By I. Cotaescu
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Extra resources for Advances in Quantum Theory
P and Q denote the paths followed by these systems. One has P (u ) = x and Q(v) = y. The double oval in the center denotes the the two systems at the point, x, of path intersection where P (t) = Q(t) = x. The interpretation maps are denoted by IP ( s ),s and IQ( s ),s for different times s. The real number structures R¯ P ( s ),s and R¯ Q( s ),s are associated with each point in the paths P and Q. Fx,t;x ,u and Fx,t:y,v are parallel transform operators that map the real number structures at the points of theory and experiment completion to the point of path intersection.
Op denotes any one of the operations, +, −, ×, ÷. Note that Fy,x is independent of paths between x and y. This follows from the requirement that for a path P from x to y and a path Q from y to z, Q∗ P Q P Fz,x = Fz,y Fy,x . (17) Here Q ∗ P is the concatenation of Q to P. If z = x then the path is cyclic and the ﬁnal structure is identical to the initial one. This gives the result that Q∗ P Fx,x = 1. (18) P is path independent so that a path label is not needed. Note that This shows that Fy,x −1 Fy,x = Fx,y .
Since the C¯ x and R¯ x are then independent of x, one can replace C¯ x and R¯ x with just one complex and real number structure, ¯ C¯ and R. 1 Scale factors for distant points The description of ry,x can be extended to points y distant from x. Let P be a path from x to P be the scale y parameterized by a real number, s, such that P (0) = x and P (1) = y. Let ry,x factor associated with the path P. If ay is a number value in C¯ y , then ay corresponds to the P a , in C ¯ x where a x = Fx,y ay is the same number value in C¯ x as ay is in C¯ y .