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

S(φ) = 1 2 dt dz z2

• d3x
• | ˙

φ|2 + |∂zφ|2 − |∇φ|2 − m2|φ|2

Singular Hamiltonians and AdS/CFT

• Free scalar field in anti-deSitter space-time

S(φ) = 1 2 dt dz z2

• d3x
• | ˙

φ|2 + |∂zφ|2 − |∇φ|2 − m2|φ|2

• Effective Hamiltonian

H = −1 2 d2 dz2 + m2 + 15

4

z2 in the half line z ∈ (0, ∞)

• H is symmetric but in this case we have three

different regimes

Singular Hamiltonians and AdS/CFT

i) 3

4 < m2 + 15 4 ⇒ unique selfadjoint extension

Singular Hamiltonians and AdS/CFT

i) 3

4 < m2 + 15 4 ⇒ unique selfadjoint extension

ii) −1

4 < m2 + 15 4 < 3 4⇒ family of bc parametrized by

α ∈ I

R lim

x=0 2xψ(x) = lim x=0

• 1 + 2g tanh[g log(α x)]
• ψ(x),

with g =

√

4 + m2

Singular Hamiltonians and AdS/CFT

i) 3

4 < m2 + 15 4 ⇒ unique selfadjoint extension

ii) −1

4 < m2 + 15 4 < 3 4⇒ family of bc parametrized by

α ∈ I

R lim

x=0 2xψ(x) = lim x=0

• 1 + 2g tanh[g log(α x)]
• ψ(x),

with g =

√

4 + m2 iii) m2 + 15

4 < −1 4⇒ family of bc parametrized by α ∈ I

R lim

x=0

• 1 + 2g cot[g log(α x)]
• ψ(x) = lim

x=0 2xψ(x)

with periodicity in α

α ≡ α e2π/g

Efimov effect

Quantum Boundary Conditions

• Ω ∈ I

RD orientable with regular boundary ∂Ω

• Physical states

φ1, φ2 =

• Ω

φ1(x)∗φ2(x) dnx

Quantum Boundary Conditions

• Ω ∈ I

RD orientable with regular boundary ∂Ω

• Physical states

φ1, φ2 =

• Ω

φ1(x)∗φ2(x) dnx

• Quantum Hamiltonian

Δ = d†d Δ is a symmetric operator but not selfadjoint ΔA = Δ†

A

Quantum Boundary Conditions

• Ω ∈ I

RD orientable with regular boundary ∂Ω

• Physical states

φ1, φ2 =

• Ω

φ1(x)∗φ2(x) dnx

• Quantum Hamiltonian

Δ = d†d Δ is a symmetric operator but not selfadjoint ΔA = Δ†

A

• Balance defect of gradient term

φ1, Δφ2 = Δφ1, φ2−

• ∂Ω

dD−1x

• (∂nφ∗

1) φ2 − φ∗ 1 ∂nφ2

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

• f L2(∂Ω,C)

φ − i∂nφ = U

• φ + i∂nφ

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 L2(∂Ω,C)

φ − i∂nφ = U

• φ + i∂nφ
• φ = 1

√ε

• −Δ∂Ω + 1

ε2I

− 1

4

φ ; ˙ φ = √ε

• −Δ∂Ω + 1

ε2I

1

4 ˙

• φ

φ, ˙ φ ∈ L2(∂Ω, CN), φ = φ0 + 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 L2(∂Ω,C)

φ − i∂nφ = U

• φ + i∂nφ
• φ = 1

√ε

• −Δ∂Ω + 1

ε2I

− 1

4

φ ; ˙ φ = √ε

• −Δ∂Ω + 1

ε2I

1

4 ˙

• φ

φ, ˙ φ ∈ L2(∂Ω, CN), φ = φ0 + φ

The problem is due to the existence of zero modes

φ0 of Δ† which are parametrized by φ = φ0|∂Ω ∈ H

− 1 2 (∂Ω,C)

String theory

One-dimensional space

Ω = [0, π] ∈ I

R

• 1. Dirichlet boundary conditions

U = −I =

• −1

−1

• φ(0) = φ(π) = 0
• 2. Neumann boundary conditions

U = I =

• 1

1

• φ(0) = φ(π) = 0
• 3. Periodic boundary conditions

U = σx =

• 1

1

• φ(0) = φ(π)

Ω = ∪N

i=1[ai, bi] ∈ I

R

• 4. One single circle

U =          

· · ·

1 1

· · ·

1

· · ·

1

· · · · · · · · · · · · ·

1

· · ·

         

• 5. N Disconected circles

UN =        1

· · ·

1

· · · · · · · · · · · · · · · ·

1

· · ·

1       

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

KT (x0, xT ) = e−TH(x0, xT ) = x(T)=xT

x(0)=x0

[δx(t)]e−SE(x(t))

Boundary conditions vs Path integrals

• Equivalence Schrödinger y Heisenberg

formulations of Quantum Mechanics [Dirac]

• Feynman formulation: Path integral

KT (x0, xT ) = e−TH(x0, xT ) = x(T)=xT

x(0)=x0

[δx(t)]e−SE(x(t))

• 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

KT (x0, xT ) = e−TH(x0, xT ) = x(T)=xT

x(0)=x0

[δx(t)]e−SE(x(t))

• 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

KT (x0, xT ) = e−TH(x0, xT ) = x(T)=xT

x(0)=x0

[δx(t)]e−SE(x(t))

• 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

KT (x0, xT ) = e−TH(x0, xT ) = x(T)=xT

x(0)=x0

[δx(t)]e−SE(x(t))

• 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

Scalar fields in a bounded domain

• Physical space Ω ∈ I

R3 with regular boundary ∂Ω

• Free scalar field

SB(φ) = 1 2

• dt
• Ω

d3x

• | ˙

φ|2 − |∇φ|2 − m2|φ|2

Scalar fields in a bounded domain

• Physical space Ω ∈ I

R3 with regular boundary ∂Ω

• Free scalar field

SB(φ) = 1 2

• dt
• Ω

d3x

• | ˙

φ|2 − |∇φ|2 − m2|φ|2

• Boundary action

Sb(φ)= 1 4

• dt
• ∂Ω

√g∂Ωd2x

• φ∗∂nφ−(∂nφ∗)φ

Scalar fields in a bounded domain

• Physical space Ω ∈ I

R3 with regular boundary ∂Ω

• Free scalar field

SB(φ) = 1 2

• dt
• Ω

d3x

• | ˙

φ|2 − |∇φ|2 − m2|φ|2

• Boundary action

Sb(φ)= 1 4

• dt
• ∂Ω

√g∂Ωd2x

• φ∗∂nφ−(∂nφ∗)φ
• In terms of boundary values of fields

φ = φ∂Ω ∂nφ = ∂nφ|∂Ω

Canonical Quantization: Free Fields

• Quantum Hamiltonian
• H = 1

2

• Ω

dx3

• π(x)∗

i

π(x)i + φ∗

i (x)(−Δ + m2)

φ(x)i

[

π(x), φ(x)] = −i δ(x − x).

Canonical Quantization: Free Fields

• Quantum Hamiltonian
• H = 1

2

• Ω

dx3

• π(x)∗

i

π(x)i + φ∗

i (x)(−Δ + m2)

φ(x)i

[

π(x), φ(x)] = −i δ(x − x).

H has to be selfadjoint and positive

−Δ + m2 has to satisfy two conditions

• −Δ + m2 selfadjoint
• −Δ + m2 positive

Consistency Conditions

The set M of selfadjoint extensions of −Δ is equivalent to the set of unitary operators in L2(∂Ω, C)

M = U

• L2(∂Ω, C)

Consistency Conditions

The set M of selfadjoint extensions of −Δ is equivalent to the set of unitary operators in L2(∂Ω, C)

M = U

• L2(∂Ω, C)
• U ∈ M

− ΔU = −Δ∗ |DU DU = { φ+φ0; φ ∈ H 2(Ω, C) y φ0 ∈ ker Δ†; φ−i ˙ φ = U(φ−i ˙ φ)} φ = 1 √ δ

• −Δ∂Ω + 1

δ2I

− 1

4

φ; ˙ φ = √ δ

• −Δ∂Ω + 1

δ2I

1

4 ˙

• φ

(φ, ˙

• φ) ∈ H− 1

2(∂Ω, C) × H 1 2(∂Ω, C)

δ = diam (Ω)

[G. Grubb ⇔ M.A. , A. Ibort, G. Marmo]

Consistency Conditions

• Selfadjointness of −ΔU ⇔ unitarity of U

φ, −ΔUφ = − → ∇φ2 + m2φ2 + i

• φ, I − U

I + U φ

• ≥ 0
• Positivity of −Δ

i

• φ, I − U

I + U φ

• ≥ 0

∀ φ ∈ L2(∂Ω, C).

Consistency Conditions

• Selfadjointness of −ΔU ⇔ unitarity of U

φ, −ΔUφ = − → ∇φ2 + m2φ2 + i

• φ, I − U

I + U φ

• ≥ 0
• Positivity of −Δ

i

• φ, I − U

I + U φ

• ≥ 0

∀ φ ∈ L2(∂Ω, C).

• New negative eigenvalues of −ΔU appear at each

crossing of ±1 eigenvalues of U

MP ≡

• U ∈ U
• L2(∂Ω, C)
• | ∀ λ = eiθ ∈ σ(U), 0 ≤ θ ≤ π
• .

[M.A. , A. Ibort, G. Marmo]

QFT Symmetries & Boundary Conditions

• Space Symmetries
• 1. must preserve the boundary

T : Ω −

→ Ω

T(∂Ω) = ∂Ω

QFT Symmetries & Boundary Conditions

2.

• Space Symmetries
• 1. must preserve the boundary

T : Ω −

→ Ω

T(∂Ω) = ∂Ω

• 2. and the boundary condition

OTφ(x) = uTφ(T −1x);

Tb ≡ T|∂Ω

• U, OTb
• = [U, uT] = 0

Symmetries and Boundary Conditions

• Homogeneous Plates. Translation Invariance

Ω = I

R2 × [0, L]

φ(y, 0) → φ(y + a, 0) φ(y, L) → φ(y + a, L)(y, a ∈ I

R2) U ∈ MP ∩ U(2).

Consistency Conditions

U = eiα I cos β + i n · σ sin β

• ∈ U(2)

Consistency Conditions

U = eiα I cos β + i n · σ sin β

• ∈ U(2)

MP ≡ {U(α, β, n) ∈ U(2) | 0 ≤ α ± β ≤ π}

F

1.5 1.0 0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Β Α

Consistency Conditions

U = eiα I cos β + i n · σ sin β

• ∈ U(2)

MP ≡ {U(α, β, n) ∈ U(2) | 0 ≤ α ± β ≤ π}

F

1.5 1.0 0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Β Α

. pseudo-periodic

Consistency Conditions

U = eiα I cos β + i n · σ sin β

• ∈ U(2)

MP ≡ {U(α, β, n) ∈ U(2) | 0 ≤ α ± β ≤ π}

F

1.5 1.0 0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Β Α

. Dirichlet

Consistency Conditions

U = eiα I cos β + i n · σ sin β

• ∈ U(2)

MP ≡ {U(α, β, n) ∈ U(2) | 0 ≤ α ± β ≤ π}

F

1.5 1.0 0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Β Α

. Neumann

Symmetries and Boundary Conditions

• Identical Parallel Plates. Reflection symmetry

Ω = {x ∈ I

R3, 0 ≤ x3 ≤ L}.R : x3 → L − x3.

ORb = σ1 :

• ORb, U
• = 0

⇔

U = U(α, β, n).

MF(R) = {U ∈ MF; n2 = n3 = 0, n1 = 1}

Symmetries and Boundary Conditions

• Identical Parallel Plates. Reflection symmetry

Ω = {x ∈ I

R3, 0 ≤ x3 ≤ L}.R : x3 → L − x3.

ORb = σ1 :

• ORb, U
• = 0

⇔

U = U(α, β, n).

MF(R) = {U ∈ MF; n2 = n3 = 0, n1 = 1}

• Boundary conditions and symmetry breaking.
• Spontaneous symmetry breaking
• Anomalies: Casimir effect
• Attractive/Repulsive Casimir forces

(non-universality)

RG and Boundary Conditions

For general boundary conditions in ∂Ω the continuum RG is defined by

δU †

δ∂δUδ = 1

2

• U †

δ − Uδ

• r

U †

t ∂tUt = 1

2

• U †

t − Ut

• for δ = δ0 et.

RG and Boundary Conditions

For general boundary conditions in ∂Ω the continuum RG is defined by

δU †

δ∂δUδ = 1

2

• U †

δ − Uδ

• r

U †

t ∂tUt = 1

2

• U †

t − Ut

• for δ = δ0 et.
• Fixed points correspond to selfadjoint bc U † = U
• Critical exponents of boundary conditions

RG flow are ±1. Conformally invariant field theories

• Dirichlet and Neumann bc are RG invariant
• For mixed boundary conditions the RG flow goes

from Dirichlet (UV) to Neumann (IR)

RG and Boundary Conditions

n =

• sin(θ) cos(γ), sin(θ) sin(γ), cos(θ)
• Renormalization Group Flow

α(L) + 1

L sin(α) cos(β) = 0; β(L) + 1 L cos(α) sin(β) = 0;

θ(L)2 + sin2(θ)γ(L)2 = 0

RG and Boundary Conditions

n =

• sin(θ) cos(γ), sin(θ) sin(γ), cos(θ)
• Renormalization Group Flow

α(L) + 1

L sin(α) cos(β) = 0; β(L) + 1 L cos(α) sin(β) = 0;

θ(L)2 + sin2(θ)γ(L)2 = 0

1.5 1.0 0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Β Α

θ(Λ) = sin(θ)γ(Λ) = 0

Vacuum Energy: The Casimir Effect

Massless Free Field (m = 0)

• Vacuum Energy

EU = 1 2

• k∈σ(−ΔU)
• λk = 1

2tr (−ΔU)

1 2 .

Vacuum Energy: The Casimir Effect

Massless Free Field (m = 0)

• Vacuum Energy

EU = 1 2

• k∈σ(−ΔU)
• λk = 1

2tr (−ΔU)

1 2 .

• Heat Kernel Regularization

Eε

U = 1

2

• k∈σ(−ΔU)
• λke−ελk = 1

2tr (−ΔU)

1 2eεΔU.

Vacuum Energy: The Casimir Effect

Massless Free Field (m = 0)

• Vacuum Energy

EU = 1 2

• k∈σ(−ΔU)
• λk = 1

2tr (−ΔU)

1 2 .

• Heat Kernel Regularization

Eε

U = 1

2

• k∈σ(−ΔU)
• λke−ελk = 1

2tr (−ΔU)

1 2eεΔU.

• Finite size contribution (ε–independent)

Eε

U = cU

L

3 S.

Vacuum Energy: The Casimir Effect

• Residue theorem
• κ∈ker hU

f (κ) = 1 2πi

• dk f (k) d

dk log (hU(k)) (1)

Vacuum Energy: The Casimir Effect

• Residue theorem
• κ∈ker hU

f (κ) = 1 2πi

• dk f (k) d

dk log (hU(k)) (2)

• Spectral function

hU(k; α, β, n1) = 2ieiα (k2 − 1) cos β + (k2 + 1) cos α

• sin kL

−

2k sin α cos kL − 2k n1 sin β

• .

Vacuum Energy: The Casimir Effect

• Finite volume contribution

cU L3

=

1 2π L4 ∞ dk k3

• 1 −

1 L3 − L3 d dk log

• h(L)

U (ik)

h(L0)

U (ik)

Vacuum Energy: The Casimir Effect

• Finite volume contribution

cU L3

=

1 2π L4 ∞ dk k3

• 1 −

1 L3 − L3 d dk log

• h(L)

U (ik)

h(L0)

U (ik)

• Periodic b.c. [cp = −π2/90],

Attractive

• Anti-periodic [cap = 7π2/720],

Repulsive

• Dirichlet [cd = −π2/1440]

Attractive

• Zaremba [cz = 7π2/11520]

Repulsive

Vacuum Energy: Analytic computations

Un = ±I cn = cd = − π2

1440

U = −σ1 cap = 7π2

720

U = σ1 cp = − π2

90

U = ±σ3 cz =

7π2 11520

U = cos θσ3 + sin θσ1 ccp =127π2

11520 − 3πθ 32 − 11θ2 96 − 4θ3+

|π−2θ|3

96π

+ θ4

48π2

U = cos φσ1 − sin φσ2 cpp(φ) = − π2

90 + φ2 12 − φ3 12π +

φ4

48π2

Casimir Energy. Numerical Results

1.5 1.0 0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Β Α n10

n1 = 0

Kenneth-Klich theorem The Casimir force between two identical bodies is always attractive

Casimir Energy. Numerical Results

n1 = 0.5

[M. A., J. M. Muñoz-Castañeda]

Casimir Energy. Numerical Results

n1 = 1

Kenneth-Klich theorem

Summary

• Vacuum energy is highly dependent on the

boundary conditions

Summary

• Vacuum energy is highly dependent on the

boundary conditions

• Atractive and repulsive regimes of boundary

conditions are separated by Casimirless boundary conditions

Summary

• Vacuum energy is highly dependent on the

boundary conditions

• Atractive and repulsive regimes of boundary

conditions are separated by Casimirless boundary conditions

• Kenneth-Klich theorem holds inside attractive

regime

Summary

• Vacuum energy is highly dependent on the

boundary conditions

• Atractive and repulsive regimes of boundary

conditions are separated by Casimirless boundary conditions

• Kenneth-Klich theorem holds inside attractive

regime

• Anomaly free boundary conditions are not

conformally invariant

Summary

• Vacuum energy is highly dependent on the

boundary conditions

• Atractive and repulsive regimes of boundary

conditions are separated by Casimirless boundary conditions

• Kenneth-Klich theorem holds inside attractive

regime

• Anomaly free boundary conditions are not

conformally invariant

• There is no anomaly free fixed point

Summary

• Vacuum energy is highly dependent on the

boundary conditions

• Atractive and repulsive regimes of boundary

conditions are separated by Casimirless boundary conditions

• Kenneth-Klich theorem holds inside attractive

regime

• Anomaly free boundary conditions are not

conformally invariant

• There is no anomaly free fixed point
• The most stable boundary condition under RG

flow is Neumann bc