Star product and its application PDF
Preview Star product and its application
...... ..... .............. ... ..... .... ........... . .. . . . . . . §1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts Star product and its application AkiraYoshioka Dept. Math. TokyoUniversityofScience,Japan 16May2014Vancouver AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan Starproductanditsapplication ...... ..... .............. ... ..... .... ........... . .. . . . . . . §1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts Abstract Weintroduceastarproduct∗ onpolynomialsandshow O someexpamles. Wedefinegeneralstarproducts∗Λ,∗ . K Thenweintroduceexponentialelementinthestarproduct algebra. Usingthestarexponentialelementswedefineseveral functionscalledstarfunctionsinthealgebra. Weshowcertainexpamplesofstarfunctions. AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan Starproductanditsapplication ...... ..... .............. ... ..... .... ........... . .. . . . . . . §1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts §1. Introduction: the idea Thecanonicalcommutationrelationisabasicidentityofquantum mechanics,whichisgivenbyapairofoperatorssuchas [pˆ,qˆ] = pˆqˆ −qˆ pˆ = i~ where pˆ = i~∂ andqˆ isamultiplicationoperatorq×actingonthe q functionsofq,and~isthePlankconstant. Thealgebrageneratedby pˆ andqˆ iscalledtheWeylalgebrawhich playsafundamentalroleinquantummechanics. AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan Starproductanditsapplication ...... ..... .............. ... ..... .... ........... . .. . . . . . . §1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts Wehaveanotherwaytogivethesamealgebrabyusingonly functions,notusingoperators. Insteadofusingoperators,weintroduceanassociativeproduct∗ O intothespaceoffunctionsof(q,p). Theproductisdifferentfromtheusualmultiplicationoffunctions, butisgivenasadeformationoftheusualmultiplicationinthe followingway. (Cf. Bayen-Flato-Fronsdal-Lichnerowicz-Sternheimer[1],Moyal). AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan Starproductanditsapplication ...... ..... .............. ... ..... .... ........... . .. . . . . . . §1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts Theproduct∗ O Forsmoothfunctions f,gonR2,wehavethecanonicalPoisson bracket {f,g}(q,p) = ∂ f∂ g−∂ f∂ g, (q,p) ∈ R2 p q q p Indeformationquantization,weveryoftenusethenotationof ←− →− ←− −→ bidifferentialopearator∂ ·∂ −∂ ·∂ suchas p q q p ( ) ←− →− ←− −→ {f,g} = f ∂ ·∂ −∂ ·∂ g = ∂ f ∂ g−∂ f ∂ g p q q p p q q p AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan Starproductanditsapplication ...... ..... .............. ... ..... .... ........... . .. . . . . . . §1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts Forpolynomials f,gweconsideraproduct f ∗ ggivenbythe O exponentialpowerseriesofthebidifferentialoperator ←− →− ←− −→ ∂ ·∂ −∂ ·∂ suchthat p q q p f∗ g = f exp i~ (←∂−·→−∂ −←∂−·−∂→) g = f ∑∞ 1 (i~)k(←∂−·→−∂ −←∂−·−∂→)k g O 2 p q q p k! 2 p q q p k=0 = fg+ i~f (←∂−·→−∂ −←∂−·−∂→) g+ 1 (i~)2 f (←∂−·→−∂ −←∂−·−∂→)2 g 2 p q q p 2! 2 p q q p +···+ 1 (i~)k f (←∂−·→−∂ −←∂−·−∂→)k g+··· k! 2 p q q p where~isapositivenumber. Theproductiswell-definedand associativeforpolynomials. AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan Starproductanditsapplication ...... ..... .............. ... ..... .... ........... . .. . . . . . . §1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts Nowwecalculatethecommutatorofthefunctions pandq. Wesee p∗ q = pexp i~ (←∂−·→−∂ −←∂−·−∂→) q = p∑∞ 1 (i~)k(←∂−·→−∂ −←∂−·−∂→)k q O 2 p q q p k! 2 p q q p ( k=0 ) ←− →− ←− −→ = pq+ i~p ∂ ·∂ −∂ ·∂ q = pq+ i~ 2 p q q p 2 Similarlywesee q ∗ p = pq− i~ O 2 Thenthefunctions pandqsatisfythecanonicalcommutation relationunderthecommutatoroftheproduct∗ O [p,q]∗ = p ∗ q−q∗ p = i~ O O AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan Starproductanditsapplication ...... ..... .............. ... ..... .... ........... . .. . . . . . . §1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts Theproduct∗ isassociativeonpolynomials,andthenweobtain O theWeylalgebragivenbytheordinarypolynomialswiththe product∗ ,(C[q,p],∗ ). O O UsingthisWeylalgebraoftheproduct∗ ,wecanobtainsome O resultsofquantummechanics,andfurtherwecandiscusssome extensions. Inthistalk,wegiveabriefreviewonthispointmainlyrelatedour investigation([4],[8]). AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan Starproductanditsapplication ...... ..... .............. ... ..... .... ........... . .. . . . . . . §1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts §2.1.EigenvaluesofHarmonicOscillator §2. Star calculation of eigenvalues §2.1.EigenvaluesofHarmonicOscillator Wecancalculatetheeigenvaluesoftheharmonicoscillatorby meansofthestarproduct∗ . O Eigenvalues TheSchrd¨ingieroperatoroftheharmonicoscillatoris ( ) Hˆ = −~2 ∂ 2+ 1q2. 2 ∂q 2 Theeigenvaluesare E = ~(n+ 1), n = 0,1,2,··· n 2 AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan Starproductanditsapplication ...... ..... .............. ... ..... .... ........... . .. . . . . . . §1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts §2.1.EigenvaluesofHarmonicOscillator Star product calculation Paralleltotheargumentsinquantummechanics,wecancalculate theeigenvalues E byusingthestarproduct∗ andthefunctions n O of pandqinthefollowingway. Theclassicalhamiltonianfunctionis H = 1(p2+q2). 2 Weputfunctionssuchas 1 1 a = √ (p+iq), a† = √ (p−iq). 2~ 2~ AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan Starproductanditsapplication
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