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§1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts
Star product and its application
AkiraYoshioka
Dept. Math. TokyoUniversityofScience,Japan
16May2014Vancouver
AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan
Starproductanditsapplication
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§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
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§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
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§1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts
Wehaveanotherwaytogivethesamealgebrabyusingonly
functions,notusingoperators.
Insteadofusingoperators,weintroduceanassociativeproduct∗
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intothespaceoffunctionsof(q,p).
Theproductisdifferentfromtheusualmultiplicationoffunctions,
butisgivenasadeformationoftheusualmultiplicationinthe
followingway. (Cf.
Bayen-Flato-Fronsdal-Lichnerowicz-Sternheimer[1],Moyal).
AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan
Starproductanditsapplication
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§1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts
Theproduct∗
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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
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§1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts
Forpolynomials f,gweconsideraproduct f ∗ ggivenbythe
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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
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§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∗
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[p,q]∗ = p ∗ q−q∗ p = i~
O O
AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan
Starproductanditsapplication
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§1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts
Theproduct∗ isassociativeonpolynomials,andthenweobtain
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theWeylalgebragivenbytheordinarypolynomialswiththe
product∗ ,(C[q,p],∗ ).
O O
UsingthisWeylalgebraoftheproduct∗ ,wecanobtainsome
O
resultsofquantummechanics,andfurtherwecandiscusssome
extensions.
Inthistalk,wegiveabriefreviewonthispointmainlyrelatedour
investigation([4],[8]).
AkiraYoshiokaDept.Math.TokyoUniversityofScience,Japan
Starproductanditsapplication
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§1.Introduction:theidea §2.Starcalculationofeigenvalues §3.Starproducts
§2.1.EigenvaluesofHarmonicOscillator
§2. Star calculation of eigenvalues
§2.1.EigenvaluesofHarmonicOscillator
Wecancalculatetheeigenvaluesoftheharmonicoscillatorby
meansofthestarproduct∗ .
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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
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§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
Description:plays a fundamental role in quantum mechanics. Akira Yoshioka Dept. Math. Tokyo University of Science, Japan. Star product and its application
