Space and Time Averaged Quantum Stress Tensor Fluctuations

We extend previous work on the numerical diagonalization of quantum stress tensor operators in the Minkowski vacuum state, which considered operators averaged in a finite time interval, to operators averaged in a finite spacetime region. Since real experiments occur over finite volumes and durations, physically meaningful fluctuations may be obtained from stress tensor operators averaged by compactly supported sampling functions in space and time. The direct diagonalization, via a Bogoliubov transformation, gives the eigenvalues and the probabilities of measuring those eigenvalues in the vacuum state, from which the underlying probability distribution can be constructed. For the normal-ordered square of the time derivative of a massless scalar field in a spherical cavity with finite degrees of freedom, analysis of the tails of these distributions confirms previous results based on the analytical treatment of the high moments. We find that the probability of large vacuum fluctuations is reduced when spatial averaging is included, but the tail still decreases more slowly than exponentially as the magnitude of the measured eigenvalues increases, suggesting vacuum fluctuations may not always be subdominant to thermal fluctuations and opening up the possibility of experimental observation under the right conditions.