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E-raamat: Path Integrals, Hyperbolic Spaces And Selberg Trace Formulae

(Univ Hamburg & Stadtteilschule Walddorfer, Germany)
  • Formaat: 292 pages
  • Ilmumisaeg: 29-Feb-1996
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814499767
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Path Integrals, Hyperbolic Spaces And Selberg Trace FormulaeSuurem pilt
  • Formaat: 292 pages
  • Ilmumisaeg: 29-Feb-1996
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814499767
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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 discussion of integrable billiards in circles and spheres (flat and hyperbolic space) 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(2)
List of Figures
xi
1 Introduction
1(7)
2 Path Integrals in Quantum Mechanics
8(26)
2.1 The Feynman Path Integral
8(5)
2.2 Defining the Path Integral
13(3)
2.3 Transformation Techniques.
16(5)
2.3.1 Point Canonical Transformations.
16(1)
2.3.2 Space-Time Transformations.
17(1)
2.3.3 Separation of Variables.
18(3)
2.4 Group Path Integration
21(3)
2.5 Klein-Gordon Particle
24(1)
2.6 Basic Path Integrals.
25(9)
2.6.1 The Quadratic Lagrangian.
25(1)
2.6.2 The Radial Harmonic Oscillator
26(1)
2.6.3 The Poschl-Teller Potential
26(2)
2.6.4 The Modified Poschl-Teller Potential.
28(1)
2.6.5 The O(2,2)-Hyperboloid
29(3)
2.6.6 Miscellaneous Results.
32(2)
3 Separable Coordinate Systems on Spaces of Constant Curvature
34(16)
3.1 Separation of Variables and Breaking of Symmetry.
34(5)
3.2 Classification of Coordinate Systems.
39(2)
3.3 Coordinate Systems in Spaces of Constant Curvature
41(9)
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.
44(2)
3.3.5 Pseudo-Euclidean Space.
46(2)
3.3.6 A Hilbert Space Model.
48(2)
4 Path Integrals in Pseudo-Euclidean Geometry
50(25)
4.1 The Pseudo-Euclidean Plane.
50(11)
4.2 Three-Dimensional Pseudo-Euclidean Space
61(14)
5 Path Integrals in Euclidean Spaces
75(11)
5.1 Two-Dimensional Euclidean Space.
75(3)
5.2 Three-Dimensional Euclidean Space.
78(8)
6 Path Integrals on Spheres
86(10)
6.1 The Two-Dimensional Sphere.
86(5)
6.2 The Three-Dimensional Sphere.
91(5)
7 Path Integrals on Hyperboloids
96(24)
7.1 The Two-Dimensional Pseudosphere.
96(8)
7.2 The Three-Dimensional Pseudosphere.
104(16)
8 Additional Results on Path Integration in Hyperbolic Spaces
120(10)
8.1 The Single-Sheeted Hyperboloid.
120(2)
8.2 The D-Dimensional Pseudosphere.
122(3)
8.3 Hyperbolic Rank-One Spaces.
125(5)
9 Billiard Systems and Periodic Orbit Theory
130(17)
9.1 Some Elements of Periodic Orbit Theory
130(3)
9.2 A Billiard System in a Hyperbolic Rectangle.
133(14)
10 The Selberg Trace Formula
147(41)
10.1 The Selberg Trace Formula in Mathematical Physics.
147(2)
10.2 Applications and Generalizations.
149(14)
10.3 The Selberg Trace Formula on Riemann Surfaces.
163(14)
10.3.1 The Selberg Zeta-Function.
171(3)
10.3.2 Determinants of Maass-Laplacians.
174(3)
10.4 The Selberg Trace Formula on Bordered Riemann Surfaces.
177(11)
10.4.1 The Selberg Zeta-Function.
184(2)
10.4.2 Determinants of Maass-Laplacians.
186(2)
11 The Selberg Super-Trace Formula
188(36)
11.1 Automorphisms on Super-Riemann Surfaces.
188(12)
11.1.1 Closed Super-Riemann Surfaces.
193(1)
11.1.2 Compact Fundamental Domain.
193(2)
11.1.3 Non-Compact Fundamental Domain.
195(5)
11.2 Selberg Super-Zeta-Functions.
200(8)
11.2.1 The Selberg Super-Zeta-Function Z(0)
201(3)
11.2.2 The Selberg Super-Zeta-Function Z(1).
204(2)
11.2.3 The Selberg Super-Zeta-Function Z(s)
206(2)
11.3 Super-Determinants of Dirac Operators.
208(2)
11.4 The Selberg Super-Trace Formula on Bordered Super-Riemann Surfaces.
210(6)
11.4.1 Compact Fundamental Domain.
212(2)
11.4.2 Non-Compact Fundamental Domain.
214(2)
11.5 Selberg Super-Zeta-Functions.
216(6)
11.5.1 The Selberg Super-Zeta-Function R(0)
217(1)
11.5.2 The Selberg Super-Zeta-Function R(1)
218(2)
11.5.3 The Selberg Super-Zeta-Function Z(s).
220(2)
11.6 Super-Determinants of Dirac Operators.
222(2)
12 Summary and Discussion
224(15)
12.1 Results on Path Integrals.
224(8)
12.2 Results on Trace Formulae.
232(1)
12.3 Miscellaneous Results, Final Remarks, and Outlook.
233(6)
Bibliography 239(38)
Index 277
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