# q-deformed Lie algebras and fractional calculus

author:
R. Herrmann
abstract:
Fractional calculus and q-deformed Lie algebras are closely related. Both concepts expand the scope of standard Lie algebras to describe generalized symmetries. For the fractional harmonic oscillator, the corresponding q-number is derived. It is shown, that the resulting energy spectrum is an appropriate tool e.g. to describe the ground state spectra of even-even nuclei. In addition, the equivalence of rotational and vibrational spectra for fractional q-deformed Lie algebras is shown and the $B_\alpha(E2)$ values for the fractional q-deformed symmetric rotor are calculated.
reference: Physica A 389 (2010) 4613

# The fractional symmetric rigid rotor

author:
R. Herrmann
abstract:
Based on the Riemann fractional derivative the Casimir operators and multipletts for the fractional extension of the rotation group SO(n) are calculated algebraically. The spectrum of the corresponding fractional symmetric rigid rotor is discussed. It is shown, that the rotational, vibrational and $\gamma$-unstable limits of the standard geometric collective models are particular limits of this spectrum. A comparison with the ground state band spectra of nuclei shows an agreement with experimental data better than 2%. The derived results indicate, that the fractional symmetric rigid rotor is an appropriate tool for a description of low energy nuclear excitations.

# Gauge invariance in fractional field theories

author:
R. Herrmann
abstract:
The principle of local gauge invariance is applied to fractional wave equations and the interaction term is determined up to order $o(\bar{g})$ in the coupling constant $\bar{g}$. As a first application, based on the Riemann-Liouville fractional derivative definition, the fractional Zeeman effect is used to reproduce the baryon spectrum accurately.
The transformation properties of the non relativistic fractional Schrödinger-equation under spatial rotations are investigated and an internal fractional spin is deduced.

reference: doi:Physics Letters A 372 (2008) 5515-5522

# Properties of a fractional derivative Schroedinger type wave equation and a new interpretation of the charmonium spectrum

author: R. Herrmann

abstract:
Based on the Caputo fractional derivative the classical, non relativistic Hamiltonian is quantized leading to a fractional Schroedinger type wave equation. The free particle solutions are localized in space. Solutions for the infinite well potential and the radial symmetric ground state solution are presented. It is shown, that the behaviour of these functions may be reproduced with a ordinary Schroeodinger equation with an additional potential, which is of the form V ~ x for $\alpha<1$, corresponding to the confinement potential, introduced phenomenologically to the standard models for non relativistic interpretation of quarkonium-spectra. The ordinary Schroedinger equation is triple factorized and yields a fractional wave equation with internal SU(3) symmetry. The twofold iterated version of this wave equation shows a direct analogy to the fractional Schroedinger equation derived. The angular momentum eigenvalues are calculated algebraically. The resulting mass formula is applied to the charmonium spectrum and reproduces the experimental masses with an accuracy better than 0.1%. Extending the standard charmonium spectrum, three additional particles are predicted and associated with $\Sigma_c^0(2455)$ and Y(4260) observed recently and one X(4965), not yet observed. The root mean square radius for $\Sigma_c^0(2455)$ is calculated to be ~0.3[fm]. The derived results indicate, that a fractional wave equation may be an appropriate tool for a description of quark-like particles.

# Continuous differential operators and a new interpretation of the charmonium spectrum

author:
R. Herrmann
abstract:
The definition of the standard differential operator is extended from integer steps to arbitrary stepsize. The classical, nonrelativistic Hamiltonian is quantized, using these new continuous operators. The resulting Schroedinger type equation generates free particle solutions, which are confined in space. The angular momentum eigenvalues are calculated algebraically. It is shown, that the charmonium spectrum may be classified by the derived angular momentum eigenvalues for stepsize=2/3.