aqora

The product form for path integrals on curved manifolds

Ch. Grosche

Physics Letters AMar 1, 198823 citations

Feynman Path Integral Using Operator Integration in Banach Space

Reza Ahangar, Erbil Çetin, Şerife Müge Ege

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.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.

Journal of Applied Mathematics and PhysicsJan 1, 20240 citations

More about path integrals for spin

Kunio Funahashi, Tarō Kashiwa, Shuji Nima +1 more

arXiv:hep-th/9504125Nuclear Physics BOct 1, 199517 citations

Spin Path Integrals and Generations

Two consecutive measurements of the position of a non relativistic free particle will give entirely unrelated results. Recent quantum information research by G. Svetlichny, J. Tolar, and G. Chadzitaskos have explained this property of position observables as a result of a path in the Feynman integral being mathematically defined as a product of incompatible states; that is, as a product of mutually unbiased bases (MUBs). On the other hand, two consecutive measurements of a particle’s spin give identical results. This raises the question “what happens when spin path integrals are computed over products of MUBs? ” We show that the usual Pauli spin is obtained in the long time limit along with three generations of particles. We propose applications to the masses and mixing matrices of the elementary fermions. 1Two consecutive measurements of the position of a non relativistic free particle will give entirely unrelated results. Recent quantum information research by G. Svetlichny, J. Tolar, and G. Chadzitaskos have explained this property of position observables as a result of a path in the Feynman integral being mathematically defined as a product of incompatible states; that is, as a product of mutually unbiased bases (MUBs). On the other hand, two consecutive measurements of a particle’s spin give identical results. This raises the question “what happens when spin path integrals are computed over products of MUBs? ” We show that the usual Pauli spin is obtained in the long time limit along with three generations of particles. We propose applications to the masses and mixing matrices of the elementary fermions. 1

Jan 1, 20090 citations

Feynman path integral for a particle with spin

W. Garczyński

Jan 1, 19730 citations

Path Integrals and Hamiltonians: Principles and Methods

Belal Ehsan Baaquie

Introduces the powerful and flexible combination of Hamiltonian operators and path integrals in quantum mathematics.

May 26, 201417 citations

Path Integrals and Product Integrals

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