where not in Hilbert Space.

He told me about this thing called http://en.wikipedia.org/wiki/Rigged_Hilbert_space" , which included both the square integrable functions in Hilbert space and their related momentum functions.

I was kind of confused about what he talked about so I was kind of curious to learn more about Rigged Hilbert Space...

Questions:
1. What are the properties of Rigged Hilbert Space compared to Hilbert Space?
2. Are Rigged Hilbert Spaces also Approximate Hilbert Space? Is a Rigged Hilbert Space what we use in Quantum Field Theory when we are talking about the quantum physics of a particle decay state that are not stable states; a problem that where we use use a Approximate Hilbert Space?
3. Are their any books on Rigged Hilbert Space if I wanted to learn more?
 
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  • #3
arkajad said:
Umm...but thanks for my question really was about mathematics linear algebra not philosophy.

So any books about math and physics of Rigged Hilbert Spaces would be nice.
 
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  • #4
Sorry, I've fixed the link.
 
  • #5
arkajad said:
Sorry, I've fixed the link.

lol. So you meant this thread:
https://www.physicsforums.com/showthread.php?t=428384

George Jones said:
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.
Thanks ^_^ ^_^
 
  • #6
There is a introductory paper on rigged Hilbert spaces and Gelfand triples: http://www.abhidg.net/RHSclassreport.pdf
 
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  • #7
tom.stoer said:
There is a introductory paper on rigged Hilbert spaces and Gelfand triples: http://www.abhidg.net/RHSclassreport.pdf

The (Riesz) Theorem in the Appendix is formulated in a way that can be misleading. It says:

"... all linear functionals have the form

[tex]F(\phi)=(f,\phi)[/tex]

where [tex]f[/tex] is a fixed vector and [tex]\phi[/tex] an arbitrary vector. "

The word "fixed" can be misleading. It should say: For every [tex]F[/tex] there is a unique [tex]f[/tex] such that for all [tex]\phi[/tex]

[tex]F(\phi)=(f,\phi).[/tex]

One may add: the one-to-one correspondence between functionals F and representing them vectors f is anti-linear.There is also a lack of consistency between Theorem 1 and Theorem 2 - they use different conventions as regards the scalar product. One is the complex conjugate of the other. (That is the result of my quick perusal of the easy part.)
 
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  • #8
I would advise anyone to look on the internet for Rafael de la Madrid's PhD thesis at the Univ of Austin to get a pretty thorough investigation of the Gelfand triples/RHS.

A use of RHS in quantum field theory is found in Bogolubov, Logunov and Todorov's 1975 text on axiomatical QFT (the 1975 edition in English is an enhanced version of the Russian 1969 one).
 

Related to What are the properties of Rigged Hilbert Space compared to Hilbert Space?

What is a Rigged Hilbert Space?

A Rigged Hilbert Space is a mathematical concept that extends the traditional Hilbert Space by adding a "rigging" or "test" function space to better describe certain phenomena in quantum mechanics. It is also known as a Gelfand triple.

How is a Rigged Hilbert Space different from a Hilbert Space?

A Rigged Hilbert Space differs from a Hilbert Space in that it includes an additional space of "test" or "rigging" functions, which allows for a more complete description of the behavior of quantum systems. This added layer of complexity is necessary for describing certain phenomena in quantum mechanics that cannot be fully captured by a traditional Hilbert Space.

What are the advantages of using a Rigged Hilbert Space?

The main advantage of using a Rigged Hilbert Space is its ability to provide a more complete and accurate description of certain quantum phenomena. It allows for a wider range of functions to be used to describe the behavior of quantum systems, leading to a more comprehensive understanding of these complex systems.

What are some applications of Rigged Hilbert Space in science?

Rigged Hilbert Space has various applications in physics, particularly in quantum mechanics. It is commonly used in the study of quantum field theory, scattering theory, and the spectral theory of operators. It has also been applied in other fields such as signal processing and control theory.

Are there any limitations to using Rigged Hilbert Space?

While Rigged Hilbert Space has many advantages, it also has its limitations. One of the main limitations is the increased complexity and mathematical rigor required for its use. It may also not be necessary for simpler systems that can be adequately described using a traditional Hilbert Space. Additionally, it may not be applicable in certain situations where the concept of rigging is not relevant.

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