  The perennial formalism is applied to the real, massive Klein-Gordon field on
a globally-hyperbolic background space-time with compact Cauchy hypersurfaces.
The parametrized form of this system is taken over from the accompanying paper.
Two different algebras ${\cal S}_{\text{can}}$ and ${\cal S}_{\text{loc}}$ of
elementary perennials are constructed. The elements of ${\cal S}_{\text{can}}$
correspond to the usual creation and annihilation operators for particle modes
of the quantum field theory, whereas those of ${\cal S}_{\text{loc}}$ are the
smeared fields. Both are shown to have the structure of a Heisenberg algebra,
and the corresponding Heisenberg groups are described. Time evolution is
constructed using transversal surfaces and time shifts in the phase space.
Important roles are played by the transversal surfaces associated with
embeddings of the Cauchy hypersurface in the space-time, and by the time shifts
that are generated by space-time isometries. The automorphisms of the algebras
generated by this particular type of time shift are calculated explicitly.
