qg signatures in the unruh effectcqc/slides/... · quantum gravity signatures in the unruh effect...

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

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

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

asinh (aτ), x =

acosh (aτ), y = y, z = z.

Klein-Gordon for m = 0: [−∂2

t + ∂2x + ∂2

y + ∂2z

]φ(t, x, y, z) = 0[

e−2aξ(−∂2τ + ∂2

ξ ) + ∂2y + ∂2

]φ(τ, ξ, y, z) = 0

t =eaξ

asinh (aτ), x =

t + ∂2x + ∂2

y + ∂2z

]φ(t, x, y, z) = 0[

e−2aξ(−∂2τ + ∂2

ξ ) + ∂2y + ∂2

]φ(τ, ξ, y, z) = 0

t =eaξ

asinh (aτ), x =

t + ∂2x + ∂2

y + ∂2z

]φ(t, x, y, z) = 0[

e−2aξ(−∂2τ + ∂2

ξ ) + ∂2y + ∂2

]φ(τ, ξ, y, z) = 0

Solutions:

uR =e−iΩτ

2π2√a

KiΩ/a

(|~p⊥|aeaξ)ei~p⊥·~x⊥

uM =1√

2(2π)3ωe−i(ωt−kxx−

~k⊥·~x⊥)

define annihilation/creation operatorsaω , bΩ

such that

aω |0M 〉 = 0, bΩ |0R〉 = 0

Solutions:

uR =e−iΩτ

2π2√a

KiΩ/a

(|~p⊥|aeaξ)ei~p⊥·~x⊥

uM =1√

2(2π)3ωe−i(ωt−kxx−

~k⊥·~x⊥)

define annihilation/creation operatorsaω , bΩ

such that

aω |0M 〉 = 0, bΩ |0R〉 = 0

Field expansion:

∫d3k

(2π)3

(aω u

Mω + a†ω (uMω )†

∫d3p

(2π)3

(bΩ u

RΩ + b†Ω (uRΩ)†

Relation between a, a† and b, b†?

⇒ Bogolyubov transformation:

∫dω

∫d~k⊥

(αΩω aω − βΩω a

)Find coefficients α, β

Field expansion:

∫d3k

(2π)3

(aω u

∫d3p

(2π)3

(bΩ u

)Relation between a, a† and b, b†?

∫dω

∫d~k⊥

Field expansion:

∫d3k

(2π)3

(aω u

∫d3p

(2π)3

(bΩ u

)Relation between a, a† and b, b†?

∫dω

∫d~k⊥

Interested in number of particles observed by accelerating observer

nΩ =〈0M | N |0M 〉

=〈0M | b†Ω bΩ′ |0M 〉

∫d3kβΩωβ

∗Ω′ω

βΩω = −1√

2πaω(e2πΩ/a − 1)

(ω + kx

ω − kx

)−iΩ/2a

nΩ =1

e2πΩa − 1

δ(Ω− Ω′)δ(~k⊥ − ~k′⊥)

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

⇒ Geometric effect

nΩ =〈0M | N |0M 〉

=〈0M | b†Ω bΩ′ |0M 〉

∫d3kβΩωβ

∗Ω′ω

βΩω = −1√

(ω + kx

ω − kx

)−iΩ/2athen

nΩ =1

e2πΩa − 1

δ(Ω− Ω′)δ(~k⊥ − ~k′⊥)

nΩ =〈0M | N |0M 〉

=〈0M | b†Ω bΩ′ |0M 〉

∫d3kβΩωβ

∗Ω′ω

βΩω = −1√

(ω + kx

ω − kx

)−iΩ/2athen

nΩ =1

e2πΩa − 1

δ(Ω− Ω′)δ(~k⊥ − ~k′⊥)

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) =

∫dτ1dτ2e

i∆E∆τG(∆τ − iε)

LI = gm(τ)φ(x)

|E2〉 |0M 〉 → |E1〉 |~k〉

Amplitude

Pi→f =

∫d3k|A|2 = g2| 〈E1|m |E2〉 |2F (∆E)

where F (∆E) =

∫dτ1dτ2e

LI = gm(τ)φ(x)

|E2〉 |0M 〉 → |E1〉 |~k〉

Amplitude

Pi→f =

∫d3k|A|2 = g2| 〈E1|m |E2〉 |2F (∆E)

where F (∆E) =

∫dτ1dτ2e

LI = gm(τ)φ(x)

|E2〉 |0M 〉 → |E1〉 |~k〉

Amplitude

Pi→f =

∫d3k|A|2 = g2| 〈E1|m |E2〉 |2F (∆E)

where F (∆E) =

∫dτ1dτ2e

LI = gm(τ)φ(x)

|E2〉 |0M 〉 → |E1〉 |~k〉

Amplitude

Pi→f =

∫d3k|A|2 = g2| 〈E1|m |E2〉 |2F (∆E)

where F (∆E) =

∫dτ1dτ2e

For massive scalar field in Minkowski space:

G(p2) =1

p2 −m2=

(p0 +√~p2 +m2)(p0 −

√~p2 +m2)

Positive frequency Wightman function encircles pole at√~p2 +m2

G+(~x, t) = −i∫

(2π)3

∮γ+

2πG(p2)e−i(p

0t−~p·~x),

in real space we find

G+(x, x′) = −im

K1(im√

(t− t′ − iε)2)− (~x− ~x′)2√(t− t′ − iε)2)− (~x− ~x′)2

m→ 0, G+(x, x′) = −1

(t− t′ − iε)2 − (~x− ~x′)2

G(p2) =1

p2 −m2=

(p0 +√~p2 +m2)(p0 −

√~p2 +m2)

G+(~x, t) = −i∫

(2π)3

∮γ+

2πG(p2)e−i(p

0t−~p·~x),

G+(x, x′) = −im

K1(im√

(t− t′ − iε)2)− (~x− ~x′)2√(t− t′ − iε)2)− (~x− ~x′)2

m→ 0, G+(x, x′) = −1

(t− t′ − iε)2 − (~x− ~x′)2

G(p2) =1

p2 −m2=

(p0 +√~p2 +m2)(p0 −

√~p2 +m2)

G+(~x, t) = −i∫

(2π)3

∮γ+

2πG(p2)e−i(p

0t−~p·~x),

G+(x, x′) = −im

K1(im√

(t− t′ − iε)2)− (~x− ~x′)2√(t− t′ − iε)2)− (~x− ~x′)2

m→ 0, G+(x, x′) = −1

(t− t′ − iε)2 − (~x− ~x′)2

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 − iε)−GR(∆τ − iε)]

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 − iε)−GR(τ − iε)]

|E2〉 |0M 〉 → |E1〉 |~k〉 , Pi→f

F (∆E) =

∫ ∞−∞

F (E) =

∫ ∞−∞

|E2〉 |0M 〉 → |E1〉 |~k〉 , Pi→f

F (∆E) =

∫ ∞−∞

F (E) =

∫ ∞−∞

|E2〉 |0M 〉 → |E1〉 |~k〉 , Pi→f

F (∆E) =

∫ ∞−∞

F (E) =

∫ ∞−∞

The detector approach: massless scalar

Wightman functions

GM = −a2

sinh2 (aτ/2− iaε)

GR = −1

(τ − iε)2

Induced response rate

F (E) =

∫ ∞−∞

= −1

4π2(2πi)(iE)

−1∑k=−∞

e−iE2πiak

e2πaE − 1

Define profile function F(E) as

F (E) ≡1

2πF(E)

e2πaE − 1

Wightman functions

GM = −a2

GR = −1

(τ − iε)2

F (E) =

∫ ∞−∞

= −1

4π2(2πi)(iE)

−1∑k=−∞

e−iE2πiak

e2πaE − 1

F (E) ≡1

2πF(E)

e2πaE − 1

Wightman functions

GM = −a2

GR = −1

(τ − iε)2

F (E) =

∫ ∞−∞

= −1

4π2(2πi)(iE)

−1∑k=−∞

e−iE2πiak

e2πaE − 1

F (E) ≡1

2πF(E)

e2πaE − 1

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) =πd−1

Γ(d−1

)(2π)d−2

Ed−3

Spectral dimension

Consider a (modified) diffusion/heat equation

∂σK(x, x′;σ) = −F (−∂2E)K(x, x′;σ)

with BC: K(x, x′; 0) = δd(x− x′)

σ diffusion time

K heat/diffusion kernel

F (−∂2E) determined by the equations of motion

⇒ F (p2E) = (G(−p2

E))−1

Spectral dimension

∂σK(x, x′;σ) = −F (−∂2E)K(x, x′;σ)

σ diffusion time

⇒ F (p2E) = (G(−p2

E))−1

Spectral dimension

∂σK(x, x′;σ) = −F (−∂2E)K(x, x′;σ)

σ diffusion time

⇒ F (p2E) = (G(−p2

E))−1

Spectral dimension

∂σK(x, x′;σ) = −F (−∂2E)K(x, x′;σ)

has as solution

K(x, x′;σ) =

∫ddp

(2π)deip(x−x

′)e−σF (p2E)

with return probability (random walk) after time σ

P (σ) =

∫ddp

(2π)de−σF (p2

Dimension as seen by diffusion process: spectral dimension

Ds(σ) = −2d lnP (σ)

d lnσ

Accessible experimentally?

Spectral dimension

∂σK(x, x′;σ) = −F (−∂2E)K(x, x′;σ)

has as solution

K(x, x′;σ) =

∫ddp

(2π)deip(x−x

′)e−σF (p2E)

P (σ) =

∫ddp

(2π)de−σF (p2

d lnσ

Spectral dimension

∂σK(x, x′;σ) = −F (−∂2E)K(x, x′;σ)

has as solution

K(x, x′;σ) =

∫ddp

(2π)deip(x−x

′)e−σF (p2E)

P (σ) =

∫ddp

(2π)de−σF (p2

d lnσ

Spectral dimension

∂σK(x, x′;σ) = −F (−∂2E)K(x, x′;σ)

has as solution

K(x, x′;σ) =

∫ddp

(2π)deip(x−x

′)e−σF (p2E)

P (σ) =

∫ddp

(2π)de−σF (p2

d lnσ

Unruh dimension

Profile function massless scalar in general dimension d

F(E) =πd−1

Γ(d−1

)(2π)d−2

Ed−3

Define Unruh dimension

DU (E) ≡d lnF(E)

d lnE+ 3

- Effective dimension of spacetime seen by Unruh effect

- Agrees with topological dimension d for m = 0

- Closely related to spectral dimension Ds

Unruh dimension

F(E) =πd−1

Γ(d−1

)(2π)d−2

Ed−3

DU (E) ≡d lnF(E)

d lnE+ 3

Unruh dimension

F(E) =πd−1

Γ(d−1

)(2π)d−2

Ed−3

DU (E) ≡d lnF(E)

d lnE+ 3

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?

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?

QG corrections: Class I

Effective Lagrangian is a polynomial Pn(z) with roots µi

2φPn(−∂2)φ where Pn(z) = c

n∏i=1

(z − µi)

Ostrogradski decomposition two-point function

G(p2) =1

n∑i=1

p2 − µiwhere Ai = (

∏j 6=i

(µi − µj))−1

Leading to

F(E) =1

n∑i=1

Ai√E2 − µiθ(E −

√µi)

- Identify µi = m2 > 0

- Restrict to polynomials with roots on positive real axis

n∏i=1

(z − µi)

G(p2) =1

n∑i=1

∏j 6=i

(µi − µj))−1

Leading to

F(E) =1

n∑i=1

√µi)

n∏i=1

(z − µi)

G(p2) =1

n∑i=1

∏j 6=i

(µi − µj))−1

Leading to

F(E) =1

n∑i=1

√µi)

n∏i=1

(z − µi)

G(p2) =1

n∑i=1

∏j 6=i

(µi − µj))−1

Leading to

F(E) =1

n∑i=1

√µi)

QG corrections: Class II

Model where Wightman function is superposition of massive contributions (Kallen-Lehmannrepresentation)

G+(x) =

∫ ∞0

dm2ρ(m2)G0(x;m)

where: ρ(m2) is the spectral density

G0 = GM the massive Wightman function

Profile function

F(E) =

∫ E2

0dm2ρ(m2)

√E2 −m2

A superposition of massive contributions, weighed by ρ(m2)

Ostrogradski: ρ(m2) sum of δ(m−√µi)

QG corrections: Class II

Model where Wightman function is superposition of massive contributions (Kallen-Lehmannrepresentation)

G+(x) =

∫ ∞0

dm2ρ(m2)G0(x;m)

where: ρ(m2) is the spectral density

G0 = GM the massive Wightman function

Profile function

F(E) =

∫ E2

0dm2ρ(m2)

√E2 −m2

A superposition of massive contributions, weighed by ρ(m2)

Ostrogradski: ρ(m2) sum of δ(m−√µi)

Multi-scale models

Consider a two scale model:

P2(p2) =1

m2p2(p2 −m2) ⇒ G(p2) =

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

Multi-scale models

P2(p2) =1

m2p2(p2 −m2) ⇒ G(p2) =

p2 −m2

scales as

p2 m2 G(p2) ∝ p−2 Ds = 4

p2 m2 G(p2) ∝ p−4 Ds = 2

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

Multi-scale models

P2(p2) =1

m2p2(p2 −m2) ⇒ G(p2) =

p2 −m2

scales as

p2 m2 G(p2) ∝ p−2 Ds = 4

p2 m2 G(p2) ∝ p−4 Ds = 2

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

Multi-scale models

P2(p2) =1

m2p2(p2 −m2) ⇒ G(p2) =

p2 −m2

scales as

p2 m2 G(p2) ∝ p−2 Ds = 4

p2 m2 G(p2) ∝ p−4 Ds = 2

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

Multi-scale models

Profile function 2-scale model (m = 1)

F(E) = E −√E2 −m2 θ(E −m)

0.01 1 100E0

1 2 3 4 5E

1.2FHEL

Multi-scale models

Profile function 2-scale model (m = 1)

F(E) = E −√E2 −m2 θ(E −m)

0.01 1 100E0

1 2 3 4 5E

1.2FHEL

Multi-scale models

Consider a three scale model:

P3(p2) =1

p2(p2 −m22)(p2 −m2

⇒ G(p2) =1

m23 −m2

p2 −m22

m23 −m2

p2 −m23

scales as

p2 m22 G(p2) ∝ p−2 Ds = 4

m22 p2 m2

3 G(p2) ∝ p−4 Ds = 2

m23 p2 G(p2) ∝ p−6 Ds =

F(E) = E −m2

m23 −m2

√E2 −m2

2 θ(E −m2) +m2

m23 −m2

√E2 −m2

3 θ(E −m3)

Multi-scale models

P3(p2) =1

p2(p2 −m22)(p2 −m2

⇒ G(p2) =1

m23 −m2

p2 −m22

m23 −m2

p2 −m23

scales as

p2 m22 G(p2) ∝ p−2 Ds = 4

m22 p2 m2

3 G(p2) ∝ p−4 Ds = 2

m23 p2 G(p2) ∝ p−6 Ds =

F(E) = E −m2

m23 −m2

√E2 −m2

2 θ(E −m2) +m2

m23 −m2

√E2 −m2

3 θ(E −m3)

Multi-scale models

P3(p2) =1

p2(p2 −m22)(p2 −m2

⇒ G(p2) =1

m23 −m2

p2 −m22

m23 −m2

p2 −m23

scales as

p2 m22 G(p2) ∝ p−2 Ds = 4

m22 p2 m2

3 G(p2) ∝ p−4 Ds = 2

m23 p2 G(p2) ∝ p−6 Ds =

F(E) = E −m2

m23 −m2

√E2 −m2

2 θ(E −m2) +m2

m23 −m2

√E2 −m2

3 θ(E −m3)

Multi-scale models

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

F(E) = E −m2

m23 −m2

√E2 −m2

2 θ(E −m2) +m2

m23 −m2

√E2 −m2

3 θ(E −m3)

0.01 0.1 1 10 100 1000E

5 10 15 20E

-0.001

0.005FHEL

Multi-scale models

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

F(E) = E −m2

m23 −m2

√E2 −m2

2 θ(E −m2) +m2

m23 −m2

√E2 −m2

3 θ(E −m3)

0.01 0.1 1 10 100 1000E

5 10 15 20E

-0.001

0.005FHEL

Relate Ds to DU

Consider two-point function of the form

G(p2) ∝ p−(2+η) (1)

in d = 4:

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

Relate Ds to DU

G(p2) ∝ p−(2+η) (1)

in d = 4:

2 + η

DU = 4− η

Du = 6−8

Relate Ds to DU

G(p2) ∝ p−(2+η) (1)

in d = 4:

2 + η

DU = 4− η

Du = 6−8

Kaluza-Klein models

Massless scalar field φ(x, x5) =∞∑

n=−∞φn(x)ei

nRx5 with x5 ∈ [0, 2πR]

gives for the action

∫d5x[(∂µφ)2 − (∂5φ)2] = 2πR

∫d4x

∞∑n=−∞

[|∂µφn|2 −

R2|φn|2

Tower of Kaluza-Klein modes φn give

G(p2) =1

∞∑n=−∞

(p2 −

Kaluza-Klein models

Massless scalar field φ(x, x5) =∞∑

n=−∞φn(x)ei

nRx5 with x5 ∈ [0, 2πR]

gives for the action

∫d5x[(∂µφ)2 − (∂5φ)2] = 2πR

∫d4x

∞∑n=−∞

[|∂µφn|2 −

R2|φn|2

]Tower of Kaluza-Klein modes φn give

G(p2) =1

∞∑n=−∞

(p2 −

Kaluza-Klein models

Profile function

F(E) =1

E + 2∞∑n=1

√E2 −

R2θ(E −

scales as

E < 1/R F(E) ∝ E DU = 4

E 1/R F(E) ∝ E2 DU = 5

1 2 3 4 5 6

350FHEL

10 20 30 40 50

10 000

15 000

20 000

25 000FHEL

Kaluza-Klein models

Profile function

F(E) =1

E + 2∞∑n=1

√E2 −

R2θ(E −

scales as

E < 1/R F(E) ∝ E DU = 4

E 1/R F(E) ∝ E2 DU = 5

1 2 3 4 5 6

350FHEL

10 20 30 40 50

10 000

15 000

20 000

25 000FHEL

Kaluza-Klein models

Profile function

F(E) =1

E + 2∞∑n=1

√E2 −

R2θ(E −

scales as

E < 1/R F(E) ∝ E DU = 4

E 1/R F(E) ∝ E2 DU = 5

1 2 3 4 5 6

350FHEL

10 20 30 40 50

10 000

15 000

20 000

25 000FHEL

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

Spectral Actions

D2 = −(1∂2 + E), E = −iγµγ5∂µφ− φ2

Spectral Actions

D2 = −(1∂2 + E), E = −iγµγ5∂µφ− φ2

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 +

3p4), µ1,2 = 3a∓

√9a2 − 24a− 12

for 2(2 +√

7)/3 < a < (3 +√

Profile function

F(E) =24π2

µ2 − µ1(√E2 − µ1θ(E −

√µ1)−

√E2 − µ2θ(E −

√µ2))

5 10 15 20E

100FHEL

Spectral Actions

χ(z) = (a+ z) θ(1− z), a > 0

P2(p2) = −1

8π2(8a+ 4− 2ap2 +

3p4), µ1,2 = 3a∓

√9a2 − 24a− 12

for 2(2 +√

7)/3 < a < (3 +√

Profile function

F(E) =24π2

µ2 − µ1(√E2 − µ1θ(E −

√µ1)−

√E2 − µ2θ(E −

√µ2))

5 10 15 20E

100FHEL

Spectral Actions

χ(z) = (a+ z) θ(1− z), a > 0

P2(p2) = −1

8π2(8a+ 4− 2ap2 +

3p4), µ1,2 = 3a∓

√9a2 − 24a− 12

for 2(2 +√

7)/3 < a < (3 +√

Profile function

F(E) =24π2

µ2 − µ1(√E2 − µ1θ(E −

√µ1)−

√E2 − µ2θ(E −

√µ2))

5 10 15 20E

100FHEL

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

Causal set theory

Wightman function causal sets

G+(x2) = −i

∫ ∞0

dξξ2K1(i√x2ξ)

√x2ξg(ξ2)

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

momentum dependence

g(ξ2) = a+ 4πξ−13∑

n!Cn∫ ∞

0s4(n+1/2)e−CsK1(ξs)ds

where a, b0, b1, b2, b3, C are numerical constants

Asymptotics g(ξ2) given by

limξ2→∞

g(ξ2)= −

ξ4+ ...

limξ2→0

g(ξ2)= −

ξ2+ ...

Causal set theory

G+(x2) = −i

∫ ∞0

dξξ2K1(i√x2ξ)

√x2ξg(ξ2)

momentum dependence

g(ξ2) = a+ 4πξ−13∑

n!Cn∫ ∞

limξ2→∞

g(ξ2)= −

ξ4+ ...

limξ2→0

g(ξ2)= −

ξ2+ ...

Causal set theory

G+(x2) = −i

∫ ∞0

dξξ2K1(i√x2ξ)

√x2ξg(ξ2)

momentum dependence

g(ξ2) = a+ 4πξ−13∑

n!Cn∫ ∞

limξ2→∞

g(ξ2)= −

ξ4+ ...

limξ2→0

g(ξ2)= −

ξ2+ ...

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) +

∫ ∞0

dm2ρ(m2)G(t, ~x;m)

ρ(m2) = e−αm2N∑n=0

bnm2n.

The profile function becomes

F(E) = E +N∑n=0

∫ E2

0dm2eαm

√E2 −m2

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) +

∫ ∞0

dm2ρ(m2)G(t, ~x;m)

ρ(m2) = e−αm2N∑n=0

bnm2n.

The profile function becomes

F(E) = E +

N∑n=0

∫ E2

0dm2eαm

√E2 −m2

Causal set theory

Profile function

F(E) = E +N∑n=0

∫ E2

0dm2eαm

√E2 −m2

For N = 1, b1 = −(b0 + 1), b0 = 1, α = 1

5 10 15 20E

1.4FHEL

0.1 1 10 100E

Causal set theory

Profile function

F(E) = E +N∑n=0

∫ E2

0dm2eαm

√E2 −m2

For N = 1, b1 = −(b0 + 1), b0 = 1, α = 1

5 10 15 20E

1.4FHEL

0.1 1 10 100E

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

Conclusions

DU connected to Ds

Conclusions

DU connected to Ds

Conclusions

DU connected to Ds

Conclusions

DU connected to Ds

qg signatures in the unruh effectcqc/slides/... · quantum gravity signatures in the unruh effect...

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