Viewing gravitational energy momentum as equal by observation, but different in essence from inertial energy-momentum requires two different symmetries to account for their independent conservations—spacetime and inner translation invariance. Gauging the latter a generalization of non-Abelian gauge theories of compact Lie groups is developed resulting in the gauge theory of the non-compact group of volume-preserving diffeomorphisms of an inner Minkowski space M⁴. As usual the gauging requires the introduction of a covariant derivative, a gauge field and a field strength operator. An invariant and minimal gauge field Lagrangian is derived. The classical field dynamics and the conservation laws for the new gauge theory are developed. Finally, the theory’s Hamiltonian in the axial gauge is expressed by two times six unconstrained independent canonical variables obeying the usual Poisson brackets and the positivity of the Hamiltonian is related to a condition on the support of the gauge fields.

Viewing gravitational energy-momentum as equal by observation, but different in essence from inertial energymomentum naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space which can describe gravitation at the classical level. This theory is quantized in the path integral formalism starting with a non-covariant Hamiltonian formulation with unconstrained canonical field variables and a manifestly positive Hamiltonian. The relevant path integral measure and weight are then brought into a Lorentz- and gauge-covariant form allowing to express correlation functions—applying the De Witt-Faddeev-Popov approach—in any meaningful gauge. Next the Feynman rules are developed and the quantum effective action at one loop in a background field approach is renormalized which results in an asymptotically free theory without presence of other fields and in a theory without asymptotic freedom including the Standard Model (SM) fields. Finally the BRST apparatus is developed as preparation for the renormalizability proof to all orders and a sketch of this proof is given.

The different roles and natures of spacetime appearing in a quantum field theory and in classical physics are analyzed implying that a quantum theory of gravitation is not necessarily a quantum theory of curved spacetime. Developing an alternative approach to quantum gravity starts with the postulate that inertial energy-momentum and gravitational energy-momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the quantum gauge field theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The classical limit of this theory coupled to a quantized scalar field is derived for an on-shell particle where inertial energy-momentum and gravitational energy-momentum coincide. In that process the symmetry under volume-preserving diffeomorphisms disappears and a new symmetry group emerges: the group of coordinate transformations of four-dimensional spacetime and with it General Relativity coupled to a classical relativistic point particle.

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