Causal sets, discrete quantum gravity, transition amplitudes, four dimensions.
One could begin a study like the present one by simply postulating that our universe is four-dimensional. There are ample reasons for doing this. Experience, observation and experiment all point to the fact that we inhabit a four-dimensional universe. Another approach would be to show that four-dimensions arise naturally from a reasonable model of the universe or multiverse. After reviewing the causal set approach to discrete quantum gravity in Section 1, we shall discuss the emergence of four-dimensions in Section 2. We shall see that certain patterns of four arise that suggest the introduction of a 4-dimensional discrete manifold. In the later sections we shall discuss some consequences of this introduced framework. In particular, we will show that quantum amplitudes can be employed to describe a multiverse dynamics. Moreover, a natural unitary operator together with energy, position and momentum operators will be introduced and their properties studied.
Gudder, S., "Emergence of Four Dimensions in the Causal Set Approach to Discrete Quantum Gravity" (2014). Mathematics Preprint Series. Paper 6.