Background independence in a nutshell: the dynamics of a tetrahedron

We study how physical information can be extracted from a background-independent quantum system. We use an extremely simple 'minimalist' system that models a finite region of 3D Euclidean quantum spacetime with a single equilateral tetrahedron. We show that the physical information can be expressed as a boundary amplitude. We illustrate how the notions of 'evolution' in a boundary proper time and 'vacuum' can be extracted from the background-independent dynamics.