A global solution to the Schrödinger equation: From Henstock to Feynman
One of the key elements of Feynman’s formulation of nonrelativistic quantum mechanics is a socalled Feynman path integral. It plays an important role in the theory, but it appears as a postulate based on intuition, rather than a welldefined object. All previous attempts to supply Feynman’s theory with rigorous mathematics underpinning, based on the physical requirements, have not been satisfactory. The difficulty comes from the need to define a measure on the infinite dimensional space of paths and to create an integral that would possess all of the properties requested by Feynman. In the present paper, we consider a new approach to defining the Feynman path integral, based on the theory developed by Muldowney [A Modern Theory of Random Variable: With Applications in Stochastic Calcolus, Financial Mathematics, and Feynman Integration (John Wiley & Sons, Inc., New Jersey, 2012)]. Muldowney uses the Henstock integration technique and deals with nonabsolute integrability of the Fresnel integrals, in order to obtain a representation of the Feynman path integral as a functional. This approach offers a mathematically rigorous definition supporting Feynman’s intuitive derivations. But in his work, Muldowney gives only local in spacetime solutions. A physical solution to the nonrelativistic Schrödinger equation must be global,more »
 Authors:

^{[1]};
^{[2]}
 School of Science and Technology, Georgia Gwinnett College, 1000 University Center Lane, Lawrenceville, Georgia 30043 (United States)
 Department of Mathematics, University of Iowa, Iowa City, Iowa 52242 (United States)
 Publication Date:
 OSTI Identifier:
 22479596
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 9; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FEYNMAN PATH INTEGRAL; MATHEMATICAL SOLUTIONS; MATHEMATICS; QUANTUM MECHANICS; RANDOMNESS; RELATIVISTIC RANGE; SCHROEDINGER EQUATION; SPACETIME; STOCHASTIC PROCESSES