  A possible avenue towards a non-perturbative Quantum Field Theory (QFT) on
Minkowski space is the constructive approach which employs the Euclidian path
integral formulation, in the presence of both ultraviolet (UV) and infrared
(IR) regulators, as starting point. The UV regulator is to be taken away by
renormalisation group techniques which in case of success leads to a measure on
the space of generalised Euclidian fields in finite volume. The IR regulator
corresponds to the thermodynamic limit of the system in the statistical physics
sense. If the resulting measure obeys the Osterwalder-Schrader axioms, the
actual QFT on Minkowski space is then obtained by Osterwalder-Schrader
reconstruction. In this work we study the question whether it is possible to
reformulate the renormalisation group non-perturbatively directly at the
operator (Hamiltonian) level. Hamiltonian renormalisation would be the natural
route to follow if one had easier access to an interacting Hamiltonian operator
rather than to a path integral, at least in the presence of UV and/or IR
cut-off, which is generically the case in complicated gauge theories such as
General Relativity. Our guiding principle for the definition of the direct
Hamiltonian renormalisation group is that it results in the same continuum
theory as the covariant (path integral) renormalisation group. In order to
achieve this, we invert the Osterwalder-Schrader reconstruction, which may be
called Osterwalder-Schrader construction of a Wiener measure from the
underlying Hamiltonian theory. The resulting correspondence between reflection
positive measures and Osterwalder-Schrader data consisting of a Hilbert space,
a Hamiltonian and a ground state vector allows us to monitor the effect of the
renormalisation flow of measures in terms of their Osterwalder-Schrader data
which motivates a natural direct Hamiltonian renormalisation scheme.
