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Hamiltonian renormalisation VI: Parametrised field theory on the cylinder

Hamiltonian Renormalisation, as defined within this series of works, was derived from covariant Wilson renormalisation via Osterwalder-Schrader reconstruction. As such it directly applies to QFT with a true (physical) Hamiltonian bounded from below. The validity of the scheme was positively tested for free QFT in any dimension with or without Abelian gauge symmetries of Yang-Mills type. The aim of this Hamiltonian renormalisation scheme is to remove quantisation ambiguities of Hamiltonians in interacting QFT that remain even after UV and IR regulators are removed as it happens in highly non-linear QFT such as quantum gravity. While not derived for that case, the renormalisation flow formulae can without change also be applied to QFT without a single true Hamiltonian but rather an infinite number of Hamiltonian constraints. Then a central question is how the constraint algebra reacts to the renormalisation flow. This question should ultimately be addressed in quantum gravity. Before one considers this interacting, constrained QFT it is well motivated to consider a free, constrained QFT where the fixed point is explicitly known. In this paper we therefore address the case of parametrised field theory for which the quantum constraint algebra coincides simultaneously with the hypersurface deformation algebra of quantum gravity (or any other generally covariant theory) and the Virasoro algebra of free, closed, bosonic string theory or other CFT to which the results of this paper apply verbatim. The central result of our investigation is that finite resolution (discretised) constraint algebras {\it must not close} and that anomaly freeness of the continuum algebra is encoded in the convergence behaviour of the renormalisation flow.