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  The marble index of a mind for ever Voyaging
through strange seas of Thought, alone. (Wordsworth)
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| 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) |
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Mach Weyl Steen Feynman Dirac Born Clark EPR Hughes Halmos |
In Göttingen in 1925-26 Werner
Heisenberg and Erwin Schrödinger
created the theory of quantum mechanics. In Heisenberg's theory the
physical fact that certain atomic observations cannot be made
simultaneously was interpreted mathematically to mean that the
operations which represented these operations were not commutative.
Since the algebra of matrices is non-commutative, Heisenberg together
with Max Born and Pascual Jordan represented each physical quantity by
an appropriate (finite or infinite) matrix, called a transformation;
the set of possible values of the physical quantity was the spectrum of
the transformation. (So the spectrum of the energy of the atom was
precisely the spectrum of the atom.)
Schrödinger,
in contrast, advanced a less unorthodox theory based on his partial
differential wave equation. Following some initial surprise that Schrödinger's "wave
mechanics" and Heisenberg's "matrix mechanics"—two
theories with substantially different hypotheses—should
yield the same results, Schrödinger
unified the two approaches by showing, in effect, that the eigenvalues
(or more generally, the spectrum) of the differential operator in Schrödinger's
wave equation determine the corresponding Heisenberg matrix. Similar
results were obtained simultaneously by the British physicist Paul A.
M. Dirac. Thus interest in spectral theory once again became quite
intense.
Steen
When a blind beetle crawls over the surface of the globe, he doesn't realize that the track he has covered is curved. I was lucky enough to have spotted it. Einstein
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
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: Now
suppose another color Y is made from the same three colors:
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: 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 Σ 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 Σ 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  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. Clark
 How are we able to engineer color? |
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The
mathematical machinery of quantum mechanics became that of spectral
analysis... Steen
It turned out that, once these foundations had been laid, symmetry could be of great help in elucidating the general character of the spectra. 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
A "hidden-variable" 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 [...] 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
In view
of the preceding section, there is a natural one-to-one correspondence
between subspaces and idempotent Hermitian operators. It is in
principle possible, therefore, to express all the geometric properties
of subspaces in terms of the algebraic properties of their projections.
Halmos
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
Well, obviously the extra dimensions have to be different somehow because otherwise we would notice them. Green
 Simple Calabi-Yau space Cao  Where does the yellow come in? |
color geometry superposition phase weight vector HVs complete dimension internal |
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