Quantization of the Simplest Grain of Space: Quantum Geometry, Picard-Fuchs Equations, and Perturbative/Non-Perturbative Relations

Quantization of the geometrical observables of spacetime is a key feature of quantum gravity. However, the area spectrum stands alone as having a complete analytic treatment. Even for the simplest, tetrahedral grain of space, research on other observables, like the volume, usually proceeds either completely numerically or via strong approximations. We methods that allow you to find the spectrum of a tetrahedral grain of space to arbitrary orders in hbar. These results are achieved through complexification of the underlying dynamical system and provide an accessible introduction to Picard-Fuchs equations and to new methods in non-perturbative quantization. In particular, this application provides a novel example of the non-perturbative quantization of a modular geometry that is not of the kinetic-plus-potential form and exhibits several features of the quantization of a nonlinear system that will be of general interest.