Fractional calculus : an introduction for physicists

Contents......Page 10
Foreword......Page 6
Acknowledgments......Page 8
1. Introduction......Page 14
2. Functions......Page 18
2.1 Gamma function......Page 19
2.2 Mittag-Leffler functions......Page 20
2.4 Miscellaneous functions......Page 22
3.1 Basics......Page 24
3.2 The fractional Leibniz product rule......Page 29
3.3.1 Orthogonal polynomials......Page 31
3.3.2 Differential representation of the Riemann fractional derivative......Page 33
4.1 Classical description......Page 36
4.2 Fractional friction......Page 39
5. Fractional Calculus......Page 46
5.1 The Fourier transform......Page 47
5.2 The fractional integral......Page 48
5.2.2 The Riemann fractional integral......Page 49
5.3 Correlation of fractional integration and differentiation......Page 50
5.3.1 The Liouville fractional derivative......Page 51
5.3.2 The Riemann fractional derivative......Page 52
5.3.3 The Liouville fractional derivative with inverted operator sequence: the Liouville–Caputo fractional derivative......Page 53
5.3.4 The Riemann fractional derivative with inverted operator sequence: the Caputo fractional derivative......Page 55
5.4 Fractional derivative of second order......Page 56
5.4.1 The Riesz fractional derivative......Page 57
5.4.2 The Feller fractional derivative......Page 59
5.5 Fractional derivatives of higher orders......Page 60
5.6 Geometric interpretation of the fractional integral......Page 64
5.7 Low level fractionality......Page 66
5.8.1 Semigroup property of the fractional integral......Page 68
6. The Fractional Harmonic Oscillator......Page 70
6.2 The harmonic oscillator according to Fourier......Page 71
6.3 The harmonic oscillator according to Riemann......Page 73
6.4 The harmonic oscillator according to Caputo......Page 75
7.1 Fractional wave equations......Page 78
7.2 Parity and time-reversal......Page 80
7.3 Solutions of the free regularized fractional wave equation......Page 81
8.1 A short history of nonlocal concepts......Page 88
8.2 From local to nonlocal operators......Page 90
8.3 Memory effects......Page 101
9. Quantum Mechanics......Page 106
9.2 Quantization of the classical Hamilton function and free solutions......Page 108
9.3 Temperature dependence of a fission yield and determination of the corresponding fission potential......Page 112
9.4 The fractional Schr¨odinger equation with an infinite well potential......Page 115
9.5 Radial solutions of the fractional Schr¨odinger equation......Page 120
10. Fractional Spin: a Property of Particles Described with the Fractional Schr¨odinger Equation......Page 124
10.1 Spin......Page 125
10.2 Fractional spin......Page 126
11.1 The Dirac equation......Page 130
11.2 The fractional Dirac equation......Page 131
11.3 The fractional Pauli equation......Page 133
12. Symmetries......Page 136
12.1 Characteristics of fractional group theory......Page 137
12.2 The fractional rotation group SO  N......Page 139
13.1 The spectrum of the fractional symmetric rigid rotor......Page 146
13.2 Rotational limit......Page 149
13.3 Vibrational limit......Page 150
13.4 Davidson potential: the so called -unstable limit......Page 151
13.5 Linear potential limit......Page 153
13.6 The magic limit......Page 154
13.7 Comparison with experimental data......Page 157
14.1 q-deformed Lie algebras......Page 166
14.2 The fractional q-deformed harmonic oscillator......Page 169
14.3 The fractional q-deformed symmetric rotor......Page 173
14.4 Half-integer representations of the fractional rotation group SO (3)......Page 175
15. Fractional Spectroscopy of Hadrons......Page 178
15.1 Phenomenology of the baryon spectrum......Page 179
15.2 Charmonium......Page 184
15.3 Phenomenology of meson spectra......Page 189
15.4 Metaphysics: About the internal structure of quarks......Page 197
16.1 The four decompositions of the mixed fractional SO (9)......Page 200
16.2 Notation......Page 202
16.3 The nine dimensional fractional Caputo–Riemann–Riemann symmetric rotor......Page 205
16.4 Magic numbers of nuclei......Page 206
16.5 Ground state properties of nuclei......Page 209
16.6 Fine structure of the single particle spectrum: the extended Caputo–Riemann–Riemann symmetric rotor......Page 214
16.7 Magic numbers of electronic clusters: the nine dimensional fractional Caputo–Caputo–Riemann symmetric rotor......Page 219
16.8 Binding energy of electronic clusters......Page 223
16.9 Metaphysics: magic numbers for clusters bound by weak and gravitational forces respectively......Page 226
17.1 Covariance for fractional tensors......Page 232
17.2 Singular fractional tensors......Page 233
18. Fractional Fields......Page 236
18.1 Fractional Euler–Lagrange equations......Page 237
18.2 The fractional Maxwell equations......Page 240
19. Gauge Invariance in Fractional Field Theories......Page 244
19.1 Gauge invariance in first order of the coupling constant ¯g......Page 245
19.2 The fractional Riemann–Liouville–Zeeman effect......Page 249
20. Outlook......Page 254
Bibliography......Page 256
Index......Page 270