like a dome of many-colored glass,
|Galileo, Newton, Weyl, Balmer, Planck, Poincaré, Wigner, Cao, Dolan, Atiyah|
It is increasingly clear that the symmetry group of nature is the deepest thing that we understand about nature today.
The immense power of group theory in physics derives from two facts. First, the laws of quantum mechanics decree that whenever a physical system has a symmetry, there is a well-defined group (G) of operations that preserve the symmetry, and the possible quantum states of the object are then in exact correspondence with the representations of G. Second, the enumeration and classification of all well-behaved groups and of their representations have been done by the mathematicians, once and for all, independently of the physical situation to which the groups may be applied.
trouble with group theory is that it leaves so much unexplained that
one would like to explain. It isolates in a beautiful way those aspects
of nature that can be understood in terms of abstract symmetry alone.
It does not offer much help of explaining the messier facts of life,
the numerical values of particle lifetimes and interaction strengths –
the great bulk of quantitative experimental data that is waiting for
explanation. The process of abstraction seems to have been too drastic,
so that many essential and concrete features of the world have been
left out of consideration.
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.
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.
Among the advances of the last fifty years in the field of geometry, the development of projective geometry occupies the first place. Although it seemed at first as if the so-called metrical relations were not accessible to this treatment, as they do not remain unchanged by projection, we have nevertheless learned recently to regard them as also from the projective point of view, so that the projective method now embraces the whole of geometry.
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.
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 Σ1 is transferred unchanged to the other Σ2. 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 [...]
Even Pythagoras knew that when strings of different lengths but of the same make, and subjected to the same tension, were used to give the perfect consonances of the Octave, Fifth, or Fourth, their lengths must be in the ratios of 1 to 2, 2 to 3, or 3 to 4 [...] Later physics has extended the law of Pythagoras by passing from the lengths of strings to the number of vibrations, and thus making it applicable to the tones of all musical instruments, and the numerical relations 4 to 5 and 5 to 6 have been added to the above for the less perfect consonances of the major and minor Thirds, but I am not aware that any real step was ever made towards answering the question: What have musical consonances to do with the ratios of the first six numbers? Musicians, as well as philosophers and physicists, have generally contented themselves with saying in effect that human minds were in some unknown manner so constituted as to discover the numerical relations of musical vibrations, and to have a peculiar pleasure in contemplating, simple ratios which are readily comprehensible.
are perhaps the most
important objects in all of mathematics.
It's the things that we most take for granted that have the tendency to come back and bite us when it really matters. The nature of space and time is generally taken for granted. But our assumptions about them seem to be inconsistent and as a result, if we are honest, theoretical physics is currently derailed at its very core.
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 phenomenological 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.
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.
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.
L Ingraham summarised some of the many contributions made by Wigner.
These include his:
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.
Conformal Field Theory
the idea of symmetry has its scientific
the Greeks’ discovery of the five regular
which are remarkably symmetrical. In the
century, this property was codified in
mathematical concept of a group invented
and then that of a continuous group by
In 1967, Victor Kac (MIT), then working in
[...], and Bob Moody (Alberta) [...] independently enlarged
the paradigm of classical Lie algebras,
in new algebras which are
representation theory of a subclass of
algebras, the affine Kac-Moody algebras,
developed into a mature mathematics.
By the 1980s, these algebras had been taken up by physicists working in the areas of elementary particle theory, gravity, and two-dimensional phase transitions as an obvious framework from which to consider descriptions of nonperturbative solutions of gauge theory, vertex emission operators in string theory on compactified space, integrability in two-dimensional quantum field theory, and conformal field theory. Recently Kac-Moody algebras have been shown to serve as duality symmetries of nonperturbative strings appearing to relate all superstrings to a single theory. The infinite-dimensional Lie algebras and groups have been suggested as candidates for a unified symmetry of superstring theory.
While a proper understanding of M-theory still eludes us, much is now known about it. In particular the various geometric results that have emerged from string theory become related in interesting but mysterious ‘dualities’ whose real meaning has yet to be discovered.
occur throughout mathematics and science. Representation theory seeks
to understand all the possible ways that an abstract collection of
symmetries can arise. Nineteenth-century representation theory helped
to explain the structure of electron orbitals, and 1920s representation
theory is at the heart of quantum chromodynamics. In number theory,
p-adic representation theory is central the Langlands program, a family
of conjectures that have guided a large part of number theory for the
past forty years.