Quantum Mechanics In The Space Of Distributions, Feynman Path Integrals, And Nonstandard Analysis (l(2) Transition Probability)

We will extend Nonrelativistic Quantum Mechanics as a theory in $L\sp2$ to a theory in the space of distributions. We will provide 3 major theories of Nonrelativist Quantum Mechanics. First, we will extend the concept of an integral kernel for the evolution operator to a distribution kernel for the $L\sp2$ transition probability amplitude. Second, we will extend the $L\sp2$ Schrodinger's equation to a distributions Schrodinger's equation. Lastly, we will rigorously prove that; Feynman's original formulation of the real time, time- sliced path integral is well defined when formulated on the $L\sp2$ transition probability amplitude