Late time behaviors of quantized scalar fields in the Unruh state in two dimensional spacetimes

It is shown that the late time behaviors for some of the modes of a scalar field in the Unruh state in two-dimensions depend on whether these modes diverge in the zero frequency or infrared limit. There is an infrared divergence for the massless minimally coupled scalar field. In this case the modes that are positive frequency with respect to the Kruskal time coordinate on the past horizon approach constant values at late times and, as has been shown previously, the two-point function grows linearly in time if the points are split in the spatial direction. Two cases are considered where there are no infrared divergences in the mode functions. One is the case of a delta function potential and the other a massive scalar field in Schwarzschild-de Sitter spacetime. In both cases the mode functions that are positive frequency with respect to the Kruskal time on the past horizon(s) vanish in the late time limit. For the delta function potential it is also shown that the two-point function does not grow linearly in time.