Fluctuations of Time Averaged Quantum Fields

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1 Fluctuations of Time Averaged Quantum Fields IOP Academia Sinica Larry Ford Tufts University January 17, 2015

2 Time averages of linear quantum fields Let E (x, t) be one component of the electric field operator, and define Ē = E (x, t) f τ (t) dt f τ (t) = sampling function of width τ Vacuum state: Ē =0 Ē2 τ 4 Gaussian probability distribution: P (Ē) e aē2 τ 4 constant of order one

3 Time averaging, with test functions of compact support, is used in rigorous quantum field theory to have well defined operators. The effect of the sampling is to suppress the contributions of modes whose wavelengths are much shorter than Are there situations which determine a specific physical choice for the form and width of the f τ (t) sampling function?

4 A nonlinear optics model for lightcone fluctuations Basic Idea: C. Bessa, V DeLorenci, N Svaiter & LF A background electric field in a nonlinear material changes the effective speed for a probe pulse. (This is analogous to the effect of a classical gravitational field on light.) If the background field has quantum fluctuations, then the effective lightcone for the probe pulse also fluctuates.

5 Consider a situation where both fields are polarized in the z-direction, but propagating in the x-direction. Electric field: E i = δ iz E = δ iz E(t, x) Polarization: P z = χ (1) E + χ (2) E 2 + χ (3) E 3 + linear term non-linear terms where χ (1) = χ (1) zz, χ (2) = χ (2) zzz, χ (3) = χ (3) zzzz Material can be anisotropic, but only z-components of the susceptibility tensors contribute here.

6 Let E = E 0 + E 1 E 1 E 0 lower frequency background field higher frequency probe field Here we assume that the material is approximately dispersionless, but that the value of χ (1) at the higher frequency is somewhat higher than that at the lower frequency.

7 Equation for E 1 (t, x) 2 E 1 x 2 1 v 2 (1 + 2ɛ 1 +3ɛ 2 ) 2 E 1 t 2 =0 where v = c 1+χ (1) Wave speed in the linear approximation ɛ 1 = χ(2) 1+χ (1) E 0(t, x), ɛ 2 = χ(3) 1+χ (1) E2 0(t, x)

8 WKB solutions of this equation have a space and time dependent phase velocity: u(t, x) = v {1 1+2ɛ1 (t, x)+3ɛ 2 (t, x) v ɛ 1 (t, x) 3 } 2 [ɛ 2(t, x) ɛ 2 1(t, x)] If dispersion can be neglected, wave packets can also have this group velocity. ɛ 1 (t, x) ɛ 2 (t, x) u(t, x) Fluctuations in and, and hence of lead to fluctuations in the flight times of pulses.

9 Consider only the effect of ɛ 1 (t, x) Assume ɛ 1 =0 The mean squared variation in flight time becomes ( t) 2 = L/v dt 1 L/v dt 2 ɛ 1 (t 1 ) ɛ 1 (t 2 ) 0 0 L = flight distance

10 Let the source of the fluctuations be the vacuum electric field fluctuations. Assume that the dominant contribution comes from modes which experience a mean index of refraction n 0 > 1. Electric field correlation function: E z (x, t)e z (x,t ) = n 0 ( x) 2 +( t) 2 /n 2 0 π 2 [( x) 2 ( t) 2 /n 2 0 ]3 Effective lightcone is along the line t = n 0 x (c = 1)

11 Assume that the probe pulse moves along the path t = n 1 x n 1 >n 0 > 1 t path of the probe pulse effective lightcone true lightcone x

12 Let the nonlinear dielectric material be in a slab with tapered density and an effective width d Let τ be the transit time for the pulse, as seen by a comoving observer, and let the sampling function f τ (t) describe the density profile seen by this observer. d

13 Now the fractional mean squared variation in flight time becomes ( t/τ) 2 = (χ (2) ) 2 dt 1 f τ (t 1 ) dt 2 f τ (t 2 ) E(x(t 1 ),t 1 )E(x(t 2 ),t 2 ) n 4 1 Result for Lorentzian sampling: ( t/τ) 2 = n5 0 (1 + n 2 0/n 2 1)(χ (2) ) 2 16π 2 (1 n 2 0 /n2 1 )3 d 4

14 Example: Mercuric sulphide (cinnabar) d = 10 µm λ probe 1 µm n n t τ

15 Some comments: 1) The anticorrelations do not prevent a physical effect of vacuum fluctuations in this case, due to the finite switching time. 2) The effect of electric field fluctuations in this model is analogous to effects in quantum gravity.

16 Effects of fluctuations of the squared electric field: Include a third-order susceptibility, which gives a fractional contribution to the flight time of the form δt/t = 3 2 χ(3) dx E 2 0 Probability distribution for E 2 0 non-gaussian. fluctuations is (C. Fewster, T. Roman and LF)

17 QED correction to quantum tunnelling Flambaum & Zelevinsky V (x) One loop correction to the potential: E 0 δv (x) = e2 3πm 2 log ( m V (x) E 0 ) 2 V (x) Typically, 2 V (x) < 0 so this effect increases the tunnelling rate. Munchhausen effect

18 Is there an alternative explanation in terms of electric field fluctuations? (H. Huang and LF) Let τ be an estimate of the time which the particle spends going through the barrier and f τ (t) be a function determined by the shape of the potential. If E 0 is of the same order as the maximum of V (x) estimate that τ d m/e 0 width of the barrier

19 Let the work done by an electric field fluctuation be W edē e d/τ 4 This seems to be of the correct order of magnitude to explain the effect found by Flambaum & Zelevinsky

20 An estimate of the (non-perturbative) effect of large electric field fluctuations: W V (x) E 0 d Correction to the tunnelling probability of order e m2 d 2 /e 2

21 For both the wavepacket traversing the dielectric slab, and the electron traversing a potential, the geometry defines a specific sampling function. f τ (t) V (x(t)) The Gaussian fluctuations of linear fields are relatively independent of the shape of the sampling function, but this is not the case for quadratic field operators, such as the squared electric field or a stress tensor component.

22 Lorentzian sampled energy density in four dimensions x = (4πτ 2 ) 2 : T tt (x,t): g(t, τ) dt g(t, τ) = τ π(t 2 + τ 2 ) Compute a finite set n 65 of moments explicitly from Wick s theorem. Result for the electromagnetic field: M n = x n CD n (3n 4)! n 1 C. Fewster, T. Roman & LF

23 Hamburger Moment Theorem There exists a unique P(x) for a given set of moments, provided that M n grow no faster than n! as n. Implications of the faster rate of growth: 1) The Hamburger moment condition is not satisfied, so we cannot uniquely determine P(x) from the moments. 2) Any probability distribution consistent with these moments must have an asymptotic form which falls more slowly than exponentially at large x.

24 Result of a fit to the larger moments: P (x) c 0 x 2 e ax1/3 x 1 c 0 a 0.96 Large positive fluctuations are more likely than one might expect, and eventually vacuum fluctuations dominate over thermal fluctuations.

25 A fit to the entire distribution: P(x) x Negative lower bound at the quantum inequality bound on expectation values in an arbitrary quantum state - lowest eigenvalue of the sampled energy density.

26 What about other sampling functions, such a compactly supported function? g x d x E.g., f τ (t) e ( 1 t + 1 τ t )

27 Rate of growth of the moments depends on the rate of decrease of the Fourier transform of the sampling function, ˆf(ωτ) = dt e iωt f τ (t) Lorentzian: ˆf(ωτ) e τω A class of compact ˆf(ωτ) e τω functions: The latter case leads to moments growing as (6n)! C. Fewster & LF

28 Now the probability distribution falls more slowly than in the Lorentzian case: P (x) e 1/x1/6 Large positive fluctuations become even more likely.

29 Possible effects of large stress tensor fluctuations: 1) Time delays in nonlinear materials with χ (3) 0 2) Enhancements of quantum tunnelling rates (including false vacuum decay). 3) Black hole and Boltzmann brain nucleation 4) Non-Gaussian perturbations in inflationary cosmology

30 Effects of quantum stress tensor fluctuations in inflationary cosmology 1) Density perturbations C.H. Wu, K.W. Ng, LF, S.P. Miao, R. Woodard Effect of electromagnetic vacuum energy density fluctuations on the expansion of geodesics in desitter space grows with the duration of inflation, leading to enhanced density perturbations. 2) Tensor perturbations Gravity waves radiated by stress tensor fluctuations J-T Hsiang, K-W Ng, C-H Wu & LF

31 Both effects lead to non-gaussian and non-scale invariant contributions. Both are calculated from time integrals of a stress tensor correlation function - form of time averaging. Time averaged quantum fields and initial conditions in cosmology: is there a natural choice of sampling function?

32 Summary 1) Time averaging is needed for well defined quantum field fluctuations 2) Natural choices for the sampling function in an analog model for lightcone fluctuations and in quantum potential scattering 3) Stress tensor fluctuations have a non-gaussian probability distribution with a long tail. Possible observable effects? 4) Stress tensor fluctuation in cosmology and the problem of initial conditions

Analog quantum gravity effects in dielectrics. Carlos H. G. Bessa (UFPB-Brazil) ICTP-SAIFR April 2017

Analog quantum gravity effects in dielectrics Carlos H. G. Bessa (UFPB-Brazil) In collaboration with V. A. De Lorenci (UNIFEI-Brazil), L. H. Ford (TUFTS-USA), C. C. H. Ribeiro (USP-Brazil) and N. F. Svaiter