Life, like a dome of many-colored glass,
Stains the white radiance of eternity. (Shelley)

Poincaré

Cao

Dolan 

Atiyah

It is increasingly clear that the symmetry group of nature is the deepest thing that we understand about nature today.

Weinberg

The two great events in twentieth century physics are the rise of relativity theory and of quantum mechanics. Is there also some connection between quantum mechanics and symmetry? Yes indeed. Symmetry plays a great role in ordering the atomic and molecular spectra, for the understanding of which the principles of quantum mechanics provide the key.

An enormous amount of empirical material concerning the spectral lines, their wave lengths, and the regularities in their arrangements had been collected before quantum mechanics scored its first success; this success consisted in deriving the law of the so-called Balmer series in the spectrum of the hydrogen atom and in showing how the characteristic constant entering into that law is related to charge and mass of the electron and Planck's famous constant of action h. [...]

It turned out that, once these foundations had been laid, symmetry could be of great help in elucidating the general character of the spectra.

To start, we shall try to make the notion of symmetry in physics clearer. The meaning of symmetry of a physical system is frequently influenced, if not shaped, by the guidelines of our investigation. It is obvious that the symmetry of a physical system is closely related to the transformations of the parameters describing the system. Notice, however, that not every transformation of parameters is linked to a symmetry of the system; such symmetries have to satisfy certain conditions. The necessary condition is that the physical system remains the same object of our perception before as well as after the transformation.

We say that a 3-dimensional sphere has a rotational symmetry because the picture of it does not change while we rotate it through an angle around an arbitrary axis through the center of this sphere [...]

Now let us turn to the central topic, the geometrization of fundamental physics. The starting-point here is the geometrization of gravity: making Poincaré symmetry local removes the flatness of space-time and requires the introduction of some geometrical structures of space-time, such as metric, affine connection, and curvature, which are correlated with gravity.

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. Then we find that the local gauge symmetries remove the 'flatness' of the fiber-bundle space since we assume that the internal space directions of a physical system at different space-times points are different.

So the local gauge symmetry also requires the introduction of gauge potentials, which are responsible for the gauge interactions, to connect internal directions at different space-time points. We also find that the role the gauge potentials play in fiber-bundle space in gauge theory is exactly same as the role the affine connection plays in curved space-time in general relativity.

Cao   

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.

Kline    

Mathematics has introduced the name isomorphic representation for the relation which according to Helmholtz exists between objects and their signs. I should like to carry out the precise explanation of this notion between the points of the projective plane and the color qualities [...] the projective plane and the color continuum are isomorphic with one another. Every theorem which is correct in the one system S₁ is transferred unchanged to the other S₂. A science can never determine its subject matter except up to an isomorphic representation. The idea of isomorphism indicates the self-understood, insurmountable barrier of knowledge. It follows that toward the "nature" of its objects science maintains complete indifference. This for example what distinguishes the colors from the points of the projective plane one can only know in immediate alive intuition [...]

Atiyah

Symmetries and apparent symmetries in the laws of nature have played a part in the construction of physical theories since the time of Galileo and Newton. The most familiar symmetries are spatial or geometric ones. In a snowflake, for example, the presence of a symmetrical pattern can be detected at a glance.

It is remarkable that in this century, symmetry has gradually come to be a central theme of microscopic physics. It attained this position through a series of subtle evolutions in the concept of symmetry itself and in its phenomeno- logical manifestations. It is my firm belief that this evolutionary process has not come to an end, and further meaning of the concept of symmetry, with perhaps new mathematical structures, will develop in the coming years.

Yang    

Let us take as an example the relativistic field theory. This theory as a whole is symmetric with respect to the Lorentz transformations. This means that independently of the choice of frame of reference, the same field theory is the object of our investigation; changing from one frame to another the fields transform covariantly according to the rule imposed by the principle of relativity.

Lopuszanski

The magical formula E = hv from which the whole of quantum theory is developed, establishes a universal relationship between the frequency v of an oscillatory process and the energy E associated with such a process.

Weyl  

For other fndamental interactions, which, it is believed, can be described as gauge interactions, we find that the theoretical structures of the corresponding theories are exactly parallel to that of gravity. There are internal symmetry spaces: phase space, for electromagnetism, which looks like a circle [...]

As far as gravity is concerned, Einstein’s General Relativity is a beautiful and complete theory. But as Einstein realized it has to be extended to account for other physical forces, the most notable being electromagnetism. It is perhaps no accident that the first and most significant step in this direction was taken by a mathematician – Hermann Weyl. He showed that, by adding a fifth dimension, electromagnetism could also be interpreted as curvature.

His idea was that the size of a particle could alter as it passed through an electro-magnetic field. In analogy with railways it was called a gauge theory, and this name has stuck through subsequent evolutions of the theory.

Unfortunately for Weyl, Einstein immediately objected on physical grounds that this would have meant different atoms of, say hydrogen, would have different sizes depending on their past history, in contradiction with observation. Given this devastating critique, it is remarkable but fortunate that Weyl’s paper was still published, with Einstein’s objection as an appendix. Clearly the beauty of the idea attracted the editor, despite the fatal flaw. In fact, beauty often wins such contests, because with the advent of quantum mechanics, with its complex wave functions, it was pointed out by Kaluza and Klein that Weyl’s gauge theory could be salvaged if one interpreted the variable as a phase rather than a length. A pure phase shift by itself is not physically observable and so Weyl’s theory avoids the Einstein objection.

Historically, the idea of symmetry has its scientific origin in the Greeks’ discovery of the five regular solids, which are remarkably symmetrical. In the nineteenth century, this property was codified in the mathematical concept of a group invented by Galois and then that of a continuous group by Sophus Lie. In 1967, Victor Kac (MIT), then working in Moscow [...], and Bob Moody (Alberta) [...] independently enlarged the paradigm of classical Lie algebras, resulting in new algebras which are infinite-dimensional. The representation theory of a subclass of the algebras, the affine Kac-Moody algebras, has developed into a mature mathematics.

Dolan

field

relativity

affine

fiber

Where does the yellow come in?