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Magnetic Weyl Quantization and Semiclassical Limit
Satoshi Okumura
Tohoku University Graduate School of Mathematics
- 2019. 9. 19.
Motivation Semiclassical Limit
Magnetic Weyl Quantization and Semiclassical Limit Satoshi Okumura - - PowerPoint PPT Presentation
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
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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 dtx = ∇ξh d dtξ = −∇xh + B ˙ x
?
← → iε∂ψ(t) ∂t = ˆ HAψ(t) ψ(0) = ψ0 ∈ L2(Rd) f : Classical Observable, i.e. real function on Rd × Rd B : Magnetic Field, A : Vector Potential of Magnetic Field,
When d = 2,
Asatisfies rotA = B.
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Motivation Semiclassical Limit
Motivation
Motivation To approximate time evolution generated by Quantum Hamiltonian ˆ
HA with time evolution generated by Classical
Hamiltonian h.
f
Quantization
- Time Evolution
- Op[f]
Time Evolution
f(t)
Quantization
Fqm(t)
What is this relation?
- Op[f(t)]
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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
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Motivation Semiclassical Limit
Magnetic Weyl Quantization(M˘ antoiu, Purice 2004)
Magnetic Weyl Quantization We consider integral of f :
OpA[f] := 1 (2π)d ∫
Rd×Rd Fσ[f](Y )W A ε (Y )dY.
We call a map f → OpA[f] as Magnetic Weyl Quantization of f.
σ((x, ξ), (y, η)) := ξ · y − x · η,
(syplectic form),
W A
ε (Y ) := exp (iσ(Y, (Q, P A ε ))),
(Y = (y, η) ∈ Rd × Rd), Fσ[f](Y ) := 1 (2π)d ∫
Rd×Rd eiσ(Y,Y ′)f(Y ′)dY ′.
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Motivation Semiclassical Limit
Classical Time Evolution
f(t) = f(x(t), ξ(t)), d dt(f(t)) =
d
∑
j=1
∂f ∂ξj ∂h ∂xj − ∂f ∂xj ∂h ∂ξj +
d
∑
j,k=1
Bjk ∂f ∂ξj ∂h ∂ξk =: {h, f(t)}B, f(0) = f
Quantum time evolution
F A
qm(t) : Quantum observable at time t ∈ R.
iε ∂ ∂t F A
qm(t) = [F A qm(t), ˆ
HA], F A
qm(0) = OpA[f]
F A
qm(t) = e
i ε t ˆ
HAOpA[f]e− i
ε t ˆ
HA.
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Motivation Semiclassical Limit
Semiclassical Limit
Egorov Type Theorem(Lein [6] 2010)
Fcl(t) := OpA[f(t)], Fqm(t) := ei t
ε ˆ
HAOpA[f]e−i t
ε ˆ
HA,
Then there exists CT > 0 such that,
∥Fcl(t) − Fqm(t)∥B(L2(Rd)) ≤ ε2CT , (∀t ∈ [−T, T])
Egorov type theorem(O. 2018) There exists gt such that
Fqm(t) = OpA[gt] + OB(L2(Rd))(ε∞)
(1)
|∂α
ξ ∂β x(gt(x, ξ) − f(x(t), ξ(t)))| ≤ CT ε2
(2)
gt(x, ξ) ≍
∞
∑
n=0
ε2ngn(x, ξ).
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Motivation Semiclassical Limit
Summary
Magnetic Weyl Quantization is good quantization when magnetic field exists :
OpA[f] := 1 (2π)d ∫
Rd×Rd Fσ[f](Y )W A ε (Y )dY.
The difference in time evolution generated by Classical Hamiltonian h and Quantum Hamiltonian OpA[h] is small. 9 / 11
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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． Marius M˘ antoiu and Radu Purice，The magnetic Weyl 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
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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,