Minkowski vacuum stress tensor fluctuations

We study the fluctuations of the stress tensor for a massless scalar field in two- and four-dimensional Minkowski spacetime in the vacuum state. Covariant expressions for the stress tensor correlation function are obtained as sums of derivatives of a scalar function. These expressions allow one to express spacetime averages of the correlation function as finite integrals. We also study the correlation between measurements of the energy density along a world line. We find that these measurements may be either positively correlated or anticorrelated. The anticorrelated measurements can be interpreted as telling us that, if one measurement yields one sign for the averaged energy density, a successive measurement with a suitable time delay is likely to yield a result with the opposite sign.

Figure 1

Some possible integration contours for Eq. (16) are illustrated. There are two simple poles on the imaginary axis, and two higher order poles on the real axis. Both types of poles are denoted by the letter X. The contours $C_1$ and $C_2$ both yield the same result for the integral. Integration around either of the poles on the real axis (dashed line circles) gives an imaginary result, so the real part of the integral is independent of whether these poles are enclosed or not.

Figure 2

Two sampling functions which initially coincide and are then gradually separated.

Figure 3

The graph of $K(t_0)$ versus $t_0$ for a Lorentzian sampling function, in units with $a=1$. Here we have chosen $b=0$, so the sampling is in time only. Here (a) shows the overall form of $K(t_0)$, but on a scale which does not reveal the final maximum. This peak is revealed on a smaller scale graph, (b).

Figure 4

The graph of $K(t_0)$ versus $t_0$ for the compactly supported sampling function given by Eq. (29), in units where $a=1$.

Figure 5

The graph of $K(t_0)$ in four dimensions as a function of $t_0$ for the compactly supported sampling function given by Eq. (56), in units where $a=1$.