A quantum picture of de Sitter spacetime Sebastian Zell Work with - PowerPoint PPT Presentation

A quantum picture of de Sitter spacetime Sebastian Zell Work with Gia Dvali and C´ esar Gomez MPP Project Review 2015 14 th December 2015 1

Corpuscular approach • Idea: The world is fundamentally quantum ⇒ Classical solution = collective effect of appropriate quanta (corpuscules) 1 1 G. Dvali and C. Gomez, Quantum Compositeness of Gravity: Black Holes, AdS and Inflation , arXiv:1312.4795 . 2

Corpuscular approach • Idea: The world is fundamentally quantum ⇒ Classical solution = collective effect of appropriate quanta (corpuscules) 1 • Tehseen’s talk: Solitons as corpuscular bound states 2 ⇒ Topological properties determined by number of corpuscules 1 G. Dvali and C. Gomez, Quantum Compositeness of Gravity: Black Holes, AdS and Inflation , arXiv:1312.4795 . 2 G. Dvali, C. Gomez, L. Gr¨ unding and T. Rug, Towards a Quantum Theory of Solitons , arXiv:1508.03074 . 2

Outline The quantum state of de Sitter 1 Application to Particle production 2 Outlook 3 3

The quantum state of de Sitter Application to Particle production Outlook De Sitter metric • Cosmological constant Λ ( ∝ H 2 ) 4

The quantum state of de Sitter Application to Particle production Outlook De Sitter metric • Cosmological constant Λ ( ∝ H 2 ) • Metric for small times: d s 2 = (1 + Λ t 2 )( d t 2 − d # x 2 ) + . . . » 4

The quantum state of de Sitter Application to Particle production Outlook De Sitter metric • Cosmological constant Λ ( ∝ H 2 ) • Metric for small times: d s 2 = (1 + Λ t 2 )( d t 2 − d # x 2 ) + . . . » • Canonically normalized Newtonian potential Φ = M p 2 Λ t 2 4

The quantum state of de Sitter Application to Particle production Outlook De Sitter metric • Cosmological constant Λ ( ∝ H 2 ) • Metric for small times: d s 2 = (1 + Λ t 2 )( d t 2 − d # x 2 ) + . . . » • Canonically normalized Newtonian potential Φ = M p 2 Λ t 2 • Goal: Obtain Φ as classical limit of a graviton bound state Λ 4

The quantum state of de Sitter Application to Particle production Outlook Bound-state gravitons • Two different Fock spaces: a † - ˆ k creates free gravitons. # » - ˆ b † k creates bound-state gravitons. # » 5

The quantum state of de Sitter Application to Particle production Outlook Bound-state gravitons • Two different Fock spaces: a † - ˆ k creates free gravitons. # » - ˆ b † k creates bound-state gravitons. # » Claim √ Bound-state graviton ( m = 0) = Free graviton ( m = Λ) 5

The quantum state of de Sitter Application to Particle production Outlook Bound-state gravitons • Two different Fock spaces: a † - ˆ k creates free gravitons. # » - ˆ b † k creates bound-state gravitons. # » Claim √ Bound-state graviton ( m = 0) = Free graviton ( m = Λ) • Conditions on the quantum state | N Λ � : - Spatially homogeneous ⇒ 0 momentum - Maximally classical ⇒ Coherent state 5

The quantum state of de Sitter Application to Particle production Outlook Bound-state gravitons • Two different Fock spaces: a † - ˆ k creates free gravitons. # » - ˆ b † k creates bound-state gravitons. # » Claim √ Bound-state graviton ( m = 0) = Free graviton ( m = Λ) • Conditions on the quantum state | N Λ � : - Spatially homogeneous ⇒ 0 momentum - Maximally classical ⇒ Coherent state 0 b † • Only free parameter left: N ∝ � N Λ | b # 0 | N Λ � » # » 5

The quantum state of de Sitter Application to Particle production Outlook Classical limit • Expectation value in Hubble patch: � N Λ | ˆ Φ | N Λ � 6

The quantum state of de Sitter Application to Particle production Outlook Classical limit • Expectation value in Hubble patch: � k t e i # » x + h.c. � � � N Λ | ˆ ˆ k # » k e − i ω # Φ | N Λ � = � N Λ | b # » | N Λ � » # » k 6

The quantum state of de Sitter Application to Particle production Outlook Classical limit • Expectation value in Hubble patch: � k t e i # » x + h.c. � � � N Λ | ˆ ˆ k # » k e − i ω # Φ | N Λ � = � N Λ | b # » | N Λ � » # » k √ � √ √ Λ t + h.c. � Ne − i = Λ 6

The quantum state of de Sitter Application to Particle production Outlook Classical limit • Expectation value in Hubble patch: � k t e i # » x + h.c. � � � N Λ | ˆ ˆ k # » k e − i ω # Φ | N Λ � = � N Λ | b # » | N Λ � » # » k √ � √ √ Λ t + h.c. � Ne − i = Λ √ � 1 + 1 � 2Λ t 2 + O (Λ 2 t 4 ) = Λ N 6

The quantum state of de Sitter Application to Particle production Outlook Classical limit • Expectation value in Hubble patch: � k t e i # » x + h.c. � � � N Λ | ˆ ˆ k # » k e − i ω # Φ | N Λ � = � N Λ | b # » | N Λ � » # » k √ � √ √ Λ t + h.c. � Ne − i = Λ √ � 1 + 1 � 2Λ t 2 + O (Λ 2 t 4 ) = Λ N ⇒ Choose N = M 2 p Λ 6

The quantum state of de Sitter Application to Particle production Outlook Classical limit • Expectation value in Hubble patch: � k t e i # » x + h.c. � � � N Λ | ˆ ˆ k # » k e − i ω # Φ | N Λ � = � N Λ | b # » | N Λ � » # » k √ � √ √ Λ t + h.c. � Ne − i = Λ √ � 1 + 1 � 2Λ t 2 + O (Λ 2 t 4 ) = Λ N ⇒ Choose N = M 2 p Λ ⇒ Quantum state | N Λ � reproduces classical metric Φ: � N Λ | ˆ Φ | N Λ � = Φ 6

The quantum state of de Sitter Application to Particle production Outlook Classical limit • Expectation value in Hubble patch: � k t e i # » x + h.c. � � � N Λ | ˆ ˆ k # » k e − i ω # Φ | N Λ � = � N Λ | b # » | N Λ � » # » k √ � √ √ Λ t + h.c. � Ne − i = Λ √ � 1 + 1 � 2Λ t 2 + O (Λ 2 t 4 ) = Λ N ⇒ Choose N = M 2 p Λ ⇒ Quantum state | N Λ � reproduces classical metric Φ: � N Λ | ˆ Φ | N Λ � = Φ • Representation of Φ independent of source 6

The quantum state of de Sitter Application to Particle production Outlook Decay constant ( E 1 , # p 1 ) » ( E 2 , # » p 2 ) N { } N ′ = N − 1 7

The quantum state of de Sitter Application to Particle production Outlook Decay constant ( E 1 , # p 1 ) » ( E 2 , # » p 2 ) N { } N ′ = N − 1 √ � 1 − 5 � Γ ∝ Λ 4 N 7

The quantum state of de Sitter Application to Particle production Outlook Decay constant ( E 1 , # p 1 ) » ( E 2 , # » p 2 ) N { } N ′ = N − 1 √ � 1 − 5 � Γ ∝ Λ 4 N • Reinterpretation (already in the semi-classical limit N → ∞ ): Energy transfer = graviton energy √ E 1 + E 2 = Λ 7

The quantum state of de Sitter Application to Particle production Outlook Decay constant ( E 1 , # p 1 ) » ( E 2 , # » p 2 ) N { } N ′ = N − 1 √ � 1 − 5 � Γ ∝ Λ 4 N • Reinterpretation (already in the semi-classical limit N → ∞ ): Energy transfer = graviton energy √ E 1 + E 2 = Λ • Quantum correction because of back-reaction ( N ′ � = N ) 7

The quantum state of de Sitter Application to Particle production Outlook Final state of the metric • Metric changes because of back-reaction (Inaccessible in semi-classical limit N → ∞ ) 8

The quantum state of de Sitter Application to Particle production Outlook Final state of the metric • Metric changes because of back-reaction (Inaccessible in semi-classical limit N → ∞ ) • Initial de Sitter metric only valid as long as N − N ′ ≪ N ⇒ Quantum break time 3 : ∆ t ≈ N Γ − 1 = M 2 p Λ 1 . 5 3 G. Dvali and C. Gomez, Quantum Exclusion of Positive Cosmological Constant? , arXiv:1412.8077 . 8

The quantum state of de Sitter Application to Particle production Outlook Final state of the metric • Metric changes because of back-reaction (Inaccessible in semi-classical limit N → ∞ ) • Initial de Sitter metric only valid as long as N − N ′ ≪ N ⇒ Quantum break time 3 : ∆ t ≈ N Γ − 1 = M 2 p Λ 1 . 5 ⇒ Final state without classical metric description? 3 G. Dvali and C. Gomez, Quantum Exclusion of Positive Cosmological Constant? , arXiv:1412.8077 . 8

The quantum state of de Sitter Application to Particle production Outlook Outlook Summary • De Sitter metric as classical limit of graviton state • Particle production because of graviton decay • 1 / N -correction of the rate caused by back-reaction • Quantum evolution of the metric 9

The quantum state of de Sitter Application to Particle production Outlook Outlook Summary • De Sitter metric as classical limit of graviton state • Particle production because of graviton decay • 1 / N -correction of the rate caused by back-reaction • Quantum evolution of the metric Future research • Minkowski as graviton state • Model final de Sitter state • Inflationary scenarios • Other metrics such as AdS 9