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Journal of Physics A Mathematical and General

Two-particle systems in relativistic Schrödinger theory

Margarete Mattes, M. Sorg

Published Jan 1, 1999• 5 citations
Publisher

Abstract

The relativistic Schrödinger theory (RST) is applied to a general 2-particle system with electromagnetic interactions. The emphasis of the investigation lies on the exchange and overlap effects in connection with the development of adequate single-particle and 2-particle concepts. It is possible to eliminate the 2-particle quantities and to formulate the 2-particle theory exclusively in terms of single-particle quantities but then there arises a new unobservable field degree of freedom: ` renormalization '.

DOI
10.1088/0305-4470/32/25/315

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The relativistic Schrödinger theory (RST) is applied to a general 2-particle system with electromagnetic interactions. The emphasis of the investigation lies on the exchange and overlap effects in connection with the development of adequate single-particle and 2-particle concepts. It is possible to eliminate the 2-particle quantities and to formulate the 2-particle theory exclusively in terms of single-particle quantities but then there arises a new unobservable field degree of freedom: ` renormalization '.
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 U(N)\mathcal U(N)U(N) of the fibre bundles must be reduced to its maximal Abelian subgroup U(1)×U(1)×...×U(1)\mathcal U(1) \times \mathcal U(1) \times ... \times \mathcal U(1)U(1)×U(1)×...×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).
Within the framework of Relativistic Schrodinger Theory (RST), the static two-particle solutions of the RST dynamics in an external binding force field (Coulomb potential) are studied in detail. The typical quantum effects occurring in bound two-particle systems (such as particle indistinguishability, entanglement, and ortho-para splitting of the energy levels) turn out to be implications of the RST dynamics itself, not implications of an additional (anti-)symmetrization postulate as in the conventional quantum theory. The RST lowest-order approximations for the 2-particle energy levels are closer to the experimental values than the corresponding predictions of the conventional theory by (roughly) one order of magnitude.
In this paper, some concepts of nonrelativistic many-particle quantum mechanics (e.g., product states, density matrix) are generalized to the relativistic domain using a framework called relativistic Schr\"odinger theory (RST). By using a general ansatz, the RST framework is simplified considerably and some of its field equations are solved directly. The RST approach is then compared with nonrelativistic quantum mechanics (QM) for the case of a product state (conventional QM) and its RST analog. It is shown that relativistic wave equations can be derived from the RST formalism, so that they coincide in the nonrelativistic limit with the well-known Hartree equations.
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).
A general method to treat the quantal two-body problem relativistically is developed on the line of Dirac's many-time theory. The case where two charged fermions interact with each other through the electromagnetic field is investigated as an example. We construct a canonical transformation which eliminates the electromagnetic field variables in the original Schrödinger equations, and derive the potential energy between two charged particles. Finally, some properties of the simultaneous wave equations for the system of two particles are discussed, and a method to treat the stationary state of that system is indicated.
Within the framework of relativistic Schr\"odinger theory (RST), the problem of bound two-particle states is studied and compared to the analogous results of conventional quantum theory. The standard dichotomy of symmetric and antisymmetric quantum states finds its RST analog in the form of the positive and negative mixtures. Similarly the conventional exchange degeneracy has its RST counterpart in the form of a certain mixture degeneracy which, however, is not broken by the interparticle interactions as in the standard quantum theory. The corresponding group of quasilinear mixing transformations turns out to be continuous [in contrast to the discrete operations of (anti)symmetrization in the conventional theory] and is closely related to the Lorentz group SO(1,1) in two dimensions. The group properties can be exploited to generate exact solutions for the mixture dynamics from the pure-state configurations where, however, certain physical quantities remain invariant: scalar densities, currents, energy eigenvalues.