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E-raamat: Path Integrals, Hyperbolic Spaces And Selberg Trace Formulae (2nd Edition)

(Univ Hamburg & Stadtteilschule Walddorfer, Germany)
  • Formaat: 388 pages
  • Ilmumisaeg: 26-Jul-2013
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814460095
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Path Integrals, Hyperbolic Spaces And Selberg Trace Formulae (2nd Edition)Suurem pilt
  • Formaat: 388 pages
  • Ilmumisaeg: 26-Jul-2013
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814460095
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In this second edition, a comprehensive review is given for path integration in two- and three-dimensional (homogeneous) spaces of constant and non-constant curvature, including an enumeration of all the corresponding coordinate systems which allow separation of variables in the Hamiltonian and in the path integral. The corresponding path integral solutions are presented as a tabulation. Proposals concerning interbasis expansions for spheroidal coordinate systems are also given. In particular, the cases of non-constant curvature Darboux spaces are new in this edition.The volume also contains results on the numerical study of the properties of several integrable billiard systems in compact domains (i.e. rectangles, parallelepipeds, circles and spheres) in two- and three-dimensional flat and hyperbolic spaces. In particular, the discussions of integrable billiards in circles and spheres (flat and hyperbolic spaces) and in three dimensions are new in comparison to the first edition.In addition, an overview is presented on some recent achievements in the theory of the Selberg trace formula on Riemann surfaces, its super generalization, their use in mathematical physics and string theory, and some further results derived from the Selberg (super-) trace formula.
List of Tables
ix
List of Figures
xi
Preface xiii
1 Introduction
1(6)
2 Path Integrals in Quantum Mechanics
7(28)
2.1 The Feynman Path Integral
7(6)
2.2 Defining the Path Integral
13(3)
2.3 Transformation Techniques
16(4)
2.3.1 Point Canonical Transformations
16(1)
2.3.2 Space-Time Transformations
16(2)
2.3.3 Separation of Variables
18(2)
2.4 Group Path Integration
20(3)
2.5 Klein-Gordon Particle
23(1)
2.6 Basic Path Integrals
24(11)
2.6.1 The Quadratic Lagrangian
24(1)
2.6.2 The Radial Harmonic Oscillator
25(1)
2.6.3 The Poschl--Teller Potential
25(2)
2.6.4 The Modified Poschl--Teller Potential
27(1)
2.6.5 Parametric Path-Integrals
27(1)
2.6.6 The O(2,2)-Hyperboloid
28(4)
2.6.7 δ-Functions and Boundary Problems
32(2)
2.6.8 Miscellaneous Results
34(1)
3 Separable Coordinate Systems on Spaces of Constant Curvature
35(16)
3.1 Separation of Variables and Breaking of Symmetry
35(4)
3.2 Classification of Coordinate Systems
39(2)
3.3 Coordinate Systems in Spaces of Constant Curvature
41(10)
3.3.1 Classification of Coordinate Systems
42(2)
3.3.2 The Sphere
44(1)
3.3.3 Euclidean Space
44(1)
3.3.4 The Pseudosphere
45(2)
3.3.5 Pseudo-Euclidean Space
47(1)
3.3.6 A Hilbert Space Model
48(3)
4 Path Integrals in Pseudo-Euclidean Geometry
51(24)
4.1 The Pseudo-Euclidean Plane
51(11)
4.2 Three-Dimensional Pseudo-Euclidean Space
62(13)
5 Path Integrals in Euclidean Spaces
75(12)
5.1 Two-Dimensional Euclidean Space
75(3)
5.2 Three-Dimensional Euclidean Space
78(9)
6 Path Integrals on Spheres
87(16)
6.1 The Two-Dimensional Sphere
87(5)
6.2 The Three-Dimensional Sphere
92(11)
7 Path Integrals on Hyperboloids
103(24)
7.1 The Two-Dimensional Pseudosphere
103(8)
7.2 The Three-Dimensional Pseudosphere
111(16)
8 Path Integral on the Complex Sphere
127(20)
8.1 The Two-Dimensional Complex Sphere
127(5)
8.2 The Three-Dimensional Complex Sphere
132(6)
8.3 Path Integral Evaluations on the Complex Sphere
138(9)
8.3.1 Path Integral Representations on S3C: Part I
138(3)
8.3.2 Path Integral Representations on S3C: Part II
141(6)
9 Path Integrals on Hermitian Hyperbolic Space
147(8)
9.1 Hermitian Hyperbolic Space HH(2)
147(3)
9.2 Path Integral Evaluations on HH(2)
150(5)
10 Path Integrals on Darboux Spaces
155(24)
10.1 Two-Dimensional Darboux Spaces
155(6)
10.2 Path Integral Evaluations
161(8)
10.2.1 Darboux Space DI
161(1)
10.2.2 Darboux Space DII
162(1)
10.2.3 Darboux Space DIII
163(3)
10.2.4 Darboux Space DIV
166(3)
10.3 Three-Dimensional Darboux Spaces
169(10)
10.3.1 The Three-Dimensional Darboux Space D3d--I
169(3)
10.3.2 The Three-Dimensional Darboux Space D3d--II
172(2)
10.3.3 Path Integral Evaluations on Three-Dimensional Darboux Space
174(5)
11 Path Integrals on Single-Sheeted Hyperboloids
179(14)
11.1 The Two-Dimensional Single-Sheeted Hyperboloid
179(14)
12 Miscellaneous Results on Path Integration
193(12)
12.1 The D-Dimensional Pseudosphere
193(2)
12.2 Hyperbolic Rank-One Spaces
195(5)
12.3 Path Integral on SU(n) and SU(n -- 1,1)
200(5)
12.3.1 Path Integral on SU(n)
200(2)
12.3.2 Path Integral on SU(n -- 1,1)
202(3)
13 Billiard Systems and Periodic Orbit Theory
205(28)
13.1 Some Elements of Periodic Orbit Theory
205(3)
13.2 A Billiard System in a Hyperbolic Rectangle
208(13)
13.3 Other Integrable Billiards in Two and Three Dimensions
221(6)
13.3.1 Flat Billiards
222(1)
13.3.2 Hyperbolic Billiards
223(4)
13.4 Numerical Investigation of Integrable Billiard Systems
227(6)
13.4.1 Two-Dimensional Systems
227(2)
13.4.2 Three-Dimensional Systems
229(4)
14 The Selberg Trace Formula
233(40)
14.1 The Selberg Trace Formula in Mathematical Physics
233(2)
14.2 Applications and Generalizations
235(13)
14.3 The Selberg Trace Formula on Riemann Surfaces
248(13)
14.3.1 The Selberg Zeta-Function
256(2)
14.3.2 Determinants of Maass-Laplacians
258(3)
14.4 The Selberg Trace Formula on Bordered Riemann Surfaces
261(12)
14.4.1 The Selberg Zeta-Function
268(2)
14.4.2 Determinants of Maass-Laplacians
270(3)
15 The Selberg Super-Trace Formula
273(38)
15.1 Automorphisms on Super-Riemann Surfaces
273(12)
15.1.1 Closed Super-Riemann Surfaces
277(1)
15.1.2 Compact Fundamental Domain
278(2)
15.1.3 Non-Compact Fundamental Domain
280(5)
15.2 Selberg Super-Zeta-Functions
285(8)
15.2.1 The Selberg Super-Zeta-Function Z0
285(3)
15.2.2 The Selberg Super-Zeta-Function Z1
288(3)
15.2.3 The Selberg Super-Zeta-Function ZS
291(2)
15.3 Super-Determinants of Dirac Operators
293(2)
15.4 The Selberg Super-Trace Formula on Bordered Super-Riemann Surfaces
295(6)
15.4.1 Compact Fundamental Domain
297(2)
15.4.2 Non-Compact Fundamental Domain
299(2)
15.5 Selberg Super-Zeta-Functions
301(6)
15.5.1 The Selberg Super-Zeta-Function R0
302(2)
15.5.2 The Selberg Super-Zeta-Function R1
304(1)
15.5.3 The Selberg Super-Zeta-Function Zs
305(2)
15.6 Super-Determinants of Dirac Operators
307(2)
15.7 Asymptotic Distributions on Super-Riemann Surfaces
309(2)
16 Summary and Discussion
311(18)
16.1 Results on Path Integrals
311(13)
16.1.1 General Results
311(1)
16.1.2 Higher Dimensions
312(5)
16.1.3 Super-Integrable Potentials in Spaces of Non-Constant Curvature
317(4)
16.1.4 Listing the Path Integral Representations
321(3)
16.2 Results on Trace Formulae
324(1)
16.3 Miscellaneous Results, Final Remarks, and Outlook
325(4)
Bibliography 329(40)
Index 369
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