A quantum theory of spacetime must be able to describe or to categorize whether spacetime vertices called spacetime charges can ever touch each other. The physical explanation can be simplified simply by making the conjecture that spacetime charges are directional vectors of zero magnitude analogous to the complex imaginary spinors of Élie Cartan. Cartan first published the French edition of his book “The Theory of Spinors” in 1937, years after Dirac equation was well accepted. The English edition came out in 1966. Dirac was the true discoverer of spinors since they are the solutions to his equation. A closer look at Chapter III of Cartan’s book, one notice that spinors resemble the pairs of Pythagorean indices for finding the infinite number of Pythagorean triples; a good example is 3, 4, and 5 in the real number domain while spinors are in the generalized complex imaginary number domain. By this analogy, the only real “spinor” of (3,4,5) is (1,2): 2˛-1˛=3, 2˛+1˛=5, and 2(2)(1)=4 such that the general forms for finding any P-triple for the index (m,n) are m˛-n˛, 2mn, and m˛+n˛. Any P-triple denotes the real distances between the three vertices of an arbitrary right triangle such that if ��= m˛+n˛, ��=2mn, and ��= m˛-n˛ then ��˛=��˛+��˛. On the other hand, for the spinor (��₀,��), ��˛+��˛+��˛=0 must always be true by the foregoing definitions found in Chapter III of Cartan’s book.

By simply comparing the computational results derived from the real index (m,n) with the spinor (��₀,��), one sees that the ��-��-�� of spinor are equal of zero length while ��-��-�� of (m,n) have no length less than the integer 3 unless the domain includes the real irrational numbers. The former implies touching vertices while the latter implies a minimum distance of 3. This min distance can be the length of the edges of an equilateral tetrahedron of spacetime charges as the spacetime charge configuration of quarks and gluons, while the irrational cubic lattices are the spacetime charge configurations for both leptons and bosons.