A Remark on Formal KMS States in Deformation Quantization

Within the framework of deformation quantization, we define formal KMS states on the deformed algebra of power series of functions with compact support in phase space as C[[]]-linear functionals obeying a formal variant of the usual KMS condition known in the theory of C^*-algebras. We show that for each temperature KMS states always exist and are up to a normalization equal to the trace of the argument multiplied by a formal analogue of the usual Boltzmann factor, a certain formal star exponential.     

Natural Nonequilibrium States in Quantum Statistical Mechanics

A quantum spin system is discussed where a heat flow between infinite reservoirs takes place in a finite region. A time-dependent force may also be acting. Our analysis is based on a simple technical assumption concerning the time evolution of infinite quantum spin systems. This assumption, physically natural but currently proved for few specific systems only, says that quantum information diffuses in space-time in such a way that the time integral of the commutator of local observables converges: ⁰ _– dt [B, ^t A]<. In this setup one can define a natural nonequilibrium state. In the time-independent case, this nonequilibrium state retains some of the analyticity which characterizes KMS equilibrium states. A linear response formula is also obtained which remains true far from equilibrium. The formalism presented here does not cover situations where (for time-independent forces) the time-translation invariance and uniqueness of the natural nonequilibrium state are broken.     

The low-temperature phase of Kac-Ising models

We analyze the low-temperature phase of ferromagnetic Kax-Ising models in dimensionsd2. We show that if the range of interactions is ^–1, then two disjoint translation-invariant Gibbs states exist if the inverse temperature satisfies –1^N, where =d(1–)/(2d+2)(d+1), for any >0. The proof involves the blocking procedure usual for Kac models and also a contour representation for the resulting long-range (almost) continuous-spin system which is suitable for the use of a variant of the Peierls argument.     

The CPT-theorem in two-dimensional theories of local observables

Let be a von Neumann algebra with cyclic and separating vector , and letU(a) be a continuous unitary representation ofR with positive generator and as fixed point. If these unitaries induce for positive arguments endomorphisms of then the modular group act as dilatations on the group of unitaries. Using this it will be shown that every theory of local observables in two dimensions, which is covariant under translation only, can be imbedded into a theory of local observables covariant under the whole Poincaré group. This theory is also covariant under the CPT-transformation.     

Representations of the CCR in the (φ4)3 model: Independence of space cutoff

The algebra of observables for the renormalized ⁴ interaction in 3-dimensional space-time is constructed. It is shown that the von Neumann algebras associated with observables in a bounded regionB are independent of the space cutoff which is used in the construction of a Hamiltonian for this interaction. This result is shown to be useful in the construction of a translation invariant ⁴ theory in three dimensions. It also gives a physical criterion for the equivalence of non-Fock representations of the canonical commutation relations.     

Pauli-Dirac matrix generators of Clifford Algebras

This article presents a Pauli-Dirac matrix approach to Clifford Algebras. It is shown that the algebra C₂ is generated by two Pauli matrices i₂ and i₃; C₃ is generated by the three Pauli matrices ₁, ₂, ₃; C₄ is generated by four Dirac matrices ₀, ₁, ₂, ₃ and C₅ is generated by five Dirac matrices i₀, i₁, i₂, i₃, i₅. The higher dimensional anticommuting matrices which generate arbitrarily high order Clifford algebras are given in closed form. The results obtained with this Clifford algebra approach are compared with the vector product method which was described in a recent article [Found. Phys. 10, 531–553 (1980) by Poole, Farach and Aharonov] and with the Dirac, Rashevskii and Ramakrishnan methods of matrix generation.Supported by the National Science Foundation under Grant ISP-80-11451.     

Phases of the number-theoretic spin chain

We present numerical and analytical evidence for a first-order phase transition of the ferromagnetic spin chain with partition functionZ()=(–1)/() at the inverse temperature _cr=2.     

Ergodic properties of infinite-dimensional stochastic systems

The ergodic properties of two stochastic models _I and _II are investigated. Each model is described by a fieldx(t),t > 0, on the lattice =Z ^d,d < . For _I,x(t) evolves according to the equations wherex _s (t) R for eachs eF. Here the {w_s(t): s } are independent, one-dimensional Wiener processes, ₂ is a bounded interaction between adjacent lattice sites, and the potentials ₁ and ₂ satisfy appropriate regularity conditions. It is shown that for each model,x(t) is a Markov process on an infinite-dimensional phase spaceX. The probability measures onX that satisfy the Dobrushin-Lanford-Ruelle (DLR) conditions are stationary for this process and have a mixing property. Moreover, for _I any stationary, time-reversal-invariant probability measure that has certain regularity properties must satisfy the DLR conditions.This paper is based on a portion of the author's Ph.D. thesis.⁽²⁾     

Damage spreading in a single-component irreversible reaction process: Dependence of the system's immunity on the Euclidean dimension

The spreading of a globally distributed damage, created in the stationary regime, is studied in a single-component irreversible reaction process, i.e., the BK model [Browne and Kleban,Phys. Rev. A 40, 1615 (1989)]. The BK model describes one variant of the A+AA₂ reaction process on a lattice in contact with a reservoir of A species. The BK model has a single parameter, namely the rate of arrival of A species to the lattice (Y). The model, exhibits an irreversible phase transition between a stationary reactive state with production of A₂ species and a poisoned state with the lattice fully covered by A species. The transition takes place at critical points (Y _C ) which solely depend on the Euclidean dimensiond. It is found that the system is immune ford=1 andd=2, in the sense that even 100% of initial damage is healed within a finite healing period (T _H ). Within the reactive regime,T _H diverges when approachingY _C according toT _H (Y _C –Y)^–, with 1.62 and 1.08 ford=1 andd=2, respectively. Ford=3 a frozen-chaotic transition is found close toY _s 0.4125, i.e., well inside the reactive regime 0YY _C 0.4985. Just atY _S the damageD(t) heals according toD(t) t ^–, with 0.71. For the frozen-chaotic transition atd=3 the order parameter critical exponent 0.997 is determined.     

Thermo field dynamics of quantum spin systems

Thermo field dynamics of quantum spin systems is formulated, which gives a new variational principle at finite temperatures. The KMS relation is reformulated as identities among thermal vacuum states. Path integral formulations of the thermal vacuum state are given, which yield a new thermo field Monte Carlo method. Thermo field dynamics of finite-spin systems are studied in detail as simple examples of the present method. Pertubational expansion methods of the thermal state and time-dependent state are also given.     

Growth of clusters in a first-order phase transition

The results of computer simulations of phase separation kinetics in a binary alloy quenched from a high temperature are analyzed in detail, using the ideas of Lifshitz and Slyozov. The alloy was modeled by a three-dimensional Ising model with Kawasaki dynamics. The temperature after quenching was 0.59T _c, whereT _c is the critical temperature, and the concentration of minority atoms was=0.075, which is about five times their largest possible single-phase equilibrium concentration at that temperature. The time interval covered by our analysis goes from about 1000 to 6000 attempted interchanges per site. The size distribution of small clusters of minority atoms is fitted approximately byc ₁(1-)³ w(t),c ₁ (1–)⁴ Q _l w(t)^l(2l10); wherec _l is the concentration of clusters of sizel;Q ₂,...,Q ₁₀ are known constants, the cluster partition functions;t is the time; andw(t)=0.015(1+7.17t ^–1/3). The distribution of large clusters (l20) is fitted approximately by the type of distribution proposed by Lifshitz and Slyozov,c _l ,(t)=–(d/dl) [lnt+p (l/t)], where is a function given by those authors and is defined by(x)=C _o e–x-C ₁ e ^–4x/3-C ₂ e ^–5x/3;C ₀,C ₁,C ₂ are constants determined by considering how the total number of particles in large clusters changes with time.Supported by the U.S. Air Force Office of Scientific Research under Grant No. 78-3522 and by the U.S. Department of Energy under Contract No. EY-76-C-02-3077*000.     

Transfer Matrix Spectrum and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We obtain new properties of general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high-temperature region (1). Each model is characterized by a single-site a priori spin distribution, taken to be even. We state our results in terms of the parameter =s ⁴–3s ²², where s ^k denotes the kth moment of the a priori distribution. Associated with the model is a lattice quantum field theory which is known to contain particles. We show that for >0, small, there exists a bound state with mass below the two-particle threshold. The existence of the bound state has implications for the decay of correlations, i.e., the 4-point functions decay at a slower rate than twice that of the 2-point function. These results are obtained using a lattice version of the Bethe–Salpeter equation. The existence results generalize to N-component models with rotationally invariant a priori spin distributions.     

The Hilbert-Space Structure of Non-Hermitian Theories with Real Spectra

We investigate the quantum-mechanical interpretation of models with non-Hermitian Hamiltonians and real spectra. After describing a general framework to reformulate such models in terms of Hermitian Hamiltonians defined on the Hilbert space L ₂(-, ), we discuss the significance of the algebra of physical observables.     

Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits

We present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter (r, t) of a binary alloy undergoing phase segregation. Our model is ad-dimensional lattice gas evolving via Kawasaki exchange with respect to the Gibbs measure for a Hamiltonian which includes both short-range (local) and long-range (nonlocal) interactions. The nonlocal part is given by a pair potential ^dJ(|x–y|), >0 x and y in ^d, in the limit 0. The macroscopic evolution is observed on the spatial scale ^–1 and time scale ^–2, i.e., the density (r, t) is the empirical average of the occupation numbers over a small macroscopic volume element centered atr=x. A rigorous derivation is presented in the case in which there is no local interaction. In a subsequent paper (Part II) we discuss the phase segregation phenomena in the model. In particular we argue that the phase boundary evolutions, arising as sharp interface limits of the family of equations derived in this paper, are the same as the ones obtained from the corresponding limits for the Cahn-Hilliard equation.     

Wetting transition on grain boundaries in Al contacting with a Sn-rich melt

The contact angle at the intersection of a grain boundary in Al bicrystals with the solid Al/liquid Al–Sn interphase boundary has been measured for two symmetric tilt <011> {001} grain boundaries with tilt angles of 32° and 38.5°. The temperature dependencies (T) present the evidence of the grain boundary wetting phase transition at T_w. The observed hysteresis is consistent with the assumption that the wetting transition is of first order. The determined discontinuity in the temperature derivative of the grain boundary energy is–5.6 J/m²K (T w1=617°C) for the boundary with a low energy (=38.5°) and –17 J/m²K (T w2=604°C) for the grain boundary with a high energy (=32°).     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

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.     

The ground state of the Bose gas

The mathematical formalism describing the Bose gas at zero temperature is analysed with the aid of methods that have recently been successful in relativistic quantum field theory. First the spectrum conditions for an infinitely extended system are given and the algebra of observables and the algebra of field operators are defined. General properties of states over these algebras are discussed and theorems are given which connect the linked cluster property, translation invariance and the purity of the states. It is proved that pure states over the algebra of observables have the property of factorisable off-diagonal long range order. The class of quasi free states is defined and of these states those which are translation invariant and possess the linked cluster property are analysed. It is shown that this class of states contains a subclass of pure states of the Bogoliubov type and a subclass of states which are mixtures of non-translationally invariant pure states. The applications of these quasi free states to the interacting Bose gas are summarized.     

Theα-α′ phase transition of hydrogen in elastically anisotropic cubic metals

The formulation of the statistical mechanics of the - phase transition in hydrogenmetal-systems is extended in such a way as to include the elastic anisotropy of the host crystal. The corresponding reduction of symmetry leads to qualitative modifications of the results. Depending on hydrogen concentration and temperature, phases with homogeneous,C ₃- andC ₂-symmetric equilibrium hydrogen distributions are formed. The transitions between these phases are investigated, and the relation of their orders and the corresponding changes in symmetry is studied. Differences between isotropic and anisotropic systems are discussed.     

Non-smoothness of event horizons of Robinson-Trautman black holes

It is shown that generic small data Robinson-Trautman space-times cannot beC ¹²³ extended beyond the r=2m Schwarzschild-like event horizon. This implies that an observer living in such a space-time can determine by local measurements whether or not he has crossed the event-horizon of the black-hole.