[Paper Review] III - Conservation of Gravitational Energy Momentum and Renormalizable Quantum Theory of Gravitation

C. Wiesendanger|arXiv (Cornell University)|Mar 5, 2011

Quantum and Classical Electrodynamics8 references4 citations

TL;DR

This paper proposes a renormalizable quantum theory of gravity based on a gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space, treating gravitational and inertial energy-momentum as distinct conserved quantities. Using a path integral quantization with a manifestly positive Hamiltonian and Lorentz- and gauge-covariant reformulation, the theory achieves one-loop renormalization and asymptotic freedom without additional fields, while including Standard Model fields breaks asymptotic freedom, with BRST symmetry enabling a full renormalizability proof to all orders.

ABSTRACT

Viewing gravitational energy-momentum as equal by observation, but different in essence from inertial energy-momentum naturally leads to the gauge theory of volume-preserving diffeormorphisms 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.

Motivation & Objective

• To develop a consistent quantum field theory of gravity by treating gravitational energy-momentum as fundamentally distinct from inertial energy-momentum, despite their observed numerical equality.
• To construct a gauge theory based on volume-preserving diffeomorphisms of an inner Minkowski space, generalizing Yang-Mills theory to non-compact internal symmetries.
• To achieve a unitary, Lorentz- and gauge-covariant quantum field theory with a positive Hamiltonian and finite renormalization via a path integral approach.
• To demonstrate one-loop renormalizability and asymptotic freedom in the absence of Standard Model fields, and identify the breakdown of asymptotic freedom when SM fields are included.
• To develop the BRST formalism as a foundation for proving renormalizability to all orders in the theory.

Proposed method

• Starting from a non-covariant Hamiltonian formulation with unconstrained canonical variables and a manifestly positive Hamiltonian, the theory ensures a positive-definite Hilbert space and unitary evolution.
• Reformulating the path integral measure and weight into a Lorentz- and gauge-covariant form enables covariant correlation function calculations in any meaningful gauge.
• Applying the DeWitt-Faddeev-Popov method to handle gauge fixing, particularly in the Minkowski-plus-axial gauge, ensures ghost-free and unitary amplitudes.
• Deriving Feynman rules from the covariant path integral allows perturbative computation of quantum corrections in the background field method.
• Computing the one-loop effective action using dimensional regularization with an inner space cutoff, preserving inner scale invariance, enables finite renormalization.
• Constructing the BRST symmetry structure prepares the ground for a non-perturbative proof of renormalizability to all orders in the theory.

Experimental results

Research questions

• RQ1 Can a quantum theory of gravity be constructed by treating gravitational and inertial energy-momentum as distinct conserved quantities arising from different symmetries?
• RQ2 How can a gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space be consistently quantized while preserving Lorentz and gauge invariance?
• RQ3 Does the resulting quantum theory remain renormalizable at one loop, and what determines its asymptotic behavior in the presence or absence of Standard Model fields?
• RQ4 Can a consistent regularization scheme be devised that respects inner scale invariance and avoids divergences from the non-compact gauge group?
• RQ5 Is the BRST formalism sufficient to prove renormalizability to all orders in this non-compact gauge theory?

Key findings

• The one-loop quantum effective action is renormalizable using dimensional regularization that respects inner scale invariance, ensuring a finite and unique renormalization procedure.
• In the absence of Standard Model fields, the theory is asymptotically free, with a divergent contribution to the effective action proportional to $\frac{1}{6}\Omega^\Lambda_1 \frac{1}{\Lambda^2} \int F_{\mu\nu}^\alpha F^{\mu\nu}_\alpha$.
• Including a complex scalar doublet contributes a divergent term with coefficient $+\frac{1}{6}\Omega^\Lambda_1 \frac{1}{\Lambda^2} \int F_{\mu\nu}^\alpha F^{\mu\nu}_\alpha$, which opposes asymptotic freedom.
• A single chiral Dirac fermion contributes half the value of a complex scalar doublet, also working against asymptotic freedom.
• The BRST symmetry is successfully constructed, providing the necessary framework for proving renormalizability to all orders in the theory.
• The path integral formulation, after covariant reorganization, yields a unitary, ghost-free, and Lorentz-covariant quantum field theory with a positive energy operator.