Volume 19, Issue 1, January 1978
Index of content:

On the first approximation of the K‐harmonics method
View Description Hide DescriptionFormulas for the calculation of two‐body central interactions in the first approximation of the K‐harmonics method are presented. These formulas are exact and account for diagonal and off‐diagonal matrix elements as well.

Self‐avoiding random walks: Some exactly soluble cases
View Description Hide DescriptionWe use the exact renormalization group equations to determine the asymptotic behavior of long self‐avoiding random walks on some pseudolattices. The lattices considered are the truncated 3‐simplex, the truncated 4‐simplex, and the modified rectangular lattices. The total number of random walksC _{ n }, the number of polygons P _{ n } of perimeter n, and the mean square end to end distance 〈R ^{2} _{ n }〉 are assumed to be asymptotically proportional to μ^{ n } n ^{γ−1}, μ^{ n } n ^{α−3}, and n ^{2ν} respectively for large n, where n is the total length of the walk. The exact values of the connectivity constant μ, and the critical exponents λ, α, ν are determined for the three lattices. We give an example of two lattice systems that have the same effective nonintegral dimensionality 3/2 but different values of the critical exponents γ, α, and ν.

On projective symmetries of dynamical systems
View Description Hide DescriptionThe present paper is concerned with symmetry transformations of a dynamical system defined on the tangent bundle of a Riemannian manifold. Of present interest are infinitesimal symmetry transformations of the vector field which defines the dynamical system on the tangent bundle. It is known that a class of such transformations entails infinitesimal projective transformations leaving the vector field invariant. Symmetry algebras formed by such projective transformations are studied. It is shown which dynamical systems admit large symmetry algebras. As a result, two kinds of dynamical systems are determined, which have the base Riemannian manifolds of constant curvature with dimensions n?4. The systems are generalizations of the classical harmonic oscillator and Kepler problem usually considered in Euclidean spaces. First integrals quadratic in the velocities are obtained, which are also generalizations of the well‐known quadratic integrals for the above classical systems.

’’IST‐solvable’’ nonlinear evolution equations and existence—An extension of Lax’s method
View Description Hide DescriptionWe present here a new and easy method, a natural extension of Lax’s method, for obtaining general ’’IST‐solvable’’ nonlinear evolution equations. These are evolution equations for the potential function(s), v, of a Hamiltonian, H, when the logarithmic t derivatives of H’s inverse scattering data are given by a t‐dependent ratio of entire functions of E, Ω (t,E). Here E is the energy variable and Ω is the ’’dispersion relation’’ of Abowitz, Kaup, Newell, and Segur (AKNS). We pose the question of existence of the evolution equation’s solution. This question is answered completely in the one‐dimensional Schrödinger case (first example). In a second example we derive the evolution equation for an n×n matrix generalization of the Zakharov–Shabat–AKNS equation. Our method displays the central role of analyticity in E in the IST method as a whole.

Dense electron‐gas response at any degeneracy
View Description Hide DescriptionAn exact expression for the linear response function of the dense electron gas valid at any temperature is worked out in the ring (RPA) approximation. The T=0 and T=∞ limits reproduce the already known results. It is used to explain the longitudinal oscillations and the screening around a test charge. The latter is either Thomas–Fermi‐like or Friedel‐like according to the values of the parameters.

Dynamics and phase transitions for a continuous system of quantum particles in a box
View Description Hide DescriptionA particular type of continuous quantum system with infinitely many particles is analyzed, and the existence of dynamics is proven in the GNS representations of certain states. The dynamics is not a group of automorphisms on the original algebra, so equilibrium states are defined in terms of the KMS condition in the representations of the states. The basic theorems about KMS states do not apply here. Nevertheless, for a special class of interactions it is proven that the central decomposition of an equilibrium state is concentrated on a Borel set of equilibrium factor states and that such factor states are precisely the extremal equilibrium states. Furthermore, the equilibrium factor states are in one‐to‐one correspondence with sets of functions satisfying a certain system of trace equations. This explicit correspondence is then used to show that there are no phase transitions for high temperature, and an example of a phase transition is constructed for low temperature. The phase transition also provides an example of continuous symmetry breaking.

Inverse Gaussian transforms: General properties and application to Slater‐type orbitals with noninteger and integer n in the coordinate and momentum representations
View Description Hide DescriptionThe use of Gaussian‐type orbitals (GTO) facilitates the evaluation of the multicenter integrals encountered in quantum chemistry by reducing all integrals of more than two centers to two‐center integrals. On the other hand Slater‐type orbitals (STO), while leading to more time‐consuming integral evaluations, provide a better approximation to variationally determined atomic orbitals. Thus, for a basis set of given size, STO’s generally give better accuracy than GTO’s. Kikuchi proposed the representation of STO’s as integral Gaussian transforms, or in effect by c o n t i n u o u s expansions in GTO’s, and Shavitt, Karplus, and Kern have applied this technique to the evaluation of multicenter integrals over STO’s. If these procedures are to be extended, it is desirable to develop a more systematic approach to the representation of a given basis function, ψ (r), as a Gaussian transform, ψ (r) =G[f (t);r] =F^{∞} _{0} f (t)exp(−r ^{2} t) d t; what this reduces to is the problem of calculating the i n v e r s e Gaussian transform,f (t) =G^{−1}[ψ (r);t]. In the present investigation it is pointed out that f (t) =L^{−1}[ψ (s ^{1/2});t], where L^{−1} represents the inverse L a p l a c etransformation. On this basis conditions on ψ (r) necessary for the existence of a unique continuous Gaussian inverse, f (t), are formulated, and general rules for the manipulation of inverse Gaussian transforms are developed. Finally, the formulas for the inverse Gaussian transforms of STO’s obtained previously by Kikuchi and Wright are generalized to noninteger principal quantum number, and angle‐dependent STO’s, in both the coordinate and momentum representations.

Nontranslationally covariant currents and associated symmetry generators
View Description Hide DescriptionWithin the framework of axiomatic field theory, the general case of a translationally noncovariant conserved local current is investigated. It is shown that the associated symmetry does not change the particle number nor the mass or the momentum of one‐particle states. There is an integer N such that the N‐fold commutators of the generator with the momentum as well as with the mass operator vanish.

On the wave‐mechanical representation of a Bose‐like oscillator
View Description Hide DescriptionA detailed study is made of the wave‐mechanical representation of a one‐dimensional Bose‐like oscillator whose canonical variables satisfy the general commutation relations first proposed by Wigner. The eigenvalue problems of the momentum and Hamiltonian operators are completely solved, and this is made possible only when wavefunctions in general are allowed to be hyperfunctions. The equivalence between the wave‐ and matrix‐mechanical representations is thereby established for any value of c (a characteristic parameter of the theory), contrary to the conclusion reached previously by Yang. It is also found that for the case −1/2<c<0 or 0<c<1/2 there exist two classes of eigenfunctions that are mutually separated by a superselection rule.

The three‐dimensional convolution of reduced Bessel functions and other functions of physical interest
View Description Hide DescriptionA method for evaluating convolution integrals over rather general functions is suggested, based on the analytical evaluation of convolution integrals over functions B ^{ M } _{ν,L }(r) = (2/π)^{1/2} r ^{ L+ν} K _{ν} (r) Y ^{ M } _{ L }(ϑ,φ), which are products of modified Bessel functions of the second kind K _{ν}(r), regular solid spherical harmonics r ^{ L } Y ^{ M } _{ L }(ϑ,φ), and powers r ^{ν}.

A generalized prolongation structure and the Bäcklund transformation of the anticommuting massive Thirring model
View Description Hide DescriptionThe prolongation structure method of Wahlquist and Estabrook is generalized to Grassmann algebra valued differential forms and used to determine a Bäcklund transformation for the equations of the anticommuting massive Thirring model.

Eigenvalues of S⋅Π for spins 1/2, 1, and 3/2
View Description Hide DescriptionThe eigenvalues of the matrix operator S⋅Π for a constant magnetic field are derived in a parallel way for spins 1/2, 1, and 3/2 using only the algebra of the spin matrices and the commutation relations of the components of Π.

Spaces of positive and negative frequency solutions of field equations in curved space–times. II. The massive vector field equations in static space–times
View Description Hide DescriptionThe space–times considered in this article are static, V _{ n }×R, with compact space‐section manifolds without boundary, V _{ n }, and such that the trajectories of the Killing vector field are geodesics. For the physical field of spin 1 and mass m≳0 in these space–times, field equations are solved in any adapted atlas, by the one‐parameter groups of unitary operators generated by scalar and vector Hamiltonians, i ^{−1} T _{ j } ^{−1}, j=0,1, in Sobolev spaces H _{ j } ^{ l−1}(V _{ n }) ×H _{ j } ^{ l } _{1}(V _{ n }), lεR. Hilbert spaces of positive energy solutions of field equations, as well as those of reduced solutions and their canonical symplectic and complex structures, are determined. The existence and the uniqueness of Lichnerowicz’s (1−1) current on space–time are established, and the corresponding frequency‐solution Hilbert spaces are constructed. Within the framework of Segal, a definition of quantum field operators is given, leading to the postulated commutator for the physical field concerned.

A variational derivation of the Bach–Lanczos identity
View Description Hide DescriptionA discussion of a modified Hilbert variational principle is presented. The Bach–Lanczos identity is then derived from this variational principle.

Uniqueness connection between charge conjugation and statistics
View Description Hide DescriptionThe charge conjugationproperties of bilinear quantum field theories are examined in considerable detail. It is shown that the connection between charge conjugation and statistics is unique. The relation between spin and statistics for a large class of these theories and the statistics of unusual fields such as Faddeev–Popov ghost fields and Gupta’s regularizing fields with negative norm are discussed.

Theory of vibrations of coated, thermopiezoelectric laminae
View Description Hide DescriptionThis study presents a theory for dynamic problems of coated laminae in which there is coupling between mechanical and electrical as well as thermal fields. The laminae is coated completely with perfectly conducting electrodes on both its faces, and it may comprise any number of bonded layers, each with a distinct but uniform thickness, curvature and electromechanical properties. First, a generalized variational theorem is derived so as to describe the complete set of the fundamental equations of thermopiezoelectricity. Next, by the use of this theorem, a system of two‐dimensional, approximate governing equations of the coated laminae is constructed for the case when the mechanical displacement, electric potential, and temperature fields vary linearly across the laminae thickness. The effects of elastic stiffnesses of, and the interactions between, layers of the laminae and its electrodes are all taken into account. Also, the uniqueness of the governing equations is examined, and a theorem which includes the conditions sufficient for the uniqueness is given.

Fluctuation theories and Gaussian stochastic processes
View Description Hide DescriptionThe theory of fluctuations for systems near equilibrium has given rise to two developments which generalize the theory in two distinct ways. One of these developments is focused on the theory of fluctuations far from equilibrium where the dynamics is nonlinear. The other development has focused on extending the class of fluctuating forces to include forces with non‐delta functioncorrelations. The near equilibrium theory corresponds with the theory of s t a t i o n a r y, G a u s s i a n, M a r k o v p r o c e s s e s; the nonlinear, far from equilibrium theory corresponds to the theory of n o n s t a t i o n a r y, G a u s s i a n, M a r k o v p r o c e s s e s; and the non‐delta function, force correlation theory corresponds to the theory of s t a t i o n a r y, G a u s s i a n, n o n‐M a r k o v i a n processes in one form, and to the theory of n o n s t a t i o n a r y, G a u s s i a n, n o n‐M a r k o v i a n processes in another form. The common feature found in all these theories is Gaussianness.

Transforming Gaussians into Wannier functions
View Description Hide DescriptionIt is clear a p r i o r i that equal Gaussian functions, spread over a lattice, can be transformed into Wannier functions. The transformation is carried out here analytically in one dimension, with the help of the theory of theta functions. The results confirm and illustrate the properties commonly assigned to these functions, with one startling exception.

A derivation of the virial expansion with application to Euclidean quantum field theory
View Description Hide DescriptionIn this paper we give a derivation of the virial expansion and some of its generalizations. Our derivation is based on the generating functional which defines a representation of the density operator ρ (x) in a nonrelativistic local current algebra. The virial expansion results from solving a functional differential equation for this quantity. We exploit the well‐known analogy between quantum field theory and classical statistical mechanics to explore the use of the virial expansion in Euclidean quantum field theory. Specifically, we show that the virial expansion can be used to derive Feynman’s rules and to provide a perturbation expansion about a static ultralocal model. The latter is worked out in detail in the case of a free neutral scalar model, and outlined in the case of a λφ^{4}model.

Multiple steady states in a simple reaction–diffusion model with Michaelis–Menten (first‐order Hinshelwood–Langmuir) saturation law: The limit of large separation in the two diffusion constants
View Description Hide DescriptionThe admissible multiple nonuniform steady states of a model bimolecular autocatalytic reaction–diffusion system with Michaelis–Menten (first‐order Hinshelwood–Langmuir) saturation law are constructed in the case of large scale separation in the two diffusion constants. Both the Dirichlet and the Neumann problems are discussed in a one‐dimensional geometry, and the corresponding bifurcation pictures are given.