The influence functional is derived for a massive scalar field in the ground state, coupled to a uniformly accelerating DeWitt monopole detector in $D+1$ dimensional Minkowski space. This confirms the local nature of the Unruh effect, and provides an exact solution to the problem of the accelerating detector without invoking a non-standard quantization. A directional detector is presented which is efficiently decohered by the scalar field vacuum, and which illustrates an important difference between the quantum mechanics of inertial and non-inertial frames. From the results of these calculations, some comments are made regarding the possibility of establishing a quantum equivalence principle, so that the Hawking effect might be derived from the Unruh effect.

32 pages