Quantum Gravity Research Considers Measures Invariant under Diffeomorphisms, Potentially Enabling Renormalization of the Stelle Model

Quantum gravity, the elusive theory uniting quantum mechanics and general relativity, presents profound challenges in defining meaningful measurements, and researchers continually explore potential solutions. O. P. Santillán investigates possible measures within this framework, analysing options that remain consistent under changes in coordinate systems, but also considering those that do not. This approach allows for the potential compensation of inconsistencies through mathematical adjustments, opening avenues for extending established quantum identities to the complex environment of curved spacetime. The work focuses specifically on the Stelle model, a theory known for its mathematical consistency in flat space, and seeks to determine if similar properties can be maintained when applied to the more challenging context of curved spacetime, potentially offering insights into the renormalizability of quantum gravity theories. Quantum Gravity, Anomalies and Effective Actions This work presents a comprehensive exploration of quantum gravity, focusing on the construction of effective actions and the handling of anomalies, which represent violations of expected symmetries at the quantum level. It delves into the core challenges of quantizing gravity and the techniques used to overcome them, employing the Vilkovisky-DeWitt formalism, a technique for constructing effective actions even in the presence of anomalies, often involving auxiliary fields and specific regularization procedures. The research investigates path integrals, emphasizing the importance of the measure used in these calculations and how it affects diffeomorphism invariance, the symmetry under coordinate transformations. Various regularization methods, including zeta regularization, heat kernel methods, and dimensional regularization, are explored as ways to manage the infinities that arise in quantum field theory. The BRST formalism, which introduces ghost fields to ensure the unitarity of the theory, is also examined in detail. The text is a highly technical review intended for graduate students and researchers already familiar with quantum field theory, differential geometry, and advanced mathematics, covering a vast amount of material and demonstrating the author’s deep knowledge of the existing literature. Pauli-Villars Regularization of Stelle Model Anomalies Scientists investigated measures used in quantum field theory, considering both those that remain unchanged and those that change under coordinate transformations, revealing that anomalies can be compensated for by redefining terms within the theoretical model. Researchers specifically examined the Stelle model, a theoretical framework known for its renormalizability in flat space, but whose behavior in curved spacetime remains an open question. To address the challenges of regularization, the team employed the Pauli-Villars method, introducing counter terms into the effective action to manage anomalous behavior. This approach demonstrates that a measure does not need to be perfectly invariant under fundamental symmetries; any deviations can potentially be absorbed by appropriate counter term redefinitions, ensuring the final result, specifically the S matrix describing particle interactions, exhibits the desired symmetry. Scientists investigated several proposed measures for quantizing scalar fields in curved spacetime, including those put forward by Liouville, Toms, Hawking, and Fujikawa, demonstrating that these measures can be related through careful redefinitions of variables and counter terms.

The team introduced a generalized measure, allowing for arbitrary parameters that control the transformation of variables, and demonstrated that anomalies arising from this generalized measure could be consistently absorbed into counter terms.

Measure Choice Impacts Renormalizability of Stelle Gravity Scientists rigorously analyzed path integral measures, investigating both those invariant and non-invariant under diffeomorphisms, demonstrating that while invariance is desirable, a non-invariant measure can be acceptable if its anomaly is precisely compensated by counter term redefinitions within the model. This allows for flexibility in quantization, particularly when dealing with gravity, where maintaining both gauge and coordinate invariance simultaneously can be challenging. Researchers focused on the Stelle gravity model, known to be renormalizable in flat space, but whose renormalizability in curved space remains an open question, revealing that the choice of measure significantly impacts the consistency of the theory. A measure exhibiting anomalous behavior can still yield physically meaningful results if properly accounted for through counter terms, challenging the strict requirement of measure invariance and opening possibilities for alternative quantization approaches. Investigations into various measures, including those proposed by Liouville, Toms, Hawking, and Fujikawa, demonstrate a fundamental tension between maintaining unitarity and canceling loop divergences, and preserving general coordinate invariance. This suggests a trade-off between mathematical consistency and physical principles, highlighting the fundamental distinction between the Liouville and Hawking-Fujikawa measures.

Curved Spacetime Measures and Quantum Field Consistency The research establishes a framework for analyzing measures within the context of quantum field theory, particularly when dealing with curved spacetime. Scientists demonstrated that seemingly divergent terms arising from non-invariant measures can be compensated by redefinitions within the theoretical model, allowing for a consistent treatment of quantum fields in curved backgrounds.

The team meticulously calculated factors arising from coordinate transformations, revealing that several terms previously considered non-trivial evaluate to unity under careful analysis, streamlining calculations and providing a clearer understanding of the relationships between different coordinate systems. The work culminates in a differential equation governing a specific function, which, when satisfied, ensures the consistency of the calculated measures and facilitates further investigation into quantum field theory in curved spacetime. Researchers acknowledge that the precise form of the governing function requires further investigation, and that their results offer a different perspective on existing calculations, potentially resolving discrepancies found in other studies. 👉 More information 🗞 About possible measures in Quantum Gravity 🧠 ArXiv: https://arxiv.org/abs/2512.09191 Tags: