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Physical Review A

Mixing group for relativistic two-particle quantum states

S. Rupp, M. Sorg

Published Jan 17, 2001• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.63.022112

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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.
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.
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.
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.
This article shows that one can consistently incorporate nonunitary representations of at least one group into the ``ordinary'' nonrelativistic quantum mechanics. This group turns out to be Lorentz group thus giving us an alternative approach to QFT for combining the quantum mechanics and special theory of relativity which keeps the concept of wave function (belonging to some representation of Lorentz group) through the whole theory. Scalar product has been redefined to take into the account the nonunitarity of representations of Lorentz group. Understanding parity symmetry turns out to be the key ingredient throughout the process. Instead of trying to guess an equation or a set of equations for some wave functions or fields (or equivalently trying to guess a Lagrangian for the same), one derives them based only on the superposition principle and properties of wave functions under Lorentz transformations and parity. The resulting model has striking similarities with the standard quantum field theory and yet has no negative energy states, no zitterbewegung effects, symmetric energy momentum tensor and angular momentum density tensor for \emph{all} representations of Lorentz group (unifying the theoretical description of all particles), as well as clear physical interpretation. It also offers a possible interpretation why particles and antiparticles have opposite quantum numbers.
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).