If one requires space to be metrically homogeneous - and a space that serve as a 'form of phenomena' is necessarily homogeneous - then one is thrown back at once from the Riemannian to the classical space concept ... But Riemann had an entirely different conception of the nature and origin of the metrical properties of space. "Therefore, either the reality on which our space is based must form a discrete manifold, or else the reason for the metrical relationships is to be looked for externally in binding forces acting on it." The metric field makes itself felt through the physical effects which it has upon rigid bodies, upon light rays, and all events in nature, and these effects alone permit us to ascertain the quantitative state of the metric field. But whatever acts must suffer too; it must be something real ...

~Weyl

Just saying.

[So] few and far between are the occasions for forming notions whose specializations make up a continuous manifold, that the only simple notions whose specializations form a multiply extended manifold are the positions of perceived objects and colors.

~Riemann

A speck in the visual field, though it need not be red must have some color; it is, so to speak, surrounded by color-space. Notes must have some pitch, objects of the sense of touch some degree of hardness, and so on.

~Wittgenstein

We shall now recall the data of a classical theory as understood by physicists and then reinterpret them in geometrical form.

Geometrically or mechanically we can interpret this data as follows. Imagine a structured particle, that is a particle which has a location at a point x of R_4 and an internal structure, or set of states, labeled by elements g of G. 

~Atiyah

You are requested to read my all articles from LinkedIn profile/ Researchgate or from the link below or attachments.

http://fqxi.org/community/forum/topic/3026

When writing about transcendental issues, be transcendentally clear.

~Descartes

Last time, I quoted Weyl on colors and projective geometry:

"Thus the colors with their various qualities and intensities fulfill the axioms of vector geometry if addition is interpreted as mixing; consequently, projective geometry applies to the color qualities."

Where does this get us?

To begin with, projective geometry is altogether fundamental, as Kline explains:

"[It] became possible to affirm that projective geometry is indeed logically prior to Euclidean geometry and that the latter can be built up as a special case. Both Klein and Arthur Cayley showed that the basic non-Euclidean geometries developed by Lobachevsky and Bolyai and the elliptic non-Euclidean geometry created by Riemann can also be derived as special cases of projective geometry. No wonder that Cayley exclaimed, 'Projective geometry is all geometry.'

The principle of duality in projective geometry states that we can interchange point and line in a theorem about figures lying in one plane and obtain a meaningful statement. Moreover, the new or dual statement will itself be a theorem — that is, it can be proven. On the basis of what has been presented here we cannot see why this must always be the case for the dual statement. However, it is possible to show by one proof that every rephrasing of a theorem of projective geometry in accordance with the principle of duality must be a theorem. This principle is a remarkable characteristic of projective geometry. It reveals the symmetry in the roles that point and line play in the structure of that geometry." (My emphasis.)

Here's a nice, easygoing article by Witten on "Duality, Spacetime and QM."

http://bit.ly/2IvcoUm

Sean Carroll widens our scope:

"In the heady days of the “first superstring revolution,” triumphalism was everywhere. String theory wasn’t just a way to quantize gravity, it was a Theory of Everything, from which we could potentially derive all of particle physics. Sadly, that hasn’t worked out, or at least not yet. (String theorists remain quite confident that the theory is compatible with everything we know about particle physics, but optimism that it will uniquely predict the low-energy world is at a low ebb.) But on the theoretical front, there have been impressive advances, including a 'second revolution' in the mid-nineties. Among the most astonishing results was the discovery by Juan Maldacena of gauge/gravity duality, according to which quantum gravity in a particular background is precisely equivalent to a completely distinct field theory, without gravity, in a different number of dimensions! String theory and quantum field theory, it turns out, aren’t really separate disciplines; there is a web of dualities that reveal various different-looking string theories as simply different manifestations of the same underlying theory, and some of those manifestations are ordinary field theories."

Weyl was the father of gauge theory, which really should be called phase theory; Weyl didn't get it quite right first time out of the gate, but the name stuck. More on that, later.

We know that colors behave like vectors, and so with photons. A little reflection should tell us, then, that the oft-repeated claim that color is just the frequency of light is a load of baloney. Frequency is a simple rate, and therefore a scalar.* Frequency is related to energy via E = hv, and the energy operator in QM is equal to the Hamiltonian: E = H. So a change in frequency gets mapped to a change in energy, which rotates the photonic vector in Hilbert space — as well as the associated color vector.

Whereas vectors are dual to differential forms, which can have the dimensions of area.

Why is this important? Because what we see are colored surfaces, which have the dimensions of area, so we have a nice fit in regard to dimensional analysis.**

http://bit.ly/1nib6IH

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* Frequency is related to energy via E = hv. The energy operator in QM is equal to the Hamiltonian: E = H. So a change in frequency gets mapped to a change in energy, which rotates the photonic vector in Hilbert space — and also rotates the associated color vector in a perfectly predictable way. So we have a nice match between theory and observation.

See, e.g., QM, by Sunil Golwala

http://bit.ly/2tSoqDV

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** "Analysis using the fact that physical quantities added to or equated with each other must be expressed in terms of the same fundamental quantities (such as mass, length, or time) for inferences to be made about the relations between them."