Open Questions: Symmetry

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See also: Mathematics and physics -- Geometry and topology


Symmetry groups

Discrete and continuous symmetries

Group representations

Lie groups and Lie algebras

Typical applications

Symmetry and conservation laws

Gauge symmetry


Recommended references: Web sites

Recommended references: Magazine/journal articles

Recommended references: Books


Recommended references: Web sites

Site indexes

Sites with general resources

Teaching Symmetry in the Introductory Physics Curriculum
This is an outstanding site, about the concept of symmetry that is so fundamental to modern physics. The site is named for Emmy Noether (the mathematician who also conceived "Noetherian rings") on account of her theorem which links symmetry to physical conservation laws.
A Timeline of Symmetry in Physics, Chemistry, and Mathematics
Chronological list of important developments, from the Mathematical Physics Group of the Department of Physics at KTH. Also contains related external links.

Surveys, overviews, tutorials

Article from Wikipedia. See also Group (mathematics), Group theory, Noether's theorem.
Noether's Theorem in a Nutshell
Short and sweet explanation of Noether's theorem, by John Baez.
Introduction to Group Theory
Very simple introduction, all in HTML.
Symmetry and Bound States
Section of an introductory physics course by David Raymond. It covers the relationship between symmetry and conservation laws -- Noether's theorem.
Group Theory
Notes for a first year graduate course by J. S. Milne, available in DVI or PDF format.
An Elementary Introduction to Groups and Representations
An elementary introduction to Lie groups, Lie algebras, and their representations, by Brian C. Hall. Prerequisites are minimized by focusing on matrix groups instead of general Lie groups (which requires some manifold theory).
Group Theory
Complete book by P. Cvitanovic, in PDF format.

Recommended references: Magazine/journal articles

Groups and Physics--Dogmatic Opinions of a Senior Citizen
A. John Coleman
Notices of the AMS, January 1997, pp, 8-17
The theory of continuous groups originated in work by Sophus Lie, Wilhelm Killing, and Isai Schur. It was significantly developed by Elie Cartan and Hermann Weyl, and quickly became incorporated in the theories of both general relativity and quantum mechanics. Since then it has established the importance of symmetry concepts in many additional branches of physics.
[Article in PDF format]
Groupoids: Unifying Internal and External Symmetry
Alan Weinstein
Notices of the AMS, July 1996, pp. 744-752
Groups provide a good way of abstracting the idea of symmetry for many purposes, but they fail to capture some subtleties. Groupoids -- which can be defined as categories in which all monomorphisms have inverses -- remedy this problem. There are applications both inside and outside of mathematics.
[Article in PDF format]
Review of Quasicrystals and Geometry by Marjorie Senechal
Charles Radin
Notices of the AMS, April 1996, pp. 416-421
Quasicrystals are solids with unexpected symmetry properties of their X-ray diffraction patterns. For example, the diffraction patterns can have 10-fold symmetry, even though no crystalline solid itself can have this symmetry. Study of quasicrystals has led to new mathematics called statistical symmetry, that also applies to aperiodic tilings of the plane.
[Article in PDF format]
Symmetry in Physics: Wigner's Legacy
David J. Gross
Physics Today, December 1995, pp. 46-50
Einstein led the way in introducing symmetry concepts into physics, but Eugene Wigner played a major role in making symmetry principles a fundamental part of physics' description of nature.
The Beacon of Kac-Moody Symmetry for Physics
Louise Dolan
Notices of the AMS, December 1995, pp. 1489-1495
A Kac-Moody algebra is a type of infinite-dimensional Lie algebra. Kac-Moody algebras has proven useful in elementary particle theory, gravitation, string theory, and conformal field theories. They have made an appearance as duality symmetries in nonperturbative superstring theory, and are also relevant to number theory and automorphic forms.
[Article in PDF format]
Symmetry and Tilings
Charles Radin
Notices of the AMS, January 1995, pp. 26-31
Certain tilings of the plane (e. g. "pinwheel" and Penrose tilings) do not have symmetry in the traditional sense of invariance under rigid motions. But they do have a more subtle sort of symmetry called statistical symmetry, and there are connections of this with ideas of logic and physics.
[Article in PDF format]

Recommended references: Books

Leon M. Lederman; Christopher T. Hill – Symmetry: And the Beautiful Universe
Prometheus Books, 2004
Symmetry, at least in its scientific applications, is a thoroughly mathematical concept. But it's a different kind of mathematics – abstract algebra – from (say) calculus or probability theory. Lederman, of course, is a physicist, and his book is a good explanation of the striking success of the symmetry concept as applied to the theory of elementary particles. Many theories in physics turn out to be expressible in terms of symmetry, and one of the most important tools, as the authors explain, is Emmy Noether's theorem which relates symmetry and conservation laws.


Copyright © 2002 by Charles Daney, All Rights Reserved