Volume 9, Issue 5, May 1968
Index of content:

Error Bounds in Equilibrium Statistical Mechanics
View Description Hide DescriptionA new method is presented for the calculation of thermodynamic properties from equilibrium statistical mechanics. Starting from the high‐temperature expansion coefficients for the canonical partition function, error bounds are obtained, which are both rigorous and optimal.

O(5) Polynomial Bases
View Description Hide DescriptionPolynomial bases are derived for the irreducible representations of the group O(5). The matrix elements of the infinitesimal generators are given.

Quantum‐Mechanical Description of a Brownian Particle
View Description Hide DescriptionIt is shown that the motion of a Brownian particle in the Smoluchowski approximation may be described by a Schrödinger‐like equation defining a complex probability amplitude whose norm is the same as the stochastic probability density. Furthermore, the quantum dynamical operators have a physical meaning which arises in a natural way, from the stochastic nature of the process. These operators satisfy the usual commutation relations and thus the uncertainty principle. Here the constant ℏ is replaced by a parameter depending on the characteristics of the system. In particular, the potential‐energy operator for a Brownian particle subject to no external forces can be interpreted as a Rayleigh dissipative function.

Green's Theorem and Invariant Transformations
View Description Hide DescriptionConservation laws are derived with the use of Green's theorem. These are studied specifically for the wave equation. A trivial class of invariant transformations exists which maps solutions into solutions. The mapping consists in the addition of a particular solution to all solutions. Because of the linearity of the wave equation, this sum will again be a solution. It is shown that this class of transformations has the Newman‐Penrose constants among its generators. Calculations are carried out explicitly for the scalar wave equation and for Maxwell's equations.

Frequency Spectra of Harmonic Lattices with Weak Long‐Range Interactions
View Description Hide DescriptionVibrational‐frequency spectra are calculated analytically for certain harmonic lattice models with weak long‐range interactions, in one and two dimensions. The force constants are chosen to decay, for large separations n, approximately as exp (−γn), where γ is an inverse range parameter. In the limit of infinite γ, standard nearest‐neighbor results are recovered. In the limit of vanishing γ, or infinite interaction range, the frequency spectra have entirely different singularities. In the examples studied here, these can be either poles or branch points at the maximum frequency. When γ is small but not zero, then the singularities are still qualitatively the same as in the corresponding nearest‐neighbor models; but the general shapes of the spectra are dominated by ``false'' singularities lying outside the allowed frequency domain. Except for a small region near the maximum frequency, spectra obtained in the limit of vanishing γ are good approximations to the correct spectra for small γ.

Electrostatic Interactions of the Configuration d ^{ n−1} sp
View Description Hide DescriptionThe matrix elements of the Coulomb interactions of the configuration d ^{ n−1} sp were obtained for L‐S coupling in the form of linear combinations of certain radial integrals. The method used can be extended to the configuration l ^{ n−1} l′ l″.

Some Inequalities Involving Traces of Operators
View Description Hide DescriptionWe prove that for arbitrary completely continuous operators A _{1}, A _{2}, ⋯, A_{n} and for positive numbers p _{1}, p _{2}, ⋯, p_{n} with , the inequalities,hold. Further if a _{1}, a _{2}, ⋯, a_{n} are the annihilation operators of an N‐dimensional harmonic oscillator,m and n are any positive integers, and ρ is a nonnegative definite operator, we prove the inequality.Some consequences of these inequalities, related results, and some applications to correlation functions of the quantized electromagnetic field are discussed.

Stability of Matter. II
View Description Hide DescriptionThe stability of a system of charged point particles is proved under the assumption that all negatively charged particles are fermions. A lower bound on the energy is found to be −Aq ^{⅔} Nme ^{4}ℏ^{−2}, where q is the number of distinct negative species, N the total number of negative particles, m an upper bound for their mass, e an upper bound for the absolute value of the charge on both negative and positive particles, and A is a numerical constant.

Orear Behavior in Potential Scattering
View Description Hide DescriptionThe object of this paper is to find and study a class of potentials for which the corresponding scattering amplitude decreases rapidly in energy at fixed (nonforward) angles. Specifically, we ask that as k → ∞ for θ fixed. It is shown here that this relation is valid for certain potentials V(r) which are even functions of r analytic in a strip about the real r axis. With further restrictions on the potentials we show that the scattering amplitude converges to its first Born approximation at high energies for fixed nonforward angles.

An Integral Equation for the Scattering Operator
View Description Hide DescriptionAn integral equation, originally derived for a perturbation expansion of the n‐point scattering function (Pugh's equation), whose (finite) solution is the renormalized perturbation result, is derived here as an exact ``strong'' equation for the nth operator derivative of the scattering operator, whose vacuum expectation value is the n‐point function.

Field‐Theoretic Formulation of Quantum Statistical Mechanics
View Description Hide DescriptionThe notions of strong convergence of state vectors, introduced by Haag in his formalism of axiomatic quantum field theory, are extended to the case of vectors with an infinite number of particles but finite densities. Some general properties of nonequilibrium distribution functions are derived without the use of power series expansions or any other simplifying assumption. An integral representation is obtained for the distribution functions which makes it possible to discuss their behavior for small and large energies and to obtain some information about the singularities of these functions when continued analytically.

Integral Representation for Systems of Interacting Particles
View Description Hide DescriptionThis is a sequel to a previous paper, designated as I, which dealt primarily with nonequilibrium systems. The ideas in I are extended and include the study of systems of interacting particles in equilibrium. An integral representation is obtained for the distribution function of interacting particles as a superposition of distribution functions for noninteracting particles. In particular, it is shown that an interacting Bose gas need not show Bose condensation and that the behavior near the point E = 0 cannot be more singular than that of a simple pole.

Some Properties of Ladder Operators
View Description Hide DescriptionThe commutation relations [H, P ± iQ] = ± (P ± iQ) for the symmetric operators H, P, Q are considered without assuming (a priori) any other relationship between H, P, Q, in particular without making any assumption concerning the commutator [P, Q]. It is shown that under certain mild restrictions the spectrum of H is integer spaced, and that in the two particular cases [P, Q] = iεH, ε = 0, ±1 and H = ½(P ^{2} + Q ^{2}) + iσ[P, Q], corresponding to Lie groups and parastatistics, respectively, it is simple. For these two cases the explicit representations of P, Q, and H are found in a simple manner. The question of the existence of a common analytic domain for P, Q, and H is investigated, and some sufficient conditions for this are found.

Quantum Corrections to the Pair Distribution Function of a Classical Plasma
View Description Hide DescriptionQuantum‐mechanical corrections to the pair distribution function of a plasma at high temperature and low density are calculated to order e ^{2} in the interaction, using standard diagram perturbation techniques. Both the effects due to quantum statistics (exchange) and the finite size of a wave packet (dynamic screening), are considered.

Solution of the Transport Equation with Anisotropic Scattering in Slab Geometry
View Description Hide DescriptionSome systematics which exist between eigenfunctions and adjoint singular integral equations arising in the solution of the transport equation in slab geometry are illustrated. The transport equation is shown to obey a singular integral equation and its relationship to the eigenfunction expansion‐method solution is shown. A new method for solving for the expansion coefficients in the eigenfunction expansion method is illustrated by solving Milne's problem. The role adjoint singular integral equations play in finding appropriate weight functions for use in orthogonality relations between the eigenfunctions of the transport equation is briefly discussed.

Two‐Dimensional Measure‐Preserving Mappings
View Description Hide DescriptionA particular area‐preserving mapping of a plane onto itself has been studied in detail with the aid of a digital computer. A large number of fixed points, finite sets of points that transform into each other, were located and classified as elliptic or hyperbolic depending on the nature of the linearized mapping in the neighborhood. A quantity called the residue was calculated for each fixed point. This quantity can be used to predict whether other nearby fixed points are elliptic or hyperbolic. The results showed that there are considerable regions in which almost all the fixed points are hyperbolic. Further calculations were made to estimate the area enclosed by the invariant curves whose existence has been established by Moser. The boundary of this region appeared to coincide with the boundary of the region in which almost all the fixed points are hyperbolic.

Weyl Transform in Nonrelativistic Quantum Dynamics
View Description Hide DescriptionThe Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase‐space distributions for noncommuting coordinate and momentum operators. The distinction is made between Weyl transforms in Schrödinger and Heisenberg pictures; the general case of time‐dependent Hamiltonians is considered. The Wigner function for the probability distribution in a phase space of Cartesian coordinates Q and momenta K propagates according to a conditional probability P(t, Q, K  t _{0}, Q _{0}, K _{0}), which is exhibited as a Feynman path integral in phase space. Properties of P(t, Q, K  t _{0}, Q _{0}, K _{0}) are developed; it is expressed in terms of the quantum generalization of the classical Liouville operator. The Weyl transform of a Heisenberg operator propagates according to P(t, Q _{0}, K _{0}  t _{0}, Q, K) which is also given as a Feynman path integral. An equation for the time evolution of Weyl transforms of Heisenberg operators is obtained, according to which the transform of Heisenberg coordinate and momentum operators obey a quantum form of Hamilton's equations of motion. If the initial density operator of the system commutes with the coordinate operator, then the state of the system is a mixture of pure coordinate states; the spectrum of the density operator in this case is continuous. For a Heisenberg operator A_{H} (t) with Weyl transformA_{H} (t, Q, K), the function is the expectation at time t of the dynamical property for a quantum system initially in a pure state of coordinate Q; it is the quantum‐mechanical generalization of the dynamical property of the system along the classical trajectory in configuration space at time t. The probability amplitude for the time dependence of can be expressed as a Feynman path integral with a Heisenberg Lagrangian. The amplitude of the conditional probability P(t, Q  t _{0}, Q _{0}) considered by Feynman is expressed as a path integral with a Schrödinger Lagrangian. The velocity in the Heisenberg Lagrangian is the negative of that in the Schrödinger Lagrangian of Feynman, but it agrees with the velocity appearing in the Hamiltonian equations. It is the Heisenberg Lagrangian that is the Lagrangian of classical dynamics. For a particle whose potential energy is a function of position, a quantum form of Newton's second law is obtained. An extension of the formalism to non‐Cartesian coordinate systems is given.

New Field‐Theory Approach to Singular Potentials
View Description Hide DescriptionAn approach similar to the methods of renormalization of the Green's functions equations of quantum field theory is adopted to the singular potential in the Lippman‐Schwinger equation. The close relation between our approach and the one used in field theory gives a method to be applied to nonrenormalizable field theories. The physical implication of this approach is discussed.

Realization of Poincaré‐Group Generators on a Light Cone
View Description Hide DescriptionThe ten generators of the proper inhomogeneous Lorentz group are explicitly constructed for spin‐0 and spin‐½ particles, in the case when wavefunctions are given on a light cone (rather than on an equal time hypersurface, as usual). The advantage of this formulation is that the generators J (spatial rotations) and K(Lorentz boosts) involve only elementary local operators. The Hamiltonian H and momentum operators P also contain the inverse radial momentum (∂/∂r)^{−1}, but do not involve any square roots. Moreover, only two‐component spinors are required for spin‐½ particles.

Properties of Higher‐Order Commutator Products and the Baker‐Hausdorff Formula
View Description Hide DescriptionThe element z = log e^{x}e^{y} , which is known to be an element of the Lie‐algebra generated by x and y, is expressed as a commutator series in x and y with coefficients given in terms of certain fixed polynomials. The result is given explicitly to sixth order. Useful recurrence relations are obtained. The method is based on certain properties of higher‐order commutator products, particularly their idempotent character.