# The Quantum Theory of Gravitation: I Quantization of Free Fields - II Proof of Renormalizability

## TL;DR

This paper develops a consistent quantum theory of gravity by quantizing a gauge field equivalent to General Relativity, proving its renormalizability through path integral methods and BRST symmetry.

## Contribution

It introduces a new SO(1,3) gauge field theory for gravity, demonstrating its quantization and renormalizability, advancing the consistent quantum formulation of gravitation.

## Key findings

- Constructed a physical Fock space for free gauge particles.
- Derived a gauge-fixed path integral representation of the quantum effective action.
- Proved the theory's renormalizability via BRST invariance and Zinn-Justin equations.

## Abstract

Quantization of Free Fields: The non-interacting field belonging to a new {\bf SO(1,3)\/} gauge field theory equivalent to General Relativity is canonically quantized in the Lorentz gauge and the physical Fock space for free gauge particles is constructed. To assure both relativistic covariance and positivity of the norm and energy expectation value for physical states restrictions needed in the construction of the physical Fock space are put consequentially on state vectors, and not on the algebra of creation and annihilation operators as usual - alltogether providing the second step in consistently quantizing gravitation Proof of Renormalizability: A new SO(1,3) gauge field theory classically equivalent to General Relativity is quantized and the gauge-fixed path integral representation of the quantum effective action (QEA) is derived. Both the gauge-fixed classical action and the QEA are shown to be invariant under nilpotent BRST variations of the gauge, matter, ghost, antighost and Nakanishi-Lautrup fields defining the theory and a Zinn-Justin equation constraining the QEA is derived. Dimensional analysis and the various linear constraints put on the QEA plus the ones from the non-linear Zinn-Justin equation are deployed to demonstrate full renormalizability such that all infinities appearing in a perturbative expansion of the QEA can be absorbed into the gauge-fixed classical action solely by field renormalizations and coupling redefinitions - alltogether providing the third step in consistently quantizing gravitation