Principle of Minimal Energy in Relativistic Schroedinger Theory

The Hamilton-Lagrange action principle for Relativistic Schr\"odinger Theory (RST) is converted to a variational principle (with constraints) for the stationary bound states. The groundstate energy is the minimally possible value of the corresponding energy functional and the relativistic energy eigenvalue equations do appear as the corresponding variational equations. The matter part of these eigenvalue equations is a relativistic generalization of the well-known Ritz principle in non-relativistic quantum mechanics which however disregards the dynamical character of the particle interactions. If the latter are included in the proposed principle of minimal energy for the bound states, one obtains a closed dynamical system for both matter and gauge fields. The new variational principle enables the development of variational techniques for solving approximately the energy eigenvalue equations. As a demonstration, the positronium groundstate is treated in great detail. Here a simple exponential trial function is sufficient in order to reproduce the (exact) result of conventional quantum mechanics where the relativistic and spin effects are neglected.Comment: 65 page

Exchange Interactions and Principle of Minimal Energy in Relativistic Schroedinger Theory

The principle of minimal energy, which has been set up in the preceding papers for systems of non-identical particles (e.g. positronium), is now generalized to include also identical particles. Since the latter kind of particles feels also the exchange forces (besides the usual electromagnetic forces), one has to deal with non-zero exchange potentials which render the theory nonlinear, according to the non-Abelian character of Relativistic Schroedinger Theory (RST). However, the present extension of the variational principle refers only to the linearized version of RST in order to keep the calculations sufficiently simple. It is also demonstrated that in RST the Dirac particles can occur in fermionic and bosonic quantum states; and the mathematical and physical consistency of the variational principle is validated for both types of states (concretely the fermionic hydrogen state 2p_3/2 and the bosonic positronium state 2^1P_1).Comment: 110 pages, 1 figur

Spherically Symmetric Approximation (and beyond) in Relativistic Schroedinger Theory

The energy eigenvalue problem of non-relativistic positronium is considered within the framework of Relativistic Schroedinger Theory (RST), and the results are compared to those of the conventional quantum theory. For the range of princi- pal quantum numbers n = 2;3;::: ;30, the RST predictions for the non-relativistic positronium energies deviate now from the corresponding predictions of the conven- tional quantum theory at an average of (roughly) 3%. These results suggest that the deviations will be further diminished in the higher orders of approximation.Comment: 150 pages, 10 figures and 4 table

Non-Relativistic Positronium Spectrum in Relativistic Schroedinger Theory

The lowest energy levels of positronium are studied in the non-relativistic approximation within the framework of Relativistic Schr\"odinger Theory (RST). Since it is very difficult to find the exact solutions of the RST field equations (even in the non-relativistic limit), an approximation scheme is set up on the basis of the hydrogen-like wave functions (i.e. polynomial times exponential). For any approximation order \NN (\NN=0,1,2,3,...) there arises a spectrum of approximate RST solutions with the associated energies, quite similarly to the conventional treatment of positronium in the standard quantum theory (Appendix). For the lowest approximation order (\NN=0) the RST prediction for the \emph{groundstate} energy exactly agrees with the conventional prediction of the standard theory. However for the higher approximation orders (\NN=1,2,3), the corresponding RST prediction differs from the conventional result by (roughly) $0,9 [eV]$ which confirms the previous estimate of the error being due to the use of the spherically symmetric approximation. The excited states require the application of higher-order approximations (\NN>>3) and are therefore not adequately described by the present orders (\NN\le 3).Comment: 67 pages and 3 figure

Magnetic Interactions in Relativistic Two Particle Systems

The magnetic interactions of the two electrons in helium-like ions are studied in detail within the framework of Relativistic Schroedinger Theory (RST). The general results are used to compute the ground-state interaction energy of some highly-ionized atoms ranging from germanium (Z=32) up to bismuth (Z=83). When the magnetic interaction energy is added to its electric counterpart resulting from the electrostatic approximation, the present RST predictions reach a similar degree of precision (relative to the experimental data) as the other theoretical approaches known in the literature. However since the RST magnetism is then treated only in lowest-order approximation, further improvements of the RST predictions seem possible.Comment: 60 pages and 1 figur

Positive and Negative Charges in Relativistic Schroedinger Theory

Relativistic Schroedinger Theory (RST), as a general gauge theory for the description of relativistic N-particle systems, is shown to be a mathematically consistent and physically reasonable framework for an arbitrary assemblage of positive and negative charges. The electromagnetic plus exchange interactions within the subset of {\it identical} particles are accounted for in a consistent way, whereas {\it different} particles can undergo only the electromagnetic interactions. The origin of this different interaction mechanism for the subsets of identical and non-identical particles is traced back to the fundamental conservation laws for charge and energy-momentum: in order that these conservation laws can hold also for different particles, the structure group $\mathcal U(N)$ of the fibre bundles must be reduced to its maximal Abelian subgroup $\mathcal U(1) \times \mathcal U(1) \times ... \times \mathcal U(1)$, which eliminates the exchange part of the bundle connection. The persisting Abelian gauge symmetry adopts the meaning of the proper gauge group for the electromagnetic interactions which apply to the identical and non-identical particles in the same way. Thus in RST there is an intrinsic dynamical foundation of the emergence of exchange effects for identical particles, whereas the conventional theory is invaded by the exchange phenomenon via a purely kinematical postulate, namely the antisymmetrization postulate for the wave functions due to Pauli's exclusion principle. As a concrete demonstration, a three-particle system is considered which consists of a positively charged particle of arbitrary rest mass and of two negatively charged particles of equal spin, mass and charge (e.g. electrons).Comment: 57 page

Geometry and Topology of Relativistic Two-Particle Quantum Mixtures

Within the framework of Relativistic Schroedinger Theory (an alternative form of quantum mechanics for relativistic many-particle systems) it is shown that a general N-particle system must occur in one of two forms: either as a ``positive'' or as a ``negative'' mixture, in analogy to the fermion-boson dichotomy of matter in the conventional theory. The pure states represent a limiting case between the two types of mixtures which themselves are considered as the RST counterparts of the entangled (fermionic or bosonic) states of the conventional quantum theory. Both kinds of mixtures are kept separated from dynamical as well as from topological reasons. The 2-particle configurations (N=2) are studied in great detail with respect to their geometric and topological properties which are described in terms of the Euler class of an appropriate bundle connection. If the underlying space-time manifold (as the base space of the fibre bundles applied) is parallelisable, the 2-particle configurations can be thought to be generated geometrically by an appropriate (2+2) splitting of the local tangent space.Comment: 35 pages and 1 figur

Positronium Groundstate in Relativistic Schroedinger Theory

The usefulness of the Relativistic Schr\"odinger Theory (RST) is studied in the field of atomic physics. As a concrete demonstration, the positronium groundstate is considered in great detail; especially the groundstate energy $E_0$ is worked out in the non-relativistic approximation and under neglection of the magnetic interactions between the positron and the electron. The corresponding RST prediction $(E_0\simeq -6,48 [eV])$ misses the analogous conventional Schr\"odinger result $(E_0\simeq -6,80 [eV])$ but is closer to the latter than the corresponding Hartree approximation $(-2,65 [eV])$. The missing binding energy of $6,80-6,48=0,32 [eV]$ can be attributed to the approximative use of an SO(3) symmetric interaction potential which in RST, however, is actually only SO(2) invariant against rotations around the z-axis. It is expected that, with the correct use of an anisotropic interaction potential due to the SO(2) symmetry, the RST predictions will come even closer to the conventional Schr\"odinger result, where however the mathematical structure of RST relies on exotic (i.e. double-valued) wave functions and on the corresponding unconventional interaction potentials (e.g. Struve-Neumann potential).Comment: 80 pages and 2 figure

Relativistic Energy Levels of Para-Helium

The practical usefulness of Relativistic Schr\"odinger Theory (RST) is tested by calculating approximately the energy difference between the excited singlet state $1s2s {}^1S_0$ and the ground state $1s^2 {}^1S_0$ of the helium-like ions with arbitrary charge number $z_{\rm ex} (2\le z_{\rm ex}\le 100)$. The results are compared to the corresponding predictions of other theoretical approaches in the literature and to the experimental data. Since the exact solutions of the RST energy eigenvalue problem are unknown, one has to resort to approximative methods. However the crudest approximation (``spherically symmetric approximation'') yields relatively accurate results so that it seems worth while to develop more powerful approximation techniquesComment: 45 pages, 2 figure

Helium Multiplet Structure in Relativistic Schr\"odinger Theory

The emergence of a multiplet structure of the helium-like ions is studied within Relativistic Schr\"odinger Theory (RST), a fluid-dynamic approach to the relativistic quantum theory of the many-particle systems. The fluid-dynamic character of RST demands to specify the electronic current densities \jmu for any $N$-particle configuration which is exemplified here by considering the helium singlet (${}^1S_0$) and triplet (${}^3S_1$) states in great detail. Since the use of densities in RST is based upon the concept of wave functions, the new theory appears as a certain kind of (relativistic) unification of the conventional wave function formalism and the density functional theory, which both are the most prominent theoretical tools in atomic and molecular physics. As a demonstration of the practical usefulness of RST, the energy difference $\Delta E_{1\backslash 2}$ of the helium singlet states $2s^2 {}^1S_0$ and $1s^2 {}^1S_0$ is calculated for a large range of nuclear charge numbers $z_{ex}$ ($2\leq z_{ex}\leq 100$), whereas the corresponding experimental values are available only up to $z_{ex}=42$ (molybdenum). The deviations of these RST results from the observational data is less than $0,3Comment: 84 pages, 2 figure