Mathematically speaking, both general relativity and a quantum field theory are considered as incomplete theories. The former lacks a description for the negative curvature of spacetime; the latter lacks the double intrinsic spin. Both shortcomings can disappear by simply introducing three topologies: Hopf for 1 dimension, Möbius for 2 dimensions and Klein for 3 dimensions.

Where and when these topologies are introduced, the quantization of the spacetime continuum can proceed according to a topological description of square of energy for the zero-point quantum vacuum fluctuations using an abstract group or ring algebra of Hadamard matrices and a double actions principle as the product of global Lagrangian and Hamiltonian with a principle of directional invariance at the local infinitesimal region of the spacetime continuum.