In an old fairy tale ("Hansel and Gretel"), one of the children left a trail of bread crumbs as they made their way through a dark forest, so that they could find their way home again by following the crumbs. (It didn't work out, but never mind.)
I'm in a similar situation -- trying to follow a number of clues (below). This is another in a series I think of as "arguably best kept to myself, but making a record anyway in case I get hit by a truck."
Here are the clues:
1) Color & sound vectors
2) Hilbert space vectors
3) Operator theory
4) Spectral theory (pdf)
5) Matrix theory
7) Dirichlet membranes
8) Harmonic relations
9) Projection operators
I've recently learned that they are, in fact, all related -- which gives me further confidence in my nose in re: sniffing out relations -- except that, in the case of #1, no one seems to have carried out the obvious analysis of the fact that we know that vibrating strings & membranes give us characteristic (eigen!) sounds and colors. This is no doubt due, again, to the "apathetic acquiescence" noted by Whitehead:
What we see depends on light entering the eye. Furthermore we do not even perceive what enters the eye. The things transmitted are waves or–as Newton thought–minute particles, and the things seen are colors. Locke met this difficulty by a theory of primary and secondary qualities. Namely, there are some attributes of the matter which we do perceive. These are the primary qualities, and there are other things which we perceive, such as colors, which are not attributes of matter, but are perceived by us as if they were such attributes. These are the secondary qualities of matter.
Why should we perceive secondary qualities? It seems an unfortunate arrangement that we should perceive a lot of things that are not there. Yet this is what the theory of secondary qualities in fact comes to. There is now reigning in philosophy and in science an apathetic acquiescence in the conclusion that no coherent account can be given of nature as it is disclosed to us in sense-awareness, without dragging in its relation to mind.
I’m delighted to report that the illustrious Alain Connes is all over this business regarding action, symmetry, spectral theory, operator theory, projective geometry—and his own great bailiwick, noncommutative geometry, which looks as though it flows from Heisenberg’s matrices.
Moreover, his writing is wonderfully clear—in the great French tradition of Pascal, Montaigne, Descartes and Voltaire.
There is, of course, quite another tendency among certain French intellectuals—but the less said about that, the better.
I continue to delve into spectral theory & related topics and, while my efforts are still quite preliminary, I now feel quite confident in a wonderful generality that flows, in my mind, from the following observation from Helmholtz, which has the simplicity of genius:
"Similar light produces, under like conditions, a like sensation of color."
Substitute "same state vector (psi)" for "similar light."
Now substitute "same operator or sequence of operators" for "like conditions."
We now have a simple, natural way to recover a vast data set from everyday experience in the standard formalism of quantum theory, to wit:
The same things, under the same conditions, look, sound, feel, taste and smell the same -- i.e., exhibit the same spectra of secondary properties.
We therefore answer the problem addressed by North Whitehead, e.g., where he wrote:
"Why should we perceive secondary qualities? It seems an unfortunate arrangement that we should perceive a lot of things that are not there. Yet this is what the theory of secondary qualities in fact comes to. There is now reigning in philosophy and in science an apathetic acquiescence in the conclusion that no coherent account can be given of nature as it is disclosed to us in sense-awareness, without dragging in its relation to mind."
I've also found a few terrific resources on spectral theory:
"Highlights in the History of Spectral Theory," (Steen)
American Mathematical Monthly 80 (1973) 359-381.
"What Does the Spectral Theorem Say?" (Halmos)
Amer. Math. Monthly 70 (1963), 241–247
Finite-dimensional vector spaces (Halmos)