``Sum over Surfaces'' form of Loop Quantum Gravity

TL;DR

The work develops a covariant spacetime formulation of quantum gravity derived from canonical loop quantum gravity by introducing a proper time evolution operator $U(T)$ and its perturbative expansion. Each term is recast as a sum over topologically inequivalent branched colored 2d surfaces in 4d, with vertex amplitudes $A_v(oldsymbol{\sigma})$ derived from the Hamiltonian constraint, yielding a finite, order-by-order, computable series. The authors show that the dynamics can be written as a sum over surfaces with weights given by a product of vertex factors, and they discuss crossing symmetry to enforce 4d diffeomorphism invariance, potentially fixing operator-ordering ambiguities. The construction aligns quantum GR with a TQFT-like structure (à la OCY) while preserving local degrees of freedom, offering a continuum counterpart to Reisenberger’s simplicial approach and a path toward a background-independent description of quantum geometry. Overall, the paper provides a finite, surface-based representation of quantum gravity dynamics, connecting LQG to covariant, observable-rich formulations and outlining directions to incorporate full diffeomorphism invariance through crossing-symmetric vertices.

Abstract

We derive a spacetime formulation of quantum general relativity from (hamiltonian) loop quantum gravity. In particular, we study the quantum propagator that evolves the 3-geometry in proper time. We show that the perturbation expansion of this operator is finite and computable order by order. By giving a graphical representation a' la Feynman of this expansion, we find that the theory can be expressed as a sum over topologically inequivalent (branched, colored) 2d surfaces in 4d. The contribution of one surface to the sum is given by the product of one factor per branching point of the surface. Therefore branching points play the role of elementary vertices of the theory. Their value is determined by the matrix elements of the hamiltonian constraint, which are known. The formulation we obtain can be viewed as a continuum version of Reisenberger's simplicial quantum gravity. Also, it has the same structure as the Ooguri-Crane-Yetter 4d topological field theory, with a few key differences that illuminate the relation between quantum gravity and TQFT. Finally, we suggests that certain new terms should be added to the hamiltonian constraint in order to implement a ``crossing'' symmetry related to 4d diffeomorphism invariance.

Figures (13)

• Figure 1: Action of $D_{iJK\epsilon\epsilon'}$. $r, q$ and $p$ are the colors of the links $I, J$ and $K$.
• Figure 2: Surface corresponding to a term of order zero.
• Figure 3: Surface corresponding to a first order term.
• Figure 4: The elementary vertex.
• Figure 5: A term of second order.