The concern of "is this thing square-integrable" arises only in infinite dimensional spaces. Inner product of all $\mathbb C^N$ where N in finite is finite. There is no point in saying "L^2" when talking about spins etc. (forcefully casting a finite sum of finite terms to integration and then introducing the concept of "square integrable" isn't helpful, if not trivial).
> This is a rationalization but it's useful.
You can do that kind of rationalization when you "know" the answer ahead. It has zero prediction power.
Newton had a rationalization about the independent nature of time. Descartes also had a rationalization about how action-at-a-distance works for gravity. Aether was also a popular rationalization once.
None of those rationalizations held any value at the end.
Such "rationalizations" are better cut by Occam's razor and left for philosophers.
At least, I prefer not waste my time with such stuff as a physicist.
The concern of "is this thing square-integrable" arises only in infinite dimensional spaces. Inner product of all $\mathbb C^N$ where N in finite is finite. There is no point in saying "L^2" when talking about spins etc. (forcefully casting a finite sum of finite terms to integration and then introducing the concept of "square integrable" isn't helpful, if not trivial).
> This is a rationalization but it's useful.
You can do that kind of rationalization when you "know" the answer ahead. It has zero prediction power.
Newton had a rationalization about the independent nature of time. Descartes also had a rationalization about how action-at-a-distance works for gravity. Aether was also a popular rationalization once. None of those rationalizations held any value at the end.
Such "rationalizations" are better cut by Occam's razor and left for philosophers.
At least, I prefer not waste my time with such stuff as a physicist.