Golden ratio discovered in a quantum world

I would like to draw attention to a few major points:

When applying a magnetic field at right angles to an aligned spin the magnetic chain will transform into a new state called quantum critical, which can be thought of as a quantum version of a fractal pattern.

Fractals are characterized by self-similarity across temporal-spatial scales. I have often argued that it makes a kind of sense to suppose that “neural form follows quantum function,” in this wise:

If Paul Churchland is correct about the neural implementation of matrix-valued operators, then that is rather interesting, since that is precisely the sort of mathematics we find at work at the quantum level of neural function. Which would seem to make a kind of sense, if, as we suggest, the form of neural networks follows the underlying function of those quantum processes which mediate neural activity. Given that that the dendritic forms of neurons are aptly captured by the mathematics of fractals, we might expect this kind of self-similarity across scales.

(from: Are Perceptual Fields Quantum Fields?)

I did not know how far down the scale we could go, however — did not know how to ground fractal behavior in the quantum realm. So this is welcome news.

By a curious coincidence, a young scholar recently sent me a rather old piece by Hofstadter, where already we see fractal patterns emerging at the quantum level.

Another point of interest in the article on the golden ratio:

“Such discoveries are leading physicists to speculate that the quantum, atomic scale world may have its own underlying order. Similar surprises may await researchers in other materials in the quantum critical state.”

Well, this kind of “underlying order” suggests “hidden variables,” and in this connection I would like to draw attention to another fascinating development:

’Loopy’ Photons Test Hidden-Variable Predictions

Again, this is welcome news to me in view of the following:

Let us attend to the simplicity of colors. For colors are so simple, we might think of them as elemental, and so perhaps count them among the proper elements of an EPR-complete quantum theory. What does this mean? Let’s remind ourselves of what Einstein & Co., said in their seminal work on the (in)completeness of QM:

In attempting to judge the success of a physical theory, we may ask ourselves two questions: (1) “Is the theory correct?” and (2) “Is the description given by the theory complete?” It is only in the case in which positive answers may be given to both of these questions, that the concepts of the theory may be said to be satisfactory. The correctness of the theory is judged by the degree of agreement between the conclusions of the theory and human experience…

Whatever the meaning assigned to the term complete, the following requirement for a complete theory seems to be a necessary one: every element of the physical reality must have a counterpart in the physical theory.

We see that our investigation naturally leads us to the question: Are the secondary properties among the “hidden variables” of QM?

I would like to begin an attempt to tie all this together in respect of spectral theory and, in particular, spectral triples.

Colors and sounds come to us in the spectra of rainbows and the notes of the scale. The explanation of atomic spectra provided the first big win for QM.

Here is what Connes writes about the explanatory power of spectral triples in a very nice collection of essays edited by Majid, On Space & Time .

The new paradigm of spectral triples passes a number of tests to qualify as a replacement of Riemannian geometry in the noncommutative world:

1. It contains the Riemannian paradigm as a special case.
2. It does not require the commutativity of coordinates.
3. It covers the spaces of leaves of foliations.
4. It covers spaces of fractal, complex or infinite dimension.
5. It applies to the analogue of symmetry groups (compact quantum groups).
6. It provides a way of expressing the full Standard Model coupled to Einstein gravity as pure gravity on a modified spacetime geometry.
7. It allows for quantum corrections to the geometry.

This all seems quite suggestive to me, as this approach looks like a natural geometry for bringing these various developments under one roof, as it were.

Thus, e.g., in one of his papers, Connes writes: “The physical action only depends on [the spectrum] Σ.”

Well, of course, the action is now known to be determined by the symmtries of nature (Weinberg, e.g.), and so we have a direct route to the Lagrangian and very nearly all of classical & quantum mechanics:

Roughly speaking, force is the space derivative of energy and the time derivative of momentum. You can take one more step up the ladder: energy and momentum are both derivatives of action: energy is its time derivative, momentum its space derivative. (Wilczek)

Finally, we see instances of field behavior influenced by number — in the transcendental golden ratio and in Hofstadter’s rational and irrational numbers. I earlier alluded to the well-known relations between musical tones and numerical ratios — and here again it seems as though Nature is hinting at wonderful things remaining to be discovered.

Now, the foregoing is obviously all very rough and preliminary, but it seems like a promising avenue for further exploration.