Volume 17, Issue 9, September 1976
Index of content:

Bifurcation properties of Dicke Hamiltonians
View Description Hide DescriptionA variation in the coupled order parameter treatment of Dicke Hamiltonians in thermodynamic equilibrium is presented. The Hamiltonian is linearized by introducing disposable c‐number parameters. These parameters are chosen to minimize the resulting free energy. This requirement leads to a system of coupled nonlinear equations whose bifurcationproperties are studied. The solution branches are labelled by the inertia of the free energy stability matrix. We prove that the parameters on the solution branch which provide the global minimum free energy also produce a linearized Hamiltonian thermodynamically equivalent to the original Hamiltonian provided only a finite number of field modes are present. This method is used to discuss the bifurcation and stability properties of the Dicke Hamiltonian with A ^{2} and counterrotating terms. We also discuss why the phase transition disappears in the presence of external currents or fields. We show how an internal gauge destroying mechanism may lead to the persistence of the phase transition even in the presence of external coupling. The method is used to discuss the phase transitions and multiplicity of ordered state phases in multilevel molecular systems. We also present a simple method for determining whether an external source will or will not destroy a second order phase transition and discuss the conditions under which such models may exhibit first order phase transitions.

Classification of all simple graded Lie algebras whose Lie algebra is reductive. I
View Description Hide DescriptionAll simple graded Lie algebras whose Lie algebra is reductive are presented, and the classification theorem is proved. Several theorems which may show up to be useful in a different context are also included.

Classification of all simple graded Lie algebras whose Lie algebra is reductive. II. Construction of the exceptional algebras
View Description Hide DescriptionThe exceptional simple graded Lie algebras whose existence is suggested by the results of the preceding paper are explicitly constructed. In this way the classification of all simple graded Lie algebras whose Lie algebra is reductive is completed.

Soluble classical spin model with competing interactions
View Description Hide DescriptionThe partition function and spin pair correlation functions have been calculated exactly for a classical linear chain model with alternate next‐nearest‐neighbor (nnn) interactions, in which the interaction energies between pairs of nearest (nn) and next‐nearest (nnn) neighbor spins are arbitrary functions of the angles between the relevant spins. Of special interest is the cosine interaction model described by the Hamiltonian H=−Σ^{ N } _{ i=1}[J _{1}(cosϑ_{2i−1,2i } +cosϑ_{2i,2i+1})+J _{2}cosϑ_{2i−1,2i+1}]. When the nnn interaction is antiferromagnetic (J _{2}<0) it competes with the nn interaction J _{1}, and there can be disorder point(s) at which nnn correlations change from monotonic to oscillatory. The ground state is ferromagnetic when the interaction ratio r≡J _{2}/‖J _{1}‖ ≳−1/2=r _{ c }, but is disordered for more negative values. The disorder point locus has been determined. It terminates at zero temperature at r _{ D }=−1/2^{1/2}, at which point the ground state energy is a maximum. The result that r _{ D } differs from r _{ c } is thought to be peculiar to one‐dimensional models. Over a limited range of values of r there can be two disorder points. The low temperature asymptotic behavior of the partition and correlation functions is analyzed in detail. Also a novel summation formula for spherical Bessel functions is obtained.

How to calculate the grand partition function in the uncorrelated jet model
View Description Hide DescriptionWe present a new technique for calculating the grand partition function and all quantities of physical interest in the uncorrelated jet model. The method, which is also valid in the large‐p _{ T } region, consists of the numerical evaluation of the appropriate integral representation in the complex plane. We analyze in detail the difficulties associated with this approach and show how to overcome them. The numerical results are checked with a new high energy expansion for the grand partition function.

On the implementability of local gauge transformations in a theory with localized states
View Description Hide DescriptionA conflict between unitary implementability of local gauge transformations (kind one) on the one hand and certain properties of sets of localized states on the other is deduced in the axiomatic framework of relativistic local quantum field theory.

Gauge theories and nonrelativistic cosmological symmetries
View Description Hide DescriptionIt is shown that the application of the locality principle in a uniformly curved space leads to the emergence of a dynamical quantum mechanical group which is precisely the Hooke group. The interaction structure is also studied.

Canonical parameters of the 3j coefficient
View Description Hide DescriptionThe 3j coefficient is expressed as a function of five new parameters which have unique properties. They are completely independent, satisfy simple validity criteria, and display the symmetry properties of the function in a particularly transparent manner. By means of the new parameters, the known 72‐element symmetry group is reduced to an eight‐element group, and the absolute symmetries are separated in a clear way from those which contain a phase factor.

Quantum two‐particle scattering in fuzzy phase space
View Description Hide DescriptionThe concepts of configuration and momentum representation space for state vectors are generalized to that of fuzzy‐phase‐space representation spaces L ^{2}(Γ_{ s }), 0<s<∞, which are interpolated in between these two standard representations. It is shown that the wavepacket in L ^{2}(Γ_{ s }) displays the familiar evanescence property from any region K _{ s } × M _{ s } in the fuzzy phase space Γ_{ s } if that region is bounded in its configuration part K _{ s }; also, that the probability of detecting the system in K _{ s } × M _{ s } has a finite asymptotic time limit if K _{ s } is a (fuzzy) cone. For scattering states the existence of free states that are asymptotic in Γ_{ s } is established, and a formula for differential cross section in Γ_{ s } is derived.

Quantum mechanical soft springs and reverse correlation inequalities
View Description Hide DescriptionVarious properties of one‐dimensional Schrödinger operators with ’’soft spring’’ potentials are derived as a consequence of the fact that the GHS and other correlation inequalities are reversed for certain general Ising modules.

Binary mixture with nearest and next nearest neighbor interaction on a one‐dimensional lattice
View Description Hide DescriptionA mixture of two kinds of molecules on a one‐dimensional lattice with free ends is considered. The energy term is assumed to consist of interactions of nearest neighbors and next nearest neighbors and interaction with the uniform external field. The partition function is evaluated by determining the degeneracies.

On the matrix representation of unbounded operators
View Description Hide DescriptionIt is shown that a matrix representation with properties analogous to the ones that hold for the bounded operators in Hilbert space is possible also for important sets of unbounded operators. These sets consist of the ‐‐algebras C _{ D } of the linear operators on any noncomplete scalar product space D, which have an adjoint in D. (These algebras have already been studied by the author, in collaboration with others, in previous papers.) Specifically it is proved that for these operators a matrix representation is possible with respect to an arbitrary orthonormal basis within D, in contrast to the situation that has been found by von Neumann for the unbounded closed symmetric operators. The matrix representation of the operators considered here also allows the usual algebraic operations. Besides, the changes of basis induced by automorphisms of D are allowed.

Exact dynamics of a model for a three‐level atom
View Description Hide DescriptionIn this paper we investigate the dynamics of a model for a three‐level atom in interaction with a radiation field. The exact solution to the spontaneous‐emission problem is derived using methods developed earlier by the authors, and expressions are obtained for the probabilities of the atom’s being in the first or second excited states at any time t. For the special case that the strengths of the coupling between each of the excited states and the λth mode of the field are proportional, detailed conclusions can be drawn concerning the effects of such factors as system size, coupling function, and level splitting on the temporal evolution of the system. The evolution of excited quantum systems having one versus two modes of decay to the ground state is also compared, and similarities and differences in the temporal behavior are noted. Finally the relevance of the theory presented in this paper to experimental problems in radiation chemistry and physics is indicated in our concluding remarks.

Fredholm determinants and multiple solitons
View Description Hide DescriptionThe discrete inverse scattering problem in one dimension is considered. Exact solutions are obtained using elementary algebraic tools. Expressions found involve determinants of infinite‐dimensional matrices. A simple, heuristic, limiting process yields the solution for the continuous problem. When the reflection coefficients do not contribute (the general N soliton case), the determinants reduce to those of given N×N matrices.

Derivation of an exact spectral density transport equation for a nonstationary scattering medium
View Description Hide DescriptionWithin the framework of the quasioptical description and the pure Markovianrandom process approximation, an exact kinetic equation is derived for the spectral density function in the case of wave propagation in a nondispersive medium characterized by large‐scale space–time fluctuations. Also, a quantity, called the d e g r e e o f c o h e r e n c e f u n c t i o n, is defined as a quantitative measure of the irreversible effects of randomness.

Closed first‐ and second‐order moment equations for stochastic nonlinear problems with applications to model hydrodynamic and Vlasov‐plasma turbulence
View Description Hide DescriptionWorking along the lines of a procedure outlined by Keller, a technique is developed for deriving closed first‐ and second‐order moment equations for a general class of stochastic nonlinear equations by performing a renormalization at the level of the second moment. The work of Weinstock, as reformulated recently by Balescu and Misguich, is extended in order to obtain two equivalent representations for the second moment using an exact, nonperturbative, statistical approach. These general results, when specialized to the weak‐coupling limit, lead to a complete set of closed equations for the first two moments within the framework of an approximation corresponding to Kraichnan’s direct‐interaction approximation. Additional restrictions result in a self‐consistent set of equations for the first two moments in the stochastic quasilinear approximation. Finally, the technique is illustrated by considering its application to two specific physical problems: (1) modelhydrodynamicturbulence and (2) Vlasov‐plasma turbulence in the presence of an external stochastic electric field.

A new analytic continuation of Appell’s hypergeometric series F _{2}
View Description Hide DescriptionThe doubly infinite series for Appell’s function F _{2}(α,a _{1},a _{2},b _{1},b _{2};x,y) is written in terms of four of Appell’s F _{3} functions. Analytic continuations are given for the F _{3} series, thereby allowing one to obtain a new analytic continuation for the F _{2} series. The new doubly infinite series are all absolutely convergent if their variables satisfy ‖x‖<1 and ‖y‖<1, whereas the F _{2} series is absolutely convergent only in the domain ‖x‖+‖y‖<1. The analytic continuations given here are very useful for evaluating the Appell F _{2} series when one of the variables is near unity. In particular, our results are useful for calculating radial matrix elements over products of Dirac–Coulomb functions and the electromagnetic interactionGreen’s function.

On the solution of nonlinear matrix integral equations in transport theory
View Description Hide DescriptionThe coupled nonlinear matrix integral equations for the matrices X (z) and Y (z) which factor the dispersion matrix Λ (z) of multigroup transport theory are studied in a Banach space X. By utilizing fixed‐point theorems we are able to show that iterative solutions converge uniquely to the ’’physical solution’’ in a certain sphere of X. Both isotropic and anisotropic scattering are considered.

A continuum theory of deformable ferrimagnetic bodies. I. Field equations
View Description Hide DescriptionThe first part of this work concerns a thorough study of both global and local field equations that govern deformable (not necessarily linear elastic) ferrimagnets and antiferromagnets from a phenomenological viewpoint. The main tool used is a generalized version of d’Alembert’s principle, valid for both reversible and irreversible phenomena, simultaneously with the invariance requirement provided by the so‐called objectivity and applied a p r i o r i to generalized internal forces which represent the various interactions. All interactions taking place in such media are thus given a phenomenological description and are introduced via the duality inherent in the method. The development follows a rational and deductive mathematical scheme in which the notion of topological linear space of velocities plays a predominant role, so that particular cases follow by selecting the appropriate member of this space. In the following companion paper the allied thermodynamics and a thorough discussion of the relevant constitutive equations that follow therefrom are given. The formulation so obtained will allow the consideration of slight perturbations superimposed on bias fields.

A continuum theory of deformable ferrimagnetic bodies. II. Thermodynamics, constitutive theory
View Description Hide DescriptionIn order to close the system of differential field equations developed in Paper I, this article proposes a rational development of the relevant macroscopic thermodynamics and of a constitutive theory. In particular, by following Coleman’s thermodynamics, exact nonlinear constitutive equations for thermoelastic antiferromagneticinsulators are formulated. According to the deductive scheme adopted in Paper I, the important case of elastically isotropic antiferromagnets with a magnetic easy axis, and possibly endowed with the property of weak ferromagnetism, is developed in detail by using approximations. In order to supplement the description of thermodynamically recoverable processes and in accordance with the Onsager–Casimir theory of irreversible processes, the constitutive equations governing phenomena such as viscosity, electric and heat conduction, and spin relaxation, the latter either for strong or weak damping, are obtained. Regarding the latter effect, it is shown, thanks to the formalism adopted in Paper I, that both viscosity and spin relaxation participate in the Cauchy equations. The relaxation term of Gilbert is thus generalized to the case of deformable antiferromagnets.