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Volume 6, Issue 8, August 1965

On the Algebraic Structure of the Cluster Expansion in Statistical Mechanics
View Description Hide DescriptionThe structure of cluster expansion which is widely used in statistical mechanics is studied from an algebraic point of view. In doing this, a commutative algebra is constructed which is generated by partitions of a finite set by regarding them as operators which divide the set into disjoint parts. Physically, these operators correspond to operations which remove interaction among certain clusters of particles. It is shown that the cluster expansion stems from the relation between two basis sets of this algebra; the first set is the set of all partitions and the second is the set of pairwise orthogonal minimal idempotents. This property enables one to demonstrate the equivalence of the product versus cluster properties of the distribution and correlation functions, respectively, in general terms. This is done by constructing a simple representation space for the partition algebra corresponding to the distribution functions. A second application of the partition algebra is considered in the case when correlations are well‐ordered with respect to the interaction strength λ, so that to a given order in λ the distribution functions are not all independent and can be expressed in terms of a finite irreducible set of functions involving smaller numbers of particles. The combinatorial problem of calculating the expansion coefficients is carried out explicitly using a graded representation space with respect to the order in λ. It is concluded that the partition algebra can be used as a mathematical tool in handling problems involving cluster expansion.

Cluster‐Star Inversion and the Möbius Formula
View Description Hide DescriptionThe cluster‐star equations are inverted by using the general Möbius inversion formula. Since Uhlenbeck and Ford have many instances of systems of equations to which general Möbius inversion can be applied, this paper first presents a general survey of Möbius inversion, then some graph terminology, and finally the cluster‐star inversion.

Cluster‐Star Inversion by Means of Generating Functionals
View Description Hide DescriptionThe cluster‐star equations are inverted. These equations, Eq. (2.1), define a set of cluster functions U_{m} in terms of star functions V_{n} for 2 ≤ n ≤ m. The inversion, Eq. (2.2) gives the V_{n} in terms of the U_{m} ; 2 ≤ m ≤ n.
The cluster‐star inversion yields the coefficients β_{ n } that appear in the number‐density expansion of the logarithm of the grand partition function Ξ of a statistical mechanical system in terms of the coefficients b_{n} that appear in its fugacity expansion, even when the system is in the presence of an external field.
In the course of obtaining the inversion, an expression for the work necessary to bring a particle from infinity to a point r inside a classical system at equilibrium is obtained in terms of the V_{m} and the one‐particle distribution function ρ(r), and also in terms of the U_{m} and ρ. The expression in terms of the V_{m} and ρ has been previously derived by others with restrictions on the form of the potential energy; the alternative expression in terms of the U_{m} seems to be new.
Operators with essentially the same algebraic structure as the V_{m} appear in a recently considered asymptotic expansion of the distribution functions of a nonequilibrium system. The cluster‐star inversion facilitates the determination of these operators in terms of the solution operators of the n‐body Liouville equation,n ≥ 2.

Decomposition of the Lorentz Transformation Matrix into Skew‐Symmetric Tensors
View Description Hide DescriptionAny matrix describing a finite proper orthochronous Lorentz transformation of the null tetrad in Minkowski space may be written as a polynomial of the second order in skew‐symmetric tensors. From a geometrical point of view these tensors describe two‐dimensional planes which are mapped by the Lorentz transformation into themselves.

Atomic Terms for Equivalent Electrons
View Description Hide DescriptionThe problem of determining the number of times, b(l N L S), a given term occurs in an atomic configuration of the form (nl)^{ N } is considered by the methods of group theory. The b are shown to be related to the classical problem of partitioning numbers. Several expressions relating b to other, simpler partitions are obtained, which result in a recursion relationship for the b. A table for b(4 N L S) is included to complement existing tables for the range 1 ≤ l ≤ 3, which are found in several places in the literature.

Trace Formalism for Quantum Mechanical Expectation Values
View Description Hide DescriptionThe trace of a positive bounded operator in a Hilbert space is defined and shown to be independent of the basis used in the definition. The trace of the product WA is defined for bounded A and positive bounded W with unit trace and is also shown to be basis‐invariant. An integral representation is derived and used to define Tr(WA) for unbounded A. Linearity, isotony, and continuity with respect to uniform covergence are proved for Tr(WA) as a function of (bounded) A. Several formulas used in statistical mechanics are derived, and it is proved that if A and B are commuting positive operators, then Tr(W(A + B)) = Tr(WA) + Tr(WB). A counter example is given which shows that the commonly used definition of Tr(WA) in terms of an orthonormal basis is not invariant under permutation of the basis vectors even in the case of a very simple unbounded operator A.
An expectation‐value function M which associates the expectation value M(A) to the operator A is assumed given and subject to certain restrictions. For bounded operators, these include positivity, normalization, additivity, and a condition which may be considered as a requirement of regularity for the probability function induced by M. Modifications of these requirements are imposed for unbounded operators, and von Neumann's statistical formula is proved: there is a unique bounded positive operator W with unit trace for which M(A) = Tr(WA). The requirements placed on M are weaker than those of von Neumann, and are in fact satisfied by Tr(WA) as a function of A.

Can the Position Variable be a Canonical Coordinate in a Relativistic Many‐Particle Theory?
View Description Hide DescriptionThe time‐symmetrical interaction of charges (Wheeler‐Feynman electrodynamics) is shown in principle to yield second‐order Newtonian‐type equations of motion under the restriction that the motions be analytic extensions of free‐particle motions. The means for explicitly generating the electrodynamic equations of motion describing invariant world‐lines are given. These physically relevant equations do not fit into Dirac's (1949) formulation of Hamiltonian relativistic particle dynamics, where either world‐line invariance is given up, or only trivially straight world lines can be described (``zero‐interaction theorem'' of Currie, Jordan, Sudarshan, 1963). The misfit is due to the requirement in Dirac's scheme that position x be canonical. Under the Lie‐Königs theorem, however, Hamiltonian statements of dynamics with invariant world lines remain possible when suitable Q(x, ẋ) are introduced instead of x as canonical position variables. A group of necessary conditions on the structure of any dynamics that permits x to be canonical are worked out to indicate how stringent is this permission in general.

Quantum Electrodynamics with Vanishing Bare Fermion Mass
View Description Hide DescriptionQuantum electrodynamics is studied in the approximation suggested by Johnson, Baker, and Willey. The (approximate) Dyson equation of this model is transformed into a second‐order, nonlinear differential equation by the application of a method due, originally, to Green. The solutions of this equation are then studied under various simplifying assumptions.

Scattering of Surface Waves by a Submerged Circular Cylinder
View Description Hide DescriptionThe effect of a rigid circular cylinder, wholly immersed within and lying parallel to the free surface of an incompressible and inviscid fluid, on straight‐crested surface waves passing overhead is investigated. A mode of analysis is developed, on the hypotheses of small amplitude and time‐periodic fluid motions, that encompasses all directions of incidence of the primary wave; and is used to extend results previously obtained in the case of normal incidence. It is shown, in particular, that the absence of surface‐wave reflection at normal incidence gives way to a partial reflection for other primary directions, which in turn verges on completeness as the direction of the incoming wave becomes more closely aligned with that of the cylinder axis.

Transformation from a Linear Momentum to an Angular Momentum Basis for Relativistic Particles of Nonzero Mass and Any Spin
View Description Hide DescriptionThe infinitesimal generators of the inhomogeneous Lorentz group have been given in a basis in which the components of the linear‐momentum operators are diagonal and in another basis in which the square of the angular momentum is diagonal for all unitary irreducible ray representations of the group. In a previous paper we showed how the two bases were related for representations corresponding to zero mass and any finite spin. In the present paper we show how the two bases are related for representations corresponding to nonzero mass and any spin. Thus this paper and the preceding one enable us to expand relativistic plane waves into relativistic spherical waves and vice‐versa for particles of any spin and any mass.
In the previous paper we used the relation between the linear‐ and angular‐momentum bases to integrate the infinitesimal generators in the angular‐momentum basis and thereby obtain closed expressions which show how the angular momentum of particles of zero mass and any finite spin transform under changes of frame of reference. Similar use could be made of the transformation of the present paper. Results of such use will be given in a later paper.

Homogeneous Solutions of the Einstein‐Maxwell Equations
View Description Hide DescriptionIn this paper the solutions of the Einstein‐Maxwell equations are investigated under the assumption that the metric of the space‐time and the electromagnetic field are invariant under the transformations of a four‐parametric, simply transitive group.
The results can be summarized as follows: In the case of null electromagnetic fields there are two different possibilities; If Λ = 0, all the solutions are Robinson waves; if Λ ≠ 0, there exists only one solution, first given here by (6.26). There exist no other solutions for null electromagnetic fields. In the case of nonnull electromagnetic fields two solutions are found. One metric is known having been first given by Robinson; we give a new solution of type I. The question as to whether there are solutions different from these remains open.

All Homogeneous Solutions of Einstein's Field Equations with Incoherent Matter and Electromagnetic Radiation
View Description Hide DescriptionIn this paper I prove the theorem that the Einstein Field equations with dust and electromagnetic null field have only two homogeneous solutions. I have given a discussion of these two solutions in in another paper.

Particlelike Solutions to Nonlinear Scalar Wave Theories
View Description Hide DescriptionAccording to a recent theorem proved by Derrick, no absolutely stable time‐independent particle‐like solution of finite energy is obtainable from a large class of Lorentz‐covariant scalar wave theories. We study a solvable nonlinear scalar wave theory and derive a rigorous metastable particlelike solution of finite energy, a quasistatic solution having a rate of dissolution which is free to be arbitrarily small relative to the associated particle rest mass. Derrick's theorem notwithstanding, the specific example presented here suggests that particlelike quasistatic solutions to a nonlinear scalar wave theory may still be of some relevancy to meson field physics, where no absolutely stable but instead metastable elementary particles are present.

On Projective Representations of Finite Groups
View Description Hide DescriptionThe algebra of projective representation, belonging to a factor system, has been presented. Exact expressions for the projection operators, Kronecker (inner) direct‐product representation, have been obtained. Formulas for obtaining the characters of all the inequivalent irreducible projective representations and the Clebsch‐Gordan coefficients have been derived.

On Irreducible Representations of Space Groups
View Description Hide DescriptionA logical extension has been made of Seitz‐Koster's method of space‐group representations to the points on the surface of the Brillouin zone, using projective representations. This method has been applied in the case of the space group D _{3} ^{4}.

Separation of the Interaction Potential into Two Parts in Treating Many‐Body Systems. I. General Theory and Applications to Simple Fluids with Short‐Range and Long‐Range Forces
View Description Hide DescriptionSystematic methods are developed for investigating the correlation functions and thermodynamic properties of a classical system of particles interacting via a pair potential v(r) = q(r) + w(r). The method is then applied to the case in which w(r) is a ``Kac potential'' w(r, γ) = γ^{ v }φ(γr) (v the dimensionality of the space) whose range γ^{−1} is very long compared to the range of q(r). Our work is related closely to the work of Kac, Uhlenbeck, and Hemmer. The main new feature of our method is the separation of the correlations, e. g., the two‐particle Ursell function F(r), into a short‐range part F^{ s }(r, γ) and a long‐range part F^{ L }(y, γ), y ≡ γ r; r the distance between the particles. The two parts of F are defined in terms of their representation by graphs with density (or fugacity) vertices and K‐and φ‐bonds, K(r) = e ^{−‐βq } − 1, Φ = −βw. A resummation of these graphs then yields a simple graphical representation for the long‐range part of the correlation functions in terms of graphs with φ‐bonds and ``hypervertices'' made up of the short‐range part of the correlations. This representation is then used in this paper to make separate expansions of F^{ s }(r, γ) and F^{ L }(y, γ) and through them of the thermodynamic parameters in powers of γ. Explicit calculations of the Helmboltz free energy is carried out to a higher order in γ than done previously by Hemmer and it is shown how to carry out the calculation, in principle, to any order. The general method is further applied (in separate articles) to lattice gases, plasmas, and to the special problem of critical phenomena.

Fokker Action Principle for Particles with Charge, Spin, and Magnetic Moment
View Description Hide DescriptionA Fokker action principle is obtained for a system of particles with charge, spin, and magnetic moment,interacting through time‐symmetric electromagnetic fields. Conservation laws are derived for the energy, and the linear and angular momentum of the system. A limiting process is performed, which involves a renormalization of mass, in order that the magnitude of the spin of each particle remain constant.

Born Reciprocity Principle and Unitary Symmetry
View Description Hide DescriptionIn view of the past attempts by Born et al. to explain elementary particles using the Born reciprocity principle as a postulate and of the recent success of unitary symmetry schemes, it is sought to establish a contact between the reciprocity principle and unitary symmetry in connection with the problem of elementary particles.

Geometric Theory of Neutrinos
View Description Hide DescriptionIt is shown that the conditions , in the limit that the scalar R _{αβ} R ^{αβ} vanishes, reproduce the physics of neutrinos. That is, these conditions ensure the existence of a two‐component spinor which obeys the Weyl equation and represents a field of completely polarized neutrinos.

General Operator Potential for the Two‐Nucleon System
View Description Hide DescriptionThe interaction between two nucleons is assumed to be described by a general Hermitian operator having certain symmetry and invariance properties.
It is shown that the usual phenomenological potentials employed in dealing with the two‐nucleon system, i.e., nonlocal, velocity‐dependent and hard‐core, are related by means of unitary transformations of the interaction operator. In addition it is shown that another such transformation of the total Hamiltonian operator leads to the effective mass formalism. Some numerical results on phase shifts are compared with most accepted values.