Showing papers in "Communications in Mathematical Physics in 1976"
TL;DR: In this paper, the notion of a quantum dynamical semigroup is defined using the concept of a completely positive map and an explicit form of a bounded generator of such a semigroup onB(ℋ) is derived.
Abstract: The notion of a quantum dynamical semigroup is defined using the concept of a completely positive map. An explicit form of a bounded generator of such a semigroup onB(ℋ) is derived. This is a quantum analogue of the Levy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained.
6,381 citations
TL;DR: In this article, the same properties can be observed in a simple mapping of the plane defined by: \({x i + 1}} = {y_i} + 1 - ax_i^2,{y i+ 1} = b{x_i}\).
Abstract: Lorenz (1963) has investigated a system of three first-order differential equations, whose solutions tend toward a “strange attractor”. We show that the same properties can be observed in a simple mapping of the plane defined by: \({x_{i + 1}} = {y_i} + 1 - ax_i^2,{y_{i + 1}} = b{x_i}\). Numerical experiments are carried out for a =1.4, b = 0.3. Depending on the initial point (x 0,y 0), the sequence of points obtained by iteration of the mapping either diverges to infinity or tends to a strange attractor, which appears to be the product of a onedimensional manifold.by a Cantor set.
2,507 citations
TL;DR: In this article, the relativistic field theories in one time and one space dimension with interactions that are entirely due to quadratic constraints are shown to be closely related to integrable Hamiltonian systems.
Abstract: O n
-invariant classical relativistic field theories in one time and one space dimension with interactions that are entirely due to quadratic constraints are shown to be closely related to integrable Hamiltonian systems.
1,027 citations
TL;DR: In this paper, new systematic approximants are proposed for exponential functions, operators and inner derivation, and remainders of the systematic approximation are evaluated explicitly, which give degrees of convergence of approximant.
Abstract: New systematic approximants are proposed for exponential functions, operators and inner derivation δ
H
. Remainders of systematic approximants are evaluated explicitly, which give degrees of convergence of approximants. The first approximant corresponds to Trotter's formula [1]: exp(A+B)=
$$\mathop {\lim }\limits_{n \to \infty } $$
[exp(A/n) exp(B/n)]
n
. Some applications to physics are also discussed.
626 citations
TL;DR: In this article, it was shown that phase transitions occur in (φ·φ) 3 2 quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions.
Abstract: We present a new method for rigorously proving the existence of phase transitions. In particular, we prove that phase transitions occur in (φ·φ) 3 2 quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions. The central element of the proof is that for fixed ferromagnetic nearest neighbor coupling, the absolutely continuous part of the two point function ink space is bounded by 0(k−2). When applicable, our results can be fairly accurate numerically. For example, our lower bounds on the critical temperature in the three dimensional Ising (resp. classical Heisenberg) model agrees with that obtained by high temperature expansions to within 14% (resp. a factor of 9%).
484 citations
TL;DR: In this paper, a class of Hamiltonian systems including perturbations of the periodic Toda lattice and homogeneous cosmological models is studied, and a separrix approximation of oscillation regimes in these systems connected with Coxeter groups is obtained.
Abstract: A class of Hamiltonian systems including perturbations of the periodic Toda lattice and homogeneous cosmological models is studied. Separatrix approximation of oscillation regimes in these systems connected with Coxeter groups is obtained. Hamiltonian systems connected with simple Lie algebras are pointed out, which generalize the system describing periodic Toda lattice and allow theL -A pair representation.
279 citations
TL;DR: In this paper, the Gel'fand-Levitan and Marchenko formalisms for solving the inverse scattering problem are applied together to a single set of scattering phase-shifts.
Abstract: The Gel'fand-Levitan and Marchenko formalisms for solving the inverse scattering problem are applied together to a single set of scattering phase-shifts. The result is an identity relating two different types of Fredholm determinant. As an application of the method, an asymptotic formula of high accuracy is derived for a particular Fredholm determinant that determines the level-spacing distribution-function in the theory of random matrices.
220 citations
TL;DR: In the Weyl quantization scheme, the δ-function at the origin of phase space corresponds to the parity operator as discussed by the authors, and the quantization of a function f(υ) on phase space is the operator efυ/2)WυdυM, whereM is the parity and Wυm is Weyl operator.
Abstract: In the Weyl quantization scheme, the δ-function at the origin of phase space corresponds to the parity operator. The quantization of a functionf(υ) on phase space is the operator ef(υ/2)W(υ)dυM, whereM is the parity andW(υ) the Weyl operator.
208 citations
Journal Article•
208 citations
TL;DR: In this article, the statistical mechanics of unbounded n-component spin systems on the lattice were developed and the existence and uniqueness of the infinite volume free energy density for a wide class of boundary conditions were established.
Abstract: We develop the statistical mechanics of unboundedn-component spin systems on the latticeZv interacting via potentials which are superstable and strongly tempered. We prove the existence and uniqueness of the infinite volume free energy density for a wide class of boundary conditions. The uniqueness of the equilibrium state (whose existence is established in general) is then proven for one component ferromagnetic spins whose free energy is differentiable with respect to the magnetic field.
157 citations
TL;DR: In this article, the canonical and grand canonical partition function in a finite volume and the existence of the thermodynamic limit for the pressure of two component classical and quantum systems of particles with charge ±e interacting via two body Yukawa forces were established.
Abstract: We estimate the canonical and grand canonical partition function in a finite volume and prove stability and existence of the thermodynamic limit for the pressure of two component classical and quantum systems of particles with charge ±e interacting via two body Yukawa — or Coulomb forces. In the case of Coulomb forces we require neutrality. For the classical system in two dimensions there exists a critical temperatureTc at and below which the system collapses. For the classical Yukawa system the correlation functions exist for arbitrary fugacity and the general structure of the pure phases can be analyzed completely.
TL;DR: In this article, a general scheme of constructing a canonical structure (i.e., Poisson bracket, canonical fields) in classical field theories is proposed, which is manifestly independent of the particular choice of an initial space-like surface in space-time.
Abstract: A general scheme of constructing a canonical structure (i.e. Poisson bracket, canonical fields) in classical field theories is proposed. The theory is manifestly independent of the particular choice of an initial space-like surface in space-time. The connection between dynamics and canonical structure is established. Applications to theories with a gauge and constraints are of special interest. Several physical examples are given.
TL;DR: In this paper, a general momentum-space subtraction procedure is proposed for the removal of both ultraviolet and infrared divergences of Feynman integrals, which allows one to define time-ordered Green functions, as tempered distributions, for a wide class of theories with zero-mass propagators.
Abstract: A general momentum-space subtraction procedure is proposed for the removal of both ultraviolet and infrared divergences of Feynman integrals. Convergence theorems are proved which allow one to define time-ordered Green functions, as tempered distributions, for a wide class of theories with zero-mass propagators.
TL;DR: The distinguished self-adjoint extension for Dirac operators with singular potentials was shown in this article. But the results were restricted to the case where the wave functions in the domain have finite mean kinetic energy, and the essential spectrum of the distinguished extension is contained in the spectrum of free Hamiltonian.
Abstract: Some general results about perturbations of not-semibounded self-adjoint operators by quadratic forms are obtained. These are applied to obtain the distinguished self-adjoint extension for Dirac operators with singular potentials (including potentials dominated by the Coulomb potential withZ<137). The distinguished self-adjoint extension, is theunique self-adjoint extension, for which the wave functions in its domain possess finite mean kinetic energy. It is shown moreover that the essential spectrum of the distinguished extension is contained in the spectrum of the free Hamiltonian.
TL;DR: In this paper, it was shown that for a large class of interactions, any canonical Gibbs state satisfying a natural temperedness condition is a mixture of Gibbs states with appropriate activities, and vice versa.
Abstract: It is shown that for a large class of interactions any canonical Gibbs state satisfying a natural temperedness condition is a mixture of Gibbs states with appropriate activities, and vice versa. Some general results on Gibbs states and canonical Gibbs states are established. In particular, a differential characterization of Gibbs states is given.
TL;DR: In this paper, a rigorous construction of new superselection selectors for the quantum sine-Gordon equation and the (φ·φ)2-quantum field models with explicitly broken isospin symmetry in two space-time dimensions is presented.
Abstract: A rigorous construction of new super-selection selectors — so-called “soliton-sectors” — for the quantum “sine-Gordon” equation and the (φ·φ)2-quantum field models with explicitly broken isospin symmetry in two space-time dimensions is presented These sectors are eigenspaces of the chargeQ≡∫dx(grad φ)(x) with non-zero eigenvalue The scattering theory for quantum solitons is briefly discussed and shown to have consequences for the physics in the vacuum sector A general theory is developed which explains why soliton-sectors may exist for theories in two but not in four space-time dimensions except possibly for non-abelian Yang-Mills theories
TL;DR: In this article, the probability estimates for classical systems of particles with superstable interactions were extended to continuous spin systems, where superstability is assumed to be non-trivial.
Abstract: Probability estimates for classical systems of particles with superstable interactions [1] are extended to continuous spin systems.
TL;DR: In this article, the interplay between contact of the volume functional and energy conditions was exploited to find that maximal surfaces are unique, they maximize volume, and they yield future and past singularities.
Abstract: Maximal surfaces and their implications for the ambient spacetime are studied. Our methods exploit the interplay between contact of the volume functional and energy conditions. Essentially, we find that in closed universes, maximal surfaces are unique; they maximize volume; and they yield future and past singularities.
TL;DR: In this article, it is shown how to construct a sequence of real numbers tending to zero in the hamiltonian system, which is a generalization of Gaussian beams, but the constant is not independent of these multi-indices.
Abstract: LetH(x,D, ɛ) be a self-adjoint partial differential operator of the form
$$H = \sum\limits_{k = 0}^K {\varepsilon ^k H_k (x,\varepsilon D),{\rm{ }}x \in R^n }$$
. Suppose the hamiltonian system
$$\dot x = \frac{{\partial H_0 }}{{\partial \xi }},{\rm{ }}\dot \xi = - \frac{{\partial H_0 }}{{\partial x}}$$
has a nondegenerate stable periodic orbit γ on which
Then it is possible to construct a sequence of real numbers ɛ m tending to zero, a sequence of functionsu m concentrated in a tube of radius ɛ m 1/2 about the projection of γ intox-space, and a polynomialE(ɛ) such that
$$\parallel (H(\varepsilon _m ) - E(\varepsilon _m ))u_m \parallel \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} C\varepsilon _m^M \parallel u_m \parallel$$
. The powerM depends on the order of stability of γ. The constructions are explicit in terms of solutions of linear O.D.E.'s, and are generalizations of “gaussian beams”. Actually, instead of just one sequence, one gets a family of sequences parametrized by the multi-indices of ordern−1, but the constantC is not independent of these multi-indices. The nondegeneracy hypothesis implies γ is part of a one-parameter family of stable periodic orbits, andC is independent of this parameter.
TL;DR: Araki and Wyss as discussed by the authors considered a mapA→Q(A) of one-particle trace-class observables on a complex Hilbert-space ℋ into the fermionC*-algebraU(ℋ) over ℓ.
Abstract: Araki and Wyss considered in 1964 a mapA→Q(A) of one-particle trace-class observables on a complex Hilbert-space ℋ into the fermionC*-algebraU(ℋ) over ℋ. In particular they considered this mapping in a quasi-free representation.
TL;DR: In this paper, the authors give a description of the mechanism of phase transitions in the Ising model, pointing out the connection between the spontaneous magnetization and the existence of infinite clusters of “up” and “down” spins.
Abstract: We give a description of the mechanism of phase transitions in the Ising model, pointing out the connection between the spontaneous magnetization and the existence of infinite clusters of “up” and “down” spins. The picture is more complete in the two-dimensional Ising model, where we can also use a generalized version of a result by Miyamoto.
TL;DR: In this paper, a special class of nonextendible maps is introduced and investigated, which is much smaller than the class of extreme maps, and is a class of positive maps of ordered vector spaces.
Abstract: Positive maps of ordered vector spaces into the algebra of all bounded operators acting on a Hilbert space are considered. A special class of so called nonextendible maps is introduced and investigated. This class is much smaller than the class of extreme maps.
TL;DR: In this paper, an existence and uniqueness theorem for the Hartree-Fock time-dependent problem in the case of a finite Fermi system interacting via a two body potential, which is supposed dominated by the kinetic energy part of the one-particle hamiltonian was proved.
Abstract: A previous result is generalized. An existence and uniqueness theorem is proved for the Hartree-Fock time-dependent problem in the case of a finite Fermi system interacting via a two body potential, which is supposed dominated by the kinetic energy part of the one-particle hamiltonian.
TL;DR: In this paper, a quantum analogue of this classical problem incorporating locality is introduced and is shown to generate those super-selection sectors whose charge can be strictly localized in a 2D space-time.
Abstract: IfA
μ is a vector field satisfying ∂μ
A
v
−∂
v
A
μ=0 can one find a scalar field φ such thatA
μ=∂μφ? A novel quantum analogue of this classical problem incorporating locality is introduced and is shown to generate those super-selection sectors whose charge can be strictly localized. In a 2-dimensional space-time there are further possibilities; in particular, soliton sectors can be generated by this procedure.
TL;DR: In this article, the quasi-spherical collapsing space-time of Szekeres is investigated and the arbitrary functions can be chosen so that it has positive density, and no Killing vectors; yet a ballr
Abstract: The quasi-spherical collapsing space-time of Szekeres is investigated. The arbitrary functions can be chosen so that it has positive density, and no Killing vectors; yet a ballr
TL;DR: In this paper, the interface profile of the two-dimensional Ising ferromagnet is obtained for all temperatures in the thermodynamic limit, and the width of the interface depends on its length as (length) 1/2.
Abstract: The interface profile of the two-dimensional Ising ferromagnet is obtained for all temperatures in the thermodynamic limit. The width of the interface depends on its length as (length)1/2.
TL;DR: In this article, the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory was proved for spin −1/2 Ising models, ϕ4 field theories, and other continuous spin models.
Abstract: We prove the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory. These include spin −1/2 Ising models, ϕ4 field theories, and other continuous spin models. The proofs are based on the properties of a classG
− of probability measures which contains all measures of the form const exp(−V(x))dx, whereV is even and continuously differentiable anddV/dx is convex on [0, ∞). A new proof of the GKS inequalities using similar ideas is also given.
TL;DR: The existence and uniqueness of global solutions to the Cauchy problem in the space of smooth density matrices for the time-dependent Hartree-Fock equations describing the motion of finite Fermi systems interacting via a Coulomb two-body potential was proved in this paper.
Abstract: The existence and uniqueness of global solutions to the Cauchy problem is proved in the space of “smooth” density matrices for the time-dependent Hartree-Fock equations describing the motion of finite Fermi systems interacting via a Coulomb two-body potential.
TL;DR: In this article, the classical concept of K-flow is generalized to cover situations encountered in non-equilibrium quantum statistical mechanics and the ergodic properties of generalized K-flows are discussed.
Abstract: The classical concept of K-flow is generalized to cover situations encountered in non-equilibrium quantum statistical mechanics The ergodic properties of generalized K-flows are discussed Several non-isomorphic examples are constructed, which differ already in the type (II1, III),, and III1) of the factor on which they are defined In particular, generalized factor K-flows with dynamical entropy either zero (singular K-flows) or infinite (special non-abelian K-flows) are constructed
TL;DR: Some points of the past Big Bang in the closed fourdimensional Friedman-model are found to be identical with points ofThe future collapse according to the bundle-boundary definition.
Abstract: Some points of the past Big Bang in the closed fourdimensional Friedman-model are found to be identical with points of the future collapse according to the bundle-boundary definition.