Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries

摘要

Computing in hypernumber arithmetics is discussed, and specifically that of M-algebra, which includes the operations of complex, quaternion, and Cayley numbers (octaves or octonions) as subsets of itself. It is shown that modern quantum gravitation theory requires minimally the 16-dimensional space of the author's M-arithmetic (announced in Appl. Math. and Comput., 1976, p. 211 f. and 1978, p. 45 f.), which has 4320 units (including positive and negative) rather than the mere 2 units of ordinary or “real” arithmetic, the 4 units of complex arithmetic, the 24 units of quaternion arithmetic, or the 240 units of octonion or octave arithmetic. Thus computer programming is the natural tool for computations in advanced quantum physics. It turns out that more than three kinds of i-type hypernumbers and more than three kinds of the \Ge-type are needed to ensure the necessary nilpotent and noncommutative algebra required in unified field theory. It is also shown that “more than three” here means “at least seven”; and it turns out that a 16-dimensional arithmetic is needed for such computations. The following paper contextualizes, characterizes, and specifies that arithmetic as the apex of a hierarchy susceptible of clear geometric definition. And the hypernumbers needed in quantized unified field theory are specified.