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
By virtue of the generalized Hellmann-Feynman theorem for the ensemble average,we obtain the internal energy and average energy consumed by the resistance R in a quantized resistance-inductance-capacitance (RLC) elect...By virtue of the generalized Hellmann-Feynman theorem for the ensemble average,we obtain the internal energy and average energy consumed by the resistance R in a quantized resistance-inductance-capacitance (RLC) electric circuit.We also calculate the entropy-variation with R.The relation between entropy and R is also derived.By the use of figures we indeed see that the entropy increases with the increment of R.展开更多
The problem in the criterion often used to judge whether the Feynman scaling in fragmentation region is violated or not is discussed.Subtracting the contamination of central region particles from the large pseudo-rapi...The problem in the criterion often used to judge whether the Feynman scaling in fragmentation region is violated or not is discussed.Subtracting the contamination of central region particles from the large pseudo-rapidity region by Monte-Carlo simulation,the reasonable pseudo-rapidity distribution in fragmentation region is obtained and a possible signal on the scaling violation in fragmentation region is seen.展开更多
Based on the solution to Bargmann-Wigner equation for a particle with arbitrary half-integral spin, a directderivation of the projection operator and propagator for a particle with arbitrary half-integral spin is work...Based on the solution to Bargmann-Wigner equation for a particle with arbitrary half-integral spin, a directderivation of the projection operator and propagator for a particle with arbitrary half-integral spin is worked out. Theprojection operator constructed by Behrends and Fronsdal is re-deduced and confirmed and simplified, the generalcommutation rules and Feynman propagator with additional non-covariant terms for a free particle with arbitraryhalf-integral spin are derived, and explicit expressions for the propagators for spins 3/2, 5/2 and 7/2 are provided.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Project supported by the National Natural Science Foundation of China (Grant Nos.10775097 and 10874174)the Research Foundation of the Education Department of Jiangxi Province of China (Grant No.GJJ10097)
文摘By virtue of the generalized Hellmann-Feynman theorem for the ensemble average,we obtain the internal energy and average energy consumed by the resistance R in a quantized resistance-inductance-capacitance (RLC) electric circuit.We also calculate the entropy-variation with R.The relation between entropy and R is also derived.By the use of figures we indeed see that the entropy increases with the increment of R.
文摘The problem in the criterion often used to judge whether the Feynman scaling in fragmentation region is violated or not is discussed.Subtracting the contamination of central region particles from the large pseudo-rapidity region by Monte-Carlo simulation,the reasonable pseudo-rapidity distribution in fragmentation region is obtained and a possible signal on the scaling violation in fragmentation region is seen.
文摘Based on the solution to Bargmann-Wigner equation for a particle with arbitrary half-integral spin, a directderivation of the projection operator and propagator for a particle with arbitrary half-integral spin is worked out. Theprojection operator constructed by Behrends and Fronsdal is re-deduced and confirmed and simplified, the generalcommutation rules and Feynman propagator with additional non-covariant terms for a free particle with arbitraryhalf-integral spin are derived, and explicit expressions for the propagators for spins 3/2, 5/2 and 7/2 are provided.
