Timelike surfaces in Lorentz covariant loop gravity and spin foam models

TL;DR

This work develops a Lorentz covariant canonical formulation of general relativity on a timelike foliation, preserving full local Lorentz symmetry and addressing second-class constraints via Dirac brackets. A unique shifted Lorentz connection diagonalizes the area operator, yielding a spectrum that, for timelike foliations, reduces to $S=8\pi\hbar G\sqrt{C(sl_{\chi}(2,\mathbb{R}))-C_1(sl(2,\mathbb{C}))}$ and is independent of the Immirzi parameter. By introducing projected spin networks and an enlarged Hilbert space that keeps the foliation data through $\chi$, the authors establish a concrete link to Lorentzian spin foam models: boundary states built from simple SL(2,\mathbb{C}) representations with singlet projections reproduce spin foam boundary states and area spectra, thereby aligning canonical and spin foam perspectives under appropriate representation restrictions. The analysis reveals that spacelike and timelike surfaces yield continuous and discrete area spectra, respectively, echoing broader themes about the role of causal structure in quantum gravity. Collectively, the results provide a covariant bridge between canonical loop quantization and spin foam approaches and suggest avenues for incorporating foliations and potential causal fluctuations into quantum gravity formalisms.

Abstract

We construct a canonical formulation of general relativity for the case of a timelike foliation of spacetime. The formulation possesses explicit covariance with respect to Lorentz transformations in the tangent space. Applying the loop approach to quantize the theory we derive the spectrum of the area operator of a two-dimensional surface. Its different branches are naturally associated to spacelike and timelike surfaces. The results are compared with the predictions of Lorentzian spin foam models. A restriction of the representations labeling spin networks leads to perfect agreement between the states as well as the area spectra in the two approaches.