Symmetry Representations in the Rigged Hilbert Space Formulation of Quantum Mechanics

We discuss some basic properties of Lie group representations in rigged Hilbert spaces. In particular, we show that a differentiable representation in a rigged Hilbert space may be obtained as the projective limit of a family of continuous representations in a nested scale of Hilbert spaces. We also construct a couple of examples illustrative of the key features of group representations in rigged Hilbert spaces. Finally, we establish a simple criterion for the integrability of an operator Lie algebra in a rigged Hilbert space

Strange magnetic moment of the nucleon and SU(3) breaking: group theoretical approach

An extended group-theoretical approach to magnetic moments of the octet baryons is proposed with the aim of extracting the strange magnetic moment of the nucleon. Special attention is given to flavor SU(3) breaking. In this approach, isoscalar and isovector magnetic moments are treated separately in view of their different behavior under SU(3) breaking. We conclude that the anomalous magnetic moment associated with the flavor singlet current is small. Together with the small isoscalar anomalous magnetic moment of the nucleon, this implies suppression of the strange magnetic moment of the proton which is found to be small and positive, mu^(s) = (0.16 \pm 0.03) mu_N in units of the nuclear magneton.Comment: 6 pages, no figure, 6 tables, use REVTeX

On Epstein's trajectory model of non-relativistic quantum mechanics

In 1952 Bohm presented a theory about non-relativistic point-particles moving along deterministic trajectories and showed how it reproduces the predictions of standard quantum theory. This theory was actually presented before by de Broglie in 1926, but Bohm's particular formulation of the theory inspired Epstein to come up with a different trajectory model. The aim of this paper is to examine the empirical predictions of this model. It is found that the trajectories in this model are in general very different from those in the de Broglie-Bohm theory. In certain cases they even seem bizarre and rather unphysical. Nevertheless, it is argued that the model seems to reproduce the predictions of standard quantum theory (just as the de Broglie-Bohm theory).Comment: 12 pages, no figures, LaTex; v2 minor improvement

Irreversible Quantum Mechanics in the Neutral K-System

The neutral Kaon system is used to test the quantum theory of resonance scattering and decay phenomena. The two dimensional Lee-Oehme-Yang theory with complex Hamiltonian is obtained by truncating the complex basis vector expansion of the exact theory in Rigged Hilbert space. This can be done for K_1 and K_2 as well as for K_S and K_L, depending upon whether one chooses the (self-adjoint, semi-bounded) Hamiltonian as commuting or non-commuting with CP. As an unexpected curiosity one can show that the exact theory (without truncation) predicts long-time 2 pion decays of the neutral Kaon system even if the Hamiltonian conserves CP.Comment: 36 pages, 1 PostScript figure include

Misleading signposts along the de Broglie-Bohm road to quantum mechanics

Eighty years after de Broglie's, and a little more than half a century after Bohm's seminal papers, the de Broglie--Bohm theory (a.k.a. Bohmian mechanics), which is presumably the simplest theory which explains the orthodox quantum mechanics formalism, has reached an exemplary state of conceptual clarity and mathematical integrity. No other theory of quantum mechanics comes even close. Yet anyone curious enough to walk this road to quantum mechanics is soon being confused by many misleading signposts that have been put up, and not just by its detractors, but unfortunately enough also by some of its proponents. This paper outlines a road map to help navigate ones way.Comment: Dedicated to Jeffrey Bub on occasion of his 65th birthday. Accepted for publication in Foundations of Physics. A "slip of pen" in the bibliography has been corrected -- thanks go to Oliver Passon for catching it

On the Incompatibility of Standard Quantum Mechanics and the de Broglie-Bohm Theory

It is shown that the de Broglie-Bohm quantum theory of multi-particle systems is incompatible with the standard quantum theory of such systems unless the former is ergodic. A realistic experiment is suggested to distinguish between the two theories.Comment: A few technical changes incorporated in section V without any change in conclusion

Entanglement and State Preparation

When a subset of particles in an entangled state is measured, the state of the subset of unmeasured particles is determined by the outcome of the measurement. This first measurement may be thought of as a state preparation for the remaining particles. In this paper, we examine how the duration of the first measurement effects the state of the unmeasured subsystem. The state of the unmeasured subsytem will be a pure or mixed state depending on the nature of the measurement. In the case of quantum teleportation we show that there is an eigenvalue equation which must be satisfied for accurate teleportation. This equation provides a limitation to the states that can be accurately teleported.Comment: 24 pages, 3 figures, submitted to Phys. Rev.

The quantitative condition is necessary in guaranteeing the validity of the adiabatic approximation

The usual quantitative condition has been widely used in the practical applications of the adiabatic theorem. However, it had never been proved to be sufficient or necessary before. It was only recently found that the quantitative condition is insufficient, but whether it is necessary remains unresolved. In this letter, we prove that the quantitative condition is necessary in guaranteeing the validity of the adiabatic approximation.Comment: 4 pages,1 figue

Quantum Einstein's Equations and Constraints Algebra

In this paper we shall address this problem: Is quantum gravity constraints algebra closed and what are the quantum Einstein equations. We shall investigate this problem in the de-Broglie--Bohm quantum theory framework. It is shown that the constraint algebra is weakly closed and the quantum Einstein's equations are derived.Comment: 13 pages, No figure, RevTeX. To appear in Pramana J. Phys., 200

Electron quantum dynamics in closed and open potentials at high magnetic fields: Quantization and lifetime effects unified by semicoherent states

We have developed a Green's function formalism based on the use of an overcomplete semicoherent basis of vortex states, specially devoted to the study of the Hamiltonian quantum dynamics of electrons at high magnetic fields and in an arbitrary potential landscape smooth on the scale of the magnetic length. This formalism is used here to derive the exact Green's function for an arbitrary quadratic potential in the special limit where Landau level mixing becomes negligible. This solution remarkably embraces under a unified form the cases of confining and unconfining quadratic potentials. This property results from the fact that the overcomplete vortex representation provides a more general type of spectral decomposition of the Hamiltonian operator than usually considered. Whereas confining potentials are naturally characterized by quantization effects, lifetime effects emerge instead in the case of saddle-point potentials. Our derivation proves that the appearance of lifetimes has for origin the instability of the dynamics due to quantum tunneling at saddle points of the potential landscape. In fact, the overcompleteness of the vortex representation reveals an intrinsic microscopic irreversibility of the states synonymous with a spontaneous breaking of the time symmetry exhibited by the Hamiltonian dynamics.Comment: 19 pages, 4 figures ; a few typos corrected + some passages in Sec. V rewritte