Von neumann entropy as information rate

Recently it has been shown that quantum theory can be viewed as a classical probability theory by treating Hilbert space as a measure space (H, B(H)) of events or hidden states. Each density operator defines a set of probability measures such that(E _n )=w _n (alln). Coding elements H by subspacesE _n entails distortion. We show that the von Neumann entropyS() = -trInequals the effective rate at which the Hilbert space produces information with zero expected distortion, and comment on the meaning of this.     

Precipitation sequences and transformation kinetics in the Al-30 wt.% Zn alloy aged below 160°C

The transformation sequences in the Al-30 wt.% Zn alloy aged below 160 °C have been investigated by X-ray diffraction and transmission electron microscopy. The sequence G.-P. zones _R has been observed between 85 and 161 °C, whereas the direct G.-P. zones to-transformation occurs below 85 °C. The identical isothermal growth kinetics of both the G.-P. zones and the _R -precipitates was found. The log-log plots yielding a set of parallel straight lines from 50 to 150 °C indicate no change in the activation energy of growth of both these precipitates. The change in the precipitation sequence led to the change in the transformation kinetics. The T-T-T curves have been measured for the formation of metastable _R -phase and for the completion of ellipsoidal G.-P. zones or of _R to-transformation. The retardation in reaching equilibrium found below 85 °C was associated with the observed change in the transformation sequence. This was explained by the different accommodation of the misfit at the interfaces between G.-P. zones or _R -precipitates and the-matrix.The authors are much indebted to Doc. Dr. V.Syneek, CSc. for stimulating discussions. Our thanks are also due to Ing. V.íma and Mr. P.Vyhlídka for the preparation and careful chemical analyses of the investigated samples. The assistance of Mr. Z.iký and Mr. V.Petr in X-ray diffraction measurements and the heat-treatment of samples is appreciated.     

More surprises in the general theory of lattice systems

I use Israel's methods to prove new theorems of ubiquitous pathology for classical and quantum lattice systems. The main result is the following: Let be any interaction and be any translation-invariant equilibrium state for (extremal or not). Then there exists a sequence { _k } of interactions converging to , having extremal (or even unique) translation-invariant equilibrium states _k , such that { _k } converges to . In certain situations the perturbations _k – can be chosen to lie in a cone of antiferromagnetic pair interactions. I discuss the connection with results of Daniëls and van Enter, and point out an application to the one-dimensional ferromagnetic Ising model with 1/r ² interaction (Thouless effect).     

Spectral superbroadening of self-focused picosecond laser pulses in D2O

Intense picosecond light pulses of a mode-locked Nd: glass laser at _L =1054 nm (fundamental wavelength) and _SH =527 nm (second harmonic wavelength) are passed through a sample of D₂O under self-focusing conditions. Spectrally structured superbroadened, spatially bell-shaped emission in the forward direction is obtained. Primary generation processes are pump-pulse-degenerate stimulated parametric four-photon interaction (₁ + _{1 3} + ₄) and stimulated Raman scattering (_{1 R} + ), which occur concurrently (₁= _L or _SH angular pump frequency, #x03C9; _R first Stokes frequency, #x03C9; signal frequency, #x03C9;₃ signal frequency, #x03C9;₄ idler frequency). The parametric four-photon interaction occurs under collinear non-phase-matched conditions and under longitudinally phase-matched, transversally non-phase-matched (erenkov-like) conditions. Subsequent interaction processes are pump-pulse-nondegenerate four-photon interaction of the type ₁ + _{R 3} + ₄, coherent antiStokes Raman scattering (CARS, ₁ + ₁ – _{4 3}), inverse Raman scattering ( _A + ₁ + ), and cascading light up-conversion of the type ₁ + _(i) – _{R (i+1)}.     

A Remark on the Use of Differential Forms for the Skyrme Problem

We introduce a formulation of the Skyrme problem using differential forms. By means of this formulation, we prove first that the homothetic map between the standard three-sphere of radius R, S³ _r R⁴, and S³ ₁ is the unique minimizer, modulo isometries, of the Skyrme energy in its homotopy class, for any R less than some critical value R₀ (3/2, 2]. We then establish a stability result for this Skyrme-form problem from which we can recover the result of M. Loss and N. S. Manton which states that this homothetic map is stable only up to R = 2.     

A noncommutative geometric approach to left-right symmetric weak interactions

By combining the generalized exterior algebra of forms over a noncommutative algebra with the gauging of discrete directions and the associated Higgs fields, we consider the construction of the bosonic sector of left-right symmetric models of the form SU(2) _L SU(2) _R U(1). We see that within this formalism maximal use can be made of the gauge connection associated with the noncommutative graded algebra.     

On the hardening in modulated structure

Hardening in modulated structure is evaluated using the periodic approximation. The critical shear stress increment due to the periodic structure is calculated in the constant line energy approximation. The results are applicable to any periodic structure (concentration waves must be neither homophase nor symmetric) exerting on the dislocation local glide forces with an amplitude smaller than ( denotes the line energy of corresponding straight dislocation directed along the concentration variations with the wave vector). In the zero approximation, the critical forceb is then simply the glide force on the straight dislocation averaged along its length in its most hardened position.     

On infinite direct products of Lorentz transformations

An infinite direct product _i =1 U _i (a) of continuous unitary representations of SL(2,C) in Hilbert spaces _i is continuous only on certain incomplete direct product subspaces of _i =1 _i . If no representations of the complementary series occur, then each of these subspaces contains a product vector almost all factors of which are SL(2, C)-invariant.     

An application of Bell's inequalities to a quantum state extension problem

Using techniques from the study of quantum violations of Bell's inequalities, we give examples of three C ^*-algebras A, B, C, and states ₁₂ on A B, and ₂₃ on B C, which agree on B, but do not have a common extension to A B C. This situation cannot occur in classical probability, i.e. for commutative algebras.     

Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

Orientation of an oxygen-induced non-axially symmetric electric field gradient at Ta in Nb

The interaction of dilute Hf in Nb with interstitially dissolved oxygen has been investigated via the --TDPAC-technique applied to the 482 keV state of¹⁸¹Ta. After melting the parent radioactivity¹⁸¹Hf with Nb the trapping of oxygen at the probe atoms leads to a quadrupole interaction frequency _Q =268(1) MHz with an asymmetry parameter of =0.68(1). The temperature dependence of this frequency between 17 K and 293 K was found to be very weak whereas varies considerably. In a single crystal experiment the orientation of the three principal axes of the electric field gradient tensor leading to _Q has been determined. TheV _zz -axis points into a 110, theV _yy -axis into a 100 and theV _x -axis into a 110 direction. The results can be explained in the point charge model by assuming that one oxygen atom is trapped on the second nearest neighbour octahedral position next to the substitutional atom and the covalent bonds between the daughter Ta and the oxygen lead to a localization of charge near the probe atom.     

Temperature in Friedmann thermodynamics and its generalization to arbitrary space-times

In recent articles we have introduced Friedmann thermodynamics, where certain geometric parameters in Friedmann models were treated like their thermodynamic counterparts (temperature, entropy, Gibbs potential, etc.). This model has the advantage of allowing us to determine the geometry of the universe by thermodynamic stability arguments. In this paper, in search for evidence for the definition of gravitational temperature, we will investigate a massless conformal scalar field in an Einstein universe in detail. We will argue that the gravitational temperature of the Einstein universe is given asT _g=1/2) (c/k) (1/R ₀), where R₀ is the radius of the Einstein universe. This is in accord with our definition of gravitational temperature in Friedmann thermodynamics and determines the dimensionless constant as 1/2. We discuss the limitations of the model we are using. We also suggest a method to generalize our gravitational temperature to arbitrary space-times granted that they are sufficiently smooth.Based on three essays awarded honorable mention in the years 1987, 1988 and 1989 by the Gravity Research Foundation—Ed.     

Gas‐Kinetic Temperature of Planar High‐Frequency Capacitive Discharge Plasma in N2/CO2/He Gas Mixtures

Using methods of emission spectroscopy, we have determined the gaskinetic temperature fields of planar highfrequency capacitive discharge plasma in N₂/CO₂/He gas mixtures depending on the excitation frequency, discharge current, pressure, and chemical composition of the mixture. It is shown that the dominant contribution to the neutral component heating in the center of the discharge chamber is made by the processes of V–T relaxation of vibrationally excited molecules N₂(X ¹, V) and CO₂(X ¹, V), whereas in the zones near the electrodes an important role in the heating is played, along with the V–T relaxation of the N₂(X ¹, V) and CO₂(X ¹, V) molecules, by the processes of deactivation of the metastable states A ³ of the N₂ molecule.     

Entropy andp-particle observables. I. The general program

An information-theoretic notion of entropy is proposed for a system ofN interacting particles which assesses an observer's limited knowledge of the state of the system, assuming that he or she can measure with arbitrary precision all one-particle observables and correlations involving some numberp of the particles but is completely ignorant of the form of any higher-order correlations involving more thanp particles. The idea is to define a generic measure of entropyS[ ] = –Tr log for an arbitrary density matrix or distribution function , and then, given the trueN-particle, to define a reduced _R ^P which reflects the observer's partial knowledge. The result, at any timet, is a chain of inequalitiesS[ _R ¹ ]S[ _R ² ]...S[ _R ^N ]S[], with true equalityS[ _R ^p ]=S[ _R ^p+1 ] if and only if the true factorizes exactly into a product of contributions involving all possiblep-particle groupings. It follows further than (1) if, at some initial timet ₀, the true factorizes in this way, thenS[ _R ^p (]S[ _R ^p (t ₀)] for all finite timest>t ₀, with equality if and only if the factorization is restored, and (2) the initial response of the system must be to increase itsp-particle entropy.     

Nelson's symmetry and the infinite volume behavior of the vacuum inP(φ)2

LetH _l be the Hamiltonian in aP()₂ theory with sharp space cutoff in the interval (–l/2,l/2). LetE _l =inf(H _l ), (l)=–E _l /l, and let _l be the vacuum forH _l . discuss properties of (l) and _l . In particular, asl, there are finite constants <0 and such that (l), ((l)–)l, and hence (l)=+/l+o(l ^–1). Moreover exp(–c ₁ l) _{l 1}exp(–c ₂ l) forc ₁,c ₂ positive constants, where _{l 1} is theL ¹(Q, d₀) norm of ₁ with respect to the Fock vacuum measure. We also present a new proof of recent estimates of Glimm and Jaffe on local perturbations ofH _l in the infinite volume limit.Research sponsored by AFOSR under Contract No. F44620-71-C-0108.On leave from Istituto di Fisica Teorica, Universitá di Napoli and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli.A. Sloan Foundation Fellow.     

An algebra related to the fusion rules of Wess-Zumino-Witten models

We introduce a new family of examples of quantum groups. Its irreducible representations are indexed by level l dominant integral weights of affine Lie algebra s(n), and their branching rules with respect to the truncated tensor product is given by the fusion rules of SU(n) _l -Wess-Zumino-Witten model in conformal field theory.     

Self-avoiding walks and trees in spread-out lattices

LetG _R be the graph obtained by joining all sites ofZ ^d which are separated by a distance of at mostR. Let (G _R ) denote the connective constant for counting the self-avoiding walks in this graph. Let (G _R ) denote the coprresponding constant for counting the trees embedded inG _R . Then asR, (G _R ) is asymptotic to the coordination numberk _R ofG _R , while (G _R ) is asymptotic toek _R. However, ifd is 1 or 2, then (G _R )-k _R diverges to –.Dedicated to Oliver Penrose on this occasion of his 65th birthday.     

The Schrödinger equation and canonical perturbation theory

LetT ₀(, )+V be the Schrödinger operator corresponding to the classical HamiltonianH ₀()+V, whereH ₀() is thed-dimensional harmonic oscillator with non-resonant frequencies =(₁, ... , _d ) and the potentialV(q ₁, ... ,q _d) is an entire function of order (d+1)^–1. We prove that the algorithm of classical, canonical perturbation theory can be applied to the Schrödinger equation in the Bargmann representation. As a consequence, each term of the Rayleigh-Schrödinger series near any eigenvalue ofT ₀(, ) admits a convergent expansion in powers of of initial point the corresponding term of the classical Birkhoff expansion. Moreover ifV is an even polynomial, the above result and the KAM theorem show that all eigenvalues _n (, ) ofT ₀+V such thatn coincides with a KAM torus are given, up to order , by a quantization formula which reduces to the Bohr-Sommerfeld one up to first order terms in .     

A Generalization of the Bott–Borel–Weil Theorem and Euler Number Multiplets of Representations

The Bott–Borel–Weil theorem (BBW) is a statement about a complex homogeneous space X=G/R where G is a compact semisimple Lie group and R is the centralizer of a torus in G. One knows that BBW is equivalent to the determination of how R operates on the cohomology of a certain nilpotent Lie algebra of antiholomorphic tangent vectors operating on an arbitrary irreducible G-module. Upon replacing the complex structure with a space S of spinors BBW is equivalent to a statement about the eigenvalues of the Casimir operator of R in the module S V where V is the irreducible G-module with highest weight . But then the complex structure is eliminated and the problem makes sense for any compact subgroup R G. We solve the problem in the case where R and G have the same rank. The very special case where R is the centralizer of a torus then yields BBW. Involved in the general case (i.e. arbitrary R) is a new (at least to mathematicians) Dirac operator with a cubic term whose square is expressed in terms of the Casimir operator. Actually the spectral resolution of the Dirac operator is determined in the general case. The kernel of the Dirac operator is only given in the equal rank case. A key feature in that case is that, associated to , there is a multiplet of representations of R, having rather striking properties, and cardinality equal to the Euler number of G/R.     

Anisotropy of the low temperature resistivity of hexagonal single crystalline spin glass systems

The impurity contribution to the resistivity in zero field (T) of dilute hexagonal single crystals of ZnMn, CdMn and MgMn has been studied in the mK range on samples cut parallel () and perpendicular () to thec-axis, using a SQUID technique for the measurements. Typical spin glass behavior is found in (T) as well as (T) for all alloys, with Kondo like logarithmic increases at higher temperatures and maxima atT _m at lower temperatures, indicating the influence of impurity interactions. The differences in the corresponding isotropic resistivity _poly(T) between the three systems can qualitatively be understood within the framework of a theoretical model by Larsen, describing (T) as a function of universal quantitiesT/T _K and _RKKY/T _K , where _RKKY is the RKKY-interaction strength andT _K the Kondo temperature. With respect to the two lattice directions studied, the behavior of (T and (T is anisotropic in the Kondo regime as well as in the range where ordering becomes important. While the anisotropy in the Kondo slope can be understood by an anisotropic unitarity limit, the understanding of the anisotropy in region where impurity interactions are important remains problematic.Dedicated to Prof. Dr. S. Methfessel on the occasion of his 60th birthday