Showing papers in "Communications in Mathematical Physics in 1971"
TL;DR: A mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed in this article, where the authors propose a mechanism for generating turbulence in a dissipative system with respect to dissipative energy.
Abstract: A mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed.
2,509 citations
TL;DR: In this article, the existence of an absolutely continuous part having spectrum [λe, ∞], where λe is the lowest many body threshold of the system, and the point spectrum is discrete.
Abstract: Quantum mechanicalN-body systems with dilatation analytic interactions are investigated. Absence of continuous singular part for the Hamiltonians is proved together with the existence of an absolutely continuous part having spectrum [λe, ∞), where λe is the lowest many body threshold of the system. In the complement of the set of thresholds the point spectrum is discrete; corresponding bound state wave-functions are analytic with respect to the dilatation group.
1,206 citations
TL;DR: In this article, it was shown that increasing functions on a finite distributive lattice are positively correlated by positive measures satisfying a suitable convexity property, and applications to Ising ferromagnets in an arbitrary magnetic field and to the random cluster model were given.
Abstract: We prove that increasing function on a finite distributive lattice are positively correlated by positive measures satisfying a suitable convexity property. Applications to Ising ferromagnets in an arbitrary magnetic field and to the random cluster model are given.
1,131 citations
TL;DR: In this paper, a class of symmetric relatively compact perturbations satisfying analyticity conditions with respect to the dilatation group was studied and the existence of an absolutely continuous part having spectrum [0, ∞] was proved.
Abstract: We study a class of symmetric relatively compact perturbations satisfying analyticity conditions with respect to the dilatation group inRn. Absence of continuous singular part for the Hamiltonians is proved together with the existence of an absolutely continuous part having spectrum [0, ∞). The point spectrum consists in R−{0} of finite multiplicity isolated energy bound-states standing in a bounded domain. Bound-state wave functions are analytic with respect to the dilatation group. Some properties of resonance poles are investigated.
1,049 citations
TL;DR: In this article, the identity and connection with the Hilbert spaces of entire functions of growth (1, 1) are studied. And the application to the evaluation of matrix elements of finite group elements and the contraction to the usual coherent states are discussed.
Abstract: Generalized “Coherent” States are the eigenstates of the lowering and raising operators of non-compact groups. In particular the discrete series of representations ofSO (2, 1) are studied in detail: the resolution of the identity and the connection with the Hilbert spaces of entire functions of growth (1, 1). Also discussed are the application to the evaluation of matrix elements of finite group elements and the contraction to the usual coherent states.
695 citations
TL;DR: In this paper, the structure of the set of superselection sectors (charge quantum numbers) and its implications for the particle aspects of the theory are studied. And the scattering states and transition probabilities are obtained in this frame.
Abstract: Starting from the principles of local relativistic Quantum Theory without long range forces, we study the structure of the set of superselection sectors (charge quantum numbers) and its implications for the particle aspects of the theory. Without assuming the commutation properties (or even the existence) of unobservable fields connecting different sectors (charge-carrying fields), one has a particle-antiparticle symmetry, an intrinsic notion of statistics for identical particles, and a spin-statistics theorem. Particles in “pseudoreal sectors” cannot be their own antiparticles (a variant of Carruthers' theorem). We also show how scattering states and transition probabilities are obtained in this frame.
634 citations
TL;DR: In this paper, it was shown that a strongly cyclic self-adjoint representation of a commutative *-algebra is standard if and only if the representation is strongly positive, i.e., the representations preserve a certain order relation.
Abstract: Unbounded *-representations of *-algebras are studied. Representations called self-adjoint representations are defined in analogy to the definition of a self-adjoint operator. It is shown that for self-adjoint representations certain pathologies associated with commutant and reducing subspaces are avoided. A class of well behaved self-adjoint representations, called standard representations, are defined for commutative *-algebras. It is shown that a strongly cyclic self-adjoint representation of a commutative *-algebra is standard if and only if the representation is strongly positive, i.e., the representations preserves a certain order relation. Similar results are obtained for *-representations of the canonical commutation relations for a finite number of degrees of freedom.
251 citations
TL;DR: In this article, a method to obtain the asymptotic forms of vertex functions at large momenta is described, with the help of Wilson operator product expansion formulas, extended to momenta where the vertex functions of the zero-mass theory underlying the vertices are infrared singular.
Abstract: A previously described method to obtain the asymptotic forms of vertex functions at large momenta is, with the help of Wilson operator product expansion formulas, extended to momenta where the vertex functions of the zero-mass theory underlying the asymptotic forms are infrared singular. To obtain from asymptotic forms information on asymptotic behaviour requires assumptions on the behaviour of the zero-mass theory in the limit of infinite dilatation. One particular set of assumptions is discussed and found to pass a simple consistency test; this set of assumptions leads to power laws, or slight modifications thereof, with coupling-constant-independent exponents. The detailed discussion is given for the ф4 model.
248 citations
TL;DR: In this paper, it was shown that the metric coefficients of such space-times depend essentially on the single variablez=r/t wherer is a radial coordinate andt is the time.
Abstract: Spherically symmetric space-times which admit a one parameter group of conformal transformations generated by a vectorξ
μ
such thatξ
μ;v
+ξ
v;μ
=2g
μv
are studied. It is shown that the metric coefficients of such space-times depend essentially on the single variablez=r/t wherer is a radial coordinate andt is the time. The Einstein field equations then reduce to ordinary differential equations. The solutions of these equations are analogous to the similarity solutions of the classical theory of hydrodynamics. In case the source of the field is a perfect fluid whose specific internal energy is a function of temperature alone, the solution of the field equations is uniquely determined by specifying data on the time-like hypersurfacez=constant and is a similarity solution. The problem of fitting a similarity solution to another solution of the field equations across a shock described by the hypersurfacez=constant is treated. A particular similarity solution for whichw=3p obtains is shown to describe a Robertson-Walker space-time. This solution is fitted to a special static solution of the Einstein field equations which has a singularity atr=0. The resulting solution of the Einstein field equations is shown to be regular everywhere except atr=0≧t and the shock. The special Robertson-Walker metric is also fitted to a particular class of collapsing dust solutions (which are also similarity solutions) across a shock. The resulting solution is regular everywhere except atr=t=0 and on the shock.
237 citations
TL;DR: In this article, a general framework for studying differential operations in renormalized perturbation theory is derived, which makes possible a simple unified derivation of the renormalization group and Callan-Symanzik equations, as well as a direct test for broken symmetries (including broken scale invariance), without the necessity of defining currents and deriving their generalized Ward identites.
Abstract: A general framework is derived for studying differential operations in renormalized perturbation theory. The method makes possible a simple, unified derivation of the renormalization group and Callan-Symanzik equations, as well as a direct test for broken symmetries (including broken scale invariance), without the necessity of defining currents and deriving their generalized Ward identites. A second-order differential equation of the Callan-Symanzik type is derived using similar methods.
234 citations
TL;DR: In this article, an infinite one-dimensional Ising ferromagnet with long-range interactions was constructed and proved to have the following properties: (1) M has an order-disorder phase transition at a finite temperature.
Abstract: An infinite one-dimensional Ising ferromagnetM with long-range interactions is constructed and proved to have the following properties. (1)M has an order-disorder phase transition at a finite temperature. (2) Any Ising ferromagnet of the same structure asM, but with interactions tending to zero with distance more rapidly than those ofM, cannot have a phase-transition. (3) The long-range-order parameter (thermal average of the spin-spin correlation at infinite distance) jumps discontinuously from zero in the disordered phase to a finite value in the ordered phase. All three properties have been conjectured by Anderson and Thouless to hold for a particular Ising ferromagnet which is relevant to the theory of the Kondo effect. AlthoughM is not identical to Anderson's model, the results proved forM support the validity of the physical arguments of Anderson and Thouless.
TL;DR: In this article, the existence and analyticity properties of the thermodynamic limit for the correlation functions are derived; in particular, the Mayer Series and the virial expansion are discussed, and the results obtained for polymer systems can be used to study classical lattice systems.
Abstract: We prove the existence of the thermodynamic limit for the pressure and show that the limit is a convex, continuous function of the chemical potential. The existence and analyticity properties of the thermodynamic limit for the correlation functions is then derived; we discuss in particular the Mayer Series and the virial expansion. In the special case of Monomer-Dimer systems it is established that no phase transition is possible; moreover it is shown that the Mayer Series for the density is a series of Stieltjes, which yields upper and lower bounds in terms of Pade approximants. Finally it is shown that the results obtained for polymer systems can be used to study classical lattice systems.
TL;DR: In this paper, the behaviour near to and far from an initial singularity in a broad subclass of the models studied in previous papers is examined and the influence of the matter on the evolution at these times is discussed.
Abstract: The behaviour near to and far from an initial singularity in a broad subclass of the models studied in previous papers [1–3] is examined. The influence of the matter on the evolution at these times is discussed. The singularity types for the various models, which are mostly of cigar or oscillatory nature, are found. It is discovered that among these models, only those of the same Bianchi type as a Robertson-Walker model can become “approximately Robertson-Walker” in a sense defined in the paper. Qualitative conclusions concerning black-body isotropy, the Hubble relation, helium abundance and horizon structure are given.
TL;DR: In this paper, the qualitative behavior of a set of non-rotating perfect-fluid cosmologies whose spatially homogeneous hypersurfaces admit a 3-parameter group of isometries of Bianchi types I, II, III, V, or VI is investigated.
Abstract: Standard geometric techniques of differential equation theory are employed to determine the qualitative behaviour of a set of non-rotating perfect-fluid cosmologies, whose spatially homogeneous hypersurfaces admit a 3-parameter group of isometries of Bianchi types I, II, III, V, or VI. In this way we are led to some new exact solutions of the field equations.
TL;DR: In this article, it was shown that in all solutions of the Einstein-Boltzmann equations in which the particle distribution function is isotropic about some 4-velocity field, the distortion of that velocity field vanishes; further, either its expansion or its rotation vanishes.
Abstract: It is shown that in all solutions of the Einstein-Boltzmann equations in which the particle distribution function is isotropic about some 4-velocity field, the distortion of that velocity field vanishes; further, either its expansion or its rotation vanishes. We discuss briefly further kinetic solutions in which the energy-momentum tensor has a perfect fluid form.
Journal Article•
TL;DR: In this paper, the authors considered a self-interacting boson field in two space-time dimensions, with interaction densities of the form:V(ϕ(x)): where ϕ (x) is a scalar Boson field, and V(α) is real positive function of exponential type.
Abstract: We consider a selfinteracting boson field in two space-time dimensions, with interaction densities of the form:V(ϕ(x)): where ϕ(x) is a scalar boson field, andV(α) is a real positive function of exponential type. We define the space cut-off interaction by
$$V_r = \int\limits_{\left| x \right| \leqq r} {:V(\varphi (x))} :dx$$
and prove thatH
r
=H
0+V
r
, whereH
0 is the free energy, is essentially self adjoint. This permits us to take away the space cut-off and we obtain a quantum field free of cut-offs.
TL;DR: Dyson's method is adopted here for the so called Gaussian ensembles and confirms the long cherished belief that the statistical properties of a small number of eigenvalues is the same for the two kinds of ensemble, the circular and the Gaussian ones.
Abstract: Dyson's method is adopted here for the so called Gaussian ensembles. Incidently this confirms the long cherished belief that the statistical properties of a small number of eigenvalues is the same for the two kinds of ensembles, the circular and the Gaussian ones.
CERN1
TL;DR: In this paper, the free energy of a system of nonrelativistic fermions which interact with attractiver−1-potentials is calculated in a suitable thermodynamic limit.
Abstract: We calculate rigorously, in a suitable thermodynamic limit, the free energy of a system of nonrelativistic fermions which interact with attractiver
−1-potentials. It is shown that the effective field approximation becomes exact in this limit and results in the temperature-dependent Thomas-Fermi equations.
TL;DR: In this article, the authors discuss the fundamentals of particle scattering of a two-body system in forces which are 0 (r−2−e) at infinity along with their Lipshitz constants.
Abstract: We discuss the fundamentals of classical particle scattering of a two body system in forces which are 0 (r−2−e) at infinity along with their Lipshitz constants. We prove asymptotic completeness for this two-body case. Of particular interest is the fact that in the absence of control on Lipshitz constants at ∞, two solutions of the interacting equation may be asymptotic to the same free solution at −∞.
TL;DR: In this paper, the authors use commutators to find classes of operators which are smooth with respect to the Hamiltonian for a system of quantum mechanical particles which repel each other.
Abstract: We use commutators to find classes of operators which are smooth with respect to the HamiltonianH for a system of quantum mechanical particles which repel each other. It follows thatH is absolutely continuous, the wave operators are complete in many cases when they exist and limits of momentum observables as time approaches ±∞ exist even in cases where the long range of the interaction precludes existence of the wave operators.
TL;DR: In this paper, a Markov process for a classical lattice spin system is considered, and it is shown that the state of the system converges weakly to the set of Gibbsian Distributions for the given interaction and that all shift invariant equilibrium states are Gibbsian distributions.
Abstract: A Markov process which may be thought of as a classical lattice spin system is considered. States of the system are probability measures on the configuration space, and we study the evolution of the free energy of these states with time. It is proved that for all initial states the free energy is nonincreasing and that it strictly decreases from any initial state which is shift invariant but not an equilibrium state. Finally we show that the state of the system converges weakly to the set of Gibbsian Distributions for the given interaction, and that all shift invariant equilibrium states are Gibbsian Distributions.
TL;DR: In this article, it was shown that the pressure is a strictly convex function of the translationally invariant interactions for classical and quantum lattice systems, under certain mild restrictions on the long-range part of these interactions.
Abstract: It is shown that the pressure is a strictly convex function of the translationally invariant interactions (under certain mild restrictions on the long-range part of these interactions) for classical and quantum lattice systems, by demonstrating that two distinct interactions can never lead to the same translationally invariant equilibrium state. This generalizes a previous result that the pressure is a continuous function of density at fixed temperature.
TL;DR: In this paper, a complete mathematical theory of two-body quantum mechanics for a class of potentials which is larger than the usual L2-classes and which includes potentials with singularities as bad as r−2+ǫ.
Abstract: We present a complete mathematical theory of two-body quantum mechanics for a class of potentials which is larger than the usualL2-classes and which includes potentials with singularities as bad asr−2+ɛ. The basic idea is to defineHo+V as a sum of quadratic forms rather than as an operator sum.
TL;DR: In this article, it was shown that a positive energy spectrum is not compatible with a decay law bounded by a decreasing exponential, and that the spectrum difficulty of wave functions for unstable, elementary particles was already recognized.
Abstract: We discuss the possibility of describing unstable systems, or dissipative systems in general, by vectors in a Hilbert space, evolving in time according to some non-unitary group or semigroup of translations. If the states of the unstable or dissipative system are embedded in a larger Hilbert space containing “decay products” as well, so that the time evolution of the system as a whole becomes unitary, we show that the infinitesimal generator necessarily has all energies from minus to plus infinity in its spectrum. This result supplements and extends the well-known fact that a positive energy spectrum is incompatible with a decay law bounded by a decreasing exponential. As an example of both facts, we discuss Zwanziger's irreducible, nonunitary representation of the Poincare group; and we find its minimal, unitary extension (the Sz.-Nagy construction). The answer provides a mathematically canonical approach to the Matthews-Salam theory of wave functions for unstable, elementary particles, where the spectrum difficulty was already recognized. We speculate on the possibility that the Matthews-Salam-Zwanziger representation might be a strong coupling approximation in the relativistic version of the Wigner-Weisskopf theory, but we have not shown the existence of a physically acceptable model where that is so.
TL;DR: In this article, the vacuum expectation value of the time-ordered product of two exponentials of free fields is defined as a distribution using minimal singularity as a criterion, and the implication of this definition for an exponentially self-coupled scalar field is studied in second order of a perturbation expansion.
Abstract: We define the vacuum expectation value of the time-ordered product of two exponentials of free fields as a distribution using minimal singularity as a criterion. The implication of this definition for an exponentially self-coupled scalar field is studied in second order of a perturbation expansion.
TL;DR: In this article, a necessary and sufficient condition for quasi-equivalence of quasi-free factor states over the Weyl algebra is proved, which is closely related to the work of Powers and Stormer on the Clifford algebra.
Abstract: A necessary and sufficient condition for quasi-equivalence of quasi-free factor states over the Weyl algebra is proved. The essential part of this paper is closely related to the work of Powers and Stormer on the Clifford algebra.
TL;DR: In this paper, it was shown that the set of cyclic vectors for a von Neumann algebra in a Hilbert space is a Gδ set, which is empty or dense.
Abstract: We prove that the set of cyclic vectors for a von Neumann algebra in a Hilbert spaceH is aGδ set, which is empty or dense. We obtain some corollaries, for instance: if (A1,A2 ...) is a sequence of von Neumann algebras inH, and if eachAn has a cyclic vector and a separating vector, then there exists a vector inH which is cyclic and separating for eachAn. For algebras of local observables, we improve the known results connecting the infinite type of the algebras and the existence of cyclic and separating vectors.
TL;DR: In this paper, the notion of non-Hausdorff space-time is introduced and it is shown that such a space time must, under very general conditions, display a kind of causal anomaly.
Abstract: Some general properties of completely separable, non-Hausdorff manifolds are studied and the notion of a non-Hausdorff space-time is introduced. It is shown that such a space-time must, under very general conditions, display a kind of causal anomaly.
TL;DR: In this paper, it was shown that the only linear kinematical groups which are compatible with the isotropy of space are the Lorentz and Galilei groups in (n+1)-dimensional space-time (n≧3).
Abstract: We prove a theorem which states that in an (n+1)-dimensional space-time (n≧3) the only linear kinematical groups which are compatible with the isotropy of space are the Lorentz and Galilei groups. The special casesn=1 andn=2 are also briefly discussed.