# QM mathematical structure

## Main Question or Discussion Point

Hello!

I have noticed that most advanced textbooks on QM start the development of the subject with a long review of linear algebra. In particular, they talk about pre-Banach, Banach, pre-Hilbert, Hilbert spaces and so on. Why is it necessary to invoke such abstract spaces in order to describe the physical reality? I mean, for example, why do you need that every Cauchy sequence converges within the space to have something physically meaningful?

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strangerep
I have noticed that most advanced textbooks on QM start the development of the subject with a long review of linear algebra. In particular, they talk about pre-Banach, Banach, pre-Hilbert, Hilbert spaces and so on. Why is it necessary to invoke such abstract spaces in order to describe the physical reality? I mean, for example, why do you need that every Cauchy sequence converges within the space to have something physically meaningful?
This is a bit like asking why we use the real numbers in physics,
instead of just using rationals. (The reals are defined via Cauchy
sequences of rationals.)

The short answer is that building models of physics is often
made easier if we accept the help that differential/integral
calculus techniques offer.

To benefit from calculus-like techniques in quantum physics,
one needs more advanced mathematical machinery like
the things you mentioned.

Fredrik
Staff Emeritus
Gold Member
I was thinking what Strangerep said. If we use a vector space that isn't a Hilbert space, we wouldn't be able to use the mathematical theorems about Hilbert spaces. For example, the theorem that says that given an arbitrary vector x and a subspace V, theres's a unique way to express x as the sum of a vector in V and a vector that's orthogonal to V. The proof of that uses the fact that Cauchy sequences are convergent, so I don't expect it to be valid for general vector spaces.

Why is it necessary to invoke such abstract spaces in order to describe the physical reality?
Why does "physical reality" match ANY (man made) mathematics?? I think we lucked out....so far.

luisgml_2000 -> The reason is that "physical reality" cannot be fully captured by "common" calculus (on manifolds). You need something more. And this something more turns out to be the mathematics of Hilbert spaces. (Functional analysis in general, to be more precise.) And thinking of quantum gravity, we might even need something more.

Another reason is the following. Most of the physicists involved in (phenomenological) exploration of "physical reality" use some "mathematical method" to do so. And these methods frequently have the taste of a "recipe". That is, "if you find this and that equation apply this particular technique and the results you'll get will be good". Essentially no one questions the validity of the "method". And the reason they are "allowed" not to question the validity of the method is because (most often) someone else has, and has proved that the specific method is, for instance, (a) consistent with the general framework and (b) yields a unique answer. And to prove this kind of results you need precise mathematical concepts because they are just mathematical results devoid of any physical meaning.

So the short answer is, you need that to make sure everything works as it intuitively should. And when it doesn't, to tell me why and how does it work instead.

(Don't know about the others, but to me advanced maths frequently isn't really that intuitive... :) )