Richard Herrmann

Gauge invariance in fractional field theories

Abstract The principle of local gauge invariance is applied to fractional wave equations and the interaction term is determined up to order o ( g ¯ ) in the coupling constant g ¯ . 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 Schrodinger-equation under spatial rotations are investigated and an internal fractional spin is deduced.

The fractional symmetric rigid rotor

Based on the Riemann fractional derivative the Casimir operators and multiplets 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 γ-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.

Common aspects of q-deformed Lie algebras and fractional calculus

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

q-deformed Lie algebras and fractional calculus

B_\alpha(E2)

Higher-dimensional mixed fractional rotation groups as a basis for dynamic symmetries generating the spectrum of the deformed Nilsson oscillator

values for the fractional q-deformed symmetric rotor are calculated.

On the origin of space

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

Fractional phase transition in medium size metal clusters

B_\alpha(E2)

Infrared Spectroscopy of Diatomic Molecules

values for the fractional q-deformed symmetric rotor are calculated.

Magic Numbers in Atomic Nuclei

Based on the Riemannand Caputo definition of the fractional derivative we use the fractional extensions of the standard rotation group SO(3) to construct a higher dimensional representation of a fractional rotation group with mixed derivative types. An extended symmetric rotor model is derived, which predicts the sequence of magic proton and neutron numbers accurately. The ground state properties of nuclei are correctly reproduced within the framework of this model. PACS numbers: 21.60.Fw, 21.60.Cs, 05.30.Pr

Magic Numbers in Metal Clusters

Within the framework of fractional calculus with variable order the evolution of space in the adiabatic limit is investigated. Based on the Caputo definition of a fractional derivative using the fractional quantum harmonic oscillator a model is presented, which describes space generation as a dynamic process, where the dimension d of space evolves smoothly with time in the range 0 ≤ d(t) ≤ 3, where the lower and upper boundaries of dimension are derived from first principles. It is demonstrated, that a minimum threshold for the space dimension is necessary to establish an interaction with external probe particles. A possible application in cosmology is suggested.