The Holst Spin Foam Model via Cubulations

TL;DR

Problem: Standard spin foam models based on the Plebanski action struggle to implement the simplicity constraints consistently and to recover a robust semiclassical limit for general relativity within a triangulation-based BF framework. Approach: The authors propose starting from the Holst action and discretizing via cubulations to regulate the path integral, deriving a generating functional for tetrad $n$-point functions and analyzing Wick-like structure and vertex amplitudes through harmonic analysis on $SO(4)$, while clarifying the link to the canonical LQG connection via a time-gauge discussion. Findings: The resulting amplitudes differ from conventional SFMs, tetrad correlators display a Wick-like structure despite an interacting covariance $G(A)^{-1}$, and representations are not automatically restricted to simple ones unless a time gauge is imposed; the vertex is encoded by an octagon diagram with non-factorizing face dependence. Significance: This offers a new covariant route toward Loop Quantum Gravity compatible with canonical $SU(2)$ variables, highlights novel nonlocal interactions, and outlines concrete open problems (continuum limit, measure, convergence, and time-gauge implementation) for future work.

Abstract

Spin foam models are an attempt for a covariant, or path integral formulation of canonical loop quantum gravity. The construction of such models usually rely on the Plebanski formulation of general relativity as a constrained BF theory and is based on the discretization of the action on a simplicial triangulation, which may be viewed as an ultraviolet regulator. The triangulation dependence can be removed by means of group field theory techniques, which allows one to sum over all triangulations. The main tasks for these models are the correct quantum implementation of the Plebanski constraints, the existence of a semiclassical sector implementing additional "Regge-like" constraints arising from simplicial triangulations, and the definition of the physical inner product of loop quantum gravity via group field theory. Here we propose a new approach to tackle these issues stemming directly from the Holst action for general relativity, which is also a proper starting point for canonical loop quantum gravity. The discretization is performed by means of a "cubulation" of the manifold rather than a triangulation. We give a direct interpretation of the resulting spin foam model as a generating functional for the n-point functions on the physical Hilbert space at finite regulator. This paper focuses on ideas and tasks to be performed before the model can be taken seriously. However, our analysis reveals some interesting features of this model: first, the structure of its amplitudes differs from the standard spin foam models. Second, the tetrad n-point functions admit a "Wick-like" structure. Third, the restriction to simple representations does not automatically occur -- unless one makes use of the time gauge, just as in the classical theory.

Figures (1)

• Figure 1: The octagon diagramme associated to vertex $v$. The eight corners correspond to the eight edges $l=l_\mu^\sigma(v)=l_\mu(v+\frac{\sigma-1}{2}\hat{\mu}),\;\sigma=\pm$ adjacent to $v$. The line between corners labelled by $l^\sigma_\mu(v),\;l^{\sigma'}_\nu(v)$ for $\mu\not=\nu$ corresponds to the face $f=f_{\mu\nu}^{\sigma\sigma'}(v)= f_{\mu\nu}(v+\frac{\sigma-1}{2}\hat{\mu}+\frac{\sigma'-1}{2}\hat{\nu})$. We should colour corners by intertwiners $\rho_l$ and lines by representations $\pi_f$ but refrain from doing so in order not to clutter the diagramme. Altogether 48 irreducible representations of $Spin(4)$ (or 96 of $SU(2)$) are involved.

Theorems & Definitions (1)

• Conjecture B.1