Hamiltonian renormalization III. Renormalisation flow of 1+1 dimensional free scalar fields: properties

This is the third paper in a series of four in which a renormalisation flow is introduced which acts directly on the Osterwalder-Schrader data (OS data) without recourse to a path integral. Here the OS data consist of a Hilbert space, a cyclic vacuum vector therein and a Hamiltonian annihilating the vacuum which can be obtained from an OS measure, that is a measure respecting (a subset of) the OS axioms. In the previous paper we successfully tested our proposal for the twodimensional massive Klein-Gordon model, that is, we could confirm that our framework finds the correct fixed point starting from a natural initial naive discretisation of the finite resolution Hamiltonians, in particular the underlying Laplacian on a lattice, and a natural coarse graining map that drives the renormalisation flow. However, several questions remained unanswered. How generic can the initial discretisation and the coarse graining map be in order that the fixed point is not changed or is at least not lost, in other words, how universal is the fixed point structure? Is the fixed point in fact stable, that is, is the fixed point actually a limit of the renormalisation sequence? We will address these questions in the present paper.