國立高雄大學圖資館 |

語系: 繁體中文

說明(常見問題)

圖資館首頁

登入

回首頁

切換: 標籤 | MARC模式 | ISBD

Rigorous time slicing approach to Fe...

Fujiwara, Daisuke.

Rigorous time slicing approach to Feynman path integrals

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Rigorous time slicing approach to Feynman path integralsby Daisuke Fujiwara.
作者:	Fujiwara, Daisuke.
出版者:	Tokyo :Springer Japan :2017.
面頁冊數:	ix, 333 p. :ill., digital ;24 cm.
Contained By:	Springer eBooks
標題:	Feynman integrals.
電子資源:	http://dx.doi.org/10.1007/978-4-431-56553-6
ISBN:	9784431565536$q(electronic bk.)

Rigorous time slicing approach to Feynman path integrals
Fujiwara, Daisuke.

Rigorous time slicing approach to Feynman path integrals[electronic resource] /by Daisuke Fujiwara. - Tokyo :Springer Japan :2017. - ix, 333 p. :ill., digital ;24 cm. - Mathematical physics studies,0921-3767. - Mathematical physics studies..

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

ISBN: 9784431565536$q(electronic bk.)

Standard No.: 10.1007/978-4-431-56553-6doiSubjects--Topical Terms:

468973
Feynman integrals.

LC Class. No.: QC174.17.F45

Dewey Class. No.: 530.12

Rigorous time slicing approach to Feynman path integrals
LDR:04107nmm a2200313 a 4500 001 517867
003 DE-He213
005 20180118095446.0
006 m d
007 cr nn 008maaau
008 180316s2017 ja s 0 eng d
020 $a 9784431565536$q(electronic bk.)
020 $a 9784431565512$q(paper)
024 7 $a 10.1007/978-4-431-56553-6 $2 doi
035 $a 978-4-431-56553-6
040 $a GP $c GP
041 0 $a eng
050 4 $a QC174.17.F45
072 7 $a PHU $2 bicssc
072 7 $a SCI040000 $2 bisacsh
082 0 4 $a 530.12 $2 23
090 $a QC174.17.F45 $b F961 2017
100 1 $a Fujiwara, Daisuke. $3 787703
245 1 0 $a Rigorous time slicing approach to Feynman path integrals $h [electronic resource] / $c by Daisuke Fujiwara.
260 $a Tokyo : $b Springer Japan : $b Imprint: Springer, $c 2017.
300 $a ix, 333 p. : $b ill., digital ; $c 24 cm.
490 1 $a Mathematical physics studies, $x 0921-3767
520 $a This book proves that Feynman's original definition of the path integral actually converges to the fundamental solution of the Schrodinger 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 Schrodinger's quantization uses the Hamiltonian function. These two methods are believed to be equivalent. But equivalence is not fully proved mathematically, because, compared with Schrodinger'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 Schrodinger 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 Schrodinger equation.
650 0 $a Feynman integrals. $3 468973
650 1 4 $a Mathematics. $3 184409
650 2 4 $a Mathematical Physics. $3 522725
650 2 4 $a Functional Analysis. $3 274845
650 2 4 $a Partial Differential Equations. $3 274075
650 2 4 $a Fourier Analysis. $3 273776
710 2 $a SpringerLink (Online service) $3 273601
773 0 $t Springer eBooks
830 0 $a Mathematical physics studies. $3 684220
856 4 0 $u http://dx.doi.org/10.1007/978-4-431-56553-6
950 $a Mathematics and Statistics (Springer-11649)