Towards exploring features of Hamiltonian renormalisation relevant for quantum gravity

Abstract We consider the Hamiltonian renormalisation group flow of discretised one-dimensional physical theories. In particular, we investigate influence choice different embedding maps has on RG and resulting continuum limit, show in which sense they are, are not equivalent as furthermore elucidating interplay algebras operators satisfy, both discrete continuum. Further, propose preferred prescriptions for operator guaranteeing to arrive at algebraic relations continuum, if suitable extension properties assumed. Finally, introduce a weaker form distributional equivalence, how unitarily inequivalent limits, arise due maps, can still be weakly that sense. expect these results have application defining an Loop Quantum Gravity.