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Volume 9, Issue 12, December 1968

Generating Functions for the Exact Solution of the Transport Equation. II. Time‐Dependent with Anisotropic Scattering
View Description Hide DescriptionPart I of this series [J. Math. Phys. 9, 1722 (1968)] introduced in detail the general method of transformingintegrodifferential transport equations to partial differential equations. The treatment there is restricted to isotropic transport in slab geometry. This part extends the method to time‐dependent anisotropic transport for slab geometry.Generating functions are used as an analytic tool to define appropriate transformations whose inverses are known. The general solution of the transport equation considered are expressed in terms of expansion modes. The expansion coefficients are determined by a combination of Fourier transforms and orthogonality relations. Fourier transforms in the time variable are used instead of the usual Laplace transforms. The solutions of the initial‐value problem and its analog with the role of space and time interchanged are given.

Inverse Wave Propagator
View Description Hide DescriptionIn a recent publication, Wolf and Shewell gave a formal solution to the inverse diffraction problem, i.e., to finding the field distribution in the plane z = 0 from the knowledge of the field in an arbitrary plane z = z _{1} > 0 in the half‐space into which the field is propagated. The solution involved the use of a singular kernel. In the present paper the inverse diffraction problem is treated in a rigorous manner. Our method makes use of the representation of the field as an angular spectrum of plane waves and demonstrates the usefulness of this type of representation. It is shown that by the use of a suitable truncation procedure one may avoid the use of a singular kernel or the generalized function theory.

Response of a Many‐Particle System to Quasistatic Changes in Volume
View Description Hide DescriptionA study is made of the linear dissipative processes associated with viscous flow in a many‐particle system. Central to this study is a direct examination of the linear‐response properties of the system when it is influenced by a dynamical perturbation which induces a time‐dependent change in the size and shape of the containing volume of the system. By examining the linear response of the pressure‐tensor operator to such a perturbation, we derive correlation‐function expressions for the coefficients of viscosity η and ζ and for the shear and bulk moduli G _{0} and K _{0}. Essential to this discussion is the careful examination of the behavior of certain spectral functions at finite volume. It is found that in order to carry through the analysis in a consistent fashion, one must require that these spectral functions exhibit a special singular behavior. In particular, it is found that the static moduli G _{0} and K _{0} are related to the singular parts of the relevant spectral functions, whereas the viscosity coefficients are related to the nonsingular parts of these same spectral functions. The expressions obtained are compared with the familiar Kubo‐Mori expressions for the coefficients of viscosity η and ζ.

Relativistic Dynamics of a Point Charge in a Magnetic‐Monopole Field
View Description Hide DescriptionThis paper derives and interprets the constants of the charge's motion. The physical meaning of these constants and their use in discussing the over‐all motion of the charge are presented.

Transmission and Reflection of Electromagnetic Waves at the Boundary of a Relativistic Collisionless Plasma
View Description Hide DescriptionThe exact solution of the transmission and reflection problem for transverse electromagnetic waves incident on a bounded plasma has been discussed to some extent by several authors. Shure considered the special cases of perpendicular incidence on nonrelativistic half‐space and slab plasmas and made use of van Kampen‐Case modes to construct the solution. For both half‐space and slab plasmas, we generalize his results to (i) arbitrary temperatures (relativistic) and (ii) arbitrary angles of incidence. For simplicity, we consider the special case where the incident electric field is perpendicular to the plane of incidence and assume that particles are reflected specularly at the interface. We proceed somewhat differently from Shure, and use a Laplace transformation in obtaining our solution. We also show that present solutions can be expressed as a superposition of van Kampen‐Case modes appropriate to a relativistic plasma.

Proof of the Fermion Superselection Rule without the Assumption of Time‐Reversal Invariance
View Description Hide DescriptionThe superselection rule which separates states with integer angular momentum from those with half‐integer angular momentum is proved using only rotational invariance.

Nonlinear Boundary‐Value Problems in One‐ and Two‐Dimensional Composite Domains
View Description Hide DescriptionSome nonlinear boundary‐value problems in one‐ and two‐dimensional composite domains have been solved by a general eigenfunction‐expansion method. The advantage of the method is that separable problems in more than one dimension can be solved almost as easily as one‐dimensional problems. An optimum eigenfunction‐expansion basis has been found that leads to accurate solutions with only a few terms in the expansion.

Variational Solutions of Nonlinear Poisson‐Boltzmann Boundary‐Value Problems
View Description Hide DescriptionVariational solutions of one‐dimensional nonlinear Poisson‐Boltzmann boundary‐value problems in the theory of colloids and plasmas are obtained. The accuracy of the solutions is measured in terms of upper and lower bounds for the field energy which result from complementary variational principles.

Reduction of Reducible Representations of the Poincaré Group to Standard Helicity Representations
View Description Hide DescriptionIn this paper we introduce realizations of the generators of the Poincaré group for real and imaginary masses which are close in form to the Lomont‐Moses realizations for zero mass. These realizations (which we call ``standard helicity realizations or representations'') are characterized by the way that the infinitesimal generators are given in terms of the helicity operator. We also give the global form of the realizations and discuss in detail the realizations for the case that they are unitary and irreducible. We then show how any reducible representation of the Poincaré group for which the infinitesimal generators of the translation and rotation subgroups are Hermitian and integrable and for which the space‐time generators are integrable (but not necessarily Hermitian) can be reduced to the standard helicity realizations. In the case that the reducible representation is unitary, this process enables one to reduce the reducible representation to irreducible unitary standard helicity representations. Finally, we show how the Foldy‐Shirokov realizations for real mass are related to the standard helicity representation.

Determination of the Amplitude from the Differential Cross Section by Unitarity
View Description Hide DescriptionBanach‐space fixed‐point theorems are used to prove two results: (a) If the differential scattering cross section is smooth and small enough, relative to the wavelength of the relative motion of the colliding particles, there always exists an amplitude function which satisfies elastic unitarity and whose squared modulus equals a given differential cross section. (b) Under somewhat stronger conditions this amplitude is uniquely determined (except for the sign of its real part) by the generalized optical theorem (unitarity) and it can be constructed by iterating the latter. The condition for (a) ensures a priori that the real part of the amplitude cannot vanish at any angle, and that for (b) implies that its real part cannot be smaller than twice its imaginary part. These results are then generalized to inelastic and production processes.

Interpretation of in a Huygens Model
View Description Hide DescriptionWe describe a classical field theory based on Huygens' principle which is characterized by an additional degree of freedom which, to our knowledge, has not been discussed previously. This additional degree of freedom asserts that the forward propagation cone is different from the backward cone. The purpose of this paper is to find a function which describes this degree of freedom, and next, to understand this effect in reference to other theories. We find that this additional degree of freedom can be described by means of . The object is then related to the torsion of a preferred‐frame geometric theory. The additional degree of freedom is of interest since it enables one to introduce in a framework involving characteristic equations, described by g_{ij} , and bicharacteristics described by , such that the role of can be understood. Also, the theory furnishes a generalized framework for gravitational theory. Paths with noncontinuous slopes appear also in Feynman's path‐integral approach. Thus, this type of discontinuity has physical interest here also, although we do not pursue this point in this paper.

On the Clebsch‐Gordan Series of a Semisimple Lie Algebra
View Description Hide DescriptionThe problem of determining the multiplicity of an irreducible representation of a semisimple Lie algebra, in the decomposition of the product of two such representations, is reduced to one of solving a system of linear equations. This is achieved by using some properties of partition functions which occur in the formulas for the multiplicity of a weight in a representation. It is shown that one need not know the partition function explicitly.

Theorems on the Ising Model with General Spin and Phase Transition
View Description Hide DescriptionThe theorem of Lee and Yang has been extended to the ferromagneticIsing model with arbitrarily mixed spin values of S_{j} = ½, 1, and , including the case of equal spin values as a special one. Namely, it has been proved that the zeros of the partition function for the above Ising model with higher spin values lie on the unit circle in the fugacity plane (or complex magnetic‐field plane). Expressions for general correlation functions in Ising ferromagnets with higher spin values have been derived in terms of the above generalized theorem. By the use of these expressions, the relations among the critical indices are discussed and the same results are obtained as those predicted by the scaling‐law approach.

Eigenvalue Problem for Lagrangian Systems. III
View Description Hide DescriptionThe quadratic Lagrangianeigenvalue problem [λ^{2} P + λQ ‐ (L + B)]ζ = 0 and the associated time‐dependent problem are investigated for the case where P, Q, and B are bounded linear Hermitian operators in Hilbert space,P is positive and invertible, L possesses a positive completely continuous Hermitian inverse, and L + B > 0. Existence and completeness theorems for the eigenvectors as well as variational characterizations of the eigenvalues are given, and the general solution of the time‐dependent problem is obtained in terms of an eigenvector expansion. Finally, these results are applied to the problem of small oscillations of a rotating elastic string.

Asymptotic Fields in Some Models of Quantum Field Theory. I
View Description Hide DescriptionA quantum field with nonlocal interaction is considered. We prove under a proper smoothness condition on the interaction that the asymptotic limits of the annihilation‐creation operators exist. The asymptotic limits are then used to prove that the state space decomposes as a tensor product of an incoming (outgoing) Fock space and a zero‐particle space.

Green's Functions for Electromagnetic Waves in Moving Lossy Media
View Description Hide DescriptionThe equations governing the potentials for electromagnetic waves in moving lossy media are obtained by using a covariant formulation. A versatile form of the time‐dependent Green's function is derived by first transforming the wave equation to the normal form and then applying the Fourier‐integral technique. The time‐harmonic Green's function is also obtained.

Class of Representations of the IU(n) and IO(n) Algebras and Respective Deformations to U(n, 1), O(n, 1)
View Description Hide DescriptionWe define directly the matrix elements of the generators of the algebra of U(n) × I _{2n } on a chosen basis. This construction, though naturally infinite‐dimensional, has a very close formal resemblance (interpretable, if so desired, in terms of a suitably defined ``contraction'' procedure) to the Gel'fand‐Zetlin (GZ) representation for U(n + 1). The representations we obtain are characterized by (n ‐ 1) integers and one continuous parameter. We then exploit the formal analogy with the GZ pattern in order to prove the necessary commutation relations and to derive the explicit expressions for some invariants. However, direct alternative methods are indicated where they are useful. Our representation is easily enlarged to that of , which we use, together with a deformation formula to obtain a class of representations of the U(n, 1) algebra. The irreducible components are characterized by (n ‐ 1) integers and two continuous parameters. We compare our deformation formula with that of Rosen and Roman. We indicate briefly the typical difficulties that arise for the case IU(p,q) (q ≥ 1). Parallel constructions, finally, are given for the IO(n) and O(n, 1) algebras

Master Analytic Representation: Reduction of O(2, 1) in an O(1, 1) Basis
View Description Hide DescriptionWe display the reduction of the pseudo‐orthogonal group O(2, 1) with respect to a noncompact O(1, 1) basis. After the explicit solution is obtained, we rederive the results using the method of master analytic representations.

Auxiliary Variables in Statistical Mechanics: Variational Principle
View Description Hide DescriptionThe variational principle of Bohm and Pines for the ground‐state energy of the electron gas with uniform neutralizing positive‐charge background, employing auxiliary variables, is reviewed as an illustration of the past use of auxiliary variables and as an example of the type of physical system to which the variational principle of this paper can be applied. We then develop this variational principle for the logarithm of the partition function of a physical system at nonzero temperatures, employing auxiliary variables. The variational principle contains a trial Hamiltonian H′ in an extended Hilbert space. For H′ equal to its optimal value H, the variational expression ln Q′ is equal to the logarithm of the partition function ln Q. For H′ ≠ H, it is shown that ln Q ≥ ln Q′ for H′ – H sufficiently small or temperatures sufficiently high, or for sufficiently low temperatures when an additional assumption is made, which reduces to one made by Bohm and Pines when applied to the electron gas. The variational expression ln Q′ contains more complicated trace formulas than are usually encountered in quantum statistical mechanics; one possible method of evaluation is sketched leading to a simpler approximate formula for ln Q′. Corrections to the variational approximation for ln Q are provided by the second‐ and higher‐order terms in a certain perturbation expansion of ln Q. The variational principle developed here implies the variational principle of Bohm and Pines in the zero‐temperature limit; in the ``no auxiliary variable'' limit it reduces to a modified form of Peierls' variational theorem. It is shown how the variational principle can be applied to any physical system containing charged particles in which the long‐range collective effects of the Coulomb interaction are important.

First‐Order Phase Transitions in Quantum‐Coulomb Plasmas
View Description Hide DescriptionA simplified model is suggested for the understanding (in principle) of the mechanism of a phase transition in a Coulomb system on a uniform neutralizing background. Quantum theory is taken into account only in so far as it provides discrete collective energy levels and, possibly, fermion statistics for the calculation of the collective modes. Other features stemming from the uncertainty relations are supposed to be irrelevant to the mechanism of the phase transition. (This is qualitatively justified in the text which goes with Fig. 1.) This procedure provides an ``effective quantum Hamiltonian'' H _{qu}, which incorporates the relevant quantum features and from which one can calculate the (approximate) quantum partition function using classical methods:.In the evaluation of this integral we use the same approximation in which the plasma modes are collective, viz., the RPA (random phase approximation). Because of the freezing‐out of the collective degrees of freedom at the relevant temperatures, and because the number of these degrees of freedom changes with temperature and density, H _{qu} describes a system with variable degrees of freedom, which seems to provide the mechanism for the phase transition. Preliminary numerical evaluations indicate a phase transition at T = 0 for an electron plasma at r_{s} = 7.6, i.e., just below the region of metallic densities. This is shown by finding a concave region in the free energy as a function of the density. The ground is then prepared for numerical evaluation of the transition at T ≠ 0. For white dwarfs we find a transition temperature of 10^{7} ∼ K°.