Iterative ultraviolet completion of the quantum-improved metric

Determine whether the deformation defined by the quantum-improved metric can be iterated, through successive mappings from the classical metric to momentum-dependent quantum improvements, to achieve ultraviolet completion.

Background

The paper concludes that the first-order quantum-improved model is not ultraviolet complete because it is restricted by a minimum proper distance and bounded energy-momentum. The authors propose an iterative sequence of metric improvements, represented schematically by a classical metric a(x) mapping to a first quantum improvement g(x,k), then to a further improvement h(x,k), and so on.

The unresolved issue is whether the specific deformation introduced in Eq. (improved-metric) can be repeatedly lifted in this manner while preserving a consistent metric structure and ultimately removing the model’s ultraviolet limitations. Establishing this would provide the ultraviolet-complete extension needed to analyze the ultraviolet regime of the quantum-improved black-hole model.