Wormhole Renormalization: The gravitational path integral, holography, and a gauge group for topology change (2407.20324v1)

Abstract: We study the Factorization Paradox from the bottom up by adapting methods from perturbative renormalization. Just as quantum field theories are plagued with loop divergences that need to be cancelled systematically by introducing counterterms, gravitational path integrals are plagued by wormhole contributions that spoil the factorization of the holographic dual. These wormholes must be cancelled by some stringy effects in a UV complete, holographic theory of quantum gravity. In a simple model of two-dimensional topological gravity, we outline a gravitational analog of the recursive BPHZ procedure in order to systematically introduce ``counter-wormholes" which parametrize the unknown stringy effects that lead to factorization. Underlying this procedure is a Hopf algebra of symmetries which is analogous to the Connes--Kreimer Hopf algebra underlying perturbative renormalization. The group dual to this Hopf algebra acts to reorganize contributions from spacetimes with distinct topology, and can be seen as a gauge group relating various equivalent ways of constructing a factorizing gravitational path integral.

• The paper introduces counter-wormholes analogous to QFT counterterms to systematically cancel wormhole contributions that break factorization.
• It employs perturbative renormalization techniques and Hopf algebra structures to analyze topology change in gravitational path integrals.
• This innovative approach clarifies the Factorization Paradox and informs future models of quantum gravity and effective field theories.

Understanding Wormhole Renormalization Through Gravitational Path Integrals

This paper discusses a novel approach to addressing the Factorization Paradox in quantum gravity. The authors propose an analogy with perturbative renormalization to understand the contributions of spacetime wormholes within gravitational path integrals. By introducing a conceptual framework similar to the BPHZ procedure in quantum field theory (QFT), the paper outlines a model that systematically factors out non-local contributions from gravitational path integrals using what are termed "counter-wormholes."

Key Concepts and Structure

The paper begins by addressing the Factorization Paradox, which arises from the conflict between the holographic principle and the sum over topologies in gravitational path integrals. This paradox highlights the issue that quantum gravity, when viewed through the lens of holography, should factorize correlation functions according to a local boundary QFT, yet wormhole contributions in the gravitational path integral often prevent this.

The authors draw from techniques in perturbative renormalization, where QFTs cancel divergences through systematic counterterms. In this gravitational setting, the counter-wormholes are analogous to counterterms: they systematically 'cancel' non-factorizing contributions from wormholes.

Marolf-Maxfield Model and Wormhole Contributions

The paper uses a simplified model introduced by Marolf and Maxfield, focusing on a two-dimensional topological gravity setup to investigate these phenomena. The model demonstrates that integrating over manifold topologies does not lead to factorization due to wormhole contributions. In this context, the wormholes act like non-local, interconnected processes that affect the entire gravitational path integral, much like loop diagrams in QFTs affecting their respective computations.

Counter-Wormholes and Hopf Algebra

To resolve the non-factorization issue, the authors propose the introduction of counter-wormholes, which act to neutralize the unwanted contributions of the actual wormholes. Their methodology follows the recursive BPHZ procedure used in the renormalization of QFTs, adapted to relate these counter-wormholes within the gravitational framework. This is underpinned by a Hopf algebra structure analogous to the Connes-Kreimer Hopf algebra seen in QFT, specifically utilizing a version often referred to as the Faà di Bruno Hopf algebra.

The core advancement here is recognizing how counter-wormholes can be systematically enumerated and handled through these algebraic structures, leading to restoration of factorization within the path integrals of this gravitational model.

Theoretical and Practical Implications

The theoretical implication of this work is profound as it provides a structured way to handle topology change in quantum gravity, potentially offering insight into more complete models of quantum gravity that encompass string theory. Practically, this could inform how effective field theories are constructed and understood in a gravitational context, hinting at a means to predict behaviors that avoid contradictions with established physical laws in quantum gravity scenarios.

Future Directions

The paper opens several avenues for future research, notably: exploring more detailed gravitational settings beyond topological models; understanding how this framework could extend to Euclidean gravity or string theories; and considering the implications of this algebraic approach in higher-dimensional theories where wormholes and topology change are vital considerations.

In summary, this paper presents a compelling, algebraically-grounded approach to resolving a key problem in quantum gravity. By employing counter-wormholes analogous to counterterms in QFT, the authors pave the way for deeper investigations into the reconciliation of gravity with quantum mechanics, particularly in the context of holographic principles.