  In the companion paper we motivated a renormalisation flow on
Osterwalder-Schrader data (OS-data) consisting of 1. a Hilbert space, 2. a
cyclic vacuum and 3. a Hamiltonian annihilating that vacuum. As the name
suggests, the motivation was via the OS reconstruction theorem which allows to
reconstruct the OS data from an OS measure satisfying (a subset of) the OS
axioms, in particular reflection positivity. The guiding principle was to map
the usual Wilsonian path integral renormalisation flow onto a flow of the
corresponding OS data.
  We showed that this induced flow on the OS data has an unwanted feature which
disqualifies the associated coarse grained Hamiltonians from being the
projections of a continuum Hamiltonian onto vectors in the coarse grained
Hilbert space. This motivated the definition of a direct Hamiltonian
renormalisation flow which follows the guiding principle but does not suffer
from the afore mentioned caveat.
  In order to test our proposal, we apply it to the only known completely
solvable model, namely the case of free scalar quantum fields. In this paper we
focus on the Klein Gordon field in two spacetime dimensions and illustrate the
difference between the path integral induced and direct Hamiltonian flow.
Generalisations to more general models in higher dimensions will be discussed
in our companion papers.
