Logically independent von Neumann lattices

Three definitions of logical independence of two von Neumann latticesP₁,P₂ of two sub-von Neumann algebras ₁, ₂ of a von Neumann algebra are given and the relations of the definitions clarified. It is shown that under weak assumptions the following notion, called logical independence is the strongest:A B 0 for any 0 A P₁, 0 B P₂. Propositions relating logical independence ofP₁,P₂ toC ^*-independence,W ^* independence, and strict locality of ₁, ₂ are presented.     

Unbounded derivations and invariant states

Let be a von Neumann algebra with a cyclic and separating vector . Let =i[H, ·] be the spatial derivation implemented by a selfadjoint operatorH, such thatH=0. Let be the modular operator associated with the pair (, ). We prove the equivalence of the following three conditions:1)H is essential selfadjoint onD(), andH commutes strongly with .2) The restriction ofH toD() is essential selfadjoint onD(^1/2) equipped with the inner product(|)_#=(|)+(^1/2|^1/2), , D(^1/2).3) exp (itH) exp (–itH)= for anyt.We show by an example, that the first part of 1),H is essential selfadjoint onD(), does not imply 3). This disproves a conjecture due to Bratteli and Robinson [3].Part of this work was done while O.B. was a member of Zentrum für interdisziplinäre Forschung der Universität Bielefeld     

Geometry of the space of triangulations of a compact manifold

In this paper we study the spaceT _M of triangulations of an arbitrary compact manifoldM of dimension greater than or equal to four. This space can be endowed with the metric defined as the minimal number of bistellar operations required to transform one of two considered triangulations into the other. Recently, this space became and object of study in Quantum Gravity because it can be regarded as a toy discrete model of the space of Riemannian structures onM.Our main result can be informally explained as follows: LetM be either any compact manifold of dimension greater than four or any compact four-dimensional manifold from a certain class described in the paper. We prove that for a certain constantC>1 depending only on the dimension ofM and for all sufficiently largeN the subsetT _M(N) ofT _M formed by all triangulations ofM with N simplices can be represented as the union of at least [C ^N] disjoint non-empty subsets such that any two of these subsets are very far from each other in the metric ofT _M. As a corollary, we show that for any functional from a very wide class of functionals onT _M the number of its deep local minima inT _M(N) grows at least exponentially withN, whenN.This work was partially supported by the New York University Research Challenge Fund Grant, by Grant ARO-DAAL-03-92-G-0143 and by NSERC Grant OGP0155879.     

Short distance scale invariance and high-energy processes in field theory

The dynamics of short distance interaction is considered from viewpoint of the quark field theory. If the effective coupling of quarks (the invariant charge(p ²)) decreases with distance (with increasingp ²) the field picture unifies different phenomenological models: reggeology in the region of smallp and partonology in the region of highp but(p ² ) lnp ² 1 and predict deviations with a subsequent increase ofp . A character of the deviations depends on the behaviour of whenp ². The variants of the asymptotic freedom (0) and scale invariance (g ₂ ⁰ ) are discussed. The strong suppression of—3 decay is considered as a support for the latter.Presented at the Symposium on Hadron-Hadron Scattering at High Energies, Liblice, Czechoslovakia, June 16–21, 1975.     

Thes-f model with Coulomb repulsion: Thermodynamics of two atomic cluster. Spin 1/2

A cluster of two atoms described by thes-f model with Coulomb repulsion has been considered. The interaction between localized 4f electrons (S=1/2) is taken in the molecular field approximation. The thermodynamic quantities like magnetization, specific heat and correlation functions n , n , S ^z n , S ^z n , S ^z (n –n ), n n and S ⁺ a ⁺ a as functions of temperature are presented for different band fillingN=0, 0.5, 1, 1.5, 2. The dependence of Curie temperature onN is calculated. The phase diagram forN=1 (T=0K) shows the possibility of existence of two phases: paramagnetic and ferromagnetic.The Curie temperature and the specific heat as functions ofN exhibit similar trends as found in experiments on doped magnetic semiconductors.     

A finite size analysis of the structure factor in the antiferromagnetic Heisenberg (AFH) model

From a finite size analysis we extract the structure factorS(p, N=) of the one dimensional AFH-model in the groundstate: The gross structure is well described byL (p) = –ln(1– ^p ). The fine structure which only contributes a few percent reveals a pronounced non-linear behavior inL(p) with a maximum atp=0.20 and a minimum atp=0.82.     

n-Dimensional representations of the unified field tensor* g λν

For the first time Hlavatý represented the tensor^*g^v, defined by (15a), in terms of the unified field tensorg in the space-timeX ₄. Recently, the representations of^* g ^v in terms ofg in two- and three-dimensional generalized Riemannian space were obtained by Chung. The purpose of the present paper is to obtain the generalized representations of^* g ^v in terms ofg in a generalizedn-dimensional Riemannian spaceX _n.     

Iterated networks and the spectra of renormalizable electromechanical systems

Beginning with anLRC network with impedance functionZ(), a sequence of iterated networksN _k with impedance functionsZ ^k(), k= 1, 2, 3,..., is introduced. The asymptotic comportment ofZ ^k() and the spectra ofN _k are analyzed in terms of the Julia set ofZ. An example is given of an iterated network associated with a cascade of period-doubling bifurcations.     

Some remarkable degenerations of quantum groups

We show that an irreducible representation of a quantized enveloping algebraU at a ^th root of 1 has maximal dimension (= ^N ) if the corresponding symplectic leaf has maximal dimension (=2N). The method of the proof consists of a construction of a sequence of degenerations ofU , the last one being aq-commutative algebraU ^(2N) . This allows us to reduce many problems concerningU to that concerningU ^(2N) .To Armand Borel on his 70^th birthdaySupported in part by the NSF grant DMS-9103792     

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.     

Bounded perturbations of dynamics

If and are one-parameter automorphism groups of a von Neumann algebraM is said to be a bounded perturbation of if _t – _t 0 ast0. We give a complete characterization of the bounded perturbations of . In particular, we show that if can be implemented by a strongly continuous one-parameter group with self-adjoint generator (Hamiltonian)H, then can be implemented in the same way and the corresponding HamiltonianH can be chosen to be of the formH=VHV ^–1+h, whereV is a unitary ofM andh=h*M.On leave of absence from II. Institut für Theoretische Physik, Universität Hamburg, D-2000 Hamburg 50, Federal Republic of Germany     

On X-ray measurements of residual stresses in materials with lattice strain gradient

The non-linear distribution of lattice strain ³¹⁰vs sin² observed on the surface of a polished steel specimen can be interpreted by tri-axial stress state with surface components of stress tensor ₁₁(0)= ₂₂(0) and gradientsg ₁₁= ₁₁/T, g ₃₃= ₃₃/T. The distribution of experimental values is duscussed from the viewpoint of various ways of calculating ₁₁.The authors would like to thank Dr. J. Musil, D. Sc. of the Institute of Physics of Czechoslovak Academy of Sciences for kindly providing the samples which made possible this study.     

Observed difference between the probabilities of recoilless resonance absorptionf′ and reemissionf″ of gamma-quanta in single crystals of Na2[Fe(CN)5NO]·2 H2O

The recoilless absorption probability factor,f, and recoilless reemission,f, both measured on Na₂[Fe(CN)₅NO]·2 H₂O single crystals using the black filter technique, were found to be different. Unexpectedly, the results found weref>f. In the calculation off, selfabsorption in the scatterer, non-ideality of the black filter and the influence of non-resonant scattering processes have all been taken into account. By varying the scattering geometry for the incoming and outgoing -beam relative to the crystallographic axes only a change in the reemitted valuesf _a, f_b, f_c could be detected because of the long lifetime of the excited nucleus (10^–7 s) relative to the lattice vibration frequencies (10¹² Hz).     

Structure factor of substitutional sequences

We study the structure factor for a large class of sequences of two elementsa andb such that longer sequences are generated from shorter ones by a simple substitution rulea ₁(a, b) andb ₂(a, b), where the's are some sequences ofa's andb's. Such sequences include periodic and quasiperiodic systems (e.g., the Fibonacci sequence), as well as systems such as the Thue-Morse sequence, which are neither. We show that there are values of the frequency at which the structure factors of these sequences have peaks that scale withL, the size of the system likeL ⁽⁾. For a given sequence a simple one- or two-dimensional dynamical iterative map of the variable can easily be abstracted from the substitution algorithm. The basin of attraction of a given fixed point or limit cycle of this map is a set of values of at which there are peaks of the structure factor all of which share the same value of. Furthermore, only those values of which are in the basin of attraction of the origin can have()=2. All other peaks will grow less rapidly withL. We show how to construct many sequences which, like the Thue-Morse sequence, have noL ² peaks. Other qualitative features of the structure factors are presented. Our approach unifies the treatment of a large class of apparently very diverse systems. Implications for the band structure of these systems as well as for the analysis of sequences with more than two elements are discussed.     

The slow passage through a steady bifurcation: Delay and memory effects

We consider the following problem as a model for the slow passage through a steady bifurcation: dy/dt = (t) y – y³ +, where is a slowly increasing function oft given by= _i + t ( ⁱ,<0). Both and are small parameters. This problem is motivated by laser experiments as well as theoretical studies of laser problems. In addition, this equation is a typical amplitude equation for imperfect steady bifurcations with cubic nonlinearities. When=0, we have found that=0 is not the point where the bifurcation transition is observed. This transition appears at a value = ^j > 0. We call ^j the delay of the bifurcation transition. We study this delay as a function of ⁱ, the initial position of, and, the imperfection parameter. To this end, we propose an asymptotic study of this equation as 0, small but fixed. Our main objective is to describe this delay in terms of the relative magnitude of and. Since time-dependent imperfections are always present in experiments, we analyze in the second part of the paper the effect of a small-amplitude but time-periodic imperfection given by (t) = cos(t).     

Magnetic moments of proton drip-line nuclei13O and9C

The magnetic moments of the proton drip-line nuclei¹³O(I = 3/2^–,T _1/2 = 8.6 ms) and 9C(I = 3/2^–,T _1/2 = 126 ms) have been determined for the first time through the combined techniques of polarized radioactive nuclear beams and-NMR detection. The observed magnetic moments are ¦(¹³O)¦ = 1.3891 ±0.0003 _N and ¦(⁹C)¦ = 1.3914 ±0.0005 _N. Spin expectation values are deduced to be 0.76 and 1.44 for¹³O and⁹C, respectively. While the of¹³O is consistent with the systematics from isospinT= 1/2 mirror pairs, the of⁹C is unusually large, even far larger than the single particle value, = 1.     

The flow of electric current through the chain of nonlinear inertial resistors and capacitors

The stability conditions for an electric current flowing through a chain consisting of nonlinear inertial resistors and capacitors have been investigated. The necessary conditions for the field dependent conductivity and the inertia of the resistors and applied voltage leading to undamped oscillations in the chain have been derived. For the threshold voltage this condition has the form ₀+V ₀(/V) _v0 cos _I ^T =0, similar to that obtained in an earlier work of one of the authors from thermodynamical considerations.Computer simulation gives results for the frequency and threshold conditions in good agreement with approximate analytical solutions.     

Width and shift of coherent spontaneous radiation line of system of identical emitters

A study is made of the influence of the retarded electromagnetic interaction of a system of identical oscillators arranged in a cubic lattice on the form of the line of spontaneous radiation. All the cases of the mutual relation of the wave-length, the lattice constanta and the length of the crystalL = Na are solved in a unified manner. In the long-wave case (a) it is shown that the line is greatly broadened and shifted (Fig. 3). Especially in the micro-wave case (a< ) the broadening is of the order ofN ³ _e and the shift of the orderN ²(/a) _e , where _e is the natural line width of an isolated emitter. In the optical case (a) the broadening is of the order ofN(/a) ² _e and the shift of the order (/a)³ _e . In the directions satisfying Bragg's condition the line loses its Lorentz form and further broadening and also shifting of the line may occur. It is shown that the vibrations of the crystal lattice influence the coherent effects studied in the same way as they influence the Mössbauer effect.     

New class of Finsler metrics

A class of Finsler spaces is introduced which is determined by the metric functionF(x, y)=[ +kB B )y y ]^1/2, whereB =B x, y andk is a constant. Various properties of these spaces are developed. A particular choice ofB is shown to produce a geodesic equation which is equivalent to the Lorentz equation of motion for a charged particle. Some general arguments for the physical applicability of Finsler spaces are also given.     

Cluster size distributions in equilibrium

Equilibrium size distributions for finite system containingM monomers in a volumeV undergoing coagulation (polymerization)/fragmentation, are determined for coagulation rateK _ij and fragmentation rateF _ij , corresponding to the processA _i +A _j A _i+j (hereA _k denotes a cluster containingk monomeric units). It is shown that microscopic detailed balance impliesF _ij /K _ij =a _i a _j /a _i+j , where anda _k are arbitrary. The mean and most probable size distributionc _k = ^–1 a _k e ^–k coincide in the thermodynamic limitM, V, =M/V fixed. If , 2<<3, then the theory predicts a (solgel) phase transition at a well defined value ofq = ^–1, with critical exponents = and = –2. The gel fraction exponent assumes its classical value unity, probably due to the neglect of spatial fluctuations. Finally it is indicated how the theory can in principle be extended to account for isomerism and cyclization.