Showing papers in "Communications in Mathematical Physics in 1967"
TL;DR: In this article, the authors studied the representation of the C*-algebra of observables corresponding to thermal equilibrium of a system at given temperature T and chemical potential μ and showed that the representation is reducible and that there exists a conjugation in the representation space, which maps the von Neumann algebra spanned by the representative of\(\mathfrak{A}\) onto its commutant.
Abstract: Representations of theC*-algebra\(\mathfrak{A}\) of observables corresponding to thermal equilibrium of a system at given temperatureT and chemical potential μ are studied. Both for finite and for infinite systems it is shown that the representation is reducible and that there exists a conjugation in the representation space, which maps the von Neumann algebra spanned by the representative of\(\mathfrak{A}\) onto its commutant. This means that there is an equivalent anti-linear representation of\(\mathfrak{A}\) in the commutant. The relation of these properties with the Kubo-Martin-Schwinger boundary condition is discussed.
763 citations
TL;DR: In this paper, a detailed study of nonrelativistic particles and their properties, as described by Galilei invariant wave equations, is devoted to a precise distinction between the specifically relativistic properties of elementary quantum mechanical systems and those which are also shared by non-relativasistic systems.
Abstract: This paper is devoted to a detailed study of nonrelativistic particles and their properties, as described by Galilei invariant wave equations, in order to obtain a precise distinction between the specifically relativistic properties of elementary quantum mechanical systems and those which are also shared by nonrelativistic systems. After having emphasized that spin, for instance, is not such a specifically relativistic effect, we construct wave equations for nonrelativistic particles with any spin. Our derivation is based upon the theory of representations of the Galilei group, which define nonrelativistic particles. We particularly study the spin 1/2 case where we introduce a four-component wave equation, the nonrelativistic analogue of the Dirac equation. It leads to the conclusion that the spin magnetic moment, with its Lande factorg=2, is not a relativistic property. More generally, nonrelativistic particles seem to possess intrinsic moments with the same values as their relativistic counterparts, but are found to possess no higher electromagnetic multipole moments. Studying “galilean electromagnetism” (i.e. the theory of spin 1 massless particles), we show that only the displacement current is responsible for the breakdown of galilean invariance in Maxwell equations, and we make some comments about such a “nonrelativistic electromagnetism”. Comparing the connection between wave equations and the invariance group in both the relativistic and the nonrelativistic case, we are finally led to some vexing questions about the very concept of wave equations.
449 citations
CERN1
TL;DR: In this paper, the equilibrium states for an infinite system of classical mechanics may be represented by states over AbelianC* algebras. And the properties of this mean entropy are investigated: linearity, upper semi-continuity, integral representations.
Abstract: The equilibrium states for an infinite system of classical mechanics may be represented by states over AbelianC* algebras. We consider here continuous and lattice systems and define a mean entropy for their states. The properties of this mean entropy are investigated: linearity, upper semi-continuity, integral representations. In the lattice case, it is found that our mean entropy coincides with theKolmogorov-Sinai invariant of ergodic theory.
177 citations
TL;DR: In this article, the authors studied the thermodynamic limit for a classical system of particles on a lattice and proved the existence of infinite volume correlation functions for a large set of potentials and temperatures.
Abstract: We study the thermodynamic limit for a classical system of particles on a lattice and prove the existence of infinite volume correlation functions for a “large” set of potentials and temperatures.
153 citations
TL;DR: In this paper, the relation between conservation laws and transformations leaving invariant the action integral is studied, where the authors present a rigorous derivation of the relation and relation between transformation and conservation laws.
Abstract: The purpose of the paper is to present a rigorous derivation of the relation between conservation laws and transformations leaving invariant the action integral. The (space-)time development of a physical system is represented by a cross section of a product bundleM. A Lagrange function is defined as a mapping\(L:\overline M \to R\) where\(\overline M \) is the bundle space of the first jet extension ofM. A symmetry transformation is defined as a bundle automorphism ofM, carrying solutions of the Euler-Lagrange equation into solutions of the same equation. An important class of symmetry transformations is that of generalized invariant transformations: they are defined by specifying their action on the Euler-Lagrange equation. The generators of generalized invariant transformations are solutions of a system of linear, homogeneous partial differential equation (Noether equations). The set of all solutions of these equations has a natural structure of Lie algebra. In a simple manner, the Noether equations give rise to differential conservation laws.
152 citations
TL;DR: In this article, the solutions of the Einstein equations which can be embedded in a flat five dimensional space are studied, without referring to the explicit form of the embedding, and invariantly characterized; in some special cases the metric is fully determined.
Abstract: Solutions of the Einstein equations which can be embedded in a flat five dimensional space are studied, without referring to the explicit form of the embedding. Those describing the gravitational field of a perfect fluid or an electromagnetic null field are invariantly characterized; in some special cases the metric is fully determined.
144 citations
TL;DR: In this paper, it was shown that the connected component of the identity i in this group contains the open ball of radius 2 with centerl and that each automorphism in ℬ extends to an inner automomorphism of ϕ(\(\mathfrak{A}))−.
Abstract: The theorem that each derivation of aC*-algebra\(\mathfrak{A}\) extends to an inner derivation of the weak-operator closure ϕ(\(\mathfrak{A}\))− of\(\mathfrak{A}\) in each faithful representation ϕ of\(\mathfrak{A}\) is proved in sketch and used to study the automorphism group of\(\mathfrak{A}\) in its norm topology. It is proved that the connected component of the identity i in this group contains the open ball ℬ of radius 2 with centerl and that each automorphism in ℬ extends to an inner automorphism of ϕ(\(\mathfrak{A}\))−.
136 citations
TL;DR: In this article, a scheme of the construction of the S-matrix according to the perturbation theory which is free of the ultraviolet divergences is suggested by the example of the one-component scalar quantized field.
Abstract: A scheme of the construction of theS-matrix according to the perturbation theory which is free of the ultraviolet divergences is suggested by the example of the one-component scalar quantized field. The causality is violated in small space-time region. The effects which are due to the causality violation at large distances are described by very high perturbation orders, and are therefore very small in the framework of the perturbation theory.
133 citations
TL;DR: In this article, the degenerate B.C.S. model was used to show how in the limit of an infinite system the exact thermal Greens-functions approach a gauge invariant average of the one's calculated with the Bogoliubov-Haag method.
Abstract: It is shown for the degenerate B.C.S.-model how in the limit of an infinite system the exact thermal Greens-functions approach a gauge invariant average of the one's calculated with the Bogoliubov-Haag method.
128 citations
TL;DR: In this paper, the consequences of axioms as physical as possible about effects and ensembles are investigated, starting from axiomatic notions of "effects" and "ensembles".
Abstract: Starting from axioms as physical as possible [1, 2, 3] about “effects” and “ensembles”, we shall investigate further consequences.
128 citations
TL;DR: Asymptotic relations for matrix elements of quasilocal operators are given in this article, which allow the simulation of a coincidence arrangement of particle detectors in the mathematical frame of the theory and thereby the expression of collision cross sections in terms of expectation values of observables.
Abstract: Asymptotic relations for matrix elements of quasilocal operators are given which generalize and extend the Lehmann-Symanzik-Zimmermann relations. These relations allow the simulation of a coincidence arrangement of particle detectors in the mathematical frame of the theory and thereby the expression of collision cross sections in terms of expectation values of observables.
TL;DR: In this article, a reformulation of the axiomatic basis of quantum mechanics with particular reference to the manner in which the usual algebraic structures arise from certain natural physical requirements is presented.
Abstract: We present a reformulation of the axiomatic basis of quantum mechanics with particular reference to the manner in which the usual algebraic structures arise from certain natural physical requirements. Care is taken to distinguish between features of physical significance and those introduced for mathematical convenience. Our conclusion is that the usual algebraic structures cannot be significantly generalised without conflicting with our current experimental picture of processes occurring at the quantum level.
TL;DR: In this paper, a general method for constructing fields in spaces with transitive group of transformations is presented, and the connection of spin and statistics in de-Sitter space of constant positive curvature is discussed.
Abstract: A general method for constructing fields in spaces with transitive group of transformations is presented. Quantum-theory of free fields with spin 0, 1/2, and the connection of spin and statistics in de-Sitter space of constant positive curvature are discussed.
TL;DR: In this paper, all the unitary continuous irreducible representations of the 4-dimensional Lie group generated by the canonical variables and a positive definite quadratic "hamiltonian" are found.
Abstract: Using the Mackey theory of induced representations all the unitary continuous irreducible representations of the 4-dimensional Lie groupG generated by the canonical variables and a positive definite quadratic ‘hamiltonian’ are found. These are shown to be in a one to one correspondence with the orbits underG in the dual spaceG′ to the Lie algebraG ofG, and the representations are obtained from the orbits by inducing from one-dimensional representations provided complex subalgebras are admitted. Thus a construction analogous to that ofKirillov andBernat gives all the representations of this group.
TL;DR: In this paper, it was shown that for an infinite lattice system, thermodynamic equilibrium is the solution of a variational problem involving a mean entropy of states introduced earlier, and a version of the Gibbs phase rule was proved.
Abstract: It is shown that for an infinite lattice system, thermodynamic equilibrium is the solution of a variational problem involving a mean entropy of states introduced earlier [2]. As an application, a version of the Gibbs phase rule is proved.
TL;DR: The two sided ideals of the C*-algebra generated by local v. Neumann algebras are investigated in this paper, where the two-sided ideals of C* are shown to be equivalent.
Abstract: The two sided ideals of theC*-algebra generated by local v. Neumann algebras are investigated.
TL;DR: In this article, an inequality relating binary correlation functions for an Ising model with purely ferromagnetic interactions is derived by elementary arguments and used to show that such a ferromagnet cannot exhibit a spontaneous magnetization at temperatures above the mean-field approximation to the Curie or critical point.
Abstract: An inequality relating binary correlation functions for an Ising model with purely ferromagnetic interactions is derived by elementary arguments and used to show that such a ferromagnet cannot exhibit a spontaneous magnetization at temperatures above the mean-field approximation to the Curie or “critical” point. (As a consequence, the corresponding “lattice gas” cannot undergo a first order phase transition in density (condensation) above this temperature.) The mean-field susceptibility in zero magnetic field at high temperatures is shown to be an upper bound for the exact result.
TL;DR: In this article, a finite form of the local field equation is checked in renormalized perturbation theory for the model of A4-coupling, and the model is shown to be stable.
Abstract: For the model ofA
4-coupling a finite form of the local field equation is proposed and checked in renormalized perturbation theory.
TL;DR: In this article, the authors prove the existence and uniqueness of a center-of-mass line as well as a center of motion line, the latter due to G. Dixon, 1964.
Abstract: We prove the existence and uniqueness of a center-of-mass line as well as a center-of-motion line, the latter due toG. Dixon, 1964. The validity of the theorems depends on some assumptions listed in § 2, whose most restrictive ones (in the sense of physics) state a certain weakness of the gravitational field. In the concluding paragraph we give some corrolaries and a very simple application to the problem of motion.
TL;DR: Galilean quantum field theories, i.e. kinematically consistent non-relativistic quantum theories with an infinite number of degrees of freedom, are considered in this paper.
Abstract: Galilean quantum field theories, i.e. kinematically consistent non-relativistic quantum theories with an infinite number of degrees of freedom, are considered. These theories transcend the frame of ordinary quantum mechanics by allowing genuine particle production processes to be described. The general structure of such theories is discussed and contrasted with the typical structure of relativistic quantum field theories which they may serve to illustrate a contrario. Despite the mass superselection rule, and due to the weakening of local commutativity conditions, galilean quantum field theories are much less constrained than relativistic ones. The CPT and spin-and-statistics theorems do not hold here, neither does Haag's theorem.
TL;DR: A group theoretical derivation of Bargmann's representation of the boson commutation rules in an Hilbert space of analytic functions is given in this paper, where the authors recover Laguerre-polynomials as spherical functions on the nilpotent Weyl group.
Abstract: A group theoretical derivation is given of Bargmann's representation of the boson commutation rules in an Hilbert space of analytic functions. Several interesting problems arise in the study of the global representation of the canonical groupSp (2n, R). As a by-product we recover Laguerre-polynomials as spherical functions on the nilpotent Weyl group.
TL;DR: In this paper, a renormalization procedure is proposed to give rigorous mathematical meaning to the infinite cancellations in this model and the renormalized Hamiltonian is rigorously defined as a bilinear form in the Fock Hilbert space.
Abstract: A renormalization procedure is proposed. It gives rigorous mathematical meaning to the infinite cancellations in this model. A space cutoff is introduced in the interaction termV and soV has the form\(\int\limits_{\left| x \right| \leqq K} {V\left( x \right)} dx\), but there are no momentum cutoffs inV. There is an infinite constant and an infinite boson mass renormalization in this model. The main result is that the renormalized Hamiltonian is rigorously defined as a bilinear form in the Fock Hilbert space.
CERN1
TL;DR: In this article, a conjecture of R. Streater on the finite covariance of functions holomorphic in the extended tube which are Laplace transforms of two tempered distributions with supports in the future and past cones was proved.
Abstract: We prove a conjecture ofR. Streater [1] on the finite covariance of functions holomorphic in the extended tube which are Laplace transforms of two tempered distributions with supports in the future and past cones. A new, slightly more general proof is given for a theorem of analytic completion of [1].
TL;DR: In this article, it was shown that the relativistic Boltzmann equation has a local solution through an initial distribution function, if the scattering cross section is bounded for high energies and if the initial distribution falls off exponentially with the energy.
Abstract: It is shown that the relativistic Boltzmann equation has a local solution through an initial distribution function, if the scattering cross section is bounded for high energies and if the initial distribution falls off exponentially with the energy.
TL;DR: In a relativistic field theory, Goldstone's theorem is proved without any assumption about the existence of covariant fields and for arbitrary expectation values as mentioned in this paper, for any expectation value.
Abstract: In a relativistic field theory Goldstone's theorem is proved without any assumption about the existence of covariant fields and for arbitrary expectation values.
TL;DR: Using properties of an embedding in a flat sixdimensional space, all solutions of the Einstein equations for a perfect fluid or an electromagnetic field are given, which are conformal to a flat space as mentioned in this paper.
Abstract: Using properties of an embedding in a flat sixdimensional space, all solutions of the Einstein equations for a perfect fluid or an electromagnetic field are given, which are conformal to a flat space.
TL;DR: In this article, it is argued that for a many-body system with short range forces the commutators between local operators at different times will be fast decreasing for large spatial separations.
Abstract: It is argued that for a many-body system with short range forces the commutators between local operators at different times will be fast decreasing for large spatial separations.
TL;DR: In this article, the authors consider a lattice of spin 1/2 ions, described by the discrete form of the current commutation relations, and prove that the Hamiltonian is non-negative in a representation of the lattice, and has a ground state Ω exhibiting ferromagnetism.
Abstract: We consider a lattice of spin 1/2 ions, described by the discrete form of the current commutation relationsJiαJ(i)α=1/2, [Jiα,Jiβ]=iδij ɛαβγJiγ where α=1, 2, 3 andi label the lattice sites. The algebra is realized as the Clifford algebra\(\mathfrak{A}\) over a Hilbert space. The equations of motion are specified by a formal Hamiltonian of the Heisenberg form:\(H = \mathop \Sigma \limits_{i,j} f_{i,j} \underline J _i \cdot \underline J _j \), wherefij≦0 and only a finite numberQ of ions are linked to any given lattice site. We prove that the Hamiltonian is non-negative in a representation of\(\mathfrak{A}\), and has a ground state Ω exhibiting ferromagnetism. The time displacement group acts continuously on\(\mathfrak{A}\), inducing automorphisms.\(\mathfrak{A}\) is asymptotically abelian with respect to the space translations of the lattice.
TL;DR: In this article, the authors studied a class of systems which are asymptotically abelian with respect to their invariant states (i.e., they are invariant under a vector cyclic).
Abstract: We study pairs {\(\mathfrak{A}\), α} for which\(\mathfrak{A}\) is aC*-algebra and α is a homomorphism of a locally compact, non-compact groupG into the group of *-automorphisms of\(\mathfrak{A}\). We examine, especially, those systems {\(\mathfrak{A}\), α} which are (weakly) asymptotically abelian with respect to their invariant states (i.e. 〈Φ |A αg(B) — αg(B)A〉 → 0 asg → ∞ for those states Φ such that Φ(αg(A)) = Φ(A) for allg inG andA in\(\mathfrak{A}\)). For concrete systems (those with\(\mathfrak{A}\)-acting on a Hilbert space andg → αg implemented by a unitary representationg →Ug on this space) we prove, among other results, that the operators commuting with\(\mathfrak{A}\) and {Ug} form a commuting family when there is a vector cyclic under\(\mathfrak{A}\) and invariant under {Ug}. We characterize the extremal invariant states, in this case, in terms of “weak clustering” properties and also in terms of “factor” and “irreducibility” properties of {\(\mathfrak{A}\),Ug}. Specializing to amenable groups, we describe “operator means” arising from invariant group means; and we study systems which are “asymptotically abelian in mean”. Our interest in these structures resides in their appearance in the “infinite system” approach to quantum statistical mechanics.
TL;DR: In this article, a theorem is derived giving sufficient conditions for a factor to be either finite or purely infinite in the Hilbert space, and the conditions are:==================i.
Abstract: A theorem is derived giving sufficient conditions for a factor to be either finite or purely infinite. These conditions are:
i.
In the Hilbert space\(\mathfrak{H}\) exists a conjugation operatorJ transforming the factor ℜ into its commutant ℜ′.
ii.
There exists a one parameter abelian group of automorphisms of ℜ implemented by unitary operatorsUt weakly continuous int and commuting withJ.
iii.
There is a cyclic and separating vector Ω, which is invariant forJ and which is the only vector in\(\mathfrak{H}\) invariant forUt.