The quantization of unimodular gravity and the cosmological constant problem (0904.4841v1)

Abstract: A quantization of unimodular gravity is described, which results in a quantum effective action which is also unimodular, ie a function of a metric with fixed determinant. A consequence is that contributions to the energy momentum tensor of the form of the metric times a spacetime constant, whether classical or quantum, are not sources of curvature in the equations of motion derived from the quantum effective action. This solves the first cosmological constant problem, which is suppressing the enormous contributions to the cosmological constant coming from quantum corrections. We discuss several forms of uniodular gravity and put two of them, including one proposed by Henneaux and Teitelboim, in constrained Hamiltonian form. The path integral is constructed from the latter. Furthermore, the second cosmological constant problem, which is why the measured value is so small, is also addressed by this theory. We argue that a mechanism first proposed by Ng and van Dam for suppressing the cosmological constant by quantum effects obtains at the semiclassical level.

• The paper proposes unimodular gravity's quantization as a novel framework to decouple vacuum energy from spacetime curvature.
• It employs Hamiltonian dynamics and a tailored path integral construction to preserve the unimodular constraint during quantization.
• The study discusses potential resolutions for dark energy puzzles and the broader implications for quantum gravity research.

Quantization of Unimodular Gravity and the Cosmological Constant Problems

The paper "The quantization of unimodular gravity and the cosmological constant problems" by Lee Smolin offers a detailed examination of unimodular gravity as a framework for addressing the notorious cosmological constant problems. The cosmological constant, symbolized as Λ, poses several persistent puzzles within the field of theoretical physics: its unexpectedly small value compared to quantum predictions, the bafflingly precise cancellation of massive symmetry-breaking contributions, and the enigmatic coincidence related to matter density. Smolin's study elucidates theoretical approaches and quantization processes within unimodular gravity that could potentially reconcile these issues.

Unimodular Gravity Overview

Unimodular gravity represents an alteration of Einstein's general relativity through a constraint imposed on the determinant of the spacetime metric, effectively reducing the gauge symmetry. The cosmological constant within unimodular gravity arises as an integration constant rather than a fixed parameter in the Lagrangian. This seemingly subtle modification decouples vacuum energy from spacetime curvature, thereby offering a means to address the "first cosmological constant problem"—the vast contributions to Λ expected from quantum corrections.

Two Key Formulations and Hamiltonian Dynamics

The paper explores variations in unimodular gravity through two primary formulations: the original unimodular constraint and the Henneaux-Teitelboim approach. The original form directly imbues the action with a fixed metric determinant while maintaining trace-free equations of motion. In contrast, Henneaux and Teitelboim introduced auxiliary fields, allowing full diffeomorphism invariance while deriving the unimodular condition as an equation of motion.

Smolin further explores the Hamiltonian dynamics associated with these formulations. Here, constrained Hamiltonian dynamics becomes vital for constructing the quantum theory. For Henneaux-Teitelboim unimodular gravity, this involves additional constraints to preserve gauge invariance during path integral quantization, a crucial component for ensuring the unimodular condition persists in the quantum effective action.

Path Integral Construction

A notable contribution of this paper is the construction of a path integral that maintains the unimodular condition through quantization. Smolin outlines the construction for the Henneaux-Teitelboim version, demonstrating that quantum corrections do not disrupt the unimodular nature of the metric—a pivotal step confirming that the quantum effective action remains a function of the unimodular metric alone. This intricacy not only assures the preservation of unimodularity post-quantization but also addresses the first cosmological constant problem by ensuring vacuum energy fluctuations do not manifest within spacetime curvature.

The Second and Third Cosmological Constant Problems

The second cosmological constant problem—concerned with the minuscule yet nonzero value of dark energy—receives attention through the lens of quantum cosmology. The paper references the proposal by Ng and van Dam which conjectures that, in the semiclassical approach, solutions of a path integral variant suggest that cosmological constant contributions nullify. Smolin extends this hypothesis, arguing that considerations within unimodular gravity echo this suppression proposal.

The third problem relates to the coincidence of matter density and dark energy and is touched upon as well. While solutions are speculative and hinge on broad assumptions, Smolin outlines potential avenues for further exploration and highlights the necessity for developing precise measurement theories in quantum cosmology.

Concluding Remarks and Future Directions

The implications of exploring unimodular gravity offer intriguing possibilities. Smolin emphasizes the theoretical validity of these findings, urging confirmations in frameworks that solidly tackle quantum gravity's ultraviolet challenges. He proposes further studies integrating unimodular principles into causal dynamical triangulations, spin foam models, and string theory to solidify these theoretical solutions.

In conclusion, Smolin's work represents an integral contribution to the dialogue surrounding the cosmological constant problems. By providing a structured pathway through unimodular gravity's quantization, this paper invites future theoretical advancements and cross-disciplinary applications, addressing long-standing puzzles in theoretical physics.