Boundaries in QuantumPhysics Manuel Asorey Universidad de Zaragoza - PowerPoint PPT Presentation

Boundaries in QuantumPhysics Manuel Asorey Universidad de Zaragoza Coloquium UC3M Madrid, October 2011

Philosophical approach Den FestKörper hat Gott geschaffen, die Oberfläche der Teufel W. Pauli

Philosophical approach God created the volumes, the Devil the boundaries W. Pauli

Quantum Boundary Effects • Casimir effect • Aharonov-Bohm effect • Edge states & conductivity quantization in QHE • Energy gap in Graphene physics • Plasmons and surface effects

Quantum Boundary Effects • Casimir effect • Aharonov-Bohm effect • Edge states & conductivity quantization in QHE • Energy gap in Graphene physics • Plasmons and surface effects • Topology change and Black Hole radiation • Cosmological effects • String Theory, D-branes, Holographic principle and AdS/CFT

Quantum Control: Boundary Shapes

Quantum Control: Boundary Shapes Integrable system

Quantum Control: Boundary Shapes Integrable system Chaotic system

Quantum Control: Boundary Shapes Attractive

Quantum Control: Boundary Shapes Attractive Repulsive

New Boundaries • Metamaterials • Monoatomic layers (Graphene, nanotubes, etc • Defects and space singularities (Black holes, domain walls, cosmic strings, ...)

New Boundaries • Metamaterials • Monoatomic layers (Graphene, nanotubes, etc • Defects and space singularities (Black holes, domain walls, cosmic strings, ...) • Open spaces and Quantum gravity

New Boundaries • Metamaterials • Monoatomic layers (Graphene, nanotubes, etc • Defects and space singularities (Black holes, domain walls, cosmic strings, ...) • Open spaces and Quantum gravity • AdS/CFT dualities. Brane World Physics

New Boundaries • Metamaterials • Monoatomic layers (Graphene, nanotubes, etc • Defects and space singularities (Black holes, domain walls, cosmic strings, ...) • Open spaces and Quantum gravity • AdS/CFT dualities. Brane World Physics

Singular Hamiltonians and AdS/CFT • Free scalar field in anti-deSitter space-time � dt dz � � φ | 2 + | ∂ z φ | 2 − |∇ φ | 2 − m 2 | φ | 2 � S ( φ ) = 1 d 3 x | ˙ z 2 2

Singular Hamiltonians and AdS/CFT • Free scalar field in anti-deSitter space-time � dt dz � � φ | 2 + | ∂ z φ | 2 − |∇ φ | 2 − m 2 | φ | 2 � S ( φ ) = 1 d 3 x | ˙ z 2 2 • Effective Hamiltonian dz 2 + m 2 + 15 d 2 H = − 1 4 z 2 2 in the half line z ∈ (0 , ∞ ) • H is symmetric but in this case we have three different regimes

Singular Hamiltonians and AdS/CFT 4 < m 2 + 15 i) 3 4 ⇒ unique selfadjoint extension

Singular Hamiltonians and AdS/CFT 4 < m 2 + 15 i) 3 4 ⇒ unique selfadjoint extension 4 < m 2 + 15 ii) − 1 4 < 3 4 ⇒ family of bc parametrized by α ∈ I R � � x = 0 2 x ψ � ( x ) = lim lim 1 + 2 g tanh[ g log( α x )] ψ ( x ) , x = 0 √ with g = 4 + m 2

Singular Hamiltonians and AdS/CFT 4 < m 2 + 15 i) 3 4 ⇒ unique selfadjoint extension 4 < m 2 + 15 ii) − 1 4 < 3 4 ⇒ family of bc parametrized by α ∈ I R � � x = 0 2 x ψ � ( x ) = lim lim 1 + 2 g tanh[ g log( α x )] ψ ( x ) , x = 0 √ with g = 4 + m 2 iii) m 2 + 15 4 < − 1 4 ⇒ family of bc parametrized by α ∈ I R � � x = 0 2 x ψ � ( x ) lim 1 + 2 g cot[ g log( α x )] ψ ( x ) = lim x = 0 with periodicity in α α ≡ α e 2 π / g Efimov effect

Quantum Boundary Conditions R D orientable with regular boundary ∂ Ω • Ω ∈ I • Physical states � φ 1 ( x ) ∗ φ 2 ( x ) d n x � φ 1 , φ 2 � = Ω

Quantum Boundary Conditions R D orientable with regular boundary ∂ Ω • Ω ∈ I • Physical states � φ 1 ( x ) ∗ φ 2 ( x ) d n x � φ 1 , φ 2 � = Ω • Quantum Hamiltonian Δ = d † d Δ is a symmetric operator but not selfadjoint Δ A � = Δ † A

Quantum Boundary Conditions R D orientable with regular boundary ∂ Ω • Ω ∈ I • Physical states � φ 1 ( x ) ∗ φ 2 ( x ) d n x � φ 1 , φ 2 � = Ω • Quantum Hamiltonian Δ = d † d Δ is a symmetric operator but not selfadjoint Δ A � = Δ † A • Balance defect of gradient term � � � d D − 1 x ( ∂ n φ ∗ 1 ) φ 2 − φ ∗ � φ 1 , Δ φ 2 � = � Δ φ 1 , φ 2 �− 1 ∂ n φ 2 ∂ Ω

Self-Adjoint Boundary Conditions ( D = 1 ) The set M of boundary conditions for • self-adjoint extensions of Δ is in one-to-one correspondence with the group of unitary operators of L 2 ( ∂ Ω , C) � � φ − i ∂ n φ = U φ + i ∂ n φ

Self-Adjoint Boundary Conditions ( D > 1 ) The set M of boundary conditions for • self-adjoint extensions of Δ is in one-to-one correspondence with the group of unitary operators U of L 2 ( ∂ Ω , C) � � φ − i ∂ n φ = U φ + i ∂ n φ � � − 1 � � 1 4 ˙ φ = √ ε φ = 1 − Δ ∂ Ω + 1 − Δ ∂ Ω + 1 4 √ ε � ε 2 I φ ; ˙ ε 2 I φ φ = φ 0 + � φ ∈ L 2 ( ∂ Ω , C N ) , φ , ˙ φ [G. Grubb ⇔ M.A. , A. Ibort, G. Marmo]

Self-Adjoint Boundary Conditions ( D > 1 ) The set M of boundary conditions for • self-adjoint extensions of Δ is in one-to-one correspondence with the group of unitary operators U of L 2 ( ∂ Ω , C) � � φ − i ∂ n φ = U φ + i ∂ n φ � � − 1 � � 1 4 ˙ φ = √ ε φ = 1 − Δ ∂ Ω + 1 − Δ ∂ Ω + 1 4 √ ε � ε 2 I φ ; ˙ ε 2 I φ φ = φ 0 + � φ ∈ L 2 ( ∂ Ω , C N ) , φ , ˙ φ The problem is due to the existence of zero modes φ 0 of Δ † which are parametrized by − 1 2 ( ∂ Ω , C) φ = φ 0 | ∂ Ω ∈ H

String theory One-dimensional space Ω = [0 , π ] ∈ I R 1. Dirichlet boundary conditions � � − 1 0 U = − I = φ (0) = φ ( π ) = 0 0 − 1 2. Neumann boundary conditions � � 1 0 φ � (0) = φ � ( π ) = 0 U = I = 0 1 3. Periodic boundary conditions � � 0 1 U = σ x = φ (0) = φ ( π ) 1 0

Ω = ∪ N i = 1 [ a i , b i ] ∈ I R 4. One single circle   0 0 0 0 0 · · · 0 1     0 0 1 0 0 · · · 0 0     0 1 0 0 0 · · · 0 0   U =   0 0 0 0 1 · · · 0 0     · · · · · · · · · ·   · · · 1 0 0 0 0 0 0 5. N Disconected circles   0 1 0 0 0 · · · 0 0   1 0 0 0 0 · · · 0 0     U N =  · · · · · · · · · ·      0 0 0 0 0 · · · 0 1 0 0 0 0 0 · · · 1 0

TOPOLOGY CHANGE ...... (a) ...... (b) (c)

Boundary conditions vs Path integrals • Equivalence Schrödinger y Heisenberg formulations of Quantum Mechanics [Dirac]

Boundary conditions vs Path integrals • Equivalence Schrödinger y Heisenberg formulations of Quantum Mechanics [Dirac] • Feynman formulation: Path integral � x ( T ) = x T K T ( x 0 , x T ) = e − T H ( x 0 , x T ) = [ δ x ( t )]e − S E ( x ( t )) x (0) = x 0

Boundary conditions vs Path integrals • Equivalence Schrödinger y Heisenberg formulations of Quantum Mechanics [Dirac] • Feynman formulation: Path integral � x ( T ) = x T K T ( x 0 , x T ) = e − T H ( x 0 , x T ) = [ δ x ( t )]e − S E ( x ( t )) x (0) = x 0 • Boundary conditions for trajectories: α : ∂ Ω × R + → ∂ Ω × R + ,

Boundary conditions vs Path integrals • Equivalence Schrödinger y Heisenberg formulations of Quantum Mechanics [Dirac] • Feynman formulation: Path integral � x ( T ) = x T K T ( x 0 , x T ) = e − T H ( x 0 , x T ) = [ δ x ( t )]e − S E ( x ( t )) x (0) = x 0 • Boundary conditions for trajectories: α : ∂ Ω × R + → ∂ Ω × R + , • Not equivalent for most general quantum boundary condition U

Boundary conditions vs Path integrals • Equivalence Schrödinger y Heisenberg formulations of Quantum Mechanics [Dirac] • Feynman formulation: Path integral � x ( T ) = x T K T ( x 0 , x T ) = e − T H ( x 0 , x T ) = [ δ x ( t )]e − S E ( x ( t )) x (0) = x 0 • Boundary conditions for trajectories: α : ∂ Ω × R + → ∂ Ω × R + , • Not equivalent for most general quantum boundary condition U • The boundary behaves as a quantum device

Boundary conditions vs Path integrals • Equivalence Schrödinger y Heisenberg formulations of Quantum Mechanics [Dirac] • Feynman formulation: Path integral � x ( T ) = x T K T ( x 0 , x T ) = e − T H ( x 0 , x T ) = [ δ x ( t )]e − S E ( x ( t )) x (0) = x 0 • Boundary conditions for trajectories: α : ∂ Ω × R + → ∂ Ω × R + , • Not equivalent for most general quantum boundary condition U • The boundary behaves as a quantum device • Feynman trajectories cannot bifurcate at the boundary [causality]

Casimir Forces at short distances Vacuum fluctuations of quantum fields ⇓ Casimir force

Casimir Forces at short distances Vacuum fluctuations of quantum fields ⇓ Casimir force • Standard Casimir force is attractive • Micro Electro-Mechanical Systems (MEMS) • Masking microgravity effects • Repulsive Casimir forces • Geometry dependence • Boundary Conditions dependence

Micro Electro-Mechanical Systems (MEMS)

MicroGravity and Casimir Effect

Repulsive Geometry

Repulsive Boundary Conditions Zaremba Boundary Conditions