Sub-Vacuum Phenomena

Transcription

1 Sub-Vacuum Phenomena Lecture II APCTP-NCTS International School/Workshop on Larry Ford Tufts University Gravitation and Cosmology January 16, 2009

2 Zero point effects for a system of quantum harmonic oscillators Zero point energy per mode: 1 2 ω Zero point fluctuations: q p = 1 2

3 An example with finite degrees of freedom - quantum density fluctuations in a fluid ˆρ(x,t) = quantum fluctuations around the mean density ρ 0 Correlation function: ˆρ(x,t)ˆρ(x,t ) = ρ 0 2π 2 c S c S = sound speed x 2 +3c 2 S t2 ( x 2 c 2 S t2 ) 3 Same functional form as in relativistic field theory, but with a natural cutoff at the interatomic spacing.

4 Scattering of light by the quantum fluctuations ˆε S =scattered polarization ˆε I =incident polarization ω =incident frequency scattered frequency V =scattering volume dσ dω = ω 5 V 32π 2 c 5 c S ρ 0 LF & N. Svaiter ω 5 = ω 4 ω Rayleigh scattering power spectrum of vacuum fluctuations

5 This zero point effect may be observable: it is about 15% of the thermal effect for liquid neon. Local phononic Casimir effects; renormalized δρ 2 R near a boundary Observable quantity: modifies light scattering Always finite due to the natural cutoff. Single plate: z δρ 2 R = ρ 0 c S 32π 2 z 4 < 0 Negative - a suppression of the zero point fluctuations

6 Quantum field theory Ultraviolet divergences: Formal expectation value of stress tensor operators diverges. Zero point energy: ω max 1 2 ω ω4 max Usual approach in Minkowski spacetime: subtract the vacuum expectation value T µν : T µν := T µν 0 T µν 0

7 Zero point fluctuations still have physical effects: Lamb shift Spontaneous emission by atoms Casimir effect Decoherence and recoherence

8 Electromagnetic vacuum fluctuations and electron interferometry Aharonov-Bohm phase fluctuations and Decoherence Gaussian fluctuations: e iϕ = e 1 2 ϕ2 Contrast decreases by a factor of: Γ = e W ϕ AB = e C dx µ A µ W = 1 2 ϕ2 AB Photon emission and vacuum Aharonov-Bohm phase fluctuations are complementary descriptions.

9 Recoherence J.T. Hsiang & LF Let the quantized EM field be in a squeezed vacuum state: ζ =e 1 2 [ζ a 2 ζ(a ) 2] 0 On average, the effect is to increase decoherence. However, if we select electrons emitted at particular times, the effect is decreased decoherence. Recoherence - suppression of the usual vacuum fluctuations. Small effect, < 10 3, but maybe observable?

10 Negative Energy Density Classical matter satisfies the weak energy condition: T µν t µ t ν 0 timelike t µ Essential for proof of the singularity theorems Quantum matter can violate this condition: Casimir effect Hawking effect Squeezed states

11 Casimir Effect ρ = cπ2 720L 4 L Effects of finite reflectivity:! p a = 50 Result: energy density can be negative, but requires high reflectivity or large separations (Sopova & LF) a 4 U ! p a = 99 ω p = plasma frequency 0.05! p a =

12 Negative energy density in a squeezed vacuum state: ρ ω sinh r[sinh r cosh r cos(2ω t)] squeeze parameter = ζ Can be made arbitarily negative at a given point by increasing the frequency of the mode

13 Null Energy Condition T µν n µ n µ 0 for all null vectors n µ Also violated by quantum stress tensors, leading to defocusing of light rays. Raychaudhuri equation: λ =affine parameter θ = n µ ;µ = expansion dθ dλ = R µν n µ n ν + R µν = 8π(T µν 1 2 g µνt ) dθ dλ = 8π T µν n µ n ν + T µν n µ n ν > 0 T µν n µ n ν < 0 defocusing

14 Some effects of negative energy: Backreaction in the Hawking Effect Negative energy flux across the horizon Singularity avoidance Violation of the weak energy conditions allows the singularity theorems to be evaded.

15 Traversable wormholes Negative energy is needed at the throat of the wormhole to cause light rays to defocus.

16 Faster-than-light travel

17 Violations of the second law of thermodynamics Shine negative energy on a black hole and reduce its horizon area.

18 Violations of cosmic censorship: shine negative energy on an extreme black hole?

19 Constraints on negative energy in Minkowski spacetime: 1) Total energy is non-negative - positivity of the Hamiltonian 2) Averaged weak energy condition - T µν u µ u ν dτ 0 Neither of these conditions is strong enough to avoid negative energy problems; the positive energy could be very far from the negative energy.

20 3) Quantum inequalities - T µν u µ u ν g(τ, τ 0 ) dτ C τ d 0 g(τ, τ 0 ) = sampling function C τ 0 = sampling time d = positive constant = spacetime dimension

21 Some Minkowski space examples with a Lorentzian sampling function: Two dimensions (1+1) τ 0 π T µν u µ u ν τ 2 + τ0 2 (Flanagan) dτ 1 48π τ 2 0 Optimum bound Four dimensions (3+1) τ 0 π T µν u µ u ν τ 2 + τ 2 0 dτ 3 32π 2 τ 4 0 (LF&Roman, Fewster&Eveson)

22 In the limit that τ 0, we recover the averaged weak energy condition as a special case. T µν u µ u ν dτ 0 A negative energy density cannot last longer than about t = ρ 1/4 (4D)

23 Physical implication: The amount of negative energy that can be absorbed by a system in time t is less than 1/t In 4D, need the collecting area < 1/t 2 This is less than the quantum energy uncertainty on this time scale.

24

25 Bound on the separation of negative and positive energy pulses A negative pulse E must be followed by an overcompensating positive pulse within time 1/ E The later the (+) pulse arrives, the larger it must be. Quantum Interest t (+) u=u f Moving mirror example: 3 2 (!) u=0 1 x

26 Non-Minkowskian spacetimes Can apply the Minkowski results on small scales E.g. if τ 0 local radius of curvature Can also prove results for specific spacetimes These are typically difference inequalities ; bound the difference between a given state & the ground state

27 We should not expect large macroscopic effects from negative energy. No macroscopic violations of the 2nd law: (Less than one bit of information) Severe constraints on the parameters of wormholes and warp drives No macroscopically observable violations of cosmic censorship

28 Summary 1) Quantum fluctuations may be modified by boundaries or choice of quantum state 2) Suppression of vacuum fluctuations may produce observable effects 3) Negative energy is a type of sub-vacuum effect 4) Negative energy has the potential to produce dramatic physical effects 5) These effects are severely constrained by quantum inequalities, but can produce observable effects.