Quantum deformation of quantum gravity

TL;DR

The paper proposes a q-deformation of the loop representation of quantum gravity by elevating loops to framed loops, with the deformation parameter $q = e^{ i \hbar^2 G^2 \Lambda /6}$ tied to the cosmological constant. It develops a framed-loop algebra $\mathcal{LA}^f$ and a $q$-spin-net basis that yields a closed, commuting operator algebra $\hat{T}_q[\alpha]$ and a $q$-deformed area operator with discrete spectra, together with eigenstates of $\hat{T}_q[\alpha]$ and potential pathways to a $q$-deformed connection representation. The framework connects Chern–Simons/Kauffman-bracket topology to nonperturbative quantum gravity, suggesting a Kodama phase where framing counts twist and offering a diffeomorphism-invariant infrared regulator. These constructions may illuminate nonperturbative vacuum structure in quantum gravity with a cosmological constant and provide practical tools for calculations via recoupling theory in the $q$-deformed setting.

Abstract

We describe a deformation of the observable algebra of quantum gravity in which the loop algebra is extended to framed loops. This allows an alternative nonperturbative quantization which is suitable for describing a phase of quantum gravity characterized by states which are normalizable in the measure of Chern-Simons theory. The deformation parameter, q, depends on the cosmological constant. The Mandelstam identities are extended to a set of relations which are governed by the Kauffman bracket so that the spin network basis is deformed to a basis of SU(2)q spin networks. Corrections to the actions of operators in non-perturbative quantum gravity may be readily computed using recoupling theory; the example of the area observable is treated here. Finally, eigenstates of the q-deformed Wilson loops are constructed, which may make possible the construction of a q-deformed connection representation through an inverse transform.

Figures (6)

• Figure 1: Examples of framing: (a.) Two unlinked unknots, $L=0$ (b.) A knot with a direction field in the plane of the diagram - "blackboard framing" - giving a linking number $-2$ between the knot $\gamma$ and its frame $\gamma'$ (c.) A pair of intersecting unknots with linking number $L=B$ [See Eq. (\ref{['blink']})].
• Figure 2: The framing on the retraced path $\eta_2^f$ in (a.) is constructed so that the linking of the closure of the two segments $\eta_2$ and $\eta'_2$ vanishes, i.e. $L\left(\bar{\eta_2}, \bar{\eta'_2} \right)=0$ for the simple tangle in (b.).
• Figure 3: For loops contract able to a point the linking number changes the limit. In this case, the limit as $\beta$ shrinks to a point is $-A^4-A^{-4}$.
• Figure 4: The trivalent vertex (a.) is decomposed into three projectors as in (b.) with $a = (j+k-l)/2$, $b=(k+l -j)/2$, and $c=(j+l-k)/2$.
• Figure 5: The decomposition of a higher valent intersection into trivalent intersections at a point. The first two incident edges are joined to a new internal edge $i_1$ at the first vertex. Then $i_1$ and $e_3$ are joined into a trivalent vertex with a new internal line $i_2$. The process continues until there are two external vertices left which are joined into the last three vertex with the last internal line, in this case $i_2$.