5 - Stress-Energy Tensor and Correlation. Point Separation

Summary

In this chapter we focus on the stress-energy bitensor and its symmetrized product, with two goals: (1) to present the point-separation regularization scheme, and (2) to use it to calculate the noise kernel that is the correlation function of the stress-energy bitensor and explore its properties. In the first part we introduce the necessary properties and geometric tools for analyzing bitensors, geometric objects that have support at two separate spacetime points. The second part presents the point-separation method for regularizing the ultraviolet divergences of the stress-energy tensor for quantum fields in a general curved spacetime. In the third part we derive a formal expression for the noise kernel in terms of the higher order covariant derivatives of the Green functions taken at two separate points. One simple yet important fact we show is that for a massless conformal field the trace of the noise kernel identically vanishes. In the fourth part we calculate the noise kernel for a conformal field in de Sitter space, both in the conformal Bunch–Davies vacuum and in the static Gibbons–Hawking vacuum. These results are useful for treating the backreaction and fluctuation effects of quantum fields.