Quantum gravity signatures in the Unruh effect N. Alkofer, G. - - PowerPoint PPT Presentation

▶
quantum gravity signatures in the unruh effect
SMART_READER_LITE
LIVE PREVIEW

Quantum gravity signatures in the Unruh effect N. Alkofer, G. - - PowerPoint PPT Presentation

Feb 09, 2024 309 likes •1.19k views

Quantum gravity signatures in the Unruh effect N. Alkofer, G. DOdorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 1 / 31 Outline: QG signatures in the Unruh effect The

slide-1
SLIDE 1

Quantum gravity signatures in the Unruh effect

  • N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 1 / 31

slide-2
SLIDE 2

Outline: QG signatures in the Unruh effect

The Unruh effect

  • Overview
  • The detector approach

Dimensional reduction

  • Spectral dimension
  • Unruh dimension

Quantum gravity corrections

  • Ostrogradski models
  • Spectral representation

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 2 / 31

slide-3
SLIDE 3

The Unruh effect: Overview

Accelerating observer sees thermal bath of particles Trajectory inertial observer (t, x, y, z) Trajectory uniformly accelerating (Rindler) observer (τ, ξ, y, z)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 3 / 31

slide-4
SLIDE 4

The Unruh effect: Overview

Accelerating observer sees thermal bath of particles Trajectory inertial observer (t, x, y, z) Trajectory uniformly accelerating (Rindler) observer (τ, ξ, y, z) Relation coordinates: t = eaξ a sinh (aτ), x = eaξ a cosh (aτ), y = y, z = z.

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 3 / 31

slide-5
SLIDE 5

The Unruh effect: Overview

Accelerating observer sees thermal bath of particles Trajectory inertial observer (t, x, y, z) Trajectory uniformly accelerating (Rindler) observer (τ, ξ, y, z) Relation coordinates: t = eaξ a sinh (aτ), x = eaξ a cosh (aτ), y = y, z = z. Klein-Gordon for m = 0:

  • −∂2

t + ∂2 x + ∂2 y + ∂2 z

  • φ(t, x, y, z)

=

  • e−2aξ(−∂2

τ + ∂2 ξ ) + ∂2 y + ∂2 z

  • φ(τ, ξ, y, z)

=

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 3 / 31

slide-6
SLIDE 6

The Unruh effect: Overview

Solutions: uR = e−iΩτ 2π2√a sinh πΩ a 1/2 × KiΩ/a | p⊥| a eaξ

  • ei

p⊥· x⊥

uM = 1

  • 2(2π)3ω

e−i(ωt−kxx−

k⊥· x⊥)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 4 / 31

slide-7
SLIDE 7

The Unruh effect: Overview

Solutions: uR = e−iΩτ 2π2√a sinh πΩ a 1/2 × KiΩ/a | p⊥| a eaξ

  • ei

p⊥· x⊥

uM = 1

  • 2(2π)3ω

e−i(ωt−kxx−

k⊥· x⊥)

define annihilation/creation operators ˆ aω, ˆ bΩ such that ˆ aω |0M = 0, ˆ bΩ |0R = 0

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 4 / 31

slide-8
SLIDE 8

The Unruh effect: Overview

Field expansion: ˆ φ =

  • d3k

(2π)3 1 √ 2ω

  • ˆ

aω uM

ω + ˆ

a†

ω (uM ω )†

=

  • d3p

(2π)3 1 √ 2Ω

  • ˆ

bΩ uR

Ω + ˆ

b†

Ω (uR Ω)†

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 5 / 31

slide-9
SLIDE 9

The Unruh effect: Overview

Field expansion: ˆ φ =

  • d3k

(2π)3 1 √ 2ω

  • ˆ

aω uM

ω + ˆ

a†

ω (uM ω )†

=

  • d3p

(2π)3 1 √ 2Ω

  • ˆ

bΩ uR

Ω + ˆ

b†

Ω (uR Ω)†

Relation between ˆ a, ˆ a† and ˆ b, ˆ b†?

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 5 / 31

slide-10
SLIDE 10

The Unruh effect: Overview

Field expansion: ˆ φ =

  • d3k

(2π)3 1 √ 2ω

  • ˆ

aω uM

ω + ˆ

a†

ω (uM ω )†

=

  • d3p

(2π)3 1 √ 2Ω

  • ˆ

bΩ uR

Ω + ˆ

b†

Ω (uR Ω)†

Relation between ˆ a, ˆ a† and ˆ b, ˆ b†? ⇒ Bogolyubov transformation: ˆ bΩ =

  • dω
  • d

k⊥

  • αΩωˆ

aω − βΩωˆ a†

ω

  • Find coefficients α, β

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 5 / 31

slide-11
SLIDE 11

The Unruh effect: Overview

Interested in number of particles observed by accelerating observer nΩ = 0M| ˆ N |0M V = 0M|ˆ b†

Ω ˆ

bΩ′ |0M V =

  • d3kβΩωβ∗

Ω′ω

where βΩω = − 1

  • 2πaω(e2πΩ/a − 1)

ω + kx ω − kx −iΩ/2a

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 6 / 31

slide-12
SLIDE 12

The Unruh effect: Overview

Interested in number of particles observed by accelerating observer nΩ = 0M| ˆ N |0M V = 0M|ˆ b†

Ω ˆ

bΩ′ |0M V =

  • d3kβΩωβ∗

Ω′ω

where βΩω = − 1

  • 2πaω(e2πΩ/a − 1)

ω + kx ω − kx −iΩ/2a then nΩ = 1 e

2πΩ a

− 1 δ(Ω − Ω′)δ( k⊥ − k′

⊥)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 6 / 31

slide-13
SLIDE 13

The Unruh effect: Overview

Interested in number of particles observed by accelerating observer nΩ = 0M| ˆ N |0M V = 0M|ˆ b†

Ω ˆ

bΩ′ |0M V =

  • d3kβΩωβ∗

Ω′ω

where βΩω = − 1

  • 2πaω(e2πΩ/a − 1)

ω + kx ω − kx −iΩ/2a then nΩ = 1 e

2πΩ a

− 1 δ(Ω − Ω′)δ( k⊥ − k′

⊥)

Thermal bath of particles with temperature T = a/2π ⇒ Geometric effect

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 6 / 31

slide-14
SLIDE 14

The Unruh effect: The detector approach

(I. Agullo, J. Navarro-Salas, G.J. Olmo and L. Parker, New J.Phys. 12 095017 (2010))

Two internal energy states E2 > E1

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31

slide-15
SLIDE 15

The Unruh effect: The detector approach

(I. Agullo, J. Navarro-Salas, G.J. Olmo and L. Parker, New J.Phys. 12 095017 (2010))

Two internal energy states E2 > E1 Interaction scalar field and detector: LI = gm(τ)φ(x)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31

slide-16
SLIDE 16

The Unruh effect: The detector approach

(I. Agullo, J. Navarro-Salas, G.J. Olmo and L. Parker, New J.Phys. 12 095017 (2010))

Two internal energy states E2 > E1 Interaction scalar field and detector: LI = gm(τ)φ(x) Spontaneous emission inertial observer (intrinsic to detector) |E2 |0M → |E1 | k

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31

slide-17
SLIDE 17

The Unruh effect: The detector approach

(I. Agullo, J. Navarro-Salas, G.J. Olmo and L. Parker, New J.Phys. 12 095017 (2010))

Two internal energy states E2 > E1 Interaction scalar field and detector: LI = gm(τ)φ(x) Spontaneous emission inertial observer (intrinsic to detector) |E2 |0M → |E1 | k Amplitude A( k) = ig E1| m(0) |E2

  • dτei(E1−E2)τ

k| φ(x(τ)) |0M

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31

slide-18
SLIDE 18

The Unruh effect: The detector approach

(I. Agullo, J. Navarro-Salas, G.J. Olmo and L. Parker, New J.Phys. 12 095017 (2010))

Two internal energy states E2 > E1 Interaction scalar field and detector: LI = gm(τ)φ(x) Spontaneous emission inertial observer (intrinsic to detector) |E2 |0M → |E1 | k Amplitude A( k) = ig E1| m(0) |E2

  • dτei(E1−E2)τ

k| φ(x(τ)) |0M Transition probability (∆E ≡ E2 − E1, ∆τ ≡ τ1 − τ2) Pi→f =

  • d3k|A|2 = g2| E1| m |E2 |2F(∆E)

where F(∆E) =

  • dτ1dτ2ei∆E∆τG(∆τ − iǫ)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31

slide-19
SLIDE 19

The Unruh effect: The detector approach

For massive scalar field in Minkowski space: ˜ G(p2) = 1 p2 − m2 = 1 (p0 +

  • p2 + m2)(p0 −
  • p2 + m2)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 8 / 31

slide-20
SLIDE 20

The Unruh effect: The detector approach

For massive scalar field in Minkowski space: ˜ G(p2) = 1 p2 − m2 = 1 (p0 +

  • p2 + m2)(p0 −
  • p2 + m2)

Positive frequency Wightman function encircles pole at

  • p2 + m2

G+( x, t) = −i

  • d3

p (2π)3

  • γ+

dp0 2π ˜ G(p2)e−i(p0t−

p· x),

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 8 / 31

slide-21
SLIDE 21

The Unruh effect: The detector approach

For massive scalar field in Minkowski space: ˜ G(p2) = 1 p2 − m2 = 1 (p0 +

  • p2 + m2)(p0 −
  • p2 + m2)

Positive frequency Wightman function encircles pole at

  • p2 + m2

G+( x, t) = −i

  • d3

p (2π)3

  • γ+

dp0 2π ˜ G(p2)e−i(p0t−

p· x),

in real space we find G+(x, x′) = − im 4π2 K1(im

  • (t − t′ − iǫ)2) − (

x − x′)2

  • (t − t′ − iǫ)2) − (

x − x′)2 m → 0, G+(x, x′) = − 1 4π2 1 (t − t′ − iǫ)2 − ( x − x′)2

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 8 / 31

slide-22
SLIDE 22

The Unruh effect: The detector approach

Total emission (evaluated on accelerated trajectory): |E2 |0M → |E1 | k , Pi→f

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 9 / 31

slide-23
SLIDE 23

The Unruh effect: The detector approach

Total emission (evaluated on accelerated trajectory): |E2 |0M → |E1 | k , Pi→f Induced transition probability Pi→f(induced) = Pi→f − Pi→f(spontaneous)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 9 / 31

slide-24
SLIDE 24

The Unruh effect: The detector approach

Total emission (evaluated on accelerated trajectory): |E2 |0M → |E1 | k , Pi→f Induced transition probability Pi→f(induced) = Pi→f − Pi→f(spontaneous) Response function induced emission F(∆E) = ∞

−∞

dτ1dτ2ei∆E∆τ [GM(∆τ − iǫ) − GR(∆τ − iǫ)] where GM(x2) = 0M| φ(x)φ(0) |0M , GR(x2) = 0R| φ(x)φ(0) |0R

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 9 / 31

slide-25
SLIDE 25

The Unruh effect: The detector approach

Total emission (evaluated on accelerated trajectory): |E2 |0M → |E1 | k , Pi→f Induced transition probability Pi→f(induced) = Pi→f − Pi→f(spontaneous) Response function induced emission F(∆E) = ∞

−∞

dτ1dτ2ei∆E∆τ [GM(∆τ − iǫ) − GR(∆τ − iǫ)] where GM(x2) = 0M| φ(x)φ(0) |0M , GR(x2) = 0R| φ(x)φ(0) |0R Induced response function per unit time (E ≡ ∆E) ˙ F(E) = ∞

−∞

dτeiEτ [GM(τ − iǫ) − GR(τ − iǫ)]

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 9 / 31

slide-26
SLIDE 26

The detector approach: massless scalar

Wightman functions GM = − a2 16π2 1 sinh2 (aτ/2 − iaǫ) GR = − 1 4π2 1 (τ − iǫ)2

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 10 / 31

slide-27
SLIDE 27

The detector approach: massless scalar

Wightman functions GM = − a2 16π2 1 sinh2 (aτ/2 − iaǫ) GR = − 1 4π2 1 (τ − iǫ)2 Induced response rate ˙ F(E) = ∞

−∞

dτeiEτ [GM(τ − iǫ) − GR(τ − iǫ)] = − 1 4π2 (2πi)(iE)

−1

  • k=−∞

e−iE 2πi

a k

= 1 2π E 1 e

2π a E − 1 Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 10 / 31

slide-28
SLIDE 28

The detector approach: massless scalar

Wightman functions GM = − a2 16π2 1 sinh2 (aτ/2 − iaǫ) GR = − 1 4π2 1 (τ − iǫ)2 Induced response rate ˙ F(E) = ∞

−∞

dτeiEτ [GM(τ − iǫ) − GR(τ − iǫ)] = − 1 4π2 (2πi)(iE)

−1

  • k=−∞

e−iE 2πi

a k

= 1 2π E 1 e

2π a E − 1

Define profile function F(E) as ˙ F(E) ≡ 1 2π F(E) 1 e

2π a E − 1

.

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 10 / 31

slide-29
SLIDE 29

The Unruh effect: The detector approach

Massless F(E) = E Massive F(E) =

  • E2 − m2 θ(E − m)

particle needs E > m for excitation Massless scalar in general dimensions F(E) = π

d−1 2

Γ

  • d−1

2

  • (2π)d−2 Ed−3

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 11 / 31

slide-30
SLIDE 30

Spectral dimension

Consider a (modified) diffusion/heat equation ∂σ K(x, x′; σ) = −F(−∂2

E) K(x, x′; σ)

with BC: K(x, x′; 0) = δd(x − x′)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 12 / 31

slide-31
SLIDE 31

Spectral dimension

Consider a (modified) diffusion/heat equation ∂σ K(x, x′; σ) = −F(−∂2

E) K(x, x′; σ)

with BC: K(x, x′; 0) = δd(x − x′) σ diffusion time K heat/diffusion kernel F(−∂2

E) determined by the equations of motion

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 12 / 31

slide-32
SLIDE 32

Spectral dimension

Consider a (modified) diffusion/heat equation ∂σ K(x, x′; σ) = −F(−∂2

E) K(x, x′; σ)

with BC: K(x, x′; 0) = δd(x − x′) σ diffusion time K heat/diffusion kernel F(−∂2

E) determined by the equations of motion

⇒ F(p2

E) = ( ˜

G(−p2

E))−1

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 12 / 31

slide-33
SLIDE 33

Spectral dimension

∂σ K(x, x′; σ) = −F(−∂2

E) K(x, x′; σ)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 13 / 31

slide-34
SLIDE 34

Spectral dimension

∂σ K(x, x′; σ) = −F(−∂2

E) K(x, x′; σ)

has as solution K(x, x′; σ) =

  • ddp

(2π)d eip(x−x′)e−σF (p2

E) Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 13 / 31

slide-35
SLIDE 35

Spectral dimension

∂σ K(x, x′; σ) = −F(−∂2

E) K(x, x′; σ)

has as solution K(x, x′; σ) =

  • ddp

(2π)d eip(x−x′)e−σF (p2

E)

with return probability (random walk) after time σ P(σ) =

  • ddp

(2π)d e−σF (p2

E) Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 13 / 31

slide-36
SLIDE 36

Spectral dimension

∂σ K(x, x′; σ) = −F(−∂2

E) K(x, x′; σ)

has as solution K(x, x′; σ) =

  • ddp

(2π)d eip(x−x′)e−σF (p2

E)

with return probability (random walk) after time σ P(σ) =

  • ddp

(2π)d e−σF (p2

E)

Dimension as seen by diffusion process: spectral dimension Ds(σ) = −2 d ln P(σ) d ln σ

Accessible experimentally?

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 13 / 31

slide-37
SLIDE 37

Unruh dimension

Profile function massless scalar in general dimension d F(E) = π

d−1 2

Γ

  • d−1

2

  • (2π)d−2 Ed−3

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 14 / 31

slide-38
SLIDE 38

Unruh dimension

Profile function massless scalar in general dimension d F(E) = π

d−1 2

Γ

  • d−1

2

  • (2π)d−2 Ed−3

Define Unruh dimension DU(E) ≡ d ln F(E) d ln E + 3

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 14 / 31

slide-39
SLIDE 39

Unruh dimension

Profile function massless scalar in general dimension d F(E) = π

d−1 2

Γ

  • d−1

2

  • (2π)d−2 Ed−3

Define Unruh dimension DU(E) ≡ d ln F(E) d ln E + 3

  • Effective dimension of spacetime seen by Unruh effect
  • Agrees with topological dimension d for m = 0
  • Closely related to spectral dimension Ds

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 14 / 31

slide-40
SLIDE 40

Non-trivial momentum dependence ˜ G(p2) gives rise to Possible QG corrections visible in profile function F(E) Dimensional reduction encoded in (Ds)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 15 / 31

slide-41
SLIDE 41

Non-trivial momentum dependence ˜ G(p2) gives rise to Possible QG corrections visible in profile function F(E) Dimensional reduction encoded in (Ds)

Connection between DU and Ds?

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 15 / 31

slide-42
SLIDE 42

QG corrections: Class I

Effective Lagrangian is a polynomial Pn(z) with roots µi L = 1 2 φ Pn(−∂2)φ where Pn(z) = c

n

  • i=1

(z − µi)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 16 / 31

slide-43
SLIDE 43

QG corrections: Class I

Effective Lagrangian is a polynomial Pn(z) with roots µi L = 1 2 φ Pn(−∂2)φ where Pn(z) = c

n

  • i=1

(z − µi) Ostrogradski decomposition two-point function ˜ G(p2) = 1 c

n

  • i=1

Ai p2 − µi where Ai = (

  • j=i

(µi − µj))−1

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 16 / 31

slide-44
SLIDE 44

QG corrections: Class I

Effective Lagrangian is a polynomial Pn(z) with roots µi L = 1 2 φ Pn(−∂2)φ where Pn(z) = c

n

  • i=1

(z − µi) Ostrogradski decomposition two-point function ˜ G(p2) = 1 c

n

  • i=1

Ai p2 − µi where Ai = (

  • j=i

(µi − µj))−1 Leading to F(E) = 1 c

n

  • i=1

Ai

  • E2 − µiθ(E − √µi)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 16 / 31

slide-45
SLIDE 45

QG corrections: Class I

Effective Lagrangian is a polynomial Pn(z) with roots µi L = 1 2 φ Pn(−∂2)φ where Pn(z) = c

n

  • i=1

(z − µi) Ostrogradski decomposition two-point function ˜ G(p2) = 1 c

n

  • i=1

Ai p2 − µi where Ai = (

  • j=i

(µi − µj))−1 Leading to F(E) = 1 c

n

  • i=1

Ai

  • E2 − µiθ(E − √µi)
  • Identify µi = m2 > 0
  • Restrict to polynomials with roots on positive real axis

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 16 / 31

slide-46
SLIDE 46

QG corrections: Class II

Model where Wightman function is superposition of massive contributions (K¨ allen-Lehmann representation) G+(x) = ∞ dm2ρ(m2)G0(x; m) where: ρ(m2) is the spectral density G0 = GM the massive Wightman function

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 17 / 31

slide-47
SLIDE 47

QG corrections: Class II

Model where Wightman function is superposition of massive contributions (K¨ allen-Lehmann representation) G+(x) = ∞ dm2ρ(m2)G0(x; m) where: ρ(m2) is the spectral density G0 = GM the massive Wightman function Profile function F(E) = E2 dm2ρ(m2)

  • E2 − m2

A superposition of massive contributions, weighed by ρ(m2) Ostrogradski: ρ(m2) sum of δ(m − √µi)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 17 / 31

slide-48
SLIDE 48

Multi-scale models

Consider a two scale model: P2(p2) = 1 m2 p2(p2 − m2) ⇒ ˜ G(p2) = 1 p2 − 1 p2 − m2

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 18 / 31

slide-49
SLIDE 49

Multi-scale models

Consider a two scale model: P2(p2) = 1 m2 p2(p2 − m2) ⇒ ˜ G(p2) = 1 p2 − 1 p2 − m2 scales as p2 ≪ m2 ˜ G(p2) ∝ p−2 Ds = 4 p2 ≫ m2 ˜ G(p2) ∝ p−4 Ds = 2

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 18 / 31

slide-50
SLIDE 50

Multi-scale models

Consider a two scale model: P2(p2) = 1 m2 p2(p2 − m2) ⇒ ˜ G(p2) = 1 p2 − 1 p2 − m2 scales as p2 ≪ m2 ˜ G(p2) ∝ p−2 Ds = 4 p2 ≫ m2 ˜ G(p2) ∝ p−4 Ds = 2 Leading to the profile function F(E) = E −

  • E2 − m2 θ(E − m)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 18 / 31

slide-51
SLIDE 51

Multi-scale models

Consider a two scale model: P2(p2) = 1 m2 p2(p2 − m2) ⇒ ˜ G(p2) = 1 p2 − 1 p2 − m2 scales as p2 ≪ m2 ˜ G(p2) ∝ p−2 Ds = 4 p2 ≫ m2 ˜ G(p2) ∝ p−4 Ds = 2 Leading to the profile function F(E) = E −

  • E2 − m2 θ(E − m)

which scales as E2 ≪ m2 F(E) ∝ E DU = 4 E2 ≫ m2 F(E) ∝ E−1 DU = 2

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 18 / 31

slide-52
SLIDE 52

Multi-scale models

Profile function 2-scale model (m = 1) F(E) = E −

  • E2 − m2 θ(E − m)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 19 / 31

slide-53
SLIDE 53

Multi-scale models

Profile function 2-scale model (m = 1) F(E) = E −

  • E2 − m2 θ(E − m)

DU Ds 0.01 1 100 E 1 2 3 4 D 1 2 3 4 5 E 0.2 0.4 0.6 0.8 1.0 1.2 FE Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 19 / 31

slide-54
SLIDE 54

Multi-scale models

Consider a three scale model: P3(p2) = 1 m2

2m2 3

p2(p2 − m2

2)(p2 − m2 3)

⇒ ˜ G(p2) = 1 p2 − m2

3

m2

3 − m2 2

1 p2 − m2

2

+ m2

2

m2

3 − m2 2

1 p2 − m2

3

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 20 / 31

slide-55
SLIDE 55

Multi-scale models

Consider a three scale model: P3(p2) = 1 m2

2m2 3

p2(p2 − m2

2)(p2 − m2 3)

⇒ ˜ G(p2) = 1 p2 − m2

3

m2

3 − m2 2

1 p2 − m2

2

+ m2

2

m2

3 − m2 2

1 p2 − m2

3

scales as p2 ≪ m2

2

˜ G(p2) ∝ p−2 Ds = 4 m2

2 ≪ p2 ≪ m2 3

˜ G(p2) ∝ p−4 Ds = 2 m2

3 ≪ p2

˜ G(p2) ∝ p−6 Ds = 4 3

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 20 / 31

slide-56
SLIDE 56

Multi-scale models

Consider a three scale model: P3(p2) = 1 m2

2m2 3

p2(p2 − m2

2)(p2 − m2 3)

⇒ ˜ G(p2) = 1 p2 − m2

3

m2

3 − m2 2

1 p2 − m2

2

+ m2

2

m2

3 − m2 2

1 p2 − m2

3

scales as p2 ≪ m2

2

˜ G(p2) ∝ p−2 Ds = 4 m2

2 ≪ p2 ≪ m2 3

˜ G(p2) ∝ p−4 Ds = 2 m2

3 ≪ p2

˜ G(p2) ∝ p−6 Ds = 4 3 Leading to the profile function F(E) = E − m2

3

m2

3 − m2 2

  • E2 − m2

2 θ(E − m2) +

m2

2

m2

3 − m2 2

  • E2 − m2

3 θ(E − m3)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 20 / 31

slide-57
SLIDE 57

Multi-scale models

Profile function 3-scale model (m2 = 0.1, m3 = 10) F(E) = E − m2

3

m2

3 − m2 2

  • E2 − m2

2 θ(E − m2) +

m2

2

m2

3 − m2 2

  • E2 − m2

3 θ(E − m3)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 21 / 31

slide-58
SLIDE 58

Multi-scale models

Profile function 3-scale model (m2 = 0.1, m3 = 10) F(E) = E − m2

3

m2

3 − m2 2

  • E2 − m2

2 θ(E − m2) +

m2

2

m2

3 − m2 2

  • E2 − m2

3 θ(E − m3)

0.01 0.1 1 10 100 1000 E 1 2 3 4 5 D 5 10 15 20 E 0.001 0.001 0.002 0.003 0.004 0.005 FE Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 21 / 31

slide-59
SLIDE 59

Relate Ds to DU

Consider two-point function of the form ˜ G(p2) ∝ p−(2+η) (1)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 22 / 31

slide-60
SLIDE 60

Relate Ds to DU

Consider two-point function of the form ˜ G(p2) ∝ p−(2+η) (1) in d = 4: Ds = 8 2 + η DU = 4 − η

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 22 / 31

slide-61
SLIDE 61

Relate Ds to DU

Consider two-point function of the form ˜ G(p2) ∝ p−(2+η) (1) in d = 4: Ds = 8 2 + η DU = 4 − η New mass-scale decreases Du by two Du = 6 − 8 Ds , DU = Ds for η = 0, 2 ⇒ Use DU to measure Ds

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 22 / 31

slide-62
SLIDE 62

Kaluza-Klein models

Massless scalar field φ(x, x5) =

∞

  • n=−∞

φn(x)ei n

R x5 with x5 ∈ [0, 2πR]

gives for the action 1 2

  • d5x[(∂µφ)2 − (∂5φ)2] = 2πR
  • d4x 1

2

∞

  • n=−∞
  • |∂µφn|2 − n2

R2 |φn|2

  • Fleur Versteegen

QG signatures in the Unruh effect April 25, 2017 23 / 31

slide-63
SLIDE 63

Kaluza-Klein models

Massless scalar field φ(x, x5) =

∞

  • n=−∞

φn(x)ei n

R x5 with x5 ∈ [0, 2πR]

gives for the action 1 2

  • d5x[(∂µφ)2 − (∂5φ)2] = 2πR
  • d4x 1

2

∞

  • n=−∞
  • |∂µφn|2 − n2

R2 |φn|2

  • Tower of Kaluza-Klein modes φn give

˜ G(p2) = 1 2πR

∞

  • n=−∞
  • p2 − n2

R2 −1

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 23 / 31

slide-64
SLIDE 64

Kaluza-Klein models

Profile function F(E) = 1 2πR  E + 2

∞

  • n=1
  • E2 − n2

R2 θ(E − n R )  

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 24 / 31

slide-65
SLIDE 65

Kaluza-Klein models

Profile function F(E) = 1 2πR  E + 2

∞

  • n=1
  • E2 − n2

R2 θ(E − n R )   scales as E < 1/R F(E) ∝ E DU = 4 E ≫ 1/R F(E) ∝ E2 DU = 5

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 24 / 31

slide-66
SLIDE 66

Kaluza-Klein models

Profile function F(E) = 1 2πR  E + 2

∞

  • n=1
  • E2 − n2

R2 θ(E − n R )   scales as E < 1/R F(E) ∝ E DU = 4 E ≫ 1/R F(E) ∝ E2 DU = 5

1 2 3 4 5 6 E 2 Π 50 100 150 200 250 300 350 FE 10 20 30 40 50 E 2 Π 5000 10000 15000 20000 25000 FE Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 24 / 31

slide-67
SLIDE 67

Spectral Actions

Action generated by trace Dirac operator Sχ,Λ = Tr [χ(D2/Λ2)]

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 25 / 31

slide-68
SLIDE 68

Spectral Actions

Action generated by trace Dirac operator Sχ,Λ = Tr [χ(D2/Λ2)] where

  • χ is a positive function specific to the model
  • Λ sets scale of the theory
  • D is a Dirac operator

D2 = −(1∂2 + E), E = −iγµγ5∂µφ − φ2

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 25 / 31

slide-69
SLIDE 69

Spectral Actions

Action generated by trace Dirac operator Sχ,Λ = Tr [χ(D2/Λ2)] where

  • χ is a positive function specific to the model
  • Λ sets scale of the theory
  • D is a Dirac operator

D2 = −(1∂2 + E), E = −iγµγ5∂µφ − φ2

  • Related to almost-commutative standard model

(A.H. Chamseddine and A. Connes, Commun. Math. Phys. 186, 731(1997)) (A.H. Chamseddine and A. Connes, Phys. Rev. Lett. 77, 4868(1995))

  • Spectral action through heat-kernel techniques
  • Setup Euclidean: Wick-rotate to get Lorentzian signature

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 25 / 31

slide-70
SLIDE 70

Spectral Actions

Ostrogradski-type model χ(z) = (a + z) θ(1 − z), a > 0

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 26 / 31

slide-71
SLIDE 71

Spectral Actions

Ostrogradski-type model χ(z) = (a + z) θ(1 − z), a > 0 Polynomial of the form P2(p2) = − 1 8π2 (8a + 4 − 2ap2 + 1 3 p4), µ1,2 = 3a ∓

  • 9a2 − 24a − 12

for 2(2 + √ 7)/3 < a < (3 + √ 15)/2

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 26 / 31

slide-72
SLIDE 72

Spectral Actions

Ostrogradski-type model χ(z) = (a + z) θ(1 − z), a > 0 Polynomial of the form P2(p2) = − 1 8π2 (8a + 4 − 2ap2 + 1 3 p4), µ1,2 = 3a ∓

  • 9a2 − 24a − 12

for 2(2 + √ 7)/3 < a < (3 + √ 15)/2 Profile function F(E) = 24π2 µ2 − µ1 (

  • E2 − µ1θ(E − √µ1) −
  • E2 − µ2θ(E − √µ2))

5 10 15 20 E 20 40 60 80 100 FE Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 26 / 31

slide-73
SLIDE 73

Causal set theory

Manifold has an underlying discrete structure (partially ordered set) Order given by causal relation between elements ⇒ Lorentzian signature by design

Credits: Lisa Glaser https://sites.google.com/site/lisaglaserphysics/research/causal-set-theory Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 27 / 31

slide-74
SLIDE 74

Causal set theory

Wightman function causal sets G+(x2) = − i 2π3 ∞ dξξ2 K1(i √ x2ξ) √ x2ξg(ξ2)

(S.Aslanbeigi, M.Saravani and R.D. Sorkin, JHEP 1406 (2014) 024) Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 28 / 31

slide-75
SLIDE 75

Causal set theory

Wightman function causal sets G+(x2) = − i 2π3 ∞ dξξ2 K1(i √ x2ξ) √ x2ξg(ξ2)

(S.Aslanbeigi, M.Saravani and R.D. Sorkin, JHEP 1406 (2014) 024)

momentum dependence g(ξ2) = a + 4πξ−1

3

  • n=0

bn n! Cn ∞ s4(n+1/2)e−CsK1(ξs)ds where a, b0, b1, b2, b3, C are numerical constants

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 28 / 31

slide-76
SLIDE 76

Causal set theory

Wightman function causal sets G+(x2) = − i 2π3 ∞ dξξ2 K1(i √ x2ξ) √ x2ξg(ξ2)

(S.Aslanbeigi, M.Saravani and R.D. Sorkin, JHEP 1406 (2014) 024)

momentum dependence g(ξ2) = a + 4πξ−1

3

  • n=0

bn n! Cn ∞ s4(n+1/2)e−CsK1(ξs)ds where a, b0, b1, b2, b3, C are numerical constants Asymptotics g(ξ2) given by lim

ξ2→∞

1 g(ξ2) = − 2 √ 6π ξ4 + ... lim

ξ2→0

1 g(ξ2) = − 1 ξ2 + ...

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 28 / 31

slide-77
SLIDE 77

Causal set theory

Use ρ(m2) to approximate G+ with one massless + continuum of massive modes

(M. Saravani, and s. Aslanbeigi, Phys. Rev. D 92, 103504(2015))

G+(t, x) = G(t, x; m = 0) + ∞ dm2ρ(m2)G(t, x; m) ρ(m2) = e−αm2

N

  • n=0

bnm2n.

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 29 / 31

slide-78
SLIDE 78

Causal set theory

Use ρ(m2) to approximate G+ with one massless + continuum of massive modes

(M. Saravani, and s. Aslanbeigi, Phys. Rev. D 92, 103504(2015))

G+(t, x) = G(t, x; m = 0) + ∞ dm2ρ(m2)G(t, x; m) ρ(m2) = e−αm2

N

  • n=0

bnm2n. The profile function becomes F(E) = E +

N

  • n=0

bn E2 dm2eαm2m2n E2 − m2

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 29 / 31

slide-79
SLIDE 79

Causal set theory

Profile function F(E) = E +

N

  • n=0

bn E2 dm2eαm2m2n E2 − m2

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 30 / 31

slide-80
SLIDE 80

Causal set theory

Profile function F(E) = E +

N

  • n=0

bn E2 dm2eαm2m2n E2 − m2 For N = 1, b1 = −(b0 + 1), b0 = 1, α = 1

5 10 15 20 E 0.2 0.4 0.6 0.8 1.0 1.2 1.4 FE 0.1 1 10 100 E 1 2 3 4 5 DU Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 30 / 31

slide-81
SLIDE 81

Conclusions

Unruh temperature unaffected by quantum corrections (geometric effect)

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 31 / 31

slide-82
SLIDE 82

Conclusions

Unruh temperature unaffected by quantum corrections (geometric effect) Profile functions not affected in low-energy regime

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 31 / 31

slide-83
SLIDE 83

Conclusions

Unruh temperature unaffected by quantum corrections (geometric effect) Profile functions not affected in low-energy regime DU connected to Ds ⇒ obtain Ds through profile function

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 31 / 31

slide-84
SLIDE 84

Conclusions

Unruh temperature unaffected by quantum corrections (geometric effect) Profile functions not affected in low-energy regime DU connected to Ds ⇒ obtain Ds through profile function Negative spectral density related to dynamical dimensional reduction

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 31 / 31

slide-85
SLIDE 85

Conclusions

Unruh temperature unaffected by quantum corrections (geometric effect) Profile functions not affected in low-energy regime DU connected to Ds ⇒ obtain Ds through profile function Negative spectral density related to dynamical dimensional reduction Opening up of dimensions (Kaluza-Klein) leads to dimensional enhancement Ds

Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 31 / 31