The Quantum theory also places emphasis on the rational numbers. There are
states with rational values of "angular momentum". Quanta come in integer
numbers. Charge, after Millikan's experiments is recognized as a rational
entity.
How do you put these physical ideas together. The first job is to formulate what is a topological property. The simplest (but not complete) definition is that a topological property is a deformation invariant. Using the Lie derivative with respect to a vector field V as the description of an evolutionary process permits the closed integrals of closed forms to be defined as deformation invariants. These closed integrals of closed forms are called deRham period integrals. Their values have rational ratios.
The article below was an attempt to utilze this idea as a marriage between
topology and quantum mechanics. In the article, the two 3 forms of A^H and A^F
were utilized, as well as the 2form H and the 1form A. Properties of these
species of period integrals were examined as applied to various physical
theories.
