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

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

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

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 = e aξ sinh ( aτ ) , x = e aξ cosh ( aτ ) , y = y, z = z. a a Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 3 / 31

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 = e aξ sinh ( aτ ) , x = e aξ cosh ( aτ ) , y = y, z = z. a a Klein-Gordon for m = 0 : − ∂ 2 t + ∂ 2 x + ∂ 2 y + ∂ 2 � � φ ( t, x, y, z ) = 0 z � � e − 2 aξ ( − ∂ 2 τ + ∂ 2 ξ ) + ∂ 2 y + ∂ 2 φ ( τ, ξ, y, z ) = 0 z Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 3 / 31

The Unruh effect: Overview Solutions: � π Ω � 1 / 2 e − i Ω τ u R = 2 π 2 √ a sinh × a � | � p ⊥ | � e aξ e i� p ⊥ · � x ⊥ K i Ω /a a 1 e − i ( ωt − k x x − � u M k ⊥ · � x ⊥ ) = � 2(2 π ) 3 ω Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 4 / 31

The Unruh effect: Overview Solutions: � π Ω � 1 / 2 e − i Ω τ u R = 2 π 2 √ a sinh × a � | � p ⊥ | � e aξ e i� p ⊥ · � x ⊥ K i Ω /a a 1 e − i ( ωt − k x x − � u M k ⊥ · � x ⊥ ) = � 2(2 π ) 3 ω define annihilation/creation operators a ω , ˆ ˆ b Ω such that ˆ ˆ a ω | 0 M � = 0 , b Ω | 0 R � = 0 Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 4 / 31

The Unruh effect: Overview Field expansion: d 3 k � 1 � ω ) † � ˆ a ω u M a † ω ( u M φ = √ ˆ ω + ˆ (2 π ) 3 2 ω d 3 p � 1 � Ω ) † � ˆ Ω + ˆ b † b Ω u R Ω ( u R = √ (2 π ) 3 2Ω Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 5 / 31

The Unruh effect: Overview Field expansion: d 3 k � 1 � ω ) † � ˆ a ω u M a † ω ( u M φ = √ ˆ ω + ˆ (2 π ) 3 2 ω d 3 p � 1 � Ω ) † � ˆ Ω + ˆ b † b Ω u R Ω ( u R = √ (2 π ) 3 2Ω a † and ˆ b, ˆ b † ? Relation between ˆ a, ˆ Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 5 / 31

The Unruh effect: Overview Field expansion: d 3 k � 1 � ω ) † � ˆ a ω u M a † ω ( u M φ = √ ˆ ω + ˆ (2 π ) 3 2 ω d 3 p � 1 � Ω ) † � ˆ Ω + ˆ b † b Ω u R Ω ( u R = √ (2 π ) 3 2Ω a † and ˆ b, ˆ b † ? Relation between ˆ a, ˆ ⇒ Bogolyubov transformation: � � � � ˆ d� a † b Ω = dω k ⊥ α Ω ω ˆ a ω − β Ω ω ˆ ω Find coefficients α, β Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 5 / 31

The Unruh effect: Overview Interested in number of particles observed by accelerating observer � 0 M | ˆ N | 0 M � n Ω = V � 0 M | ˆ b † Ω ˆ b Ω ′ | 0 M � = V � d 3 kβ Ω ω β ∗ = Ω ′ ω where � ω + k x � − i Ω / 2 a 1 β Ω ω = − � ω − k x 2 πaω ( e 2 π Ω /a − 1) Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 6 / 31

The Unruh effect: Overview Interested in number of particles observed by accelerating observer � 0 M | ˆ N | 0 M � n Ω = V � 0 M | ˆ b † Ω ˆ b Ω ′ | 0 M � = V � d 3 kβ Ω ω β ∗ = Ω ′ ω where � ω + k x � − i Ω / 2 a 1 β Ω ω = − � ω − k x 2 πaω ( e 2 π Ω /a − 1) then 1 δ (Ω − Ω ′ ) δ ( � k ⊥ − � k ′ n Ω = ⊥ ) 2 π Ω e − 1 a Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 6 / 31

The Unruh effect: Overview Interested in number of particles observed by accelerating observer � 0 M | ˆ N | 0 M � n Ω = V � 0 M | ˆ b † Ω ˆ b Ω ′ | 0 M � = V � d 3 kβ Ω ω β ∗ = Ω ′ ω where � ω + k x � − i Ω / 2 a 1 β Ω ω = − � ω − k x 2 πaω ( e 2 π Ω /a − 1) then 1 δ (Ω − Ω ′ ) δ ( � k ⊥ − � k ′ n Ω = ⊥ ) 2 π Ω e − 1 a Thermal bath of particles with temperature T = a/ 2 π ⇒ Geometric effect Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 6 / 31

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 E 2 > E 1 Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31

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 E 2 > E 1 Interaction scalar field and detector: L I = gm ( τ ) φ ( x ) Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31

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 E 2 > E 1 Interaction scalar field and detector: L I = gm ( τ ) φ ( x ) Spontaneous emission inertial observer (intrinsic to detector) | E 2 � | 0 M � → | E 1 � | � k � Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31

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 E 2 > E 1 Interaction scalar field and detector: L I = gm ( τ ) φ ( x ) Spontaneous emission inertial observer (intrinsic to detector) | E 2 � | 0 M � → | E 1 � | � k � Amplitude � dτe i ( E 1 − E 2 ) τ � � A ( � k ) = ig � E 1 | m (0) | E 2 � k | φ ( x ( τ )) | 0 M � Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31

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 E 2 > E 1 Interaction scalar field and detector: L I = gm ( τ ) φ ( x ) Spontaneous emission inertial observer (intrinsic to detector) | E 2 � | 0 M � → | E 1 � | � k � Amplitude � dτe i ( E 1 − E 2 ) τ � � A ( � k ) = ig � E 1 | m (0) | E 2 � k | φ ( x ( τ )) | 0 M � Transition probability ( ∆ E ≡ E 2 − E 1 , ∆ τ ≡ τ 1 − τ 2 ) � d 3 k |A| 2 = g 2 | � E 1 | m | E 2 � | 2 F (∆ E ) P i → f = � dτ 1 dτ 2 e i ∆ E ∆ τ G (∆ τ − iǫ ) where F (∆ E ) = Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 7 / 31

The Unruh effect: The detector approach For massive scalar field in Minkowski space: 1 1 G ( p 2 ) = ˜ p 2 − m 2 = ( p 0 + � p 2 + m 2 )( p 0 − � p 2 + m 2 ) � � Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 8 / 31

The Unruh effect: The detector approach For massive scalar field in Minkowski space: 1 1 G ( p 2 ) = ˜ p 2 − m 2 = ( p 0 + � p 2 + m 2 )( p 0 − � p 2 + m 2 ) � � � p 2 + m 2 Positive frequency Wightman function encircles pole at � d 3 � dp 0 � p � G ( p 2 ) e − i ( p 0 t − � ˜ p · � x ) , G + ( � x, t ) = − i (2 π ) 3 2 π γ + Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 8 / 31

The Unruh effect: The detector approach For massive scalar field in Minkowski space: 1 1 G ( p 2 ) = ˜ p 2 − m 2 = ( p 0 + � p 2 + m 2 )( p 0 − � p 2 + m 2 ) � � � p 2 + m 2 Positive frequency Wightman function encircles pole at � d 3 � dp 0 � p � G ( p 2 ) e − i ( p 0 t − � ˜ p · � x ) , G + ( � x, t ) = − i (2 π ) 3 2 π γ + in real space we find ( t − t ′ − iǫ ) 2 ) − ( � � x ′ ) 2 − im K 1 ( im x − � G + ( x, x ′ ) = ( t − t ′ − iǫ ) 2 ) − ( � 4 π 2 � x ′ ) 2 x − � − 1 1 G + ( x, x ′ ) m → 0 , = ( t − t ′ − iǫ ) 2 − ( � 4 π 2 x − � x ′ ) 2 Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 8 / 31

The Unruh effect: The detector approach Total emission (evaluated on accelerated trajectory): | E 2 � | 0 M � → | E 1 � | � k � , P i → f Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 9 / 31

The Unruh effect: The detector approach Total emission (evaluated on accelerated trajectory): | E 2 � | 0 M � → | E 1 � | � k � , P i → f Induced transition probability P i → f ( induced ) = P i → f − P i → f ( spontaneous ) Fleur Versteegen QG signatures in the Unruh effect April 25, 2017 9 / 31