Unitary cocycle representations of the Galilean line group: Quantum mechanical principle of equivalence

We present a formalism of Galilean quantum mechanics in non-inertial reference frames and discuss its implications for the equivalence principle. This extension of quantum mechanics rests on the Galilean line group, the semidirect product of the real line and the group of analytic functions from the real line to the Euclidean group in three dimensions. This group provides transformations between all inertial and non-inertial reference frames and contains the Galilei group as a subgroup. We construct a certain class of unitary representations of the Galilean line group and show that these representations determine the structure of quantum mechanics in non-inertial reference frames. Our representations of the Galilean line group contain the usual unitary projective representations of the Galilei group, but have a more intricate cocycle structure. The transformation formula for the Hamiltonian under the Galilean line group shows that in a non-inertial reference frame it acquires a fictitious potential energy term that is proportional to the inertial mass, suggesting the equivalence of inertial mass and gravitational mass in quantum mechanics.     

Interior symmetries of hadrons:SO(3,2) as a spectrum-generating group

In this work, we study some applications of the spectrum-generating group (SGG) formalism to obtain the mass-spin spectra of hadrons. The possibility of classifying (medium-energy) hadrons in terms of a symmetry group defining the center-of-mass motion and an SGG defining the interior motion is discussed. After considering the defining commutation relations and equations of motion of the generators of the SGG, it is shown how the hadronic spectral information is obtained through a Hamiltonian that is a constraint relation between the generators of the symmetry group and those of the SGG.     

Quantum mechanics in noninertial reference frames: Violations of the nonrelativistic equivalence principle

In previous work we have developed a formulation of quantum mechanics in non-inertial reference frames. This formulation is grounded in a class of unitary cocycle representations of what we have called the Galilean line group, the generalization of the Galilei group that includes transformations amongst non-inertial reference frames. These representations show that in quantum mechanics, just as is the case in classical mechanics, the transformations to accelerating reference frames give rise to fictitious forces. A special feature of these previously constructed representations is that they all respect the non-relativistic equivalence principle, wherein the fictitious forces associated with linear acceleration can equivalently be described by gravitational forces. In this paper we exhibit a large class of cocycle representations of the Galilean line group that violate the equivalence principle. Nevertheless the classical mechanics analogue of these cocycle representations all respect the equivalence principle.     

Spatial contraction of the Poincaré group and Maxwell’s equations in the electric limit

The contraction of the Poincaré group with respect to the space translations subgroup gives rise to a group that bears a certain duality relation to the Galilei group, that is, the contraction limit of the Poincaré group with respect to the time translations subgroup. In view of this duality, we call the former the dual Galilei group. A rather remarkable feature of the dual Galilei group is that the time translations constitute a central subgroup. Therewith, in unitary irreducible representations (UIRs) of the group, the Hamiltonian appears as a Casimir operator proportional to the identity H = EI, with E (and a spin value s) uniquely characterizing the representation. Hence, a physical system characterized by a UIR of the dual Galilei group displays no non-trivial time evolution. Moreover, the combined U(1) gauge group and the dual Galilei group underlie a non-relativistic limit of Maxwell’s equations known as the electric limit. The analysis presented here shows that only electrostatics is possible for the electric limit, wholly in harmony with the trivial nature of time evolution governed by the dual Galilei group.     

Rapid detection of bacterial endotoxins in ophthalmic viscosurgical device materials by direct analysis in real time mass spectrometry

Bacterial endotoxins are lipopolysaccharides bound to the bacterial cell wall and released when bacteria rupture or disintegrate. Possible contamination of endotoxin in ophthalmic devices can cause a painful eye inflammation or result in toxic anterior segment syndrome after cataract surgery. Measurement of bacterial endotoxin in medical device materials is difficult since endotoxin binds with polymer matrix and some of the materials are very viscous and non-water soluble, where traditional enzyme-based Limulus amebocyte lysate (LAL) assay cannot be applied. Here we propose a rapid and high throughput ambient ionization mass spectrometric (MS) method using direct analysis in real time (DART) for the evaluation of endotoxin contamination in medical device materials. Large and structurally complex endotoxin instantaneously breaks down into low-mass characteristic fragment ions using DART and is detected by MS in both positive and negative ion modes. This method enables the identification and separation of endotoxin from medical materials with a detection limit of 0.03 ng mL⁻¹ endotoxins in aqueous solution. Ophthalmic viscosurgical device materials including sodium hyaluronate (NaHA), non-water soluble perfluoro-n-octane (PFO) and silicone oil (SO) were spiked with different known concentrations of endotoxin and analyzed by DART MS, where the presence of endotoxin was successfully detected and featured small mass fragment ions were generated for NaHA, PFO and SO as well. Current findings showed the feasibility of measuring endotoxin contamination in medical device materials using DART-MS, which can lead to a one-step analysis of endotoxins in different matrices, avoiding any potential contamination during sample pre-treatment steps.     

Time asymmetric quantum theory and the ambiguity of the Z-boson mass and width

Relativistic Gamow vectors emerge naturally in a time asymmetric quantum theory as the covariant kets associated to the resonance pole in the second sheet of the analytically continued S-matrix. They are eigenkets of the self-adjoint mass operator with complex eigenvalue and have exponential time evolution with lifetime . If one requires that the resonance width (defined by the Breit-Wigner lineshape) and the resonance lifetime always and exactly fulfill the relation , then one is lead to the following parameterization of in terms of resonance mass and width : . Applying this result to the -boson implies that and $\Gamma_R \approx \Gamma_Z-1.2\mbox{MeV}$ are the mass and width of the {\it Z}-boson and not the particle data values or any other parameterization of the Z-boson lineshape. Furthermore, the transformation properties of these Gamow kets show that they furnish an irreducible representation of the causal Poincaré semigroup, defined as a semi-direct product of the homogeneous Lorentz group with the semigroup of space-time translations into the forward light cone. Much like Wigner's unitary irreducible representations of the Poincaré group which describe stable particles, these irreducible semigroup representations can be characterized by the spin-mass values . Received 8 June 2000 / Published online: 27 November 2000     

Time asymmetric quantum theory – II. Relativistic resonances from S‐matrix poles

Relativistic resonances and decaying states are described by representations of Poincaré transformations, similar to Wigner's definition of stable particles. To associate decaying state vectors to resonance poles of the S‐matrix, the conventional Hilbert space assumption (or asymptotic completeness) is replaced by a new hypothesis that associates different dense Hardy subspaces to the in‐ and out‐scattering states. Then one can separate the scattering amplitude into a background amplitude and one or several “relativistic Breit‐Wigner” amplitudes, which represent the resonances per se. These Breit‐Wigner amplitudes have a precisely defined lineshape and are associated to exponentially decaying Gamow vectors which furnish the irreducible representation spaces of causal Poincaré transformations into the forward light cone.