Quantum geometry with intrinsic local causality

TL;DR

This work proposes a background-independent quantum gravity framework in which spin networks are generalized to labelled 2-manifolds and quantum-group invariants, yielding a kinematical space $H_{G_q}$ that aggregates invariant tensors over all finite-genus surfaces. Dynamics are built from tubular evolution moves that generate discrete histories with intrinsic local causality, allowing a discrete path-integral–like treatment of causal evolution and a connection to spin foams and membranes. The paper introduces tube operators (surface, bulk, substitution) to encode geometric data relationally, and develops a geometrical interpretation of area and volume observables while showing how coarse graining leads to a holographic, Bekenstein-bound–consistent boundary theory for SU(2). It argues that with appropriate choices of $G_q$ and dynamics, one can recover classical general relativity in the continuum limit, while naturally accommodating holography and entropy notions in a fully background-free, non-perturbative setting.

Abstract

The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU(2) spin networks of quantum general relativity are replaced by labelled compact two-dimensional surfaces. The space of states of the theory is the direct sum of the spaces of invariant tensors of a quantum group G_q over all compact (finite genus) oriented 2-surfaces. The dynamics is background independent and locally causal. The dynamics constructs histories with discrete features of spacetime geometry such as causal structure and multifingered time. For SU(2) the theory satisfies the Bekenstein bound and the holographic hypothesis is recast in this formalism.

Figures (8)

• Figure 1: (a) A genus 4 surface cut into six trinions $B^3_I$ by circles $c_\alpha$. (b) The same surface in a different trinion decomposition.
• Figure 2: This trinion decomposition is degenerate because the two trinions have two circles in common.
• Figure 3: A trinion decomposition of a genus 5 surface reduced to a spin network graph.
• Figure 4: The tubular 4-simplex ${\cal P}$, a genus 6 surface decomposed to 5 4-punctured spheres.
• Figure 5: An elementary substitution move.