Magnetic Weyl Quantization and Semiclassical Limit Satoshi Okumura - PowerPoint PPT Presentation

Magnetic Weyl Quantization and Semiclassical Limit Satoshi Okumura Tohoku University Graduate School of Mathematics 2019. 9. 19.

Motivation Semiclassical Limit Table of contants 1 Introduction 2 Semiclassical Limit What is Semiclassical Limit? Motivation 2 / 11

Motivation Semiclassical Limit What is Semiclassical Limit? Semiclassical Limit ・ Approximate quantum mechanics using classical mechanics. ・ Introducing a semiclassical parameter ε > 0 . ・ When the limit of ε → 0 is taken, we expect classical mechanics and quantum mechanics to match.  d  iε∂ψ ( t ) dtx = ∇ ξ h = ˆ H A ψ ( t )   ?  ← → ∂t d ψ (0) = ψ 0 ∈ L 2 ( R d ) dtξ = −∇ x h + B ˙ x    f : Classical Observable, i.e. real function on R d × R d B : Magnetic Field , A : Vector Potential of Magnetic Field , When d = 2 , A satisfies rot A = B. 3 / 11

� � � � Motivation Semiclassical Limit Motivation Motivation To approximate time evolution generated by Quantum H A with time evolution generated by Classical Hamiltonian ˆ Hamiltonian h . Time Evolution � F qm ( t ) Op [ f ] Quantization f What is this relation? Time Evolution Quantization � f ( t ) Op [ f ( t )] 4 / 11

Motivation Semiclassical Limit Table of contants 1 Introduction 2 Semiclassical Limit Definition of Magnetic Weyl Quantization Time Evolution in Classical System and Quantum System Egorov type Theorem 5 / 11

Motivation Semiclassical Limit Magnetic Weyl Quantization(M˘ antoiu, Purice 2004) Magnetic Weyl Quantization We consider integral of f : 1 ∫ Op A [ f ] := R d × R d F σ [ f ]( Y ) W A ε ( Y ) dY. (2 π ) d We call a map f �→ Op A [ f ] as Magnetic Weyl Quantization of f . σ (( x, ξ ) , ( y, η )) := ξ · y − x · η, (syplectic form) , ( Y = ( y, η ) ∈ R d × R d ) , W A ε ( Y ) := exp ( iσ ( Y, ( Q, P A ε ))) , 1 ∫ R d × R d e iσ ( Y,Y ′ ) f ( Y ′ ) dY ′ . F σ [ f ]( Y ) := (2 π ) d 6 / 11

Motivation Semiclassical Limit Classical Time Evolution f ( t ) = f ( x ( t ) , ξ ( t )) , d d d ∂f ∂h − ∂f ∂h ∂f ∂h ∑ ∑ dt ( f ( t )) = + B jk ∂ξ j ∂x j ∂x j ∂ξ j ∂ξ j ∂ξ k j =1 j,k =1 =: { h, f ( t ) } B , f (0) = f Quantum time evolution F A qm ( t ) : Quantum observable at time t ∈ R .  iε ∂ qm ( t ) , ˆ ∂t F A qm ( t ) = [ F A H A ] , F A qm (0) = Op A [ f ]  ε t ˆ i H A Op A [ f ] e − i ε t ˆ H A . F A qm ( t ) = e  7 / 11

Motivation Semiclassical Limit Semiclassical Limit Egorov Type Theorem(Lein [6] 2010) ε ˆ ε ˆ F qm ( t ) := e i t H A Op A [ f ] e − i t H A , F cl ( t ) := Op A [ f ( t )] , Then there exists C T > 0 such that, ∥ F cl ( t ) − F qm ( t ) ∥ B ( L 2 ( R d )) ≤ ε 2 C T , ( ∀ t ∈ [ − T, T ]) Egorov type theorem(O. 2018) There exists g t such that F qm ( t ) = Op A [ g t ] + O B ( L 2 ( R d )) ( ε ∞ ) (1) x ( g t ( x, ξ ) − f ( x ( t ) , ξ ( t ))) | ≤ C T ε 2 | ∂ α ξ ∂ β (2) ∞ ∑ ε 2 n g n ( x, ξ ) . g t ( x, ξ ) ≍ n =0 8 / 11

Motivation Semiclassical Limit Summary Magnetic Weyl Quantization is good quantization when magnetic field exists : 1 ∫ Op A [ f ] := R d × R d F σ [ f ]( Y ) W A ε ( Y ) dY. (2 π ) d The difference in time evolution generated by Classical Hamiltonian h and Quantum Hamiltonian Op A [ h ] is small. 9 / 11

Motivation Semiclassical Limit References A.Arai ， "Mathematical Structure of Quantum Mecanics I ， II" ， Asakura ， 1999 in Japanese ． A.Arai ， "Mathematical Aspects of Quantum Phenomenon" ， Asakura ， 2006 in Japanese ． Helmut Abels ， " Pseudodifferential and Singular Integral Operators ， " De Gruyter ， 2012 ． Maciej Zworski ， " Semiclassical Analysis " ， AMS ， 2012 ． antoiu and Radu Purice ， The magnetic Weyl Marius M˘ calculus Journal of Mathematical Physics 45, 1394 (2004) Max Lein ， " Semiclassical Dynamics and Magnetic Weyl Calculus " PhD thesis Technische Universität München, Germany, 2010 10 / 11

Motivation Semiclassical Limit References Max Lein , “ Two-parameter Asymptotics in Magnetic Weyl Calculus ” ， Journal of Mathematical Physics 51, p. 123519, 2010 ． M.W.Wong ， " Weyl Transformations " ， Springer ， 1999 ． D. Robert," de l’Approximation Semi-Classique| " Birkhäuser, 1987 antoiu and Radu Purice ， "Magnetic Viorel Iftimie, Marius M˘ Pseudodifferential Operators" " ， RIMS,Kyoto Univ ， 43 ， 585-623 ， 2007 ． 11 / 11