The general relativistic hydrogen atom

The general relativistic Dirac equation is formulated in an arbitrary curved space-time using differential forms. These equations are applied to spherically symmetric systems with arbitrary charge and mass. For the case of a black hole (with event horizon) it is shown that the Dirac Hamiltonian is self-adjoint, has essential spectrum the whole real line and no bound states. Although rigorous results are obtained only for a spherically symmetric system, it is argued that, in the presence of any event horizon there will be no bound states. The case of a naked singularity is investigated with the results that the Dirac Hamiltonian is not self-adjoint. The self-adjoint extensions preserving angular momentum are studied and their spectrum is found to consist of an essential spectrum corresponding to that of a free electron plus eigenvalues in the gap (–mc ², +mc ²). It is shown that, for certain boundary conditions, neutrino bound states exist.Supported in part by the National Science Foundation     

Free states of the canonical anticommutation relations

Each gauge invariant generalized free state _A of the anticommutation relation algebra over a complex Hilbert spaceK is characterized by an operatorA onK. It is shown that the cyclic representations induced by two gauge invariant generalized free states _A and _B are quasi-equivalent if and only if the operatorsA ^1/2–B ^1/2 and (I–A)^1/2–(I–B)^1/2 are of Hilbert-Schmidt class. The combination of this result with results from the theory of isomorphisms of von Neumann algebras yield necessary and sufficient conditions for the unitary equivalence of the cyclic representations induced by gauge invariant generalized free states.Work supported in part by US Atomic Energy Commission, under Contract AT (30-1)-2171 and by the National Science Foundation.     

Lattice theory,quadratic spaces,and quantum proposition systems

A quadratic space is a generalization of a Hilbert space. The geometry of certain kinds of subspaces (closed, splitting, etc.) is approached from the purely lattice theoretic point of view. In particular, theorems of Mackey and Kaplansky are given purely lattice theoretic proofs. Under certain conditions, the lattice of closed elements is a quantum proposition system (i.e., a complete orthomodular atomistic lattice with the covering property).     

Light quarks and instantons

Instantons and anti-instantons can profoundly influence the structure of a non-Abelian gauge theory involving N flavors of massless quarks. Interactions of the quarks with these pseudoparticles can spontaneously generate a quark mass, break the theory's SU(N) × SU(N) chiral symmetry and bind quark-antiquark pairs to form N² ? 1 Goldstone bosons. If the spontaneously generated quark mass is small, multipseudoparticle configurations can be treated in a dilute gas approximation.     

Conductivity variations in composites of α-zirconium phosphate and fumed silica

Composite proton-conducting solid electrolytes have been formed from α-zirconium phosphate (α-Zr(HPO₄)₂·H₂O, α-ZrP) and high surface area non-porous fumed silica in varying mole ratios. Conductivity variations as a function of temperature and relative humidity have been characterised and compared to that for a similarly prepared delaminated ZrP (no silica). Conductivities increase with relative humidity and E_a decreases at high silica mole fractions. Conduction is via exposed lamellar surfaces and other microstructural imperfections in the composites.     

Linewidths of HCl broadened by CO2 and N2 and CO broadened by CO2

High resolution measurement of the linewidths of HCl broadened by CO₂ and N₂ and CO broadened by CO₂ have been performed in both the 1-0 and 2-0 bands of HCl and the 2-0 band of CO. The data were analyzed by the direct and the peak absorption methods. Values of the linewidths obtained by the two methods are in good agreement. For |m| ≤ 3, for the case of HCl + CO₂, the agreement is good for the values obtained in both bands of HCl. However for |m| > 3, the HCl + CO₂ linewidths in the 1-0 band are smaller than the corresponding lines in the 2-0 band by as much as 11% for |m| = 9. Lines (|m| ≤ 3) of the 1-0 and 2-0 bands of HCl broadened by CO₂ were also analyzed in terms of the super-Lorentzian line profile proposed by Varanasi, S. K. Sarangi, and G. D. T. Tejwani (J. Quan. Spectr. Radiative Transfer12, 857 (1972)) and the Lorentzian profile. The results indicate that near the line center (within 3γ), the shape of HCl + CO₂ lines are Lorentzian.     

Microwave spectrum of 2-chloropyridine

The microwave spectrum of 2-chloropyridine, C₅H₄NCl, has been studied in the frequency range from 26.5–40.0 GHz. The spectrum is characterized by strong parallel type transitions of a near-prolate asymmetric top. The assigned transitions have been used to evaluate the ground state rotational constants of the two chlorine isotopes. The rotational constants are (in MHz): A = 5872.52, B = 1637.83, C = 1280.48 for the ³⁵Cl isotopic species and A = 5872.16, B = 1591.76, C = 1252.17 for the ³⁷Cl isotopic species. The small inertial defect indicates the molecule is planar. In addition an excited vibrational state of C₅H₄N³⁵Cl has been observed and analyzed. The chlorine quadrupolar coupling constants were determined for the ground state and are: χ_aa = ?71.9 MHz for ³⁵Cl and χ_aa = ?54.9 MHz for ³⁷Cl. By assuming the pyridine ring structure the CCl bond length is found to be 1.72 Å.     

Self-adjoint algebras of unbounded operators

Unbounded *-representations of *-algebras are studied. Representations called self-adjoint representations are defined in analogy to the definition of a self-adjoint operator. It is shown that for self-adjoint representations certain pathologies associated with commutant and reducing subspaces are avoided. A class of well behaved self-adjoint representations, called standard representations, are defined for commutative *-algebras. It is shown that a strongly cyclic self-adjoint representation of a commutative *-algebra is standard if and only if the representation is strongly positive, i.e., the representations preserves a certain order relation. Similar results are obtained for *-representations of the canonical commutation relations for a finite number of degrees of freedom.Work supported in part by U.S. Atomic Energy Commission under Contract AT(30-1)-2171 and by the National Science Foundation.Alfred P. Sloan Foundation Fellow.     

Young modules and filtration multiplicities for Brauer algebras

We define permutation modules and Young modules for the Brauer algebra B _k (r,δ), and show that if the characteristic of the field k is neither 2 nor 3 then every permutation module is a sum of Young modules, respecting an ordering condition similar to that for symmetric groups. Moreover, we determine precisely in which cases cell module filtration multiplicities are well-defined, as done by Hemmer and Nakano for symmetric groups. Supported by the European Community through Marie Curie fellowship MCFI 2002-01325 Supported by EPSRC grant GR/S18151/01