Probability distributions for space and time averaged quantum stress tensors

Abstract

We extend previous work on quantum stress tensor operators which have been averaged over finite time intervals to include averaging over finite regions of space as well. The space and time averaging can be viewed as describing a measurement process for a stress tensor component, such as the energy density of a quantized field in its vacuum state. Although spatial averaging reduces the probability of large vacuum fluctuations compared to time averaging alone, we find that the probability distribution decreases more slowly than exponentially as the magnitude of the measured energy density increases. This implies that vacuum fluctuations can sometimes dominate over thermal fluctuations and potentially have observable effects. I. INTRODUCTIONAlthough the vacuum state of a quantum field theory is an eigenstate of the Hamiltonian, the integral of the energy density over all space, it is not an eigenstate of the local energy density or of other components of the stress tensor.This implies the existence of vacuum fluctuations of the energy density and other quadratic operators. For these fluctuations to be finite, and hence physically meaningful, these operators must be averaged over a finite spacetime region. We can view the averaging process as representing the outcome of a measurement of the operator. The energy density at a single spacetime point is not measurable, and hence not meaningful. However, the spacetime average is meaningful, and will have finite fluctuations described by a probability distribution.The study of the probability distributions for quantum stress tensors was begun in Ref.[1] for conformal field theory (CFT) in two spacetime dimensions, and continued in Refs.[2] and [3] for quantum fields in flat four dimensional spacetime. Further results on CFT appear in [4]. Let x denote a dimensionless measure of the averaged stress tensor component T . If τ is a measure of the size of the sampling region, then in units where = c = 1, we may take x = τ d T , where d is the dimension of the spacetime. Let P (x) denote a probability distribution so that P (x) dx is the probability in a measurement of finding an outcome in the interval [x, x + dx]. There are two key features of P (x) for a quadratic operator, such as the energy density, which have emerged in the papers just cited: 1) There is a negative lower bound on the region where P (x) = 0 if T ≥ 0 at the classical level, and 2) P (x) can fall more slowly than exponentially, leading to an enhanced probability for large positive fluctuations relative to thermal fluctuations.By contrast, the probability distribution for the spacetime average of a linear operator, such as the electric field, is a Gaussian function.If T is a non-negative quantity in classical physics, such as the energy density, its quantization typically admits quantum states for which its expectation value is below the vacuum value. In particular, if the vacuum expectation value vanishes there exist states for which its expectation value is negative, T < 0, so regions where the mean energy d...

Large zero point density fluctuations in fluids

“…The quantum fluctuations of a quadratic field operator, such as energy density, are especially subtle, with a probability distribution which falls more slowly than exponentially [1][2][3]. This leads to an enhanced probability for very large vacuum fluctuations for quantum stress tensors, which can in turn drive large fluctuations of the gravitational field, a variety of quantum gravity effect [4].…”

“…A class of such functions was constructed in Refs. [2,3], and may be characterized by their asymptotic forms of their Fourier transforms:…”

“…As discussed in Ref. [3], time averaging is essential for the fluctuations to be finite. If we were to let f (t ) = δ(t ), so that f (ω) = 1, then μ 2 would diverge due to a large contribution from modes with q 1 ≈ −q 2 .…”

“…Here η is the index of refraction of the fluid at the peak wavelength, and is assumed to be of order one. We wish to consider photons scattered from this packet in a time of τ = η /c, during which the packet moves a distance of order , so the scattering volume V will be of order 3 . The continuum approximation which we use seems to require that both λ 0 and c s τ be larger than the interatomic spacing.…”

“…For a specific estimate, we setf (ω) =ĥ fit (ω) andĝ(k) =ĝ fit (k), whereĥ fit (ω) andĝ fit (k) are defined in Appendix A of Ref. [3], and correspond to α = λ = 1/2 in Eqs. (6) and (7).…”