Insights Hilbert Spaces and Their Relatives - Definitions

[fresh_42]

Language first: There is no such thing as the Hilbert space.

Hilbert spaces can look rather different, and which one is used in certain cases is by no means self-evident. To refer to Hilbert spaces by a definite article is like saying the moon when talking about Jupiter, or the car on an automotive fair. The is either related to a unique subject or a context-sensitive certain one if there is no doubt about which one. The least should be a notation like e.g. Hilbert space of wave functions, of square-integrable functions, of Hilbert-Schmidt operators, or any other example, although even these neglect crucial information to some extend.

Hilbert spaces on the other hand are inevitably interwoven with quantum physics, be it the classical or the relativistic formalism. Whereas Gauß law and Maxwell's equations could be viewed as expressions in the language of differential geometry, Schrödinger's equations brought us wave functions and Hilbert spaces. I like to point out the interesting observation of the timely parallels between mathematical and physical developments, which led to all of them. It is a fascinating give and take which did not start with and continues up today in realms as gauge theories, Lie theory, representation theory, string theories or homological algebra, and the topological questions in cosmology.

Continue reading ...

 

[Mark44]

Nice job, fresh!

 

[bhobba]

Yes indeed a very nice job. Very important those that wish to progress in QM understand it, plus of course it has many other applications as well. When I learned it at uni my teacher said he could spend a 2 semester course on just the applications alone and still just scratch the surface. Thanks for being careful and mentioning its only isomorphic to its continuous dual - not its dual - I keep forgetting that one when explaining it to others which I do via the concept of Rigged Hilbert Spaces. If I remember correct its the same as demanding its bounded. Nice for people reading Ballentine QM - A Modern Approach. He gives his own proof of the Fréchet-Riesz Representation Theorem (page 10) but as he says it ignores convergence issues - in other words its wrong - but I will let others sort that one out (he is not careful with some manipulations he does on infinite series). I remember when first reading Ballentine all those years ago I thought naughty, naughty.

For those that do not know the link to Rigged Hilbert Spaces see:
https://www.univie.ac.at/physikwiki/images/4/43/Handout_HS.pdf

As I said I keep forgetting the continuous bit when I explain it - the above corrects it. Oh and the test space must be dense as well. Damn I am getting sloppy in my old age .

Thanks
Bill