The twistorial structure of loop-gravity transition amplitudes

TL;DR

This work develops a twistorial description of the covariant phase space for loop quantum gravity, recasting SL(2,C) holonomies and fluxes in terms of twistors and revealing a canonical area–dihedral angle pairing. It shows how reality and simplicity constraints reduce twistors to SU(2) spinors, identifies the lattice Ashtekar-Barbero connection as the SU(2) holonomy in a covariant embedding, and derives the EPRL spinfoam amplitude from a twistor-space path integral with a bilinear BF action. The paper then constructs a twistor-based measure and provides a complete twistor-space derivation of the EPRL model, including the Y-map linking SU(2) boundary data to Lorentz representations, and analyzes semiclassical behavior, showing wedge-flatness and Regge-like large-spin limits with a curvature tensor that carries off-shell torsion. The results illuminate how secondary constraints implement a non-trivial embedding of T^*SU(2) into the covariant phase space, connect covariant and canonical formalisms, and offer a semi-coherent picture where areas are quantized while boundary data encode directional information.

Abstract

The spin foam formalism provides transition amplitudes for loop quantum gravity. Important aspects of the dynamics are understood, but many open questions are pressing on. In this paper we address some of them using a twistorial description, which brings new light on both classical and quantum aspects of the theory. At the classical level, we clarify the covariant properties of the discrete geometries involved, and the role of the simplicity constraints in leading to SU(2) Ashtekar-Barbero variables. We identify areas and Lorentzian dihedral angles in twistor space, and show that they form a canonical pair. The primary simplicity constraints are solved by simple twistors, parametrized by SU(2) spinors and the dihedral angles. We construct an SU(2) holonomy and prove it to correspond to the (lattice version of the) Ashtekar-Barbero connection. We argue that the role of secondary constraints is to provide a non trivial embedding of the cotangent bundle of SU(2) in the space of simple twistors. At the quantum level, a Schroedinger representation leads to a spinorial version of simple projected spin networks, where the argument of the wave functions is a spinor instead of a group element. We rewrite the Liouville measure on the cotangent bundle of SL(2,C) as an integral in twistor space. Using these tools, we show that the Engle-Pereira-Rovelli-Livine transition amplitudes can be derived from a path integral in twistor space. We construct a curvature tensor, show that it carries torsion off-shell, and that its Riemann part is of Petrov type D. Finally, we make contact between the semiclassical asymptotic behaviour of the model and our construction, clarifying the relation of the Regge geometries with the original phase space.

Figures (5)

• Figure 1: Primary constraint structures between twistor and holonomy-flux spaces. $F$ and $C$ schematically denote the simplicity and area matching constraints, and arrows include division by gauge orbits, when relevant.
• Figure 2: More detailed constraint structures and the role of the secondary constraints. In the presence of secondary constraints, the rightmost part of the diagram becomes irrelevant, as the orbits of $D$ are no longer pure gauges. In the final step, we have reintroduced the graph structure, as a proper definition of the secondary constraints should not be local on the links.
• Figure 3: In a given 4-simplex two tetrahedra share a triangle $t$, that is in turn dual to a wedge $w_{tv}$. The wedge is bounded by a loop. Half of the loop lies in the boundary of the 4-simplex, and connects the two tetrahedra $\tau$ and $\undertilde{\tau}$, piercing through the triangle $t$. The other half enters the bulk and passes through the center $v$ of the 4-simplex.
• Figure 4: Left: a triangle $t$ in the spatial hypersurface bounds two tetrahedra. Twistors $Z$ and $\undertilde{Z}$ are attached to the underlying spinnetwork graph. Right: The same triangle seen from a 4-dimensional perspective. The triangle $t$ is dual to a spinfoam face $f$ consisting of several wedges $w_{tv}$ one for each adjacent vertex $v$.
• Figure 5: The integration domain is bounded by a Hankel contour around the branch cuts in the complex plane.