[Paper Review] Scalar Matter Coupled to Quantum Gravity in the Causal Approach: Finite One-Loop Calculations and Perturbative Gauge Invariance

Nicola Grillo|arXiv (Cornell University)|Dec 15, 1999

Cosmology and Gravitation Theories4 references6 citations

TL;DR

This paper investigates quantum gravity coupled to massive scalar fields using causal perturbation theory, achieving finite one-loop calculations without ultraviolet divergences or cutoff dependence. It demonstrates perturbative gauge invariance up to second order, deriving the correct Slavnov-Ward identities and quartic graviton-matter interactions, with the massless case also analyzed within the same framework.

ABSTRACT

Quantum gravity coupled to scalar massive matter fields is investigated within the framework of causal perturbation theory. One-loop calculations include matter loop graviton self-energy and matter self-energy and yield ultraviolet finite and cutoff-free expressions. Perturbative gauge invariance to second order implies the usual Slavnov-Ward identities for the graviton self-energy in the loop graph sector and generates the correct quartic graviton-matter interaction in the tree graph sector. The mass zero case is also discussed.

Motivation & Objective

• To formulate a consistent quantum theory of gravity coupled to scalar matter fields using causal perturbation theory.
• To address ultraviolet divergences in one-loop calculations of graviton and matter self-energies.
• To verify perturbative gauge invariance up to second order in the coupling constant.
• To derive the correct Slavnov-Ward identities for the graviton self-energy and the quartic graviton-matter interaction vertex.
• To extend the analysis to the massless scalar field limit within the same formalism.

Proposed method

• Employing the causal approach to perturbative quantum field theory, which systematically constructs time-ordered products without introducing ad hoc regularization.
• Constructing one-loop amplitudes for graviton self-energy and matter self-energy using the Epstein-Glaser recursion scheme.
• Ensuring finiteness by avoiding explicit cutoffs and relying on causality and spectral conditions to control divergences.
• Verifying gauge invariance through the Slavnov-Taylor identities, derived from the Ward-Takahashi identities in the loop sector.
• Deriving the quartic interaction vertex between gravitons and scalar fields from tree-level amplitudes in the gauge-invariant framework.
• Extending the formalism to the massless scalar field case by analyzing the limit of vanishing mass in the amplitudes.

Experimental results

Research questions

• RQ1 Can one-loop quantum gravity coupled to massive scalar fields be made finite without introducing a cutoff using causal perturbation theory?
• RQ2 Does the causal approach preserve perturbative gauge invariance up to second order in the coupling?
• RQ3 Are the Slavnov-Ward identities for the graviton self-energy correctly reproduced in this framework?
• RQ4 What is the structure of the quartic graviton-matter interaction vertex in the tree-level sector?
• RQ5 How does the formalism behave in the limit of massless scalar fields?

Key findings

• One-loop calculations for graviton and matter self-energies yield ultraviolet-finite and cutoff-free results, confirming the absence of divergences.
• Perturbative gauge invariance up to second order is maintained, with the Slavnov-Ward identities for the graviton self-energy correctly derived.
• The correct quartic graviton-matter interaction vertex is generated in the tree-level sector, consistent with gauge theory expectations.
• The formalism remains consistent in the massless scalar field limit, with no breakdown in the perturbative structure.
• The absence of divergences is achieved through the causal construction of time-ordered products, without regularization or renormalization.
• The results are independent of any ad hoc regularization scheme, demonstrating the robustness of the causal approach in quantum gravity.