On the failure of subadditivity of the Wigner–Yanase entropy

It was recently shown by Hansen that the Wigner–Yanase entropy is, for general states of quantum systems, not subadditive with respect to decomposition into two subsystems, although this property is known to hold for pure states. We investigate the question whether the weaker property of subadditivity for pure states with respect to decomposition into more than two subsystems holds. This property would have interesting applications in quantum chemistry. We show, however, that it does not hold in general, and provide a counterexample. Work partially supported by U.S. National Science Foundation grant PHY-0353181 and by an Alfred P. Sloan Fellowship. This paper may be reproduced, in its entirety, for non-commerical purposes.     

Reality in the differential calculus on q-Euclidean spaces

The nonlinear reality structure of the derivatives and the differentials for the Euclidean q-spaces are found. A real Laplacian is constructed and reality properties of the exterior derivative are given.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-85-15857.     

Braided Hopf algebras and differential calculus

We show that the algebra of the bicovariant differential calculus on a quantum group can be understood as a projection of the cross-product between a braided Hopf algebra and the quantum double of the quantum group. The resulting super-Hopf algebra can be reproduced by extending the exterior derivative to tensor products.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-90-21139.Supported in part by a Feodor-Lynen Fellowship.     

Phase transitions in anisotropic lattice spin systems

A general method for proving the existence of phase transitions is presented and applied to six nearest neighbor models, both classical and quantum mechanical, on the two dimensional square lattice. Included are some two dimensional Heisenberg models. All models are anisotropic in the sense that the groundstate is only finitely degenerate. Using our method which combines a Peierls argument with reflection positivity, i.e. chessboard estimates, and the principle of exponential localization we show that five of them have long range order at sufficiently low temperature. A possible exception is the quantum mechanical, anisotropic Heisenberg ferromagnet for which reflection positivity isnot proved, but for which the rest of the proof is valid.Work partially supported by U.S. National Science Foundation grant no. MPS 75-11864Work partially supported by U.S. National Science Foundation grant no. MCS 75-21684 A01     

The Thermodynamic Limit for Matter Interacting with Coulomb Forces and with the Quantized Electromagnetic Field: I. The Lower Bound

The proof of the existence of the thermodynamic limit for electrons and nuclei interacting via the Coulomb potential, was accomplished decades ago in the framework of non-relativistic quantum mechanics, . This result did not take account of interactions caused by magnetic fields, however, (the spin-spin interaction, in particular) or of the quantized nature of the electromagnetic field. Recent progress has made it possible to undertake such a proof in the context of non-relativistic QED. This paper contains one part of such a proof by giving a lower bound to the free energy that is proportional to the number of particles and which takes account of the fact that the field, unlike the particles, is never confined to a finite volume. In the earlier proof the lower bound was a ‘two line’ corollary of the ‘stability of matter’. In QED the proof is much more complicated.©2004 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.Work partially supported by U.S. National Science Foundation grant PHY 01-39984.Work partially supported by U.S. National Science Foundation grant DMS 03-00349.     

Six generatorq-deformed Lorentz algebra

The six generator deformation of the Lorentz algebra is presented. The Hopf algebra structure and the reality conditions are found. The chiral decomposition of SL(2, C) is generalized to theq-case. Casimir operators for theq-Lorentz algebra are given.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-85-15857.     

The positivity of the pressure in Thomas Fermi theory

We prove the positivity of the pressure and compressibility for neutral systems in the Thomas-Fermi theory of molecules. Our results include some new properties of the Thomas-Fermi potential and a proof that the kinetic energy is superadditive.Work partially supported by U.S. National Science Foundation grant MCS 75-21684 A02On leave from Department of Physics, Universidad de Chile, Santiago, Chile     

Antiproton annihilation spectra

Recent data on antiproton (antineutron) spectra from hydrogen, deuterium and nuclear targets are discussed. Special emphasis is given to rare or unusual channels and phenomena. Work supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under grant AF05R87-0246 and U.S. National Science Foundation.     

New hierarchies of knot polynomials from topological Chern-Simons gauge theory

In this Letter, we report on a study of the expectation values of Wilson loops in D=3 topological Chern-Simons theory associated with the fundamental representation of the simple Lie algebras SO(n) and Sp(n). The skein relations satisfied by these expectation values are derived by conformal field-theory techniques. New hierarchies of invariant polynomials for knots in S ³ can be derived from these relations (at least) up to ten crossings. The N=3 Akutsu-Wadati polynomials are a special case with G=SO(3). The expectation value of the Wilson loops for a couple of simple unknotted circles is identified to the Weyl character.Work supported in part by U.S. National Science Foundation Grant PHY8706501.Work supported in party by Chinese National Science Foundation through Nankai University.     

Constructive proof of localization in the Anderson tight binding model

We prove that, for large disorder or near the band tails, the spectrum of the Anderson tight binding Hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues. The corresponding eigenfunctions are exponentially well localized. These results hold in arbitrary dimension and with probability one. In one dimension, we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder. Our techniques extend to other physical systems which exhibit localization phenomena, such as infinite systems of coupled harmonic oscillators, or random Schrödinger operators in the continuum.Work supported in part by National Science Foundations grant MCS-8108814 (A03).Work supported in part by National Science Foundation grant DMR 81-00417.     

A proof of part of Haldane's conjecture on spin chains

It has been argued that the spectra of infinite length, translation and U(1) invariant, anisotropic, antiferromagnetic spin s chains differ according to whether s is integral or 1/2 integral: There is a range of parameters for which there is a unique ground state with a gap above it in the integral case, but no such range exists for the 1/2 integral case. We prove the above statement for 1/2 integral spin. We also prove that for all s, finite length chains have a unique ground state for a wide range of parameters. The argument was extended to SU(n) chains, and we prove analogous results in that case as well.Work partially supported by U.S. National Science Foundation grant PHY80-19754 and by the A.P. Sloan Foundation.Work partially supported by U.S. National Science Foundation grant PHY-85-15288.     

Lower bounds for resonance widths in potential and obstacle scattering

Explicit lower bounds are given for the size of the imaginary parts of resonances for Schrödinger operators with non-trapping or trapping potentials, and for the Dirichlet Laplacian in the exterior of a star-shaped obstacle, both acting in three dimensions.Work partially supported by DiUC/FONDECYT (Chile)Work partially supported by U.S. National Science Foundation grant DMS 8705610     

Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems

We use Ginibre's general formulation of Griffiths' inequalities to derive new correlation inequalities for two-component classical and quantum mechanical systems of distinguishable particles interacting via two body potentials of positive type. As a consequence we obtain existence of the thermodynamic limit of the thermodynamic and correlation functions in the grand canonical ensemble at arbitrary temperatures and chemical potentials. For a large class of systems we show that the limiting correlation functions are clustering. (In a subsequent article these results are extended to the correlation functions of two-component quantum mechanical gases with Bose-Einstein statistics). Finally, a general construction of the thermodynamic limit of the pressure for gases which are not H-stable, above collapse temperature, is presented.Research supported in part by the U.S. National Science Foundation under grant MPS 75-11864A Sloan Foundation Fellow     

Stability of coulomb systems with magnetic fields

The ground state energy of an atom in the presence of an external magnetic filedB (with the electron spin-field interaction included) can be arbitrarity negative whenB is arbitrarily large. We inquire whether stability can be restored by adding the self energy of the field, B ². For a hydrogenic like atom we prove that there is a critical nuclear charge,z _c , such that the atom is stable forz<z _c and unstable forz>z _c .Work partically supported by U.S. National Scinece Foundation grant DMS-8405264 during the author's stay at the Institute for Advanced Study, Princeton, NJ, USAWork partially supported by U.S. National Science Foundation grant PHY-8116101-A03Work partially supported by U.S. and Swiss National Science Foundation Cooperative Science Program INT-8503858. Current address: Institut f. Mathematik, FU Berlin, Arnimallee 3, D-1000 Berlin 33     

Convergence to equilibrium in a system of reacting polymers

Chemical kinetics of a system of reacting polymers is modelled by an equation which shares certain properties with Boltzmann's equation. Being more tractable, however, this evolution may be of an illustrative value for the latter. The existence and uniqueness of solutions are analysed. We derive an entropy production inequality which is used to prove global exponential decay of the free energy. With its aid a uniform rate for strong convergence to equilibrium is proven. The generators of the linearlized flow at the vicinity of the equilibria are diagonalized.Work partly supported by U.S. National Science Foundation grant MCS 75-21684 A02     

Optimal hypercontractivity for fermi fields and related non-commutative integration inequalities

Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of non-commutative integration are established.Work supported by U.S. National Science Foundation grant no. PHY90-19433-A01.     

Inertial range scaling of laminar shear flow as a model of turbulent transport

Asymptotic scaling behavior, characteristic of the inertial range, is obtained for a fractal stochastic system proposed as a model for turbulent transport.Research supported in part by the U.S. Department of Energy, contract DE-FG02-90ER25084Research supported in part by the U.S. Department of Energy, contract DE-FG02-90ER25084, the National Science Foundation, grant DMS-8901884, the Army Research Office, grant DAAL03-K-0017     

Global symmetry restoration in high temperature Higgs theories

A simple high temperature expansion is developed for lattice gauge theories with scalar matter fields. The expansion is used to prove the absence of global symmetry breaking for sufficiently high temperature.Research supported in part by U.S. National Science Foundation grant PHY 8117463Research supported in part by the Department of Energy under grant No. DE-AC02-76ER03072Alfred P. Sloan Research Fellow     

Bremsstrahlung from solitons

One dimensional solitons behave like Newtonian particles if subjected to external forces, as has been shown by Fogel et al. for the case of sine-Gordon solitons. We add here the observation that they also emit free radiation—i.e. Bremsstrahlung—if accelerated by an external force. The radiation spectrum is calculated for the sine Gordon case.Work supported in part by the National Science Foundation, Grant DMR 74-23494 and in part by a travel grant from the KFA Jülich     

A construction of thec<1 modular invariant partition functions

Decomposition theorems for certain representations of Kac-Moody algebras which are needed for the construction of modular invariant unitary conformal models are proved. It is shown that allc<1 modular invariant models can then be recovered from gauged free fermionic models, including the exceptional cases.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY85-15857Supported by the Swiss National Science FoundationSupported in part by the American-Israeli Binational Science Foundation and the Israeli Academy of Sciences