[Paper Review] Super-renormalizable Higher-Derivative Quantum Gravity

Leonardo Modesto|arXiv (Cornell University)|Jan 31, 2012

Black Holes and Theoretical Physics1 references20 citations

TL;DR

This paper proposes a ghost-free, super-renormalizable higher-derivative quantum gravity theory using entire function form factors in the action to suppress ultraviolet divergences. By replacing polynomial higher-derivative terms with nonlocal entire functions of the d'Alembertian, the theory remains unitary and microcausal, with finite amplitudes from two loops onward and a regular gravitational potential at r=0, resolving spacetime singularities perturbatively.

ABSTRACT

In this paper we study perturbatively an extension of the Stelle higher derivative gravity involving an infinite number of derivative terms. We know that the usual quadratic action is renormalizable but is not unitary because of the presence of a ghost in the theory (pole with negative residue in the propagator). The new theory is instead ghost-free since an entire function (or form factor) is introduced in the model without involving new poles in the propagator. The local high derivative theory is recovered expanding the entire functions to the lowest order in the mass scale of the theory. Any truncation of the entire function gives rise to unitarity violation. The theory is divergent at one loop and finite from two loops upwards: the theory is then super-renormalizable. Using the modified graviton propagator, we demonstrate the regularity of the gravitational potential in r=0.

Motivation & Objective

• To construct a perturbatively quantum-gravitational theory that is unitary, renormalizable, and free of ghosts.
• To resolve the singularity problem in classical gravity by modifying the graviton propagator via nonlocal form factors.
• To ensure UV finiteness beyond one loop while preserving Lorentz invariance and a single spacetime continuum.
• To demonstrate that the gravitational potential remains finite and regular at r=0, avoiding the 1/r divergence of Einstein gravity.
• To provide a nonlocal extension of Stelle's higher-derivative gravity that avoids the unitarity violation of truncated local theories.

Proposed method

• Introduces a nonlocal action with entire functions h₂ and h₀ as form factors γ₁(□Λ) and γ₂(□Λ), replacing polynomial terms in the Stelle theory.
• Uses entire functions h₂(−□Λ) and h₀(−□Λ) to define the propagator, ensuring no poles and thus no ghosts in the spectrum.
• Applies harmonic gauge fixing and computes the graviton propagator in momentum space as D(k) = V(k²/Λ²)/k², with V(z) analytic and of finite order.
• Imposes microcausality and unitarity conditions on V(z): entire, rapidly decreasing at infinity, Hermitian, V(0)=1, and non-negative on the real axis.
• Derives the gravitational potential via Fourier transform of the modified propagator, leading to Φ(r) = −(G_NM/π) ∫₀^∞ dp J₀(p) V(p²/r²Λ²).
• Analyzes loop amplitudes using topological identity I = V + L − 1, showing UV finiteness for L > 1 and 1-loop divergence only.

Experimental results

Research questions

• RQ1 Can a higher-derivative quantum gravity theory be made ghost-free while preserving perturbative renormalizability?
• RQ2 Does the inclusion of entire function form factors in the action lead to a UV-finite theory beyond one loop?
• RQ3 Is the gravitational potential regular at r=0 in nonlocal quantum gravity, avoiding the 1/r singularity of general relativity?
• RQ4 Can a nonlocal theory maintain unitarity and microcausality without introducing new poles or violating Lorentz invariance?
• RQ5 How do different classes of entire form factors (e.g., exponential or power-law) affect the behavior of the gravitational potential at short distances?

Key findings

• The theory is super-renormalizable: divergent only at one loop, finite from two loops onward, due to the exponential suppression in the form factor V(z) = e^−z^n.
• The graviton propagator is regular at r=0, with Φ(r) ≈ −2G_NM const. Λ²^γ⁺² r²^γ⁺¹ for small r, which is finite and non-singular.
• For the case V(z) = e^−z, the exact potential is Φ(r) = −G_NM/ r Er(rΛ/2), with Φ(0) = −G_NMΛ/√π, confirming regularity at the origin.
• The theory satisfies unitarity and microcausality because the form factor V(z) is entire, of finite order, and decreases rapidly at infinity.
• Truncating the entire function to a polynomial leads to ghost states and unitarity violation, proving the necessity of nonlocality.
• The gravitational potential remains regular for all n ≥ 1 in V(z) = e^−z^n, with Φ(0) ∝ −G_NMΛ, indicating a universal short-distance behavior.