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Geometrical Formulation of Quantum Mechanics

By Abhay Ashtekar and Troy A. Schilling

Abstract

..., has a very different appearance. In particular, states are now represented bypointsofasymplecticmanifold (whichhappenstohaveinadditionacomplecticowgeneratedbya Hamiltonianfunction. Thereisthusaremarkablepatible Riemannian metric), observablesarerepresentedbycertain real-valued functionsonthisspaceandthe Schrödingerevolutioniscapturedbythesym- similaritywiththestandardsymplecticformulationofclassicalmechanics. Features|suchasuncertaintiesandstatevectorreductions|whicharespeclassicalconsiderationsandtheWKBapproximation.Moreimportantly,it Thegeometricalformulationshedsconsiderablelightonanumberofissues cictoquantummechanicscanalsobeformulatedgeometricallybutnowrefer totheRiemannianmetric|astructurewhichisabsentinclassicalmechanics. ture.Thegeometricalreformulationprovidesauniedframeworktodiscuss suggestsgeneralizationsofquantummechanics. Thesimplestamongtheseare suchasthesecondquantizationprocedure,theroleofcoherentstatesinsemi- theseandtocorrectamisconception. Finally,italsosuggestsdirectionsin equivalenttothedynamicalgeneralizations thathaveappearedinthelitera-hasanastonishingrangeofapplications|fromquarksandleptonstoneutronstarsand Quantummechanicsisprobablythemostsuccessfulscientictheoryeverinvented.It whichmoreradicalgeneralizationsmaybe found

Year: 1997
OAI identifier: oai:CiteSeerX.psu:10.1.1.153.333
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