ESI 2250
Probability distributions of smeared quantum stress tensors
arXiv:1004.0179v2 [quant-ph] 22 May 2010
Christopher J. Fewster∗ Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom L. H. Ford† Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA Thomas A. Roman‡ Department of Mathematical Sciences, Central Connecticut State University, New Britain, Connecticut 06050, USA (Dated: May 25, 2010)
Abstract We obtain in closed form the probability distribution for individual measurements of the stressenergy tensor of two-dimensional conformal field theory in the vacuum state, smeared in time against a Gaussian test function. The result is a shifted Gamma distribution with the shift given by the previously known optimal quantum inequality bound. For small values of the central charge it is overwhelmingly likely that individual measurements of the sampled energy density in the vacuum give negative results. For the case of a single massless scalar field, the probability of finding a negative value is 84%. We also report on computations for four-dimensional massless scalar fields showing that the probability distribution of the smeared square field is also a shifted Gamma distribution, but that the distribution of the energy density is not. PACS numbers: 03.70.+k,04.62.+v,05.40.-a,11.25.Hf
∗ † ‡
Electronic address:
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1
The expectation value of the stress tensor of a quantum field, when suitably renormalized, can be used to give a description of the gravitational effects of quantum fields. However, this semiclassical theory clearly has limitations in that it does not describe fluctuations around the mean value. In recent years, several authors have discussed possible physical effects of these fluctuations. For recent reviews with further references, see Refs. [1, 2]. Much of the work on this topic has involved seeking observables which can be related to integrals of the stress tensor correlation function. However, this amounts to calculating a variance, or a second moment, of the probability distribution for measurement results of the smeared stress-energy tensor. It is desirable to go further and determine the complete distribution function. In this paper, we will report results for such a function. Here we will be concerned with Minkowski spacetime in the vacuum state and seek the probability distribution function for a stress tensor operator which has been averaged with a sampling function. This averaging is crucial to having a well-defined probability distribution. (A recent attempt has been made to assign a distribution to a local, unsmeared, stress tensor operator [3]. However, in our view this is not a meaningful quantity.) There is a deep connection between the probability distribution for vacuum fluctuations, and the quantum inequalities (QIs) which place lower bounds on the expectation values of smeared stress tensors in arbitrary physically reasonable quantum states [4–10]. These lower bounds represent the lowest eigenvalues of the smeared operators. As a consequence of the Reeh– Schlieder theorem (see, e.g., [11]), they are also the lower bounds of the support of the corresponding probability distributions in the vacuum state [18]. In this paper, we will concentrate on conformal field theory (CFT) in two spacetime dimensions, but will also report a result for the square of a massless scalar field in four spacetime dimensions. Further details will be given in a later paper. We consider a general unitary, positive energy CFT with central charge c in twodimensional Minkowski space (with +− signature and inertial coordinates xa ). A general reference is [12]. Our notation is as follows: TL and TR are the left- and right-moving components of the (renormalized) stress-energy tensor, so that T00 (x0 , x1 ) = −T11 (x0 , x1 ) = TR (u) + TL (v) T01 (x0 , x1 ) = T10 (x0 , x1 ) = −TR (u) + TL (v)
(1)
where u = x0 − x1 , v = x0 + x1 . As TR and TL commute, we can find the probability distribution for a smearing of T00 from the probability distributions for corresponding smearings of TL and TR . We begin by considering TR only and dropping the subscript (identical results hold for TL ). The vacuum n-point functions will be denoted Gn (un , . . . , u1 ) = hT (un ) · · · T (u1)i, with the convention that G0 = h1i = 1; where one or more of the uj is replaced by the letter f , this indicates smearing against a test function f (uj ), and we write Gn [f ] = Gn (f, . . . , f ) = hT (f )n i for the n-th moment of T (f ) in the vacuum state. All test functions will be assumed real-valued and rapidly decaying at infinity. Our goal is to determine an expression for the moment generating function M[µf ] =
∞ X µn Gn [f ] n=0
n!
(2)
from which the underlying probability distribution can be determined under suitable conR ditions. Here, we take f (u) du to be dimensionless and µ with dimensions of length2 . We 2
start with the Ward identities in CFT (see, e.g., Sect. 3 of [12]), which entail the recurrence relation n−1 X c Gn−2 (un−1 , . . . , uˆj , . . . , u1 ) Gn (un , . . . , u1 ) = 8π 2 (un − uj − i0)4 j=1 ∂j Gn−1 (un−1 , . . . , u1) Gn−1 (un−1 , . . . , u1) − − , (3) 2π(un − uj − i0) π(un − uj − i0)2 where the hat denotes an omitted variable. Since G0 = 1 and G1 (u1 ) = hT (u1 )i ≡ 0, it follows immediately that c G2 (u2 , u1 ) = 2 . (4) 8π (u2 − u1 − i0)4
Smearing Eq. (3) against n copies of f and noting that the first line of the right-hand side provides n − 1 identical contributions, we have Gn [f ] = (n − 1)G2 [f ]Gn−2 [f ] +
n−1 X
In,j ,
(5)
j=1
where it may be shown that In,j = Gn−1 (f, . . . , f ⋆ f , . . . , f ), | {z }
(6)
j
with the underbrace indicating that f ⋆ f is inserted in the j’th position from the right, and f ⋆ f given after some computation by Z ∞ f (w)f ′(u) − f ′ (w)f (u) . (7) f ⋆ f (u) = dw 2π(w − u) −∞ Note that the integrand is nonsingular at w = u. In deriving this formula several steps of integration by parts have been used; no boundary terms are introduced provided f decays sufficiently fast at infinity. Now suppose that a one-parameter family fλ (u) (λ ∈ R) can be found, satisfying the differential equation dfλ = fλ ⋆ fλ (8) dλ and initial condition f0 = f (λ has dimensions of length2 ). Then, replacing f by fλ in Eq. (5), the sum over j can be rewritten as a derivative with respect to λ [19], yielding Gn [fλ ] = (n − 1)G2 [fλ ]Gn−2 [fλ ] +
d Gn−1 [fλ ]. dλ
(9)
Multiplying by µn−1 /(n − 1)! and summing over n ≥ 2, this becomes a partial differential equation ∂ ∂ M[µfλ ] = µG2 [fλ ]M[µfλ ] + M[µfλ ] (10) ∂µ ∂λ
3
for M[µfλ ], on using the facts G0 [fλ ] = 1, G1 [fλ ] = 0, and Gn [µfλ ] = µn Gn [fλ ]. To solve this, we write M[f ] = exp(W [f ]), whereupon Eq. (10) becomes ∂ ∂ W [µfλ] = µG2 [fλ ] + W [µfλ ] . ∂µ ∂λ
(11)
As W [µfλ ] = 0 for µ = 0, the unique solution is W [µfλ ] =
Z
λ+µ
dλ′ (µ + λ − λ′ )G2 [fλ′ ] ,
λ
(12)
and we set λ = 0 to obtain W [µf ] =
Z
µ
dλ (µ − λ)G2 [fλ ].
0
(13)
Equation (13) is similar to a formula derived in the Euclidean holomorphic picture (in contrast to our Lorentzian treatment) by Haba in Eq. (9) of [13]. The differential equation Eq. (8) can be solved explicitly in the case of a Gaussian sam√ 2 2 pling function f (u) = e−u /τ /(τ π). Making the ansatz fλ (u) = A(λ)f (u), the calculation (fλ ⋆ fλ )(u) =
A(λ) A(λ)2 −u2 /τ 2 e = 2 fλ (u), 3 3/2 τ π τ π
(14)
reduces Eq. (8) to dA/dλ = A(λ)2 /(τ 2 π), whose unique solution with A(0) = 1 is πτ 2 . πτ 2 − λ
A(λ) =
(15)
We now have G2 [fλ ] = A(λ)2 G2 [f ], so Z
µ
dλ (µ − λ)A(λ)2 0 πτ 2 2 4 2 − µτ π = G2 [f ] τ π log πτ 2 − µ cµ πτ 2 c − log , = 24 πτ 2 − µ 24πτ 2
W [µf ] = G2 [f ]
where we have used
to compute
1 1 = 4 (v − iǫ) 6 c G2 [f ] = 48π 2
Z
∞
Z
∞
dω ω 3 e−iω(v−iǫ)
(16)
(17)
0
dω ω 3e−ω
2 τ 2 /2
=
0
c . 24π 2 τ 4
(18)
The moment generating function for the Gaussian-averaged right-moving flux is then M[µf ] = eW [µf ]
"
2
e−µ/(πτ ) = 1 − µ/(πτ 2 ) 4
#c/24
.
(19)
It remains to determine the corresponding probability distribution P (ω) such that Z ∞ M[µf ] = dω P (ω)eµω .
(20)
−∞
As the moments obey the uniqueness conditions in the Hamburger moment theorem [14], there can be only one such distribution. (One needs |an | ≤ CD n n! for the n’th moment an = dn M[µf ]/dµn |µ=0 , which is satisfied for our case.) It is a shifted Gamma distribution, of form β α (ω + ω0 )α−1 P (ω) = ϑ(ω + ω0 ) exp(−β(ω + ω0 )), (21) Γ(α) with parameters
c c , α= , β = πτ 2 , (22) 2 24πτ 24 and where ϑ is the Heaviside function. Note that Eq. (21) is singular at its lower limit provided that c < 24, but the singularity is integrable, so that the integrated probability is unity, as required. As described above, the lower bound of this distribution is also the best possible quantum inequality bound on the expectation value in a normalized state ψ, so ω0 =
hT (f )iψ ≥ −ω0
(23)
for all physically reasonable ψ. This can be checked against the optimal bound given in Theorem 4.1 of [10], 2 Z p d c c = −ω0 . (24) hT (f )iψ ≥ − f (u) du = − 12π du 24πτ 2
Thus we have the expected agreement between the two methods. The nonzero negative lower bound should be a general feature of stress tensor probability distributions, and is required by the vanishing mean value in the vacuum state, in contrast to the result claimed in Ref. [3]. The cumulative distribution function of our probability distribution is an incomplete Gamma function. Perhaps surprisingly, it is overwhelmingly likely that an individual measurement will yield a negative result: the probability is 0.89 for c = 1 (independent of τ0 and τ ) and decreases as c increases, tending to 0.5 as c → ∞. When both TR and TL are combined to obtain the overall probability distribution for the energy density T00 (t, 0), the effect is to replace c/24 by c/12 in the probability distribution R∞ 2 2 and definition of ω0 . The averaged energy density operator is ρ = √1π τ −∞ Ttt (t, 0) e−t /τ dt . Then x = ρτ 2 is distributed according to the shifted Gamma distribution with probability density function π c/12 (x + x0 )c/12−1 −π(x+x0 ) P (x) = ϑ(x + x0 ) e , (25) Γ(c/12) where −x0 /τ 2 is the quantum inequality bound on expectation values of Ttt in arbitrary quantum states. The probability distribution is plotted in Fig. 1 for the case of a free massless scalar field in two-dimensional Minkowski spacetime, for which c = 1. In this case, the probability of a negative result is 0.84, and the corresponding QI bound coincides with that of [8]. 5
FIG. 1: The probability distribution P (x) for the smeared energy density of a massless scalar field in two-dimensional Minkowski spacetime is plotted. Here x = ρ τ 2 , where ρ is the energy density operator averaged in time with a Gaussian function of width τ . The lower limit of P (x) occurs at x = −x0 = −1/(12π), illustrated by the vertical line. n 012 3
4
5
6
7
8
Mn 1 0 2 48 1740 83904 5051640 364724928 30707616912 TABLE I: Computed moments Mn = (4πτ )2n Gn [L] for the Lorentzian averaged squared field operator.
This is a remarkable result which sheds light on the nature of vacuum energy fluctuations, at least in two-dimensional spacetime. The zero average value of the smeared energy density in the vacuum state arises, in a sequence of measurements, from many negative values being balanced by rarer, but larger positive values. Although the probability distribution for stress tensor operators in four-dimensional spacetime is not yet known, we have been able to construct distributions for the square of a massless scalar field by empirical methods. This amounts to calculating several moments, and then showing that they fit to a shifted Gamma distribution to high accuracy. Using the Lorentzian sampling function L(t) = τ /(π(t2 + τ 2 )), we computed the first twenty R 2 moments of :Φ :(t, x)L(t) dt in the vacuum state, using symbolic integration in Maple. After suitable normalization, the computed moments are integers; the first eight of which are presented in Table I. Remarkably, all twenty moments are fitted exactly by the shifted Gamma distribution Eq. (21) with parameters α=
1 , 72
β=
4π 2 τ 2 , 3
ω0 =
1 . 96π 2 τ 2
(26)
Because the form in Eq. (21) is already normalized, we need the n = 1, 2, and 3 moments to fit the above parameters. However, as the remaining 17 moments are reproduced exactly, this provides strong evidence that this is the correct distribution. The lower bound of the probability density support is −ω0 , which provides a conjectured 6
optimal QI bound
Z
1 (27) 96π 2 τ 2 for all normalized ψ in the relevant domain. By comparison, the methods of [9] yield a QI [20] 2 Z Z ∞ p 1 d 2 f (t) dt. (28) hψ | :Φ :(t, x)ψif (t)dt ≥ − 2 8π −∞ dt Substituting the Lorentzian L in place of f , the lower bound is −1/(64π 2τ 2 ), which is weaker by a factor of 3/2 than (and therefore does not contradict) the conjectured bound just given. Just as in the case of fluxes and energy densities in two dimensions, the probability distribution diverges integrably at its lower limit. In this case, the probability of a negative outcome for a measurement of the averaged squared field is 0.95. The first fifteen moments of :Φ2 : smeared against the squared Lorentzian S(t) = 2τ 3 /(π(t2 + τ 2 )2 ) have also been computed and are exactly fitted by the shifted Gamma distribution (21) with parameters α = 1/45, β = 8π 2 τ 2 /15, and ω0 = 1/(24π 2 τ 2 ). The corresponding conjectured optimal QI bound is again stronger by a factor of 3/2 than that of Eq. (28); it is tempting to conjecture that the optimal bound for general sampling functions is given by (28) with a 12 replacing the 8. As in the case of the stress tensor in CFT with c < 24, these distributions for the squared scalar field are singular at the lower limit, although with an integrable singularity. The physical effect of this singularity is to favor negative fluctuations close to the lower bound. Note that there nothing unphysical about this singularity, as the observable quantity, the probability of an outcome in a finite interval, is finite. We have also attempted a similar fit to the Lorentzian average of the squared time derivative of a massless scalar field and to the squared electric field. In both cases, a shifted Gamma distribution which is fit to the lower moments underestimates the higher moments. This suggests that the correct probability distribution for these quantities, and for the fourdimensional stress tensor, has a positive tail which falls off more slowly than is described by the shifted Gamma distribution. The probability distribution for the energy density in four-dimensional theories is of considerable interest. One application is to inflationary cosmology, where quantum stress tensor fluctuations might contribute a potentially observable component to the cosmological density fluctuations [15]. This component would be non-Gaussian in a way which is associated with the skewness of the quantum stress tensor probability distribution. Another, perhaps more exotic, application to cosmology arises in models which employ anthropic reasoning to attempt to compute probabilities of various observables. These models require a counting of observers, possibly including “Boltzmann brains” which have nucleated from the vacuum in deSitter or even Minkowski spacetime. If this is the more prevalent type of observer, it would greatly complicate attempts at anthropic prediction. (For further discussion and references, see, for example Refs. [16, 17].) The key to studying this question is in the details of the long positive tail of the probability distribution. hψ | :Φ2 :(t, 0)ψiL(t) dt ≥ −
Acknowledgments
This work was supported in part by the National Science Foundation under Grants PHY0855360 and PHY-0652904. CJF thanks the Erwin Schr¨odinger Institute, Vienna, and the 7
organizers of the program Quantum Field Theory on Curved Space-times and Curved Target Spaces for support and hospitality during the final stages of this work.
[1] B.L. Hu and E. Verdaguer, Living Rev. Rel. 7, 3 (2004), gr-qc/0307032. [2] L.H. Ford and C.H. Wu, AIP Conf.Proc. 977 145 (2008), arXiv:0710.3787. [3] G. Duplancic, D. Glavan, and H. Stefancic, arXiv:1002.1846. [4] L. H. Ford, Proc. Roy. Soc. Lond. A364, 227 (1978). [5] L. H. Ford, Phys. Rev. D43, 3972 (1991). [6] L.H. Ford and T.A. Roman, Phys. Rev. D 51, 4277 (1995), gr-qc/9410043. [7] L.H. Ford and T.A. Roman, Phys. Rev. D 55, 2082 (1997), gr-qc/9607003. [8] E.E. Flanagan, Phys. Rev. D, 56, 4922 (1997), gr-qc/9706006. [9] C.J. Fewster and S.P. Eveson, Phys. Rev. D 58, 084010 (1998), gr-qc/9805024. [10] C.J. Fewster and S. Hollands, Rev. Math. Phys. 17, 577 (2005), math-ph/0412028. [11] R. Haag, Local quantum physics (Springer-Verlag, Berlin, 1996). [12] P. Furlan, G.M. Sotkov, and I.T. Todorov, Riv. Nuovo Cimento 12, 1 (1989). [13] Z. Haba, Phys. Rev. D 41, 724 (1990). [14] M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis, selfadjointness (Academic Press, New York, 1975), p 205. [15] C.-H. Wu, K.-W. Ng, and L.H. Ford, Phys. Rev. D 75, 103502 (2007), gr-qc/0608002. [16] J. Garriga and A. Vilenkin, Phys. Rev. D 77, 043526 (2008), arXiv:0711.2559. [17] A. De Simone, A.H. Guth, A. Linde, M. Noorbala, M.P. Salem, and A. Vilenkin, arXiv:0808.3778. [18] The connection begins with the observation that the following three statements are equivalent, where A is a self-adjoint operator (with domain D(A) and spectrum σ(A)) representing a quantum observable: (a) hψ | Aψi ≥ −Q for all normalised ψ ∈ D(A); (b) σ(A) ⊂ [−Q, ∞); (c) the probability measure µψ that describes the distribution of individual measurement results of A in state ψ has support supp µψ ⊂ [−Q, ∞). [The implication (a) =⇒ (b) is the Rayleigh–Ritz variational principle; (b) =⇒ (c) because measurements always give values in the spectrum; (c) =⇒ (a) because any random variable taking values in [−Q, ∞) has its expected value in that range.] In general, supp µψ (for a fixed choice of ψ) can be a proper
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subset of σ(A), and in particular the infimum of the support of µψ can exceed inf σ(A). For example, this happens if inf σ(A) is an isolated eigenvalue whose eigenfunction is orthogonal to ψ. However, the spectrum and the support of the probability measure coincide if A is a local observable, such as a smeared field, and ψ is the vacuum state Ω of a Minkowski space QFT obeying the standard assumptions of algebraic quantum field theory [11]. To see this, note first that if λ ∈ σ(A) then Pǫ , the spectral projection of A corresponding to the interval (λ − ǫ, λ + ǫ), is nonzero for any ǫ > 0. As Pǫ is a local observable (because A is) the Reeh– Schlieder theorem entails that it cannot annihilate the vacuum, and hence there is a nonzero probability kPǫ Ωk2 of obtaining a measurement of A in the interval (λ − ǫ, λ + ǫ), which must therefore intersect the support of µΩ . Taking ǫ → 0 we conclude that λ ∈ supp µΩ (which is a closed subset of R). Varying λ, this argument establishes that σ(A) ⊂ supp µΩ , and we already know that the reverse inclusion holds [by the implication (b) =⇒ (c) above]. In particular, σ(A) and suppµΩ have a common infimum, which provides a sharp quantum inequality bound hψ | Aψi ≥ inf supp µΩ for all normalized ψ ∈ D(A) by our earlier observations. [19] For example, use fλ+δ = fλ + δfλ ⋆ fλ + O(δ2 ) and linearity of Gn−1 in each slot. √ [20] Compare with the QI given in Eq. (5.5) of [9]; the starting point for Eq. (28) is to set p = 1/ ω in Eq. (3.11) of [9].
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