Imaginary action, spinfoam asymptotics and the 'transplanckian' regime of loop quantum gravity

TL;DR

This work shows that the imaginary part of the gravitational action, tied to boundary and corner structures, persists in first-order formulations of GR. In the Lorentzian EPRL/FK spinfoam, the correct imaginary part emerges only after analytically continuing the Barbero-Immirzi parameter γ to ±i in the large-spin limit, linking to a transplanckian regime of LQG. The results align with a recent black hole entropy calculation and suggest two distinct classical GR limits (IR continuum vs. UV discrete) connected by RG-like running of γ. While the γ→±i procedure is formal for the quantum theory, it provides a coherent picture connecting semiclassical GR, spinfoam amplitudes, and black-hole thermodynamics within a high-energy LQG framework.

Abstract

It was recently noted that the on-shell Einstein-Hilbert action with York-Gibbons-Hawking boundary term has an imaginary part, proportional to the area of the codimension-2 surfaces on which the boundary normal becomes null. We discuss the extension of this result to first-order formulations of gravity. As a side effect, we settle the issue of the Holst modification vs. the Nieh-Yan density by demanding a variational principle with suitable boundary conditions. We then set out to find the imaginary action in the large-spin 4-simplex limit of the Lorentzian EPRL/FK spinfoam. It turns out that the spinfoam's effective action indeed has the correct imaginary part, but only if the Barbero-Immirzi parameter is set to +/- i after the quantum calculation. We point out an agreement between this effective action and a recent black hole state-counting calculation in the same limit. Finally, we propose that the large-spin limit of loop quantum gravity can be viewed as a high-energy 'transplanckian' regime.

Figures (5)

• Figure 1: A purely spacelike closed boundary, composed of two intersecting hypersurfaces. The full circles denote the corner surface. The arrows indicate the two boundary normals at each intersection point. A continuous boost between these two normals involves two signature flips. As a result, the "corner angle" has an imaginary part with magnitude $\pi$.
• Figure 2: A smooth closed boundary in Lorentzian spacetime. The arrows indicate the normal direction at various points. The normal's sign is chosen so that it has a positive scalar product with outgoing vectors. Empty circles denote "signature flips", where the normal becomes momentarily null.
• Figure 3: Two types of corners between spacelike tetrahedra: thick (a) and thin (b). The arrows denote ingoing unit normals. These correspond to outgoing covectors, i.e. have a positive scalar product with outgoing vectors. For each type of corner, the content of eqs. \ref{['eq:Theta']}-\ref{['eq:Pi']} is summarized.
• Figure 4: An assignment of boost angles to points in a Lorentzian plane. The points represent values of the boundary normal $n^\mu$ in the 1+1d plane transverse to a corner (2d face) of the 4-simplex. The angles are defined up to integer multiples of $2\pi i$.
• Figure 5: Two classical-GR limits of loop quantum gravity. At high energy, discrete classical geometries are described by coherent states with large spins (no coarse graining implied). At low energy, continuum classical geometries are supposed to emerge through coarse graining.