  In the context of an extended General Relativity theory with boundary terms
included, we introduce a new nonlinear quantum algebra involving a quantum
differential operator, with the aim to calculate quantum geometric alterations
when a particle is created in the vicinity of a Schwarzschild black-hole by the
Hawking radiation mechanism. The boundary terms in the varied action give rise
to modifications in the geometric background, which are investigated by
analyzing the metric tensor and the Ricci curvature within the framework of a
renormalized quantum theory of gravity.
