对易和非对易平面上的Chern-Simons力学分析

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1、北京化工人学硕士论文Chern-SimonsQuantumMechanicsintheGeneralizedCommutativeandNon．commutativePlaneABSTRACTChern-Simonsquantummechanicsinthegeneralizedcommutativeandnon-commutativeplaneisinvestigatedinthisthesis．Wemainlyadoptthepathintegralformulationinthisartic

2、le，becauseallthenumbersinthepathintegralareC·number,itmayavoidtheproblemofoperatorordering，Inthecommutativeplane，wegettheenergyspectrabythepathintegralandfindthatinthelimitofm—÷0．oneofthefrequencieswillbedivergent．Inordertogetaphysicalresultinthislimi

3、t，wemustregularizethespectraandaffordaphysicalexplanation．Atlast，weverifythattheregulationwemadeiscorrect．Thenwegeneralizethemodeltothenon-commutativeplane．Inthenon-commutativeplane，wefindsomethingmoreinteresting．Respondingtothetwodifferentwaysincoupl

4、ingofthemagneticfield，therearetwodifferentChem—Simonsmodelsinthenon—commutativeplane．Wefindthatbesidesthequantityoftheparticle，thereisalsoanotherparameterinthefulltheory；theymayleadtotwodifferentreducedtheorieswhenthesetwoparameterstakezerovalue，respe

5、ctively．InthefulltheorytheclassicalaspectIII北京化T人学硕．1：论文hasthecontinuouslimitwhentheparameterxtakeszerovalueinbothmodels．However,inthequantumaspectthepropertiesofthesetwomodelsbehaveinterestingly．Inthefirstmodel，thespectraofthefulltheoryhavethecontinu

6、ouslimitwhenoneoftheparameterstakeszerovaluewhiletheydonotwhentheotherparameter,say,themassoftheparticle，takesthesamevalue．Thespectraofthefulltheorywillbedivergentwhenthemasstendstozero．Inordertogetafiniteresult，wemustregularizethespectraproperlyandgi

7、veareasonablephysicalexplanation．Finally,weresorttotheDiracTheorytoverifythattheregularizationwemadeisreasonable．Forthesecondmodel，wefindthatthespectrawillbedivergentwhenthetwoparameterstakezerovalue，respectively．Weproposeregularizationstodealwiththes

8、pectra．Andatlast，weresorttotheDiractheoryandtheFaddeev—Jackiwreductiontoshowthattheregularizationwemadeiscorrect．Keywords：non··commutability；Chern··Simonsquantummechanics；pathintegral；DiracTheoryIV北京化工大学位论文原创性声明本人郑重声明：所呈交的学位论文，是本人在导师的指导下，独立进行研