## Table of Contents

Foreword v

Acknowledgments vii

1 Introduction 1

2 Functions 5

2.1 Gamma function 6

2.2 functions Mittag-Leffler 7

2.3 Hypergeometric functions 9

2.4 Miscellaneous functions 9

3 The Fractional Derivative 11

3.1 Basics 11

3.2 The fractional Leibniz product rule 16

3.3 Discussion 18

3.3.1 Orthogonal polynomials 18

3.3.2 Differential representation of the Riemann fractional derivative 20

4 Friction Forces 23

4.1 Classical description 23

4.2 Fractional friction 26

5 Fractional Calculus 33

5.1 The Fourier transform 34

5.2 The fractional integral 35

5.2.1 The Liouville fractional integral 36

5.2.2 The Riemann fractional integral 36

5.3 Correlation of fractional integration and differentiation 37

5.3.1 The Liouville fractional derivative 38

5.3.2 The Riemann fractional derivative 39

5.3.3 The Liouville fractional derivative with inverted operator sequence: the Liouville-Caputo fractional derivative 40

5.3.4 The Riemann fractional derivative with inverted operator sequence: the Caputo fractional derivative 42

5.4 Fractional derivative of second order 43

5.4.1 The Riesz fractional derivative 44

5.4.2 The Feller fractional derivative 46

5.5 Fractional derivatives of higher orders 47

5.6 Geometric interpretation of the fractional integral 51

5.7 Low level fractionality 53

5.8 Discussion 55

5.8.1 Semigroup property of the fractional integral 55

6 The Fractional Harmonic Oscillator 57

6.1 The fractional harmonic oscillator 58

6.2 The harmonic oscillator according to Fourier 58

6.3 The harmonic oscillator according to Riemann 60

6.4 The harmonic oscillator according to Caputo 62

7 Wave Equations and Parity 65

7.1 Fractional wave equations 65

7.2 Parity and time-reversal 67

7.3 Solutions of the free regularized fractional wave equation 68

8 Nonlocality and Memory Effects 75

8.1 A short history of nonlocal concepts 75

8.2 From local to nonlocal operators 77

8.3 Memory effects 88

9 Quantum Mechanics 93

9.1 Canonical quantization 95

9.2 Quantization of the classical Hamilton function and free solutions 95

9.3 Temperature dependence of a fission yield and determination of the corresponding fission potential 99

9.4 The fractional Schrödinger equation with an infinite well potential 102

9.5 Radial solutions of the fractional Schrödinger equation 107

10 Fractional Spin: a Property of Particles Described with the Fractional Schrödinger Equation 111

10.1 Spin 112

10.2 Fractional spin 113

11 Factorization 117

11.1 The Dirac equation 117

11.2 The fractional Dirac equation 118

11.3 The fractional Pauli equation 120

12 Symmetries 123

12.1 Characteristics of fractional group theory 124

12.2 The fractional rotation group SO^{α}_{N} 126

13 The Fractional Symmetric Rigid Rotor 133

13.1 The spectrum of the fractional symmetric rigid rotor 133

13.2 Rotational limit 136

13.3 Vibrational limit 137

13.4 Davidson potential: the so called γ-unstable limit 138

13.5 Linear potential limit 140

13.6 The magic limit 141

13.7 Comparison with experimental data 144

14 q-deformed Lie Algebras and Fractional Calculus 153

14.1 q-deformed Lie algebras 153

14.2 The fractional q-deformed harmonic oscillator 156

14.3 The fractional q-deformed symmetric rotor 160

14.4 Half-integer representations of the fractional rotation group SO^{α}(3) 162

15 Fractional Spectroscopy of Hadrons 165

15.1 Phenomenology of the baryon spectrum 166

15.2 Charmonium 171

15.3 Phenomenology of meson spectra 176

15.4 Metaphysics: About the internal structure of quarks 184

16 Higher Dimensional Fractional Rotation Groups 187

16.1 The four decompositions of the mixed fractional SO^{α} (9) 187

16.2 Notation 189

16.3 The nine dimensional fractional Caputo-Riemann-Riemann symmetric rotor 192

16.4 Magic numbers of nuclei 193

16.5 Ground state properties of nuclei 196

16.6 Fine structure of the single particle spectrum: the extended Caputo-Riemann-Riemann symmetric rotor 201

16.7 Magic numbers of electronic clusters: the nine dimensional fractional Caputo-Caputo-Riemann symmetric rotor 206

16.8 Binding energy of electronic clusters 210

16.9 Metaphysics: magic numbers for clusters bound by weak and gravitational forces respectively 213

17 Fractors: Fractional Tensor Calculus 219

17.1 Covariance for fractional tensors 219

17.2 Singular fractional tensors 220

18 Fractional Fields 223

18.1 Fractional Euler-Lagrange equations 224

18.2 The fractional Maxwell equations 227

19 Gauge Invariance in Fractional Field Theories 231

19.1 Gauge invariance in first order of the coupling constant g 232

19.2 The fractional Riemann-Liouville-Zeeman effect 236

20 Outlook 241

Bibliography 243

Index 257