Quantum (quadratic) gravity: replacing the massive tensor ghost with an inverted harmonic oscillator-like instability

Abstract

The quadratic theory of gravity is the unique renormalizable theory of quantum gravity in 4 dimensions, as proved by K. S. Stelle in 1977. Over the decades, the theory has been understood to contain a massive tensor ghost, and several attempts have been made to evade its adverse effects by proposing new quantization prescriptions and interpretations. In this paper, we show that the additional spin--2 of quadratic gravity can be turned into a healthy inverted harmonic oscillator (IHO)-like instability, which can be quantized consistently with direct-sum quantum field theory (DQFT), which incorporates geometric superselection sectors. Such modes possess a well-defined quantum description yet do not admit a particle interpretation and are not part of the asymptotic spectrum, being characterized by hyperbolic evolution and spacelike momentum support. We argue that, as a consequence, the extra spin--2 degree of freedom remains off-shell and effectively decoupled from ordinary matter fields, avoiding unitarity violations in observable processes. We argue that this IHO instability is a prevalent feature of fundamental physics, whether it concerns quantum fields on curved spacetimes or the Higgs $\mathbb{Z}_2$ symmetry breaking in the Standard Model of particle physics. Thus, we demonstrate that our new understanding of quadratic gravity offers a fundamental pathway to quantum gravity and a safe beginning for the Universe. Furthermore, we derive key observational predictions of this construction in the view of primordial gravitational waves with new bounds on the tensor-to-scalar ratio and the parity asymmetric features on the large angular scales.

Figures (4)

• Figure 1: Phase space of the IHO (left panel) and the dual IHO (right panel) representing doubly degenerate positive and negative energy solutions in \ref{['sol2BK']}. The negative energy trajectories are given by $Q>0,\, P<0$ and $Q<0,\, P>0$ whereas the positive energy trajectories are $Q>0,\,P>0$ and $Q<0,\,P<0$. These double degenerate trajectories are related by \ref{['discreteTPQ']} whereas the positive and negative energy regions are related by \ref{['BKPE']}.
• Figure 2: The figure illustrates a new formulation of the quantum harmonic oscillator in a direct-sum Hilbert space. In quantum theory, time appears as a parameter, whereas spatial position is represented by an operator. Within this framework, a quantum state is described as a direct-sum of two components localized at parity-conjugate points in physical space.
• Figure 3: Diagrammatic optical theorem / cutting rule.
• Figure 4: Dual-IHO spin--2: absent from physical unitarity cuts (top), but improves UV behavior via $1/q^4$ falloff (bottom).