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Volume 7, Issue 4, April 1966

A Perturbation Method for the Classical Time‐Dependent Pair Correlation Function
View Description Hide DescriptionA perturbation technique useful for computing many‐time thermal averages of classical quantities is developed. The canonical distribution function for the system is shown to evolve isothermally from that for a free‐particle system as the interaction is switched on slowly. This permits convenient use of an interaction picture in which to perform thermal averaging. The technique is applied to the calculation of the time‐dependent pair correlation function in position for a uniform gas. The correlation function is shown to be a sum of two components, one the solution of a kinetic equation and essentially a generalization of the autocorrelation in equilibrium, the other a generalization of the mutual correlation function in equilibrium. The equations arise as sums over diagrams. The equations resulting from the random phase approximation, valid for the short‐time behavior, are solved exactly. It is shown directly that the generalized dielectric function given in terms of the correlation function is identical with that found by solution of the kinetic equation in the random phase approximation for all frequencies.

Asymptotic Renormalizability Conditions on Field Propagators
View Description Hide DescriptionHigh‐energy boundary conditions upon the basic Green's functions which limit the types of divergence arising in any approximational method of solution of a field theory are derived and shown to be related to the recently defined stability criterion.

Empty Space‐Times of Embedding Class Two
View Description Hide DescriptionThe null tetrad notation of Newman and Penrose is used to investigate empty space‐times of embedding class two. Necessary conditions are found for algebraically special empty space‐times to have this property.

Symmetry Properties of the 3j‐Symbols for an Arbitrary Group
View Description Hide DescriptionThe symmetry properties of the 3j‐symbols are studied for an arbitrary compact group. It is shown that when the three j's are all inequivalent it is possible to choose 3j‐symbols which are invariant under any permutation of the j's and of the corresponding m's (generalized magnetic quantum numbers). When two of the three j's are equivalent, the 3j‐symbols can be chosen in such a way that at most a minus sign appears when the j's and m's are permuted. It is also shown that when the three j's are equivalent it is in general not possible to choose the 3j‐symbol such that its absolute value is invariant under every permutation of the m's.

Statistical Mechanics of High‐Temperature Quantum Plasmas Beyond the Ring Approximation
View Description Hide DescriptionA quantum mechanical perturbation expansion of the partition function is used to evaluate the free energy of the electron gas and multicomponent plasmas to logarithmic accuracy in the particle number density, thus including the next important contribution beyond the ring approximation. The quantum generalization of Abe's work on the classical electron gas is given for the ladder interactions with the dynamic screened Coulomb potential, and each ladder is shown to be separately finite because of the finite size of the wave packets describing point electrons [of the order of the thermal de Broglie wavelength ƛ = h̄(β/2m)^{1/2}]. The results show that quantum effects due to the uncertainty principle persist at high temperature, and that when kT > Ryd plasmas are quantum systems, rather than classical, because ƛ is greater than the average distance of closest approach, βe ^{2}. Results are also obtained for the Wigner‐Kirkwood wave mechanical diffraction corrections to the classical electron‐gas free energy which are valid for low temperature (kT < Ryd). The connection between the high‐ and low‐temperature formula is discussed, and it is shown how the logarithmic divergence in the free energy that is cut off at βe ^{2} in the low‐temperature electron gas in the Abe theory is cut off at ƛ in the high‐temperature case. Also it is shown that the quantum diffraction effects contained in the Montroll‐Ward ring sum formula are valid only for kT > Ryd, even though the quantum ring sum formula contains the classical Debye‐Hückel result.

Limiting Forms of the Screened Coulomb T Matrix
View Description Hide DescriptionIn the complex energy plane, the pure Coulomb T matrix possesses branch points which would not appear if the force were properly defined. This is demonstrated by a study of the screened Coulomb T matrix in the limit as the screening radius R tends to infinity. No branch points develop if the proper order of limiting processes is observed and the results agree with previous calculations; however, the T matrix is discontinuous in the limit. A formula for the screened Coulomb T matrix is given which is valid to order 1/R for all energies.

Multipole Theory in the Time Domain
View Description Hide DescriptionSpherical outgoing waves of arbitrary time dependence are first written in the usual way as a Fourier integral of a sinusoidally time‐varying multipole expansion. It is then shown that the integrals over ω of the r‐ and t‐dependent part of the multipole terms can be replaced by differential operators operating on arbitrary functions of retarded time. Thus a form of the multipole expansion is obtained that does not explicitly contain the frequency spectrum of the multipoles. Given the value of E_{r} (for electric multipoles, or B_{r} for magnetic multipoles) as a function of time on the surface of a sphere, expressions for the multipole expansions of all the spherical field components are derived. The method employs a convolution integral and is useful in problems involving a very broad frequency spectrum.

A Continuous Representation of an Indefinite Metric Space
View Description Hide DescriptionAn overcomplete family of states (OFS) is constructed for a countably infinite linear vector space with an indefinite metric for the case that the metric is diagonal with eigenvalues (−1)^{ n }, where n is an integer. A continuous representation is indicated and the properties of a semiclassical description of a quantum mechanical system (the pseudo oscillator whose creation and destruction operators ā and ā ^{+} satisfy [ā, ā ^{+}] = −1) defined in this vector space are studied. It is found that a consistent OFS z> can be constructed if the operator G(z) which generates the state z> from the vacuum is unitary. Furthermore, with the statistical state of this system specified by a bounded pseudo‐Hermitian density matrix ρ̄, the related semiclassical complex function ρ_{ A }(z) for antinormal ordering of operators in the indefinite metric space is found to be bounded, with ρ_{ A }(z) and [ρ_{ A }(z)]^{2} integrable, continuous, and a boundary value of an entire analytic function of two complex variables. The semiclassical function ρ_{ N }(z) for normal ordering is associated with a sequence of functions ρ_{ N(ν)}(z) whose square is integrable and related to a sequence of tempered distributions ρ_{ N(ν)} such that the corresponding sequence of density matrices ρ̄_{(ν)} converges to ρ̄ in the norm.

Leading Landau Curves of a Class of Feynman Diagrams
View Description Hide DescriptionIn a previous paper it was shown that the leading Landau curves of some Feynman diagrams do not give singularities on the physical sheet if some of the internal and external masses satisfy certain simple inequalities. In the present paper it is shown that a similar property is satisfied by a class of Feynman diagrams. The inequalities involve a fixed number of masses for the whole class.

Effective Dielectric Constant, Permeability, and Conductivity of a Random Medium and the Velocity and Attenuation Coefficient of Coherent Waves
View Description Hide DescriptionRandom media are considered in which the dielectric constant, permeability, and conductivity are random functions of position. For them, the average electric field,electric current,dielectric displacement, magnetic field, and magnetic induction are determined, assuming that these average quantities are time‐harmonic plane waves. The proportionality factors between appropriate pairs of these quantities are found and defined to be the effective dielectric constant, permeability, and conductivity of the random medium. These effective parameters depend upon the frequency and propagation constant of the field in addition to the two‐point auto‐ and cross‐correlation functions of the random dielectric constant, permeability, and conductivity. For transverse fields thay are all scalars. In addition the dispersion equation for the propagation constant of the average or coherent field, derived previously, is analyzed and solved for high‐ and low‐frequency fields. From the propagation constant, the phase velocity and attenuation coefficient can be found.

Talmi Transformation for Unequal‐Mass Particles and Related Formulas
View Description Hide DescriptionThree‐dimensional polynomials which occur as coefficients of the exponential in the wavefunctions of the harmonic oscillator are used in nuclear physics and kinetic theory of gases. A generating function for these polynomials is used to simplify the calculation of several integrals. These include the integrals involving products of two and three polynomials and the coefficients of the Talmi transformation. Explicit formula in terms of recoupling coefficients of angular momentum theory are obtained.

Analyticity Properties of the Scattering Amplitude
View Description Hide DescriptionAnalyticity properties in three variables, energy, momentum transfer, and ``squared mass'' are deduced from single‐variable dispersion relations.

Continuity of Bound and Unbound States in a Fermi Gas: A Soluble Example
View Description Hide DescriptionConsidering a gas of independent fermions in the presence of an attractive localized potential, one can show that the properties of the system as a whole are smooth (analytic) functions of the strength parameter of the potential, even at those values where new single‐particle bound states appear. Thus for the system as a whole, the transition from ``unbound'' to ``bound'' states is continuous and the concept of a bound state cannot be made precise. This is illustrated here for a simple mathematically soluble model—noninteracting spinless fermions moving in the presence of a delta‐function potential in one dimension. Some related physical ideas are also presented.

Gelfand States and the Irreducible Representations of the Symmetric Group
View Description Hide DescriptionThe set of Gelfand states corresponding to a given partition [h _{1} … h_{n} ] form a basis for an irreducible representation of the unitary group U_{n} . The special Gelfand states are defined as those for which [h _{1} … h_{n} ] is a partition of n and the weight is restricted to (11 … 1). We show that the special Gelfand states constitute basis for the irreducible representations of the symmetric group S_{n} and use this property to construct explicitly states in configuration and spin‐isospin space with definite permutational symmetry.

Poles of the Proper Vertex Function in the Bethe‐Salpeter Formalism
View Description Hide DescriptionA formal general proof of the statement of Goebel and Sakita is presented on the basis of the Bethe‐Salpeter formalism; namely, it is shown that the poles of a proper vertex function cannot appear in the corresponding scattering amplitude. Some related conjectures are also verified. An exactly solvable example is presented and discussed in this connection.

Analytic Continuations of Higher‐Order Hypergeometric Functions
View Description Hide DescriptionSolutions in powers of 1 −x of the differential equation associated with the hypergeometric function _{p}F _{ p−1} are derived and the function is continued analytically in terms of these solutions. The analytic continuation is derived in a simple way from an expansion which is well suited for the purpose and which is valid for all values of the argument x. The usefulness of the _{3} F _{2} function in studying certain hypergeometric functions of two variables is emphasized.

Permutation‐Algebraic Formulation of Spin‐Free Transition Density Matrices
View Description Hide DescriptionSpin‐free transition density matrices are derived from spin‐free kets which are symmetry‐adapted to the symmetric group and its algebra. The Dirac identity establishes that these spin‐free density matrices are identical to those obtained by integrating the spin from the full‐spin density martices. Derivations are first given for arbitrary primitive kets which may be geminals of higher polymals, after which we consider products of orbitals, either orthonormal or nonorthonormal.
Correlation in the spin‐free space is discussed and we show the influence of permutational symmetry on the probability of coincidence of pairs. A special case of this correlation is the well‐known Fermi hole.

Solution of Schrödinger Equation Involving Time
View Description Hide DescriptionThe general solution of the Schrödinger equation involving the time‐dependent perturbation is presented in a compact and manageable form both for periodic and aperiodic perturbations. The method includes as a special case the solution of the Schrödinger equation involving the time‐independent perturbation. Formulas ready for practical uses are explicitly described. The essential part of the procedure is the determinantal method in the Laplace‐transformed space. The solution is applicable to strong perturbations as well as weak perturbations.

Generalizations of the Virial and Wall Theorems in Classical Statistical Mechanics
View Description Hide DescriptionA generalized virial theorem which expresses inverse compressibility in terms of integrals of virials and canonical distribution functions through the four‐particle distribution is transformed to the grand canonical ensemble and becomes an expression for compressibility in terms of the same integrals formed with grand canonicaldistribution functions. The integrals are of a mixed (virial and fluctuation) type.
While the thermodynamic functions expressed by the same integrals with canonical and grand canonical distribution functions are quite different, the two formulas agree in the thermodynamic limit because of the different asymptotic behavior of canonical and grand canonical distribution functions. We also derived an alternative form of the second virial theorem which expresses compressibility in terms of integrals over virials and grand canonical distribution functions through the three‐particle distribution function only. It is shown that this form and its generalizations to higher derivatives of the density, as well as the hierarchy of fluctuation theorems and the fugacity expansions of distribution and correlation functions can all be very simply derived from a set of integro‐differential equations satisfied by the grand canonical distribution functions. A generalization of the wall theorem (P/kT = ρ_{wall}) is derived and shown to be equivalent to the generalized virial theorem (canonical form).

On the Propagation of Gravitational Fields in Matter
View Description Hide DescriptionA purely covariant treatment is made of those solutions of the Einstein field equations which represent pure gravitational radiation propagating in fluid and electromagnetic media. The analysis involves a discussion of the full Bianchi identities in carefully selected tetrad frames. In this way the interaction between the gravitational field and the medium is transferred to a coupling between a preferred frame for the gravitational field and one for the matter field. The gravitational radiation no longer propagates along shear‐free null geodesics, as it does in vacuum, and the shear and ray curvature of the propagation vector are shown to depend directly on the properties of the medium. Some new solutions of the field equations, representing transverse gravitational waves propagating in an electromagnetic field, are exhibited and discussed in some detail. It is shown that no such solutions exist, at least in simple cases, for perfect fluids. Finally, the treatment presented here is compared with the more usual electromagnetictreatment, and it is shown why the theories require basically different approaches.