Chapter 10
Symmetry and Degeneracy
Weliveinaworldwhichisalmost,butnotquite,symmetric. TheEarthisround,
nearly,andmovesinanorbitaroundtheSunwhichiscircular,almost. Ourgalaxy
looksroughlylikeaspiral.Humanbeingsandmostanimalshavealeft-rightsymme-
try;theleft-handsidelooksthesameastheright-handside,moreorless. Starfish
haveapentagonalsymmetry,tablesalthasacubicsymmetry,andingeneralNature
aboundsingeometricalshapesofvariouskinds.
Thesymmetrywhichisapparentinlargestructures(animals,planets,galaxies...)
isevenmoreevidentattheverysmallscalesthatarethedomainofquantumphysics.
Symmetries appearin thearrangementof atomsinsolids, inthe arrangementof
electronsinatoms, andinthe structureof the nucleus. Symmetry principlesare
especiallyimportantforunderstandingthevariety andbehaviorof theelementary
particles,outofwhichalllargerstructuresarebuilt.
Themathematical expressionofageometricalsymmetryisaninvarianceunder
sometransformationof coordinates. Letusconsider someexamples whichwillbe
important:
- ReflectionSymmetry Astick-figuresketchisshowninFig.[10.1].Suppose
weassigntoeverypointonthesketchasetofcoordinates(x,y)inthecoordinate
systemshown,andthensupposethateverypointonthesketchismovedtoanother
point
(x,y)→(x′,y′) (10.1)
accordingtotherule
x′=x+a y′=y+b (10.2)
whereaandbareconstants. Thisisanexampleofatransformationofcoordinates,
knownasaTranslation. Underthistransformation,thewholefigureismovedto
somewhereelseintheplane,asshowninFig.[10.2].Thisisnotaninvarianceofthe
sketch,sincethetransformedsketchcanbedistinguished,byitsnewposition,from
theoriginalsketch.Sowesaythatthetranslations(10.2)arenotasymmetryofthe