Bread crumbs

[Brian J Flanagan]

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 "arguable best kept to myself, but making a record anyway in case I get hit by a truck."

Here are the clues:

1) Spectra of color & sound vectors and the characteristic Hilbert space vectors associated with their respective wavelengths; and

2) Operator theory in re: quantum theory and neural nets and color & sound; and

3) Spectral theory in re: #2 and in re: common mathematical origins & later application to quantum theory; and

3) Matrix theory in re: #2 and M-theory and Dirichlet membranes; and

4) Harmonic relations vis-a-vis all the above; and

5) Projection operators in re: all of the above.

I've recently discovered 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.

If anyone has suggestions, please advise!

[Brian J Flanagan]

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.

www.alainconnes.org/en/downloads.php

Moreover, his writing is wonderfully clear—in the great French tradition of Pascal, Montaigne, Descartes and Voltaire.

[Brian J Flanagan]

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."

OK:

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)

http://www.amazon.com/Finite-Dimensional-Vector-Spaces-P-R-Halmos/d...