  Pair production in a constant electric field is closely analogous to bubble
nucleation in a false vacuum. The classical trajectories of the pairs are
Lorentz invariant, but it appears that this invariance should be broken by the
nucleation process. Here, we use a model detector, consisting of other
particles interacting with the pairs, to investigate how pair production is
seen by different Lorentzian observers. We focus on the idealized situation
where a constant external electric field is present for an infinitely long
time, and we consider the in-vacuum state for a charged scalar field that
describes the nucleating pairs. The in-vacuum is defined in terms of modes
which are positive frequency in the remote past. Even though the construction
uses a particular reference frame and a gauge where the vector potential is
time dependent, we show explicitly that the resulting quantum state is Lorentz
invariant. We then introduce a "detector" particle which interacts with the
nucleated pairs, and show that all Lorentzian observers will see the particles
and antiparticles nucleating preferentially at rest in the detector's rest
frame. Similar conclusions are expected to apply to bubble nucleation in a
sufficiently long lived vacuum. We also comment on certain unphysical aspects
of the Lorentz invariant in-vacuum, associated with the fact that it contains
an infinite density of particles. This can be easily remedied by considering
Lorentz breaking initial conditions.
