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Research Article, J Comput Eng Inf Technol Vol: 5 Issue: 1

Causal and Acausal Sets Based on a Quotient Lattice

Gunji YP1*, Haruna T2 and Uragami D3
1Department of Intermedia Art and Science, School of Fundamental Science and Technology, Waseda University, Ohkubo 3-4-1, Shinjuku, Tokyo 169-0072, Japan
2Department of Planetology, Faculty of Science, Kobe University, Nada, Kobe 657-8501, Japan
3College of Industrial Technology, Nihon University, Izumi 1-2-1, Narashino, Chiba, 275-8575, Japan
Corresponding author : Yukio-Pegio Gunji
Department of Intermedia Art and Science, School of Fundamental Science and Technology, Waseda University, Ohkubo 3-4-1, Shinjuku, Tokyo 169-0072, Japan
Tel: +81 788035759
E-mail: [email protected], [email protected], [email protected]
Received: January 27, 2016 Accepted: March 07, 2016 Published: March 14,2016
Citation: Gunji YP, Haruna T, Uragami D (2016) Causal and Acausal Sets Based on a Quotient Lattice. J Comput Eng Inf Technol 5:1. doi:10.4172/2324-9307.1000143

Abstract

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.

Keywords: Causal history; Acausal sets; Category theory; Lattice; Congruence; Bicontinuous poset.

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