The Feynman path integral is defined over the space
RT of all possible paths; it has been a powerful tool to develop Quantum Mechanics. The absolute value of Feynman's integrand is not integrable, then Lebesgue integration theory could not be used by Feynman. However, it exists formally as a Henstock integral (which does not require the measure concept) and is a suitable alternative to the ordinary integrals that normally appear in path integrals. Feynman proved the equivalence of his theory with the traditional formulation of Quantum Mechanics, since his path integral satisfies Schrödinger's equation. On the other hand, Feynman's path integral is related to the diagrams of Feynman. For the application of this integral in Feynman's diagrams it is necessary to exchange the integral
∫RT and the series. We discuss the impossibility to exchange the integral and the sum, considering integral of Henstock and the version of Dominated Convergence Theorem. Even it has not been proved through the several mathematical formalisms that have been used.
Abstract In Chapter 9 we answer Zeilinger’s question, “What’s really going on?” concerning how Nature works. In other words, we present an ontology over which the constraints NPRF + h and NPRF + c range and operate. We contrast contextuality with reductionism and relationalism, then discuss relational quantum mechanics. We explain our version of multiscale contextual emergence called quantum-classical contextuality. Accordingly, bodily (classical) objects interact via the exchange of quanta of energy-momentum. NPRF + c constrains the distribution of worldtubes of bodily objects in spacetime while NPRF + h constrains the distribution of quanta exchanged between bodily objects. Thus, NPRF + h is an “all-at-once” or “atemporal” constraint on the distribution of quanta among bodily objects in spacetime. We call such a constraint an adynamical global constraint (AGC). Feynman’s path integral formalism for QM is contrasted with Schrödinger’s equation. We present our Poincaré symmetric ontology.
The influence of the process of exchange interaction during the lifetime of luminous atoms and molecules was studied. The opportunity of stationary states existence of interacting luminous atoms and molecules was established. With the help of Feynman integrals paths method, the mechanism of occurrence of stationary states was determined, on condition of exchange interaction taking place. The way of definition of specific conditions at which such states occur is demonstrated. Dependence of a matrix element of potential accounting exchange interaction on coordinates of electrons and their wave functions was obtained.