0
Your cart

Your cart is empty

Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations

Buy Now

Rigorous Time Slicing Approach to Feynman Path Integrals (Hardcover, 1st ed. 2017) Loot Price: R4,350
Discovery Miles 43 500
Rigorous Time Slicing Approach to Feynman Path Integrals (Hardcover, 1st ed. 2017): Daisuke Fujiwara

Rigorous Time Slicing Approach to Feynman Path Integrals (Hardcover, 1st ed. 2017)

Daisuke Fujiwara

Series: Mathematical Physics Studies

 (sign in to rate)
Loot Price R4,350 Discovery Miles 43 500 | Repayment Terms: R408 pm x 12*

Bookmark and Share

Expected to ship within 12 - 19 working days

This book proves that Feynman's original definition of the path integral actually converges to the fundamental solution of the Schroedinger 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 Schroedinger's quantization uses the Hamiltonian function. These two methods are believed to be equivalent. But equivalence is not fully proved mathematically, because, compared with Schroedinger'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 Schroedinger 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 Schroedinger equation.

General

Imprint: Springer Verlag,Japan
Country of origin: Japan
Series: Mathematical Physics Studies
Release date: July 2017
First published: 2017
Authors: Daisuke Fujiwara
Dimensions: 235 x 155mm (L x W)
Format: Hardcover
Pages: 333
Edition: 1st ed. 2017
ISBN-13: 978-4-431-56551-2
Categories: Books > Science & Mathematics > Physics > General
Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Functional analysis
Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations
Books > Science & Mathematics > Mathematics > Applied mathematics > General
Promotions
LSN: 4-431-56551-5
Barcode: 9784431565512

Is the information for this product incomplete, wrong or inappropriate? Let us know about it.

Does this product have an incorrect or missing image? Send us a new image.

Is this product missing categories? Add more categories.

Review This Product

No reviews yet - be the first to create one!

You might also like..

Differential Equations with…
Dennis Zill Paperback R1,373 R1,277 Discovery Miles 12 770
Differential Equations with…
Warren Wright, Dennis Zill Paperback  (1)
R1,424 R1,320 Discovery Miles 13 200
Differential Equations with Linear…
Matthew R. Boelkins, Jack L. Goldberg, … Hardcover R3,047 Discovery Miles 30 470
Multigrid
Ulrich Trottenberg, Cornelius W. Oosterlee, … Hardcover R2,387 Discovery Miles 23 870
Calculus and Ordinary Differential…
David Pearson Paperback R845 Discovery Miles 8 450
Differential Equations with Boundary…
Dennis Zill Paperback R1,394 R1,294 Discovery Miles 12 940
Applied Partial Differential Equations…
Richard Haberman Paperback R2,548 Discovery Miles 25 480
Schaum's Outline of Differential…
Richard Bronson, Gabriel B Costa Paperback R502 Discovery Miles 5 020
Partial Differential Equations
Osamu Sano Hardcover R2,698 Discovery Miles 26 980
Dissipative Lattice Dynamical Systems
Xiaoying Han, Peter Kloeden Hardcover R3,581 Discovery Miles 35 810
Handbook of Differential Equations
Daniel Zwillinger Hardcover R3,767 R2,729 Discovery Miles 27 290
Qualitative Theory Of Odes: An…
Henryk Zoladek, Raul Murillo Hardcover R2,575 Discovery Miles 25 750

See more

Partners