The marble index of a mind for ever
Voyaging through strange seas of Thought, alone. ~Wordsworth
Weyl, Feynman, Dirac, EPR, Clark, Hughes, Bohm, Green, Cao, Langlands
The science of colors becomes a speculation as truly
mathematical as any other part of physics. ~Newton
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.
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
color is a physical object a soon as we consider its dependence, for
instance, upon its luminous source, upon temperatures, and so forth.
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:
This paper presents a new exposition of the foundations of trichromatic color measurement. There are three closely interrelated reasons that mandate a reexposition of this material. First, both the language of mathematics and the standards of mathematical rigor have progressed greatly in the past century. The subject of color measurement can be communicated more clearly to new generations of students if it is cast in modern mathematical concepts. Second, many points that are obscure or difficult in traditional treatments are much clearer if a slightly more abstract standpoint is assumed. Many modern ideas in linear algebra — especially the concept of linear functional and the resulting distinction between a linear space and its dual space — seem to be specially designed to clarify thinking about color theory.
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.
Simple Calabi-Yau space
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.
The world as described by natural science has no obvious place for colors, 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 [...]
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—colors,
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
The mathematical machinery of quantum.
mechanics became that of spectral analysis..
It turned out that, once these foundations had been laid, symmetry could be of great help in elucidating the general character of the spectra.
The physical action only depends on [the spectrum] Σ.Connes
take a physical state and do
something to it — like
rotating it, or like waiting for some time t — you get a different state. We
say, "performing an operation on a state
produces a new state." We can express the same idea by an equation:
|> = A|>.
An operation on a state produces another state. The operator A stands for some particular operation. When this operation is performed on any state, say |>, it produces some other state
Similar light produces,
under like conditions, a like sensation of color.
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 [...]
[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.
Many of the foundations of wave mechanics are based on the
analyses and equations that Rayleigh
derived for the theory of acoustics in his book The Theory of Sound.
Erwin Schrödinger, a pioneer in quantum mechanics, studied this book
and was familiar with the perturbation methods it describes.
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.
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.
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.
Now it may be asked why these hidden variables should have so long remained undetected.
Well, obviously the extra dimensions have to be different somehow because otherwise we would notice them.
It was not until the advent of quantum mechanics in the twentieth century that absorbtion spectra were given a satisfactory theoretical explanation. They were shown to correspond with eigenvalues of appropriate Schrödinger operators. A given atom could absorb or emit light only at certain frequencies, corresponding to the energy levels of bound states represented by different eigenvalues. The mathematical spectra of differential operators thus carried fundamental information about the physical world, which even now seems almost magical.
The analogy with number theory is through spectra of other differential operators. These are Laplace-Beltrami operators (and variants of higher degree) attached to certain Riemannian manifolds. The spectra of these and other operators are expected to carry fundamental information about the arithmetic world, a possibility that also seems quite magical. [...]
The spectral data that is believed to be related to number theory is framed in the language of automorphic forms.