Finite-dimensional relativistic quantum mechanics

A finite-dimensional relativistic quantum mechanics is developed by first quantizing Minkowski space. Two-dimensional space-time event observables are defined and quantum microscopic causality is studied. Three-dimensional colored even observables are introduced and second quantized on a representation space of the restricted Poincaré group. Creation, annihilation, and field operators are introduced and a finite-dimensional Dirac theory is presented.     

Current density in relativistic quantum mechanics

We extend to relativistic theories the concepts of probability density and probability current density of nonrelativistic quantum mechanics, together with the charge and current densities that are used as sources of the electromagnetic field in the semi-classical theory of radiation. There are some limitations in the procedure, especially in the case of several particles.     

Advances in relativistic molecular quantum mechanics

A quantum mechanical equation HΨ=EΨ$H\Psi =E\Psi$ is composed of three components, viz., Hamiltonian H$H$, wave function Ψ$\Psi$, and property E(λ)$E\left(\lambda \right)$, each of which is confronted with fundamental issues in the relativistic regime, e.g., (1) What is the most appropriate relativistic many-body Hamiltonian? How to solve the resulting equation? (2) How does the relativistic wave function behave at the coalescence of two electrons? How to do relativistic explicit correlation? (3) How to formulate relativistic properties properly?, to name just a few. It is shown here that the charge-conjugated contraction of Fermion operators, dictated by the charge conjugation symmetry, allows for a bottom-up construction of a relativistic Hamiltonian that is in line with the principles of quantum electrodynamics (QED). Various approximate but accurate forms of the Hamiltonian can be obtained based entirely on physical arguments. In particular, the exact two-component Hamiltonians can be formulated in a general way to cast electric and magnetic fields, as well as electron self-energy and vacuum polarization, into a unified framework. While such algebraic two-component Hamiltonians are incompatible with explicit correlation, four-component relativistic explicitly correlated approaches can indeed be made fully parallel to the nonrelativistic counterparts by virtue of the ‘extended no-pair projection’ and the coalescence conditions. These findings open up new avenues for future developments of relativistic molecular quantum mechanics. In particular, ‘molecular QED’ will soon become an active and exciting field.     

Stochastic microcausality in relativistic quantum mechanics

A recently formulated concept of stochastic localizability is shown to be consistent with a concept of stochastic microcausality, which avoids the conclusions of Hegerfeldt's no-go theorem as to the inconsistency of sharp localizability of quantum particles and Einstein causality. The proposed localizability on quantum space-time is shown to lead to strict asymptotic causality. For finite time evolutions, upper bounds on propagation to the exterior of stochastic light cones are derived which show that the resulting probabilities are too small to be actually observable in a realistic context.Supported by an NSERC Fellowship.Suported in part by NSERC research grant No. A5206.     

Nonstandard extension of quantum logic and Dirac's bra-ket formalism of quantum mechanics

An extension of the quantum logical approach to the axiomatization of quantum mechanics usingnonstandard analysis methods is proposed. The physical meaning of a quantum logic as a lattice of propositions is conserved by its nonstandard extension. But not only the usual Hubert space formalism of quantum mechanics can be derived from the nonstandard extended quantum logic. Also the Dirac bra-ket quantum mechanics can be derived as a consequence of such an extended quantum logic.     

Spectator-model operators in point-form relativistic quantum mechanics

We address the construction of transition operators for electromagnetic, weak, and hadronic reactions of relativistic few-quark systems along the spectator model. While the problem is of relevance for all forms of relativistic quantum mechanics, we specifically adhere to the point form, since it preserves the spectator character of the corresponding transition operators in any reference frame. The conditions imposed on the construction of point-form spectator-model operators are discussed and their implications are exemplified for mesonic decays of baryon resonances within a relativistic constituent-quark model.     

Causal green function in relativistic quantum mechanics

The causal Green function or Feynman propagator for the free-field Klein-Gordon equation and related singular functions, defined as distributions, are related to the causal time-boundary data. Probability densities and amplitudes are defined in terms of the solutions of the Klein-Gordon equation for a complex scalar field interacting with an electromagnetic field. The convergence of the perturbation expansion of the solution of the Klein-Gordon equation for a charged scalar particle in an external field is shown for well-behaved electromagnetic potentials. Other relativistic wave equations are discussed briefly.     

Derivation of quantum mechanics from the Boltzmann equation for the Planck aether

The Planck aether hypothesis assumes that space is densely filled with an equal number of locally interacting positive and negative Planck masses obeying an exactly nonrelativistic law of motion. The Planck masses can be described by a quantum mechanical two-component nonrelativistic operator field equation having the form of a two-component nonlinear Schrödinger equation, with a spectrum of quasiparticles obeying Lorentz invariance as a dynamic symmetry for energies small compared to the Planck energy. We show that quantum mechanics itself can be derived from the Newtonian mechanics of the Planck aether as an approximate solution of Boltzmann's equation for the locally interacting positive and negative Planck masses, and that the validity of the nonrelativistic Schrödinger equation depends on Lorentz invariance as a dynamic symmetry. We also show how the many-body Schrödinger wave function can be factorized into a product of quasiparticles of the Planck aether with separable quantum potentials. Finally, we present a possible explanation of wave function collapse as a kind of enhanced gravitational collapse in the presence of the negative Planck masses.     

Spin of Dirac's relativistic membrane

The Hamiltonian of Dirac's relativistic membrane is linearized by means of a gauge transformation. It is pointed out that the membrane spin is +/- (planck)/2. Furthermore, in a quantum relativistic framework neutrinolike particles are obtained when the membrane charge vanishes.     

The Wigner function in the relativistic quantum mechanics

A detailed study is presented of the relativistic Wigner function for a quantum spinless particle evolving in time according to the Salpeter equation.     

Localizability and causal propagation in relativistic quantum mechanics

We show that, in a relativistic quantum theory in which the mass shell is not sharp, and positive and negative energy states are admissable, causal propagation is possible, and Hegerfeldt's theorem can be avoided. The conditions under which this is true have simple physical interpretation.1. On sabbatical leave from School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Israel. This work was supported in part by a grant from the Ambrose Monell Foundation.     

Completeness of wave operators in relativistic quantum mechanics

A combination of geometric and algebraic methods is used to prove asymptotic completeness for Schrödinger-type equations with potential not vanishing at infinity along hyperboloids (in spacetime), and with the free Hamiltonian given by the (not bounded below) relativistic (mass)² operator. The proof is based on the use of a modified form of local compactness and additional geometric properties of asymptotic scattering states which are needed to distinguish them from states trapped inside some hyperboloid for all times.Supported in part by the Fund for Basic Research administered by the Israeli Academy of Sciences and Humanities Basic Research Foundation.     

Gibbs ensembles in relativistic classical and quantum mechanics

Classical and quantum Gibbs ensembles are constructed for equilibrium statistical mechanics in the framework of an extension to many-body theory of a relativistic mechanics proposed by Stueckelberg. In addition to the usual chemical potential in the grand canonical ensemble, there is a new potential corresponding to the mass degree of freedom of relativistic systems. It is shown that in the nonrelativistic limit the relativistic ensembles we have obtained reduce to the usual ones, and mass fluctuations for the free-particle gas approach the fluctuations in N. The ultrarelativistic limit of the canonical ensemble for the free-particle gas differs from the corresponding limit of the ensemble proposed by Jüttner and Pauli. Due to the mass degree of freedom, the quantum counting of states is different from that of the nonrelativistic theory. If the mass distribution is sufficiently sharp, the thermodynamical effects of this multiplicity will not be large. There may, however, be detectable effects such as a shift in the Fermi level and the critical temperature for Bose-Einstein condensation, and some change in specific heats.     

Identical motion in relativistic quantum and classical mechanics

The Klein-Gordon equation for the stationary state of a charged particle in a spherically symmetric scalar field is partitioned into a continuity equation and an equation similar to the Hamilton-Jacobi equation. There exists a class of potentials for which the Hamilton-Jacobi equation is exactly obtained and examples of these potentials are given. The partitionAnsatz is then applied to the Dirac equation, where an exact partition into a continuity equation and a Hamilton-Jacobi equation is obtained.     

On the electromagnetic interaction in relativistic quantum mechanics

A fundamental problem in the construction of local electromagnetic interactions in the framework of relativistic wave equations of Klein-Gordon or Dirac type is discussed, and shown to be resolved in a relativistic quantum theory of events described by functions in a Hilbert space on the manifold of space-time. The relation, abstracted from the structure of the electromagnetic current, between sequences of events, parametrized by an evolution parameter (historical time), and the commonly accepted notion of particles is reviewed. As an illustration of these ideas, a perturbative calculation is made for photon emission from a charged two-body system in which the electromagnetic field is quantized in the usual way. The result is in essential agreement with calculations in which the charged particles are treated in the framework of nonrelativistic quantum mechanics, and provides them with a relativistic interpretation. In particular, we obtain a relativistically invariant form of the Bohr radiation condition.Supported in part by the Fund for Basic Research administered by the Israeli Academy of Sciences and Humanities Basic Research Foundation.     

On path integrals and relativistic quantum mechanics

A proposal for formulation of relativistic quantum mechanics in terms of path integrals is presented.We are deeply indebted to Dr. M.Petrá for many stimulating discussions.