Substance, and accident, and their operations,
All interfused together in such wise
That what I speak of is one simple light. ~Dante
|Einstein, Helmholtz, Wigner, Sherrington, Mach, Schrodinger, Bohr, Tomonaga, Aharanov, Bohm|
All the fifty years of conscious brooding have brought me no closer to the answer to the question: "What are light quanta?" Of course today every rascal thinks he knows the answer, but he is deluding himself.
[If] one entity is influenced by another entity, in all known cases the latter one is also influenced by the former. The most striking and originally the least expected example for this is the influence of light on matter, most obviously in the form of light pressure. That matter influences light is an obvious fact — if it were not so, we could not see objects. The influence of light on matter is, however, a more subtle effect and is virtually unobservable under the conditions which surround us [...] Since matter clearly influences the content of our consciousness, it is natural to assume that the opposite influence also exists, thus demanding the modification of the presently accepted laws of nature which disregard this influence.
One can only admire the colossal boldness
which led Weyl, onthe basis of a purelyformal correspondence, to his gauge-geometrical interpretation of electromagnetism. (London)
When a beam of light falls on the human eye, certain sensations are produced, from which the possessor of that organ judges of the color and luminance of the light. Now, though everyone experiences these sensations and though they are the foundation of all the phenomena of sight, yet, on account of their absolute simplicity, they are incapable of analysis, and can never become in themselves objects of thought. If we attempt to discover them, we must do so by artificial means and our reasonings on them must be guided by some theory.
It turned out that, once these foundations had been laid, symmetry could be of great help in elucidating the general character of the spectra.
If you ask a physicist what is his idea of yellow light, he will tell you that it is transversal electromagnetic waves of wavelength in the neighborhood of 590 millimicrons. If you ask him: But where does yellow come in? he will say: In my picture not at all, but these kinds of vibrations, when they hit the retina of a healthy eye, give the person whose eye it is the sensation of yellow.
We now proceed to the derivation of the laws of geometrical optics from the laws of wave optics. A rigorous and general derivation, however, requires a considerable amount of advanced mathematics so that we shall not go into details here. We shall be satisfied with observing the essential points by taking the simple example given above.
The problem is again that of the refraction of monochromatic light [...]
holds, as is apparent from the figure. In other words, the refractive index n is inversely proportional to the wavelength of the light wave in the respective media; i.e.,
n = k''/l
The proportionality constant k'', however, may depend on the frequency or, in other words, on the color of the light.
The science of colors becomes a speculation as truly mathematical as any other part of physics. ~Newton
Hermann Grassmann (along with William Rowan Hamilton) invented vector and tensor algebra in 1844, but was unable to attract attention to the new method. He subsequently applied the methods of vector analysis to color in order to calculate the outcomes of color mixtures. In so doing, Grassmann proved that for every spectral color there exists some other opponent color in the spectrum which when mixed with the first color in the correct proportions will produce white light.
In the early 1990s, it was shown that the various superstring theories were related by dualities, which allow physicists to relate the description of an object in one superstring theory to the description of a different object in another superstring theory. These relationships imply that each of the superstring theories is a different aspect of a single underlying theory, proposed by Witten, and named "M-theory".
The Langlands programme has, along these lines, something called the geometric Langlands programme, which replaces the number fields by Riemann surfaces. It is a very interesting theory which is much easier than the number field case, but not trivial and still quite big. It is developed by using the theory of vector bundles on Riemann surfaces. This theory has been going on for quite a long time, has nice results and in it there are geometrical analogues of the Langlands conjectures.
For instance a star which we perceive. The energy scheme deals with it, describes the passing of radiation thence into the eye, the little light image of it formed at the bottom of the eye, the ensuing photo- chemical action in the retina, the trains of action potentials traveling along the nerve to the brain, the further electrical disturbance in the brain, the action potentials streaming thence to the muscles of eyeballs and of the pupil, the contraction of them sharpening the light image and placing the best seeing part of the retina under it. The best 'seeing'? That is where the energy scheme forsakes it. It tells us nothing of any 'seeing'. Everything but that.
Erwin ("Mad Dog") Schrödinger
[It] was found possible to account for the atomic stability, as well as for the empirical laws governing the spectra of the elements, by assuming that any reaction of the atom resulting in a change of its energy involved a complete transition between two so-called stationary quantum states and that, in particular, the spectra were emitted by a step-like process in which each transition is accompanied by the emission of a monochromatic light quantum of an energy just equal to that of an Einstein photon.
E = hv
In the hands of others, notably Issai Schur, Elie Cartan, and Hermann Weyl, the notion of representation was extended to continuous groups, the most important example being the group of rotations about a given point in three-dimensional space. This group has dimension three. For example, if we are thinking of rotations of the earth about its center, we have first to decide to what point we shall rotate the north pole. To describe this point takes two coordinates. We have then to add a third, because we are still free to rotate around the new pole through an angle, whose size can be between 0◦ and 360◦. It is the representations of this group to which the oracular pronouncement of Hermann Weyl about quantum numbers and group representations applies.
The understanding of the fundamental role of this group and its representations in atomic spectroscopy and, ultimately, not only of representations of this group but also of many other groups as well in various domains of quantum physics led to a vigorous interest of the representation theory of finite and of continuous groups on the part of both physicists and mathematicians. The mathematical development remained for some time largely independent of the questions raised by spectroscopy and the later, more sophisticated theories of fundamental particles. There is not even, so far as I can see, a close relation between Weyl’s reflections on group theory and quantum mechanics and his fundamental contributions to the mathematics of group representations. Before describing how the concepts introduced by the physicists found their way back into mathematics, I recall briefly the issues that faced spectroscopists in the period after 1885, when the structure of the frequencies or wavelengths, thus the color in so far as the light is visible and not in the infrared or ultraviolet regime, of the light emitted by hydrogen atoms was examined by Balmer, and in the period after 1913, when N. Bohr introduced his first quantum-mechanical explanations.
In classical mechanics, we recall that potentials cannot have such significance because the equation of motion involves only the field quantities themselves. For this reason, the potentials have been regarded as purely mathematical auxiliaries, while only the field quantities were thought to have a direct physical meaning.
In quantum mechanics, the essential difference is that the equations of motion of a particle are replaced by the SchrÖdinger equation for a wave. This SchrÖdinger equation is obtained from a canonical formalism, which cannot be expressed in terms of the fields alone, but which also requires the potentials. Indeed, the potentials play a role, in SchrÖdinger equation, analogous to that of the index of refraction in optics. [...]
The above discussion suggests that some further development of the theory is needed. Two possible directions are clear. First, we may try to formulate a nonlocal theory in which, for example, the electron could interact with a field that was some finite distance away. Then there would be no trouble in interpreting these results, but, as is well known, there are severe difficulties in the way of doing this. Secondly, we may retain the present theory and, instead, we may give a further new interpretation to the potentials.
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
Is there such a thing as a 'natural history of colors' and to what
extent is it analogous to a natural history of plants? Isn't the latter temporal, the former non-temporal? ~Wittgenstein
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.
Surprisingly, the Langlands dual group also appears in quantum physics in what looks like an entirely different context; namely, the electromagnetic duality. Looking at the Maxwell equations describing classical electromagnetism, one quickly notices that they are invariant under the exchange of the electric and magnetic fields. It is natural to ask whether this duality exists at the quantum level.
E = hv