What is 'Hilbert Space'?

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I hate to be 'that guy', but I've heard so-called "Hilbert Space" referenced many times. I can imagine that it's derived from physicist David Hilbert. I'd guess that you'd learn about it in an Undergrad course.
 

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  • #2
A. Neumaier
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You should have guessed that definitions can be looked up in many places, for example in wikipedia. Please do your work before asking others to work for you.
 
  • #3
dextercioby
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Actually David Hilbert was was above all a mathematician who became (like many mathematicians starting with G.W. Leibniz in the 17th century) interested in the foundations of physics, once he attended lectures by A. Einstein on relativity theory a little after 1910 (a search by Einstein to generalize special relativity, a few years before formulating general relativity). It was Hilbert who coached Einstein in tensor calculus on manifolds, who had just been developed in Italy by Gregorio Ricci-Curbastro and later by Tulio Levi-Civita at the turn of the century.
As anyone should know, the Lagrangian action and the field equations for the gravitational field were officially published by D. Hilbert a few days (8 if I recall correctly) before Einstein. His famous article contains however an erroneous interpretation of the equations derived based on some infamous ideas of Gustav Mie.
Then Hilbert turned his attention to the mathematical foundations of quantum mechanics, about 10 years later where he positively influenced the work of the great Janos (Johann/John) von Neumann.
I wouldn't call Hilbert a physicist, he's a mathematician, just like Weyl and von Neumann.

In the same vein, one notices that Arnold Neumaier (mathematician from Vienna/Austria - 2nd poster of this thread) also got very interested in the foundations of quantum mechanics and learned so much over the years, that he even constructed an interpretation of the quantum mechanics formalism more in the spirit of mathematical statistics.
 
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Hopefully you have looked it up.

But purely as a historical note check out the following:
http://www.lajpe.org/may08/09_Carlos_Madrid.pdf

Von Neumann coined the term Hilbert space and used it to develop a rigorous theory of QM as a counter to Dirac's beautiful but mathematically dubious formulation. The mathematicians were not idle though and it motivated a lot of work in functional analysis from such luminaries as Gelfland, Grothendieck and Schwartz to fix it. Dirac's formulation is now rigorous using Rigged Hilbert Spaces which are also interesting to look into:
https://www.univie.ac.at/physikwiki/images/4/43/Handout_HS.pdf

Here is the history:
http://plato.stanford.edu/entries/qt-nvd/
'A rigorous definition of the delta function became possible in distribution theory, which was developed by Schwartz from the mid-1940s to the early 1950s. Distribution theory inspired Gelfand and collaborators during the mid-to-late 1950s to formulate the notion of a rigged Hilbert space, the firm foundation for Dirac's formal framework. This development was facilitated by Grothendiek's notion of a nuclear space, which he introduced in the mid-1950s. The rigged Hilbert space formulation of quantum mechanics was then developed independently by Böhm and by Roberts in 1966.'

As is often the case in math it now has application beyond QM eg white noise theory.

Thanks
Bill
 
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he's a mathematician, just like Weyl and von Neumann.
While officially a mathematician, and one of the greatest that ever lived, likely in the top 10 of all time, von Neumann was in fact much much more. He was an even rarer beast, a polymath
http://www.thehindu.com/2000/09/21/stories/08210005.htm

He could penetrate to the heart of a problem which such frightening ease people said he was the only person fully awake. Even the great Feynman freely admitted Von Neumann was above him. That's not quite true (ie the only person fully awake) - Einstein was considered better at penetrating issues, likely he was the greatest that ever lived. Interestingly, while Einstein was a competent mathematician, he was nowhere near the class of Von Neumann. Despite that Einstein is considered greater because the ability to penetrate is much much more important.

Thanks
Bill
 
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You should have guessed that definitions can be looked up in many places, for example in wikipedia. Please do your work before asking others to work for you.
I figured that I'd get a better answer from this website, what with it being solely dedicated to physics, and being primarily filled with physics professors and students.
 
  • #7
A. Neumaier
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I figured that I'd get a better answer from this website, what with it being solely dedicated to physics, and being primarily filled with physics professors and students.
So which prior efforts did you make after my suggestion? Nothing is obtained for free, and nobody here likes to repeat stuff that you can read in many places.

On standard mathematics, wikipedia is quite good and far more thorough for a first orientation than anything you'd be explained here - just remain aware that nothing you read (whether on wikipedia or here) is guaranteed to be correct. You must do your own checking anyway.

Then when you have questions that wikipedia leaves open, or when you get inconsistent messages from different sources, its the time to come here to PF and ask. (About Hilbert spaces in the math section, about the underlying physics in the present one.)
 
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So which prior efforts did you make after my suggestion? Nothing is obtained for free, and nobody here likes to repeat stuff that you can read in many places.

On standard mathematics, wikipedia is quite good and far more thorough for a first orientation than anything you'd be explained here - just remain aware that nothing you read (whether on wikipedia or here) is guaranteed to be correct. You must do your own checking anyway.

Then when you have questions that wikipedia leaves open, or when you get inconsistent messages from different sources, its the time to come here to PF and ask. (About Hilbert spaces in the math section, about the underlying physics in the present one.)
Good to know, I'll do my research first beforehand, next time. Thanks for the advice.
 
  • #9
A. Neumaier
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I wouldn't call Hilbert a physicist, he's a mathematician
Strange. What do you think makes a person a physicist? Are a paper ''The foundations of physics'' where general relativity was derived from an action principle, and a 2-volume treatise on ''Methods of mathematical physics" not enough? Many with a Ph.D. in physics (and thus certified physicists) don't even achieve that much.
 
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  • #10
radium
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The physicist answer would be a complex vector space which is self dual and is endowed with an inner product between vectors and dual vectors. In quantum mechanics these would be kets (vector) and bras (dual vector). A ket is the complex conjugate of a bra etc.
 
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A. Neumaier
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The physicist answer would be a complex vector space which is self dual and is endowed with an inner product between vectors and dual vectors. In quantum mechanics these would be kets (vector) and bras (dual vector). A ket is the complex conjugate of a bra etc.
Dirac defined what bras and kets are. They need not be in the Hilbert space - this accounts for part of their usefulness.
 
  • #12
radium
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I just used this as an example about how you would use it in QM. The general definition is much more abstract so I thought providing an example in physics would be useful.
 
  • #13
martinbn
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Strange. What do you think makes a person a physicist? Are a paper ''The foundations of physics'' where general relativity was derived from an action principle, and a 2-volume treatise on ''Methods of mathematical physics" not enough. Many with a Ph.D. in physics (and thus certified physicists) don't even achieve that much.
A mathematician can contribute to physics.
 
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A. Neumaier
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martinbn
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Sure, and a person contributing significantly to physics is a physicist. What else should make a person a physicist?
It;s a matter of semantics, but I don't think that significant contribution to physics is neither necessary nor sufficient. Most physicists don't contribute significantly. And there are plenty of examples of mathematicians contributing significantly to physics (Newton one of them), but they are still mathematicians. Why would they be anything else?!
 
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A. Neumaier
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there are plenty of examples of mathematicians contributing significantly to physics (Newton one of them), but they are still mathematicians. Why would they be anything else?!
Why should a person be only one thing? This is very unnatural!

Would Einstein have been no physicist if he had remained at the patent office?

All people I listed were regarded by Wikipedia as physicists (among other designations).
And I consider myself as a mathematician and physicist, too.
 

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