Causal structure of spacetime and geometric algebra for quantum gravity

We construct a background independent model of spacetime from a minimal set of postulates of causality. The topology and geometry of spacetime can be derived from the model, instead of being postulated. We define a measure of causality and relate it to the noncommutative geometric algebra of spacetime coordinates. The geometric algebra formalism leads to the definition of a Euclidean action. The Euclidean action is positive definite and allows the path integral formulation of quantum gravity to be treated as statistical physics of a causal stochastic Markov field.

Figure 1

Two-dimensional poset.

Figure 2

Three-dimensional poset.

Figure 3

Poset without supremum or infimum.

Figure 4

Uniformly two-dimensional lattice.

Figure 5

Two-dimensional lattice with one-dimensional intervals.

Figure 6

Lattices $M_3$ and $N_5$.

Figure 7

The nonsemidistributive lattice $L_3$.

Figure 8

The nonsemidistributive lattice $L_4$ and its dual.

Figure 9

Locally distributive lattice $S_7$ and its dual.

Figure 10

Two- and three-dimensional Boolean lattice.

Figure 11

Hasse diagram of a nondistributive lattice.

Figure 12

Alternative Hasse diagram of the lattice in Fig. 11.