Neighboring cells in different states, that are simultaneously neighbors in time and in space represent the possibility of force, BL-1T-1. Force is the thing that changes momentum. The DM rule results in a change in momentum going from the past to the future, based on the present. Force involves 2 space-time neighbors that differ in state.
E. g. S2x, 2y, 2z, t= 1 and S2x 1, 2y, 2z, t 1= –1 the force vector may be (–1, –1, 1) but the fact that it is acting on a system with angular momentum needs to be taken into account.
The sign of a force vector remains the same under Time reversal, while the sign of a momentum vector is complemented. When we give a particular DM rule as an example, (on page 38) it is based on a configuration of cells in the present that cause 2 spatial nearest neighboring cells in the past to swap places. However, the rule that we give coincides with the concept of force as the changer of momentum despite the fact that force must necessarily involve cells both in the present and the past, while the rule that decides swap or no swap must do so based only on the state of cells in the present. This is in line with the goal of the consistent representation of properties of physics as step one in the exposition of Digital Philosophy.
The relationship between Force and Momentum is interesting. In the Salt Array there are 6 possible atomic force vectors; one associated with each face of a smallest cube in the lattice. There are 12 possible atomic momentum vectors, one associated with each of the 12 edges of the cube.
The six force vectors are:
±{(0,0,1), (0,1,0), (1,0,0)}, each is associated with a face of the cube..
The 12 momentum vectors are:
Notice that there is no microscopic momentum vector that is aligned with a force vector. On the other hand, every momentum vector is the vector sum of 2 force vectors. Every edge of the cube is associated with the 2 faces that have the edge in common. We take these facts into account in formulating our best DM rule.
