Journal article
A global solution to the Schrödinger equation: From Henstock to Feynman
Journal of mathematical physics, Vol.56(9), p.1
09/01/2015
DOI: 10.1063/L4930250
Abstract
One of the key elements of Feynman's formulation of non-relativistic quantum mechanics is a so-called Feynman path integral. It plays an important role in the theory, but it appears as a postulate based on intuition, rather than a well-defined 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 non-absolute 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 space-time solutions. A physical solution to the non-relativistic Schrodinger equation must be global, and it must be given in the form of a unitary one-parameter group in L 2(R n ). The purpose of this paper is to show that a system of one-dimensional local Muldowney's solutions may be extended to yield a global solution. Moreover, the global extension can be represented by a unitary one-parameter group acting in L 2(R n ).
Details
- Title: Subtitle
- A global solution to the Schrödinger equation: From Henstock to Feynman
- Creators
- Ekaterina S NathansonPalle ET Jørgensen
- Resource Type
- Journal article
- Publication Details
- Journal of mathematical physics, Vol.56(9), p.1
- Publisher
- American Institute of Physics
- DOI
- 10.1063/L4930250
- ISSN
- 0022-2488
- eISSN
- 1089-7658
- Language
- English
- Date published
- 09/01/2015
- Academic Unit
- Mathematics
- Record Identifier
- 9984240876502771
Metrics
11 Record Views