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1、11Preface:WhyCli ordAlgebraisUsefulWebeginbydiscussingwhyweshouldcareaboutCli ordAlgebra.(IfyouwantanoverviewofhowCli ordAlgebraactuallyworks,skiptosection2.)(1)TherealnumbersareasubalgebraofCli ordalgebra:justthrowawayallelementswithgrade>0.Alasthis
2、doesn'ttellusmuchbeyondwhatwealreadyknew.(2)OrdinaryvectoralgebraisanothersubalgebraofCli ordalgebra.Alas,again,thisdoesn'ttellusmuchbeyondwhatwealreadyknew.(3)ThecomplexnumbersareanothersubalgebraofCli ordalgebra,asdiscussedinref-erence1.Thisgivesus
3、efulinsightintocomplexnumbersandintorotationsintwodimensions.(4)QuaternionscanbeunderstoodintermsofanothersubalgebraofCli ordalgebra,namelythesubalgebracontainingjustscalarsandbivectors.Thisistremendouslyusefulfordescribingrotationsinthreeormoredimen
4、sions(includingfour-dimensionalspacetime).Seereference2.NotethatthePaulispinmatricesareisomorphictoquaternions.(5)Itisadvantageoustogetridofcrossproductsandreplacethemwithwedgeproducts.Thecrossproductonlymakessenseinthreedimensions,becausetheresultpo
5、intsinthe"directionperpendiculartothetwomultiplicands.Intwodimensions,therecannotbeanysuchdirection,andinfourdimensions(orhigher),thereisanin nitudeofdirectionsperpendiculartoanytwovectors,sotalkingaboutthe"perpendiculardirectionissounderspeci edas
6、tobemeaningless.Incontrast,thewedgeproductiswellde nedandwellbehavedinalldimensionsfromtwodimensionsonup.Thewedgeproductoftwovectorsisantisymmetric,andinvolvesthesineoftheanglebetweentwovectors...andthesamecanbesaidofthecrossproduct.However,thesimila
7、ritiesmore-or-lessendthere.Theresultofacrossproductisavector,whereastheresultofawedgeproductisabivector.Thecrossproductisde nedintermsofarighthandrule."Incontrast,awedgeproductiswellde nedwithoutanynotionofhandedness,withoutanynotionofchirality.This
8、ismoreimportantthanitmightseem,becauseitchangeshowweperceivetheapparentsymmetryofthelawsofphysics,asdiscussedinreference3.(6)Inallofphysics,wheneveryouseeanideaexpressedasthecrossproductofvectors,youwillusuallybemuchbettero ifyoure-expresstheideainte
