Symplectic geometry for classical and quantum magnetic fields San V - PowerPoint PPT Presentation

Symplectic geometry for classical and quantum magnetic fields San V˜ u Ngo .c Univ. Rennes 1 & Institut Universitaire de France Symplectic Techniques in Dynamical Systems ICMAT, Madrid November 11-15, 2013 joint work with Nicolas Raymond (Rennes) and Fr´ ed´ eric Faure (Grenoble) arXiv:1306.5054 San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 1/30

The magnetic Laplacian Consider a charged particle in a domain X ⊂ R n (or Riemannian manifold) moving in a non-vanishing, time-independent magnetic field. San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 2/30

The magnetic Laplacian Consider a charged particle in a domain X ⊂ R n (or Riemannian manifold) moving in a non-vanishing, time-independent magnetic field. Magnetic field: B ∈ Ω 2 ( X ) is a differential 2-form. San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 2/30

The magnetic Laplacian Consider a charged particle in a domain X ⊂ R n (or Riemannian manifold) moving in a non-vanishing, time-independent magnetic field. Magnetic field: B ∈ Ω 2 ( X ) is a differential 2-form. Magnetic potential: A ∈ Ω 1 ( X ) , s.t. dA = B . San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 2/30

The magnetic Laplacian Consider a charged particle in a domain X ⊂ R n (or Riemannian manifold) moving in a non-vanishing, time-independent magnetic field. Magnetic field: B ∈ Ω 2 ( X ) is a differential 2-form. Magnetic potential: A ∈ Ω 1 ( X ) , s.t. dA = B . Classical Hamiltonian: H ( q, p ) = � p − A ( q ) � 2 on T ∗ X . San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 2/30

The magnetic Laplacian Consider a charged particle in a domain X ⊂ R n (or Riemannian manifold) moving in a non-vanishing, time-independent magnetic field. Magnetic field: B ∈ Ω 2 ( X ) is a differential 2-form. Magnetic potential: A ∈ Ω 1 ( X ) , s.t. dA = B . Classical Hamiltonian: H ( q, p ) = � p − A ( q ) � 2 on T ∗ X . Gauge transformation: p �→ p + d f . San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 2/30

The magnetic Laplacian Consider a charged particle in a domain X ⊂ R n (or Riemannian manifold) moving in a non-vanishing, time-independent magnetic field. Magnetic field: B ∈ Ω 2 ( X ) is a differential 2-form. Magnetic potential: A ∈ Ω 1 ( X ) , s.t. dA = B . Classical Hamiltonian: H ( q, p ) = � p − A ( q ) � 2 on T ∗ X . Gauge transformation: p �→ p + d f . � 2 � � � ∂ Quantum Hamiltonian: H = − a j . i ∂q j Here X = R n , A = a 1 dq 1 + · · · a n dq n . San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 2/30

The magnetic Laplacian Consider a charged particle in a domain X ⊂ R n (or Riemannian manifold) moving in a non-vanishing, time-independent magnetic field. Magnetic field: B ∈ Ω 2 ( X ) is a differential 2-form. Magnetic potential: A ∈ Ω 1 ( X ) , s.t. dA = B . Classical Hamiltonian: H ( q, p ) = � p − A ( q ) � 2 on T ∗ X . Gauge transformation: p �→ p + d f . � 2 � � � ∂ Quantum Hamiltonian: H = − a j . i ∂q j Here X = R n , A = a 1 dq 1 + · · · a n dq n . Gauge transformation: unitary conjugation by e if/ � . San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 2/30

Motivations Maths: much less studied than electric field ∆ + V San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 3/30

Motivations Maths: much less studied than electric field ∆ + V (some similarities, sometimes mysterious) San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 3/30

Motivations Maths: much less studied than electric field ∆ + V (some similarities, sometimes mysterious) Earth’s magnetic field San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 3/30

Motivations Maths: much less studied than electric field ∆ + V (some similarities, sometimes mysterious) Earth’s magnetic field Superconductors [Fournais-Helffer, Lu-Pan, etc.] San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 3/30

The Semiclassical Credo Quantum Mechanics is related to Classical Mechanics Quantum Classical Quantum state Classical particle Hilbert space Phase space (symplectic manifold) L 2 ( X ) T ∗ X San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 4/30

The Semiclassical Credo Quantum Mechanics is related to Classical Mechanics Quantum Classical Quantum state Classical particle Hilbert space Phase space (symplectic manifold) L 2 ( X ) T ∗ X Quantum observable: Classical observable: selfadjoint operator ˆ H Hamiltonian H ∈ C ∞ ( M ) San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 4/30

The Semiclassical Credo Quantum Mechanics is related to Classical Mechanics Quantum Classical Quantum state Classical particle Hilbert space Phase space (symplectic manifold) L 2 ( X ) T ∗ X Quantum observable: Classical observable: selfadjoint operator ˆ H Hamiltonian H ∈ C ∞ ( M ) � 2 ∆ g � ξ � 2 � ∂ ξ j i ∂x j � 2 � � � ∂ H ( q, p ) = � p − A ( q ) � 2 H = − a j i ∂q j San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 4/30

The Semiclassical Credo Quantum Mechanics is related to Classical Mechanics Quantum Classical Quantum state Classical particle Hilbert space Phase space (symplectic manifold) L 2 ( X ) T ∗ X Quantum observable: Classical observable: selfadjoint operator ˆ H Hamiltonian H ∈ C ∞ ( M ) � 2 ∆ g � ξ � 2 � ∂ ξ j i ∂x j � 2 � � � ∂ H ( q, p ) = � p − A ( q ) � 2 H = − a j i ∂q j Op w � ( p ) with p 0 = principal symbol p ∼ p 0 + � p 1 + � 2 p 2 + · · · . . . . . . San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 4/30

The Semiclassical Credo Quantum Mechanics is related to Classical Mechanics Quantum Classical Quantum state Classical particle Hilbert space Phase space (symplectic manifold) L 2 ( X ) T ∗ X Quantum observable: Classical observable: selfadjoint operator ˆ H Hamiltonian H ∈ C ∞ ( M ) � 2 ∆ g � ξ � 2 � ∂ ξ j i ∂x j � 2 � � � ∂ H ( q, p ) = � p − A ( q ) � 2 H = − a j i ∂q j Op w � ( p ) with p 0 = principal symbol p ∼ p 0 + � p 1 + � 2 p 2 + · · · . . . . . . The Hamiltonian dynamics of H can be used rigorously to get information on the spectrum of ˆ H , in the regime � → 0 . San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 4/30

The Semiclassical Credo Quantum Mechanics is related to Classical Mechanics Quantum Classical Quantum state Classical particle Hilbert space Phase space (symplectic manifold) L 2 ( X ) T ∗ X Quantum observable: Classical observable: selfadjoint operator ˆ H Hamiltonian H ∈ C ∞ ( M ) � 2 ∆ g � ξ � 2 � ∂ ξ j i ∂x j � 2 � � � ∂ H ( q, p ) = � p − A ( q ) � 2 H = − a j i ∂q j Op w � ( p ) with p 0 = principal symbol p ∼ p 0 + � p 1 + � 2 p 2 + · · · . . . . . . The Hamiltonian dynamics of H can be used rigorously to get information on the spectrum of ˆ H , in the regime � → 0 . (and vice-versa !) San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 4/30

Semiclassical analysis for magnetic fields asymptotics, as � → 0 , of eigenfunctions, eigenvalues, gaps, tunnel effect, etc. (many authors !) San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 5/30

Semiclassical analysis for magnetic fields asymptotics, as � → 0 , of eigenfunctions, eigenvalues, gaps, tunnel effect, etc. (many authors !) constant magnetic field variable magnetic field, non zero possibly vanishing magnetic field various geometries San V˜ u Ngo .c, Univ. Rennes 1 & Institut Universitaire de France Symplectic geometry for classical and quantum magnetic fields 5/30