Krein space regularization and the lagrange multiplier approach to quantum gravity equivalence demonstration for the einstein-hilbert action

We establish a rigorous correspondence between the Krein space regularization method and the Lagrange multiplier (LM) approach to quantum gravity, extending both to the full Einstein-Hilbert action within the Unified Standard Model with Emergent Gravity-Effective Field Theory (USMEG-EFT) framework. Through explicit calculation of the one-loop effective action using the heat kernel expansion in the LM framework and the modified propagator structure in Krein quantization, we demonstrate that both methods yield identical finite results. The one-loop effective action takes the form Formula: see text, with coefficients derived from the Seeley-DeWitt heat kernel expansion. The parameter correspondence Formula: see text emerges naturally from the regularization structure, with both encoding the finite domain of validity characteristic of an effective field theory. We demonstrate that both methods independently restrict quantum gravitational corrections to one-loop order through structurally isomorphic cancellation mechanisms: opposite-sign propagators in the LM formalism correspond to negative-norm states in Krein space, with higher-loop contributions vanishing identically via binomial summation. We provide detailed analysis of why standard Hilbert space quantization fails due to positive-definiteness constraints, and how the indefinite metric structure of Krein spaces enables systematic divergence cancellation while preserving unitarity in the physical sector. This equivalence provides mutual theoretical support for the USMEG-EFT framework representing the first successful unification of quantum gravity with the Standard Model.