[Paper Review] Simplifying Quantum Gravity Calculations

Safi Rafie-Zinedine|arXiv (Cornell University)|Aug 18, 2018

Cosmology and Gravitation Theories14 references5 citations

TL;DR

This thesis simplifies quantum gravity calculations by introducing a generalized field redefinition and parameterized gauge conditions, reducing the number of terms in triple and quadruple graviton vertices from 40 to 4 and from 113 to 12, respectively. The method preserves physical amplitudes while streamlining Feynman rules, verified at tree level and applied to one-loop scalar-graviton scattering with significant computational advantages.

ABSTRACT

The Einstein-Hilbert Lagrangian for gravity is non-renormalizable at loop level. However, it can be treated in the effective field theory framework which means that gravity as an effective theory can be renormalized when a proper expansion of the effective Lagrangian is made. At the same time, the Feynman rules for gravity are very complicated, although the resulting amplitudes do not have the same complications. Therefore, in this thesis we want to simplify the Feynman rules as much as possible by using the most general parameterized gauge condition, adding all possible parameterized total derivative terms and redefining the gravitational, ghosts and scalar fields in a general parameterization way. By choosing the parameters in a specific way, we obtain simplified Feynman rules, especially the triple and quadruple graviton vertices are simplified. In addition, we verify our simplified rules by calculating the amplitudes of scalar-graviton and graviton-graviton scattering at tree level using the simplified and standard Feynman rules. Finally, we show the utility of these simplified rules by calculating some one-loop diagrams for scalar-graviton scattering and comparing to the standard Feynman rules.

Motivation & Objective

• To address the extreme complexity of Feynman rules in quantum gravity, especially for graviton self-interactions.
• To develop a systematic method for simplifying the gravitational vertex structures without altering physical amplitudes.
• To verify the simplification through explicit tree-level scattering amplitudes and one-loop calculations.
• To demonstrate computational efficiency gains in quantum gravity calculations using the simplified rules.

Proposed method

• Employing a general parameterized gauge condition to fix the gravitational field's degrees of freedom.
• Including all possible parameterized total derivative terms to maintain equivalence under field redefinitions.
• Performing a general field redefinition of the metric, gravitons, ghosts, and scalar fields to simplify vertex structures.
• Optimizing parameters to minimize the number of terms in triple and quadruple graviton vertices.
• Using dimensional regularization and Passarino-Veltman reduction for one-loop amplitude calculations.
• Implementing symbolic computation via the FORM program to automate and verify amplitude computations.

Experimental results

Research questions

• RQ1 Can the number of terms in triple and quadruple graviton vertices be significantly reduced without altering physical amplitudes?
• RQ2 Does the simplified Feynman rule set yield identical tree-level scattering amplitudes compared to the standard approach?
• RQ3 Can the simplified rules reduce computational complexity in one-loop scalar-graviton scattering?
• RQ4 What parameter choices in the generalized field redefinition lead to the minimal vertex structure?

Key findings

• The triple graviton vertex is reduced from 40 to 4 terms using the optimized parameterization.
• The quadruple graviton vertex is reduced from 113 to 12 terms through the same simplification procedure.
• Tree-level scalar-graviton and graviton-graviton scattering amplitudes computed with the simplified rules exactly match those from the standard Feynman rules.
• One-loop corrections for scalar-graviton scattering computed with the simplified rules show consistent results with standard methods, confirming equivalence.
• The simplified rules significantly reduce the number of contributing terms in vertex and propagator structures, streamlining both analytical and computational workflows.
• The method is robust and general, applicable to higher-order calculations in effective quantum gravity.