Fractional Calculus

The following sections are included:

• Foreword

• Acknowledgments

• Contents

During recent years the interest of physicists in nonlocal field theories has been steadily increasing. The main reason for this development is the expectation that the use of these field theories will lead to a much more elegant and effective way of treating problems in particle and high-energy physics as it has been possible up to now with local field theories…

The following sections are included:

• Gamma function

• Mittag-Leffler functions

• Hypergeometric functions

• Miscellaneous functions

The following sections are included:

• Basics

• The fractional Leibniz product rule

• Discussion
  • Orthogonal polynomials
  • Differential representation of the Riemann fractional derivative

The following sections are included:

• Classical description

• Fractional friction

The following sections are included:

• The Fourier transform

• The fractional integral
  • The Liouville fractional integral
  • The Riemann fractional integral

• Correlation of fractional integration and differentiation
  • The Liouville fractional derivative
  • The Riemann fractional derivative
  • The Liouville fractional derivative with inverted operator sequence: the Liouville–Caputo fractional derivative
  • The Riemann fractional derivative with inverted operator sequence: the Caputo fractional derivative

• Fractional derivative of second order
  • The Riesz fractional derivative
  • The Feller fractional derivative

• Fractional derivatives of higher orders

• Geometric interpretation of the fractional integral

• Low level fractionality

• Discussion
  • Semigroup property of the fractional integral

The following sections are included:

• The fractional harmonic oscillator

• The harmonic oscillator according to Fourier

• The harmonic oscillator according to Riemann

• The harmonic oscillator according to Caputo

The following sections are included:

• Fractional wave equations

• Parity and time-reversal

• Solutions of the free regularized fractional wave equation

The following sections are included:

• A short history of nonlocal concepts

• From local to nonlocal operators

• Memory effects

The following sections are included:

• Canonical quantization

• Quantization of the classical Hamilton function and free solutions

• Temperature dependence of a fission yield and determination of the corresponding fission potential

• The fractional Schrödinger equation with an infinite well potential

• Radial solutions of the fractional Schrödinger equation

The following sections are included:

• Spin

• Fractional spin

The following sections are included:

• The Dirac equation

• The fractional Dirac equation

• The fractional Pauli equation

The following sections are included:

• Characteristics of fractional group theory

• The fractional rotation group

The following sections are included:

• The spectrum of the fractional symmetric rigid rotor

• Rotational limit

• Vibrational limit

• Davidson potential: the so called γ-unstable limit

• Linear potential limit

• The magic limit

• Comparison with experimental data

The following sections are included:

• q-deformed Lie algebras

• The fractional q-deformed harmonic oscillator

• The fractional q-deformed symmetric rotor

• Half-integer representations of the fractional rotation group SO^α(3)

The following sections are included:

• Phenomenology of the baryon spectrum

• Charmonium

• Phenomenology of meson spectra

• Metaphysics: About the internal structure of quarks

The following sections are included:

• The four decompositions of the mixed fractional SO^α(9)

• Notation

• The nine dimensional fractional Caputo–Riemann–Riemann symmetric rotor

• Magic numbers of nuclei

• Ground state properties of nuclei

• Fine structure of the single particle spectrum: the extended Caputo–Riemann–Riemann symmetric rotor

• Magic numbers of electronic clusters: the nine dimensional fractional Caputo–Caputo–Riemann symmetric rotor

• Binding energy of electronic clusters

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

The following sections are included:

• Covariance for fractional tensors

• Singular fractional tensors

The following sections are included:

• Fractional Euler–Lagrange equations

• The fractional Maxwell equations

The following sections are included:

• Gauge invariance in first order of the coupling constant

• The fractional Riemann–Liouville–Zeeman effect

In this book we have given a concise introduction to the methods and strategies used within fractional calculus. We have demonstrated, that this specific non-local approach may be a powerful alternative for a description of phenomena in high energy and particle physics. In addition we have shown that the application of fractional group theory leads to a vast amount of intriguing and valuable results…

The following sections are included:

• Bibliography

• Index