Volume 5, Issue 1, January 1964
Index of content:

Von Neumann Algebras of Local Observables for Free Scalar Field
View Description Hide DescriptionBasic properties of von Neumann algebras of local observables for the free scalar field are investigated. In particular, the duality theorem for regions at a fixed time is proved. It is also shown that von Neumann algebras for regions at a fixed time are factors of type II_{∞} or III_{∞}. An argument which excludes the possibility of a factor of type I for a general class of theory and for a certain class of regions is given.

Stationary Entropy Principle and Renormalization in Normal and Superfluid Systems. I. Algebraic Formulation
View Description Hide DescriptionThis paper shows how the dynamical and thermodynamicalproperties of an interacting quantum mechanical system with many degrees of freedom may be expressed and calculated solely in terms of renormalized propagators and renormalized vertices or interactions. The formulation employed is sufficiently general to encompass systems which have several components, with Fermi or Bose statistics, whether or not they exhibit superfluidity or superconductivity. The process of renormalization is the functional generalization of the thermodynamic transformation from the chemical potential and temperature to the energy and matter densities. With each set of variables (here, functions) is associated a natural thermodynamic function (here a functional). The natural functional for the unrenormalized potentials which occur in the Hamiltonian is the logarithm of the grand partition function; the natural functional for the fully renormalized variables, the distribution functions, is the entropy. In particular, a stationarity principle for a functional F ^{(2)} of distribution functions subject to constraints is shown to provide a fully renormalized description of the system. The numerical value of this functional, at the stationarity point at which the distribution functions take their actual value, is the entropy of the system. The equations of stationarity are expressions for the unrenormalized ν‐body potentials v _{ν} in terms of the ν′‐body distribution functionsG _{ν}′. The functionals F ^{(2)} and v _{ν} (of the distribution functionsG _{ν}′) are expressed as the solutions of closed functional differential equations which may be used to generate their power‐series expansions. For a superfluid Bose system, as for the electromagnetic field interacting with matter, it is necessary to consider expectation values of odd, as well as even, numbers of field operators. In particular it is necessary to employ the expectation values G _{ν} for 2ν = 1, 2, 3, 4 field operators. For a fermion system, even if it is superconducting, only the functions G _{ν} for 2ν = 2, 4 are required. In contrast to other thermodynamical functionals, the entropy functional F ^{(2)} makes no reference to equilibrium parameters such as temperature and chemical potential.

Stationary Entropy Principle and Renormalization in Normal and Superfluid Systems. II. Diagrammatic Formulation
View Description Hide DescriptionQuantum statistical mechanics is renormalized, that is, the thermodynamical functions and the ``bare'' ν‐body potentials v _{ν} occurring in the Hamiltonian are expressed as functionals of the ν‐body distribution functionsG _{ν} (the expectation values of 2ν field operators). The field operators are considered as having two components (creation and annihilation), and G _{ν} thus has (2)^{2ν} components. The renormalization is carried out for superfluid Bose systems for which it is necessary to consider the expectation values of 2ν = 1, 2, 3, 4 operators. Superfluid Fermi systems and normal systems appear as particular cases, described by expectation values of 2ν = 2, 4 operators.
This problem, dealt with by algebraic methods in part I, is attacked here by diagrammatic methods. These methods are more dependent on convergence properties but the resulting functionals are obtained explicitly as power series of the G _{ν}′s, the general term being represented by a class of diagrams characterized by their topological structure.
In the final result, the entropy is exhibited as an explicit functional of the distribution functionsG _{ν}, (2ν = 1, 2, 3, 4), or more precisely, of functions directly related to them. This functional no longer involves the potentials (nor the equilibrium parameters). It is stationary under independent variations of the G _{ν}′s, subject to the constraints of constant energy and particle number. The four equations of stationarity exhibit each function v _{ν} as a functional of the G _{ν}′s, self‐consistently defining these distribution functions.

On the van der Waals Theory of the Vapor‐Liquid Equilibrium. III. Discussion of the Critical Region
View Description Hide DescriptionThe discussion of the properties of the Kac one‐dimensional fluid model presented in Parts I and II of this series of papers breaks down near the critical point. In Sec. II of the present paper we develop a new successive‐approximation method for the eigenvalues and eigenfunctions of the Kac integral equation which is valid in the critical region and which connects smoothly with the developments in the one‐ and two‐phase regions given in Part I. The perturbation parameter is (γδ)^{½} where γδ is the ratio of the ranges of the repulsive and attractive forces. The main physical consequence is that in the critical region the long‐range behavior of the two‐point distribution function is represented by an infinite series of decreasing exponentials with ranges all of order 1/γ(γδ)^{½}, and with amplitudes of order (γδ)^{⅔}. This leads to deviations from the Ornstein‐Zernike theory and to a specific heat anomaly which are discussed in Sec. V. We conclude with some comments on the possible relevance of our results for the three‐dimensional problem.

On the van der Waals Theory of the Vapor‐Liquid Equilibrium. IV. The Pair Correlation Function and Equation of State for Long‐Range Forces
View Description Hide DescriptionA fluid where the pair interaction potential between the particles consists of a hard core and a weak attraction of long range is considered. The pair correlation function and the equation of state in the one‐phase region are calculated by a perturbation method with the ratio of the volume of the hard core to the interaction volume as perturbation parameter. The technique used is to expand the well‐known density expansion in this perturbation parameter, and finally resum to all orders in the density. To lowest order, the equation of state is of the van der Waals type, and the asymptotic behavior of the pair correlation function near the critical point is that described by the Ornstein‐Zernike theory. Higher approximations to these results are given.

Generalized Master Equations for the Density Matrix
View Description Hide DescriptionGeneralized master equations determining the time dependence of both diagonal and off‐diagonal density matrix elements are derived. The equations are valid for systems of arbitrary size and initial conditions, and have all the generality of the von Neumann equation for the density matrix. They are general enough for use in determining all higher‐order contributions to transport coefficients. Their advantage over the von Neumann equation is that they are in forms suited to reduction by time‐independent many‐body approximation techniques.

Applicability of the Pfaffian Method to Combinatorial Problems on a Lattice
View Description Hide DescriptionFor the solution of certain problems in statistical mechanics, a formulation in terms of the combinatorial problem of counting open or closed polygons drawn on a lattice has been found useful. The solution of such combinatorial problems has been given as a generating function which can be expressed as a Pfaffian. Necessary and sufficient conditions are given here to determine when such a Pfaffian representation is possible. If certain consistency conditions are satisfied, then the solution to the combinatorial problem can be written down immediately. The method makes some use of the ideas of emission and absorption operators for fermions.

Integral Representations for Vertex Functions
View Description Hide DescriptionThe third‐order Feynman graph is studied as a function of the three external masses squared for arbitrary real values of the internal masses. Single and double ``dispersion'' relations are derived which, for arbitrary real values of the undispersed variable(s), involve integrations only over real contours. Tables list the spectral functions for both the single and double integral formulas. In several cases, a non‐Landau singularity (on the forward scattering curve) appears on the ``physical sheet'', but not as a singularity of the physical boundary value.

Position and Polarization Operators in Relativistic and Nonrelativistic Mechanics
View Description Hide DescriptionThe author shows how the position operator and the polarization operator are obtained in nonrelativistic quantum mechanics from their equivalents in relativistic quantum mechanics. This problem is investigated with the aid of group methods; it is furthermore shown that the two operators cannot be studied separately.

Motion of Test Bodies in General Relativity
View Description Hide DescriptionTwo distributions in the sense of Schwartz are introduced to describe a test body. Papapetrou's equations for a pole‐dipole test body are derived in a coordinate‐independent manner. The supplementary condition S ^{σμ} U _{μ} = 0 arises in a natural fashion in the course of the derivation. The method for treating test bodies with higher‐order poles is indicated.

Poles in Coupled Scattering Amplitudes
View Description Hide DescriptionElastic scattering amplitudes which have the analytic structure of the Mandelstam representation and which satisfy the unitarity condition and substitution law are severely restricted. Because of these restrictions, poles cannot occur independently and arbitrarily in amplitudes coupled by the unitarity condition. Instead, the locations and residues of poles in coupled amplitudes must satisfy the same relations as do poles in perturbation theory amplitudes.

Approach to a Phase Transition in a One‐Dimensional System
View Description Hide DescriptionThe manner in which a one‐dimensional system approaches a phase transition as the range of its attractive pairwise interactions becomes large is analyzed, as a continuation of studies on such systems by Kac, Uhlenbeck, Hemmer, and the author. Explicitly, consideration is given to an Ising ferromagneticmodel and the corresponding lattice gas with interaction −αγ exp (−γ i − j) as γ → 0. One finds that the fluid isotherms are actually analytic, but that in a certain region they approach the zero slope characteristic of a phase transition in the essentially singular fashion exp (−const/γ).

Tensor Artificial Viscosity for Numerical Hydrodynamics
View Description Hide DescriptionArguments are presented to the effect that a tensor artificial viscosity is a more suitable quantity to use than the usual scalar viscosity. A specific tensorviscosity is constructed for the case of a two‐dimensional Lagrangian system.

Motion of a Relativistic Damped Oscillator
View Description Hide DescriptionQuantitative effects of damping in a nonlinear system are illustrated for the case of a relativistic oscillator. The elementary techniques employed furnish details of the decay phenomena which are more extensive than those normally furnished by stability theorems.

Excitation Spectrum of an Impurity in the Hard‐Sphere Bose Gas
View Description Hide DescriptionThe impurity excitation spectrum is calculated for the hard‐sphere Bose gas with dilute impurity concentration, using the pseudopotential method of Lee, Huang, and Yang. The spectrum for the ideal case, in which the mass and interaction of impurity are the same with those of the bosons in the gas, is k ^{2} in the unit of 2m_{a} = h/ = 1 (m_{a} being the mass of boson, and k the momentum of impurity), with an effective mass correction of , where ρ is the density of the boson gas, and a is the hard‐sphere diameter of the boson. The general impurity case is also calculated and found to have an impurity spectrumrk ^{2}, with a similar effective mass correction, where r is the mass ratio of the boson to the impurity.

Approximate Calculations of Scattering Phase Shifts
View Description Hide DescriptionThe nonrelativistic scattering problem is reformulated by transforming the partial‐wave equation into similarity with the corresponding equation for a known or soluble problem. It is shown that if the potentials for the two cases are sufficiently close (within about 10% typically), a particularly simple and accurate expression for the desired phase shifts can be written. Numerical examples for a Gaussian potential are given. The method is also extended to the calculation of bound‐state energy levels, but no numerical examples are given.