Quantum theory from quantum gravity

We provide a mechanism by which, from a background-independent model with no quantum mechanics, quantum theory arises in the same limit in which spatial properties appear. Starting with an arbitrary abstract graph as the microscopic model of spacetime, our ansatz is that the microscopic dynamics can be chosen so that (i) the model has a low-energy limit which reproduces the nonrelativistic classical dynamics of a system of $N$ particles in flat spacetime, (ii) there is a minimum length, and (iii) some of the particles are in a thermal bath or otherwise evolve stochastically. We then construct simple functions of the degrees of freedom of the theory and show that their probability distributions evolve according to the Schrödinger equation. The nonlocal hidden variables required to satisfy the conditions of Bell’s theorem are the links in the fundamental graph that connect nodes adjacent in the graph but distant in the approximate metric of the low-energy limit. In the presence of these links, distant stochastic fluctuations are transferred into universal quantum fluctuations.

Figure 1

A sketch of how locality in the graph may not be preserved in the embedding. In the abstract graph on the left, the nearest neighbors of $i$ are $j$, $k$, $l$, and $m$. In the embedded graph on the right, a local neighborhood of $i$ defined by the metric of $R^3$ is drawn, and $k$ is far outside it.