Here I find myself in the middle of an example of how hard it can be to learn mathy things without having gone through years of training. You might say “So, go through years of training.” And then I would say “No, that training comes embedded with a way of seeing the world that confines native curiousity into the sword-like shape of a polarizing ego pursuit oriented around competition driven by greed and power as an end instead of a means, and I’m studying math in part because I want to deconstruct that approach, not so I can strengthen it and get lost like every other mathematician except Grigori Perelman who escaped the trap only by extreme asceticism.” And then you might say “Who is Grigori Perelman?” And then I might tell you a fun story.
But first, this present story, which begins as I randomly find myself here.
Because I sometimes think of quantization as though it was some kind of chopping up space into a grid or something, I find myself intrigued by the first paragraph’s bold “You are thinking of quantization as though it was some kind of chopping up space into a grid or something. But that’s not the way quantization works.” So I’m drawn in. In the 2nd paragraph I find these words: “Quantization fundamentally amounts to switching from ordinary numbers (representing a classical universe) to noncommuting variables (Dirac called these q-numbers) that obey different rules of arithmetic.”
Wait, what? I want to know more about a new kind of number I’ve never heard of before; I thought there was only one kind of number, and I have already found some issues with how numbers work. What’s a noncommuting variable? What’s a q-number? (I already see a hint of useful information toward my ongoing pursuit to define ternary logic in a way that is more ternary than binary, since our current understanding of all logic, even ternary, is essentially binary in nature.)
I search on these terms “noncommuting variables (Dirac called these q-numbers)” and find only a few salient hits. Refine the search: “Dirac q-numbers” still hoping I get a quick idea of them before continuing with the conversation. While those pages are loading, I also search “non-commuting” and thankfully there is a simple definition for this term. Perhaps I finally have a place to apply this basic mathematical concept (commutability = ability to change order) I’ve seen hundreds of times and promptly forgot because: 1) it had no immediate practical value and 2) could be easily found in any math textbook if ever the time came to use it.
At this point I don’t want a simple, trivial, dictionary idea of non-commuting but “why” and “what” such things are — albeit quickly, because I’m already in a tangent from the original discussion on measuring space. I find this: https://physics.stackexchange.com/questions/240543/is-there-something-behind-non-commuting-observables
Cool, but within moments I’m wondering “now what is a Hilbert space?” Another term I’ve seen mentioned dozens of times but never needed to know… let’s track it down.
Soon I have a dozen tabs open in my browser, and Hilbert space is proving to be even more elusive than non-commuting q-numbers. So I add “intuitive” to my query on Hilbert spaces and finally find some answers that I can begin to understand. However, scrolling through a dozen of the intuitive answers, I’m fast discovering Hilbert spaces are so abstract that even simplified descriptions are indecipherable to me. Lots of terms I will have to search the same way I’m searching “Hilbert space.” But along the way I did see the line “once you discover Hilbert spaces, you’ll never go back,” so maybe it’s something I do want to understand finally. Keep digging.
Finally I find this:
Well it’s tough to explain in complete layman’s terms. You require some mathematical knowledge. It helps particularly if you know some linear algebra and the notion of linearity. 3D space has 3 independent dimensions, which we can combine in different amounts to address any point in space. So to address the point (2,4,5) we say go two units in the X-dimension, go 4 units in the Y-dimension and 5 units in the Z-dimension. Now with Hilbert spaces the dimensions can be arbitrary. Consider the set of all sine waves which fit an integer amount of times on a string. You can create an arbitrary function by combining these together with appropriate coefficients. So we can consider each sine wave to each be an independent dimension in a Hilbert space. The function is now a point in this abstract space.
Ok so now I can see a vague outline of what I’m looking for, and can also see the oft-mentioned affinity to Fourier transforms, something I learned about long ago but appears to be unrelated to my current inquiry. Let’s go deeper… except now Quora won’t let me surf the question further until I log in. Bother, I’d rather end the browser session so Quora thinks I’m a stranger again, and continue. Meanwhile, the next answer to the layman question helps a little more, but this one is an example of what DOESN’T work:
Basically it’s an Euclidean space generalized to infinite dimensions. The same operations with vectors that can be done in Euclidean space (measuring distance, angles, scalar product) can also be done in Hilbert space. It is also complete, which means that we can work with limits of sequences and stuff like that.
Avoiding “intuitive answers” that have, in the first 3 sentences, a half-dozen words I need to define is part of the art of finding what I’m seeking. Remember, I’m an artist, not a mathematician… yet.
Ok, time to go back to the original query and see if the vague ideas are enough for me to move forward. Back to Is there something behind non-commuting observables? only to quickly find I still don’t know enough about Hilbert spaces, but I do remember this: “If you look at this condition the right way, the resulting uncertainty principle becomes very intuitive” and still in skim-quickly mode, I see this comment: “this is a truly excellent answer,” and this: “As a lay enthusiast, I think this is one of the best answers I’ve seen on PhysicsSE” so I know that if I do figure out what is being said here, I’m on solid ground. Looks like I should stay in this thread…
But then it hits me. I’ve just spent 30 minutes getting through a couple sentences, at this rate it’ll be another couple hours before I’m done. I’ll have to figure out what Hilbert spaces are some other day. Let’s see if I can muddle through Viktor Toth’s answer on measuring space without knowing about q-variables, Hilbert space, completeness, or scalar products.
Conceptually, Wikipedia or Quora or PhysicsSE or any of the others should have made this much easier (they have, but I mean even easier) but Wikipedia for example only adds fuel to the fire with answers that seem intentionally obscure even if technically accurate, as though written by committee afraid of liability for speaking in simple metaphors.