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arXiv:hep-th/0609164

Positive and Negative Charges in Relativistic Schroedinger Theory

Beck, T., M. Sorg

Published Sep 23, 2006• 0 citations
arXivHTMLPDF

Abstract

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).

DOI
10.48550/arxiv.hep-th/0609164

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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).
We show that the concept of particles and antiparticles is related to the Weyl-Heisenberg group W ⁣H3W\!H_3WH3​. Its generators are the position operators x^a\hat x^ax^a, the momentum operators p^a\hat p_ap^​a​, and the central element QvQ_{\sf{v}}Qv​, which defines a subgroup U(1)v_{\sf v}v​ of W ⁣H3W\!H_3WH3​. We show that the wave functions have a charge relative to the group U(1)v_{\sf{v}}v​ equal to either q+=1/ℏq_+=1/\hbarq+​=1/ℏ (particle ψ+ψ_+ψ+​) or q−=−1/ℏq_-=-1/\hbarq−​=−1/ℏ (antiparticle ψ−ψ_-ψ−​). We call these eigenvalues of the operator QvQ_{\sf{v}}Qv​ quantum charges: Qvψ±=q±ψ±Q_{\sf{v}}ψ_\pm^{}=q_\pm^{}ψ_\pm^{}Qv​ψ±​=q±​ψ±​. The wave functions ψ+ψ_+ψ+​ and ψ−ψ_-ψ−​ belong to the Hilbert spaces H+\mathcal{H}_+H+​ and H−\mathcal{H}_-H−​, on which unitarily inequivalent representations πℏπ_\hbarπℏ​ and π−ℏπ_{-\hbar}π−ℏ​ of the Weyl-Heisenberg group W ⁣H3W\!H_3WH3​ are defined. These representations are related to each other by the antilinear charge conjugation operator C^\hat CC^, which maps H+,q+\mathcal{H}_+, q_+H+​,q+​ to H−,q−\mathcal{H}_-, q_-H−​,q−​. Charges q±q_\pm^{}q±​ are intrinsic characteristics of the wave functions ψ±ψ_\pm^{}ψ±​, similar to spin and electric charge. The wave function possesses one of these charges, q+q_+q+​ or q−q_-q−​, prior to the specification of any equations. We show that negative probabilities/energies and the Klein paradox in Klein-Gordon and Dirac theories are consequences of the violation of U(1)v_{\sf v}v​-invariance in these theories. Restoring this invariance separates particle and antiparticles and eliminates all paradoxes at the level of first-quantized relativistic theories.
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 '.
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.
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.
We describe here the coherent formulation of electromagnetism in the nonrelativistic quantummechanical many-body theory. We use the mathematical frame of the field theory and its quantization in the spirit of the QED. This is necessary because of the manifold of misinterpretations emerging from the hystorical development of quantum mechanics, starting from the Schrödinger equation of a single particle in the presence of given electromagnetic fields, followed by the many-body theories of many charged identical particles having just Coulomb interactions inspired from the classical electromagnetic theory of point-like charges. However, this later is known to be inconsistent due to the self-interaction. This way could not be continued further to include properly the magnetic forces between the charged particles and lead to a lot of confusion about the interpretation of the magnetic field in the Hamiltonian, as well as about the gauge invariance. We emphasize the importance of the distinction between the applied (external fields) and the field in the matter. All these problems are length properly solved within the non-relativistic QED, nevertheless the confusion dominates in all the problems related to the magnetic properties of the solid state.
Solitary-particle quantum mechanics' inherent compatibility with special relativity is implicit in Schroedinger's postulated wave-function rule for the operator quantization of the particle's canonical three-momentum, taken together with his famed time-dependent wave-function equation that analogously treats the operator quantization of its Hamiltonian. The resulting formally four-vector equation system assures proper relativistic covariance for any solitary-particle Hamiltonian operator which, together with its canonical three-momentum operator, is a Lorentz-covariant four-vector operator. This, of course, is always the case for the quantization of the Hamiltonian of a properly relativistic classical theory, so the strong correspondence principle definitely remains valid in the relativistic domain. Klein-Gordon theory impairs this four-vector equation by iterating and contracting it, thereby injecting extraneous negative-energy solutions that are not orthogonal to their positive-energy counterparts of the same momentum, thus destroying the basis of the quantum probability interpretation. Klein-Gordon theory, which thus depends on the square of the Hamiltonian operator, is as well thereby cut adrift from Heisenberg's equations of motion. Dirac theory confuses the space-time symmetry of the four-vector equation system with such symmetry for its time component alone, which it fatuously imposes, thereby breaching the strong correspondence principle for the free particle and imposing the starkly unphysical momentum-independence of velocity. Physically sensible alternatives, with external electromagnetic fields, to the Klein-Gordon and Dirac equations are derived, and the simple, elegant symmetry-based approach to antiparticles is pointed out.