Creation of a localised source in quantum field theory Jorma Louko - - PowerPoint PPT Presentation

Creation of a localised source in quantum field theory

Jorma Louko

School of Mathematical Sciences, University of Nottingham

RQIN 2017, YITP, Kyoto University, Japan, 4–7 July 2017

• E. G. Brown and JL JHEP 1508, 061 (2015)
• L. J. Zhou, M. E. Carrington, G. Kunstatter, JL PRD 95, 085007 (2017)
• W. M. H. Wan Mokhtar and JL in preparation

r t

Plan

• 1. Motivation: Firewalls

− → Correlation breakdown in quantum field theory

• 2. Wall for scalar field in 1 + 1
• 3. Wall for spinor field in 1 + 1
• 4. Pointlike source for scalar field in 3 + 1
• 5. Summary

• 1. Motivation: Firewall proposal

Almheiri et al 2013 singularity

I

_

I

+

S1 S 2

B’ C’

S 0 Suppose BH evaporates fully and the process preserves unitarity

◮ Pure state on Σ2 ⇒

B′ and C ′ strongly correlated

• 1. Motivation: Firewall proposal

Almheiri et al 2013 singularity

I

_

I

+ B

S1 S 2

C B’ C’

S 0 Suppose BH evaporates fully and the process preserves unitarity

◮ Pure state on Σ2 ⇒

B′ and C ′ strongly correlated

◮ Evolution ⇒

B and C strongly correlated

• 1. Motivation: Firewall proposal

Almheiri et al 2013 singularity

I

_

I

+ B

A

S1 S 2

C B’ C’

S 0 Suppose BH evaporates fully and the process preserves unitarity

◮ Pure state on Σ2 ⇒

B′ and C ′ strongly correlated

◮ Evolution ⇒

B and C strongly correlated

◮ Hawking ⇒

B and A strongly correlated

• 1. Motivation: Firewall proposal

Almheiri et al 2013 singularity

I

_

I

+ B

A

S1 S 2

C B’ C’

S 0 Suppose BH evaporates fully and the process preserves unitarity

◮ Pure state on Σ2 ⇒

B′ and C ′ strongly correlated

◮ Evolution ⇒

B and C strongly correlated

◮ Hawking ⇒

B and A strongly correlated Contradicts entanglement monogamy theorem ?!?

• 1. Motivation: Firewall proposal

Almheiri et al 2013 singularity

I

_

I

+ B

A

S1 S 2

C B’ C’

S 0 Suppose BH evaporates fully and the process preserves unitarity

◮ Pure state on Σ2 ⇒

B′ and C ′ strongly correlated

◮ Evolution ⇒

B and C strongly correlated

◮ Hawking ⇒

B and A strongly correlated Contradicts entanglement monogamy theorem ?!? Almheiri at al (AMPS) 2013 resolution proposal: A–B correlations broken by “drama” at the shrinking horizon even for macroscopic BH

“Firewall”

Cf Fuzzball Mathur 2002 Energetic Curtain Braunstein 2009 et al 2013

• 2. Wall for scalar field in 1 + 1

Brown and JL 2015

1+1 Minkowski φ(t, x) massless ∂2

t φ − ∂2 xφ = 0

t

wall Dirichlet

x

• 2. Wall for scalar field in 1 + 1

Brown and JL 2015

1+1 Minkowski φ(t, x) massless ∂2

t φ − ∂2 xφ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

t

duration forming wall, Dirichlet wall

x

1/λ

• 2. Wall for scalar field in 1 + 1

Brown and JL 2015

1+1 Minkowski φ(t, x) massless ∂2

t φ − ∂2 xφ = 0

→ ∂2

t φ − ∆θ(t)φ = 0 forming wall, Dirichlet wall

t

duration

x

1/λ

• 2. Wall for scalar field in 1 + 1

Brown and JL 2015

1+1 Minkowski φ(t, x) massless ∂2

t φ − ∂2 xφ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

µ infrared cutoff (required)

forming wall, Dirichlet wall t0

t

duration

x

1/λ

◮ Total energy radiated: Etot = +∞, from |x| → (t0 − λ−1)+

• 2. Wall for scalar field in 1 + 1

Brown and JL 2015

1+1 Minkowski φ(t, x) massless ∂2

t φ − ∂2 xφ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

µ infrared cutoff (required)

Dirichlet wall t0 near− duration forming wall,

t

1/λ

x

◮ Total energy radiated: Etot = +∞, from |x| → (t0 − λ−1)+ ◮ Softer: to λ−1 from Dirichlet ⇒ Etot ∝ λ ln(λ/µ)

• 2. Wall for scalar field in 1 + 1

Brown and JL 2015

1+1 Minkowski φ(t, x) massless ∂2

t φ − ∂2 xφ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

µ infrared cutoff (required)

Dirichlet wall t0 near− duration forming wall,

t

1/λ

x

◮ Total energy radiated: Etot = +∞, from |x| → (t0 − λ−1)+ ◮ Softer: to λ−1 from Dirichlet ⇒ Etot ∝ λ ln(λ/µ) −

− − →

λ→∞ ∞

Divergent for sharp wall formation

Cf Anderson and DeWitt 1986

• 2. Wall for scalar field in 1 + 1

Brown and JL 2015

1+1 Minkowski φ(t, x) massless ∂2

t φ − ∂2 xφ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

µ infrared cutoff (required)

Dirichlet wall t0 near− duration forming wall,

x t

detector

1/λ

◮ Total energy radiated: Etot = +∞, from |x| → (t0 − λ−1)+ ◮ Softer: to λ−1 from Dirichlet ⇒ Etot ∝ λ ln(λ/µ) −

− − →

λ→∞ ∞

Divergent for sharp wall formation

Cf Anderson and DeWitt 1986 ◮ Atom coupled to φ: use Unruh-DeWitt detector

• 2. Wall for scalar field in 1 + 1

Brown and JL 2015

1+1 Minkowski φ(t, x) massless ∂2

t φ − ∂2 xφ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

µ infrared cutoff (required)

Dirichlet wall t0 near− duration forming wall,

x t

detector

1/λ

◮ Total energy radiated: Etot = +∞, from |x| → (t0 − λ−1)+ ◮ Softer: to λ−1 from Dirichlet ⇒ Etot ∝ λ ln(λ/µ) −

− − →

λ→∞ ∞

Divergent for sharp wall formation

Cf Anderson and DeWitt 1986 ◮ Atom coupled to φ: use Unruh-DeWitt detector

Transition probability finite for sharp wall formation

• 2. Wall for scalar field in 1 + 1

Brown and JL 2015

1+1 Minkowski φ(t, x) massless ∂2

t φ − ∂2 xφ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

µ infrared cutoff (required)

Dirichlet wall t0 near− duration forming wall,

x t

detector

1/λ

◮ Total energy radiated: Etot = +∞, from |x| → (t0 − λ−1)+ ◮ Softer: to λ−1 from Dirichlet ⇒ Etot ∝ λ ln(λ/µ) −

− − →

λ→∞ ∞

Divergent for sharp wall formation

Cf Anderson and DeWitt 1986 ◮ Atom coupled to φ: use Unruh-DeWitt detector

Transition probability finite for sharp wall formation Moral: sharp wall formation singular gravitationally but nonsingular for a matter coupling

• 3. Wall for spinor field in 1 + 1

Wan Mokhtar and JL in preparation

1+1 Minkowski ψ(t, x) massless / Dψ = 0

wall

x t

MIT bag

• 3. Wall for spinor field in 1 + 1

Wan Mokhtar and JL in preparation

1+1 Minkowski ψ(t, x) massless / Dψ = 0 → / D

θ(t),nψ = 0 wall duration forming wall, MIT bag

1/λ

x t

• 3. Wall for spinor field in 1 + 1

Wan Mokhtar and JL in preparation

1+1 Minkowski ψ(t, x) massless / Dψ = 0 → / D

θ(t),nψ = 0 forming wall, wall duration MIT bag

1/λ

x t

• 3. Wall for spinor field in 1 + 1

Wan Mokhtar and JL in preparation

1+1 Minkowski ψ(t, x) massless / Dψ = 0 → / D

θ(t),nψ = 0

(no infrared cutoff)

forming wall, wall duration MIT bag

1/λ

x t

◮ Total energy radiated: Etot −

− − →

λ→∞ ∞

Divergent for sharp wall formation

• 3. Wall for spinor field in 1 + 1

Wan Mokhtar and JL in preparation

1+1 Minkowski ψ(t, x) massless / Dψ = 0 → / D

θ(t),nψ = 0

(no infrared cutoff)

forming wall, wall duration

x t

detector MIT bag

1/λ

◮ Total energy radiated: Etot −

− − →

λ→∞ ∞

Divergent for sharp wall formation

◮ Atom coupled to ψ: Unruh-DeWitt detector

Transition probability diverges for sharp wall formation

• 3. Wall for spinor field in 1 + 1

Wan Mokhtar and JL in preparation

1+1 Minkowski ψ(t, x) massless / Dψ = 0 → / D

θ(t),nψ = 0

(no infrared cutoff)

forming wall, wall duration

x t

detector MIT bag

1/λ

◮ Total energy radiated: Etot −

− − →

λ→∞ ∞

Divergent for sharp wall formation

◮ Atom coupled to ψ: Unruh-DeWitt detector

Transition probability diverges for sharp wall formation Moral: sharp wall formation singular both gravitationally and for a matter coupling

• 4. Pointlike source for scalar field in 3 + 1

Zhou et al 2016

3+1 Minkowski φ(t, x) massless ∂2

t φ − ∇2φ = 0

r

Formed source

t

• 4. Pointlike source for scalar field in 3 + 1

Zhou et al 2016

3+1 Minkowski φ(t, x) massless ∂2

t φ − ∇2φ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

θ(t): origin boundary condition (spherically symmetric sector)

r

Formed

1/λ

duration source, forming

t

source

• 4. Pointlike source for scalar field in 3 + 1

Zhou et al 2016

3+1 Minkowski φ(t, x) massless ∂2

t φ − ∇2φ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

θ(t): origin boundary condition (spherically symmetric sector)

1/λ

duration source, forming source Formed

t r

◮ T00 well defined; time-dependent even for t > r + λ−1

• 4. Pointlike source for scalar field in 3 + 1

Zhou et al 2016

3+1 Minkowski φ(t, x) massless ∂2

t φ − ∇2φ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

θ(t): origin boundary condition (spherically symmetric sector)

1/λ

duration source, forming

t0

r

Formed

t

source ◮ T00 well defined; time-dependent even for t > r + λ−1 ◮ t = t0 > λ−1: T00 →

∞,

r→0+ −∞, r→t0−

⇒

Etot = “∞ − ∞” not defined

• 4. Pointlike source for scalar field in 3 + 1

Zhou et al 2016

3+1 Minkowski φ(t, x) massless ∂2

t φ − ∇2φ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

θ(t): origin boundary condition (spherically symmetric sector)

1/λ

duration source, forming

t0

r

Formed

t

source ◮ T00 well defined; time-dependent even for t > r + λ−1 ◮ t = t0 > λ−1: T00 →

∞,

r→0+ −∞, r→t0−

⇒

Etot = “∞ − ∞” not defined

◮ λ → ∞: T00 → ∞ at t > r

• 4. Pointlike source for scalar field in 3 + 1

Zhou et al 2016

3+1 Minkowski φ(t, x) massless ∂2

t φ − ∇2φ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

θ(t): origin boundary condition (spherically symmetric sector)

1/λ

duration source, forming

t0

detector Formed

t r

source ◮ T00 well defined; time-dependent even for t > r + λ−1 ◮ t = t0 > λ−1: T00 →

∞,

r→0+ −∞, r→t0−

⇒

Etot = “∞ − ∞” not defined

◮ λ → ∞: T00 → ∞ at t > r ◮ Unruh-DeWitt detector at t > r:

Transition probability diverges as λ → ∞

• 4. Pointlike source for scalar field in 3 + 1

Zhou et al 2016

3+1 Minkowski φ(t, x) massless ∂2

t φ − ∇2φ = 0

→ ∂2

t φ − ∆θ(t)φ = 0

θ(t): origin boundary condition (spherically symmetric sector)

1/λ

duration source, forming

t0

detector Formed

t r

source ◮ T00 well defined; time-dependent even for t > r + λ−1 ◮ t = t0 > λ−1: T00 →

∞,

r→0+ −∞, r→t0−

⇒

Etot = “∞ − ∞” not defined

◮ λ → ∞: T00 → ∞ at t > r ◮ Unruh-DeWitt detector at t > r:

Transition probability diverges as λ → ∞ Moral: sharp source formation (quite) singular both gravitationally and for a matter coupling

Summary

◮ Rapid creation of a localised source tends to be singular!

◮ Both gravitationally and for a model atom’s response ◮ 1+1 scalar field exceptional (and needs an infrared cutoff)

◮ Model for a black hole firewall?

◮ Spacetime will react. How? ◮ Gµν = 8πTµν ? May or may not suffice. . .

◮ Fully-developed firewall?

◮ Quantum theory of spacetime needed

Appendix: pointlike detector in quantum field theory

(Unruh-DeWitt)

Quantum field Two-state detector (atom)

D spacetime dimension

• state with energy 0

φ real scalar field 1

• state with energy ω

|0 (initial) state x(τ) detector worldline, τ proper time

Appendix: pointlike detector in quantum field theory

(Unruh-DeWitt)

Quantum field Two-state detector (atom)

D spacetime dimension

• state with energy 0

φ real scalar field 1

• state with energy ω

|0 (initial) state x(τ) detector worldline, τ proper time

Interaction: one of

H(0)

int (τ) = cχ(τ)µ(τ)φ

• x(τ)
• ←

− usual UDW H(1)

int (τ) = cχ(τ)µ(τ) d dτ φ

• x(τ)
• ←

− derivative-coupling c coupling constant χ switching function, C ∞ µ detector’s monopole moment operator

Probability of transition ⊗ |0 − → 1 ⊗ |anything in first-order perturbation theory: P(ω) = c2

• 0µ(0)1
• 2
• detector internals only:

drop!

× F(ω)

trajectory and |0: response function

F (0)(ω) = ∞

−∞

dτ ′ ∞

−∞

dτ ′′ e−iω(τ ′−τ ′′) χ(τ ′)χ(τ ′′) W (τ ′, τ ′′) F (1)(ω) = ∞

−∞

dτ ′ ∞

−∞

dτ ′′ e−iω(τ ′−τ ′′) χ(τ ′)χ(τ ′′) ∂τ ′∂τ ′′W (τ ′, τ ′′) W (τ ′, τ ′′) = 0|φ

• x(τ ′)
• φ
• x(τ ′′)
• |0

Wightman function

Probability of transition ⊗ |0 − → 1 ⊗ |anything in first-order perturbation theory: P(ω) = c2

• 0µ(0)1
• 2
• detector internals only:

drop!

× F(ω)

trajectory and |0: response function

F (0)(ω) = ∞

−∞

dτ ′ ∞

−∞

dτ ′′ e−iω(τ ′−τ ′′) χ(τ ′)χ(τ ′′) W (τ ′, τ ′′) F (1)(ω) = ∞

−∞

dτ ′ ∞

−∞

dτ ′′ e−iω(τ ′−τ ′′) χ(τ ′)χ(τ ′′) ∂τ ′∂τ ′′W (τ ′, τ ′′) W (τ ′, τ ′′) = 0|φ

• x(τ ′)
• φ
• x(τ ′′)
• |0

Wightman function (distribution!)