field vision light spectra
spectra
I could behold ... Newton with his prism and silent face,
The marble index of a mind for ever
Voyaging through strange seas of Thought, alone. (Wordsworth)

symmetry action manifold
Mach, Weyl, Feynman, Dirac, EPR, Clark, Hughes, Bohm, Green, Cao


[The] science of colours becomes a speculation
as truly mathematical as any other part of physics.
(Newton)








Mach







Weyl














Feynman

Dirac

























Hughes
















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Clark

EPR

physical object

A color is a physical object a soon as we consider its dependence, for instance, upon its luminous source, upon temperatures, and so forth.

Mach
    

fundamental operation


The second principle of color mixing of lights is this: any color at all can be made from three different colors, in our case, red, green, and blue lights. By suitably mixing the three together we can make anything at all, as we demonstrated [...]

Further, these laws are very interesting mathematically. For those who are interested in the mathematics of the thing, it turns out as follows. Suppose that we take our three colors, which were red, green, and blue, but label them A, B, and C, and call them our primary colors. Then any color could be made by certain amounts of these three: say an amount a of color A, an amount b of color B, and an amount c of color C makes X:

X = aA + bB + cC.

Now suppose another color Y is made from the same three colors:
Y = a'A + b'B + c'C.

Then it turns out that the mixture of the two lights (it is one of the consequences of the laws that we have already mentioned) is obtained by taking the sum of the components of X and Y:

Z = X + Y = (a + a')A + (b + b')B + (c + c')C.

It is just like the mathematics of the addition of vectors, where (a, b, c ) are the components of one vector, and (a', b', c' ) are those of another vector, and the new light Z is then the "sum" of the vectors. This subject has always appealed to physicists and mathematicians. In fact, Schrödinger wrote a wonderful paper on color vision in which he developed this theory of vector analysis as applied to the mixing of colors.

Feynman    


What we learn from our whole discussion and what indeed has become a guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure endowed entity S try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed.
You can expect to gain a deep insight into the constitution of S in this way. After that you may start to investigate symmetric configurations of elements, i.e., configurations which are invariant under a certain subgroup of the group of all automorphisms [...]
Weyl      

automorphisms
 

The world as described by natural science has no obvious place for colours, tastes, or smells. Problems with sensory qualities have been philosophically and scientifically troublesome since ancient times, and in modern form at least since Galileo in 1623 identified some sensory qualities as characterizing nothing real in the objects themselves [...]

The qualities of size, figure (or shape), number, and motion are for Galileo the only real properties of objects. All other qualities revealed in sense perception--colours, tastes, odours, sounds, and so on--exist only in the sensitive body, and do not qualify anything in the objects themselves. They are the effects of the primary qualities of things on the senses. Without the living animal sensing such things, these 'secondary' qualities (to use the term introduced by Locke) would not exist.

Much of modern philosophy has devolved from this fateful distinction. While it was undoubtedly helpful to the physical sciences to make the mind into a sort of dustbin into which one could sweep the troublesome sensory qualities, this stratagem created difficulties for later attempt to arrive at some scientific understanding of the mind. In particular, the strategy cannot be reapplied when one goes on to explain sensation and perception. If physics cannot explain 
secondary qualities, then it seems that any science that can explain secondary qualities must appeal to explanatory principles distinct from those of physics. Thus are born various dualisms.

Clark      


spectral sensitivity



















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similar light
It seems useful to me to develop a little more precisely the "geometry" valid in the two- dimensional manifold of perceived colors. For one can do mathematics also in the domain of these colors. The fundamental operation which can be performed upon them is mixing: one lets colored lights combine with one another in space [...]

Weyl    
[When] a state is formed by the superposition of two other states, it will have properties that are in some vague way intermediate between those of the original states and that approach more or less closely to those of either of them according to the greater or less 'weight' attached to this state in the superposition process. The new state is completely defined by the two original states when their relative weights in the superposition process are known, together with a certain phase difference, the exact meaning of weights and phases being provided in the general case by the mathematical theory.
Dirac      

 

wave superposition


A "hidden-variable" [HV] theory, as the name implies, postulates that alongside (or, more graphically, beneath) the measurable quantities dealt with by the theory (position, momentum, spin, and so on) there are further quantities inaccessible to measurement, whose values determine the values yielded by individual measurements of the observables. The quantum mechanical statistics are to be obtained by "averaging" over the values of the hidden variables. The inaccessibility of these variables may be a contingent and temporary matter, to be remedied as we develop new experimental procedures, or these quantities may be in principle inaccessible [...]

The suggestion that there may be such "hidden variables" is as old as the probabilistic interpretation of the state vector. It was made by Born (1926b, p. 825) a few months after he first proposed that interpretation: "Anyone dissatisfied with these ideas may feel free to assume that there are additional parameters not yet introduced into the theory which determine the individual event." But almost as old is the denial that such hidden variables can exist. 

Hughes     

manifest variable


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.

EPR            


Now it may be asked why these hidden variables should have so long remained undetected.

Bohm    

different somehow

Well, obviously the extra dimensions have to be different somehow because otherwise we would notice them.

Green     

Calabi-Yau manifold

The internal space defined at each space-time point is called a fiber, and the union of this internal space with space-time is called fiber-bundle space.

Cao
   
     











color


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superposition


phase

weight

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