Quantization of Space and Time in 3 and in 4 Space-time Dimensions

TL;DR

This paper demonstrates that the quantization of space and time in Minkowski space can be derived from the interplay of General Relativity and Quantum Mechanics, resulting in a non-commutative lattice structure in both 3+1 and lower dimensions.

Contribution

It shows how space and time quantization naturally emerge from combining General Relativity and Quantum Mechanics without additional postulates.

Findings

Space and time form a non-commutative lattice structure.

Quantization mechanisms differ between 2+1 and 3+1 dimensions.

Derived outcomes are similar across different dimensional models.

Abstract

The fact that in Minkowski space, space and time are both quantized does not have to be introduced as a new postulate in physics, but can actually be derived by combining certain features of General Relativity and Quantum Mechanics. This is demonstrated first in a model where particles behave as point defects in 2 space dimensions and 1 time, and then in the real world having 3+1 dimensions. The mechanisms in these two cases are quite different, but the outcomes are similar: space and time form a (non-cummutative) lattice. These notes are short since most of the material discussed in these lectures is based on two earlier papers by the same author (gr-qc/9601014 and gr-qc/9607022), but the exposition given in the end is new.