Frank Antonsen

Random graphs as a model for pregeometry

A toy model for pregeometry based on random graphs is proposed, and it is shown how it is related to other models in the literature. A prediction of the dimensionality of space is given and we obtain bounds on the Euler number.

Relativistic Sturmian basis functions

The properties of relativistic one-electron Sturmian basis sets are discussed using Goscinskis definition of Sturmians rather than Rotenbergs. The potential-weighted orthonormality relations obeyed by the members of such a set are discussed. Weighted orthonormality relations in momentum space are also derived and used to construct a Sturmian expansion of the kernel of the momentum-space Dirac equation. The special case of spherical symmetry is discussed, and, as an illustrative example, relativistic Coulomb Sturmian basis sets are used to obtain solutions to the Dirac equation for an electron moving in a non-Coulomb potential.

Logics and quantum gravity

We consider the logic needed for models of quantum gravity, taking as our starting point a simple pregeometric toy model based on graph theory. First a discussion of quantum logic seen in the light of canonical quantum gravity is given, then a simple toy model is proposed and the logical structure underlying it exposed. It is then shown that this logic is nonclassical and in fact contains quantum logics as special cases. We then go on to show how Yang-Mills theory and quantum mechanics fits in. A single mathematical structure is proposed capable of containing all these subjects in a natural and elegant way. Causality plays an important role. The mere presence of a causal relation almost inevitably yields this kind of logic.

Coherent States on Lie Algebras: A Constructive Approach

We generalize the notion of coherent states toarbitrary Lie algebras by making an analogy with the GNSconstruction in C*-algebras. The method is illustratedwith examples of semisimple and nonsemisimple finite-dimensional Lie algebras as well as loopand Kac–Moody algebras. A deformed addition on theparameter space is also introduced simplifying someexpressions and some applications to conformal field theory are pointed out, e.g., differentialoperator and free field realizations found.

Representation of cusps in a hyperspherical basis set

The wave functions of Coulomb systems have cusps at points corresponding to two particle coelescences. In this paper, we derive series representing the cusps in terms of hyperspherical harmonics multiplied by functions of the hyperradius. When the hyperspherical method is applied to Coulomb systems, the harmonics which appear in these series should be included in the hyperangular basis set.

Magnetic interactions and 4-currents

Abstract The 4-current formalism is reviewed and applied to the treatment of magnetic effects in quantum chemistry. The electromagnetic interaction operator is expressed in terms of 4-currents and in terms of electron creation and annihilation operators; and a relativistic analogue of Roothaans equations is derived. Finally, the 4-current formalism is used to treat resonance transfer of excitation energy through magnetic interactions.

Zero-Point Energy of Quantum Fields in a Schwarzschild Geometry

The effective Lagrangian and the zero-point (or Casimir) energy is calculated from the zeta-function which is obtained by the heat kernel method using the expansion of (Bormann and Antonsen, 1995). Calculated this way this unavoidable energy contribution is automatically regularised and ready for further investigation. Interesting observations include a large energy contribution (from scalar field and fermionic zero-point fluctuations) that is non-zero as the mass goes to zero, perhaps indicating a topological origin. Also, plots of the contribution of gauge boson fields to the zero-point energy, as a function of radial distance (gravitational field strength) and the size of the gauge boson coupling (gauge field strength) shows great variation, notably the occurrence of ‘resonances.’