# Rigorous Time Slicing Approach to Feynman Path Integrals

English | PDF | 2017 | 333 Pages | ISBN : 4431565515 | 3.7 MB

This book proves that Feynman's original definition of the path integral actually converges to the fundamental solution of the Schr?dinger equation at least in the short term if the potential is differentiable sufficiently many times and its derivatives of order equal to or higher than two are bounded. The semi-classical asymptotic formula up to the second term of the fundamental solution is also proved by a method different from that of Birkhoff.

A bound of the remainder term is also proved.The Feynman path integral is a method of quantization using the Lagrangian function, whereas Schr?dinger's quantization uses the Hamiltonian function. These two methods are believed to be equivalent. But equivalence is not fully proved mathematically, because, compared with Schr?dinger's method, there is still much to be done concerning rigorous mathematical treatment of Feynman's method. Feynman himself defined a path integral as the limit of a sequence of integrals over finite-dimensional spaces which is obtained by dividing the time interval into small pieces. This method is called the time slicing approximation method or the time slicing method.This book consists of two parts. Part I is the main part. The time slicing method is performed step by step in detail in Part I. The time interval is divided into small pieces. Corresponding to each division a finite-dimensional integral is constructed following Feynman's famous paper. This finite-dimensional integral is not absolutely convergent. Owing to the assumption of the potential, it is an oscillatory integral. The oscillatory integral techniques developed in the theory of partial differential equations are applied to it. It turns out that the finite-dimensional integral gives a finite definite value. The stationary phase method is applied to it. Basic properties of oscillatory integrals and the stationary phase method are explained in the book in detail.Those finite-dimensional integrals form a sequence of approximation of the Feynman path integral when the division goes finer and finer. A careful discussion is required to prove the convergence of the approximate sequence as the length of each of the small subintervals tends to 0. For that purpose the book uses the stationary phase method of oscillatory integrals over a space of large dimension, of which the detailed proof is given in Part II of the book. By virtue of this method, the approximate sequence converges to the limit. This proves that the Feynman path integral converges. It turns out that the convergence occurs in a very strong topology. The fact that the limit is the fundamental solution of the Schr?dinger equation is proved also by the stationary phase method. The semi-classical asymptotic formula naturally follows from the above discussion.A prerequisite for readers of this book is standard knowledge of functional analysis. Mathematical techniques required here are explained and proved from scratch in Part II, which occupies a large part of the book, because they are considerably different from techniques usually used in treating the Schr?dinger equation.**Download:**

http://longfiles.com/lqvoy4zuhpcp/Rigorous_Time_Slicing_Approach_to_Feynman_Path_Integrals.pdf.html

**[Fast Download] Rigorous Time Slicing Approach to Feynman Path Integrals**

**Ebooks related to "Rigorous Time Slicing Approach to Feynman Path Integrals" :**

Mathematica Beyond Mathematics: The Wolfram Language in the Real World

Free Probability and Random Matrices

Optimization in Engineering: Models and Algorithms

Regularity and Stochasticity of Nonlinear Dynamical Systems

Stochastic Communities: A Mathematical Theory of Biodiversity

Complex Analysis and Geometry: KSCV10, Gyeongju, Korea, August 2014

Lie Sphere Geometry

Elementary Topology

Applied Regression Analysis

Soliton Equations and Hamiltonian Systems, 2nd Edition

**Copyright Disclaimer:**

This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.

**Categories**

Astronomy and Cosmology | Physics |

Philosophy | Medicine |

Mathematics |
DSP |

Cryptography | Chemistry |

Biology and Genetics | Psychology and Behavior |

**Popular ebooks**

An Introduction to Fourier Analysis(3690)

Introductory Statistics, 10th Global Edi(3559)

Introduction to Probability(2995)

Game Theory: A Playful Introduction(2823)

C++ for Financial Mathematics(2624)

Essential Statistics, Global Edition(2577)

Higher Engineering Mathematics(2541)

Simulation and the Monte Carlo Method, Thi(2518)

Discrete Mathematics, 7th Edition(2412)

A Friendly Introduction to Differential Eq(2331)

Calculus Problems(2241)

Probability: For the Enthusiastic Beginner(2087)

An Introduction to Statistical Methods and(2074)

Idiot's Guides: Calculus I(2073)

Statistics: Informed Decisions Using Data(1991)