Applications of Bohmian mechanics in quantum gravity

Transcription

1 Applications of Bohmian mechanics in quantum gravity Ward Struyve LMU, Munich, Germany

2 Outline: Why do we need a quantum theory for gravity? What is the quantum theory for gravity? Problems with quantum gravity Bohmian quantum gravity Applications of Bohmian quantum gravity: Semi-classical gravity Space-time singularities

3 Why do we need a quantum theory for gravity? Classically gravity - Einstein s theory of general relativity - Gravity from space-time geometry, described by metric g µν - Dynamics: G µν (g) = T µν But matter should be described quantum mechanically, so T µν T µν Hence G µν Ĝµν and g µν ĝ µν i.e., gravity must be quantized.

4 What is the quantum theory for gravity? Different proposals: Wheeler-DeWitt theory, loop quantum gravity, string theory,... (22 mentioned on Wikipedia)

5 Wheeler-DeWitt theory Canonical quantization of Einstein s theory for gravity: In functional Schrödinger picture: g (3) (x) ĝ (3) (x) Ψ = Ψ(g (3) ) Satisfies the Wheeler-De Witt equation: i Ψ t (g(3) ) = ĤΨ(g(3) ) = 0 Loop quantum gravity Canonical quantization of Einstein s theory for gravity in a different representation. States are functionals of loops: Satisfies the Wheeler-De Witt equation: Ψ = Ψ(γ) i Ψ (γ) = ĤΨ(γ) = 0 t

6 Problems with Wheeler-DeWitt theory and loop quantum gravity Technical problems: regularizing the equations, finding solutions,... Conceptual problems: Measurement problem. Is especially severe in cosmological context. The theory is supposed to describe the whole universe, there are no outside observers or measurement devices. Problem of time. The wave function is static. How can we tell whether the universe is expanding or contracting?

7 Bohmian mechanics solves the conceptual problems E.g. Wheeler-DeWitt theory, see: Pinto-Neto, Goldstein & Teufel,... In a Bohmian approach there is an actual 3-metric g (3) which satisfies: ġ (3) = v Ψ (g (3) ) Solves the measurement problem. Also solves the problem of time. We can tell whether the universe is expanding or not: It depends on the actual metric.

8 Bohmian mechanics solves the conceptual problems E.g. Wheeler-DeWitt theory, see: Pinto-Neto, Goldstein & Teufel,... In the Bohmian formulation there is an actual 3-metric g (3) which satisfies: ġ (3) = v Ψ (g (3) ) Solves the measurement problem. Also solves the problem of time. We can tell whether the universe is expanding or not: It depends on the actual metric. We can also derive a time-dependent Schrödinger equation for subsystems using the conditional wave function. E.g. suppose gravity and scalar field. Conditional wave functional for scalar field Ψ s (φ, t) = Ψ(φ, g (3) (t)) is time-dependent if g (3) (t) is time-dependent.

9 Applications of Bohmian quantum gravity/quantum cosmology Cosmological fluctuations Origin of seeds of structure (with N. Pinto-Neto & G. Santos, Phys. Rev. D 89, (2014)) Boltzmann brain problem (with S. Goldstein & R. Tumulka, arxiv: [gr-qc])

10 Semi-classical gravity (arxiv: ) Space-time singularities (with F.T. Falciano & N. Pinto-Neto, Phys. Rev. D 91, (2015))

11 Semi-classical gravity Matter is treated quantum mechanically, as quantum field on curved space-time. E.g. scalar field: i t Ψ(φ, t) = Ĥ(φ, g)ψ(φ, t) Grativity is treated classically, described by G µν (g) = 8πG c Ψ T 4 µν (φ, g) Ψ G µν = R µν 1 2 Rg µν

12 Is a good approximation when matter and gravity both behave approximately classically. Breaks down when the matter is in a macroscopic superposition. E.g. ψ = 1 2 (ψ 1 + ψ 2 ) Energy densities corresponding to ψ 1 and ψ 2 : Energy density corresponding to ψ: Page & Geilker experiment (1981)

13 Is there a better semi-classical approximation based on Bohmian mechanics? In Bohmian mechanics matter is described by Ψ(φ) and actual scalar field φ B (x, t). Proposal for semi-classical theory: G µν (g) = 8πG c 4 T µν(φ B, g) We have: ρ(φ B ) or ρ(φ B ) x x

14 Is there a better semi-classical approximation based on Bohmian mechanics? In Bohmian mechanics matter is described by Ψ(φ) and actual scalar field φ B (x, t). Proposal for semi-classical theory: G µν (g) = 8πG c 4 T µν(φ B, g) In general doesn t work because µ T µν (φ B, g) 0! (In non-relativistic Bohmian mechanics energy is not conserved.)

15 Is there a better semi-classical approximation based on Bohmian mechanics? In Bohmian mechanics matter is described by Ψ(φ) and actual scalar field φ B (x, t). Proposal for semi-classical theory: G µν (g) = 8πG c 4 T µν(φ B, g) In general doesn t work because µ T µν (φ B, g) 0! (In non-relativistic Bohmian mechanics energy is not conserved.)

16 Semi- Similar situation in scalar electrodynamics: Quantum matter field described by Ψ(φ) and actual scalar field φ B (x, t). classical theory: µ F µν = j ν (φ B ) In general doesn t work because ν j ν (φ B ) 0!

17 Semi- Similar situation in scalar electrodynamics: Quantum matter field described by Ψ(φ) and actual scalar field φ B (x, t). classical theory: µ F µν = j ν (φ B ) In general doesn t work because ν j ν (φ B ) 0! Quantum correction needs to be included: µ F µν = j ν +j ν Q, j ν Q depends on quantum potential Is consistent since: µ (j µ + j µ Q ) = 0

18 Semi-classical approximation to non-relativistic quantum mechanics System 1: quantum mechanical. System 2: classical Usual approach (mean field): ( ) i t ψ(x 1, t) = V (x 1, X 2 (t)) ψ(x 1, t) 2m 1 m 2 Ẍ 2 (t) = ψ F 2 (x 1, X 2 (t)) ψ = dx 1 ψ(x 1, t) 2 F 2 (x 1, X 2 (t)), F 2 = 2 V backreaction through mean force

19 Semi-classical approximation to non-relativistic quantum mechanics System 1: quantum mechanical. System 2: classical Usual approach (mean field): ( ) i t ψ(x 1, t) = V (x 1, X 2 (t)) ψ(x 1, t) 2m 1 m 2 Ẍ 2 (t) = ψ F 2 (x 1, X 2 (t)) ψ = dx 1 ψ(x 1, t) 2 F 2 (x 1, X 2 (t)), F 2 = 2 V backreaction through mean force Bohmian approach: i t ψ(x 1, t) = ( ) V (x 1, X 2 (t)) ψ(x 1, t) 2m 1 Ẋ 1 (t) = v ψ 1 (X 1(t), t), m 2 Ẍ 2 (t) = F 2 (X 1 (t), X 2 (t)) backreaction through Bohmian particle

20 Prezhdo and Brookby (2001): Bohmian approach yields better results than usual approach:

21 Results so far: Consistent Bohmian semi-classical approximation for: Non-relativistic systems Quantum electrodynamics Simplified models of quantum gravity (mini-superspace models) Investigation of applications in: Quantum optics (Jaynes-Cummings model), with D.-A. Deckert, L. Kellers, T. Norsen To do: Find semi-classical approximation for full quantum gravity Develop higher order corrections Cosmological inflation theory Find more applications (black-hole physics,...)

22 Space-time singularities Einstein s general relativity: space-time singularities (such as Big Bang) are unavoidable (under some mild assumptions) Signals breakdown of the theory? Does quantum gravity eliminate the singularities? Depends on approach to quantum gravity. E.g. Wheeler-DeWitt quantization, loop quantum gravity,... Recent results for mini-superspace (Ashtekar, Corichi, Singh,... ): Wheeler-DeWitt quantization: singularities for a certain class of states Loop quantum gravity: no singularities for a certain class of states

23 What does it mean to have a space-time singularity in quantum gravity? Ψ has support on singular metrics? Ψ is peaked around singular metrics? Ψ ĝ Ψ is singular?

24 What does it mean to have a space-time singularity in quantum gravity? Ψ has support on singular metrics? Ψ is peaked around singular metrics? Ψ ĝ Ψ is singular? In the Bohmian approach: singularities if the actual metric is singular! Bohmian mechanics for mini-superspace: Wheeler-DeWitt quantization: possibly singularities Loop quantum gravity: no singularities

25 Classical mini-superspace Friedman-Lemaître-Robertson-Walker space-time. Restriction to homogeneous and isotropic metrics and fields: Gravity: ds 2 = dt 2 a(t) 2 dx 2 Matter: φ = φ(t) Singularity if scale factor a = 0 Equations of motion (4πG/3 = 1): φ = ± c e 3α, α = c e 3α with a = e α c constant

26 If c = 0, α constant, i.e. Minkowski space-time; no singularities. Otherwise singularity (a = 0 or α ) a In (α, φ)-space φ t α

27 Wheeler-DeWitt theory Wheeler-DeWitt equation ( φ 2 α)ψ 2 = 0, a = e α Solutions ψ = ψ R + ψ L ; ψ R = ψ R (α φ), ψ L = ψ L (α + φ): φ α ψ R ψ L Bohmian equations: φ = 1 e 3α φs, α = 1 e 3α αs, ψ = ψ e is

28 Examples: ψ real, i.e., S = 0: α is constant, i.e. Minkowski space-time; no singularities. ψ = ψ R (α φ) = e (α φ)2 i(α φ) classical trajectories always singular (big bang) φ α ψ = ψ L (α φ) = e (α+φ)2 +i(α+φ) classical trajectories always singular (big crunch) φ α

29 Superposition ψ = e (α φ)2 i(α φ) + e (α+φ)2 +i(α+φ) Is symmetric: ψ(φ, α) = ψ(φ, α) φ α Big bang and big crunch for trajectories on the left; bounce for trajectories on the right

30 Loop quantum cosmology Application of loop quantum gravity ideas to mini-superspace Different from Wheeler-DeWitt quantization Scale factor a takes discrete values. States ψ(ν, φ), where ν a 3 and ν = 4λn, n N Wave equation becomes difference equation: ( φ 2 + Θ)ψ(ν, φ) = 0 with Θψ(ν, φ) ν(ν + 4λ) ν + 2λ ψ(ν + 4λ, φ) 2ν 2 ψ(ν, φ) + ν(ν 4λ) ν 2λ ψ(ν 4λ, φ)

31 Bohmian approach. There is an actual value for the scale factor a and scalar field φ. a takes discrete values, with a 3 ν = 4λn, n N Bohmian dynamics is stochastic (a la Bell for QFT). T ν,ν transition probability to jump from ν to ν per unit time, with Equation for φ: ) + ( T ν,ν+4λ ν2/3 (ν + 2λ) ψ Im ν (ν + 4λ) 2/3 ψν+4λ ( (ν+4λ) 2/3 (ν+2λ) ψ ) + Im ν+4λ T ν+4λ,ν ν 2/3 ψ if ν 0 ν 0 if ν = 0 φ = 1 a 3 φs

32 T 0,4λ = 0: Probability for a to make the jump 4λ 0 is zero: a non-singular metric never becomes singular T 4λ,0 = 0: Probability for a to make the jump 0 4λ is zero a singular metric never becomes non-singular No singularities

33 Conclusions: With Bohmian mechanics we can unambiguously formulate the question of singularities in quantum gravity We found: Wheeler-DeWitt quantization: Possibly singularities. It depends on the wave function. For a generic wave function there both solutions with and without singularity Loop quantum gravity: No singularities Question: Results are only for mini-superspace. What happens in the more general quantum space-times?