qg signatures in the unruh effectcqc/slides/... · quantum gravity signatures in the unruh effect...
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![Page 1: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/1.jpg)
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
![Page 2: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/2.jpg)
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
![Page 3: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/3.jpg)
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 =
eaξ
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
z
]φ(τ, ξ, y, z) = 0
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 3 / 31
![Page 4: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/4.jpg)
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 =
eaξ
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
z
]φ(τ, ξ, y, z) = 0
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 3 / 31
![Page 5: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/5.jpg)
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 =
eaξ
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
z
]φ(τ, ξ, y, z) = 0
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 3 / 31
![Page 6: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/6.jpg)
The Unruh effect: Overview
Solutions:
uR =e−iΩτ
2π2√a
sinh
(πΩ
a
)1/2
×
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
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 4 / 31
![Page 7: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/7.jpg)
The Unruh effect: Overview
Solutions:
uR =e−iΩτ
2π2√a
sinh
(πΩ
a
)1/2
×
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
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 4 / 31
![Page 8: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/8.jpg)
The Unruh effect: Overview
Field expansion:
φ =
∫d3k
(2π)3
1√
2ω
(aω u
Mω + a†ω (uMω )†
)=
∫d3p
(2π)3
1√
2Ω
(bΩ u
RΩ + b†Ω (uRΩ)†
)
Relation between a, a† and b, b†?
⇒ Bogolyubov transformation:
bΩ =
∫dω
∫d~k⊥
(αΩω aω − βΩω a
†ω
)Find coefficients α, β
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 5 / 31
![Page 9: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/9.jpg)
The Unruh effect: Overview
Field expansion:
φ =
∫d3k
(2π)3
1√
2ω
(aω u
Mω + a†ω (uMω )†
)=
∫d3p
(2π)3
1√
2Ω
(bΩ u
RΩ + b†Ω (uRΩ)†
)Relation between a, a† and b, b†?
⇒ Bogolyubov transformation:
bΩ =
∫dω
∫d~k⊥
(αΩω aω − βΩω a
†ω
)Find coefficients α, β
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 5 / 31
![Page 10: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/10.jpg)
The Unruh effect: Overview
Field expansion:
φ =
∫d3k
(2π)3
1√
2ω
(aω u
Mω + a†ω (uMω )†
)=
∫d3p
(2π)3
1√
2Ω
(bΩ u
RΩ + b†Ω (uRΩ)†
)Relation between a, a† and b, b†?
⇒ Bogolyubov transformation:
bΩ =
∫dω
∫d~k⊥
(αΩω aω − βΩω a
†ω
)Find coefficients α, β
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 5 / 31
![Page 11: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/11.jpg)
The Unruh effect: Overview
Interested in number of particles observed by accelerating observer
nΩ =〈0M | N |0M 〉
V
=〈0M | b†Ω bΩ′ |0M 〉
V
=
∫d3kβΩωβ
∗Ω′ω
where
βΩω = −1√
2πaω(e2πΩ/a − 1)
(ω + kx
ω − kx
)−iΩ/2a
then
nΩ =1
e2πΩa − 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
![Page 12: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/12.jpg)
The Unruh effect: Overview
Interested in number of particles observed by accelerating observer
nΩ =〈0M | N |0M 〉
V
=〈0M | b†Ω bΩ′ |0M 〉
V
=
∫d3kβΩωβ
∗Ω′ω
where
βΩω = −1√
2πaω(e2πΩ/a − 1)
(ω + kx
ω − kx
)−iΩ/2athen
nΩ =1
e2πΩa − 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
![Page 13: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/13.jpg)
The Unruh effect: Overview
Interested in number of particles observed by accelerating observer
nΩ =〈0M | N |0M 〉
V
=〈0M | b†Ω bΩ′ |0M 〉
V
=
∫d3kβΩωβ
∗Ω′ω
where
βΩω = −1√
2πaω(e2πΩ/a − 1)
(ω + kx
ω − kx
)−iΩ/2athen
nΩ =1
e2πΩa − 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
![Page 14: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/14.jpg)
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)
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31
![Page 15: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/15.jpg)
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)
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31
![Page 16: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/16.jpg)
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)
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31
![Page 17: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/17.jpg)
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)
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31
![Page 18: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/18.jpg)
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)
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31
![Page 19: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/19.jpg)
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
2πG(p2)e−i(p
0t−~p·~x),
in real space we find
G+(x, x′) = −im
4π2
K1(im√
(t− t′ − iε)2)− (~x− ~x′)2√(t− t′ − iε)2)− (~x− ~x′)2
m→ 0, G+(x, x′) = −1
4π2
1
(t− t′ − iε)2 − (~x− ~x′)2
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 8 / 31
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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
2πG(p2)e−i(p
0t−~p·~x),
in real space we find
G+(x, x′) = −im
4π2
K1(im√
(t− t′ − iε)2)− (~x− ~x′)2√(t− t′ − iε)2)− (~x− ~x′)2
m→ 0, G+(x, x′) = −1
4π2
1
(t− t′ − iε)2 − (~x− ~x′)2
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 8 / 31
![Page 21: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/21.jpg)
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
2πG(p2)e−i(p
0t−~p·~x),
in real space we find
G+(x, x′) = −im
4π2
K1(im√
(t− t′ − iε)2)− (~x− ~x′)2√(t− t′ − iε)2)− (~x− ~x′)2
m→ 0, G+(x, x′) = −1
4π2
1
(t− t′ − iε)2 − (~x− ~x′)2
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 8 / 31
![Page 22: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/22.jpg)
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 − iε)−GR(∆τ − iε)]
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 − iε)−GR(τ − iε)]
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 9 / 31
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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 − iε)−GR(∆τ − iε)]
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 − iε)−GR(τ − iε)]
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 9 / 31
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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 − iε)−GR(∆τ − iε)]
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 − iε)−GR(τ − iε)]
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 9 / 31
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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 − iε)−GR(∆τ − iε)]
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 − iε)−GR(τ − iε)]
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 9 / 31
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The detector approach: massless scalar
Wightman functions
GM = −a2
16π2
1
sinh2 (aτ/2− iaε)
GR = −1
4π2
1
(τ − iε)2
Induced response rate
F (E) =
∫ ∞−∞
dτeiEτ [GM (τ − iε)−GR(τ − iε)]
= −1
4π2(2πi)(iE)
−1∑k=−∞
e−iE2πiak
=1
2πE
1
e2πaE − 1
Define profile function F(E) as
F (E) ≡1
2πF(E)
1
e2πaE − 1
.
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 10 / 31
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The detector approach: massless scalar
Wightman functions
GM = −a2
16π2
1
sinh2 (aτ/2− iaε)
GR = −1
4π2
1
(τ − iε)2
Induced response rate
F (E) =
∫ ∞−∞
dτeiEτ [GM (τ − iε)−GR(τ − iε)]
= −1
4π2(2πi)(iE)
−1∑k=−∞
e−iE2πiak
=1
2πE
1
e2πaE − 1
Define profile function F(E) as
F (E) ≡1
2πF(E)
1
e2πaE − 1
.
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 10 / 31
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The detector approach: massless scalar
Wightman functions
GM = −a2
16π2
1
sinh2 (aτ/2− iaε)
GR = −1
4π2
1
(τ − iε)2
Induced response rate
F (E) =
∫ ∞−∞
dτeiEτ [GM (τ − iε)−GR(τ − iε)]
= −1
4π2(2πi)(iE)
−1∑k=−∞
e−iE2πiak
=1
2πE
1
e2πaE − 1
Define profile function F(E) as
F (E) ≡1
2πF(E)
1
e2πaE − 1
.
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 10 / 31
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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) =πd−1
2
Γ(d−1
2
)(2π)d−2
Ed−3
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 11 / 31
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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
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 12 / 31
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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
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 12 / 31
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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
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 12 / 31
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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
E)
Dimension as seen by diffusion process: spectral dimension
Ds(σ) = −2d lnP (σ)
d lnσ
Accessible experimentally?
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 13 / 31
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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
E)
Dimension as seen by diffusion process: spectral dimension
Ds(σ) = −2d lnP (σ)
d lnσ
Accessible experimentally?
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 13 / 31
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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
E)
Dimension as seen by diffusion process: spectral dimension
Ds(σ) = −2d lnP (σ)
d lnσ
Accessible experimentally?
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 13 / 31
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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
E)
Dimension as seen by diffusion process: spectral dimension
Ds(σ) = −2d lnP (σ)
d lnσ
Accessible experimentally?
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 13 / 31
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Unruh dimension
Profile function massless scalar in general dimension d
F(E) =πd−1
2
Γ(d−1
2
)(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
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 14 / 31
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Unruh dimension
Profile function massless scalar in general dimension d
F(E) =πd−1
2
Γ(d−1
2
)(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
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 14 / 31
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Unruh dimension
Profile function massless scalar in general dimension d
F(E) =πd−1
2
Γ(d−1
2
)(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
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 14 / 31
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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
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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
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QG corrections: Class I
Effective Lagrangian is a polynomial Pn(z) with roots µi
L =1
2φPn(−∂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 − µiwhere Ai = (
∏j 6=i
(µi − µj))−1
Leading to
F(E) =1
c
n∑i=1
Ai√E2 − µiθ(E −
õi)
- 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
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QG corrections: Class I
Effective Lagrangian is a polynomial Pn(z) with roots µi
L =1
2φPn(−∂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 − µiwhere Ai = (
∏j 6=i
(µi − µj))−1
Leading to
F(E) =1
c
n∑i=1
Ai√E2 − µiθ(E −
õi)
- 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
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QG corrections: Class I
Effective Lagrangian is a polynomial Pn(z) with roots µi
L =1
2φPn(−∂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 − µiwhere Ai = (
∏j 6=i
(µi − µj))−1
Leading to
F(E) =1
c
n∑i=1
Ai√E2 − µiθ(E −
õi)
- 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
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QG corrections: Class I
Effective Lagrangian is a polynomial Pn(z) with roots µi
L =1
2φPn(−∂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 − µiwhere Ai = (
∏j 6=i
(µi − µj))−1
Leading to
F(E) =1
c
n∑i=1
Ai√E2 − µiθ(E −
õi)
- 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
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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)
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 17 / 31
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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)
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 17 / 31
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Multi-scale models
Consider a two scale model:
P2(p2) =1
m2p2(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
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Multi-scale models
Consider a two scale model:
P2(p2) =1
m2p2(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
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Multi-scale models
Consider a two scale model:
P2(p2) =1
m2p2(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
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Multi-scale models
Consider a two scale model:
P2(p2) =1
m2p2(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
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Multi-scale models
Profile function 2-scale model (m = 1)
F(E) = E −√E2 −m2 θ(E −m)
DU
Ds
0.01 1 100E0
1
2
3
4
D
1 2 3 4 5E
0.2
0.4
0.6
0.8
1.0
1.2FHEL
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 19 / 31
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Multi-scale models
Profile function 2-scale model (m = 1)
F(E) = E −√E2 −m2 θ(E −m)
DU
Ds
0.01 1 100E0
1
2
3
4
D
1 2 3 4 5E
0.2
0.4
0.6
0.8
1.0
1.2FHEL
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 19 / 31
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Multi-scale models
Consider a three scale model:
P3(p2) =1
m22m
23
p2(p2 −m22)(p2 −m2
3)
⇒ G(p2) =1
p2−
m23
m23 −m2
2
1
p2 −m22
+m2
2
m23 −m2
2
1
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 =
4
3
Leading to the profile function
F(E) = E −m2
3
m23 −m2
2
√E2 −m2
2 θ(E −m2) +m2
2
m23 −m2
2
√E2 −m2
3 θ(E −m3)
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 20 / 31
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Multi-scale models
Consider a three scale model:
P3(p2) =1
m22m
23
p2(p2 −m22)(p2 −m2
3)
⇒ G(p2) =1
p2−
m23
m23 −m2
2
1
p2 −m22
+m2
2
m23 −m2
2
1
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 =
4
3
Leading to the profile function
F(E) = E −m2
3
m23 −m2
2
√E2 −m2
2 θ(E −m2) +m2
2
m23 −m2
2
√E2 −m2
3 θ(E −m3)
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 20 / 31
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Multi-scale models
Consider a three scale model:
P3(p2) =1
m22m
23
p2(p2 −m22)(p2 −m2
3)
⇒ G(p2) =1
p2−
m23
m23 −m2
2
1
p2 −m22
+m2
2
m23 −m2
2
1
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 =
4
3
Leading to the profile function
F(E) = E −m2
3
m23 −m2
2
√E2 −m2
2 θ(E −m2) +m2
2
m23 −m2
2
√E2 −m2
3 θ(E −m3)
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 20 / 31
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Multi-scale models
Profile function 3-scale model (m2 = 0.1, m3 = 10)
F(E) = E −m2
3
m23 −m2
2
√E2 −m2
2 θ(E −m2) +m2
2
m23 −m2
2
√E2 −m2
3 θ(E −m3)
0.01 0.1 1 10 100 1000E
0
1
2
3
4
5D
5 10 15 20E
-0.001
0.001
0.002
0.003
0.004
0.005FHEL
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 21 / 31
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Multi-scale models
Profile function 3-scale model (m2 = 0.1, m3 = 10)
F(E) = E −m2
3
m23 −m2
2
√E2 −m2
2 θ(E −m2) +m2
2
m23 −m2
2
√E2 −m2
3 θ(E −m3)
0.01 0.1 1 10 100 1000E
0
1
2
3
4
5D
5 10 15 20E
-0.001
0.001
0.002
0.003
0.004
0.005FHEL
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 21 / 31
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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
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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
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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
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Kaluza-Klein models
Massless scalar field φ(x, x5) =∞∑
n=−∞φn(x)ei
nRx5 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
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Kaluza-Klein models
Massless scalar field φ(x, x5) =∞∑
n=−∞φn(x)ei
nRx5 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
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Kaluza-Klein models
Profile function
F(E) =1
2πR
E + 2∞∑n=1
√E2 −
n2
R2θ(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
350FHEL
10 20 30 40 50
E
2 Π
5000
10 000
15 000
20 000
25 000FHEL
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 24 / 31
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Kaluza-Klein models
Profile function
F(E) =1
2πR
E + 2∞∑n=1
√E2 −
n2
R2θ(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
350FHEL
10 20 30 40 50
E
2 Π
5000
10 000
15 000
20 000
25 000FHEL
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 24 / 31
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Kaluza-Klein models
Profile function
F(E) =1
2πR
E + 2∞∑n=1
√E2 −
n2
R2θ(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
350FHEL
10 20 30 40 50
E
2 Π
5000
10 000
15 000
20 000
25 000FHEL
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 24 / 31
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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
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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
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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
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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
3p4), 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 −
õ2))
5 10 15 20E
20
40
60
80
100FHEL
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 26 / 31
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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
3p4), 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 −
õ2))
5 10 15 20E
20
40
60
80
100FHEL
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 26 / 31
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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
3p4), 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 −
õ2))
5 10 15 20E
20
40
60
80
100FHEL
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 26 / 31
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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
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Causal set theory
Wightman function causal sets
G+(x2) = −i
2π3
∫ ∞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=0
bn
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→∞
1
g(ξ2)= −
2√
6π
ξ4+ ...
limξ2→0
1
g(ξ2)= −
1
ξ2+ ...
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 28 / 31
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Causal set theory
Wightman function causal sets
G+(x2) = −i
2π3
∫ ∞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=0
bn
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→∞
1
g(ξ2)= −
2√
6π
ξ4+ ...
limξ2→0
1
g(ξ2)= −
1
ξ2+ ...
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 28 / 31
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Causal set theory
Wightman function causal sets
G+(x2) = −i
2π3
∫ ∞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=0
bn
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→∞
1
g(ξ2)= −
2√
6π
ξ4+ ...
limξ2→0
1
g(ξ2)= −
1
ξ2+ ...
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 28 / 31
![Page 77: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/77.jpg)
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
bn
∫ E2
0dm2eαm
2m2n
√E2 −m2
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 29 / 31
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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
bn
∫ E2
0dm2eαm
2m2n
√E2 −m2
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 29 / 31
![Page 79: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/79.jpg)
Causal set theory
Profile function
F(E) = E +N∑n=0
bn
∫ E2
0dm2eαm
2m2n
√E2 −m2
For N = 1, b1 = −(b0 + 1), b0 = 1, α = 1
5 10 15 20E
0.2
0.4
0.6
0.8
1.0
1.2
1.4FHEL
0.1 1 10 100E
1
2
3
4
5DU
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 30 / 31
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Causal set theory
Profile function
F(E) = E +N∑n=0
bn
∫ E2
0dm2eαm
2m2n
√E2 −m2
For N = 1, b1 = −(b0 + 1), b0 = 1, α = 1
5 10 15 20E
0.2
0.4
0.6
0.8
1.0
1.2
1.4FHEL
0.1 1 10 100E
1
2
3
4
5DU
Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 30 / 31
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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
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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
![Page 83: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/83.jpg)
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
![Page 84: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/84.jpg)
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
![Page 85: QG signatures in the Unruh effectcqc/Slides/... · Quantum gravity signatures in the Unruh effect N. Alkofer, G. D’Odorico, F. Saueressig and FV PRD 94, 104055 (2016) Fleur Versteegen](https://reader030.vdocument.in/reader030/viewer/2022040205/5edb9471ad6a402d6665dec0/html5/thumbnails/85.jpg)
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