Showing papers in "Communications in Mathematical Physics in 1972"

The Finite Group Velocity of Quantum Spin Systems

Abstract: It is shown that if Ф is a finite range interaction of a quantum spin system, τ t Ф the associated group of time translations, τ x the group of space translations, and A, B local observables, then $$ \mathop {{\text{lim}}}\limits_{\mathop {\left| t \right| \to \infty }\limits_{\left| x \right| > v\left| t \right|} } \left\| {\left[ {\tau _t^\varphi \tau \left( A \right),B} \right]} \right\|e^{\mu \left( v \right)t} = 0$$ (1) whenever v is sufficiently large (ν > V Ф ,) where μ(ν) > 0. The physical content of the statement is that information can propagate in the system only with a finite group velocity.

Black holes in general relativity

Abstract: It is assumed that the singularities which occur in gravitational collapse are not visible from outside but are hidden behind an event horizon. This means that one can still predict the future outside the event horizon. A black hole on a spacelike surface is defined to be a connected component of the region of the surface bounded by the event horizon. As time increase, black holes may merge together but can never bifurcate. A black hole would be expected to settle down to a stationary state. It is shown that a stationary black hole must have topologically spherical boundary and must be axisymmetric if it is rotating. These results together with those of Israel and Carter go most of the way towards establishing the conjecture that any stationary black hole is a Kerr solution. Using this conjecture and the result that the surface area of black holes can never decrease, one can place certain limits on the amount of energy that can be extracted from black holes.

Coherent states for arbitrary Lie group

Abstract: The concept of coherent states originally closely related to the nilpotent group of Weyl is generalized to arbitrary Lie group. For the simplest Lie groups the system of coherent states is constructed and its features are investigated.

Theory of monomer-dimer systems

Abstract: We investigate the general monomer-dimer partition function, P(x), which is a polynomial in the monomer activity, x, with coefficients depending on the dimer activities. Our main result is that P(x) has its zeros on the imaginary axis when the dimer activities are nonnegative. Therefore, no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomer density is minimal (i.e. x = 0). Elaborating on this theme we prove the existence and analyticity of correlation functions (away from x = 0) in the thermodynamic limit. Among other things we obtain bounds on the compressibility and derive a new variable in which to make an expansion of the free energy that converges down to the minimal monomer density. We also relate the monomer-dimer problem to the Heisenberg and Ising models of a magnet and derive Christoffell-Darboux formulas for the monomer-dimer and Ising model partition functions. This casts the Ising model in a new light and provides an alternative proof of the Lee-Yang circle theorem. We also derive joint complex analyticity domains in the monomer and dimer activities. Our considerations are independent of geometry and hence are valid for any dimensionality.

Quadratic form techniques and the Balslev-Combes theorem

Abstract: We extend the theorem of Balslev and Combes on the absence of singular continuous spectrum to a class of interactions includingr −α(3/2≦α<2) local potentials. In addition, we note that the theory of sectorial operators allows a simplification of their proof and allows one to push the cuts through angles larger than the π/2 restriction employed by Balslev-Combes.

Black holes in the brans--dicke theory of gravitation.

Abstract: It is shown that a stationary space containing a black hole is a solution of the Brans-Dicke field equations if and only if it is a solution of the Einstein field equations. This implies that when the star collapses to form a black hole, it loses that fraction (about 7%) of its measured gravitational mass that arises from the scalar interaction. This mass loss is in addition to that caused by emission of scalar or tensor gravitational radiation. Another consequence is that there will not be any scalar gravitational radiation emitted when two black holes collide.

Solutions of the Einstein-Maxwell equations with many black holes

Abstract: In Newtonian gravitational theory a system of point charged particles can be arranged in static equilibrium under their mutual gravitational and electrostatic forces provided that for each particle the charge,e, is related to the mass,m, bye=G1/2m. Corresponding static solutions of the coupled source free Einstein-Maxwell equations have been given by Majumdar and Papapetrou. We show that these solutions can be analytically extended and interpreted as a system of charged black holes in equilibrium under their gravitational and electrical forces.

The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system. I

Abstract: A systematic presentation of the quasi-linear first order symmetric hyperbolic systems of Friedrichs is presented A number of sharp regularity and smoothness properties of the solutions are obtained The present paper is devoted to the case ofRn with suitable asymptotic conditions imposed As an example, we apply this theory to give new proofs of the existence and uniqueness theorems for the Einstein equations in general relativity, due to Choquet-Bruhat and Lichnerowicz These new proofs usingfirst order techniques are considerably simplier than the classical proofs based onsecond order techniques Our existence results are as sharp as had been previously known, and our uniqueness results improve by one degree of differentiability those previously existing in the literature

Operator product expansions and composite field operators in the general framework of quantum field theory

Abstract: The short distance behavior of field operator products is analyzed. It is shown that under certain conditions operator product expansions can be derived which give complete information on the short distance behavior and lead to the construction of composite field operators.

On factor representations and the c*-algebra of canonical commutation relations

Abstract: A newC*-algebra,A, for canonical commutation relations, both in the case of finite and infinite number of degrees of freedom, is defined. It has the property that to each, not necessarily continuous, representation of CCR there corresponds a representation ofA. The definition ofA is based on the existence and uniqueness of the factor type II1 representation. Some continuity properties of separable factor representations are proved.

On the uniqueness of the equilibrium state for Ising spin systems

Abstract: We show that for an Ising spin system of arbitrary spin with a ferromagnetic pair interaction and a “periodic” external magnetic field there is a unique equilibrium state if and only if the magnetization is continuous with respect to a uniform change in the external field. Hence, if the critical temperatureTc is defined as the temperature where the spontaneous magnetization (which is a non-increasing function of the temperature) becomes positive, then the equilibrium state is unique forT>Tc and is non-unique forT Tc.

The phase separation line in the two-dimensional Ising model

Abstract: We prove that, at low temperature, the line of separation between the two pure phases shows large fluctuations in shape. This implies the translation invariance of the correlation functions associated with some non translation invariant boundary conditions and should be a peculiarity of the dimensionality of the model.

Energy and angular momentum flow into a black hole

Abstract: The area of the event horizon round a rotating black hole will increase in the presence of a non-axisymmetric or time dependent perturbation. If the perturbation is a matter field, the area increase is related to the fluxes of energy and of angular momentum into the black hole in such a way as to maintain the formula for the area in the Kerr solution. For purely gravitational perturbations one cannot define angular momentum locally but one can use the area increase and the expression for area in terms of mass and angular momentum to calculate the slowing down of a black hole caused by a non-axisymmetric distribution of matter at a distance. It seems that the coupling between the rotation of a black hole and the orbit of a particle going round it can be significant if the angular momentum of the black hole is close to its maximum possible value and if the angular velocity of the particle is nearly equal to that of the black hole.

Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems

Abstract: We consider a ferromagnetic Ising spin system isomorphic to a lattice gas with attractive interactions. Using the Fortuin, Kasteleyn and Ginibre (FKG) inequalities we derive bounds on the decay of correlations between two widely separated sets of particles in terms of the decay of the pair correlation. This leads to bounds on the derivatives of various orders of the free energy with respect to the magnetic fieldh, and reciprocal temperature β. In particular, if the pair correlation has an upper bound (uniform in the size of the system) which decays exponentially with distance in some neighborhood of (β′,h′) then the thermodynamic free energy density ψ(β,h) andall the correlation functions are infinitely differentiable at (β′,h′). We then show that when only pair interactions are present it is sufficient to obtain such a bound only ath=0 (and only in the infinite volume limit) for systems with suitable boundary conditions. This is the case in the two dimensional square lattice with nearest neighbor interactions for 0≦β<β0, where β 0 −1 is the Onsager temperature at which ψ(β,h=0) has a singularity. For β>β0, ∂ψ(β,h)/∂h is discontinuous ath=0, i.e. β0=βc, where β −1 is the temperature below which there is spontaneous magnetization.

The inverse $s$-wave scattering problem for a class of potentials depending on energy

Abstract: The inverse scattering problem is considered for the radials-wave Schrodinger equation with the energy-dependent potentialV+(E,x)=U(x)+2\(\sqrt E \)Q(x). (Note that this problem is closely related to the inverse problem for the radials-wave Klein-Gordon equation of zero mass with a static potential.) Some authors have already studied it by extending the method given by Gel'fand and Levitan in the caseQ=0. Here, a more direct approach generalizing the Marchenko method is used. First, the Jost solutionf+(E,x) is shown to be generated by two functionsF+(x) andA+(x,t). After introducing the potentialV−(E,x)=U(x)−2\(\sqrt E \)Q(x) and the corresponding functionsF−(x) andA−(x,t), fundamental integral equations are derived connectingF+(x),F−(x),A+(x,t) andA−(x,t) with two functionsz+(x) andz−(x);z+(x) andz−(x) are themselves easily connected with the binding energiesEn+ and the scattering “matrix”S+(E),E>0 (the input data of the inverse problem). The inverse problem is then reduced to the solution of these fundamental integral equations. Some specific examples are given. Derivation of more elaborate results in the case of real potentials, and applications of this work to other inverse problems in physics will be the object of further studies.

On a quadratic first integral for the charged particle orbits in the charged Kerr solution

Abstract: Associated with the charged Kerr solution of the Einstein gravitational field equation there is a Killing tensor of valence two. The Killing tensor, which is related to the angular momentum of the field source, is shown to yield a quadratic first integral of the equation of the motion for charged test particles.

Feynman's path integral. definition without limiting procedure

Abstract: Feynman's integral is defined with respect to a pseudomeasure on the space of paths: for instance, letC be the space of pathsq:T⊂ℝ → configuration space of the system, letC be the topological dual ofC; then Feynman's integral for a particle of massm in a potentialV can be written where $$S_{\operatorname{int} } (q) = \mathop \smallint \limits_T V(q(t)) dt$$ and wheredw is a pseudomeasure whose Fourier transform is defined by for μ∈C′. Pseudomeasures are discussed; several integrals with respect to pseudomeasures are computed.

The time symmetric initial value problem for black holes

Abstract: The time symmetric initial value problem for black holes is discussed. It is shown that if a solution contains marginally trapped surfaces these correspond to minimal surfaces lying inside the black holes. Such minimal surfaces must have spherical topology. These minimal surfaces are used to obtain lower bounds for the areas of event horizons and upper bounds for the efficiency for radiating gravitational radiation. It is shown that moving black holes closer together reduces the energy available and that a single initially distorted black hole (perhaps formed just after a very assymetric collapse) cannot radiate more than 65% of its rest mass away. “Wormholes” are also briefly discussed.

Contractions of representations of de Sitter groups

Abstract: In order to construct the quantum field theory in a curved space with no “old” infinities as the curvature tends to zero, the problem of contraction of representations of the corresponding group of motions is studied. The definitions of contraction of a local group and of its representations are given in a coordinate-free manner. The contraction of the principal continuous series of the de Sitter groupsSO0(n, 1) to positive mass representations of both the Euclidean and Poincare groups is carried out in detail. It is shown that all positive mass continuous unitary irreducible representations of the resulting groups can be obtained by this method. For the Poincare groups the contraction procedure yields reducible representations which decompose into two non-equivalent irreducible representations.

Inequalities for traces on von Neumann algebras

Abstract: A number of useful inequalities, which are known for the trace on a separable Hilbert space, are extended to traces on von Neumann algebras. In particular, we prove the Golden rule, Holder inequality, and some convexity statements.

Variational principles and spatially-homogeneous universes, including rotation

Abstract: The validity of imposing spatial homogeneity on the variations in the usual action principle for Einstein's equations is studied. It is proved that with this procedure the standard and ADM Lagrangians give correct Einstein equations if and only if the space belongs to Class A of Ellis and MacCallum [1], i.e., the structure constants of the simply transitive group satisfy Cfgf=0. The possibility of overcoming this difficulty in the Class B spaces is examined.

Fields, statistics, and non-Abelian gauge groups.

Abstract: We examine field theories with a compact groupG of exact internal gauge symmetries so that the superselection sectors are labelled by the inequivalent irreducible representations ofG. A particle in one of these sectors obeys a parastatistics of orderd if and only if the corresponding representation ofG isd-dimensional. The correspondence between representations of the observable algebra and representations ofG extends to a mapping of the intertwining operators for these representations preserving linearity, tensor products and conjugation. Although we assume no explicit commutation property between fields, the commutation relations of fields of the same irreducible tensor character underG at spacelike separations are largely determined by the statistics parameter of the corresponding sector. For fields of conjugate irreducible tensor character the observable part of the commutator (anticommutator) vanishes at spacelike separations if the corresponding sector has para-Bose (para-Fermi) statistics.

An entropy inequality for quantum measurements

Abstract: It is proved that for an ideal quantum measurement the average entropy of the reduced states after the measurement is not greater than the entropy of the original state.

The thermodynamic limit for an imperfect Boson gas

Abstract: We give a rigorous treatment in the infinite volume limit of a model Hamiltonian representing an imperfect Boson gas. In particular we obtain the exact expression for the mean particle density in the infinite volume limit as a function of the chemical potential, and show that the density function has a singularity at the critical density for Bose-Einstein condensation. We prove that, unlike the ideal Boson gas, the imperfect Boson gas has the same behaviour in the infinite volume limit for the grand canonical ensemble as for the canonical ensemble, and is moreover stable under small perturbations. We finally exhibit the possibility of ordinary condensation and prove that a system in an intermediate situation between two pure phases consists of a simple mixture of the two phases involved.

Currents, stress tensor and generalized unitarity in conformal invariant quantum field theory

Abstract: Matrix elements of internal symmetry currents and energy momentum density tensor are constructed in Migdal Polyakov conformal invariant bootstrap field theory. Their 3-point functions satisfy Bethe Salpeter equations which determine any free coefficients that may still occur in the conformal invariant Ansatz. Ward identities are verified for alln-point functions. They imply correct equal time current commutation relations. A proof of generalized unitarity is also given. Various equivalent forms of the propagator bootstrap are discussed. Our algebraic techniques also yield an eigenvalue equation for first order correction to the exactly conformal invariant theory, assuming the latter is Gell-Mann Low large momentum asymptote of a renormalizable finite mass theory.

Qualitative magnetic cosmology

Abstract: The technique of phase plane analysis, which was used in a previous paper [4] to study the behaviour of a class of perfect-fluid anisotropic cosmological models, is applied to some simple anisotropic models that contain a uniform magnetic field A formal correspondence is established between these magnetic models (of Bianchi type I) and certain perfect fluid models (of Bianchi type II), and new exact solutions are consequently discovered

Thermodynamic functions for fermions with gravostatic and electrostatic interactions

Abstract: The energy as function of entropy and the free energy as function of temperature is calculated rigorously for nonrelativistic fermions with interactions. It is shown that in the appropriate thermodynamic limit the corresponding Thomas-Fermi equation becomes exact.

The structure of space-time transformations

Abstract: LetT be a one-to-one mapping ofn-dimensional space-timeM onto itself. IfT maps light cones onto light cones and dimM≧3, it is shown thatT is, up to a scale factor, an inhomogeneous Lorentz transformation. Thus constancy of light velocity alone implies the Lorentz group (up to dilatations). The same holds ifT andT−1 preserve (x−y)2>0. This generalizes Zeeman's Theorem. It is then shown that ifT maps lightlike lines onto (arbitrary) straight lines and if dimM≧3, thenT is linear. The last result can be applied to transformations connecting different reference frames in a relativistic or non-relativistic theory.

Nelson's symmetry and the infinite volume behavior of the vacuum in P (φ) 2

Abstract: LetHl be the Hamiltonian in aP(φ)2 theory with sharp space cutoff in the interval (−l/2,l/2). LetEl=infσ(Hl), α(l)=−El/l, and let Ωl be the vacuum forHl. discuss properties of α(l) and Ωl. In particular, asl→∞, there are finite constants β∞<0 and α∞ such that α(l)↑α∞, (α(l)−α∞)l↓β∞, and hence α(l)=α∞+β∞/l+o(l−1). Moreover exp(−c1l)≦∥Ωl∥1≦exp(−c2l) forc1,c2 positive constants, where ∥Ωl∥1 is theL1(Q, dμ0) norm of Ω1 with respect to the Fock vacuum measure. We also present a new proof of recent estimates of Glimm and Jaffe on local perturbations ofHl in the infinite volume limit.

Distributions on Minkowski space and their connection with analytic representations of the conformal group

Abstract: Unitary analytic representations of the conformal group are relized on Hilbert spaces of holomoprhic or antiholomorphic functions over a tube domain in complex Minkowski space. The distributional boundary values of these functions are tempered distributions on real Minkowski space. The representations are characterized by an integral scale dimension labeln and two spin labelsj 1 andj 2. The connection between the dimensionn and the degree of singularity of the tempered distribution is investigated. We propose an application to inclusive reactions of elementary particles.