Hamiltonian renormalization. VI. Parametrized field theory on the cylinder - Thiemann, Thomas; Zwicknagel, Ernst-Albrecht (Physical Review D, 2023)
Hamiltonian renormalization, as defined within this series of works, was derived from covariant Wilson renormalization via Osterwalder-Schrader reconstruction. As such it directly applies to quantum field theory (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 renormalization scheme is to remove quantization ambiguities of Hamiltonians in interacting QFT that remain even after UV and IR regulators are removed as it happens in highly nonlinear QFT such as quantum gravity. Also, while not derived for that case, the renormalization flow formulas can without change also be applied to QFT without a single true Hamiltonian but rather an infinite number of Hamiltonian constraints. In that case a number of interesting questions arise: (1) Does the flow reach the correct fixed point also for an infinite number of "Hamiltonians"simultaneously (2) As the constraints are labeled by test functions, which in the presence of a regulator are typically regularized (discretized and of compact support), how do those test functions react to the flow (3) Does the quantum constraint algebra, which in the presence of a regulator is expected to be anomalous, close at the fixed point These questions 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 parametrized 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 conformal field theories to which the results of this paper apply verbatim. The central result of our investigation is that the finite resolution (discretized) constraint algebras typically do not close, that there is not necessarily anything wrong with that, and that anomaly freeness of the continuum algebra is encoded in the convergence behavior of the renormalization flow.
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