Causal and Acausal Sets Based on a Quotient Lattice

Causal and Acausal Sets Based on a Quotient Lattice

The spacetime viewed from the inside by using a lattice is here described as a causal set, and a quotient lattice is described as a causal history. It is shown that this description is consistent with general relativity and quantum theory. In particular, the influence of measurement on a causal set is considered and the interaction between a causal set and history is defined. The transitions from a causal set to a causal history and vice versa are both defined in a category theory and they are shown to be an adjunction. This adjunction leads to the idea of a relative casual set, and it results in the evolution and superposition of causal sets and histories. It is shown that the superposed casual set is a bicontinuous poset equipped with a compact interval. Relative causal sets are also equipped with the operation of self-embedding. A causal set is continually transformed into a degenerate distributive lattice due to the self-embedding. When an acausal set in a causal set can be transformed into a tensor product of underlying Hilbert spaces, the distributive lattice reveals a unitary evolution of composite states that is consistent with quantum theory.