  The massive, real scalar field described by the Klein-Gordon equation in one
spatial dimension is the most elementary example of a bosonic quantum field
theory, and has been investigated for many decades either as a simple academic
theory or as a realistic emergent many-body theory in low-dimensional systems.
Despite this, the space and time behavior of its propagators have rarely been
in the foreground, and although exact results are known, there remain gaps in
the description and a lack of an in-depth physical analysis. The aim of this
paper is to address the deficits by providing a comprehensive discussion of the
results, and to show that this old theory still allows for several new results
and insights. To start, known results are rederived in full detail, with an
added discussion on how exactly space and time variables need to be extended to
complex values to ensure analyticity throughout spacetime. This procedure shows
also how singularities on the lightcone need to be regularized to remain
compatible with the analyticity and the physical limit of a vanishing mass. An
extension to nonzero temperatures is provided by considering the contact of the
field to a nonrelativistic thermal reservoir, such as is necessary for emerging
field theories in condensed matter systems. Subsequently, it is shown that the
transient, short spacetime propagation can be understood in the context of the
modern development of a generalized Gibbs ensemble, which describes a massless
theory with an effective temperature that is set by the Klein-Gordon mass and
the physical temperature. Finally, an approximation scheme is presented that
captures the non-trivial mass dependence of the propagators throughout all
spacetime but involves only elementary functions.
