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The effective field theory of the gravitational functional measure

Henrique Knopki, Iberê Kuntz

TL;DR

This work uses effective field theory to analyze the gravitational functional measure by constructing a configuration-space metric for metrics with a one-parameter DeWitt family $G_{IJ}$. At leading order, the metric introduces a running DeWitt parameter $\lambda$, whose RG flow exhibits a UV fixed point at $\lambda=-1$ and an IR singular point at $\lambda=-1/2$, with unitarity requiring $2\lambda+1<0$. The leading measure correction adds a cosmological-constant-like term to the effective action, and the RG analysis shows that physically consistent flows generically approach the UV fixed point, providing a principled justification for the commonly used DeWitt value in gravity. The results imply a UV-complete, unitary measure sector and highlight how higher-order corrections could refine or alter the fixed-point structure, suggesting directions for future work in EFT-inspired gravitational quantization.

Abstract

The gravitational path integral measure has been the subject of an increasing interest lately, and no conclusive answer yet exists for its correct form. In this paper, we adopt effective field theory techniques to shed light on this issue. We build the configuration-space metric as an energy expansion, including all possible terms that satisfy the underlying symmetries, and use it to define a Riemannian measure. We study the running of the free parameters that show up in this expansion at leading order, which corresponds to the DeWitt metric with parameter $λ$. We show that a flat configuration space is excluded on unitarity grounds. The renormalization group contains one UV fixed point at $λ=-1$, thus allowing for the UV completion of the measure sector. This fixed point corresponds to the value obtained by identifying DeWitt's metric from the kinetic term of general relativity, a standard procedure in the literature that otherwise lacks physical motivation. Our results provide such a justification from first principles.

The effective field theory of the gravitational functional measure

TL;DR

This work uses effective field theory to analyze the gravitational functional measure by constructing a configuration-space metric for metrics with a one-parameter DeWitt family . At leading order, the metric introduces a running DeWitt parameter , whose RG flow exhibits a UV fixed point at and an IR singular point at , with unitarity requiring . The leading measure correction adds a cosmological-constant-like term to the effective action, and the RG analysis shows that physically consistent flows generically approach the UV fixed point, providing a principled justification for the commonly used DeWitt value in gravity. The results imply a UV-complete, unitary measure sector and highlight how higher-order corrections could refine or alter the fixed-point structure, suggesting directions for future work in EFT-inspired gravitational quantization.

Abstract

The gravitational path integral measure has been the subject of an increasing interest lately, and no conclusive answer yet exists for its correct form. In this paper, we adopt effective field theory techniques to shed light on this issue. We build the configuration-space metric as an energy expansion, including all possible terms that satisfy the underlying symmetries, and use it to define a Riemannian measure. We study the running of the free parameters that show up in this expansion at leading order, which corresponds to the DeWitt metric with parameter . We show that a flat configuration space is excluded on unitarity grounds. The renormalization group contains one UV fixed point at , thus allowing for the UV completion of the measure sector. This fixed point corresponds to the value obtained by identifying DeWitt's metric from the kinetic term of general relativity, a standard procedure in the literature that otherwise lacks physical motivation. Our results provide such a justification from first principles.

Paper Structure

This paper contains 6 sections, 76 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Configuration space: notations and conventions
  3. DeWitt's geometry: a metric over metrics
  4. The gravitational functional measure
  5. RG analysis
  6. Conclusions

Figures (3)

  • Figure 1: The configuration space $\mathcal{C}$ is a fiber bundle where the spacetime $\mathcal{M}$ is the base manifold. All fibers $\mathcal{F}(x_i)$ are isomorphic to one another.
  • Figure 2: Phase portrait for the RG equation \ref{['RGa']} with the beta function $\beta_\lambda$ (orange curve) given by Eq. \ref{['eq:beta']}. The fixed points are marked with green blobs, with the arrows (pointed towards the UV) showing their (in)stability. The figure also shows the unitary (in green) and the non-unitary (in red) regions of parameter space. In the non-unitary region, $\beta_\lambda$ is complex and the orange curve illustrates its real part.
  • Figure 3: RG trajectories given by the solution \ref{['eq:RGsol']} for different choices of initial data $(\Lambda_0,\lambda_0)$. Different colors correspond to different choices of $\lambda_0$, whereas different linestyles correspond to different choices of $\Lambda_0$. All trajectories asymptotically flow to the UV fixed point at $\lambda=-1$.