Semiclassical Limits of Extended Racah Coefficients

TL;DR

This work develops semiclassical limits of extended Racah coefficients by linking the extended SU(2) $6j$ symbol to the positive discrete series of $SU(1,1)$ and providing a geometric interpretation in terms of Lorentzian tetrahedra with timelike faces. It derives an explicit Racah formula for $SU(1,1)$, relates it to the extended SU(2) construction via a transformation $S$, and proves a BE-type identity in this setting. The geometry reveals two regimes determined by the Cayley determinant sign: $V^2>0$ corresponds to Lorentzian tetrahedra with real dihedral angles, while $V^2<0$ yields timelike faces with complex angles and two boost-type configurations, both connected by the same algebraic maps. Finally, the paper obtains asymptotic expressions reminiscent of the Ponzano–Regge formula, yielding oscillatory behavior for $V^2>0$ and exponential decay for $V^2<0$, thereby supporting a Lorentzian 3D quantum gravity state-sum framework using extended Racah data.

Abstract

We explore the geometry and asymptotics of extended Racah coeffecients. The extension is shown to have a simple relationship to the Racah coefficients for the positive discrete unitary representation series of SU(1,1) which is explicitly defined. Moreover, it is found that this extension may be geometrically identified with two types of Lorentzian tetrahedra for which all the faces are timelike. The asymptotic formulae derived for the extension are found to have a similar form to the standard Ponzano-Regge asymptotic formulae for the SU(2) 6j symbol and so should be viable for use in a state sum for three dimensional Lorentzian quantum gravity.

Figures (6)

• Figure 1: A graphic representation of equations \ref{['eq1']} - \ref{['eq8']}
• Figure 2: The Pentagon relation
• Figure 3: A Lorentzian tetrahedron with all edges and all faces timelike. Time increases vertically up the page.
• Figure 4: A plot of $\sum^4_{h,k=0} j_{hk}\theta_{hk}$ (x-axis) versus $\sqrt{12\pi V} \{6j\}$ (y-axis)
• Figure 5: A plot of $-\sum^4_{h,k=0}j_{hk}^\prime \theta_{hk}^\prime +\left(j^\prime_{12} +j_{14}^\prime +j_{34}^\prime+ 2j^\prime_{24} +j_{23}^\prime\right)\pi$ (x-axis) versus $\sqrt{12\pi V}\{6j\}$ (y-axis)

Theorems & Definitions (16)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Definition 2.5
• Proposition 2.6
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Proposition 3.4: Biedenharn-Elliot relation for SU(1,1)