Introduction to Hilbert Space (2022) [pdf]
cphysics.org
cphysics.org
Pretty sure the name quantum comes from the fact that some physical phenomena (eg absorption spectrum) get discrete allowed values. That in principle has nothing to do with separability (eg you can come up with non separable spaces which have operators with discrete spectrum). In fact presenting things this way is pretty confusing since separable Hilbert spaces do support operators with continuous spectrum (which is not obvious!) As far as I know separability is mostly technical, and often added to make life a bit simpler, since it's pretty hard to come up with useful non separable Hilbert spaces.
Been on HN before, got very positive comments. From the author of ANTLR no less.
A: Just enough to describe all your independent physical states, which is proportional to the number of your particles.
Q: So if some interaction creates more particles I have to increase the dimension of my Hilbert Space?
A: Yes, I'm afraid so.
Q: But I am used to physics providing some constant background to goings-on here and now. It was absolute Newtonian space and time, then it was warped Einsteinian spacetime, now you say it depends on how many particles I create?
A: Yes.
Q: But that's insane?
A: Yes.
A: OK, I'll allow you to create or destroy as many particles as you like, especially if you have an really big Large Hardon Collider.
Q: Are you sure I can keep my Hilbert Space the same?
A: Yes, aha, I have thought of a special number that does not change when you add or remove finite integers...
Q: There is no such number ... oh wait, you mean infinite dimensional?
A: Yes, infinite dimensional, and complex, of course.
Q: Of course.
Q: Does not sound in the least bit plausible. How do you add particles?
A: It's called the second quantization of Quantum Field Theory using Fock Spaces.
Q: Makes perfect Focking sense.
> Complete means that every sequence of vectors |a1>, |a2>, ... satisfying lim ...
How are the elements of this sequence related? And why are we only interested in the elements where the index n/m goes to infinity? What does that even mean if the sequence is arbitrary?
That's probably why I also can't make sense of this:
> Loosely speaking, saying that a Hilbert space is complete means that it contains all of its limits.
Example: if the space is all real numbers except 0, then any sequence of real numbers accumulating around 0 (for howsoever small a distance, there is always infinite number of points closer to 0), the sequence is a Cauchy sequence, but not convergent (because 0 is not present). So that space is not complete (has a hole).
If the space is all real numbers, then the same sequence is also convergent, and the space is complete (no holes).
Completeness is required for nice results like the spectral theorem for self-adjoint operators [1] to hold, which is pretty essential for Quantum Mechanics.
Turns out, rational numbers are the classic example of a space that is not complete. A sequence of rational numbers can approximate pi but pi itself doesn't exist in the space (since its irrational). So the rational numbers that get closer and closer to pi form a limit to a value that's not in the space.
What does it mean to say that the sequence looks like it is converging?
Naively, it means that any two elements far in the sequence are always very close together. We make that intuition more precise by turning it into a challenge-response, "You tell me how close you want the elements to be, I'll tell you how far out to pick your elements." And then we write that mathematically as
∀ ϵ > 0 # You tell me how close
∃ N # I'll tell you how far
such that if N < n, m # so that any 2 elements
then || a_n - a_m || < ϵ # will always be that close
Joseph Fourier discovered that the following functions are orthonormal (each has length 1, their dot product with each other is 0).
1/sqrt(2π),
sin(x)/sqrt(π), sin(2x)/sqrt(π), sin(3x)/sqrt(π), ...
cos(x)/sqrt(π), cos(2x)/sqrt(π), cos(3x)/sqrt(π), ...
Given any function, he could compute a series that we now call the Fourier series that looks like it added up to that function. But there were complications. It added up perfectly for smooth functions with the same value at the edges like x(2π-x). But it also added up except at a few points for things like square waves. This caused a crisis in mathematics because at the time square waves were not considered functions, and nobody had ever realized that adding up infinite series of smooth functions could do such weird things. Their idea of functions was essentially what we would call analytic functions today. Basically things that look like power series. And Mr. Fourier had just shown that the analytic functions are not complete.
In addition to revealing problems in how we understood math, his technique was very, very useful. Because now you just had to figure out the physics of how, say, heat spreads out or vibrating strings vibrate just for those those sin and cos terms, and then you could figure out heat and vibration for ANY function.
Resolving the math problems started many decades of research. The results of which included better definitions of the real numbers, the ϵ-δ definition of a limit (or ϵ-N for a series), new theories of integration, and the idea of Hilbert spaces.
They are not necessarily related
> And why are we only interested in the elements where the index n/m goes to infinity?
If the sequence is finite, then we don’t really care to discuss the “limit” of the sequence.
> Loosely speaking, saying that a Hilbert space is complete means that it contains all of its limits.
For a set S to not contain all of its limits means you can have an infinite sequence of points (a_n) where each a_n is in S and there is no point a in S so that the sequence is eventually as close to a as you’d like.
More formally, there does not exist a in S so that for any e > 0 we can pick an M so that m > M implies | a_m - a | < e.
You can see how “m > M” gives a formal meaning to “eventually,” and “for any e > 0 … | a_m - a | < e” gives a formal meaning to “as close to a as you’d like.”
I prefer Halmos' book.
Here's a link: https://archive.org/details/introductiontohi0000halm
The story goes that once Neumann was giving a lecture in Germany on such spaces with Hilbert in attendance. Hilbert had supposedly raised his hands to ask, Dr. von Neumann, I would very much like to know, what after all is a Hilbert space? Hilbert was quite underwhelmed by the definition, "is that all" he remarked.