Introduction

What is more natural than constructing space from elementary geometric building blocks? It is not as easy as one might think, based on our intuition of playing with Lego blocks in three-dimensional space. Imagine the building blocks are d-dimensional flat simplices all of whose side lengths are a, and let d > 2. The problem is that if we glue such blocks together carelessly we will with probability one create a space of no extension, in which it is possible to get from one vertex to any other in a few steps, moving along the one-dimensional edges of the simplicial manifold we have created. We can also say that the space has an extension which remains at the “cut-off” scale a. Our intuition coming from playing with Lego blocks is misleading here because it presupposes that the building blocks are embedded geometrically faithfully in Euclidean ℝ3, which is not the case for the intrinsic geometric construction of a simplicial space.

By contrast, let us now be more careful in our construction work by assigning to a simplicial space T – which we will interpret as a (Euclidean) spacetime – the weight e−S(T), where S(T) denotes the Einstein action associated with the piecewise linear geometry uniquely defined by our construction. As long as the (bare) gravitational coupling constant GN is large, we have the same situation as before.