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Volume 18, Issue 3, March 1977

A new type of soliton with particle properties
View Description Hide DescriptionThis paper describes stable field configurations of two scalar fields ϑ (x,y,z,t) and φ (x,y,z,t). The field configurations follow from a simple least action principle based on an energy density which is a function of ϑ, φ, and their first derivatives. The description is Lorentz‐invariant. The structures are of a stringlike type and are characterized by several integers. It is shown, that the simplest closed strings, described by the integers N=1, M=1, P±1, are stable. The structures P=1 and P=−1 are related by mirror symmetry. Three constants enter in the basic action principle: a length l, a constant E with the dimension of energy time length, and a dimensionless parameter γ. All properties of these field configurations have discrete values, which is a direct consequence of the nonlinearity of the basic expression for the energy density. An attempt is made to identify these structures with elementary particles, the electron and the positron in the simplest case P=1 and P=−1. To this aim, the total energy of the field structures is equated to the rest energy of the particles. The constants E, l, and γ are related to the fundamental physical constants h, m, e. The model proposed represents a classical field structure with quantized properties.

Convergence of lattice approximations and infinite volume limit in the (λφ^{4}−σφ^{2}−μφ)_{3} field theory
View Description Hide DescriptionBy unified method we prove the convergence of the lattice approximation of the (λφ^{4}−σφ^{2}−ψφ)_{3} field model with periodic, Dirichlet and Neumann boundary conditions in a finite box. This then allows us to take the inifinite volume limit of the Dirichlet states by the Nelson’s monotonicity argument. The model under consideration satisfies all the Wightman axioms except possibly the uniqueness of vacuum for μ=0 and the mass gap.

Equilibrium properties of fluids in the semiclassical limit
View Description Hide DescriptionThe problem of calculating the equilibrium properties of dense fluids in the semiclassical limit when the quantum effects are small is studied. Expressions are given for the pressure, free energy, and the radial distribution function in terms of the properties and correlation functions of the classical system and s‐body ’’modified’’ Mayer functions f ^{ s } _{1,2,...,s }. It is shown that the correct radial distribution function of a fluid in the semiclassical limit is generated from the classical radial distribution function if we replace in turn each f ^{0} bond (f ^{0} _{12}=e ^{−βφ} ^{(1,2)}−1) by an effective f ^{eff} bond, wheref ^{eff}=f ^{0}+(1+f ^{0}) f ^{II} +(1+f ^{0})(1+f ^{II}) L and where L is subset of the line‐irreducible graphs each of which contain one f ^{III} bond. The effective pair bond correct to the second order in thermal wavelength λ (={2πh/^{2}β/m}^{1/2}) for a fluid of hard spheres is calculated for λ/d=0.1, and 0.2 at reduced densities ρ=0.3 and 0.6. The most striking effect of the quantum mechanics on the structure of a hard‐sphere fluid is found at and near the point of contact of the hard spheres.

Half‐space analysis basic to the time‐dependent BGK model in the kinetic theory of gases
View Description Hide DescriptionThe elementary solutions of the linearized time‐dependent BGKequation are shown to have, for the case of no discrete eigenvalues, the half‐range expansion property necessary for half‐space analysis. Also the partial indices corresponding to the basic matrix Riemann problem encountered are shown, for the general case, to be nonnegative, as required for the half‐space analysis.

The lattice of verifiable propositions of the spin‐1 system
View Description Hide DescriptionA lattice of verifiable propositions L _{ V } for a spin‐1 system is constructed by admitting only propositions which correspond to appropriate Stern–Gerlach filters. L _{ V } is a complete, orthocomplemented, weakly modular lattice, and it satisfies the first part of the atomicity axiom of Jauch and Piron, but not the second part (the covering law) nor related axioms of Zierler and MacLaren. Doubt is therefore thrown upon the program of recovering the Hilbert space formulation of quantum mechanics from empirically justified axioms. The class of admissible states on L _{ V } is exhaustively characterized, and it is shown that there exist some nonquantal states but none that are dispersion free.

Asymptotic approximations, with error estimates, of the scattering matrix for quantal Coulomb excitation by means of a nonlinear (Riccati) matrix differential equation
View Description Hide DescriptionA scattering matrix function is defined, which obeys a nonlinear (Riccati) matrix differential equation, containing two coupling potential matrices U and W, which are slowly vanishing, and which are mildly oscillatory and rapidly oscillatory, respectively. The scattering matrix is the limiting value of this scattering function. The equation is first transformed to separate the effects of U and W, thereby yielding separate equations in each. The long range effects of U and W are included in approximations for the scattering matrix, errors are assessed, and a prescription is outlined for the numerical computation of these approximations. In the case where the effect of W is entirely neglected beyond a certain point, the approximation obtained by Alder and Pauli [Nucl. Phys. 128, 193 (1969)] is recovered. An assessment of the error in this approximation is obtained.

Ordering of the exponential of a quadratic in boson operators. I. Single mode case
View Description Hide DescriptionThe Weyl ordered form of the operator exp[α ^{2}+β ^{2+}+γ ( ^{+} + ^{+})] is derived in a very simple way. Using this result, we also obtain the normal and antinormal ordered forms as well as the diagonal coherent states representation of this operator.

Ordering of the exponential of a quadratic in boson operators. II. Multimode case
View Description Hide DescriptionWe derive the Weyl, the normal, and the antinormal ordered forms of the exponential of a multimode quadratic expression in boson operators. The trace of this exponential operator is also evaluated.

Unitary analytic representations of SL(3,R)
View Description Hide DescriptionUnitary, analytic representations of SL(3,R) are studied by operator formalism. It is found that SL(3,R) has two different principal series of representations. Analytic representations are labeled by an integer n and a real number a. The Hilbert space of analytic functions f (z,x) is constructed, and an invariant scalar product is formed.

Evolution of isometries in the Bondi formalism
View Description Hide DescriptionIt is shown that if an additional symmetry, assumed part of the BMS group, is imposed in the Bondi formalism at one retarded time, and if gravitational radiation is absent, then the symmetry will evolve to fill the region of space–time where the Bondi metric is nonsingular. Furthermore, that region will admit a static Weyl metric. There is no necessary evolution if there is radiation present. The evolution of vector field which are nearly isometric is then examined: These evolve as small perturbations off a Weyl metric. A simple and nonlinear but approximate energy formula is written in terms of a quadrupole moment.

Time‐dependent dynamical symmetry mappings and associated constants of motion for classical particle systems. II
View Description Hide DescriptionThis paper is a continuation of a previous paper with a similar title [J. Math. Phys. 17, 1345 (1976)]. In this paper we develop further properties of time‐dependent symmetries of dynamical systems expressible in the form (a) E ^{ i }(χ̈,χ̇,x,t) ≡ E ^{ i }(χ̈^{1},...,χ̈^{ n }; χ̇^{1},...,χ̇^{ n }; x ^{1},...,x ^{ n };t) = 0. Such dynamical symmetries are based upon infinitesimal transformations of the form (b) χ̄^{ i }=x ^{ i } +δx ^{ i }, δx ^{ i }≡ξ^{ i }(x,t) δa, (c) =t +δt, δt≡ξ^{0}(x,t) δa, which satisfy the condition (d) δE ^{ i }=0 whenever E ^{ j }=0. It is shown that if (ξ^{ i } _{ A }, ξ^{0} _{ A }), A=1,...,ρ, is a complete set of solutions of the symmetry equations as determined by (d), then these solutions generate a ρ‐parameter complete group of symmetry mappings, and the group structure implies linear dependency relations between first and second derived time‐dependent constants of motion as obtained by a related integral theorem. The complete groups of time‐dependent symmetry mappings are obtained for all conservative systems (n≳1) with spherically symmetric potentials. These groups are classified into six types according to the associated form of the potential. A similar analysis leads to three types of Noether symmetries. In the case where (a) takes the form (e) E ^{ i }(χ̈,χ̇,x) =0, it is shown that if (ξ^{ i }, ξ^{0}) defines a symmetry mapping then in general (∂^{ K }ξ^{ i }/∂t ^{ K }, ∂^{ K }ξ^{ o }/∂t ^{ K }), K=1,2,..., will also define symmetry mappings; similar properties are shown for Noether symmetries. These results when applied to a large class of time‐dependent constant of motion defined in terms of (ξ^{ i }, ξ^{0}) lead to further contants of motion.

An extended Levinson’s theorem
View Description Hide DescriptionWe investigate the form Levinson’s theorem takes when the two‐body scattering amplitude is not decomposed into partial waves. It is found that the theorem changes its structure in this case and is not merely the sum over angular momentum of the well‐known partial wave results. The energy dependent quantity that replaces the partial wave phase shift turns out to be the trace of the two‐body time delay operator. This extended version of the theorem remains valid for scattering by nonspherically symmetric potentials.

Spin‐frame independent variables in general relativity
View Description Hide DescriptionThe purpose of this paper is to present the spin‐frame independent variables in general relativity. The work is based on the fact that the tetrad Newman–Penrose form of Einstein’s equations can be put into the Yang–Mills form with the group SL(2,C) as the gauge group. The set of Mandelstam path‐dependent dynamical variables for such a theory forms spin‐frame independent variables in general relativity. The empty‐space field equations for spin‐frame independent variables have formally the same form as Maxwell’sequations. In addition, the full set of field equations for spin‐frame independent variables have formally the same form as the equations of nonlinear electrodynamics.

Global operator product expansions for free fields of arbitrary mass m⩾0
View Description Hide DescriptionA set (more than countably many) of global operator expansions—’’on and off the vacuum’’—are proved to hold for free fields of any mass m⩾0. Conformal invariance (m=0) singles out exactly one of them in the case of the ’’vacuum expansion.’’ There does not exist any termwise conformal covariant expansion ’’off the vacuum.’’

Wigner–Eckart theorem for tensor operators of graded Lie algebras
View Description Hide DescriptionAn invariant functional, analog to the group integral associated with a Lie group, is defined for the graded Lie algebras. A sufficient condition for the vanishing of the group volume is given. Orthogonality relations of the matrix elements of the representations are obtained, and the Wigner–Eckart theorem is proved for a class of graded Lie algebras.

Wightman distributions on conformal space
View Description Hide DescriptionWe show that every tempered distribution T _{ n }εS′ (M ^{ n } _{4}) which is the boundary value of a function f _{ n }(z) holomorphic in the field theoretic tube domain T_{4} ^{ n } can be uniquely continued to a distribution _{ n } on the universal covering space ^{ n } _{4} of the conformally compactified Minkowski space M ^{4n } _{ c }. It can be shown that _{ n } is the boundary value of a function _{ n } holomorphic in a certain domain ^{ n } _{4} of the complex manifoldC ^{ n } _{4}= (C×Σ_{3})^{ n }, where Σ_{3} denotes the affine complex three‐dimensional unit sphere.

Approximating functions with a given singularity
View Description Hide DescriptionIn this paper, assuming that one knows one of the singularities s _{1} of the function f (z) and its power series expansion on a domain D of the complex plane, we introduce some sequences of Gammel–Baker generalized Padé approximants with the same type of singularity s _{1}. Two examples are given: One concerns the convergence acceleration of approximations for functions with a logarithmic singularity; in the other one, a physical application to optical polarizability is discussed.

A gauge invariant formulation of quantum electrodynamics using local currents
View Description Hide DescriptionIn this paper we formulate nonrelativistic quantum electrodynamics in a local and manifestly gauge invariant manner. This is accomplished by using the electromagnetic field strengths, rather than potentials, to describe the electromagnetic field and local currents, rather than canonical fields, to describe the matter. The exponentiated currents and field strengths form a group, whose representations can be studied using the Gel’fand–Vilenkin formalism. The currents and electromagnetic field strengths can be represented on a physical Hilbert space having positive norm. (The necessity for an indefinite metric does not arise here.) Furthermore, the classical equations of motion hold as operator equations on this Hilbert space. In this formulation, the requirement of gauge invariance is essentially replaced by imposing the Maxwell initial value equtions, which in turn lead to constraints on systems of Gel’fand–Vilenkin multipliers.

Meta universe
View Description Hide DescriptionFoster and Ray have pointed out that since tachyons are particles having energy and momentum, they should hence contribute to the gravitational field through the energy–momentum tensor. Following them, a spacelike metric for the tachyon dust model has been proposed, and the problem of condensation of tachyons, after a slight perturbation of the model, has been examined. The tetrad technique has been frequently used for the purpose.

Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. VII. Comparison of different approaches
View Description Hide DescriptionA general analysis is given for the theoretical evaluation of the effective permittivity of random heterogeneous materials that are statistical homogeneous. The perturbative and variational formulations presented in Papers I–VI in this series of work are reconstructed from a slightly different point of view and compared with recent approaches developed by other authors. Perturbation expansions for the effective permittivity are derived through electric field, electric displacement, Lorentz field, and T matrix. The validity of various approximate solutions involving the effective‐medium approximation and the cumulant expansion method is discussed with the aid of a diagrammatic representation of the perturbation series. It is confirmed that, at the present stage, the cumulant theory is the best approximation for a three‐dimensional system, while the effective‐medium theory is the best for a two‐dimensional system. The meaning and applicability of variational approaches are also reviewed.