Showing papers in "Communications in Mathematical Physics in 1973"
TL;DR: This article derived expressions for the mass of a stationary axisymmetric solution of the Einstein equations containing a black hole surrounded by matter and for the difference in mass between two neighboring such solutions.
Abstract: Expressions are derived for the mass of a stationary axisymmetric solution of the Einstein equations containing a black hole surrounded by matter and for the difference in mass between two neighboring such solutions. Two of the quantities which appear in these expressions, namely the area A of the event horizon and the “surface gravity” κ of the black hole, have a close analogy with entropy and temperature respectively. This analogy suggests the formulation of four laws of black hole mechanics which correspond to and in some ways transcend the four laws of thermodynamics.
3,494 citations
TL;DR: In this article, it was shown that in two dimensions the Goldstone phenomenon can not occur; Goldstone's theorem does not end with two alternatives (either manifest symmetry or Goldstone bosons) but with only one (manifest symmetry).
Abstract: In four dimensions, it is possible for a scalar field to have a vacuum expectation value that would be forbidden if the vacuum were invariant under some continuous transformation group, even though this group is a symmetry group in the sense that the associated local currents are conserved. This is the Goldstone phenomenon, and Goldstone's theorem states that this phenomenon is always accompanied by the appearance of massless scalar bosons. The purpose of this note is to show that in two dimensions the Goldstone phenomenon can not occur; Goldstone's theorem does not end with two alternatives (either manifest symmetry or Goldstone bosons) but with only one (manifest symmetry).
1,069 citations
TL;DR: In this paper, the necessary and sufficient conditions for Euclidean Green's functions to have analytic continuations to a relativistic field theory were given, extending and correcting a previous paper.
Abstract: We give new (necessary and) sufficient conditions for Euclidean Green's functions to have analytic continuations to a relativistic field theory. These results extend and correct a previous paper.
1,033 citations
TL;DR: In this article, the authors derived a classical integral representation for the partition function, Z Q, of a quantum spin system and obtained upper and lower bounds to the quantum free energy in terms of two classical free energies (or ground state energies).
Abstract: We derive a classical integral representation for the partition function, Z Q , of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentum J → ∞ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality is Z C (J) ≦ Z Q (J) ≦ Z C (J + 1).
474 citations
TL;DR: The entropy density of spin lattice systems is known to be a weak upper semi-continuous functional on the set of the lattice invariant states, and it is even weak discontinuous as mentioned in this paper.
Abstract: The entropy density of spin lattice systems is known to be a weak upper semi-continuous functional on the set of the lattice invariant states. (It is even weak discontinuous.) However we prove here that it is continuous with respect to the norm topology on those states.
468 citations
TL;DR: In this article, O'Connor's approach to spatial exponential decay of eigenfunctions for multiparticle Schrodinger Hamiltonians is developed from the point of view of analytic perturbations with respect to transformation groups.
Abstract: O'Connor's approach to spatial exponential decay of eigenfunctions for multiparticle Schrodinger Hamiltonians is developed from the point of view of analytic perturbations with respect to transformation groups. This framework allows an improvement of his results in some directions; in particular if interactions are dilation analytic, exponential fall-off is shown to hold for any bound-state wave-function corresponding to an eigenvalue distinct from thresholds; it is shown that the exponential decay rate depends on the distance from the bound-state energy to the nearest threshold. Applications include non existence of positive energy bound-states.
390 citations
TL;DR: In this paper, the conditional entropy between two states of a quantum system is shown to be nonincreasing when a complete measurement is performed on the system, and the information between two quantum systems is defined and bounded above by the logarithmic correlation.
Abstract: The conditional entropy between two states of a quantum system is shown to be nonincreasing when a complete measurement is performed on the system. The information between two quantum systems is defined and is shown to be bounded above by the logarithmic correlation. This inequality is then applied to the measurement process. The entropy changes in the observed system and the measuring apparatus are compared with the information gain in the measurement.
343 citations
TL;DR: In this article, the authors examined spatially homogeneous cosmological models in which the matter content of space-time is a perfect fluid, and the fluid flow vector is not normal to the surfaces of homogeneity.
Abstract: We examine spatially homogeneous cosmological models in which the matter content of space-time is a perfect fluid, and in which the fluid flow vector is not normal to the surfaces of homogeneity. In such universes, the matter may move with non-zero expansion, rotation and shear; we examine the relation between these kinematic quantities and the Bianchi classification of the symmetry group. Detailed characterizations of some of the simplest such universe models are given.
312 citations
TL;DR: In this article, an ensemble of spin 1/2 Ising spins with ferromagnetic pair interactions was used to prove a Lee-Yang theorem and GHS type correlation inequalities for the (φ4)2 theory.
Abstract: We approximate a (spatially cutoff) (φ4)2 Euclidean field theory by an ensemble of spin 1/2 Ising spins with ferromagnetic pair interactions. This approximation is used to prove a Lee-Yang theorem and GHS type correlation inequalities for the (φ4)2 theory. Application of these results are presented.
203 citations
TL;DR: In this paper, a canonical formalism based on the geometrical approach to the calculus of variations is given, which enables to define whole the canonical structure (physical quantities, Poisson bracket, canonical fields) without use of functional derivatives.
Abstract: A canonical formalism based on the geometrical approach to the calculus of variations is given. The notion of multi-phase space is introduced which enables to define whole the canonical structure (physical quantities, Poisson bracket, canonical fields) without use of functional derivatives. All definitions are of pure geometrical (finite dimensional) character.
194 citations
TL;DR: In this article, a time dependent scattering theory for a quantum mechanical particle moving in an infinite, three dimensional crystal with impurity is given, and it is shown that the Hamiltonian for the particle in the crystal without impurity has only absolutely continuous spectrum.
Abstract: A time dependent scattering theory for a quantum mechanical particle moving in an infinite, three dimensional crystal with impurity is given. It is shown that the Hamiltonian for the particle in the crystal without impurity has only absolutely continuous spectrum. The domain of the resulting wave operators is therefore the entire Hilbert space.
TL;DR: In this article, the classical spin models where the Hamiltonians are small modifications of the Hamiltonian of Dyson's hierarchical models are considered and the neighbourhood of the critical point is investigated rigorously.
Abstract: We consider the classical spin models where the Hamiltonians are small modifications of the Hamiltonians of Dyson's hierarchical models. Under some assumptions we investigate rigorously the neighbourhood of the critical point and find the critical indices. It follows that in the cases under consideration phenomenological Landau's theory of phase transitions is valid.
TL;DR: In this article, a general criterion for the uniqueness of a classical statistical mechanical system in terms of a given system of correlation functions is derived, where the criterion is ( √ √ k + 1/k) for allj and all bounded sets.
Abstract: A general criterion is derived which assures the uniqueness of the state of a classical statistical mechanical system in terms of a given system of correlation functions. The criterion is\(\sum\limits_k {(m_{k + j}^A )} ^{ - 1/k} = \infty\) for allj and all bounded setsA, where
$$m_k^A = (k!)^{ - 1} \int\limits_A \cdots \int\limits_A {\varrho _k } (x_1 , \ldots ,x_k )dx_1 \ldots dx_1 .$$
TL;DR: In this paper, the Cauchy problem associated with the Einstein field equations is solved under the assumption that the source of the gravitational field is a perfect fluid with pressure,p, equal to energy density,w, and the space-time admits the three parameter group of motions of the Euclidean plane, that is, the space time is plane symmetric.
Abstract: Solutions of the Cauchy problem associated with the Einstein field equations which satisfy general initial conditions are obtained under the assumptions that (1) the source of the gravitational field is a perfect fluid with pressure,p, equal to energy density,w, and (2) the space-time admits the three parameter group of motions of the Euclidean plane, that is, the space-time is plane symmetric. The results apply to the situation where the source of the gravitational field is a massless scalar field since such a source has the same stress-energy tensor as an irrotational fluid withp=w. The relation between characteristic coordinates and comoving ones is discussed and used to interpret a number of special solutions. A solution involving a shock wave is discussed.
TL;DR: In this article, it was shown that spherically symmetric collapse can lead to singularities which are not hidden within black holes, and that singularities can be hidden within a black hole.
Abstract: It is shown that spherically symmetric collapse can lead to singularities which arenot hidden within “black holes”.
TL;DR: In this paper, the Weyl operators are obtained as a quotient of a universal C*-algebra by some of its ideals, and characterized all these ideals are shown to be degenerate when the symplectic form which defines the C.R. is possibly degenerate.
Abstract: We consider theC*-algebras which contain the Weyl operators when the symplectic form which defines the C.C.R. is possibly degenerate. We prove that the C.C.R. are all obtained as a quotient of a universalC*-algebra by some of its ideals, and we characterize all these ideals.
TL;DR: In this article, it was shown that two translation invariant positive almost Markovian random fields have the same finite set conditional probabilities if and only if one minimizes the specific free energy of the other.
Abstract: A positive almost Markovian random field is a probability measure on a lattice gas whose finite set conditional probabilities are continuous and positive. We show that each such random field has a potential and in the translation invariant case an absolutely convergent potential. We give a criterion for determining which random fields correspond to pair potentials, or in generaln-body potentials. We show that two translation invariant positive almost Markovian random fields have the same finite set conditional probabilities if and only if one minimizes the specific free energy of the other.
TL;DR: In this paper, the infrared-singularity structure of the vertex functions of massless-particle φ4 theory is studied and the parquet approximation introduced by Diatlov, Sudakov, and Martirosian is used.
Abstract: The infrared-singularity structure of the vertex functions of massless-particle φ4 theory is studied. This allows to construct the asymptotic forms of the vertex functions of massive-particle φ4 theory in a simpler and more explicit fashion than in a previous paper. With the help of the parquet approximation introduced by Diatlov, Sudakov, and Martirosian we show that the infrared-singularity structure in a theory with besides the massless particles, massive ones is the same as in the theory with massless particles only. All these results in φ4 theory have analoga in other renormalizable theories.
TL;DR: In this article, the authors considered the problem of return to equilibrium in terms of a C*-algebraU, and two one-parameter groups of automorphisms τ, τ P corresponding to the unperturbed and locally perturbed evolutions.
Abstract: The problem of return to equilibrium is phrased in terms of aC*-algebraU, and two one-parameter groups of automorphisms τ, τ P corresponding to the unperturbed and locally perturbed evolutions. The asymptotic evolution, under τ, of τ P -invariant, and τ P -K.M.S., states is considered. This study is a generalization of scattering theory and results concerning the existence of limit states are obtained by techniques similar to those used to prove the existence, and intertwining properties, of wave-operators. Conditions of asymptotic abelianness provide the necessary dispersive properties for the return to equilibrium. It is demonstrated that the τ P -equilibrium states and their limit states are coupled by automorphisms with a quasi-local property; they are not necessarily normal with respect to one another. An application to theX−Y model is given which extends previously known results and other applications, and examples, are given for the Fermi gas.
TL;DR: In this article, perturbation analysis is applied to the theory of a General Relativistic perfectly elastic medium as developed by Carter and Quintana (1972), where the perturbations are induced both by infinitesimal displacements of the medium and by the variations of the metric tensor.
Abstract: Perturbation analysis is applied to the theory of a General Relativistic perfectly elastic medium as developed by Carter and Quintana (1972). Formulae are derived for the Eulerian variations of the principal fields (density, pressure tensor, etc.) on which the description of such a medium is based, where the perturbations are induced both by infinitesimal displacements of the medium and by infinitesimal variations of the metric tensor. These formulae will be essential for problems such as the study of torsional vibration modes in a neutron star.
TL;DR: In this article, strong restrictions on the solutions of the initial value constraints of General Relativity when the spatial hypersurface is closed are presented, which limit perturbations of non-flat closed initial solutions.
Abstract: There are strong restrictions on the solutions of the initial value constraints of General Relativity when the spatial hypersurface is closed. In particular, closed flat space is unstable: not all solutions of the linearized constraints correspond to nearby solutions of the constraints themselves. For example, no nearby solutions whatever exist which are time symmetric. Other restrictions, which limit perturbations of non-flat closed initial solutions, are also exhibited.
TL;DR: In this article, the spatial decay properties of the wave functions of multiparticle systems are investigated, where the particles interact through pair potentials in the class R+L Ã −L ǫ Ã Ã · Ã ≥ 0.
Abstract: The spatial decay properties of the wave functions of multiparticle systems are investigated. The particles interact through pair potentials in the classR+L
ɛ
∞
. The bound states lie below the bottom of the continuous spectrum of the system. Exponential decay, in anL
2 sense, is proven for these wave functions. The result is the best possible one which will cover every potential in this class.
TL;DR: In this paper, the concavity of two Herglotz functions, Tr exp(B + log A) and TrArKApK* (whereB=B* and K are fixed matrices), was proved by using the theory of Herglots functions.
Abstract: The concavity of two functions of a positive matrixA, Tr exp(B + logA) and TrArKApK* (whereB=B* andK are fixed matrices), recently proved by Lieb, can also be obtained by using the theory of Herglotz functions.
TL;DR: In this paper, a general mathematical framework called a convex structure is introduced, which generalizes the usual concept of convex set in a real linear space and shows that mappings which preserve the structure are contractions.
Abstract: A general mathematical framework called a convex structure is introduced. This framework generalizes the usual concept of a convex set in a real linear space. A metric is constructed on a convex structure and it is shown that mappings which preserve the structure are contractions. Convex structures which are isomomorphic to convex sets are characterized and for such convex structures it is shown that the metric is induced by a norm and that structure preserving mappings can be extended to bounded linear operators.
TL;DR: It is shown that a torsion free linear connection is determined by a metric of given signature if and only if its holonomy group is a subgroup of the orthogonal group corresponding to the signature.
Abstract: It is shown that a torsion free linear connection is determined by a metric of given signature if and only if its holonomy group is a subgroup of the orthogonal group corresponding to the signature.
TL;DR: In this paper, the existence and uniqueness theorem of the solution of the Cauchy problem for the coupled Einstein-Maxwell-Boltzman system is proven, in an appropriate Sobolev space for the potentials, and weighted Soboleve space for distribution function.
Abstract: An existence and uniqueness theorem of the solution of the Cauchy problem for the coupled Einstein-Maxwell-Boltzman system is proven, in an appropriate Sobolev space for the potentials, and weighted Sobolev space for the distribution function. The proof relies on a priori estimates for the collision operator previously established by D.B., and for the solution of the Einstein-Maxwell-Liouville system by Y.C.B. It is also proved here that the solution depends continuously on the data.
TL;DR: In this paper, the authors consider quantum field theoretical models in which the interaction densities are bounded functions of an ultraviolet cut-off boson field and construct the infinite volume imaginary and real time Wightman functions as limits of corresponding quantities for the space cutoff models.
Abstract: We consider quantum field theoretical models inn dimensional space-time given by interaction densities which are bounded functions of an ultraviolet cut-off boson field. Using methods of euclidean Markov field theory and of classical statistical mechanics, we construct the infinite volume imaginary and real time Wightman functions as limits of the corresponding quantities for the space cut-off models. In the physical Hilbert space, the space-time translations are represented by strongly continuous unitary groups and the generator of time translationsH is positive and has a unique, simple lowest eigenvalue zero, with eigenvector Ω, which is the unique state invariant under space-time translations. The imaginary time Wightman functions and the infinite volume vacuum energy density are given as analytic functions of the coupling constant. The Wightman functions have cluster properties also with respect to space translations.
TL;DR: In this paper, the authors studied the time evolution of a quantum-mechanical harmonic oscillator in interaction with an infinite heat bath, which is supposed to be initially in the canonical equilibrium at some temperature.
Abstract: We study the time evolution of a quantum-mechanical harmonic oscillator in interaction with an infinite heat bath, which is supposed to be initially in the canonical equilibrium at some temperature. We show that the oscillator relaxes from an arbitrary initial state to its canonical state at the same temperature, and that in the weak coupling limit the relaxation is Markovian, that is exponential. In contrast to earlier treatments of the problem [4, 5], the results are obtained without assuming any particular special form for the self-interaction of the heat bath. No use is made of coarse graining, finite memory assumptions or randomly varying Hamiltonians.
TL;DR: In this paper, the distribution of zeros of the partition function for various classes of classical lattice systems was investigated using techniques which generalize the Lee-Yang circle theorem to investigate the distribution.
Abstract: We use techniques which generalize the Lee-Yang circle theorem to investigate the distribution of zeroes of the partition function for various classes of classical lattice systems.