Doing Physics with Quaternions

《Doing Physics with Quaternions》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、DoingPhysicswithQuaternionsDouglasB.Sweetser©2005dougAllrighsreserved.1INDEXIntroduction2WhatareQuaternions?3UnifyingTwoViewsofEvents4ABriefHistoryofQuaternionsMathematics6MultiplyingQuaternionstheEasyWay7Scalars,Vectors,TensorsandAllThat11InnerandOuterProductsofQuat

2、ernions13QuaternionAnalysis23TopologicalPropertiesofQuaternions28QuaternionAlgebraToolSetClassicalMechanics32Newton'sSecondLaw35OscillatorsandWaves37FourTestsofforaConservativeForceSpecialRelativity40RotationsandDilationsCreatetheLorentzGroup43AnAlternativeAlgebraforLorentzBoostsElectromag

3、netism48ClassicalElectrodynamics51ElectromagneticFieldGauges53TheMaxwellEquationsintheLightGauge:QED?56TheLorentzForce58TheStressTensoroftheElectromagneticFieldQuantumMechanics62ACompleteInnerProductSpacewithDirac'sBracketNotation67MultiplyingquaternionsinPolarCoordinateForm69Commutatorsan

4、dtheUncertaintyPrinciple74UnifyingtheRepresentationsofIntegralandHalf−IntegralSpin79DerivingAQuaternionAnalogtotheSchrödingerEquation83IntroductiontoRelativisticQuantumMechanics86TimeReversalTransformationsforIntervalsGravity89UnifiedFieldTheorybyAnalogy101Einstein'svisionI:Classicalunifie

5、dfieldequationsforgravityandelectromagnetismusingRiemannianquaternions115Einstein'svisionII:Aunifiedforceequationwithconstantvelocityprofilesolutions123StringsandQuantumGravity127AnsweringPrimaFacieQuestionsinQuantumGravityUsingQuaternions134LengthinCurvedSpacetime136ANewIdeaforMetrics138T

6、heGravitationalRedshift140ABriefSummaryofImportantLawsinPhysicsWrittenasQuaternions155Conclusions2WhatAreQuaternions?Quaternionsarenumbersliketherealnumbers:theycanbeadded,subtracted,multiplied,anddivided.Thereissomethingoddaboutthem,hintedatbytheLatin.Quaternionsarecomposedoffournumbersth

7、atworktogetherasone.Theywerediscoveredbyseveralpeoplebackintheeighteenhundreds.Someenthusiaststhoughtquaternionswouldbeabletoexpresseverythingthatcouldhappeninourthreedimensionsofspaceandonefortimebecausequaternionsnaturallyhadthatformtoo.Mathaccidentsdonothap