Matter coupled to 3d Quantum Gravity: One-loop Unitarity

TL;DR

This paper investigates whether matter-coupled 3d quantum gravity theories are unitary by analyzing one-loop corrections, revealing the need for an additional propagator term to preserve unitarity, which impacts quantum gravity models.

Contribution

It explicitly checks unitarity at one-loop in matter-coupled 3d quantum gravity and identifies a necessary extra term in the propagator for unitarity preservation.

Findings

Unitarity requires an extra massless mode term in the propagator.

The extra term corresponds to a representation with zero Plancherel measure.

Implications for the inclusion of matter in quantum gravity models.

Abstract

We expect quantum field theories for matter to acquire intricate corrections due to their coupling to quantum fluctuations of the gravitational field. This can be precisely worked out in 3d quantum gravity: after integrating out quantum gravity, matter fields are effectively described as noncommutative quantum field theories, with quantum-deformed Lorentz symmetries. An open question remains: Are such theories unitary or not? On the one hand, since these are effective field theories obtained after integrating out high energy degrees of freedom, we may expect the loss of unitarity. On the other hand, as rigorously defined field theories built with Lorentz symmetries and standing on their own, we naturally expect the conservation of unitarity. In an effort to settle this issue, we explicitly check unitarity for a scalar field at one-loop level in both Euclidean and Lorentzian space-time signatures. We find that unitarity requires adding an extra-term to the propagator of the noncommutative theory, corresponding to a massless mode and given by a representation with vanishing Plancherel measure, thus usually ignored in spinfoam path integrals for quantum gravity. This indicates that the inclusion of matter in spinfoam models, and more generally in quantum gravity, might be more subtle than previously thought.