资源描述:
《Clifford Algebra 克利福德代数介绍.pdf》由会员上传分享,免费在线阅读,更多相关内容在教育资源-天天文库。
1、11Preface:WhyCliordAlgebraisUsefulWebeginbydiscussingwhyweshouldcareaboutCliordAlgebra.(IfyouwantanoverviewofhowCliordAlgebraactuallyworks,skiptosection2.)(1)TherealnumbersareasubalgebraofCliordalgebra:justthrowawayallelementswithgrade>0.Alasthis
2、doesn'ttellusmuchbeyondwhatwealreadyknew.(2)OrdinaryvectoralgebraisanothersubalgebraofCliordalgebra.Alas,again,thisdoesn'ttellusmuchbeyondwhatwealreadyknew.(3)ThecomplexnumbersareanothersubalgebraofCliordalgebra,asdiscussedinref-erence1.Thisgivesus
3、efulinsightintocomplexnumbersandintorotationsintwodimensions.(4)QuaternionscanbeunderstoodintermsofanothersubalgebraofCliordalgebra,namelythesubalgebracontainingjustscalarsandbivectors.Thisistremendouslyusefulfordescribingrotationsinthreeormoredimen
4、sions(includingfour-dimensionalspacetime).Seereference2.NotethatthePaulispinmatricesareisomorphictoquaternions.(5)Itisadvantageoustogetridofcrossproductsandreplacethemwithwedgeproducts.Thecrossproductonlymakessenseinthreedimensions,becausetheresultpo
5、intsinthe"directionperpendiculartothetwomultiplicands.Intwodimensions,therecannotbeanysuchdirection,andinfourdimensions(orhigher),thereisaninnitudeofdirectionsperpendiculartoanytwovectors,sotalkingaboutthe"perpendiculardirectionissounderspeciedas
6、tobemeaningless.Incontrast,thewedgeproductiswelldenedandwellbehavedinalldimensionsfromtwodimensionsonup.Thewedgeproductoftwovectorsisantisymmetric,andinvolvesthesineoftheanglebetweentwovectors...andthesamecanbesaidofthecrossproduct.However,thesimila
7、ritiesmore-or-lessendthere.Theresultofacrossproductisavector,whereastheresultofawedgeproductisabivector.Thecrossproductisdenedintermsofarighthandrule."Incontrast,awedgeproductiswelldenedwithoutanynotionofhandedness,withoutanynotionofchirality.This
8、ismoreimportantthanitmightseem,becauseitchangeshowweperceivetheapparentsymmetryofthelawsofphysics,asdiscussedinreference3.(6)Inallofphysics,wheneveryouseeanideaexpressedasthecrossproductofvectors,youwillusuallybemuchbetteroifyoure-expresstheideainte