In this monograph I want to give an overview and summary of two lines of research I have carried out: These are the theory of path integrals on the one hand, and Selberg trace formulae on the other. The first topic, the study of path integrals I started with my Diploma Thesis which was entitled "Das Coulombpotential im Pfadintegral" [195]. I calculated the radial path integial for the Coulomb potential, however, in a somewhat complicated way: I used a two-dimensional analogue of the Kustaanheimo- Stiefel transformation. My Diploma Thesis was the starting point for an intensive investigation of path integral formulations on curved manifolds.
Later on, I could generalize these results to the path integral formulation for some specific coordinate systems in spaces of constant curvature on the sphere and the pseudosphere, for general hyperbolic spaces of rank one, for hermitian spaces (later also fbr hermitian hyperbolic spaces), and for single-sheeted hyperboloids. I started a systematic investigation of the path integral formulation (and evaluation if possible) in spaces of constant curvature, where all coordinate systems which separate the Schrodinger equation or the path integral, respectively, were taken into account.
Motivated by string theory, in particular by the Polyakov approach to string perturbation theory which is a path integral formulation, and quantum mechanics in spaces of constant negative curvature, I started an investigation in the theory of the Selberg trace formula, i.e., quantum field theory on Riemann surfaces. The first principal achievement I presented in my Dissertation [203]. It included a thorough discussion of the Selberg super-trace formula on super-Riemann surfaces, and I could derive the trace formula for super-automorphic forms of integer weight. Analytic properties of the Selberg super-zeta-functions could be discussed by a proper choice of testfunctions in the trace formula, and super-determinants of Laplacians on superRiemann surfaces could be expressed in terms of the Selberg super-zeta-functions, thus giving well-defined expressions for all genera in the integrand of the Polyakov partition function. It is interesting to note that the Selberg trace formula can be derived by a path integration. This is true for the usual as well as the super-hyperbolic plane.

List of Tables ix

List of Figures xi

Preface xiii

1Introduction 1

2 Path Integrals in Quantum Mechanics 7

2.1 The Feynman Path Integral 7

2.2 Defining the Path Integral 13

2.3 Transformation Techniques 16

      2.3.1 Point Canonical Transformations 16

      2.3.2 Space-Time Transformations 16

      2.3.3 Separation of Variables 18

2.4 Group Path Integration 20

2.5 Klein-Gordon Particle 23

2.6 Basic Path Integrals 24

      2.6.1 The Quadratic Lagrangian 24

      2.6.2 The Radial Harmonic Oscillator 25

      2.6.3 The Poschl-Teller Potential 25

      2.6.4 The Modified Poschl-Teller Potential 27

      2.6.5 Parametric Path-Integrals 27

      2.6.6 The O(2,2)-Hyperboloid 28

      2.6.7 δ-Functions and Boundary Problems 32

      2.6.8 Miscellaneous Results 34

3 Separable Coordinate Systems on Spaces of Constant Curvature 35

3.1 Separation of Vaiiables and Breaking of Symmetry 35

3.2 Classification of Coordinate Systems 39

3.3 Coordinate Systems in Spaces of Constant Curvature 41

      3.3.1 Classification of Coordinate Systems 42

      3.3.2 The Sphere 44

      3.3.3 Euclidean Space 44

      3.3.4 The Pseudosphere 45

      3.3.5 Pseudo-Euclidean Space 47

      3.3.6 A Hilbert Space Model 48

4 Path Integrals in Pseudo-Euclidean Geometry 51

4.1 The Pseudo-Euclidean Plane 51

4.2 Three-Dimensional Pseudo-Euclidean Space 62

5 Path Integrals in Euclidean Spaces 75

5.1 Two-Dimensional Euclidean Space 75

5.2 Three-Dimensional Euclidean Space 78

6 Path Integrals on Spheres 87

6.1 The Two-Dimensional Sphere 87

6.2 The Three-Dimensional Sphere 92

7 Path Integrals on Hyperboloids 103

7.1 The Two-Dimensional Pseudosphere 103

7.2 The Three-Dimensional Pseudosphere 111

8 Path Integral on the Complex Sphere 127

8.1 The Two-Dimensional Complex Sphere 127

8.2 The Three-Dimensional Complex Sphere 132

8.3 Path Integral Evaluations on the Complex Sphere 138

      8.3.1Path Integral Representations on S₃C: Part I 138

      8.3.2 Path Integral Representations on S₃C: Part II 141

9 Path Integrals on Hermitian Hyperbolic Space 147

9.1Hermitian Hyperbolic Space HH(2) 147

9.2 Path Integral Evaluations on HH(2) 150

10 Path Integrals on Darboux Spaces 155

10.1 Two-Dimensional Darboux Spaces 155

10.2 Path Integral Evaluations 161

      10.2.1 Darboux Space D_(Ⅰ) 161

      10.2.2 Darboux Space D_(Ⅱ) 162

      10.2.3Darboux Space D_(Ⅲ)163

      10.2.4Darboux Space D_(Ⅳ) 166

10.3 Three-Dimensional Darboux Spaces 169

      10.3.1The Three-Dimensional Darboux Space D_(3d-Ⅰ) 169

      10.3.2The Three-Dimensional Darboux Space D_(3d-Ⅱ) 172

      10.3.3 Path Integral Evaluations on Three-Dimensional Darboux Space 174

11 Path Integrals on Single-Sheeted Hyperboloids 179

11.1 The Two-Dimensional Single-Sheeted Hyperboloid 179

12 Miscellaneous Results on Path Integration 193

12.1 The D-Dimensional Pseudosphere 193

12.2 Hyperbolic Rank-One Spaces 195

12.3 Path Integral on SU(n) and SU(n—1,1) 200

      12.3.1 Path Integral on SU(n) 200

      12.3.2 Path Integral on SU(n — 1,1) 202

13 Billiard Systems and Periodic Orbit Theory 205

13.1 Some Elements of Periodic Orbit Theory 205

13.2 A Billiard System in a Hyperbolic Rectangle 208

13.3 Other Integrable Billiards in Two and Three Dimensions 221

      13.3.1 Flat Billiards 222

      13.3.2 Hyperbolic Billiards 223

13.4 Numerical Investigation of Integrable Billiard Systems 227

      13.4.1 Two-Dimensional Systems 227

      13.4.2 Three-Dimensional Systems 229

14 The Selberg Trace Formula 233

14.1The Selberg Trace Formula in Mathematical Physics 233

14.2 Applications and Generalizations 235

14.3 The Selberg Trace Formula on Riemann Surfaces 248

      14.3.1 The Selberg Zeta-Function 256

      14.3.2 Determinants of Maass-Laplacians 258

14.4 The Selberg Trace Formula on Bordered Riemann Surfaces 261

      14.4.1 The Selberg Zeta-Function 268

      14.4.2 Determinants of Maass-Laplacians 270

15 The Selberg Super-Trace Formula 273

15.1 Automorphisms on Super-Riemann Surfaces 273

      15.1.1 Closed Super-Riemann Surfaces 277

      15.1.2 Compact Fundamental Domain 278

      15.1.3 Non-Compact Fundamental Domain 280

15.2 Selberg Super-Zeta-Functions 285

      15.2.1The Selberg Super-Zeta-Function Z_(0) 285

      15.2.2The Selberg Super-Zeta-Function Z_(1) 288

      15.2.3The Selberg Super-Zeta-Function Z_(S) 291

15.3 Super-Determinants of Dirac Operators 293

15.4 The Selberg Super-TYace Formula on Bordered Super-Riemann Surfaces 295

      15.4.1Compact Fundamental Domain 297

      15.4.2 Non-Compact Fundamental Domain 299

15.5 Selberg Super-Zeta-Functions 301

      15.5.1 The Selberg Super-Zeta-Function R_(0) 302

      15.5.2 The Selberg Super-Zeta-Function R_(1) 304

      15.5.3 The Selberg Super-Zeta-Function Z_(S) 305

15.6 Super-Determinants of Dirac Operators 307

15.7 Asymptotic Distributions on Super-Riemann Surfaces 309

16 Summary and Discussion 311

16.1Results on Path Integrals 311

      16.1.1 General Results 311

      16.1.2 Higher Dimensions 312

      16.1.3 Super-Integrable Potentials in Spaces of Non-Constant Curvature 317

      16.1.4 Listing the Path Integral Representations 321

16.2 Results on TYace Formulae 324

16.3 Miscellaneous Results, Final Remarks, and Outlook 325

Bibliography 329

Index 369

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