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Inertial and gravitational mass or energy-momentum need not be the same forvirtual quantum states. Separating their roles naturally leads to the gaugetheory of volume-preserving diffeomorphisms of an inner four-dimensional space.The gauge-fixed action and the path integral measure occurring in thegenerating functional for the quantum Green functions of the theory are shownto obey a BRST-type symmetry. The related Zinn-Justin-type equation restrictingthe corresponding quantum effective action is established. This equation limitsthe infinite parts of the quantum effective action to have the same form as thegauge-fixed Lagrangian of the theory proving its spacetime renormalizability.The inner space integrals occurring in the quantum effective action which aredivergent due to the gauge group's infinite volume are shown to beregularizable in a way consistent with the symmetries of the theorydemonstrating as a byproduct that viable quantum gauge field theories are notlimited to finite-dimensional compact gauge groups as is commonly assumed.

A Renormalizable Theory of Quantum Gravity: Renormalization Proof of the Gauge Theory of Volume Preserving Diffeomorphisms