The WKB Wavefunctions of Simplicial Grains of Space

Quantization of space is a key theme in quantum gravity. Here, we discuss the dynamics, volume spectrum, and the wavefunctions of the simplest grain of space---a tetrahedron. We take a semiclassical approach to the problem, which gives a more intuitive perspective on the quantum nature of space. In particular, we apply some classical geometrical results and a Poisson structure on the phase space to derive the dynamics of a Euclidean tetrahedron, and then derive the WKB eigenfunctions of these Euclidean simplices (which are of the form exp(iS/h), where S is the action integral) and compare them with the standard results. The action integral on our phase space naturally gives rise to elliptic integrals, which hints towards a very rich mathematical structure in the problem.