Open Questions: Symmetry
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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

Symmetry
 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.
 Groups and PhysicsDogmatic Opinions of a Senior Citizen
A. John Coleman
Notices of the AMS, January 1997, pp, 817
 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. 744752
 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. 416421
 Quasicrystals are solids with unexpected symmetry properties
of their Xray diffraction patterns. For example, the diffraction
patterns can have 10fold 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. 4650
 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 KacMoody Symmetry for Physics
Louise Dolan
Notices of the AMS, December 1995, pp. 14891495
 A KacMoody algebra is a type of infinitedimensional Lie
algebra. KacMoody 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. 2631
 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]
 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.
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Copyright © 2002 by Charles Daney, All Rights Reserved