Reconstructing Quantum Geometry from Quantum Information: Spin Networks as Harmonic Oscillators

TL;DR

The paper addresses how to recast loop quantum gravity's quantum geometry in quantum-information terms by mapping spin networks to harmonic oscillators and identifying boundary degrees of freedom with a matrix-model structure. It introduces a two-oscillator representation of SU(2) to describe elementary surfaces, shows that intertwiners form a U(N) algebra acting on the N-puncture boundary, and connects this to non-commutative geometry via fuzzy sphere constructions with a Dirac operator controlling the geometry. It then proposes a boundary matrix model capturing LQG dynamics holographically and discusses spin-network dynamics as quantum circuits, relational spacetime localization, and quantum reference frames. The work lays groundwork for bridging LQG with NC geometry and quantum information, offering new tools to study semiclassical limits, holography, and possible links to string theory.

Abstract

Loop Quantum Gravity defines the quantum states of space geometry as spin networks and describes their evolution in time. We reformulate spin networks in terms of harmonic oscillators and show how the holographic degrees of freedom of the theory are described as matrix models. This allow us to make a link with non-commutative geometry and to look at the issue of the semi-classical limit of LQG from a new perspective. This work is thought as part of a bigger project of describing quantum geometry in quantum information terms.

Figures (3)

• Figure 1: Using the presentation of $\mathrm{SU}(2)$ in terms of two harmonic oscillators, we replace all lines of the spin network by double lines symbolizing the harmonic oscillators $a$ and $b$. Then, considering an intertwiner with $N$ legs, which can be interpreted as dual to a ${\cal S}^2$ surface punctured by $N$ spin networks edges, we look for $\mathrm{SU}(2)$ invariant states in the tensor product of the $2\times N$ harmonic oscillators.
• Figure 2: Considering a bounded region of space, its boundary defined as a $N$ punctured sphere ${\cal S}^2$, the LQG bulk degrees of freedom are described as all spin networks states with support on arbitrary graphs puncturing the surface $N$ times. Then one can coarse-grain these states down to a single vertex: the state is then described by a single intertwiner and is interpreted as the boundary state.
• Figure 3: Four-valent intertwiners between spin $1/2$ representations can be considered as the basic building block of spin networks. They are generated by two states, of spin $j=0$ and of spin $j=1$, and can be interpreted as two-qubit gates of a quantum circuit.