Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits...Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.展开更多
The Aharonov-Bohm(AB)effect is an important discovery of quantum theory.It serves as a surprising quantum phenomenon in which an electrically charged particle can be affected by an electromagnetic potential,despite be...The Aharonov-Bohm(AB)effect is an important discovery of quantum theory.It serves as a surprising quantum phenomenon in which an electrically charged particle can be affected by an electromagnetic potential,despite being confined to a region in which both the magnetic field and electric field are zero.This fact gives the electromagnetic potentials greater significance in quantum physics than in classical physics.The original AB effect belongs to an“electromagnetic type”.A certain vector potential is crucial for building a certain type of AB effect.In this work,we focus on the“spin”,which is an intrinsic property of microscopic particles that has been widely accepted nowadays.First,we propose the hypothesis of spin vector potential by considering a particle with a spin operator.Second,to verify the existence of such a spin vector potential,we present a gedanken double-slit interference experiment(i.e.,the spin AB effect),which is possible to be observed in the lab.Third,we apply the spin vector potential to naturally explain why there were the Dzyaloshinsky-Moriya-type interaction and the dipole-dipole interaction between spins,and also predict a new type of spin-orbital interaction.展开更多
文摘Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.
基金supported by the National Natural Science Foundation of China(12275136 and 12075001)the 111 Project of B23045supported by the Nankai Zhide Foundations.
文摘The Aharonov-Bohm(AB)effect is an important discovery of quantum theory.It serves as a surprising quantum phenomenon in which an electrically charged particle can be affected by an electromagnetic potential,despite being confined to a region in which both the magnetic field and electric field are zero.This fact gives the electromagnetic potentials greater significance in quantum physics than in classical physics.The original AB effect belongs to an“electromagnetic type”.A certain vector potential is crucial for building a certain type of AB effect.In this work,we focus on the“spin”,which is an intrinsic property of microscopic particles that has been widely accepted nowadays.First,we propose the hypothesis of spin vector potential by considering a particle with a spin operator.Second,to verify the existence of such a spin vector potential,we present a gedanken double-slit interference experiment(i.e.,the spin AB effect),which is possible to be observed in the lab.Third,we apply the spin vector potential to naturally explain why there were the Dzyaloshinsky-Moriya-type interaction and the dipole-dipole interaction between spins,and also predict a new type of spin-orbital interaction.
